Analysis of different approaches for evaluation of surface energy of microbial cells by contact angle goniometry

Advances in Colloid and Interface Science 98 Ž2002. 341᎐463

Analysis of different approaches for evaluation of surface energy of microbial cells by contact angle goniometry P.K. Sharma, K. Hanumantha RaoU Di¨ ision of Mineral Processing, Lulea ˚ Uni¨ ersity of Technology, SE-97187, Lulea, ˚ Sweden

Abstract Microbial adhesion on solid substrate is important in various fields of science. Mineral᎐microbe interactions alter the surface chemistry of the minerals and the adhesion of the bacterial cells to mineral surface is a prerequisite in several biobeneficiation processes. Apart from the surface charge and hydrophobic or hydrophilic character of the bacterial cells, the surface energy is a very important parameter influencing their adhesion on solid surfaces. There were many thermodynamic approaches in the literature to evaluate the cells surface energy. Although contact angle measurements with different liquids with known surface tension forms the basis in the calculation of the value of surface energy of solids, the results are different depending on the approach followed. In the present study, the surface energy of 140 bacterial and seven yeast cell surfaces has been studied following Fowkes, Equation of state, Geometric mean and Lifshitz᎐van der Waals acid᎐base ŽLW᎐AB. approaches. Two independent issues were addressed separately in our analysis. At first, the surface energy and the different components of the surface energy for microbial cells surface are examined. Secondly, the different approaches are evaluated for their internal consistency, similarities and dissimilarities. The Lifshitz᎐van der Waals component of surface energy for most of the microbial cells is realised to be approximately 40 mJrm2 " 10%. Equation of state and Geometric mean approaches do not possess any internal consistency and yield different results. The internal consistency of the LW᎐AB approach could be checked only by varying the apolar liquid and it evaluates coherent surface energy parameters by doing so. The electron-donor surface energy component remains exactly the

U

Corresponding author. Tel.: q46-920-917-05; fax: q46-920-973-64. E-mail address: [email protected] ŽK. Hanumantha Rao..

0001-8686r02r$ - see front matter 䊚 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 1 - 8 6 8 6 Ž 0 2 . 0 0 0 0 4 - 0

342 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

same with the change of apolar liquid. This parameter could differentiate between the Gram-positive and Gram-negative bacterial cells. Gram-negative bacterial cells having higher electron-donor parameter had lower nitrogen, oxygen and phosphorous content on their cell surfaces. Among the four approaches, LW᎐AB was found to give the most consistent results. This approach provides more detailed information about the microbial cell surface and the electron᎐donor parameter differentiates different type of cell surfaces. 䊚 2002 Elsevier Science B.V. All rights reserved. Keywords: Microbial surface energy; Contact angle; Fowkes; Equation of state; LW᎐AB

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 2. Thermodynamic approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 2.1. Berthelot’s combining rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 2.2. Antonow’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 2.3. Zisman approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 2.4. Good and Garifalco approach w45x . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 2.5. Fowkes approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 2.6. Polar component of the solid surface energy . . . . . . . . . . . . . . . . . . . . . 356 2.7. Equation of state approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 2.8. Objections on equation of state approach . . . . . . . . . . . . . . . . . . . . . . . 370 2.9. Lifshitz᎐van der Waalsracid᎐base approach ŽLW᎐AB. . . . . . . . . . . . . . . 384 2.10. Objections on Lifshitz-van der Waals acidrbase approach . . . . . . . . . . . . . 389 2.11. Spreading pressure estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 2.12. Comparison of the thermodynamic approaches . . . . . . . . . . . . . . . . . . . . 405 3. Measurement of contact angle on microbial cell surface . . . . . . . . . . . . . . . . . . . 405 3.1. Axisymmetric drop shape analysis-contact diameter ŽADSA-CD. . . . . . . . . . 407 3.2. The contact angle data on bacterial lawns . . . . . . . . . . . . . . . . . . . . . . . . 408 4. Analysis of thermodynamic approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 4.1. Contact angle of the bacterial cells with apolar liquids . . . . . . . . . . . . . . . . 418 4.2. Lifshitz van der Waals Ždispersion. component of surface energy using Fowkes and equation of state approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 4.3. Contact angle with polar liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 4.4. Total and acid᎐base Žpolar. component of surface energy evaluated by Geometric mean and Equation of state approach . . . . . . . . . . . . . . . . . . . . . . . . . . 426 4.5. Polar and dispersion component of surface energy using Least square method to fit the contact angle data with four liquids to Geometric mean approach . . . . 437 4.6. Total surface energy and its components using Lifshitz᎐van der Waalsracid᎐base ŽLW᎐AB. approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 4.7. Comparison of LW᎐AB approach to Equation of state and Geometric mean approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 343

1. Introduction The adhesion of microbial cells on solid surfaces and then the formation of biofilms is important in many diverse areasᎏbiocorrosion w1᎐5x, biofouling w2,6᎐8x, biodeterioration w9,10x, opthalmology w11x, odontology w12,13x, thrombosis of biomaterial implants w14x and biobeneficiation w15,16x. The separation of minerals in biobeneficiation, which comprises bioflotation and bioflocculation as sub-processes, is governed by selective adhesion of microbial cells on the mineral surface ŽFig. 1.. In order to fully understand and control the biobeneficiation processes, it is important to understand the adhesion process. The attachment of microbial cells on mineral surface is influenced by several properties, for example, surface charge, surface hydrophobicity, presence and configuration of surface polymers. Any theory that attempts to explain the bacterial adhesion must incorporate all these parameters. In general, the bacterial adhesion can be illustrated by the surface thermodynamics and by the extended DLVO theory of calculating the interaction energy between cells and substrate as a function of separation distance w15,17x. These methods accommodate Lifshitz van der Waals interactions, electrostatic interactions and hydrophobicrhydrophilic force interactions. These interactions are very well understood and formulated in mathematical equations. The most important input, in aforementioned calculations, is the surface energy and its different components Žpolar, apolar, electron donating and electron accepting. of the bacterial cell surface and the solid substrate. Although the solid surface tensions Ženergy. can be estimated using different independent approaches, i.e. direct force measurement, contact angles, capillary penetration into columns of powders, sedimentation of particles, solidification front interactions with particles, gradient theory, Lifshitz theory of van der Waals forces and theory of molecular interactions, the contact angle is believed to be the simplest and hence widely used. In the literature, various different approaches were mentioned which makes it possible to evaluate the solid surface tension using measured contact angles by liquids with known or pre-characterised surface energy parameters. Depending on the theoretical basis of the approaches, contact angles with one or more than one liquid are required on the solid surface for which surface energy is required. Three major approaches namely, Equation of state, Geometric mean and Lifshitz᎐van der Waalsracid᎐base, respectively, needs contact angles with one, two and three liquids ŽFig. 1.. Fig. 2 chronologically presents the development of the field of theoretical evaluation of solid surface energy from contact angle goniometry. Approximately 200 years ago, Thomas Young w18x proposed contact angle of a liquid as mechanical equilibrium of the drop resting on a plane solid surface at the three-phase boundary. The three forces at the interface are the surface tensions at liquid᎐vapour

344 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 1. Theoretical estimation of the possibility and extent of microbial adhesion on solid surfaces. Approaches to convert contact angle data into solid surface energy.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 345

interface, ␥ l ¨ , solid᎐liquid interface, ␥sl , and solid᎐vapour interface, ␥s ¨ , which in equilibrium gives the following relation ŽFig. 3a.. ␥ l ¨ cos␪ s ␥s ¨ y ␥sl

Ž1.

In 1937, Bangham and Razouk w19,20x pointed out the importance of not neglecting the adsorption of vapour on the surface of the solid phase in deriving the equilibrium relation concerning the contact angle. They suggested the following equation with the spreading pressure ␲e term ŽFig. 3b.: ␥ l ¨ cos␪ s ␥s y ␥sl y ␲e

Ž2.

where ␲e is the reduction of ␥s resulting from vapour adsorption on the solid surface. Young’s equation contains the known parameters of ␥ l ¨ and cos␪ and it is understood that this equation provides the unknown solid surface energy, ␥s ¨ . The only drawback being the unknown solid᎐liquid interfacial energy, ␥sl . If this parameter could be represented or expressed in terms of the solid and liquid surface energies then the problem is solved. Three methods have come into existence to tackle this problem. The first method is to express ␥sl in terms of solid and liquid surface energy using some mathematical formulation and then evaluating the unknown ␥s ¨ . The first attempts were made in 1898 where the solid᎐liquid interfacial energy was expressed in terms of a geometric mean of solid and liquid surface energies. This method is followed to date in the form of the Equation of state approach for which the latest formulation has been published as recent as 2000. The use of the second method dates back to 1952, where approximation of solid surface energy is done using the concept of critical contact angle, which is the surface tension that divides the liquids forming zero contact angles on the solid from those forming contact angles greater than zero. This concept is used extensively in the fields of paints, adhesives, etc. In the third method, the total surface energy is divided into different components and then the solid᎐liquid interfacial energy is expressed in terms of solid and liquid surface energy components. Fowkes pioneered this approach in 1962 w21x and this method is still used in the form of the Lifshitz᎐van der Waalsracid᎐base approach. The following section describes each approach successively in detail with the criticisms by various workers and clarifications provided by the proponent authors. Finally, an analysis has been presented for the various approaches for the microbial cells surface using the contact angle data from the work of van der Mei and co-workers w22x. Chemical composition and structural arrangement of the microbial cell surface is very complex due to the presence of a large variety of chemical groups and surface appendages of different lengths. Bacterial species belonging to prokaryotic microorganisms can be divided into two different groups on the basis of the Gramstaining reaction on the cell wall distinguishing Gram-positive and Gram-negative

346 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 2. Chronological development of different approaches for evaluation of surface energy of solids from contact angle goniometry.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 347

Fig. 3. Contact angle of a sessile drop Ža. neglecting the spreading pressure Žb. accounting for spreading pressure.

bacteria. The Gram-positive bacteria usually have a well-defined rigid cell wall composed of peptidoglycan and an underlying phospholipid bilayer of the cytoplasmic membrane. The cell wall constitutes 60᎐70% of the weight of the cell wall. On the contrary, Gram-negative bacteria have a very thin, 1᎐2-nm thick, peptidoglycan layer sandwiched between outer and inner cytoplasmic membranes. Surface appendages, often containing proteins and lipoteichoic acids as well as polysaccharides rich capsules, w23x are found as the outermost material on the bacterial cell surface which probably determines the surface energy and the other surface properties. Yeast Ž Candida, etc., species. represents a different class of microorganisms belonging to unicellular eukaryotics. The major difference is the size Ž3᎐10 ␮m compared to 1᎐2 ␮m diameter for bacteria .. These organisms possess a thick cell wall consisting mainly of manna, insoluble glucan, protein and a small amount of chitin.

2. Thermodynamic approaches 2.1. Berthelot’s combining rule Brethelot w24x used the geometric mean combining rule for the first time in obtaining the interfacial tension from the surface tensions of the two phases. The relationship was based on the way dispersion energy coefficients C6i j can be written in terms of C6i i and C6j j in the treatment of London theory of dispersion forces: C6i j s

'C

ii j j 6 C6

Ž3.

348 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

This relation forms the basis of the Berthelot combining rule wEq. Ž4.x where ␧ i j is the potential parameter Žwell depth. of unlike-pair interactions, ␧ ii and ␧ j j are the potential energy parameters Žwell depth. of like-pair interactions w25x: ␧i j Ž ␧ i i ␧ j j . 1r2

s 1 or

␧i j s

'␧

ii␧ j j

Ž4.

This equation was written in terms of work of adhesion between two phases Ži.e. Wsl . and work of cohesion of the two phases Ži.e. Ws s and Wl l .. Wsl s Ws sWl l

'

Ž5.

Putting the relevant values, i.e. Ws s s 2 ␥s ¨ , Wl l s 2 ␥ l ¨ and Wsl s ␥ l ¨ q ␥s ¨ y ␥sl in Eq. Ž5. and rearranging we get the final relation of Berthelot: ␥sl s ␥s ¨ q ␥ l ¨ y 2 ␥s ¨ ␥ l ¨

'

Ž6.

2.2. Antonow’s rule In 1907 Antonow w26x related ␥S L in terms of ␥S V and ␥LV in a simple manner as in Eq. Ž7.. There was no theoretical background behind this relationship according to Kwok and Neumann w25x. ␥sl s < ␥ l ¨ y ␥s ¨ <

Ž7.

2.3. Zisman approach Zisman w27x has introduced the concept of critical surface tension ␥c as an empirical method of determining the ‘wettability’ of solid surfaces by plotting the cosine of the contact angle ␪ vs. the surface tensions of a series of liquids. The point at which the resulting curve intercepts the line at cos␪ s 1 is called the critical surface tension ␥c . The ␥c is the surface tension that divides the liquids forming zero contact angle on the solid surface from those forming a contact angle greater than zero. The liquids with surface tension ␥ l below the ␥c value of the solid simply spread on the solid. This approach has been extensively used to determine the critical surface tension ␥c of various low energy solids and organic films deposited on high energy solids like glass and metals. In general, a rectilinear relation is established empirically between the cosine of the contact angle and the liquid surface tension, for each homologous series of organic liquids. Fig. 4a illustrates the results with the n-alkanes or polytetrafluoroethylene. Even when cos␪ is plotted against ␥l ¨ for a variety of non-homologous liquids, the results fall close to a straight line or collect around it in a narrow band like that shown in Fig. 4b. Certain low energy solids exhibit curvature of this band for liquids with surface tension above 50 dynesrcm2 ŽFig. 4c., this results because weak hydrogen bonds form between the molecules of

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 349

Fig. 4. Ža. Wettability of polytetrafluoroethylene ŽTeflon. by n-alkanes; Žb. wettability by various liquids on the surface of polyvenylchloride; and Žc. wettability of polytetrafluoroethylene ŽTeflon. by various liquids. Taken from w134x.

liquid and those in the solid. This is most likely to happen with liquids of high surface tension, because they are always hydrogen bond forming liquids Žpolar liquids.. If critical surface tension is considered to give an indication of the surface tension of the solid then by using this method: 1. it is possible to obtain the total solid surface energy of an apolar solid by using series of homologous apolar liquids, e.g. n-alkanes; 2. it is possible to find only the dispersion force component Ž ␥sd . of the total surface energy of a polar solid by using series of homologous apolar liquids, e.g. n-alkanes, and; 3. deviation from rectilinear relation is observed when polar liquids are used on polar and apolar solids by using the Zisman method. Hence, it is not possible to determine any component of the solid surface tension by using polar liquids using the Zisman method.

350 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 5. Cos␪ vs. ␥1 curve for polyŽethylene terephthalate .: ŽA. homologous series of hydrocarbon liquids; ŽB. calculated for the ethanolrwater series; ŽC. observed for ethanolrwater series Žfrom Dann w29x..

In 1970 using the Good᎐Garifalco᎐Fowkes approach, Dann w28,29x demonstrated, it is expected to obtain different values of ␥c for a particular solid surface, depending upon what liquid series is used in the determination of the critical angle by the Zisman method. He used a homologous series of ethanolrwater, mixed glycols and ASTM series of liquids prepared from mixtures of 2-ethoxyethanol and formamide on apolar, polar and monopolar solids. The results were compared with the ones obtained by Zisman and co-workers using a homologous series of n-alkanes. Considerable difference was found between the measured values of ␥c and the generally accepted values of Zisman. The deviation can be regarded as a difference between the plots between cos␪ vs. ␥ l for different homologous series of liquids on the same solid as shown in Fig. 5. The difference between the critical surface tension ␥c found by hydrocarbon series ŽA. and ethanolrwater ŽB. series was explained using the Good᎐Garifalco᎐Fowkes᎐Young equation wEq. Ž14.x. It was demonstrated that with some precaution ␥sd values of the polymers can be accurately determined from contact angle measurements with standardised series of polar liquids by the use of conversion curves ŽFig. 6.. As an example, let ␥s obtained by ethanolrwater homologous series be 24 mJrm2 . A vertical line is drawn from the point where a

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 351

Fig. 6. Critical surface tension conversion curves Žfrom Dann w28x..

horizontal line at ␥s s 24 cuts the curve C. The point at which this vertical line cuts curve A gives the ␥sd value, which is 33.5 mJrm2 in this case. The deviation of the curve C, measured for ethanolrwater from the curve B, which is theoretical ethanolrwater, is attributed to the contributions due to polar interactions between the solid and the liquid. This was very clearly explained and proved by Owens and Wendt w30x, by introducing the terms for polar contribution in interfacial energy in the Fowkes equation wEq. Ž13.x. 2.4. Good and Garifalco approach [45] Accepting the Berthelot relation for the attractive constants between molecules, A aa and A b b , and between unlike molecules A ab , i.e. A ab Ž A aa A b b . 1r2

s1

Ž8.

the authors arrived at an almost similar relationship as Brethelot but with a constant ␾. Taking analogy from Eq. Ž8. they set up the corresponding ratio, involving the energies Žor free energy. of adhesion and cohesion of two phases which was taken to equal a constant ␾: a ⌬Gab

⌬Gac⌬G bc

'

s

Ž ␥a q ␥ b y ␥ab . 2 ␥a␥ b

'

s␾

Ž9.

352 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 a Using free energy of adhesion ⌬Gab s ␥ab y ␥a y ␥ b and free energy of cohec sion ⌬Gn s 2 ␥n , the following relation was attained:

␥sl s ␥s ¨ q ␥ l ¨ y 2␾ ␥s ¨ ␥ l ¨

'

Ž 10 .

and ␾ was evaluated for different systems. The value of ␾ was found to be 1 for ‘regular’ interfaces, i.e. systems for which the cohesive forces of the two phases and the adhesive forces across the interface are of the same type. When the predominant forces within the separate phases are unlike, e.g. London᎐van der Waals vs. metallic or ionic or dipolar, then low values of ␾ are expected. Clearly this applies to pairs such as non-hydrogen bonding in organic compound vs. water; organic compound vs. metal and salt vs. metal. When there are specific interactions between the molecules forming the two phases, the energy of adhesion is greater than the value it should have in the absence of specific interactions. 2.5. Fowkes approach [21,31,32] This approach forms a basis of all the surface tension component approaches used today and the dispersion component of the total surface energy is still calculated by using this approach. Fowkes considered the surface tension Ž ␥ . to be a measure of the attractive force between surface layer and liquid phase, and that such forces and their contribution to the free energy are additive. Therefore, the surface tension of liquid metals, polar liquids, hydrocarbons, low energy solids and other solids is considered to be made up of independent additive terms. ␥ s ␥ d q ␥ h q ␥ m ......

Ž 11.

He attributed the ␥ d term only to the London dispersion interactions because it has been shown for macroscopic condensed systems in aqueous media that out of the three electrodynamic interactions only London’s dispersion interaction is predominant w33,34x. ␥ h is due to hydrogen bonding and ␥ m due to metallic bonding, etc. The intermolecular attractions, which cause surface tension, arise from a variety of well-known intermolecular forces. Most of these forces, such as metallic bonding and hydrogen bonding, are a function of specific chemical nature. However, London dispersion forces exist in all types of matters and always give an attractive force between adjacent atoms or molecules, no matter how dissimilar their nature may be. The London dispersion forces arise from the interaction of fluctuating electronic dipoles with the induced dipoles in neighbouring atoms or molecules. The effect of fluctuating dipoles cancel out, but not that of the induced dipoles. These dispersion forces contribute to the cohesion in all substances and are independent of other intermolecular forces, but their magnitude depends on the

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 353

Fig. 7. Diagram of the two neighbouring monolayers at an interface in which tension resides. Taken from w134x.

type of material and density. Therefore, the ␥ d term includes only the London dispersion force contribution and, Keesom and Deby force contributions are included in the ␥ h term. The interface between two phases with only dispersion force interactions is composed of two monolayers as indicated in Fig. 7. At the interface, the adjacent layers of dissimilar molecules are in a different force field than the bulk phase and consequently, the molecules or atoms in these layers have a different pressure, intermolecular spacing and chemical potential. If the molecules in one of these monolayers are less strongly attracted by the adjacent phase than by its bulk phase, the molecules in the interfacial layer have an increased intermolecular distance and are in tension. However, if the attraction by the adjacent phase is greater than that of the bulk phase, the molecules of the interfacial monolayer have a shorter intermolecular distance and are under two-dimensional pressure. The measured tension of the interface is always the sum of the tensions in the two interfacial monolayers. The surface monolayer of phase 1 has a tension ␥1 resulting from the unopposed attraction of the bulk liquid. An interfacial monolayer of phase 1 is attracted by its bulk in the identical manner, but this attraction is opposed by the attraction of phase 2. When the interacting forces are entirely dispersion forces Žsuch as in between saturated hydrocarbons and water or mercury. then the decrease in tension in the interfacial monolayer of phase 1 resulting from the presence of phase 2 is ␥ 1d ␥ 2d . Consequently, the tension in the interfacial monolayer of phase

' y '␥ ␥

d d 1 is ␥ 1 1 2 and in the interfacial monolayer of phase 2 is ␥ 2 y leads to the equation

␥ 12 s ␥1 q ␥ 2 y

'␥ ␥ d 1

d 2

'␥ ␥ d 1

d 2

which

Ž 12.

354 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

By measuring the interfacial tension at the saturated hydrocarbonrwater interd to be 21.8 " 0.7 face and using the above equation, Fowkes calculated ␥water 2 dynesrcm , since only dispersive interactions between water and hydrocarbons d s 200 " 7 dynesrcm2 by using a saturated exists. Similarly, he arrived at ␥ Hg hydrocarbonrmercury interface. Since mercury and water interacts only by dispersion forces, Fowkes was able to predict the interfacial tension to be 425 " 4 dynesrcm2 which agrees well with the experimental value of 426 " 4 dynesrcm2 for waterrmercury interface. This demonstrates the usefulness of the geometric mean approach and helpful for both the liquid᎐liquid and liquid᎐solid interfaces. At the solid᎐liquid interface the Fowkes relation looks as follows ␥sl s ␥s q ␥ l y 2 ␥sd ␥ ld

'

Ž 13.

Combining this with the Young equation wEq. Ž2.x, we get an equation of the form: cos␪ s y1 q 2 ␥sd

'

␥ ld

ž / ␥l

y

␲e

Ž 14.

␥l

If the spreading pressure term, ␲e , is neglected then a plot of cos␪ vs. ␥ ld r␥l gives a straight line ŽFig. 8. with the origin at cos␪ s y1 and slope 2 ␥sd . Since the origin is fixed, one contact angle measurement is sufficient to determine the dispersion force component of the surface energy of the solid Ž ␥s .. The spreading pressure term, ␲e , can be assumed to be zero only for the system where high energy liquids are brought in contact with low energy solids. The basic reason for this assumption is that all theoretical and experimental evidence predicts that adsorption of high-energy materials cannot reduce the surface energy of a low energy material. For example, adsorbing water never reduces the surface tension of a liquid hydrocarbon. The fact that a given liquid has a contact angle greater than 0⬚ on a given low energy solid asserts that the liquid possesses a higher energy and therefore ␲e should be zero. This holds true only for the solids interacting by dispersion forces only. It does not apply for the high-energy solids such as metals, graphite; water does not wet these solids but it does absorb and produce appreciable ␲e . Zettlemoyer w35x corroborated the Fowkes approach of using geometric mean for estimating the interfacial forces for hydrophobic solids, but points out an interesting work by his co-worker Lavelle, who used arithmetic mean of the dispersion force attraction to estimate the magnitude of the interaction between dissimilar materials. Zettlemoyer agrees that ‘the arithmetic mean approach does not ha¨ e a great scientific basis for its uses as the geometric mean has’. Lavelle used the following equation and got some interesting results.

'

'

␥ 12 s ␥1 q ␥ 2 y Ž ␥ 1d q ␥ 2d .

Ž 15 .

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 355

Fig. 8. Relation of contact angle ␪ to surface tension of liquid ␥1 . Žfrom Fowkes w21x..

The surface energy of the two phases diminishes by ␥1d q ␥ 2dr2 at the interface due to the attractive dispersion interactions. The internal consistency was demonstrated by calculating waterrmercury interfacial tension of 426 ergsrcm 2 and the d to be 22 ergsrcm 2 which were in agreement with the value found by ␥water d was found to be 108 ergsrcm 2 instead of 200 by geometric mean. However, ␥ Hg this averaging technique. Based on just the thermodynamic point of view there are some objections on the usage of geometric mean to combine the dispersion force contribution of the two phases to arrive at the interfacial energy. Lyklema w36x put forward two reservationsᎏwhether it is thermodynamically allowable to split the interfacial tensions in components and secondly, if geometric mean is the most appropriate mathematical form to combine them? The first objection stems from the fact that the interfacial tensions are Helmholtz energies, whereas Fowkes wEq. Ž13.x approach treats them as energies. The entropy contributions are totally neglected and when entropy

356 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

contributions are combined, they do not follow the geometric mean law but go with the logarithm of the composition. If only the energetic contribution, Ua␣ , to ␥ is considered, the geometric average is a very acceptable choice because dispersion forces prevail obeying Berthelot’s rule very well. Keeping this in mind if the energetic and entropic contributions of the work of adhesion are separated, then Wad h s ⌬ ad h Ua␴ y T⌬ S a␴

Ž 16 .

⌬ ad h Ua␴ s Ua␴ , s q Ua␴ ,l y Ua␴ , sl

Ž 17 .

⌬ ad h S a␴ s S a␴ , s q S a␴ ,l y S a␴ , sl

Ž 18 .

with a good approximation ␴ , s ␴ ,l . ⌬ ad h Ua␴ f 2 Ž Ua,d Ua,d

1r2

Ž 19 .

and therefore the solid᎐liquid interfacial energy should be ␴ , s ␴ ,l . ␥sl s ␥s q ␥ l y T⌬ ad h S a␴ y 2 Ž Ua,d Ua,d

1r2

Ž 20 .

␣ where Ua,d stands for the dispersion part of the corresponding surface tensions. The purely energetic distribution contributions are independent of the temperature and the entropic term now accounts for the temperature dependence. The Fowkes relation wEq. Ž13.x has a term of spreading pressure, which is often neglected, but the present approach wEq. Ž20.x that term is not necessary as it is already accounted for in the equation. Here the T⌬ S term cannot be neglected as it accounts for approximately 20᎐30% of the Helmholtz energy. Lyklema w36x also emphasises that there is no physical ground that this geometric mean law is applicable to the acid᎐base or hydrogen bond interactions, which has been done by various workers and presented in the following sections. Except for the equation of state approach the Fowkes relation is used in all the surface tension component approaches to determine the dispersion Žsometimes termed as ‘apolar’ and ‘Lifshitz van der Waals’. component of the total surface energy, Since this equation takes only apolar interaction into account, the contact angle data with an apolar liquid Že.g. methyleneiodide, ␣-bromonapthalene. must be used as this liquid would only have dispersionrvan der Waals interaction with the bacterial cell surface.

2.6. Polar component of the solid surface energy Tamai and co-workers w37x introduced a term ‘I’ quantifying the interaction energy of the non-dispersive forces at the interface in Fowkes’ Eq. Ž13.: ␥sl s ␥s q ␥ l y 2 ␥sd ␥ ld y Isl

'

Ž 21.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 357

Fig. 9. Contact angle of a sessile drop in the two-liquid system of water and a hydrocarbon.

They used a two-liquid system to measure the water contact angle on the solid. Here the solid was immersed in a water-saturated hydrocarbon, the water drop was formed on the solid᎐hydrocarbon interface and then the contact angle was measured ŽFig. 9.. This was done in order to have minimum effect of the spreading pressure of water on the solid. The Young equation becomes ␥sh s ␥s w q ␥w h cos␪

Ž 22 .

using Tamai’s Eq. Ž21. we get ␥sh s ␥s q ␥ h y 2 ␥sd ␥ hd y Ish

Ž 23.

␥s w s ␥s q ␥w y 2 ␥sd ␥wd y Is w

Ž 24.

'

'

Combining Eqs. Ž22. ᎐ Ž24. and because Ish s 0, we get ␥ h y 2 ␥sd ␥ hd s ␥w y 2 ␥sd ␥wd y Is w q ␥ hw cos␪

'

'

Ž 25.

In this equation we have two unknowns and hence contact angle data of water on solid is required in presence of two different hydrocarbons as the third phase. The polar component of surface energy is not limited to dipole interactions but includes all of the non-dispersive forces such as hydrogen bonding. In fact, in condensed phases, dipole interactions are small and Fowkes w34x has concluded that ‘polarity’ as measured by dipole moments is not a significant factor in intermolecular interactions in liquids and solids. If the ionization potentials of the two phases are assumed to be equal and the polar component is assumed to be dominated by dipole᎐dipole interactions, one obtains the geometric mean equation w30,38x. However, if the polarizability of the two phases is assumed to be equal Žand assuming that the polar component has the same form as the dispersive component., one obtains the harmonic mean equation w39,40x.

358 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Owens and Wendt w30x proposed the division of the total surface energy of a solid or liquid in two componentsᎏdispersion force component and hydrogen bonding component Ž ␥ l s ␥ ld q ␥ lh and ␥s s ␥sd q ␥sh .. The interaction energy of the non-dispersive forces at the interface was quantified and included as geometric mean of the non-dispersive components of solid and liquid. The equation proposed was the extension of the equation proposed by Fowkes wEq. Ž12.x. ␥sl s ␥s ¨ q ␥ l ¨ y 2 ␥sd ␥ld y 2 ␥sh ␥ lh

'

'

Ž 26.

This can be alternatively expressed as ␥sl s

ž '␥

d s

y

'␥ / d l

2

q

ž '␥

h s

y

'␥ / h l

2

Ž 27.

The geometric mean combination of ␥sh and ␥ lh was used even though the fact that hydrogen bonding interactions are more specific in nature. The main reason was that the authors believed that hydrogen bonding is similar to the dipole᎐dipole interactions that take the form of a geometric mean. Using the Young equation and neglecting the spreading pressure wEq. Ž1.x, the following expression is obtained: 1 q cos␪ s 2 ␥sd

'

'␥

d l

ž /

q 2 ␥sh

'

␥l ¨

'␥

h l

ž /

Ž 28.

␥l ¨

Since this equation has two unknowns, ␥sd and ␥sh , the contact angle data from two liquids are needed. In general the liquids used are water and methyleneiodide. Owens and Wendt evaluated the surface energy of many polymers and then compared them to the values obtained by Zisman using the critical surface tension approach. A fairly good agreement was observed between the values from both the approaches. The differences, if any, were explained using the following arguments. According to Zisman the critical surface tension is the value of ␥ l ¨ at the intercept of the plot cos␪ vs. ␥ l ¨ with the horizontal line, cos␪ s 1, i.e. when ␥ l ¨ s ␥c then cos␪ s 1 putting this in Young’s Eq. Ž1., we get cos␪␥ l ¨ s Ž 1 . ␥c s ␥s ¨ y ␥sl

Ž 29 .

Although many workers have been inclined to identify ␥c with ␥s ¨ , Zisman has been careful to point out that ␥c is symbatic with, but not necessarily equal to the solid surface free energy because it is not certain that ␥sl and ␲ s 0 when ␪ s 0. In fact it has been shown that ␥sl is usually not equal to zero when ␪ s 0. Combining Eq. Ž27. and Eq. Ž29. we get ␥c s ␥ s y

ž '␥

d s

y

'␥ / d l

2

q

ž '␥

h s

y

'␥ / h l

2

Ž 30.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 359

From Eq. Ž30. it is apparent that ␥s y ␥c G 0. The following four cases follow and should be recognised: Use of non-polar liquid Ž ␥ lh s 0. to determine ␥c of non-polar solids Ž ␥sh s 0.. Here ␥sh s ␥lh s 0 and ␥ l s ␥ ld s ␥c substituting these in Eq. Ž30. we get ␥s s ␥sd s ␥c , i.e. the critical surface tension gives the dispersion surface energy Ž ␥sd . of the solid, which is equal to ␥s . II. Use of non-polar liquid Ž ␥ lh s 0. to determine ␥c of polar solids Ž ␥sh / 0.. Here ␥ lh s 0 and ␥ l s ␥ ld s ␥c substituting this in Eq. Ž30. we again get ␥sd s ␥c / ␥s , i.e. the critical surface tension gives only the dispersion part of the total surface free energy of the solid. III. Use of polar liquid to determine ␥c of non-polar solids Ž ␥sh s 0.. Here ␥ l s ␥ ld q ␥lh and ␥s s ␥sd. Substituting these in Eq. Ž30. we get ␥c s ␥s y w ␥ lh q Ž ␥sd y ␥ld . 2 x this means that the use of polar liquids to determine ␥c of non-polar solids leads to a value considerably less than ␥s . IV. Use of polar liquid Ž ␥ lh / 0. to determine ␥c of polar solids Ž ␥sh / 0.. Here ␥ l s ␥ ld q ␥lh and ␥s s ␥sd q ␥sh. Substituting these values in Eq. Ž30. we get ␥c s ␥s y wŽ ␥sh y ␥ lh . 2 q Ž ␥sd y ␥ ld . 2 x which means that the use of polar liquids to determine ␥c of polar solids leads to a value considerably less than ␥s . I.

'

'

'

'

'

'

Wu w39,40x proposed the harmonic mean to combine the polar and dispersion components of the solid and liquid surface energies in order to obtain the solid᎐liquid interfacial energy and proposed the following expression wEq. Ž31.x: ␥sl s ␥s q ␥ l y 4

␥ ld ␥sd ␥ld q ␥sd

q

␥ lp ␥sp ␥ lp q ␥sp

Ž 31 .

On combining this to Young’s equation, the following relation is obtained: 1 q cos␪ s

4

␥ ld ␥sd

␥ l ¨ ␥ld q ␥sd

q

␥ lp ␥sp ␥ lp q ␥sp

Ž 32 .

Similar to the Geometric mean approach, the contact angle data with two liquids are required in order to obtain the polar and dispersion components of the solids surface energy. Dalal w41x performed a comparative study of the two approaches, Geometric mean and Harmonic mean, on 12 common polymers using the published data with six liquids. It is found that the total surface energy of the solid obtained by the two methods are generally quite close, and neither of the two conceptually different equations is clearly incompatible with the available experimental data. However, the more widely used Geometric mean approach is preferable because it consistently fits the data better. Since Geometric mean approach is widely used and is found to be consistent, we have used only the geometric mean in the analysis.

360 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Dalal w41x developed a method for the best-fit solution of simultaneous equations, which are obtained when more than two contact angle liquids are used. This method is developed for both the Geometric mean and Harmonic mean systems. Later during the analysis of the Geometric mean approach the aforementioned minimisation method has not been used but least square method is used to solve the over-determined systems of equation. Since the contact angle data with four liquids is available for each microbial surface, the following matrix has been solved using least square method by Matlab 5.3.

'␥ '␥ '␥ '␥

d W

'␥ '␥ '␥ '␥

p W

 0 d F

d M

d Br

p F

p M

Ž 1 q cos␪W . r2 ␥sd

'

 0 ␥sp

'

s

p Br



Ž 1 q cos␪ F . r2 Ž 1 q cos␪ M . r2 Ž 1 q cos␪ B r . r2

0

Ž 33.

2.7. Equation of state approach [42,43] Considering the surface thermodynamics of a two component three-phase solid᎐liquid᎐vapour system and assuming ideal solid, i.e. smooth, homogeneous, rigid with no appreciable vapour pressure, Ward and Neumann w52x showed that an equation-of-state type relation exists between the solid᎐vapour, the solid᎐liquid and the liquid᎐vapour interfacial tensions. The solid᎐liquid interfacial tension is a unique function of liquid and solid surface tensions. The Gibbs᎐Duhem equation for the three interfaces can be written as s¨ d␥s ¨ s yS1s ¨ dT y ⌫2Ž1. d␮ 2

Ž 34 .

s¨ d␥sl s yS1sl dT y ⌫2Ž1. d␮ 2

Ž 35 .

s¨ d␥ l ¨ s yS1l ¨ dT y ⌫2Ž1. d␮ 2

Ž 36 .

where: 䢇 䢇 䢇 䢇

S1s ¨ is the surface entropy of the solid᎐vapour interface; T is the absolute temperature; ␮ 2 is the chemical potential of the liquid components; s¨ ⬘ ⌫2Ž1. is the surface excess concentration of component 2 Žliq.. at solid liquid interface.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 361

The above equations indicate that each of the surface tensions is a function of T and ␮ 2 , i.e. ␥s ¨ s ␥s ¨ Ž T ,␮ 2 .

Ž 37.

␥sl s ␥sl Ž T ,␮ 2 .

Ž 38 .

␥ l ¨ s ␥l ¨ Ž T ,␮ 2 .

Ž 39 .

Therefore, there are three equations in terms of two variables and hence, any one of them can be expressed in terms of the other two, i.e. ␥s ¨ s f Ž ␥l ¨ ,␥s ¨ .

Ž 40 .

In 1989 Li and co-workers w44x gave another proof of the existence of equation of state by using modified phase rule for capillary system with curved surface. Classical Gibbs phase rule states, FsCq2yP

Ž 41.

where, F is degree of freedom, C is independent chemical components and P is the number of phases. This equation is not universally applicable. If the concentration of any set of components can be related by an equilibrium constant, then the number of degrees of freedom is reduced by 1. Also explicit in the derivation of Eq. Ž41. is the assumption that if the temperature and pressure in one phase is known, they are determined in all other phases. Naturally having temperature and pressure the same in all phases satisfies this condition. If a system contains curved interfaces, such as sessile drop on a substrate, then the pressure will not be the same in all phases. The change in pressure between the two phases can be calculated by the Laplace equation: P ␣ y P␤ s ␥␣␤J ␣␤

Ž 42 .

where ␥ ␣ ␤ is the interfacial tension and J ␣ ␤ the curvature. The curvature must be known to calculate the pressure difference across the interface. This apparently introduces a new degree of freedom for each curved interface, the curvature of the interface, and hence that the Gibbs phase rule has to be modified. However, the system must have negligible boundary effects, i.e. all boundaries must be thermally conducting, deformable and permeable to all components, no chemical reaction must occur and volume is the only work ordinate ŽW s Pd¨ .. These conditions are not satisfied by the system under consideration ŽFig. 1. therefore, a different form of phase rule has to be used to determine the number of independent intensive variables or degree of freedom. Essentially, the number of degrees of freedom or variance of any composite system is evaluated by subtracting the equilibrium constraint equations from the variables used to describe the composite system.

362 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

For a multicomponent, multiphase system in equilibrium if P of the total number of bulk or volume phases with P independent chemical components. Each ␣ , where x i␣ for bulk phase ␣ can be described by variables T ␣ , P ␣ , x 1␣ , x 2␣ ,..... x ry1 i s 1,2,3,.....,P y 1 is the bulk mole fraction of the ith component in the phase ␣. Consequently, for all P phases the required number of intensive variables will be P Ž C q 1.. Since these coexisting phases are all in equilibrium, the intensive variables are constrained to satisfy thermal, mechanical and chemical equilibrium conditions. Thermal equilibrium T ␣ s T ␤ s ........s T n

P y 1 equations

Chemical equilibrium ␮␣i s ␮ ␤I s ........s ␮nI

C Ž P y 1 . equations

Mechanical equilibrium are of three types: I. Laplace equations, P ␣ y P ␤ s ␥␣ ␤ J ␣ ␤

Ž 43 .

Where ␣ and ␤ represent adjacent bulk phases separated by a curved liquid᎐fluid interface and J ␣ ␤ is the mean curvature of the ␣␤ interface. If the interface is planar, i.e. J ␣ ␤ s 0, then this equation reduces to P ␣ s P ␤ the equilibrium conditions used in the original derivation of Gibbs phase rule. II. Young equation. III. Neumann triangle relations. 2

2

2 ␥ 12 ␥ 23 cos␪ s Ž ␥ 12 . y Ž ␥ 23 . y Ž ␥ 13 .

2

Ž 44.

Therefore, the total constraint equations equal to Ž P y 1. q C Ž P y 1. q N

Ž 45 .

where, N is the total number of distinct P ␣ s P ␤ type relations. Therefore, the surface system phase rule is F s P Ž C q 1 . y wŽ P y 1 . q C Ž P y 1 . q N x FsCq1yN

Ž 45a . Ž 46 .

Applying this to the system in consideration ŽFig. 3a. where we have three bulk phases, three surface phases, two components Ž C s 2. and if the solid phase is isotropic i.e. P s s P ¨ Ž N s 1. so F s 2. This means that any two of the intensive variables describing the system can independently vary and any other variable is then a function of the other two, i.e. an equation of the form wEq. Ž40.x ␥s ¨ s f Ž ␥l ¨ ,␥s ¨ . .

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 363

Fig. 10. General liquid᎐liquid lens᎐fluid system with Ža. three curved interfaces and Žb. two curved interfaces.

Li w46x applied the modified phase rule to liquid᎐liquid lens fluid system consisting of three bulk fluid phases of different densities as shown in Fig. 10a. Along with the three bulk phases, there are also three surface phases. If all the surfaces are curved, the surface mechanical equilibrium conditions will be P 3 y P 1 s ␥ 31 J 31

Ž 47 .

P 3 y P 2 s ␥ 32 J 32

Ž 48 .

P 2 y P 1 s ␥ 21 J 21

Ž 49 .

There are no pressure equality relations of the type P ␣ s P ␤ since all the interfaces are curved. As a consequence, N s 0 and the phase rule wEq. Ž46.x yields F s C q 1. For a two-component system the degrees of freedom are 3 and the equation of state of the form 40 does not exist. Only under very special conditions shown in Fig. 10b where the interface between fluid 1 and liquid 2 is planar the degrees of freedom are 2 and the relation of the type 40 may exist. This is because J 12 s 0 and the mechanical equilibrium condition given in Eq. Ž49. is replaced by a pressure equality relation P ␣ s P ␤ and hence the phase rule yields F s C. For the two-component system the degrees of freedom come out to be 2. This is important because liquid-liquid contact angle, which belongs to Fig. 10a condition, cannot be used to either validate or question the usefulness of equation of state approach. Even without having an explicit formulation of this equation of state and just with the knowledge of its existence, a test for determining whether adsorption at the solid᎐vapour interface occurs, was proposed in the aforementioned paper w44x. If a system is considered where adsorption at the solid᎐liquid interface is absent then the solid᎐vapour interfacial tension, ␥s ¨ , is independent of vapour in contact

364 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

and has a value of ␥s o . The surface tension of the solid᎐liquid interface, ␥sl , is given by ␥slo s f Ž ␥s o ,␥ l ¨ .

Ž 50 .

where ␥slo denotes the solid᎐liquid interfacial tension in the absence of adsorption. Substitution of ␥slo in the Young equation wEq. Ž1.x we get ␥s o y f Ž ␥s o ,␥ l ¨ . s ␥ l ¨ cos␪

Ž 51 .

If now, succession of liquids are brought in contact with the solid, and if there is no adsorption and temperature is maintained the same in each case, then ␾Ž ␥l ¨ . s ␥ l ¨ cos␪. That is, if there is no adsorption of the vapour of any liquids, then from Young’s equation and existence of equation of state one concludes that there must be a correlation between ␥ l ¨ and ␥l ¨ cos␪, i.e. a plot of ␥ l ¨ vs. ␥ l ¨ cos␪ will give a smooth curve as shown in Fig. 11. The difference between the Gibbs phase rule wEq. Ž41.x and the reformulated phase rule wEq. Ž46.x for the capillary systems is the term for the number of phases. For the Gibbs phase rule, the number of phases is equal to that of the bulk phase, the interfacial phase is not a phase in the same sense. For the reformulated phase rule; the number of phases is not taken into account as long as all the interfaces are significantly curved. For every interface insignificantly curved, the mean curvature is no longer a variable, and the number of degrees of freedom is actually reduced by 1. Li w46x suggested the moderately curved interface to be a radius of 1 mm. Neumann w42x developed the first form of equation of state, which allowed the determination of surface tension of low-energy solids from a single contact angle formed by a liquid which is chemically inert with respect to the solid and its surface tension is known. Experimental values obtained by measurement of contact angles with a series of liquids on eight different low energy solids are plotted on a graph ␥ l ¨ cos␪ vs. ␥ l ¨ . If there is no adsorption of vapour taking place at the solid᎐liquid interface then the points follow smooth curves ŽFig. 11.. Two general conclusions were drawn from the plots Žsecond conclusion was more a hypothesis.. ␥ l ¨ decreases as ␥ l ¨ cos␪ increases, this is true from the Young equation when ␥s ¨ is constant. 2. The slope dŽ ␥l ¨ cos␪.rd␥ l ¨ is zero at ␪ s 0, i.e. at ␥ l ¨ cos␪ s ␥s ¨ , hence a line at 45⬚ is plotted.

1.

A second-order polynomial of the type ␥ l ¨ cos␪ s a␥ l2¨ q b␥ l ¨ q c is fitted to the data. The intercept of the fitted curve to the 45⬚ line and the slope dŽ ␥ l ¨ cos␪.rd␥ l ¨ at the point of intersection are determined by solving a␥ l2¨ q Ž b y 1. ␥ l ¨ q c s 0 and then putting the value of ␥ l ¨ Žpoint of intersection . into dŽ ␥ l ¨ cos␪.rd␥ l ¨ s 2 a␥l ¨ q b. The average limiting angle of inclination calculated from the average

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 365

Fig. 11. Plot of ␥l ¨ cos␪ vs. ␥l ¨ of various liquids on: Ž1. methacrylic polymer A with fluorinated side chains; Ž2. methylacrylic polymer S with fluorinated side chains; Ž3. 17-Žperfluoropropylene.-heptadecanoicacid; Ž4. 17-Žperfluoroethyl.-heptadecanoic acid; Ž5. polytetrafluoroethylene; Ž6. 80-20 copolymer of tetrafluoroethylene and chlorotrifluoroethylene; Ž7. 60-40 copolymer of tetrafluoroethylene and chlorotrifluoroethylene; and Ž8. copolymer of tetrafluoroethylene and polyethylene. Taken from w135x.

slope of eight solids came out to be 0.1 " 4.2, here the standard deviation includes the value of zero and hence the aforementioned hypothesis is accepted. lim

d Ž ␥ l ¨ cos␪ .

␪ª0

d␥ l ¨

s0

Ž 52 .

Since ␥s ¨ is assumed to be constant d Ž ␥l ¨ cos␪ . d␥ l ¨

s

d Ž ␥s ¨ y ␥sl . d␥ l ¨

sy

d␥sl d␥ l ¨

Ž 53 .

366 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

which, implies that d␥sl

lim

␪ª0

d␥ l ¨

s 0.

Ž 54 .

From the fact that ␥s ¨ decreases as ␥l ¨ decreases and from Eq. Ž54. it is concluded that ␥sl has its minima, when ␪ s 0. From the knowledge of liquid᎐liquid interface where lower limit of interfacial tension between two liquids in equilibrium is zero, it was concluded that ␥sl has zero as its minimum value. lim ␥sl s ␥sl) s 0

Ž 55 .

␪ª0

The formulation of the equation of state was empirical curve fit to contact angle data in terms of the Girifalco and Good interaction parameter w45x. ␾s

␥s ¨ q ␥ l ¨ y ␥sl

Ž 56.

2 ␥ l ¨ ␥s ¨

'

The method was as follows: 1.

␥s ¨ is constant and ␥Usl s 0, so ␥s ¨ is determined from the plots by using ␥s ¨ s lim ␥ l ¨ s ␥Ul ¨

Ž 57 .

␪ª0

2. Using this value of ␥s ¨ and experimental values of ␥ l ¨ and cos␪, value of ␥sl is obtained as a function of ␪ from the Young equation wEq. Ž1.x. 3. Using values of ␥s ¨ , ␥ l ¨ and ␥sl , the interaction parameter, ␾ is evaluated. 4. ␾ vs. ␥sl is plotted for the eight solids ŽFig. 12., as can be seen from the plots that ␾ is a linear function of ␥sl and so straight lines of the type ␾ s ␣␥sl q ␤ were fitted to the plots. 5. Values of the two constants were found from the curve fitting and were found to be

␣s

d␾ d␥sl

s y0.0075 m2rmJ and

␤ s 1.000

Ž 58.

Hence, the equation of state is obtained

␥sl

ž '␥

y ␥l ¨

' / s 1 y 0.015'␥ ␥ s¨

2

s¨ l¨

Ž 59.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 367

Fig. 12. Good and Garifalco interaction parameter, ␾, as a function of ␥sl for the eight solids in Fig. 11. Taken from w135x.

Combining this to Young’s equation, we get cos␪ s

Ž 0.015␥s ¨ y 2.00. ␥s ¨ q ␥ l ¨ q ␥ l ¨

'

␥ l ¨ 0.015 ␥s ¨ ␥ l ¨ y 1

ž

'

Ž 60 .

/

Li and Neumann w46x gave another form of this equation of state using a totally different approach. Agreeing and continuing the Berthelots’s geometric mean combining rule for the attractive constant in van der Waals equation of state, the work of adhesion, Wsl , is taken as a geometric mean of the work of cohesion of the solid, Ws s , and liquid, Wl l , i.e. Wsl s Ws sWl l

Ž 61 .

where, Ws s s 2 ␥s ¨ ;Wl l s 2 ␥ l ¨

Ž 62 .

'

368 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

hence, Wsl s 2 ␥s ¨ ␥ l ¨ s ␥s ¨ q ␥ l ¨ y ␥sl

Ž 63 .

or, ␥sl s ␥s ¨ q ␥ l ¨ y 2 ␥s ¨ ␥ l ¨

Ž 64 .

'

'

This equation works only when ␥s ¨ values are very close to the value of ␥ l ¨ , because the function form is the result of Berthelot’s geometric mean combining rule, Wsl s Ws sWl l , which is valid only when Ws s f Wl l . To modify this combining rule, Good and Girifalco introduced the interaction parameter w45x.

'

Wsl s ␾ Ws sWl l

'

Ž 65 .

where ␾ is found to be below 0, this means that the geometric mean combining rule over estimates the value of Wsl . In general, the geometric mean combining rule is applicable to bulk phases also. In theory of intermolecular interaction and the theory of mixtures the combining rule is used to evaluate the parameters of unlike-pair interactions in terms of the like pair interactions. Berthelot’s geometric mean combining rule wEq. Ž66.x is only a useful approximation. ␧i j s

'␧

ii ␧ j j

Ž 66 .

where, ␧ i j is the energy parameter of unlike-pair interaction and ␧ ii and ␧ j j are energy parameters for like-pair interactions. By London’s theory of dispersion forces it has been shown w47x that geometric mean combining rule is applicable only for similar molecules, because implicit in this rule is the condition that the two energy parameters of the like-pair interaction must be very close to each other, i.e. ␧ i i f ␧ j j . However, for the interaction between two very dissimilar molecules of the material, where there is an apparent difference between ␧ i i and ␧ j j , it has been demonstrated that the geometric mean combining rule generally overestimates the strength of the unlike-pair interactions. In the study of mixtures commonly a factor like, Ž1 y k i j . is introduced in the combining rule, i.e. ␧ i j s Ž 1 y k i j . ␧ ii ␧ j j

'

Ž 67 .

where k i j is an empirical parameter quantifying deviation from the geometric mean. The factor should decrease with the difference, Ž ␧ ii y ␧ j j ., and be equal to 1 when Ž ␧ ii y ␧ j j . is zero. Based on this Li and Neumann w46x introduced a modified combining rule ␧i j s

'␧

ii ␧ j j

ey␣ Ž␧ i iy␧ j j .

2

Ž 68 .

where, ␣ is an empirical parameter and the square of Ž ␧ ii y ␧ j j . is used rather than the difference itself for taking into account the symmetry of the combining rule.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 369

Correspondingly, for the case of large differences between Wl l and Ws s ; ␥ l ¨ and ␥s ¨ , the combining rule for the work of adhesion of a solid᎐liquid pair is written as: Wsl s Ws sWl l ey␣ ŽW l lyW s s .

'

2

Ž 69 .

Introducing values from Eqs. Ž62. and Ž63. we get ␥sl s ␥s ¨ q ␥ l ¨ y 2 ␥s ¨ ␥ l ¨ ey␤ Ž␥ s¨ y␥ l¨ .

'

2

Ž 70 .

Combining this with the Young equation wEq. Ž1.x we get

(

cos␪ s y1 q 2

␥s ¨ ␥l ¨

ey ␤Ž␥ s¨ y␥ l¨ .

2

Ž 71 .

For the set of ␥ l ¨ and ␪ data obtained by different liquids on the same solid, the constants ␤ and the surface energy of solid-␥s ¨ can be determined by least-square analysis method. While, starting with arbitrary values of ␥s ¨ and ␤, iterative procedure is used to estimate those values of ␥s ¨ and ␤, which best fit the experimental data. The value of ␤ s 0.000115 Žm2rmJ. 2 was obtained w46x by fitting Eq. Ž71. to the data presented in Fig. 11 and used by Neumann w42x in the derivation of the first form of the equation of state wi.e. Eq. Ž59.x. However, later a value of ␤ s 0.0001247 Žm2rmJ. 2 was obtained w48x by fitting Eq. Ž71. to accurate contact angle data obtained on very smooth and homogeneous solid surfaces of polyethylene terephthalate ŽPET., fluorinated ethyl propylene ŽFEP. and glass coated with fluoropolymer FC-721. The contact angle was measured by ADSA-P Žasymmetric drop shape analysis-profile . and capillary rise techniques. Kwok w49x; Kwok and Neumann w25,50x modified the Berthelot’s rule by introducing another modifying factor of the form 1 y ␬ 1Ž ␧ ii y ␧ j j . 2 which is a decreasing function of the difference Ž ␧ ii y ␧ j j . and is equal to 1 when ␧ ii s ␧ j j . Here ␬ 1 is an unknown constant. Therefore, after modification the Berthelot’s combining rule Eq. Ž66. becomes ␧ i j s 1 y ␬ 1Ž ␧ ii y ␧ j j .

ž

2

/ '␧

ii ␧ j j

Ž 72 .

The square of the difference Ž ␧ i i y ␧ j j ., rather than the difference itself is used which reflects the symmetry of this combining rule. Since free energy is directly proportional to the energy parameter ␧, then for the cases of large differences, i.e. < Wl l y Ws s < or < ␥ l ¨ y ␥s ¨ < the combining rule for the energy of adhesion of a solid liquid pair is written as Wsl s Ž 1 y ␣ 1Ž Wl l y Ws s .

2

. 'Wl lWs s

Ž 73 .

370 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

where, Wl l s 2 ␥ l ¨ and Ws s s 2 ␥s ¨ Wsl s 2 Ž 1 y ␤ 1Ž ␥l ¨ y ␥s ¨ .

2

. '␥l ␥s ¨

Ž 74.

¨

where, ␤ 1 s 4␣ 1 ␥sl s ␥l ¨ q ␥s ¨ y 2 ␥l ¨ ␥s ¨ Ž 1 y ␤ 1Ž ␥ l ¨ y ␥s ¨ .

'

2

.

Ž 75.

Combining Eq. Ž75. with Young’s equation Eq. Ž1. we arrive at

(

cos␪ s y1 q 2

␥s ¨ ␥l ¨

Ž 1 y ␤1Ž ␥l

¨

y ␥s ¨ .

2

.

Ž 76.

A value of ␤ s 0.0001057 Žm2rmJ. 2 was obtained by using the same least square method Žas used by Li and Neumann w48x. to fit the contact angle data obtained by various liquids on 15 homogeneous and smooth solids. The contact angles were measured using both ADSA-P and capillary rise techniques. The solids and techniques used for generating the data were FluoropolymersFC-721 on mica by capillary rise, FC-722 on mica by ADSA-P, FC-722 on silicon wafer by ADSA-P, FC-725 on silicon wafer by ADSA-P, Teflon by capillary rise, hexatriacontane by capillary rise, cholestery acetate by capillary rise, polywpropanealt-N-Ž n-hexyl.maleimidex by ADSA-P, polyŽ n-butyl methacrylate . by ADSA-P, polystyrene by ADSA-P, polywstyrene-Žhexylr10-carboxydecyl 90:10.-maleimidex by ADSA-P, polyŽmethyl methacrylatern-butyl methacrylate . by ADSA-P, polywpropene-alt-N-Ž n-propyl.maleimidex by ADSA-P, polyŽmethyl methacrylate . by ADSAP and polyŽpropene-alt-N-methyl maleimide. by ADSA-P. More detailed data can be found in Table 4 of Kwok and Neumann w25x. 2.8. Objections on equation of state approach The initial formulation of equation of state w42x was critically analysed by Van de Ven w51x and some objections were presented. Before the formulation of equation of state w42x some basic assumptions were made: 1. There exists a relation of the form ␥s ¨ s f Ž ␥ l ¨ , ␥s ¨ . w44x. ŽTheoretical assumption.. 2. For all non-spreading liquids Ž ␪ ) 0., as the contact angle approaches zero Ž ␪ ª 0q. , ␥sl approaches a minimum value ␥sl) which depends only on the properties of the solid ŽExperimental assumption.. 3. For ␪ ) 0, plot of the Goods interaction parameter, ␾, wEq. Ž56.x as a function of ␥sl yield an indeterminate series of straight lines with slopes and intercepts determined by the choice of ␥sl) Žor ␥s ¨ . ŽExperimental assumption.. 4. Among all conceivable liquids there exists at least one for which ␥sl s 0 equivalently ŽGoods interaction parameter. ␾ s 1. ŽPhilosophical assumption..

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 371

From the second and fourth assumptions, it was concluded w42x that ␥sl) s 0, i.e. lim ␥sl s ␥Usl s 0

Ž 77a .

lim ␾ s ␾) s 1

Ž 77b .

␪ª0 q

␪ª0 q

but the authors point out that this does not logically follow from assumptions 2 and 4. There are many spreading liquids Ž ␪ s 0. for which no information about ␥sl exists because Young’s equation wEq. Ž1.x does not apply. The ␥sl for spreading liquids can be less than the ␥sl) , therefore no conclusion about the absolute minima of ␥sl for all the liquids can be made by considering only the experimental evidence for non-spreading liquids. It cannot be assumed that the absolute minima of ␥sl for the spreading liquids as ␪ ª 0q is zero, i.e. ␥sl) s 0, in other words the minimum value of ␥sl can be negative. In the same paper the plots of ␾ vs. ␥sl , Fig. 13 ŽFig. 3 of w42x. with lines passing through ␾ s 1 and ␥sl s 0 were accepted but the other lines which had points for which ␾ ) 1 or when ␾U - l were discarded. This means that the applicability of the equation of state was within a short range of ␾ or only when ␥sl q ␥sl F 130 ergsrcm 2 which is not satisfied by some systems like blood cells in aqueous media. It is implicit from assumption 4 that ␥sl - 0 is impossible, thus zero is taken as the lowest possible value. However, from statistical mechanics ␥sl consists of two parts ␥sl s ␥s q ˜ ␥l

Ž 78 .

where ␥s is solid surface energy and ˜ ␥ l is that of liquid in the external field created by the solid. Although ␥s ) 0 but ˜ ␥ l can be negative; physically which means that the liquid density increases near the solid᎐liquid interface, it is difficult to see why ˜ ␥ l ) y␥s . In principle, ␥sl can have any value and ␥sl s 0 deserves no special status. Thus, assumption 4, although probably true, becomes irrelevant. However, Johnson and Dettre w53x doubted if arbitrarily small Žpositive. solidrliquid interfacial energies were not possible when the contact angle approaches zero. They tested the assumption on hexane᎐water system, where the equivalent contact angle is zero even when the interfacial tension is approximately 51 mJrm2 . The test assumes that the equation of state at liquid᎐liquid interface of the form 70 also exists. ␥ l 1 l 2 s f Ž ␥ l 1¨ ,␥ l 2 ¨ .

Ž 79 .

The equivalent contact angle is calculated using the following equation and the liquid᎐liquid interfacial tensions are evaluated using the equation of state. cos␪ Eq s

␥ l 1 l 2 y ␥ l 1¨ ␥l 2 ¨

Ž 80 .

372 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 13. Goods parameter Ž ␾ . vs. solid᎐liquid interfacial tension for different ␥s ¨ values such that ŽI. ␥s ¨ - 0 Ž`. ␥sl s 0 and Ž^. ␥sl ) 0. Taken from w42x.

The second test depends on the corollary, deduced from the equation of state approach, that if one liquid has the same contact angle on two different substrates, Zisman plots for the surfaces should superimpose. Since the PTFE, water and stearic acid, as pointed out by Johnson and Dettre w53x cross but do not superimpose, they conclude that the equation of state approach is neither applicable to liquids and nor to solids. Li w54x refuted Johnson and Dettre’s w53x assumption that the thermodynamics of equation of state does not require the substrate to be solid. Almost the same period when Johnson and Dettre’s comments were published, Li and Neumann w44x proposed an alternate proof for the existence of the Equation of state using the phase rule for heterogeneous system containing moderately curved surfaces wEq. Ž46.x. This phase rule wEq. Ž46.x also predicts that the number of degrees of freedom for two-component liquid᎐liquid lens᎐fluid system ŽFig. 10a. is generally three; only for the special case where one of the interface in the liquid lens system is planar is the number of degrees of freedom 2. These results tell us that generally for a two-component liquid᎐liquid lens fluid system an equation of state of the form Eq. Ž79. does not exist.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 373

Neumann w55x, defend their previous w42x assumption of taking ␥sl s 0 as the least possible value and further discussed the possibility of negative solid᎐liquid interfacial tension. The first argument is that it is generally accepted interfacial tension between liquid᎐liquid and liquid᎐solid under conditions of stable equilibrium cannot be negative. The negative tension if it exists would cause mixing of two fluids or dissolution of solid in the liquid phase leading to formation of a homogeneous phase. Pairs of polymer melts which are insoluble to a degree comparable with that of polymer solids and low-molecular-weight liquids are considered w55,56x. In the plots of ␾ vs. ␥ 12 a straight line is achieved with limiting values of ␾ s 1 and ␥ 12 s 0, even when the free energy of spreading is negative for some and positive for some pairs, and condition for negative interfacial tension is conducive for some pairs. This is in excellent agreement with the assumption that the minimum possible value of ␥sl is zero. Neumann w55x also considered the interaction of small particles initially suspended in liquid, with an advancing solidification front, i.e. solidification front experiment. When the particle encounters the solidification front, it is engulfed in the solid or is swept along by the solidification front. Particle engulfment or rejection depends upon the free energy of adhesion or engulfment. ⌬G Adhesion s ␥p s y ␥p l y ␥sl

Ž 81a .

⌬G En gulfment s ␥p s y ␥p l

Ž 81b .

Negative free energies mean that the particle will be engulfed by the solidification front otherwise not. The surface tension of the melt, ␥ l ¨ , and the contact angles on the solid matrix material and particle material are measured. From these contact angles the solid surface tension, ␥s ¨ , is evaluated using the equation of state approach and then the interfacial tensions required in Eqs. Ž81a. and Ž81b. are also evaluated using the equation of state approach. The straight lines in Fig. 12 can be represented by ␾ s ␤ y ␣␥sl

Ž 82 .

The solidification front observations are used to test the validity of ␤ s 1 which means that ␥sl s 0 is the minimum possible value of ␥sl . The free energy of adhesion and engulfment wEqs. Ž81a. and Ž81b.x are evaluated using first ␤ s 1 and then by using 5% higher value of ␤ s 1.05, which means negative values of ␥sl . The free energy values are reported in Table 1 of Neumann et al. w55x for pairs of polymer particles suspended in polymer melts. Only when ␤ is taken to be 1 the free energies predict the engulfment or rejection correctly. When ␤ s 1.05 is taken the free energy values fail to predict the engulfment. Morrison w57x critically examined the thermodynamic proof given by Ward and Neumann w52x for the existence of Equation State wEq. Ž40.x at the solid᎐liquid

374 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

interface. The work draws attention to three errors committed during the derivation. The first error is the error of omission, while writing the Gibbs᎐Duhem equations for the three interfaces, i.e. Eqs. Ž34. ᎐ Ž36. and then transforming them into Eqs. Ž37. ᎐ Ž39.. The equations written show that the interfacial energies to depend on the properties of only one of the two components, the chemical potential of the fluid, ␮ 2 , and the temperature, T. Since the three equations wEqs. Ž34. ᎐ Ž36.x for the interfacial energies are shown as function of T and ␮ 2 only, the interfacial energies appear to be independent of the solid. Ward and Neumann w52x have gone from the relations between differentials d␥, dT and d␮ to undefined relations between integral quantitiesᎏ␥, T and ␮ 2 without adequate consideration of the dependence of the thermodynamic functions on the properties of both the fluid and the solid. The relations between integral quantities must be as follows and not Eqs. Ž34. ᎐ Ž36. s¨ ␥ s ¨ s u1s ¨ y TS1s ¨ y ⌫2Ž1. ␮2

Ž 34a .

sl ␥ sl s u1sl y TS1sl y ⌫2Ž1. ␮2

Ž 35a .

␥ l ¨ s u l ¨ y TS l ¨ y ⌫2l ¨ ␮ 2

Ž 36a .

Each of the interfacial energies is a function of T and ␮ 2 but they are also functions of specific internal energy, u, specific entropy, S, and specific adsorption, ⌫, all depend both upon the liquid and the solid. The significance of being explicit about the dependence of interfacial energy on each of the thermodynamic functions if that it makes clear the dependence of the interfacial energy on chemistry of both of the components, something which is lacking in the equation used by Ward and Neumann w52x saying that ␥ s ␥ Ž T ,␮ 2 .

Ž 37b .

is true but incomplete, as it does not show the material dependence of the internal energy on both the fluid and solid. Showing the explicit dependence of the surface tension on material-dependent quantities point out that the equations for interfacial energy will be different for every pair of materials. Therefore, Eq. Ž37b. is not only incomplete but also incorrectly implies that the equations are same for all materials. The more exact representation is as follows s¨ s¨ Ž ␥ 2Ž1. s ␥ 2Ž1. T ,␮ 2 .

Ž 37a .

sl sl Ž ␥ 2Ž1. s ␥ 2Ž1. T ,␮ 2 .

Ž 38a .

l¨ l¨ Ž ␥ 2Ž1. s ␥ 2Ž1. T ,␮ 2 .

Ž 39a .

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 375

Here the extra set of subscripts show that each equation will be different for different pairs of fluids and solids. The rearrangement would lead to a different function for every pair of materials. No universal expression for any interfacial energies, ␥s ¨ , ␥sl and ␥s ¨ can be shown to exist. Johnson and Dettre w53x also pointed out this discrepancy and supports that the set of Eq. 37a᎐39a will be different for different sets of solids and liquids. Otherwise two liquids having same surface tension will have the same contact angle on the same solid surface, which does not have any scientific justification. The dependence of interfacial energy on the chemical nature of the components can be made even more explicit through statistical mechanics, where the interfacial energy can be derived from an analysis of the grand canonical ensemble for this system. ␥ s Ž kT ln⌶ . rA

Ž 83 .

where the grand partition function, ⌶, is given by, N

⌶s

Ý e ŽyE Ž N .r kT . eŽ N␮ j

2

r kT .

Ž 84 .

j

Ej is the total energy of N molecules in the interface of area A. Note that by combining Eqs. Ž83. and Ž84. the functional form of Eq. Ž37b. can be written. The possible energy level of each interface depends on the detail of the intermolecular interactions and are different for every pair of materials. The total energy of the molecules in the interface depends on the chemistry of both fluid and solid. The second error is an error in thermodynamics. Ward and Neumann w52x claim that at equilibrium the interfacial energies are function of only two independent variables, T and ␮ 2 which is not true. The confusion is about the number of independent quantities necessary to analyse the sessile drop, that is; between the number of degree of freedom of the system and the number of independent parameters necessary to calculate a thermodynamic property of the system. Considering the ideal system of a sessile drop of liquid and its vapour contacting a uniform solid substrate in which the liquid and solid are insoluble in each other. The numbers of degree of freedom can be calculated by applying phase rule, Eq. Ž41., here P s 3, C s 2, therefore, d.f.s 1, most conveniently temperature. This means that the fundamental equations of Ward and Neumann w52x could just be written as function of only temperature. ␥ s ␥ŽT .

Ž 85 .

If one argues that the curvature of a phase can introduce a new degree of freedom hence making T and ␮ 2 as the factors. However, this is only true when the curvature is less than a micron. For observable systems, such as the sessile drop being considered by Ward and Neumann and for systems to which the equation of state approach has been applied, the usual phase rule applies and the number of degrees of freedom is 1.

376 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

The third error is the error in mathematics and is consequence of the first two errors. The statement that the three equations depend on only two variables and therefore can be solved to eliminate one variable is incorrect, i.e. ␥sl / f Ž ␥s ¨ ,␥ l ¨ .

Ž 86 .

In fact these three equations are functions of unknown number of material variables as well as temperature. None of the interfacial energies is necessarily a function of just the other two. Considering that all interfacial energies are fixed at thermal equilibrium, and hence the three interfacial energies for the sessile drop wEqs. Ž37a., Ž38a. and Ž39a.x would be determined when the chemical difference between the fluid and the solid can be characterised, at least approximately, by two material variables, one for the fluid and one for solid. The energy levels, Ej , in the grand partition function wEq. Ž84.x would then be functions of only two material-dependent parameters. Examples of such systems are fluids and solids that interact purely by dispersion forces, where each material is characterised by single London constant and the intermolecular distances of closest approach are similar. According to Gaydos et al. w59x the above three criticisms by Morrison w57x are unfounded. The error of omission is unfounded because ␮ 2 used in the equations are interfacial chemical potentials: ␮Ž2l ¨ . s

⭸U Ž l ¨ .

ž / ⭸N2Ž l ¨ .

Ž 87 . S Ž l ¨ . , AŽ l ¨ . , N1Ž l ¨ .

where NlŽ l ¨ . s 0 by definition and not the bulk liquid chemical potential defined by ␮Ž2l . s

⭸U Ž l .

ž / ⭸N2Ž l .

Ž 87a . S Ž l . ,V Ž l . , N1Ž l .

and ␮Ž2l ¨ . is dependent upon both the liquid and vapour phases. It is not surprising that ␮Ž2l ¨ . has properties which reflect the explicit nature of its interface as similar effects occur with surface tension. For example, of oil᎐water and nitrogen᎐water interface considered, then it is not expected that ␥ o w s ␥ nw and similarly ␮Ž2o w . / ␮Ž2nw .. This suggests that the value of ␮Ž2l ¨ . is a reflection of the material properties of the interface’s adjacent phases. The difference between the ‘surface of tension’ dividing surface and the dividing surface of ‘zero mass’ is not understood by Morrison w57x. For any planar interface it is possible to shift these dividing surfaces to the same location. Consequently, in the relations like Eqs. Ž34a., Ž35a. and Ž36a. one discovers that the ␮ 2 terms are evaluated at a dividing surface position such that ⌫1 ' 0 Žzero mass of solid.. It is also explicitly stated by Ward and Neumann w52x that ‘there is no dissolution of the solid nor is there any adsorption of any components from the liquid or gaseous phase by the solid’. Taken together, it is not surprising that the solids properties do not appear explicitly in the final expression.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 377

Lee w58x categorically rejects this claim. He states that all writers on equilibrium thermodynamics point out that, at equilibrium, the chemical potential of each component has a constant value everywhere in the closed system. There can only be one chemical potential whether in bulk or at any interface, whether curved or flat. There is no such thing as a distinct ‘interfacial chemical potential’ that is not identical with that in each of the bulk phases. Of course, there exists a concentration gradient across any interface. Definitely, however, there is no corresponding gradient of any chemical potential; thus, ␮Ž2l ¨ . ' ␮l2 ' ␮¨2 at equilibrium. The second error is in thermodynamics, where according to Morrison only one thermodynamic variable is necessary to define the system, which would most conveniently be the temperature. However, Gaydos w59x points out that the normal Gibbs phase rule is not applicable in the present case but the phase rule for heterogeneous system Eq. Ž46. is applicable according to which the degrees of freedom are two-temperature and interfacial chemical potentials. The third criticism of Morrison w57x is error of mathematics, while proceeding from differential relations Eqs. Ž34. ᎐ Ž36. to integral relations Eqs. Ž37. ᎐ Ž39.. According to Gaydos w59x the requirements necessary to proceed from the differential relation to the integral relations are fulfilled in this case. Morrison w60x analysed the second proof given by Li w44x for the existence of the equation of state using the modified phase rule. Morrison agrees that the Gibbs phase rule needs to be modified for the systems with curved surfaces. Li w44x correctly maintains that Laplace pressures are created across all curved interfaces no matter how slight is the curvature so that the pressure cannot be constant throughout the system containing curved interfaces. However, Morrison is of the opinion that, Laplace pressures are often significant enough to have mechanical effects, such as capillary rise. However, Li w44x is at error when they claim that these Laplace pressures influence the thermodynamics. Morrison w60x illustrates the insignificance of Laplace pressures on the thermodynamics of observable systems by two examples. A hemispherical liquid drop on a solid substrate, the Laplace pressure is ⌬ PL s 2 ␥rR

Ž 88.

where ⌬ PL is the Laplace pressure, ␥ is the interfacial tension and R is the drop radius. Apart from the Laplace pressure there is another pressure is the hydrostatic pressure at the bottom of the drop. ⌬ PH s ␳ gR

Ž 89 .

where ␳ is the density and g is the acceleration due to gravity. The relative magnitude of these pressures can be compared by calculating the radius of the drop when the two pressures are the same R s Ž 2 ␥r␳ g .

1r2

Ž 90.

378 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

For water this radius is 4 mm. A full analysis shows that the two pressures differ only by a small Laplace pressure across the top of a drop. For usual kind of studies by interfacial tensions and contact angles, Laplace pressures are no more significant that gravitational pressures. Gravitational effects are insignificant for liquids until the depth is greater than several meters, similarly Laplace pressures are insignificant for the liquid drops are three orders of magnitude smaller, or on the order of a few micrometers. Laplace pressures exist in all systems containing curved interface; they are just insignificant to the thermodynamics until the curvatures are less than a micrometer. The size of drops necessary to make capillary pressures significant on the thermodynamics can be estimated by Kelvin equation for the effect of an increase in curvature on the vapour pressure of a liquid. A numerical example is given by Defay and Prigogine w61x and presented in Table I of w60x. The pressure of a drop with a radius of 1 ␮m is only 0.1% higher than a drop with infinite radius, i.e. a flat surface, the pressure is three times higher only when the drop is 10y3 ␮m in size. Therefore, the increase in vapour pressure of water drop due to curvature is not significant until the drop is much less than a micrometer in radius. A 1-␮m drop is barely visible with a light microscope and features like contact are indistinct. Liquids with lower surface tensions or liquid᎐liquid interface of lower tension must be even smaller. Therefore, all liquids for which interfacial tensions and contact angles have been measured directly are too large for capillary pressures to change the thermodynamics of the phase rule. Morrison w60x re-derived the phase rule for capillary systems following the treatment by Defay and Progogine w61x. The capillary system is where the pressure difference caused by the curved interfaces is significant enough to change the thermodynamics properties of the system. F⬘ s 1 q C

Ž 91 .

where C is the number of components in bulk phase. For the Gibbs phase rule wEq. Ž41.x, the number of phases is just the number of bulk phases, surface phases are not counted. For the modified phase rule for capillary systems, the number of phases is not needed as long as all the interfaces are significantly curved. For every interface insignificantly curved, the mean curvature is no longer a variable, and the number of degrees of freedom is reduced by one. Morrison w60x also studied the two systems Žas studied by Li w46x., i.e. sessile drop on solid substrate and liquid᎐liquid lens᎐fluid system. Considering a sessile drop on a solid substrate as shown in Fig. 3a. If the drop is large Žradius greater than 1 ␮m. then its curvature is insignificant and the classical Gibbs phase rule wEq. Ž41.x applies. For two components and three phases the number of degrees of freedom is one. Specifying the temperature completely specifies the intensive state of this system. Considering the sessile drop on fluid surface as shown in Fig. 10a. If the sessile drop is large and the vapour consists only of molecules of liquid 2 and liquid 3, then the number of phases is three, the number of components is two and by

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 379

applying the classical Gibbs phase rule wEq. Ž41.x we get d.f.s 1. Specifying the temperature completely specifies all the intensive variables of the system. If all the interfaces are highly curved then the modified phase rule wEq. Ž91.x applies and d.f.s 3. Specifying the curvature of the two phases and the temperature specifies the system. If the substrate liquid is plentiful then its interface with the vapour is flat and hence one can reduce the numbers of degrees of freedom by 1 because one more pressure equality equation is possible. This makes d.f.s 2 and hence, the equation of state like Eq. Ž40. possible. Therefore, Morrison w60x concludes that the classical phase rule does not need modifications simply because a system contains curved surfaces. It need be modified only when the curvatures are sufficient to cause significant pressure gradient, which happens for drops less than 1 ␮m. The number of degrees of freedom for a sessile drop on a substrate is the same Ž F s 1. whether the substrate is flat solid or extended liquid and hence a equation of state of the type 40 cannot exist. Since the system of sessile drop on both the substrates ᎏflat solid or extended liquid are similar hence, the experimental evidence Žused by Johnson and Dettre w53x. taken on liquid᎐liquid systems to analyse and disprove the equation of state approach is well justified. The equation of state relating to the interfacial tensions exist when the curvature of the interface is high Ždrop size - 1 ␮m. but is different for different materials and is by no means a universal function. Both Li and Neumann w62x and Gaydos and Neumann w63x defended the original proof for the equation of state w44x from the criticisms of Morrison w60x. Li and Neumann w62x used the experimental data to justify and defend the existence of equation of state. Fig. 11 shows the variation of the liquid surface tension, ␥l ¨ , with the ␥ l ¨ cos␪ term. Different solids Ž1᎐8. have different curves with the most hydrophobic solid lying in the topmost regions of the ␥ l ¨ vs. ␥ l ¨ cos␪ plots and the least hydrophobic in the lowermost region of the ␥ l ¨ vs. ␥ l ¨ cos␪ plots. Changing continuously the liquid surface tension from low surface tension to higher surface tension causes the data points to move along one smooth curve ŽFig. 11. depending on the solid under consideration. It is quite clear that the contact angle can vary, and because of Young’s equation, ␥sl can vary by changing the ␥ l ¨ and ␥s ¨ through simply changing the liquid and solid. If there was only one degree of freedom, according to Morrison w60x, then this would not have been possible. Other approaches like Fowkes approach w31,64x, LW᎐AB w65x approach divide the surface energy into components. Which means that ␥sl depends not only the ␥¨ l and ␥s ¨ but also on intermolecular forces or the different components of the surface energies of solid and liquid. In other words the equation like 40 will have more than two variables on the right-hand-side. This would require more than two degrees of freedom, but Morrison w60x does not allow more than one degree of freedom. Therefore, Morrison’s claim that there is only one degree of freedom does not contribute to a real debate. Morrison’s w60x most important argument is the insignificance of Laplace pressure on the thermodynamics. He has tried to justify by considering two examplesᎏcomparison of Laplace pressure to hydrostatic pressure at the bottom of a sessile drop and the increase in equilibrium vapour pressure of a drop in

380 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

comparison to flat surface. Though, Li and Neumann w62x agree that the increase in equilibrium vapour pressure of a 1-␮m drop is only 0.1% higher in comparison to flat surface, but this is only one aspect of curvature effect. If the effect of curvature on the increase in boiling is considered then the scene is totally different. Taking water as an example at 1 atm it has been shown by Lupis w66x, that a bubble of 1 ␮m in radius will require a boiling point of 121 ⬚C. In comparison with a flat surface Ž100 ⬚C. this is a 21% increase. Clearly, a dimension of the order of 1 ␮m or any other dimension should not be used to justify the significance or insignificance of curvature or Laplace pressure effects in general. Li and Neumann w62x also pointed out that it is wrong when Morrison reduces the total degrees of freedom by 1 when the substrate liquid is plentiful and has almost flat interface with the vapour. It is true that far enough from the drop; the liquid᎐vapour interface is flat. However, a drop interacts with the liquid substrate only in drop’s immediate vicinity, where the interface is curved. Also, it is wrong to consider the liquid substrate ᎐vapour interface to be flat. Gaydos and Neumann w63x claim that classical Gibbs phase rule under predicts the actual degrees of freedom for capillary systems and most importantly the phase rule in general makes no statement regarding the significance of the influence which capillary pressure has on other intensive variables. The Kelvin relation is taken as an example because it has been discussed by Morrison w60x. The Kelvin relation gives an expression for the manner by which the bulk saturation pressure must change in order to keep a spherical drop of radius, R, in equilibrium. If a two-phase water᎐steam system is considered where water is in the form of a sphere suspended in the vapour phase and the system is in a gravitationally free zone. The Gibb’s phase rule predicts for C s l and P s 2 then F s 1 wEq. Ž41.x. However, if the Clausius᎐Clapeyron equation is written for one-component, two-phase system Žwater drop suspended in water vapour. Ž ¨ Ž␣ . y ¨ Ž␤. . d P Ž␤. q ¨ Ž␣ . d

2

ž / R

␥ Ž␣ ␤. s Ž s Ž␣ . y s ␤ . dT

Ž 92 .

where Ž ␣ . represents water and Ž␤ . represents vapour phase, ¨ Ž␣ . s V Ž␣ .rN Ž␣ ., ¨ Ž␤. s V Ž␤.rN Ž␤., V and N are volume and mole number, R is the radius of the drop, T is the equilibrium temperature. Even if the temperature is fixed, so that one degree of freedom is removed from the system, that the saturation pressure P Ž␤. outside the drop will need to acquire different values for drops of different radii so as to maintain equilibrium. Obviously, this demonstrates in a theoretical sense that the classical phase rule does not correctly predict the two degrees of freedom which, exists in this system. Furthermore, it should be understood that the conclusion that there are two degrees of freedom in this droplet᎐vapour system is independent of any decision regarding the significance or magnitude of the influence of the droplet’s radius upon the saturation pressure P Ž␤. which is needed to maintain the droplet in equilibrium. Therefore, to claim that capillary effects are significant to thermodynamics until the curvature is less than 1 ␮m w60x or so is meaningless without specifying what thermodynamic properties are of interest.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 381

On the same lines as Li and Neumann w62x, Gaydos and Neumann w63x also point out that Morrison w60x took only the intensive property which are hardly influenced by the presence of capillary pressure, it would make far more sense to study those quantities which are. They site an example from Defay and Prigogine w61x which, pertains to the existence of small vapour bubbles in water. Accordingly, they calculated that at a temperature of 18 ⬚C the saturation pressure P Ž␤. required to maintain the water᎐vapour equilibrium across the planar interface is approximately 0.02033 atm, while the corresponding vapour pressure inside a bubble of radius l ␮m is just slightly smaller at 0.02032 atm. However, to assume that there are no significant thermodynamic changes in the system w60x is to ignore the fact that there is a significant ‘difference between the pressures in the bubble and in the liquid which, is approximately 1.441 atm. It follows that ‘for bubble to exist under these conditions the liquid must be under negative pressure’ w61x of 1.412 atm. Both Li and Neumann w62x and Gaydos and Neumann w63x maintain that the case of a sessile drop on the curved liquid surface in the form of lens is different from the system of a sessile drop resting on flat solid surface. Both maintain that the Classical Gibbs phase rule is applicable in this liquid᎐liquid lens fluid system ŽFig. 10a. and accept the Gibbs phase rule’s prediction of three degrees of freedom in this system. Accordingly, they point out that that the two systems, the sessile drop on solid substrate and sessile drop on liquid substrate, are different and that the latter has one higher degree of freedom than the former. Therefore, in a theoretical sense the data taken from liquid᎐liquid lens system cannot be used to disprove the potential existence or validity of a two-variable equation of state Žas done by Johnson and Dettre w53x.. Moy and Neumann w67x attempted to use direct force measurement data carried out by Claesson w68x, on surface and interfacial tensions for comparison with their data calculated with equation of state approach. Despite the problem of the surface force formula, in terms of 3␲ ␥ s ¨ R vs. 4 ␲ ␥ s ¨ R, they do not realise that the force equation was originally derived on the basis of the van der Waals interactions between sphere and flat surface. Thus, the data obtained by equation of state are comparable with those directly derived from apolar, van der Waals interactions. Hence, Lee w58x agrees with the conclusion of Morrison w57x and Fowkes w69x, that the equation of state approach is applicable to apolar systems involving solely physisorption. Although physisorption between a liquid and solid is universal and more common than chemisorption, it is generally much weaker in magnitude. Without considering chemisorption the equation of state, as persistantly asserted and vigorously defended by Neumann and his coworkers, is incomplete and definitely not universal for interfacial tensions. Furthermore, the equation of state approach cannot be extended to the system involving a thin liquid film at the interface when the disjoining pressure consists of at least three components: van der Waals; structural; and electrostatic. However, Lee w58x supports the LW᎐AB approach as it is applicable to both apolar and polar systems involving physisorption andror chemisorption at the interface.

382 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Drelich and Miller w70x used the advancing contact angles of water, glycerol, diidomethane and ethylene glycol on Teflon, polyethylene and different bitumen, bitumen asphaltenes and bitumen resins. The surface tensions were evaluated using the equation of state approach. They observed a significant scatter in the surface tension values of the solids when contact angle values with varying surface tension and polarity were used. It was found that the surface tensions of Teflon and polyethylene could be successfully estimated with the Equation of state, but only for systems with an apolar liquid Ždiiodomethane.. A lack of agreement, for other systems, in which contact angles of polar liquids were used, points out the limitation. They conclude that the equation of state approach is applicable to polymer᎐apolar liquid systems Žpolymers with surface tension from a range of approx. 21᎐42 mJrm2 were considered in their study. involving physisorption. Gaydos w71x used thermodynamics to prove that the equation of state type of relation is only possible for a sessile drop resting on ideal solid and not for the liquid᎐liquid lens system. Equation of state is formulated using the appropriate Gibb’s adsorption equation which does not apply to the liquid᎐liquid capillary system. For an interface between two immiscible bulk phases, denoted by ␣ and ␤ the equation is as follows, r

Ý ␳Ž␣i ,␤. d␮ i q s Ž␣ ␤. dT q d␥ Ž␣ ␤. s 0

Ž 93 .

is1 ␤. is the surface density of component-i with corresponding chemical where ␳Ž␣ i potential ␮ i , s Ž␣ ␤. is the specific surface entropy, T is the temperature and ␥ Ž␣ ␤. is the surface tension. In the original derivation Ward and Neumann w52x the position of the dividing surface chosen in such a way so that one of the quantities, e.g. s Ž␣ ␤. ␤. or ␳Ž␣ becomes zero, this process is convenient and often desirable. The effect of i such a choice is that Ž␣ ␤. d␥ Ž␣ ␤. s ysŽ1. dT y

r

Ý ␳Ž␣iŽ1.␤.d␮ i

Ž 94 .

is2

where the subscript Ž1. implies that the dividing surface position is selected such ␤. s 0. Because of this, two results follow that ␳Ž␣ i ⭸␥ Ž␣ ␤.

ž / ⭸␮ 2

s T ,␮ 3 . . . ␮ r

␤. y␳Ž␣ 2

y

Ž␣ . ␳Ž␤. 2 y ␳2 . Ž␤. ␳Ž␣ 1 y ␳1

␤. ␳Ž␣ 1

Ž 95 .

and ⌬␻Ž␣ ␤. s ⌬␥ Ž␣ ␤. s ⌬␭ Ž P Ž␣ . y P ␤ .

Ž 96 .

where, ␻Ž␣ ␤. is the specific grand canonical potential for the surface, i.e. the surface energy when the surface is in a state of thermal and chemical equilibrium, ␭ is the distance into Ž␤ . phase and P Ž␣ . and P Ž␤. are the pressures on adjacent sides of the interface.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 383

From Eq. Ž95. it follows that if the shift is performed to a new location such that ␤. s 0 then the distance of the shift is ␳Ž␣ 1 ␭s

␤. ␳Ž␣ 1

Ž 97 .

. Ž␤. ␳Ž␣ 1 y ␳1

␤. and the value of ␳Ž␣ will change to 2

␤. Ž␣ ␤. ␳Ž␣ q 2Ž1. s ␳ 2

Ž␣ . ␳Ž␤. 2 y ␳2

␳Ž␣ ␤. so that Ž␤. 1

. ␳Ž␣ 1 y ␳1

⭸␥ Ž␣ ␤.

ž / ⭸␮ 2

␤. s y␳Ž␣ 2Ž1.

Ž 98 .

T ,␮ 3 . . . ␮ r

However, it is impossible to shift the dividing surface from the surface tension . Ž␤. position to any other position if ␳Ž␣ 1 s ␳1 . From Eq. Ž96. we see that corresponding to the shift wEq. Ž97.x one has a change in energy as given by Eq. Ž92.. However, for any system Žcapillary or otherwise. to be in equilibrium, the energy must be stationary. The requirement of a stationary state can only be accomplished in one way for these systems i.e. by choosing P Ž␣ . s P Ž␤. so that ⌬␻Ž␣ ␤. s ⌬␥ Ž␣ ␤. s 0. This means that Gibb’s moderately curved theory of capillarity only permits one to shift the dividing surface if the surface is planar. Thus, when the surface is planar, the pressures on the adjacent sides of the surface will be equal and any shift in dividing surface position will not change the value of the energy. Therefore, it is possible to arbitrarily move the dividing surface for solid᎐vapour and solid᎐liquid surfaces as these are assumed to be planar a priori during the derivation of the equation of state. It is apparent that the equation of state derivation will not apply to liquid᎐liquid lens systems since the corresponding conditions P Ž␣ . s P Ž␧ . and P Ž␧ . s P Ž␤. Žwhere P Ž␣ ., P Ž␤. and P Ž␧ . denote pressures in three contacting bulk phases with densities subjected to the condition ␳Ž␧ . ) ␳Ž␤. ) ␳Ž␣ . is rarely met. One can say that two component liquid᎐liquid systems with three degrees of freedom are not equivalent to two component liquid᎐solid systems with two degrees of freedom and hence data taken with liquid᎐liquid lens system cannot be used to disprove the potential existence or validity of a two variable equation of state. It is also obvious from the form of these equations and the presence of surface densities beyond i s 2 that there is nothing which restricts the equation of state to exclusively binary systems with just two components. To this Lee w72x replied stating various points. The interfacial tensions between liquids, for moderately curved surfaces, have never been found to be dependent on curvature. Drops of the third liquid between two immiscible liquids can be of any size, including diameters approaching infinity. Therefore, the argument that such surfaces are curved cannot be used to object to the employment of liquid᎐liquid interfacial tension to test the equation of state approach. This fact raises strong questions about any thermodynamic derivation that implies otherwise. The consequence of having a curved interface is creation of small Laplace pressure that is

384 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

thought, by proponents of equation of state approach, to increase the degree of freedom which has been proved untrue by Morrison w57,60x. It is simple to show that the Laplace pressure is less than the gradient caused by gravity, but no one have suggested adjusting contact angle measurements for that. The idea of displacement of the dividing surface, as used by Ward and Nuemann w52x is unacceptable as shown and proved by Markin and Kozlov w73x on account of the energy conservation principle. 2.9. Lifshitz᎐¨ an der Waals r acid᎐base approach (LW᎐AB) This approach came into existence when the thermodynamic nature of interface was re-examined by van Oss w65x in the light of Lifshitz theory of forces. The role of van der Waals forces and hydrogen bonds was studied in order to explain the strong attachment of biopolymers Žhuman serum albumin, human immunoglobulin. on low energy solids Žpolytetrafluoroethylene, polystyrene. which was previously attributed to the hydrophobic interactions. The apolar interaction between protein and low energy solid is repulsive and hence only the apolar interaction cannot explain the strong attachment of biopolymer on the low energy solids. A new polar termᎏLewis acidrbase interaction, AB Žpreviously referred to as short-range ŽSR. interactions in van Oss et al., 1986 w65x. was introduced in order to explain the attraction between biopolymers and low energy solids. The definition of the Lewis AB term has been ambiguous until van Oss w74,75x defined them properly. The importance of distinguishing apolar and polar contributions to the surface tension of liquids and solids has been stressed already in 1962 w21x. In the beginning only the London dispersion force contribution was considered as apolar and the rest of the following contributions were considered as polar: 1. Dipolar compounds, i.e. substances that have appreciable dipole moment. 2. Hydrogen bonding compounds: treated under the Brønsted Žproton donor᎐proton acceptor. theory of acids and bases. 2.1. Substances that are both proton-donors Žacids. and proton acceptors Žbases., e.g. water, termed as bipolar, the proton donor and acceptor functionalities of a molecule may not be of equal strength. 2.2. Substances that are very much more effective as proton donors than as proton acceptors, e.g. CHCl 3 . 2.3. Substances that are very much more effective as proton acceptors than as proton donors, e.g. ketones. 3. Compounds that interact as Lewis acids Želectron acceptors. and bases Želectron donors.. 3.1. Substances that have both kinds of functionalities, electron acceptors and donors, termed as bipolar- the same as the class 2.1, because the Lewis acid᎐base theory encompasses the Brønsted acid᎐base theory.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 385

3.2. Substances that are much more effective as electron acceptors than electron donors. 3.3. Substances that are much more effective as electron donors than electron acceptors. The three electrodynamic interactions present at the interface areᎏKeesom Žrandomly oriented dipole᎐dipole also called ‘orientation’. interactions, Debye Žrandomly oriented dipole-induced dipole also called ‘induction’. interactions and London Žfluctuating dipole-induced dipole also called ‘dispersion’. interactions. It is known that a possession of a large dipole moment is not necessarily a condition for hydrogen bond formation w76x, e.g. nitriles and nitro compounds both have large dipole moments, yet, nitriles form hydrogen bonds and not nitro compounds. Moreover, in the condensed, macroscopic system using the Lifshitz approachᎏChaudhury w77x showed that surface tension components arising from the three electrodynamic interactions must be treated in the same manner and should be grouped together. When grouped together, these electrodynamic interactions are alluded to as Lifshitz᎐van der Waals interactions ŽLW.. ␥ LW s ␥ Keesom q ␥ Debye q ␥ London

Ž 99 .

The rest of the interactions, i.e. two and three constitute the Lewis acidrbase contribution. The symbol ␥y is used to indicate the parameter of surface tension that is due to proton acceptor or electron donor functionality and ␥q is used to indicate the surface tension parameter due to proton donor and electron acceptor functionality. The joining together of the two kinds of basic behaviour, ŽBrønsted bases and Lewis bases, using the symbol ␥y. is justified because the proton acceptor group is necessarily an electron donor group. The grouping together of Brønsted acids and Lewis acids for the purpose of surface tension under the term of ␥q is done for operational reasons and because during the surface chemical measurements, i.e. contact angle, adsorption and interfacial tension, Brønsted and Lewis acid behaviour are not easily distinguished. The clear subdivision between apolar ŽLW. and polar ŽAB. interactions, also has made it possible to arrive at the quantitative definition of hydrophobic interactions which is also referred to as the total interfacial interaction ⌬G Total s ⌬G LW q ⌬G AB

Ž 100.

The apolar part ␥ LW , follows the Fowkes treatment wEq. Ž12.x w74,75,78᎐81x ␥slLW s ␥sLW q ␥ lLW y 2 ␥sLW ␥ lLW

'

Ž 101a .

or ␥slLW s

ž '␥

LW s

y

'␥ / LW l

2

Ž 101b .

386 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Then the cohesion energy of a liquid is expressed as ⌬GlLW s y2␥ lLW l

Ž 102.

Using Vissers notations where 1 and 2 denotes solids and liquids and 3 only liquids and Dupre ´ work of adhesion w82x WA12 s ␥ 1 q ␥ 2 y ␥ 12

Ž 103.

translated in terms of free energy change ⌬G12 s ␥ 12 y ␥1 y ␥ 2

Ž 103a .

The apolar Žor LW. energy of adhesion between 1 and 2 is expressed as LW LW LW ⌬G12 s ␥ 12 y ␥ 1LW y ␥ 2LW s ⌬G12 s y2 ␥ 1LW ␥ 2LW

'

Ž 104.

and the LW interaction of 1 and 2 in 3 is expressed as LW LW LW LW s ␥ 12 y ␥ 13 y ␥ 23 s ⌬G132 LW ⌬G132 s2

ž '␥

LW LW 2 ␥3

q

'␥

LW LW 1 ␥3

y

'␥

LW LW 1 ␥2

y ␥ 3LW

/

Ž 105.

Unlike the LW interactions, which are mathematically symmetrical, the acidrbase interactions are essentially asymmetrical according to van Oss w74,75x in the sense that for a polar substance i the electron acceptor and the electron donor parameters are quite different. Also, one parameter is not manifested at all, unless the other parameter is either present in another part of the same molecule of the substance i or in another polar molecule j with which molecule i can interact. Thus, at the solid᎐liquid interface the electron acceptors of solid will interact with the electron donors of liquid, and vice versa. van Oss w74,75,78᎐81x expressed the acidrbase interactions in the following manner AB ⌬G12 s y2

q y 1 ␥2

ž '␥

y q ␥q 2 ␥1

'

/

Ž 106.

This was based on Kollman w83x where he analysed the non-covalent interactions in wide variety of intermolecular complexes Žvan der Waals molecules, H-bonded complexes, charge-transfer complexes, ionic association, radical complexes and three body interactions . and used an expression like Eq. Ž106. for the asymmetrical interactions. In anology with the Drago’s approach in solution thermodynamics w84x ␥ AB is expressed in a more rigorous manner as ␥ AB s 2 ␥q ␥y

'

Ž 107.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 387

Dupre ´ equation wEq. Ž103a.x is valid for acidrbase interactions also hence, ⌬GslAB s ␥slAB y ␥ lAB y ␥sAB

␥slAB s ⌬GslAB q ␥ lAB q ␥sAB

or

Ž 108.

Putting the value of ⌬GslAB from Eq. Ž106. and expanding ␥sAB and ␥ lAB in terms of ␥q and ␥y wEq. Ž107.x we get, q y s ␥s

y y q ␥q ␥q l ␥l y s ␥l

q s

' / ž '␥

␥slAB s 2

ž '␥

␥slAB s 2

ž '␥

'

'

y q s ␥l

'␥

Ž 109.

/

or y ␥q l

y s

y ␥y l

' /

Ž 109a .

Therefore, for a binary system like the solid᎐liquid interface the total free energy of interaction is ⌬G Total s y2

ž '␥

LW LW s ␥l

y q q ␥q ␥y s ␥l q s ␥l

'

'

Ž 110.

/

and the total interfacial tension is ␥slTotal s

ž '␥

y

LW s

'␥ / LW l

2

q2

q s

ž '␥

y ␥q l

y s

y ␥y l

' / ž '␥

' /

Ž 111.

Combining this with the Young equation wEq. Ž1.x we get Ž 1 q cos␪ . ␥ l s 2

ž '␥

y q q ␥q ␥y s ␥l q s ␥l

LW LW s ␥l

'

'

Ž 112.

/

For a ternary system the acidrbase interaction energy is q 3

AB ⌬G132 s2

y 1

y 2

'␥ ž '␥ q '␥ y'␥ ␥ y '␥ ␥ q y 1 2

y ␥y q ␥y 3 3

q 1

' / ' ž '␥

q ␥q ␥q 2 y 3

'

' /

y q 1 2

Ž 113.

and the total interaction energy is AB ⌬G132 s2

ž '␥

q2

LW LW 2 ␥3

␥q 3

q ␥y 1

' ž'

'␥ q

LW LW 1 ␥3

␥y 2

'

y

y

'␥

␥y 3

LW LW 1 ␥2

y ␥ 3LW

␥y 3

␥q 1

' / q ' ž'

/ Ž 114. q ␥q ␥q 2 y 3

'

' /

y q y ␥q ␥y 1 ␥2 y 1 ␥2

'

'

Later it has been clearly demonstrated that, while the apolar-␥ LW and polar-␥ AB surface components are additive, the Lewis acid᎐base electron acceptor-␥q and donor-␥y surface tension parameters are not additive w85x. Eq. Ž112. contains three unknownsᎏ␥sLW ; ␥sq; ␥sy, hence contact angle measure-

388 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

ment has to be performed using three standard Ž2 polar and 1 apolar. liquids Žwell characterised . on the solid surface in order to totally evaluate the surface energy of the solid. Hence, by means of contact angle measurements with the number of different well-characterised, standard, polar and apolar liquids, the apolar ŽLifshitz᎐van der Waals. surface tension component Ž ␥ LW ., the polar ŽLewis acid᎐base. surface tension component Ž ␥ AB . and the latter’s electron-acceptor Ž ␥q. and electron-donor Ž ␥y. parameters, can be determined for a number of polar surfaces. y The standard liquids are pre-characterised with respect to their ␥1LW , ␥q 1 and ␥ 1 Total values. The absolute value of ␥ 1 is available from the surface tension measurements and the absolute value of ␥1LW is obtained by using the Fowkes approach w31x. The ␥ 1AB is obtained from the difference of ␥Total and ␥ 1LW . Relatively few 1 such well-characterised polar, or apolar contact angle liquids are present at our disposal. One requirement that all such liquids must fulfill is to have a rather high surface tension Ž ␥1 ., in order not to spread on most polar surfaces. Usually this means a ␥ 1 ) 44 mJrm2 . This especially limits the number of available apolar liquids, but in practice, ␣-bromonapthalene Ž ␥ 1 s 44.4 mJrm2 . and diiodomethane Ž ␥ 1 s 50.8 mJrm2 . are quite adequate for most purposes. Among polar, hydrogen bonded liquids, water Ž ␥ 1 s 72.8 mJrm2 , ␥ 1LW s 21.8 mJrm2 , ␥ 1AB s 51 mJrm2 . is extremely suitable. Other high tension liquids available are glycerol Ž ␥ 1 s 64 mJrm2 , ␥ 1LW s 34 mJrm2 , ␥ 1AB s 30 mJrm2 . and formamide Ž ␥ 1 s 58 mJrm2 , ␥ 1LW s 39 mJrm2 , ␥ 1AB s 19 mJrm2 .. y Ž No absolute value of ␥q 1 or ␥ 1 are known at present for any compound not even for water.. One is reduced to making an arbitrary estimate of the ratio of ␥q and ␥y for a reference compound Že.g. for water.. However, for obtaining the AB AB and ⌬G132 , it is not necessary to know the absolute values of ␥slAB , ⌬GslAB , ⌬G121 q y absolute values of ␥i and ␥i of any substance i. It is sufficient to use the polarity y ratio of ␥q and ␥y relative to the ␥q i i R and ␥R of the reference compound. The q y ␥ r␥ ratio for water is assumed to be 1 because of its equivalence to the pH convention, and hence ␥q and ␥y take the values of 25.5 mJrm2 each as the ␥ AB for water is 51 mJrm2 w74x. Although Gutman w86x has reported the H-donor and acceptor number in the ratio 1:3, which makes water a Lewis base, however, the relation between Gutman’s scale of ‘electron acceptor numbers’ and ‘electron donor numbers’ and the relative values of ␥q and ␥y for water has not yet been fully explored. The polarity ratioᎏ␥qr␥y for other standard liquids Že.g. glycerol and formamide. is obtained by measuring contact angle with water, glycerol and formamide on various monopolar ŽLewis. basic solid surfaces, e.g. poly Žmethylmethacrylate . ŽPMMA., poly Žethylene oxide. ŽPEO., clay films, corona-treated poly Žpropylene. ŽCPPL., dried agarose gel, dried zein Ža water insoluble corn protein, cellulose acetate and dried film of human serum albumin ŽHAS. w87x. Eq. Ž112. for the water contact angle on the solids is of the form ␥w Ž 1 q cos␪ s w . s 2

ž '␥

LW LW s ␥w

y q q ␥q ␥y s ␥w q s ␥w

'

'

/

Ž 115.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 389

and similarly for glycerol the equation is of the following form ␥g Ž 1 q cos␪ w g . s 2

ž '␥

LW LW s ␥g

q

q y s ␥g

'␥

q

y q s ␥g

'␥

/

Ž 116.

The Lifshitz van der Waals parameter of the solids is determined before hand by measuring contact angle using apolar liquids. Since only monopolar ŽLewis bases. solids are used meaning the term ␥q s s 0, hence, the above two equations reduce to y q s ␥w

s ␥w Ž 1 q cos␪ s w . r2 y

'␥

y q s ␥g

s ␥g Ž 1 q cos␪ w g . r2 y

'␥

'␥

'␥

LW LW s ␥w

LW LW s ␥g

Ž 115a . Ž 116a .

q The ratioᎏ␥q is evaluated from the above two equations which yields w r␥ g also. Using this method on 10 monopolar solids, the averages thus obtained 2 Ž 2. were Žglycerol. ␥q which yields ␥y and ŽForg s 3.92 " 0.7 mJrm g s 57.4 mJrm y 2 q 2 mamide. ␥ f s 2.28 " 0.6 mJrm Žwhich yields ␥ f s 39.6 mJrm . w87x. y ␥y w r␥ g

2.10. Objections on Lifshitz-¨ an der Waals acid r base approach The LW᎐AB approach appears successful to separate the surface polar components and determine negative interfacial tension as an indication of solubility. The LW᎐AB approach has been successfully applied by many. However, unavoidably there have been some criticisms. Li w44x applied phase rule for capillary systems and showed that only two degrees of freedom are allowed in the system where a sessile drop is sitting on a solid substrate ŽFig. 3a.. They expressed solid᎐liquid interfacial energy in the form of expression 40 where the solid᎐vapour and liquid᎐vapour interfacial energies are the two degrees of freedom. However, according to the LW᎐AB approach the solid᎐liquid interfacial energy is expressed in the form of Eq. Ž111., which means that the expression like 40 will have six variables on the right-hand-side. This is not allowed according to the modified phase rule. Fowkes w88x pointed out that the use of single parameter for expression of acidity and basicity ignores the hard and soft character of acids and bases, which is accommodated in the Gutmann’s acceptor and donor numbers w86x. He also expressed scepticism about the accuracy of the electron donor and electron acceptor values obtained for the solid surfaces as the degree of acidity and basicity of the standard liquids are still under investigation Žassumption: ␥qr␥ys l for water.. Use of the LW᎐AB approach to evaluate the acidic and basic components of the solid surface free energy shows that most of the solid surfaces are overwhelmingly basic, with a small or negligible acidic component. Berg w89x doubts this to be a general law for the solid surfaces and suggests the checking of the consistency of the method. Van der Mei and Busscher w22x reason that biosurfaces are predomi-

390 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

nantly electron-donating as a consequence of prevalence of oxygen in the Earth’s lower atmosphere and the hydration of the microbial cell surface. Lee w90x reported a correlation between the Lewis acid᎐base surface interaction components and linear solvation energy relation ŽLSER. solvatochromic parameters ␣ and ␤. The solvatochromic approach gives separate subfactors of hydrogenbond-accepting and hydrogen-bond-donating abilities for many acids and bases. Where ␣ is an empirical quantitative measure of the hydrogen-bond-donating ŽHBD. ability of a bulk solvent towards a solute and ␤ is an empirical quantitative measure of the hydrogen-bond-accepting ŽHBA. or electron-pair-donating ŽEPD. ability of a bulk solvent towards a solute for a hydrogen bond or a Lewis coordination bond. For non-HBD solvents, such as apolar, aliphatic and aromatic hydrocarbons, the ␣ value is zero, for polar aliphatic-alcohols, 0.5 - ␣ - 1.0, while for fluoro-substituted alcohols and phenols, ␣ ) l, reaching a maximum of 1.96 for hexafluoro-2-propanol. In contrast the ␤ scale is fixed by setting ␤ s 0 for cyclohexane and is the same for all apolar aliphatic hydrocarbons. However, for aromatic hydrocarbons the ␤ value is 0.1, for aliphatic ethers, ␤ f 0.7᎐0.9, for aliphatic amines ␤ s 0.5᎐0.7, ␤ s 1 for methylphosphoric triamide and maximum value of ␤ is 1.43 for 1,2-diaminoethane. Lee further comments that ‘... if the interaction is limited only to hydrogen bonding instead of the broadly defined acid᎐base interaction, then ␥q resembles HBD parameter ␣ and ␥y the HBA parameter ␤’. Marcus w91x has compiled a list of solvatochromic parameters for approximately 170 liquids. It is interesting to note that there are many more HBA than HBD compounds in the list, coincidentally in terms of Lewis acid᎐base classification, this means that there are many more Lewis bases than Lewis acids. This observation, in a way, supports the LW᎐AB approach’s evaluation of most of the solid surfaces as basic. For liquid water at ambient temperature, the HBD ability is stronger than HBA ability. Thus, unlike other Lewis acid᎐base standards like Gutmann donor᎐acceptor scale, water appears to be acidic with ␣r␤ s 1.8. Various polymers has been reported as predominantly basic by the LW᎐AB approach w92x, especially the case of polyvinyl chloride ŽPVC. is doubted by many workers w89,93x. PVC is accepted to be a monofunctionally acidic polymer w93x. The acidic behaviour has been reported by measurement of work of adhesion against PVC of monofuncationally acidic and basic liquids w94x, by peeling experiment involving a polymeric film of PVC on glass and strengthening of PVC polymer by acidic filler Žglass powder. ŽFowkes, 1987 w95x.. However, PVC is evaluated to be a basic polymer using LW᎐AB approachᎏ␥qs 0.04 and ␥ys 3.5 mJrm2 w92x. Lee w58x used ␥qr␥ys 1.8 for water and reevaluated the surface energies of many polymers using the LW᎐AB approach. With the new ␥qr␥y ratio the major improvement was in lowering of the surface Lewis base component for all the polymers, but still PVC came out to be predominantly basic with ␥qs 0.1 and ␥ys 2.4 mJrm2 . It is important to stress that the problem is broader, and by no means limited to the PVC surface alone. Van Oss w80x finds, by contact angle measurement, that

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 391

cellulose is basic, while results of Berg w89x on the characterisation of cellulose surface by inverse gas chromatography ŽIGC. experiments show that it has primary acidic sites on its surface. Kwok w96,97x tested the internal consistency of the LW᎐AB approach by evaluating the surface energy of fluorocarbon ŽFC721., Teflon ŽFEP. and polyethylenetetrapthalate ŽPET. using different sets of polar wŽwater, formamide, glycerol, ethylene glycol, ŽDMSO. dimethyl sulfoxidex and apolar liquids. Eq. Ž112. needs the contact angle data from three standard liquids in order to be solved and to evaluate ␥ LW , ␥q and ␥y parameters for the solid. Kwok tested the two different strategies to solve the equation. First, to use contact angle data from three polar liquids and then solve the three simultaneous equations to get the parameters. Second, to determine the ␥sLW for the solid first by using contact angle with a high-energy apolar liquid and then the ␥q and ␥y parameters using contact angle data from two polar liquids. In the second case the value of ␥sLW showed a strong dependence on the apolar liquid surface tensions. In both the cases a wide range of values were obtained for the solid surface parameters when different sets of liquids were used, negative values for Ž ␥q. 1r2 and Ž ␥y. 1r2 were obtained which is anomalous and the negative values of total solid surface energy, ␥s , is not allowed in the LW᎐AB approach. For Teflon, which, is most hydrophobic, it was expected that the polar component of the surface energy should be negligible but the ␥sAB component range from y64 to 0.38 mJrm2 and the best value is 0.14 for the water᎐formamide pair. It was also shown that the anomalies of large variations in the solid surface energy and occurrence of negative solid surface tension and square roots of surface tension components were not due to errors in the contact angle data. Therefore, Kwok et al. concluded that the LW᎐AB approach could not give a consistent value of ␥s . Solid surface tension components are not unique properties of the solid. The contact angle does not contain information about putative surface tension components and contact angle is a function of only the total solid surface tension Žas in equation of state approach. and not of its components. w98x partially attributed the lack of internal consistency of the LW᎐AB Hollander ¨ approach to the mathematical instability of the model and proposed a selection criterion for the standard liquids for evaluation of the acid᎐base properties of solid surfaces by contact angle goniometry. He agrees with the proponents of the LW᎐AB approach that the most severe constraint for the use of the approach is the lack of broad spectrum of test liquids with sufficiently high surface tension, which are well characterised. While the choice of basic liquids is adequate, this is not the case for predominantly acidic liquids Žmonopols.. Strong monopoles like chloroform or nitromethane, have much too low LW interaction capabilities that they spread on most of the surfaces of interest. He observed that the solid surface energy values obtained by a pair of widely different liquids, with respect to their acidic and basic nature, like water᎐formamide and water᎐glycerol were similar but the values obtained by the combina-

392 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 14. Surface energy components ␥q and ␥y as a function of the contact angles of Ža. water, Žb. w98x.. formamide and Žc. glycerol Žfrom Hollander ¨

tion of basic liquids, i.e. formamide᎐glycerol, exhibit large differences. The reason becomes obvious from Fig. 14, where the parameters ␥q and ␥y are plotted as a function of one contact angle, when the other contact angle in the pair is kept constant. The functional variation for the glycerol᎐formamide pair are seen to be very steep, and small deviations of the contact angles Žwithin the range of experimental error. results in large changes in the values of ␥q and ␥y derived from these contact angles. The data of the pairs water᎐formamide and water᎐glycerol are less critical. Therefore, the liquids used in the pair must have a large difference in their ␥yr␥q Ž Q r . ratios. In order that the simultaneous equations be least sensitive towards experimental errors then the difference in their Q r numbers Ži.e. ⌬Q r . should be more than 3 Ž ⌬Q r G 3.. In order to obtain reliable results from the contact angle data, ⌬Q r G 15 seems to be appropriate. However, if the data from Kwok w96,97x is re-examined in the light of Hollander’s ¨ selection criterion, then we see that the criterion works but not completely. Table 1 shows the solid surface energy data for two solids arranged according to decreasing ⌬Q r of the pair of the polar liquids used for the determination of solid surface energy parameters. It is safe only if ⌬Q r G 15 because for both water᎐formamide and water᎐glycerol pairs we get positive solid surface energy and square roots of the surface energy components Žgood parameters.. Though, water᎐ethylene glycol Ž ⌬ Q r s 9. and water᎐DMSO Ž ⌬ Q r s 6.7. gives good parameters but formamide᎐DMSO, with a higher ⌬Q r Žs 9.6., evaluates bad surface energy parameters for both the solids. By Hollander’s criterion it is expected that ¨

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 393

DMSO᎐ethylene glycol with ⌬Q r s 2.3 should evaluate bad surface energy parameters, however, for FC721 the pair of liquids give good parameters and the total surface energy is only 0.04 mJrm2 off the average which is not bad when the standard deviation is 1.5 mJrm2 . Since the two solids are apolar hence negligible values for ␥sAB is expected. The best value for ␥sAB is obtained when the water᎐formamide pair is used ŽFC721ᎏ0.04 and FEPᎏ0.14 mJrm2 . though, it is unexpected that fluorocarbon have lower ␥sAB than Teflon. Another important point, which becomes obvious, is that all the liquid pairs, which contain water give good parameters even if their ⌬Q r is poor. Although the above analysis is based on only two apolar solids, however, we can see that Hollander’s selection criterion further narrows down the choice of liquids ¨ which can be used to determine the solid surface energy using LW᎐AB approach. The water᎐formamide pair Ž ⌬Q r s 16.4. seems to be the best. The water᎐glycerol pair Ž ⌬Q r s 14.6. is a borderline case both according to Hollander’s criterion ¨ Ž ⌬Q r G 15. and that the ␥sAB values jumped from 0.04 to 0.24 and 0.14 to 0.34 for fluorocarbon and Teflon Žapolar solids., respectively, when glycerol was used with water instead of formamide. The proponents of the LW᎐AB approach w99x refuted the statement put by Kwok w96x. Firstly, it is pointed out that while determining the solid surface energy of FEP and FC721 by using three polar liquids the choice of liquids was not properly done. Good and van Oss w100x pointed out thatᎏ‘mathematically, it is possible to use three polar liquids and a set of three equations in the form of Eq. Ž112.. Such tactics work if the values of the parameters Že.g. ␥ . for the three liquids are not too close together. If they are close, the calculated values of the three parameters for the solid will be unduly sensitive to small errors in the values of the parameters of the liquids, and in measured contact angles’. However, neither Wu w99x nor Good and van Oss w100x suggest the selection of those three polar liquids which are most suitable or a selection criterion to select the three polar liquids for the purpose of determining the solid surface energy parameters accurately. For the second method where ␥sLW is pre-determined using an apolar liquid and then two polar liquids are used to determine the remaining two parameters Ž ␥q and ␥y .. w98x for the failure and suggests Van Oss w99x gives the same reason as Hollander ¨ that the two polar liquids used must be very different in their acidic and basic behaviour. Differences in the solid surface energy parameters are still obtained when different sets of liquids are used even according to the selection criterion of w98x and Wu w99x. In absence of any guidelines from the proponents of Hollander ¨ the LW᎐AB approach and other supporters it is difficult to judge which liquid set gives the most accurate solid parameters. Though, van Oss w87x mentions that ‘␥sq and ␥sy can be entirely defined with two polar liquids, the availability of a third liquid could yield a useful set of control values’ but this does not help in arriving at the most accurate set of parameters. Wu w99x used the least square method to solve the contact angle data for liquid pairsᎏwater᎐formamide, water᎐glycerol, water᎐ethylene glycol and water᎐DMSO, the results of which are presented in

FC721ᎏFluorocarbon Liquids

⌬Qr

␥LW a

Ž ␥q .1r2

Ž ␥y .1r2

␥q

␥y

␥AB

␥Total

WaFo Wa-Gl Fo-DM Wa-Eg Fo-Eg Gl-DM Wa-DM Gl-Eg Fo-Gl DM-Eg

16.4 13.6 9.6 9 7.4 6.9 6.7 4.6 2.8 2.3

9.07 9.07 9.07 9.07 9.07 9.07 9.07 9.07 9.07 9.07

0.02 0.16 0.53 0.5 2.44 0.39 0.3 1.36 y1.1 0.04

0.9 0.76 y1.19 0.42 y9.16 y0.13 0.63 y3.82 5.58 2.7

0.0004 0.0256 0.2809 0.25 5.9536 0.1521 0.09 1.8496 1.21 0.0016

0.81 0.5776 1.4161 0.1764 83.9056 0.0169 0.3969 14.5924 31.1364 7.29

0.04 0.24 0.04 0.42 y44.7 y0.1 0.38 y10.39 y12.28 0.22

9.11 9.31 7.81 9.49 y35.63 8.97 9.45 y1.32 y3.21 9.29

A¨ g. S.D.

9.07

0.0735 0.1051

1.8501 3.0498

0.26 0.150

9.33 0.150

Least Sq.b

9.15

0.07

0.44

0.34

9.40

0.14 0.34 y0.74 0.38 y64.18 0.25

15.43 15.63 14.55 15.67 y48.89 15.54

FEPᎏTeflon Wa-Fo Wa-Gl Fo-DM Wa-Eg Fo-Eg Gl-DM

16.4 13.6 9.6 9 7.4 6.9

15.29 15.29 15.29 15.29 15.29 15.29

0.08 0.24 0.47 0.64 2.92 0.33

0.86 0.7 y0.79 0.3 y10.9 0.38

0.0064 0.0576 0.2209 0.4096 8.5264 0.1089

0.7396 0.49 0.6241 0.09 118.81 0.1444

394 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Table 1 Polar liquid sets arranged according to decreasing ⌬Q r , data from Kwok w96,97x

FEPᎏTeflon Liquids

⌬Qr

␥LW a

Ž ␥q .1r2

Ž ␥y .1r2

␥q

␥y

␥AB

Wa-DM Gl-Eg Fo-Gl DM-Eg

6.7 4.6 2.8 2.3

15.29 15.29 15.29 15.29

0.29 1.63 y1.31 y0.15

0.65 y4.6 6.63 4.2

0.0841 2.6569 1.7161 0.0225

0.4225 21.16 43.9569 17.64

0.38 y15 y17.37 y1.26

␥Total 15.67 0.29 y2.08 14.03

A¨ g. S.D.

15.29

0.1333 0.1590

0.3773 0.2658

0.298 0.103

15.58 0.103

Least Sq.b

15.25

0.13

0.34

0.41

15.66

⌬Q r s Ž ␥yr␥q . liquid 1 y Ž ␥yr␥q . liquid 2 ŽHollanders selection criterion.. ¨ Rows in bold letters are for liquid pairs, which give positive solid surface energy and square roots of the surface energy components for FEP and FC721. All values in mJrm2 . Notes. Wa, Water; Fo, Formamide; Gl, Glycerol; DM, DMSO; Eg, Ethylene glycol. Average and S.D. only for the pairs which give positive surface energy and square root of the surface energy components. a Predetermined using ␣-bromonapthalene. b Wa-Gl, Wa-Fo, Wa-Eg and Wa-DM contact angle data solved using least square method ŽFrom Wu et al. w99x.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 395

Table 1.1 Ž Continued.

396 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Table 1. The least square values are very different from the ones obtained using a single set of liquids. If the situation is seen in totality, both FEP and FC721 are apolar and hence y AB components must be negligible. This happens only when their ␥q s and ␥s and ␥s a water᎐formamide pair is used for the determination of the solid surface energy. This pair gives the minimum ␥sTotal out of all the liquid pairs. Volpe and Siboni w101x investigated the acid᎐base solid surface free energy theories in detail. They also studied the mathematical instability inherent in the simultaneous equations to be solved for the solid surface energy parameters using LWAB approach. The simultaneous equations of the form of Eq. Ž112a. are written in matrix form A ⭈ x s B wEq. Ž117.x with the contact angle data of various liquids on the solid. The system can be over determined with more than three liquid contact angles.

'␥

LW LW s ␥l i

' '␥ '␥

q

q y s ␥l i

'␥ q l1

'␥ '␥ '␥

LW l1

q l2

LW l2

q l3

LW l3

q y l1

'␥ '␥ '␥

y l2 y l3

y q s ␥l i

'␥

'␥

q s

Ž 112a .

Ž 1 q cos␪1 . ␥l r2

LW s

␥y s

' '␥

s Ž 1 q cos␪ i . ␥ l ir2

1

s Ž 1 q cos␪ 2 . ␥ l 2r2 Ž 1 q cos␪ 3 . ␥ l r2

Ž 117.

3

While solving the matrix its condition number governs whether meaningful solid surface energy parameters can be obtained from that matrix Žor the liquid system.. The condition number of the A matrix determines its nearness to singularity Ži.e. the norm of matrix is almost equal to a norm of some singular matrix.. The condition number is defined as a non-negative real scalar given by CondŽ A . s 5 A 5 ⭈ 5 Ay1 5

Ž 118.

where 5 A 5 is the norm of a matrix is a scalar that gives some measure of the magnitude of the elements of the matrix. High condition number means that the system is ill-conditioned, i.e. the system is more prone to the errors in the contact angle and measured liquid parameters. In other words, the condition number of a matrix measures the sensitivity of the solution of a system of linear equations to errors in the data. It gives an indication of the accuracy of the results from matrix inversion and the linear equation solution w102x. The norm of the matrix is defined as 5 A5 p s

1rp

n

ž

Ý 5 Ai 5 p is1

/

Where p G 1

Ž 119.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 397

Many functions qualify to be called as matrix norms according to the following rules 5 A 5 1 s max j

5 A5 2 s

Ý < ai j <

Ý < ai < is1

5 A 5 ⬁ s max i

Ž 119a .

1r2

n

ž

Ž Column norm.

i

2

/

Ý < ai j <

Ž Euclidean norm.

Ž Row norm.

Ž 119b .

Ž 119c .

j

Table 2 lists the liquid systems according to increasing condition numbers. Three different set of liquids are presented, Two polar liquid system where ␥sLW is pre-determined using contact angle of an apolar liquid. II. Three liquids out of which two are polar and the third one is apolar, all three y parameters ␥sLW , ␥q s and ␥s are determined using the liquid set. y III. Three polar liquids, all three parameters ␥sLW , ␥q s and ␥s are determined using the liquid set. IV. More than three liquidsᎏa combination of polar and apolar liquids, least square method is used to solve the system of equations. I.

The first two groups, i.e. II and I are essentially the same as they give the same and ␥y values for the parameters ␥sLW , ␥q s s , but their condition numbers are different. Kiely w103x reported contact angle data with three polar and two apolar liquids, for two strains of Bre¨ ibacterium linens used as a taste enhancer in the Danish Cheddar cheese ŽTable 3.. The cell surface energy parameters are analysed from liquid systems using Matlab 5.3 and presented in Table 4. Case-1 is when the ␥sLW is predetermined using contact angle data with some apolar liquid and then polar liquids are used in pairs or in over-determined system y when all three liquids are used for the determination of ␥q s and ␥s . Out of the Ž . four possible liquid groups only water᎐formamide cond. no. 3.5 group gives positive solid surface energy and square roots of the surface energy components. Even the over-determined system with water᎐formamide᎐Glycerol Žcond. no. 3.9. does not give all positive parameters. The condition numbers for the formamide᎐glycerol pair is very high, i.e. 106 and the system is ill-conditioned and should never be used. Case-II is when two polar and one apolar liquid is used. This case is the same as the first case since the solid surface energy parameters obtained are essentially the same. However, the condition numbers are higher than the respective pairs in case 1. In the three liquid system also only water᎐formamide᎐bromonapthalene Žcond.

398 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 Table 2 Condition numbers of the system of simultaneous equation required to be solved in evaluation of solid surface energy parameters using the LW᎐AB approach Two liquid systems Condition numbers using norms 5 A 51 column

5 A 5⬁ row

⌬Qr

Two liquid system where ␥LW is known before hand Wa-Gl 4.5 3.6 Wa-Fo 4.7 3.5 Wa-Eg 5.6 4.1 Wa-DM 6.3 4.7 Fo-DM 23.4 18.5 Gl-DM 31.9 23.7 Fo-Eg 35.1 28.6 Gl-Eg 54.1 40.9 Eg-DM 62.2 49.9 Fo-Gl 136.5 106.4

4.5 4.7 5.6 6.3 23.4 31.9 35.1 54.1 62.2 136.5

13.6 16.4 9 6.7 9.6 6.9 7.4 4.6 2.3 2.8

Three liquid system with one apolar and two polar liquids Wa-Gl-Br 6.12 4.85 Wa-Fo-Br 7.35 5.14 Wa-Eg-Br 8.91 5.91 Wa-DM-Br 10.95 7.06 Fo-DM-Br 39.3 29 Gl-DM-Br 47 34.6 Fo-Eg-Br 54.6 42.8 Gl-Eg-Br 74.1 56.7 Eg-DM-Br 107.2 79.3 Fo-Gl-Br 184.7 148.9

7.35 7.12 8.38 10.19 45.7 61.2 63.6 95.9 117.4 245.3

13.6 16.4 9 6.7 9.6 6.9 7.4 4.6 2.3 2.8

Liquid system

5 A 52 Euclidean

Three liquid system with all polar liquids Wa-Gl-DM 17.2 Wa-Fo-Gl 25.5 Wa-Gl-Eg 31.8 Wa-Fo-DM 34.5 Wa-Eg-DM 35.2 Fo-Gl-DM 53.4 Fo-Gl-Eg 67.1 Fo-Eg-DM 76.4 Gl-Eg-DM 872.5 Wa-Fo-Eg 1.55 = 103

11.6 18.7 21 25.5 26.5 43.1 50.1 47.8 674 1.14 = 103

13.3 20.3 24.6 28.7 30.6 68.9 69.9 61.6 1.09 = 103 1.24 = 103

Rows in bold letters are for liquid pairs, which give positive solid surface energy and square roots of the surface energy components both for FEP and FC-721 Žin Table 1.. ⌬Q r s Ž ␥yr␥q . Liquid 1 y Ž ␥yr␥q . Liquid 2 . Žfor Hollander’s Selection Criterion.. Condition number cond ŽA. s 5 A 5 5 Ay1 5, 5 A 5 1 s ¨ < a i j <. max j

Ý i

5 A5 2 s

1r2

n

5 ai 5

2

žÝ / is1

, 5 A 5 ⬁ s max i

Ý < ai j < j

Abbre¨ iations: Wa, Water; Fo, Formamide; Gl, Glycerol; DM, DMSO; Eg, Ethylene glycol; Br, Bromonapthalene.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 399 Table 3 Mean contact angles Ždegrees. for two strains of Ber¨ ibacterium linens ŽBL. strains BL-MGE

BL-9174

Mean ␪Wa ter ␪Formamide ␪Gl ycerol ␪␣ ᎐ bromonaphthalene ␪Me thyleneiodide

S.D.

30 13 65 43 48

2 1 1 3 1

Mean 18 11 65 35 49

S.D. 1 1 1 3 1

Data taken from Table 1 of Kiely w103x.

no. 5.14. and water᎐formamide᎐methyleneiodide Žcond. no. 5.17. give positive solid surface energy and square roots of the surface energy components. Similar to the two liquid system the formamide ᎐ glycerol᎐ bromonapthalene and formamide᎐glycerol᎐methyleneiodide have very high condition numbers of 148 and 151, respectively, and hence are highly ill-conditioned systems and should never be used. Case-III is when three polar liquids are used to determine all the solid surface energy parameters, i.e. without any apolar liquid in the group then the condition number is high Ž18.5.. Highly negative ␥q values are obtained. s In the fourth case overdetermined system of equation is used with polar and apolar liquids. Even though the condition numbers are 5᎐6, negative parameters are obtained. Therefore, we can see that when biological surfaces are involved then the choice of liquid systems, which can be used for the evaluation of solid surface energy from contact angle data, is very restricted. Only the water᎐formamide pair along with bromonapthalene or methyleneiodide gives positive surface energy and square root of surface energy parameters. Errors in the contact angle measurements can be one of the reasons for this. The bacterial layers, deposited on cellulose nitrate membranes, used for contact angle determination are not smooth, an inherent surface roughness of the order of bacterial cell size is involved. The spreading pressure, moisture content of the bacterial layer, layer thickness and liquid purity are the other factors, which influence the accuracy of contact angle measurements. Volpe and Siboni w101x are of the opinion that the large inconsistencies of the experimental data in current literature are not acceptable without coherent explanation. Published contact angle data frequently show kinetic effects interpreted as thermodynamic ones Žreceding angles greater than advancing ones., lack of any statistical analysis, of whether the angles are advancing or receding, of the temperature values at which they were taken, etc. Many years ago Padday w104x defined the situation of contact angles data as ‘the comedy of errors’. Volpe and Siboni w101x compared the LW᎐AB approach with other scales available to quantify the acid᎐base properties. They concluded that from mathematical point of view the LW᎐AB theory can be classified in the same realm as

'

Liquid system

Condition number

BL-MGE 6␥s

q

BL-9174 6␥s

y

␥s

LW

␥s

q

␥s

y

6␥sq

6␥sq

␥sLW

␥sq

␥sy

I ␥sLW predetermined using ␪Br , only ␥sq and ␥sy are determined using the polar liquid systems Wa-Fo 3.5 1.87 6.27 33.2 3.5 39.32 1.42 Wa-Gl 3.6 y0.74 8.88 33.2 0.55 78.86 y1.15 Fo-Gl 106 24.9 y89.6 33.2 620.01 8028.16 24.19 Wa-Fo-Gla 3.9 0.37 7.71 33.2 0.14 59.45 y0.05

7.05 9.64 y87.8 8.48

36.7 36.7 36.7 36.7

2.02 1.33 585.16 0.01

49.71 92.93 7708.84 71.92

II Two polar and one apolar liquid, evaluation of ␥sLW , ␥sq Wa-Fo-Br 5.14 1.87 6.27 Wa-Gl-Br 4.85 y0.74 8.88 Fo-Gl-Br 148 24.9 y89.6 Wa-Fo-Mi 5.17 1.6 6.3 Wa-Gl-Mi 4.88 y0.93 8.83 Fo-Gl-Mi 151 23.9 y86.7

and ␥sy 33.3 33.3 33.3 36.4 36.4 36.4

36.7 36.7 36.7 34.7 34.7 34.7

2.02 1.33 585.16 2.53 1.09 610.09

49.71 92.93 7708.84 49.57 93.51 8028.16

III Three polar liquids Wa-Fo-Gl 18.7

212

54.47

61.16

39.4 37.1 36.9

0.06 0.01 0.01

71.75 71.92 71.92

36

2

71

y7.03

7.04

IV Overdetermined system of polar and apolar liquids Wa-Fo-Gl-Bra 5.25 0.18 7.69 Wa-Fo-Gl-Mia 5.27 y0.01 7.68 Wa-Fo-Br-Mia 5.48 0.17 7.69 Mean values reported by Kiely et al., 1997 w103x The liquid systems used are not clear from the paper

3.5 0.55 620.01 2.56 0.87 571.21

39.32 78.86 8028.16 39.69 77.97 7516.89

1.42 y1.15 24.19 1.59 y1.04 24.7

7.05 9.64 y87.8 7.04 9.67 y89.6

49.43

49.57

y7.38

7.82

35.8 38.6 36.26

0.04 0.01 0.03

59.14 58.99 59.14

y0.24 y0.083 y0.07

8.47 8.48 8.48

34

2

59

217

The determination of surface energy parameters using 2,3 or over-determined set of equations. Condition number is reported along with the liquid 1r2

n

systems. Condition number cond ŽA. s 5 A 5 5 Ay1 5, the norm used is 5 A 5 2 s a

ž

Ý < ai < 2 is1

/

.

Are overdetermined system of equations, and least square method is used for solid parameter determination. Abbre¨ iations: Wa, Water; Fo, Formamide; Gl, Glycerol; DM, DMSO; Eg, Ethylene glycol; Br, Bromonapthalene; Mi, Methyleneiodide.

400 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Table 4 Surface energy parameter of two strains of Ber¨ ibacterium linens ŽBL. using LW᎐AB approach

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 401

Drago w105x, Taft w106,107x and Abraham w108x scales. They commented that the use of the LW᎐AB approach is easy but results must be considered with attention because 1. The base component of the analysed surface has systematically higher base components. 2. The results strongly depend on the choice of three solvents used for measurements Žas seen in Table 4.. 3. The unknowns of the calculation are the square roots of the surface tension components, and in some cases they assume negative values Žas seen in Table 4.. 4. The values of the coefficients used for acid᎐base properties of test liquids are completely dependent on the choice made for water: the acid᎐base ratio at 20 ⬚C is set at 1, i.e. each components contributes 25.5 mJrm2 . 4.1. This does not mean that water is equally strong as a Lewis acid as a Lewis base. Actually no definition of strength of acids and bases is given y 2 in the LW᎐AB approach, ␥q w s ␥w s 25.5 mJrm is just an assumption. 4.2. One cannot compare the acid and base components of the same solvent, but, eventually, the acid Žor base. components of different solvents can be compared. This is similar to Drago’s theory w105x. However, different to Taft scale w106,107x where the electron acceptor parameters of many solvents are put in such a form that one can compare the two parameters of the same solvent; in that scale, water is considered an electron acceptor or Lewis acid approximately 6.5 times stronger. 4.3. An ambiguity exists with regards to the monopolarity, when a component Žacid᎐base. is very small as compared to the other. According to LW᎐AB approach a substance is monopole if its component ratio with respect to water is less than 25.5, corresponding to the value of 1 mJrm2 . However, this argument is not sufficient because of 4.1. 5. In the set of solvents usually employed, water, glycerol, ethylene glycol, formamide, dimethyl sulfoxide, diiodomethane and bromonapthalene, and on the scale proposed by LW᎐AB, none is prevalently acid and all the polar ones have base components stronger than water, so the situation is strongly asymmetric, with a pronounced role for base functions. These choices are probably at the origin of the systematic base predominance in tabulated LW᎐AB coefficients. 6. The equation system, whose solution provides the surface tension components can be ill-conditioned for an improper choice of solvents, unfortunately the LW᎐AB approach does not examine this subject closely. Different liquid systems are studied before and presented in Table 4. A consequence is that no evaluation of the goodness of liquid and solid parameters has been done in the LW᎐AB approach, so no standard deviation or error estimate of reckoned values is known. Keeping all the above considerations Volpe and Siboni w101x suggested another

402 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

method in order to arrive at the surface energy parameters of solids and solvents. A wider set of equation containing Eq. Ž112a. for a large set of solid᎐liquid couples and the following equations for every liquid i and solid j. q y ␥Total s ␥ lLW l ,i ,i q 2 ␥ l ,i ␥ l ,i

Ž 120a .

q y ␥s,Total s ␥s,LW j j q 2 ␥s, j ␥s, j

Ž 120b .

'

'

The symbols have the same meaning as in Eq. Ž112a. with an additional index j to distinguish the various solids. Similarly, the contact angle of ith liquid on jth solid is denoted by ␪ i j . For L liquids and S solids we have a set of SL q S q L equation and 4S q 3 L unknowns; the four unknowns for each solid include the total surface free energy because for liquids the total surface energy is experimentally measured but there is no commonly accepted technique for solids. The equation system Eqs. Ž112a., Ž120a. and Ž120b.x is non-linear and overdetermined Žexcept for very low values of S and L.. The procedure of solution is similar to Drago w105x for his set of equations leading to the well-known acid᎐base scale. The system has no exact solution but can be solved by best-fit criterion Žleast squares in our case. searching for the minimum of the objective function wEq. Ž121.x. This system has an infinite number of best-fit solutions. L

Us

X iT RX i y ␥Total l ,i

Ý is1

2

S

q

Ý

Total Yi T RYi y ␥s,i

2

js1 L

S

qÝ

Ý

X iT RYj y 12 Ž 1 q cos␪ i j . ␥Total l ,i

2

Ž 121.

is1 js1

where R is a 3 = 3 symmetric orthogonal matrix, and

'␥

Xi s

'␥

LW l ,i

LW s, j

 0  0 q l ,i

'␥ '␥

y l ,i

and Yj s

q s, j

'␥ '␥

Ž 121a .

y s, j

Among the infinitely many solutions, one can eventually choose that which is based on some conventionally assigned components of reference solvents. Infinitely many equivalent scales are possible, but a scale based on realistic ratio between acid᎐base components of a reference solvent Žas for water acid᎐base ratio of 1 in LW᎐AB or 6.5 in Taft scale or 5.5 in Abrahams scale.. This procedure is difficult to realise, due to the disagreement in the values of contact angles of common non-polar and basic solvents on common polymers and to the lack of data about

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 403

acid solvents, however, in their opinion this proposed method represents the most correct way to apply the LW᎐AB approach. A less restrictive, but very useful, constraint would be simply to impose acidic and basic components of a solvent to be equal, without assigning a particular value to them. In this case the scale is not completely determined. However, in the present form this choice is incorrect due to reliable values of contact angles and solvent surface energy parameters. The reference solvents and components have to be judiciously chosen because a wrong choice could be inconsistent with the experimental data. The possibility of estimating the errors of the coefficients, we can confirm that the location of minima for the non-linear best-fit function is obviously influenced by the knowledge of the constant parameters, which comes from experiments. Errors in the estimate of parameters result in a certain amount of uncertainty about the minima positions: owing to non-linear coupling of variables. Even a small change in even a single parameter will yield a global rearrangement of minima, all of whose location will generally be modified, as will, therefore, the estimated value of the acid᎐base components for all the chemical species involved. Volpe and Siboni w101x claim that the necessity of a wide, accurate and homogeneous file of experimental data, collect according to standard procedures for methods and materials employed, specimen treatment, working temperature, etc. The number of available data for each solid᎐solvent pair should be sufficiently large to clearly provide significant statistics, to ensure good reproducibility, and to allow a reasonable estimate of the experimental error. A round robin test is proposed where a wide group of laboratories should work together, realising a round robin with solvents and polymer surfaces chosen by mutual consent; the mean results of all laboratories, both advancing and receding, could be used as a base set of proper contact angles to calculate the most satisfactory acid᎐base components of liquids and solid surface free energies. Kwok w50,97x once again put the LW᎐AB approach to test using another method, i.e. by comparing the liquid᎐liquid interfacial tension from calculations using the LW᎐AB approach and by measurement using the ADSA᎐P Žasymmetric drop shape analysis-profile . technique. Though, Van Oss w75x has reported that by the LW᎐AB approach some interfacial tensions for the solutersolvent systems with good solubility are negative. Lee w90x supports this, where he reported the solubilities and interfacial tensions of many polymers, carbohydrates and proteins, and found that indeed the interfacial tensions were negative for those with good solubility. Kwok w109x reported the interfacial tensions of liquidsᎏglycerol, formamide, diiodomethane, bromonapthalene, ethylene glycol and DMSO against pentane, decane, dodecane, tetradecane and hexadecane. The percentage error between the measured and calculated Žby LW᎐AB approach. interfacial tensions for the liquid pairs which are immiscible range from 34% lower to 112% higher but the scatter diagram between the measured and calculated show a definite correlation. Kwok w50,97x also reported a previous work where they studied the miscible liquid systems w110x. The LW᎐AB approach reported negative interfacial tensions for the miscible system of water᎐glycerol, water᎐formamide, water᎐ethyleneglycol

404 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

and water᎐DMSO. However, it did not give negative interfacial tension values for many other miscible liquid systems like glycerol᎐formamide, glycerol᎐ethylene glycol, glycerol᎐DMSO, formamide᎐ethylene glycol, formamide᎐DMSO, diiodomethane᎐bromonapthalene, diiodomethane᎐DMSO, ethylene glycol᎐DMSO and bromonapthalene᎐DMSO. They concluded that the LW᎐AB approach does not predict the correct interfacial tensions of a large number of arbitrarily chosen miscible and immiscible liquid᎐liquid system hence, caution should be exercised when this approach is used to determine solid surface tension components from contact angles. 2.11. Spreading pressure estimation Spreading pressure is the reduction of the solid surface energy due to the vapour adsorption. In general the approaches assume this term to be negligible, which is true for low energy solids in contact with high-energy liquids w31,64x. The spreading pressure term is significant for high-energy surfaces when the low energy liquid spontaneously spreads and forms a very thin layer in order to reduce the total energy of the system. It has been made possible to quantify the spreading pressure on polymer surfaces but spreading pressure quantification due to water᎐bacterial surface has not yet been reported. Busscher w111x used the Geometric mean approach in order to obtain the spreading pressure. First the ␥sd is determined by using a contact angle with an apolar liquid, then the contact angle with a series of water q propanol mixtures is obtained. A plot of ␥ l ¨ Žcos␪ q 1. y 2 ␥ ld ␥sd vs. ␥lp gives a straight line the slope of 2 ␥sp and intercept on the Y axis of y␲e according to Eq. Ž122..

'

'

'

␥ l ¨ Ž cos␪ q 1 . y 2 ␥ ld ␥sd s 2 ␥ lp

'

' '␥

p s

y ␲e

Ž 122.

The water q propanol mixture surface tension parameters are obtained by contact angle measurement on paraffin wax surface Ž ␥sd s 25.5 mJrm2 and ␥sp s 0.. Bellon-Fontain and Cerf w112x evaluated the spreading pressure using the fact that the work of adhesion, WA , is a maximum value when there is no vapour adsorption on the solid surface. A plateau region is observed in the plot between work of adhesion WA s ␥l ¨ Žcos␪ q 1. and ␥ l ¨ for different liquids on the solid surface. The spreading pressure is determined from the difference between the plateau work of adhesion and the work of adhesion of the liquid. Erbil w113x combined one-liquid and two-liquid ŽFig. 9. Žwhere the contact angle with one liquid is measured in presence of another immiscible liquid and not air. contact angle data and the Geometric mean approach to obtain the spreading pressure of water᎐polymer interactions. Later Erbil w114,115x used the Lifshitz᎐van der Waals acidrbase approach to determine the spreading pressure by combining the one-liquid and two-liquid contact angle methods. Most of the methods use either the Geometric mean approach or Lifshitz᎐van

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 405

der Waals acidrbase approach. The Equation of state approach applies to ideal solids, which does not absorb vapours Žessentially true for low energy solids, e.g. Teflon, etc... However, deviation of some of the points from the ␥ l ¨ vs. ␥ l ¨ cos␪ ŽFig. 11. curves gives a possibility to determine the spreading pressure w116,117x. 2.12. Comparison of the thermodynamic approaches Studies have been performed in different fields to find out the best approach for evaluation of solid surface energy from contact angle data. Bellon᎐Fontaine w118x studied the adhesion behaviour of dairy microorganisms Ž L. mesenteroides and S. thermophilus. on FEP, polypropylene, PMMA and glass, and used the different approaches to predict the possibility and extent of adhesion. They concluded that all the approaches yield similar surface energies, but equation of state approach with different liquids did not give consistent values for high surface energy surfaces. The Geometric mean approach accounting for spreading pressure was the best approach, which could predict the adhesion phenomenon. Gindl w119x compared the different approaches to evaluate the surface energy of wood samples. Surface energy of 10 Norway spruce were evaluated using different approaches and was shown that the LW᎐AB approach delivered results very similar to geometric mean and, to a lesser degree, to the equation of state approach. The results achieved with harmonic mean and Zisman approach deviated clearly from the group. It was concluded that if the right group of liquid contact angles are used then the LW᎐AB approach delivers a maximum information about the chemical composition, in giving the values for electron-donor and electron-acceptor components of surface, which is especially valuable with a chemically heterogeneous material like natural polymer wood, containing cellulose, lignin and a variety of hemicellulose. The LW᎐AB approach is most suitable to explain the coating properties Žadhesion. of wood surfaces. Schneider w120x studied the adhesion of Gram-negative bacterium SW8 on various substratumᎏgermanium, stainless steel, polypropylene, perspex coated with conditioning films. The surface energy of the bacterium and the solid substrates were evaluated using the different approaches. The bacterium᎐substrate interfacial energy and free energy of adhesion were calculated from the solid surface energies. He concludes that the LW᎐AB approach provides the most consistent treatment of acid᎐base and apolar interfacial interactions. Results obtained with LW᎐AB were not in conflict with its theoretical framework but the fact that negative square roots were obtained with some regularity for the determination of ␥q and hence suggested to be cautious while interpreting the results obtained from this theory.

3. Measurement of contact angle on microbial cell surface The contact angle of bacterial cells is measured by producing a uniform layer of cells on agar or the bacterial lawn is deposited on membrane filters w121᎐123x.

406 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

The membrane technique is preferred over the agar techniques as the cells are less apt to detachment in the liquid phase, secondly by depositing the bacterial lawns on the membrane they can be dried of that only bound water is present on the bacterial surface. The membrane method is described in detail by Busscher w122x and van der Mei w23,124x. The bacterial substrate for measuring contact angles are prepared by depositing bacterial cells, suspended in water, on cellulose triacetate filter Žpreferable pore diameter of 0.45 ␮m. by applying negative pressure. The bacterial lawns are deposited to a density of 10 8 cellsrmm 2 w122x or approximately 50 layers of bacteria w124x, 800᎐900 layers w16x. To establish constant moisture content the filters with bacteria are placed in a petri dish on the surface of a layer of 1% Žwt.rvol.. agar in water containing 10% Žvol.rvol.. glycerol. The filters are left in the petri dish until they can be used for contact angle measurement. This serves two purposesᎏagar acts as a moisture buffer for approximately 3᎐5 h and it does not allow the filter to dry, secondly the moisture content in all the filters with bacterial lawns are brought to the same level before they are dried under controlled conditions Žthe moisture content left after the filtration process may not be the same.. The filter with bacterial lawns is cut in strips of appropriate width Žapprox. 1 cm. and fixed on the sample holder with the help of double-sided adhesive tape. The bacterial lawns are allowed to air-dry until a physiologically relevant state w121x is achieved, where only bound water is present on the bacterial surface. This physiologically relevant state is characterised by attainment of a plateau region in the water contact vs. drying time curves, this state lasts for 30᎐60 min w23x. The contact angle behaviour after the drop has been positioned on the bacterial lawn and is different for polar and apolar liquids. For the apolar liquids contact angle along with the drop volume, height and base diameter stays constant with time under the period of monitoring the drop, i.e. 30 s. However, the contact angle changes with time for polar liquids ŽFig. 15.. There are three different regions in the contact angle vs. time curve. In the beginning the contact angle drops until approximately 0.1᎐0.3 s this happened because the drop spreads on the surface as can be inferred from the fact that the drop volume stays constant, the drop height decreases and the base diameter increases. After this stage the drop comes in equilibrium where there is no change in drop contact angle, volume, height or base diameter. The equilibrium stage lasts until approximately 1᎐3 s. Again after some time the contact angle starts decreasing this is because the polar liquid wets the bacterial lawns and absorption of the liquid takes place. This stage is characterised by a steep decrease of volume and height, the drop keeps on spreading as seen from the increase in the drop diameter. The relative spans of the three stages depend on the bacterial lawn thickness and probably on the bacterial strain hydrophobicity. Sharma w16x reported that 800᎐900 bacterial layers were required to get a reasonably long equilibrium stage for wild and sulfide mineral adapted Paenibacillus polymyxa. If too less bacterial layers are taken then the spreading can directly merge with the absorption stage and hence no equilibrium stage is available to measure the contact angle value. Though other

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 407

Fig. 15. A typical curve for polar liquids-contact angle vs. time lapsed after positioning of drop over the bacterial lawns. Measured by the FIBRO DAT 1100 dynamic absorption tester.

authors have also observed this decrease in contact angle after positioning of the drop w125x, they have considered it be an anomaly and have proposed extrapolation of the curve to t s 0 in order to account for the stability problems. In our view the initial drop in contact angle takes place for the drop to attain equilibrium and van der Mei w124x are also of the same opinion that it takes approximately 5᎐7 s for the contact angle equilibrium. 3.1. Axisymmetric drop shape analysis-contact diameter (ADSA-CD) Neumann and co-workers has been using the ADSA-CD technique to measure the contact angles on bacterial lawnsᎏLin w126x; Drumm w127x and Duncan-Hewitt w128x. ADSA-CD is a modified version of ADSA-P, which was developed by Rotenberg and implemented by Skinner w129x. This technique was initially developed to measure low contact angles. The technique requires the contact diameter, the drop volume and the liquid surface tension, the density difference across the liquid᎐liquid interface, and the gravitational constant as inputs to calculate the contact angle by means of numerical integration of the Laplace equation of capillarity. As the contact angle decreases, the profile of sessile drop becomes increasingly flat about the apex, and the accuracy of direct methods, such as goniometry, is adversely affected. The lack of curvature in the profile also presents a problem for all methods that rely on the profile of a drop to determine the contact angle. The

408 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

success of drop profile methods is dependent on the existence of comparable surface tension and gravitational effects. For very flat drops, the effect of gravity dominates, and the surface tension has negligible effect on the shape of the interface. Thus, using the shape of the interface to determine the surface tension is not an effective method for such experimental situations. Therefore, the ADSA-P technique and goniometry are not suitable. ADSA-CD circumvents this problem by utilising top view of the drop instead of the side view. Essentially, the contact angle is computed by numerically minimising the difference between the volume of the drop, as predicted by Laplace equation of capillarity and the experimentally measured volume. Although, ADSA-CD was originally developed to measure the contact angles of very flat drops but since it uses the top view of the drop, it is found to be very useful for measurement of drops on non-ideal surfaces, which are relatively rough and heterogeneous. It is practically impossible to form axisymmetric drop on such surfaces. The irregularities in the three-phase contact line are averaged to an average diameter of the drop by a least-square fit of a circle to the experimentally measured points along the three-phase boundary and then an average contact angle is determined. All these facts have made ADSA-CD methodology particularly useful for biological materials, e.g. layers of bacterial cells which are necessarily rough and absorb water and other liquids so that the drops sink into the layer of cells, in addition the hydrophilic, bacterial layers produce small time dependent contact angles. 3.2. The contact angle data on bacterial lawns Ven der Mei and Busscher w22x has compiled a list of contact angles measured on bacterial lawns with water, formamide, methyleneiodide and ␣-bromonapthalene. This review provides a reference guide to microbial cell surface hydrophobicity based on contact angles with the above diagnostic liquids and involves Acinebacter, Actinobacillus actinomycetemcomitans, Actinomyces, Bre¨ ibacterium linens, various Candida species, Capnocytophaga gingi¨ alis, Enterococci, Escherichia coli, Lactobacilli, Leuconostoc mesenteroides, Peptostreptococci, Porphyromonas gingi¨ alis, Pre¨ otella intennedia, Pseudomonads, Serratia marcescens, Staphylococci, and Streptococci adding up to 142 isolates. Only those strains or species are included in the reference guide where the plateau contact angles with the four diagnostic liquids were available in the literature. This compiled list of contact angles is used along with the contact angles of five wild and sulfide mineral adapted Paenibacillus polymyxa w16x cells.

4. Analysis of thermodynamic approaches In the following discussion two issues have been addressed independently without mixing them wherever possible: Ž1. the surface energy and its components for bacterial cell surface; and Ž2. the similarities, dissimilarities and internal

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 409

Fig. 16. Distribution of water, formamide, methyleneiodide and ␣-bromonapthalene contact angles on 147 different microbial isolates.

consistency of the three approaches, namelyᎏEquation of state, Geometric mean and Lifshitz᎐van der Waals acidrbase ŽLW᎐AB.. The contact angles of 147 different isolates with four liquids have very different distribution, contact angles with polar and apolar liquids have totally different distribution ŽFig. 16.. Except the Equation of state approach, the other approaches ŽFowkes, Geometric mean and LW᎐AB. allow that the total surface energy can be divided in two separate partsᎏthe apolar, so called dispersion or Lifshitz van der Waals and polar or acid᎐base part. The surface energy using the Equation of state approach is evaluated using the conversion tables from contact angle to surface energy by Neumann w130x. Since the conversion is available only at fixed liquid surface tensions therefore, ␥ 1 s 73 is used for water Ž72.8., ␥ 1 s 58 is used for formamide Ž58., ␥1 s 44 is used for ␣-bromonapthalene Ž44.4. and ␥ 1 s 51 is used for methyleneiodide Ž50.8.. The detailed parameters for the standard liquids are presented in Table 6. Recently, Balkenende w131x measured liquid contact angles on FC722 and have reported different surface energy parameters for methyleneiodide Ž ␥ 1 s 50.8 and ␥ 1LW s 33.5 mJrm2 . and ␣-bromonapthalene Ž ␥ 1 s 43.9 and ␥ 1LW s 37.6 mJrm2 ., but since they are not universally accepted yet, the widely accepted parameters ŽTables 5 and 6. has been used in the analysis. The contact angle values for the 147 microbial isolates along with the surface energies evaluated by the different approaches have been presented in Table 5.

Bacterial isolate

Acenobactor calcoacenticus MR-481 Acenobactor calcoacenticus RAG-1 Actinobacillus actinomycetemcomitans HG1098 Actinobacillus actinomycetemcomitans HG1099 Actinomyces israelii PK16WT Actinomyces naeslundii 147 Actinomyces naeslundii 5519 Actinomyces naeslundii 5951 Actinomyces naeslundii PK29 Actinomyces naeslundii T14V-J1 Bre¨ ibacterium linens BL-9174 Bre¨ ibacterium linens BL-MGE Candida albicans ATCC10261 (30C), DYM Candida albicans ATCC10261 (30C ), GSB Candida albicans ATCC10261 (37C ), DTM Candida albicans ATCC10261 (37C ), GSB Capnocytophaga gingi¨ alis PC1000 Capnocytophaga gingi¨ alis PC1000 r 1᎐7

Gram

␪W

␪F

␪M

␪ Br

Equation of state approach

Geometric mean approach

LW ᎐AB approach using ␪ W, ␪F and ␪ Br

␥Total By ␪ W

␥LW by ␪ Br

␥LW by ␪ B r

␥AB by ␪W , ␪B r

␥ Total

␥q

␥Total y␥ LW

␥y

␥ AB

␥ Total

᎐

32

25

46

37

63.44

41.83

21.61

35.91

30.28

66.19

1.63

42.51

16.64

52.55

᎐

48

49

43

38

54.57

41.39

13.18

35.49

21.28

56.77

0.04

40.15

2.54

38.03

᎐

93

36

76

61

27.24

30.40

y3.16

24.48

2.55

27.02

14.48

2.39

11.75

36.22

᎐

87

47

41

54

31.07

33.87

y2.80

27.99

3.63

31.62

5.98

0.07

1.25

29.23

q q q q q q

60 53 62 62 90 64

35 45 51 45 41 37

44 45 48 45 43 43

30 31 32 31 31 29

47.51 51.64 46.34 46.34 29.16 45.17

44.68 44.3 43.9 44.3 44.3 45.06

2.83 7.34 2.44 2.04 y15.14 0.11

38.66 38.29 37.91 38.29 38.29 39.01

12.83 16.97 12.01 11.87 1.11 10.55

51.48 55.26 49.92 50.15 39.39 49.56

1.73 0.19 0.09 0.54 4.39 1.70

14.14 29.24 21.33 17.21 0.37 10.90

9.88 4.66 2.69 6.08 2.54 8.59

48.54 42.94 40.60 44.37 40.82 47.60

q

18

11

49

35

69.62

42.68

26.94

36.74

35.79

72.53

2.05

49.67

20.15

56.88

q

30

13

48

43

64.45

39.14

25.31

33.28

32.96

66.23

3.53

39.15

23.48

56.76

59

29

33

46

48.10

37.74

10.36

31.88

16.25

48.13

4.83

12.36

15.45

47.32

78

40

44

30

36.74

44.68

y7.94

38.66

4.38

43.03

2.76

1.71

4.34

42.99

21

᎐

49

24

68.49

46.82

21.67

40.65

32.35

73.00

᎐

᎐

᎐

36

34

49

27

61.34

45.79

15.55

39.70

26.03

65.73

0.34

44.17

7.75

47.45

᎐

32

59

68

57

63.44

32.39

31.05

26.49

36.67

63.16

0.06

77.83

4.28

30.77

᎐

37

74

67

60

60.80

30.90

29.90

24.98

34.82

59.80

2.06

95.93

28.08

53.06

᎐

410 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Table 5 Summary of microbial cell surface propertiesᎏ type of Gram staining, contact angle with water, formamide, methyleneiodide and ␣-bromonapthalene w22,23 x and surface energy parameters evaluated using different thermodynamic approaches

Table 5

Ž Continued .

E. coli 917 E. coli C1212 E. coli C1214 E. coli O111K38 E. coli O157 KE. coli O161 KE. coli O2K2 E. coli O2K7 E. coli O39K1 E. coli O8K (A )28 E. coli Col E. coli Hu734 E. coli O83K Enterococci faecalis 1131 Enterococci faecalis 29212 Enterococci faecalis C1030 Enterococci faecalis IC14 Lactobacillus acidophilus 68 Lactobacillus acidophilus 75 Lactobacillus acidophilus RC14 Lactobacillus acidophilus T13 Lactobacillus casei 36

Gram

␪W

␪F

␪M

␪ Br

Equation of state approach

Geometric mean approach

LW ᎐AB approach using ␪ W, ␪F and ␪ Br

␥Total By ␪ W

␥LW by ␪ Br

␥Total y␥ LW

␥LW by ␪ B r

␥AB by ␪W , ␪B r

␥ Total

␥q

␥y

␥ AB

␥ Total

᎐ ᎐ ᎐ ᎐ ᎐ ᎐ ᎐ ᎐ ᎐ ᎐ ᎐ ᎐ ᎐ q

23 25 25 18 19 26 57 29 21 46 21 17 54 23

22 23 23 23 17 37 30 28 25 29 25 18 36 30

52 50 50 60 56 57 59 60 54 57 52 48 57 42

38 34 34 50 46 51 35 40 40 36 38 33 39 27

67.67 66.80 66.80 69.62 69.26 66.35 49.28 64.94 68.49 55.73 68.49 69.96 51.05 67.67

41.39 43.09 43.09 35.82 37.74 35.36 42.68 40.51 40.51 42.26 41.39 43.50 40.95 45.79

26.28 23.71 23.71 33.80 31.52 30.99 6.60 24.43 27.98 13.47 27.10 26.46 10.10 21.88

35.49 37.14 37.14 29.96 31.88 29.47 36.74 34.62 34.62 36.33 35.49 37.53 35.06 39.70

34.75 32.87 32.87 40.54 38.76 37.58 15.30 32.59 36.11 22.07 35.54 35.61 17.83 32.15

70.24 70.00 70.00 70.50 70.64 67.04 52.03 67.21 70.73 58.40 71.03 73.13 52.89 71.85

1.63 1.27 1.27 2.74 2.93 1.17 2.74 1.41 1.42 1.96 1.25 1.36 1.83 0.31

50.11 48.95 48.95 54.27 50.58 57.86 15.10 47.68 53.63 27.64 53.73 53.13 21.44 55.97

18.03 15.76 15.76 24.38 24.31 16.41 12.87 16.35 17.45 14.71 16.37 16.99 12.53 8.28

53.52 52.90 52.90 54.33 56.19 45.88 49.60 50.97 52.07 51.04 51.86 54.51 47.58 47.97

q

30

68

46

21

64.45

47.75

16.70

41.50

28.09

69.59

6.02

99.25

48.88

90.38

q

100

91

46

39

22.82

40.95

y18.13

35.06

0.12

35.18

3.73

5.83

9.32

44.38

q

35

75

23

26

61.88

46.14

15.74

40.03

26.39

66.41

8.60

104.25

59.88

99.990

q

74

39

52

35

39.22

42.68

y3.46

36.74

6.45

43.19

3.00

3.50

6.47

43.20

q

66

56

50

39

44.00

40.95

3.05

35.06

10.89

45.94

0.07

19.60

2.20

37.25

q

102

47

55

38

21.57

41.39

y19.82

35.49

0.03

35.51

5.58

4.67

10.21

45.70

q

80

39

46

27

35.49

45.79

y10.30

39.70

3.47

43.17

2.96

0.85

3.17

42.86

q

19

29

44

33

69.26

43.50

25.76

37.53

34.95

72.48

0.53

58.54

11.10

48.63

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 411

Bacterial isolate

Ž Continued .

Bacterial isolate

Lactobacillus casei 43 Lactobacillus casei 55 Lactobaciflus casei 62 Lactobacillus casei 65 Lactobacillus casei 70 Lactobacillus casei 8 Lactobacillus casei subsp. rhamnosus 81 Lactobacillus casei subsp. rhamnosus ATCC7469 Lactobacillus casei subsp. rhamnosus GR1 Lactobacillus casei subsp. rhamnosus RC15 Lactobacillus casei subsp. rhamnosus RC17 Lactobacillus fermentum A60 Lactobacillus fermentum B54 Lactobacillus gasseri 56 Lactobacillus gasseri 60 Lactobacillus jensenii RC28 Lactobacillus plantarum RC20 Lactobacillus plantarum RC6

Gram

␪W

␪F

␪M

␪ Br

Equation of state approach

Geometric mean approach

LW ᎐AB approach using ␪ W, ␪F and ␪ Br

␥Total By ␪ W

␥LW by ␪ Br

␥LW by ␪ B r

␥AB by ␪W , ␪B r

␥ Total

␥q

␥y

␥ AB

␥ Total

␥Total y␥ LW

q q q q q q q

46 36 19 58 43 30 86

29 27 30 32 26 33 37

55 44 50 46 44 48 52

38 33 32 33 23 31 37

55.73 61.34 69.26 48.69 57.46 64.45 31.70

41.39 43.50 43.90 43.50 47.14 44.3 41.83

14.34 17.84 25.36 5.19 10.32 20.15 y10.13

35.49 37.53 37.91 37.53 40.95 38.29 35.91

22.50 27.21 34.71 14.39 21.55 29.88 2.36

57.99 64.74 72.62 51.92 62.49 68.17 38.27

2.17 1.30 0.42 2.30 1.18 0.40 5.77

27.57 39.11 59.38 14.91 30.17 50.68 0.06

15.44 14.22 9.89 11.69 11.92 8.92 1.09

50.93 51.75 47.80 49.22 52.86 47.20 37.00

q

34

28

51

31

62.4

44.30

18.10

38.29

27.86

66.14

0.96

42.23

12.73

51.02

q

33

38

44

26

62.93

46.14

16.79

40.03

27.42

67.44

0.06

51.55

3.45

43.46

q

52

29

48

27

52.23

45.79

6.44

39.70

16.95

56.64

1.70

20.53

11.80

51.49

q

54

39

47

30

51.05

44.68

6.37

38.66

16.23

54.88

0.74

23.53

8.33

46.98

q

29

27

54

33

64.94

43.50

21.44

37.53

30.81

68.34

0.98

47.30

13.58

51.10

q

105

46

55

38

19.71

41.39

y21.68

35.49

0.02

35.51

6.54

7.41

13.91

49.40

q

90

46

47

29

29.16

45.06

y15.90

39.01

1.03

40.04

2.80

0.06

0.79

39.80

q

67

43

47

24

43.42

46.82

y3.40

40.65

8.53

49.18

0.76

10.69

5.69

46.33

q

87

40

47

30

31.07

44.68

y13.61

38.66

1.69

40.34

4.06

0.04

0.70

39.35

q

79

43

45

30

36.12

44.68

y8.56

38.66

4.02

42.67

2.22

1.84

4.03

42.68

q

25

31

49

24

66.80

46.82

19.98

40.65

30.79

71.44

0.21

54.86

6.75

47.39

412 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Table 5

Table 5

Ž Continued .

Leuconostoc mesenteroides NCDO 523 Paenibacillus polymyxa NCIM 2539 Chalcopyrite adapted Paenibacillus polymyxa NCIM 2539 Galena adapted Paenibacillus polymyxa NCIM 2539 Pyrite adapted Paenibacillus polymyxa NCIM 2539 Sphalerite adapted Paenibacillus polymyxa NCIM 2539 Peptostreptococci micros ATCC 33270 Peptostreptococci micros HG1108 Peptostreptococci micros HG1109 Peptostreptococci micros HG1111 Per¨ otella intermedia HG1103 Per¨ otella intermedia HG1104 Per¨ otella intermedia HG1105 Porphyromonas gingi¨ alis HG1101

Gram

␪W

␪F

␪M

␪ Br

Equation of state approach

Geometric mean approach

LW ᎐AB approach using ␪ W, ␪F and ␪ Br

␥Total By ␪ W

␥LW by ␪ Br

␥Total y␥ LW

␥LW by ␪ B r

␥AB by ␪W , ␪B r

␥ Total

␥q

␥y

␥ AB

␥ Total

q

17

39

52

36

69.96

42.26

27.70

36.33

36.38

72.71

0.05

69.51

3.48

39.81

q

40.81

56

66.68

51.84

58.59

34.85

23.74

29.06

29.50

58.56

0.01

59.43

0.21

29.27

q

15.85

21.05

70.6

55.6

70.29

32.88

37.41

27.19

43.46

70.64

3.89

54.31

29.04

56.23

q

17.55

19.37

51

48.3

69.79

36.79

33.00

30.79

40.07

70.85

2.94

52.67

24.85

55.63

q

19.85

18.42

71.8

56.66

68.88

32.39

36.49

26.66

42.47

69.12

4.69

49.95

30.61

57.26

q

29.97

27.63

69.94

57

64.45

32.39

32.06

26.49

37.81

64.29

3.89

45.45

26.57

53.05

q

39

49

72

51

59.70

35.36

24.34

29.47

30.36

59.83

0.25

53.53

7.18

36.64

q

66

69

51

34

44.00

43.09

0.91

37.14

10.16

47.29

1.32

32.07

13.01

50.14

q

33

49

58

43

62.93

39.14

23.79

33.28

31.37

64.65

0.01

62.91

0.07

33.34

q

58

70

55

34

48.69

43.09

5.60

37.14

14.55

51.68

2.34

48.08

21.21

58.34

᎐

38

᎐

75

71

60.25

25.46

34.79

19.51

38.78

58.28

᎐

᎐

42

55

65

46

58.02

37.74

20.28

31.88

26.98

58.86

0.03

56.46

2.33

34.21

᎐

36

39

67

58

61.34

31.89

29.45

25.99

34.68

60.66

2.22

46.98

20.42

46.40

᎐

30

᎐

60

30

64.45

44.68

19.77

38.66

29.67

68.32

᎐

᎐

᎐

᎐

᎐

᎐

᎐

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 413

Bacterial isolate

Ž Continued .

Bacterial isolate

Gram

Porphyromonas gingi¨ alis ᎐ HG1102 Porphyromonas gingi¨ alis W83 ᎐ Pseudomonas aeruginosa AK1 ᎐ Pseudomonas fluorescens ᎐ Serratia marcescens 3162 ᎐ Serratia marcescens 3164 ᎐ Serratia marcescens RZ30 ᎐ Serratia marcescens RZ37 ᎐ Staphylococci epidermidis 169 q Staphylococci epidermidis 236 q Staphylococci epidermidis 242 q Staphylococci epidermidis 252 q Staphylococci epidermidis q 26512 Staphylococci epidermidis q 26585 Staphylococci epidermidis q 26741 Staphylococci epidermidis 298 q Staphylococci epidermidis 3059 q Staphylococci epidermidis 3081 q Staphylococci epidermidis 3294 q Staphylococci epidermidis 3399 q Staphylococci epidermidis q ATCC35893 Staphylococci epidermidis q ATCC35984 Staphylococci epidermidis q HBH171 Staphylococci epidermidis HBH2 q

␪W

␪F

␪M

␪ Br

Equation of state approach

Geometric mean approach

LW ᎐AB approach using ␪ W, ␪F and ␪ Br

␥Total By ␪ W

␥LW by ␪ Br

␥LW by ␪ B r

␥AB by ␪W , ␪B r

␥ Total

␥q

␥y

␥Total y␥ LW

␥ AB

␥ Total

37

51

71

46

60.80

37.74

23.06

31.88

30.00

61.88

0.01

59.33

0.89

32.77

61 106 38 50 21 54 37 37 32 30 29 77

57 52 38 55 18 55 41 31 31 29 21 55

59 52 50 47 58 88 97 58 52 54 47 53

50 39 32 39 49 91 90 45 35 29 19 37

46.92 19.09 60.25 53.40 68.49 51.05 60.80 60.80 63.44 64.45 64.94 37.37

35.82 40.95 43.90 40.95 36.31 16.18 16.63 38.21 42.68 45.06 48.31 41.83

11.10 y21.86 16.35 12.45 32.18 34.87 44.17 22.59 20.76 19.39 16.63 y4.46

29.96 35.06 37.91 35.06 30.45 10.72 11.10 32.35 36.74 39.01 42.02 35.91

15.92 0.04 25.9 20.28 39.05 35.45 48.87 29.70 29.79 29.47 28.28 5.41

45.88 35.09 63.80 55.33 69.49 46.17 59.97 62.05 66.52 68.48 70.29 41.32

0.19 4.63 0.28 0.05 3.37 6.82 10.05 1.97 0.83 0.62 0.79 0.46

27.23 5.78 44.66 43.35 49.26 33.93 45.35 39.96 46.58 47.69 44.23 6.86

4.55 10.34 7.05 2.70 25.76 30.41 42.70 17.70 12.40 10.85 11.76 3.55

34.50 45.40 44.96 37.75 56.20 41.13 53.80 50.05 49.13 49.86 53.78 39.46

41

57

49

49

58.59

36.31

22.28

30.45

28.49

58.93

0.06

60.66

3.53

33.97

77

47

45

33

37.37

43.50

y6.13

37.53

5.01

42.54

1.45

3.89

4.74

42.27

68 31 87 107 29 66

49 53 50 44 56 45

51 46 49 53 56 50

22 37 32 35 39 34

42.83 63.95 31.07 18.48 64.94 44.00

47.75 41.83 43.90 42.68 40.95 43.09

y4.92 22.12 y12.83 y24.2 23.99 0.91

41.23 35.91 37.91 36.74 35.06 37.14

7.89 30.79 1.79 0.13 32.32 10.16

49.12 66.70 39.70 36.87 67.37 47.29

0.17 0.36 1.76 7.17 0.68 0.95

12.80 71.52 0.33 10.12 78.78 12.56

2.88 10.08 1.52 17.03 14.59 6.89

44.11 45.98 39.43 53.77 49.65 44.02

39

25

77

38

59.70

41.39

18.31

35.49

26.67

62.16

2.18

34.14

17.23

52.71

33

26

53

27

62.93

45.79

17.14

39.70

27.60

67.29

0.89

42.30

12.22

51.91

28

17

49

34

65.42

43.09

22.33

37.14

31.52

68.65

1.96

42.99

18.33

55.46

414 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Table 5

Table 5

Ž Continued . Gram

Staphylococci epidermidis q HBH2102 Staphylococci epidermidis q HBH2169 Staphylococci epidermidis q HBH2277 Staphylococci epidermidis q HBH23 Staphylococci epidermidis q HBH276 Staphylococci epidermidis q HBH45 Staphylococci epidermidis q NCTC 100835 Staphylococci epidermidis q NCTC 100892 Staphylococci epidermidis q NCTC 100894 Staphylococci epidermidis q SL58 Staphylococci hominus SL33 q Staphylococci saprophyticus q SAP1 Streptococci cricetus AHT q Streptococci cricetus E49 q Streptococci cricetus HS6 q Streptococci gordonii q NCTC 7869 Streptococci mitis 244 q Streptococci mitis 272 q Streptococci mitis 357 q Streptococci mitis 398 q

␪W

␪F

␪M

␪ Br

Equation of state approach

Geometric mean approach

LW ᎐AB approach using ␪ W, ␪F and ␪ Br

␥Total By ␪ W

␥LW by ␪ Br

␥Total y␥ LW

␥LW by ␪ B r

␥AB by ␪W , ␪B r

␥ Total

␥q

␥y

␥ AB

␥ Total

69

51

50

29

42.24

45.06

y2.82

39.01

8.07

47.08

0.21

12.75

3.25

42.26

28

34

46

32

65.42

43.90

21.52

37.91

31.05

68.96

0.32

53.82

8.21

46.12

42

40

64

36

58.02

42.26

15.76

36.33

24.45

60.78

0.40

40.68

8.02

44.34

36

32

48

22

61.34

47.75

13.59

41.23

25.23

66.46

0.34

42.79

7.54

48.77

29

23

51

34

64.94

43.09

21.85

37.14

31.04

68.18

1.44

44.83

16.06

53.19

28

31

48

36

65.42

42.26

23.16

36.33

32.01

68.34

0.75

51.17

12.34

48.67

27

54

50

37

65.89

41.83

24.06

35.91

32.73

68.64

0.58

78.57

13.44

49.35

19

10

73

50

69.26

35.82

33.44

29.96

40.18

70.14

4.24

47.93

28.50

58.45

27

43

46

13

65.89

49.71

16.18

43.27

228.50

71.77

0.24

64.70

7.73

51.00

22

45

49

35

68.09

42.68

25.41

36.74

34.35

71.09

0.04

72.06

3.14

39.88

37 20

28 19

57 49

32 12

60.80 68.88

43.90 49.90

16.9 18.98

37.91 43.44

26.45 31.12

64.36 74.55

1.17 0.49

38.53 51.93

13.41 10.06

51.32 53.49

28 33 33 93

32 31 51 46

51 45 51 51

35 35 36 32

65.42 62.93 62.93 27.24

42.68 42.68 42.26 43.9

22.74 20.25 20.67 y16.66

36.74 36.74 36.33 37.91

31.76 229.27 29.51 0.65

68.49 66.00 65.83 38.56

0.60 0.87 0.18 3.54

52.01 45.37 65.89 0.55

11.15 12.52 6.84 2.79

47.88 49.26 43.17 40.70

60 54 53 59

38 39 31 33

47 51 42 48

30 30 33 26

47.51 51.05 51.64 48.10

44.68 44.68 43.50 46.14

2.83 6.37 8.14 1.96

38.66 38.66 37.53 40.03

12.83 16.23 17.31 12.86

51.48 54.88 54.83 52.88

1.27 0.74 1.98 1.66

15.60 23.53 20.16 14.40

8.89 8.33 12.62 9.77

47.54 46.98 50.14 49.79

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 415

Bacterial isolate

Ž Continued .

Bacterial isolate

Streptococci mitis A TCC33399 Streptococci mitis A TCC9811 Streptococci mitis BA Streptococci mitis BMS Streptococci mitis T9 Streptococci mutans HG1003 Streptococci mutans HG979 Streptococci mutans HG982 Streptococci mutans HG983 Streptococci mutans HG985 Streptococci oralis 34 Streptococci oralis H1 Streptococci oralis J22 Streptococci oralis PK1317 Streptococci rattus ATCC19645 Streptococci rattus BHT Streptococci rattus BMsm Streptococci rattus FA1 Streptococci sali¨ arius HB Streptococci sali¨ arius HBC12

Gram

␪W

␪F

␪M

␪ Br

Equation of state approach

Geometric mean approach

LW ᎐AB approach using ␪ W, ␪F and ␪ Br

␥Total By ␪ W

␥LW by ␪ Br

␥LW by ␪ B r

␥AB by ␪W , ␪B r

␥ Total

␥q

␥y

␥Total y␥ LW

␥ AB

␥ Total

q

56

40

45

31

49.87

44.30

5.57

38.29

15.23

53.51

0.79

21.56

8.23

46.51

q

68

53

48

31

42.83

44.30

y1.47

38.29

8.77

47.06

0.1

15.11

2.41

40.69

q q q q

103 100 91 22

55 43 47 27

49 50 55 49

35 34 36 38

20.94 22.82 28.52 68.09

42.68 43.09 42.26 41.39

y21.74 y20.27 y13.74 26.70

36.74 37.14 36.33 35.49

0.01 0.06 1.14 35.15

36.74 37.19 37.47 70.64

2.68 5.98 3.45 1.08

2.74 4.51 0.12 54.20

5.42 10.38 1.28 15.29

42.15 47.51 37.6 50.78

q

26

48

54

38

66.35

41.39

24.96

35.49

33.46

68.94

0.07

71.25

4.15

39.63

q

33

34

51

35

62.93

42.68

20.25

36.74

29.27

66.00

0.58

47.70

10.48

47.21

q

27

28

48

35

65.89

42.68

23.21

36.74

32.23

68.96

0.94

50.08

13.68

50.42

q

23

30

61

43

67.67

39.14

28.53

33.28

36.22

69.49

1.20

55.26

16.26

49.53

q q q q

24 90 24 92

7 37 31 37

54 43 49 45

33 28 33 34

67.24 29.16 67.24 27.88

43.50 45.43 43.50 43.09

23.74 y16.27 23.74 y15.21

37.53 39.36 37.53 37.14

33.05 0.99 33.05 0.88

70.58 40.35 70.58 38.01

2.30 5.18 0.48 6.44

43.96 0.75 55.55 1.47

20.09 3.94 10.32 6.15

57.62 43.29 47.85 43.29

q

20

22

48

38

68.88

41.39

27.49

35.49

35.91

71.40

1.52

52.73

17.86

53.34

q q q q q

23 20 19 42 21

31 25 29 48 28

50 56 50 56 55

37 36 38 33 32

67.67 68.88 69.26 58.02 68.49

41.83 42.26 41.39 43.50 43.90

25.84 26.62 27.87 14.52 24.59

35.91 36.33 35.49 37.53 37.91

34.48 35.37 36.27 23.82 34.00

70.39 71.69 71.76 61.34 71.91

0.66 1.07 0.81 0.01 0.60

56.36 54.68 58.31 48.66 56.10

12.19 15.26 13.70 0.86 11.56

48.10 51.58 49.18 38.38 49.47

416 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Table 5

Table 5

Ž Continued . Gram

Streptococci sanguis q ATCC10556 Streptococci sanguis q CR311 Streptococci sanguis q CR311VAR1 Streptococci sanguis q CR311VAR2 Streptococci sanguis q CR311VAR3 Streptococci sanguis PK q 1889 Streptococci sanguis q PSH1b Streptococci sobrinus q HG1025 Streptococci sobrinus q HG970 Streptococci sobrinus q HG977 Streptococci thermoq philus B

␪W

␪F

␪M

␪ Br

Equation of state approach

Geometric mean approach

LW ᎐AB approach using ␪ W, ␪F and ␪ Br

␥Total By ␪ W

␥LW by ␪ Br

␥Total y␥ LW

␥LW by ␪ B r

␥AB by ␪W , ␪B r

␥ Total

␥q

␥y

␥ AB

␥ Total

22

17

54

27

68.09

45.79

22.30

39.70

32.54

72.23

1.19

48.94

15.21

54.90

74

67

55

47

42.83

37.27

5.56

31.41

8.06

39.46

0.03

17.44

1.30

32.70

75

59

58

46

38.60

37.74

0.86

31.88

7.43

39.31

0.36

10.66

3.90

35.78

75

63

55

51

38.6

35.36

3.24

29.47

8.22

37.69

0.20

13.08

3.23

32.70

31

39

55

46

63.95

37.74

26.21

31.88

33.36

65.23

0.68

54.18

12.11

43.98

28

᎐

50

33

65.42

43.50

21.92

37.53

31.28

68.81

᎐

60

60

53

36

47.51

42.26

5.25

36.33

13.73

50.06

0.19

32.3

4.93

41.25

29

33

47

34

64.94

43.09

21.85

37.14

31.04

68.18

0.50

51.73

10.09

47.23

26

45

54

38

66.35

41.39

24.96

35.49

33.46

68.94

0.01

67.40

0.25

35.73

27

43

53

33

65.89

43.50

22.39

37.53

31.74

69.27

0.01

64.06

0.34

37.87

23

34

50

35

67.67

42.68

24.99

36.74

33.96

70.69

0.33

59.05

8.79

45.52

W

F

M

Equation of state approach ␪

␪

␪

Br

Total

LW

␥ by ␪ W

␥ by ␪ Br

Bacterial isolate

␪

A¨ erage S.D. Minimum

45.81 38.41 24.92 14.43 15.85 7.00

52.63 9.33 23.00

37.06 11.17 12.00

55.1 14.29 18.48

41.54 4.99 16.18

Maximum

107.00 91.00

97.00

91.00

70.29

49.9

Total

␥ y␥ LW

Geometric mean approach LW

AB

Total

␥ by ␪ B r

␥ by ␪W , ␪B r

␥

13.55 15.38 I24.20

35.58 4.89 10.72

23.28 12.83 0.00017

58.85 12.30 27.02

44.17

43.44

48.87

74.55

᎐

᎐

᎐

LW᎐AB approach using ␪ W and ␪ Br ␥q

␥y

1.93 37.89 2.37 23.31 1.8 = 0.03 10 5 14.48 104.3

␥ AB

␥ Total

11.8 9.38 0.07

47.37 8.86 29.23

59.88

99.90

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 417

Bacterial isolate

418 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 Table 6 Surface energy parameters ŽmJrm2 . of the standard liquids used for contact angle measurements Žvan der Mei et al. w22x; Chapter XIII van Oss w80x; Bellon-Fountaine et al. w118x. Liquid

␥Total

Water Formamide ␣-Bromonapthalene Methyleneiodide

72.8 58 44.4 50.8

␥dr␥LW 21.8 39 44.4 50.8

␥h r␥AB

␥q

␥y

51 19 f0 f0

25.5 2.3 - 0.1 - 0.1

25.5 39.6 - 0.1 - 0.1

The total surface energy by Equation of state approach and Geometric mean approach Žalong with the acid᎐base component. can be evaluated by water or formamide contact angle and least square method. Similarly either methyleneiodide or ␣-bromonapthalene can be used to evaluate the Lifshitz᎐van der Waals component of surface energy. However, due to lack of space, the surface energies evaluated by using water and ␣-bromonapthalene is only presented in Table 5. Data evaluated using formamide and methyleneiodide is only presented in the figures. For the evaluation of surface energy Excel 97 spreadsheet is used and for the over-determined system of equations, Matlab 5.3 is used. 4.1. Contact angle of the bacterial cells with apolar liquids Majority of the contact angles with the apolar liquids for the bacterial isolates fall in a very narrow region. The contact angle values with methyleneiodide have a mean of 53⬚ and standard deviation of 9.3; 95% of the contact angles fall between 36 and 72⬚. The contact angle values with ␣-bromonapthalene have a mean of 37⬚ and standard deviations of 11.17; 95% values fall between 20 and 60⬚. This is evident from Fig. 16 and the cluster formation in Fig. 17a. This means that majority of the bacterial cells have very similar interaction with the apolar liquids. 4.2. Lifshitz ¨ an der Waals (dispersion) component of surface energy using Fowkes and equation of state approach When the contact angles of bacterial surfaces with apolar liquids are used in the Fowkes equation then we arrive at the dispersion component of the surface energy of the bacterial lawns. These apolar liquids, namely methyleneiodide and ␣bromonapthalene interact with the bacterial cell surfaces through apolar interactions only. Because of the fact that the contact angle values with apolar liquids group together the dispersion component of the surface energy also forms cluster around an average value ŽFig. 17a.. There is no straight-line correlation between the dispersion energies evaluated from methyleneiodide and ␣-bromonapthalene. Ninety-five percent of the bacterial cells have ␥ LW values in the range of 20᎐40 mJrm2 with a mean of 32.7 and standard deviations 5.1 when methyleneiodide contact angle is used. Similarly, when ␣-bromonapthalene contact angle is used

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 419

Fig. 17. For the 147 microbial isolates Ža. correlation of methyleneiodide and ␣-bromonapthalene contact angle and the correlation of the dispersion component of surface energy evaluated by using methyleneiodide and ␣-bromonapthalene contact angles. Correlation shown when either Fowkes or Equation of state approach is used. Žb. Correlation of dispersion component of surface energy evaluated using Fowkes and Equation of state approach, correlations shown when either methyleneiodide or ␣-bromonapthalene contact angles are used.

420 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 18. Distribution of Lifshitz᎐van der Waals component of surface energy evaluated by using apolar liquid contact angles on 147 different bacterial isolates, with Equation of state, pair-wise and least squared Geometric mean approach.

then, 95% of bacterial cell ␥ LW lies between 25 and 45 mJrm2 with a mean of 35.6 and standard deviation of 4.88 mJrm2 ŽFig. 18.. When the contact angles of apolar liquids are used to evaluate the bacterial surface energy by Equation of state approach then it follows the same behaviour as the dispersion energy, ␥sLW , evaluated by the Fowkes approach ŽFig. 17a.. The evaluated surface energies form clusters with slightly higher average value than the dispersion energies evaluated by Fowkes approach. Ninety-five percent of the bacterial cells have ␥ values in the range of 20᎐40 mJrm2 with a mean of 34.5 and S.D.s 4.48 when methyleneiodide contact angle is used. Similarly, when ␣bromonapthalene contact angle is used then 95% of bacterial cell ␥ lies between 30 and 50 mJrm2 with a mean of 41.5 and S.D.s 4.98 mJrm2 ŽFig. 18.. The average values are in accordance with the observation made by van Oss w80x that the Lifshitz van der Waal’s component of surface energy of biopolymers is close to 40 mJrm2 " 10%. Since the bacterial cells surfaces are composed of biopolymers like peptidoglycan, phospholipids Žbilayer., lipopolysaccharides, lipoproteins, teichoic acid, other proteins, etc., bacterial cells also have Lifshitz van der Waals component of surface energy in this range. For the approaches to have internal consistency, it is expected that the apolar component of surface energy of bacterial cell surface evaluated by using two different liquids must be the same, i.e. instead of forming a cluster in Fig. 17a the points are expected to lie on the 45⬚ line. Which does not happen, because the

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 421

contact angle for different bacterial cells with the two liquids just forms a cluster and does not follow the expected trend by either equation of state or Fowkes approach. The lack of internal consistency is also evident from Fig. 18. The expected correlation between the contact angle with methylene iodide and ␣-bromonapthalene by Fowkes approach wEq. Ž14. without the spreading pressure termx is evaluated using the following relations and presented in Table 7. First, ␪ Me thyleneiodide is assumed to be 35, 40, 45...75⬚, then ␥sLW is evaluated using 123 and, then ␪␣ -Bromonapthalene is calculated using 124.

'␥

LW s

s

'␥

LW s

s

'50.8 2

'44.4 2

Ž cos␪ Methyleneiodide q 1 .

Ž 123.

Ž cos␪␣ -Bromonaphthalene q 1 .

Ž 124.

Similar values are presented in Table 5 for Equation of state approach, which is evaluated using the conversion tables w130x. The difference in ␪␣ -Bromonapthalene values for the two approaches is in the range of 5.46⬚᎐3.32⬚. Fig. 17b correlates the surface energies evaluated using the two approaches for different bacterial strains. The curves are plotted for methyleneiodide and ␣bromonapthalene contact angles. Equation of state approach consistently gives higher surface energy than Fowkes approach by 5.24 mJrm2 for the whole range of ␣-bromonapthalene contact angles. The energies evaluated are well correlated by a straight line fit with a slope of 1.02 Ž45.56⬚. and an intercept of 5.42 mJrm2 the Equation of state axis. Surface energies evaluated by the two approaches using methyleneiodide contact angle also correlate well with each other by a straight line fit. The intercept of the straight line fit is 5.76 mJrm2 , i.e. similar to the case of ␣-bromonapthalene but now the dispersion energy evaluated from Fowkes approach increase at a slower rate than the surface energy evaluated by the Equation of state approach Žslope s 0.87, i.e. 41⬚.. The reason for this difference from the case of using the ␣-bromonapthalene contact angle becomes clearer from Fig. 19. Plots of surface energy vs. contact angle for two different approaches and the two contact angles are shown in Fig. 19. The plots fit very well with polynomials of third degree w.r.t ␪. We can see that when the Equation of state approach is used then the surface energies evaluated from contact angles with two different apolar liquids follow exactly the same curve, though the individual points do not overlap. However, this is not true for ␥ LW , as evaluated from Fowkes approach, the plot with respect to contact angles of two different apolar liquids. The ␥ LW vs. ␣-bromonapthalene contact angle plot is almost parallel to the ␥ vs. contact angle plots for the Equation of state approach, with an offset of 5.24 mJrm2 Žsame as Fig. 17b.. The ␥ LW vs. methyleneiodide contact angle plot has a totally different slope. At low contact angles Ž- 40⬚. when Fowkes approach is used the meth-

Fowkes approach

␪F Br omonapthalene ᎐

Equation of state approach LW

␪Me thyleneiodide

␥s

35 40 45 50 55 60 65 70 75

41.95 39.53 36.94 34.21 31.39 28.52 25.65 22.83 20.09

LW

␪Bromonapthalene

␪Methyleneiodide

␥s

19.46 27.61 34.59 41.02 47.11 52.99 58.71 64.32 69.85

35 40 45 50 55 60 65 70 75

42.68 40.51 38.21 35.82 33.38 30.9 28.42 25.95 23.53

␪Bromonapthalene

␪E Bromonapthalene

14 23.5 31 37 43 49 55 61 66

5.46 4.11 3.59 4.02 4.11 3.99 3.71 3.32 3.85

422 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Table 7 Expected correlation between ␪ Me thyleneiodide and ␪ Bromonapthalene according to Fowkes and Equation of state approach

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 423

Fig. 19. Correlation of the Lifshitz᎐van der Waals component of surface energy as evaluated by Fowkes and Equation of state approaches vs. ␪ Ža. and cos␪ where the contact angles used are with methyleneiodide and ␣-bromonapthalene Žb..

424 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

yleneiodide contact angles give similar ␥ LW as the Equation of state surface energies and at high contact angles the evaluated ␥ LW comes close to the ␥ LW vs. ␣-bromonapthalene contact angle plot. Plots of ␥ Bacterial cell surface vs. cos␪ for the two approaches and the two different apolar liquid contact angles are shown in Fig. 19b. As expected the ␥ LW Fowkes vs. cos␪ plots follow the quadratic relation of type Eq. Ž14. without the spreading pressure term, which can also be written as ␥sLW s

␥l 4

q

␥l 2

cos␪ q

␥l 4

cos 2 ␪

Ž 125.

Since ␥ 1LW s ␥ 1 for apolar liquids, the actual second degree polynomial fits of vs. cos␪ for two different liquids are:

␥ LW Fowkes

2 ␥FLW o s 12.7 q 25.4cos␪ M q 12.7cos ␪ M

for Methyleneiodide

Ž 125a .

for ␣᎐Bromonaphthalene

Ž 125b .

2 ␥FLW o s 11.1 q 22.2cos␪ B r q 11.1cos ␪ B r

Surprisingly the plots for ␥ Equation of state vs. cos␪ also follow the quadratic relation of the type 125, the quadratic fit for the plots are: 2 ␥ELW q. st s 16.6 q 23.8cos␪ M q 9.53cos ␪ M

2 ␥ELW q. st s 16.7 q 21.95cos␪ B r q 11.9cos ␪ B r

for Methyleneiodide for ␣᎐Bromonaphthalene

Ž 125c . Ž 125d.

Surprisingly, the coefficients of cos␪ and cos 2 ␪ in both Eqs. Ž125c. and Ž125d. . are similar to Eq. Ž125b. Ž ␥ LW Fowkes vs. cos␪ ␣ -Bromonapthalene in Fig. 19b . However, 2 LW these equations give the value of ␥, higher by 5.5 mJrm than ␥ evaluated by Eq. Ž125b.. w Ž .x The ␥ LW Fowkes vs. cos␪ Methyleneiodide plot follows the quadratic fit Eq. 125a , as it 2 should, but the coefficients of cos␪ and cos ␪ are very different from Eqs. Ž125b., Ž125c. and Ž125d., hence the slope of the curve is very different from the others. The plots of Fig. 19b can also be interpreted as reverse Zisman plots. Instead of plotting ␥ 1 vs. cos␪ for different liquids on a solid surface, here ␥sLW vs. cos␪ is plotted for one liquid on many solids. The quadratic fits cross cos␪ s 1, i.e. ␪ s 0 line at a critical ␥sLW . The liquid will give zero contact angle for the bacterial cell surface Žor solid surface . which have the critical ␥sLW . These ␥sLW values are not measured but calculated ones using the two approaches therefore, the two approaches are compared based on this critical ␥sLW . According to the Fowkes approach the apolar liquids will spread on that bacterial cell surface which have the same ␥sLW as the liquid surface tension Ž ␥ 1 s ␥ 1LW .. Therefore, methyleneiodide will spread on the bacterial cell surface

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 425

with ␥sLW s; 51 mJrm2 Ž ␥ 1 s 50.8 mJrm2 . and ␣-bromonapthalene will spread on the bacterial cell surface with ␥sLW s; 44 mJrm2 Ž ␥ 1 s 44.4 mJrm2 .. When the Equation of state approach is used then irrespective of the liquid used the critical ␥sLW is the same, i.e. ; 50 mJrm2 . If the analysis with two liquids can be generalised then we can say that all the apolar liquids will spread on the bacterial cell surface with ␥sLW s; 50 mJrm2 . Since the equation of state approach does not believe on the division of surface energy into different components. Although the ␥ vs. ␪ ŽFig. 19a. and cos␪ vs. ␥ ŽFig. 19b. curves for equation of state approach overlap each other, the surface energy evaluated by the two liquids are different. This is evident from Fig. 18 where the distribution of ␥ is different for the two liquids. There are few points, which become very clear by the above analysis of the bacterial cell surface energy evaluated by the two approaches and when contact angle with apolar liquids are used. 1. Microbial cell surfaces Ž147. have very similar contact angles to each other with an apolar liquid. This is same for the two-tested apolar liquids. This indicates that the Lifshitz van der Waals component of their surface energy Ž ␥sLW . is not very different from each other and lies in the vicinity of 35᎐40 mJrm2 ŽFigs. 16 and 18.. 2. Although, the proponents of equation of state approach reject the division of solid surface energy into components, but when apolar liquid contact angles are used then the equation of state approach evaluates the surface energy very close to the Lifshitz van der Waals component of surface energy Ž ␥sLW . evaluated by the Fowke’s approach. The difference is only 0᎐6 mJrm2 depending on the liquids used ŽFig. 17b.. 3. When ␣-bromonapthalene is used then the surface energy obtained from equation of state approach is consistently higher than the ␥sLW by Fowkes approach by 5᎐6 mJrm2 ŽFig. 17b.. 4. The proponents of equation of state approach have totally refuted the use of the geometric mean for combining the components of solid᎐liquid surface energies. However, the ␥ vs. cos␪ plots follow a quadratic relation of the type Eq. Ž125. which originates from geometric mean and is very different from the proposed Eq. Ž60. or Eq. Ž71. or Eq. Ž76. ŽFig. 19b.. 4.3. Contact angle with polar liquids Fig. 16 shows the distribution of water and formamide contact angles on the bacterial cell surface. As can be seen the contact angle values are spread over a large range from close to zero to approximately 120⬚. Since water and formamide interact with the bacterial cells both by polar and apolar interactions we can say that the polar characteristics of different bacterial cells is very different, as we have

426 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

already seen that the apolar characteristics of bacterial cells are not very different from each other. 4.4. Total and acid᎐base (polar) component of surface energy e¨ aluated by Geometric mean and Equation of state approach Geometric mean approach needs contact angle values with one polar and one apolar liquid in order to give the polar and apolar component of the solid surface energy. The data available are with two polar liquids Žwater and formamide. and two apolar liquids. As seen before, the apolar-Lifshitz van der Waals component of the surface energy is found using Fowkes approach and we get different values when the contact angle values with two different apolar liquids are used. The polar or acid᎐base component of the surface energy can be evaluated using either water or formamide contact angle. For further analysis the apolar component ␥sLW of the bacterial cell surface evaluated by using ␣-bromonapthalene contact angle is used, which have an average value of 35.6 mJrm2 . Apart from using the liquid contact angles pair-wise, the acid᎐base and Lifshitz᎐van der Waals component of solid surface energy are evaluated by solving the over-determined system of four equations by Least square method. Fig. 20a shows the total and acid᎐base component of the surface energy for the 147 bacterial isolates decreases as a function of water contact angle. The acid᎐base component of the surface energy for the bacterial cell surface varies between 0 and 50 mJrm2 Žcan also be seen from Fig. 34b.. ␥ AB vs. ␪ plot follows a straight relation until the contact angle of 90⬚ after which ␥ AB approaches zero value and becomes plateau, i.e. the straight-line behaviour does not continue into negative values of ␥ AB . The ␥ LW values are distributed between 30 and 40 mJrm2 and are independent of the water contact angle. Total surface energy,␥Total , which is the sum of acid᎐base component and the Lifshitz᎐van der Waals component, follows the same behaviour as ␥ AB vs. ␪ and range between 75 and 35 mJrm2 . For the bacterial cells having water contact angle above 90⬚ the acid᎐base component is zero and the total surface energy is entirely due to apolar contribution. Fig. 21a shows that the total and acid᎐base component of the surface energies for the 147 bacterial isolates decrease as a function of formamide contact angle. The acid᎐base component of surface energy for bacterial cells range from 0 to 30 mJrm2 . For the bacterial cells having formamide contact angle above 60⬚, the acid᎐base component of surface energy approaches zero. The Lifshitz᎐van der Waals component of surface energy is independent of the formamide contact angle. The total surface energy also follows the same trend as the acid᎐base component and ranges between 35 and 60 mJrm2 and after a formamide contact angle of 60⬚, it is entirely composed of the Lifshitz᎐van der Waals component of surface energy. As expected the ␥Total and ␥ AB evaluated by Geometric mean approach follows a quadratic relation with cos␪ like Eq. Ž28.. The acidrbase Žpolar. component of surface energy for the bacterial cell surface follow a relation of the type 126

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 427

␥sAB s

␥ l2

cos 2 ␪ q AB

4␥ l q

ž

␥ l2 4␥ lAB

q

ž

␥ l2 2 ␥ lAB

␥sLW ␥ lLW ␥ lAB

␥ ␥sLW ␥ lLW

'

y y

␥ lAB ␥ l ␥sLW ␥ lLW

'

␥ lAB

/ /

cos␪ Ž 126.

where ␥sLW is previously known from the Fowkes approach. For further calculations the ␥sLW value is taken to be an average of 35.6 mJrm2 for ␣-bromonapthalene contact angle. Using Eq. Ž126., average ␥sLW and the liquid parameters the coefficients for cos 2 ␪, cos␪ and the constant can be evaluated. For water the coefficients are 25.97, 12.19 and 1.42; for formamide the coefficients are 44.26, y25.21 and 3.58, respectively. The total surface energy is evaluated using two contact angles one with polar liquid Žwater or formamide. and one apolar liquid therefore the relation for total surface energy is the same as Eq. Ž126. but higher by the Lifshitz van der Waals part of the solid surface energy wEq. Ž127.x ␥sTotal s

␥ l2

cos 2 ␪ q AB

4␥ l q

ž

␥ l2 4␥ lAB

q

ž

␥ l2 2 ␥ lAB

␥sLW ␥ lLW ␥ lAB

y y

␥l ␥sLW ␥ lLW

'

␥ lAB ␥ l ␥sLW ␥ lLW

'

␥lAB

/ /

cos␪ Ž 127. q ␥sLW

Therefore, the expected coefficients for cos 2 ␪, cos␪ remain the same as for ␥sAB but the constant will be higher by the ␥sLW value Žon an average by 35.6 mJrm2 .. The real quadratic fit for ␥sAB vs. cos␪W is as follows AB ␥GM s 1.4 q 13cos␪W q 25.4cos 2 ␪W

Ž 126a .

with R 2 of 0.937 and S.D. of 3.25. The R 2 statistics indicates that the model Žsecond degree polynomial fit. explains 93.7% of the variability in ␥sAB and S.D. gives the standard deviation of the experimental points from the model. The quadratic fit for ␥sAB vs. cos␪ F is as follows AB ␥GM s 5.9 y 29cos␪ F q 46cos 2 ␪ F

Ž 126b .

with R 2 of 0.514 and S.D. of 5.17. The quadratic fit for ␥sTotal vs. cos␪W are as follows Total ␥GM s 37.5 q 12.8cos␪W q 24.8cos 2 ␪W

with R 2 of 0.973 and S.D. of 2.01.

Ž 127a .

428 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 20. Polar, apolar and total surface energy of bacterial cell surfaces as a function of water contact angle Ža. and cosines water contact angle Žb., while using the Geometric mean approach.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 429

Fig. 21. The polar, apolar and total surface energy of bacterial cell surfaces as a function of formarnide contact angle Ža. and cosines formamide contact angle Žb., while the using Geometric mean approach.

430 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

The quadratic fit for ␥sTotal vs. cos␪ F are as follows Total ␥GM s 37.9 y 22.5cos␪ F q 43.8cos 2 ␪ F

Ž 127b .

with R 2 of 0.914 and S.D. of 1.82. The coefficients of the real fit are close to the expected coefficients, difference is due to the fact that the expected coefficients are calculated by using an average ␥sLW value of 35.6 mJrm2 . Fig. 24a shows the plots of ␥sTotal vs. ␪ evaluated by the geometric mean approach using water and formamide contact angles. The figure also contains the surface energy evaluated by Equation of state approach and by using water and formamide contact angle. A clear conclusion which can be drawn from the plots is that when contact angle of polar liquids are used for evaluation of surface energy of bacterial cell surface then the values are similar to the ␥sTotal evaluated by that particular polar liquid contact angle by the Geometric mean approach. This conclusion is valid for water contact angles below 70⬚ and for formamide contact angles of 55⬚. Figs. 22a and Fig. 23a show the bacterial surface energy evaluated by using equation of state approach and contact angle values with polar liquids. They are designated as ␥Total because the surface energies are similar to the ␥sTotal evaluated by the geometric mean approach. Since Lifshitz᎐van der Waals component of the surface energy is already known from apolar liquid contact angles, acid᎐base component of surface energy can be evaluated from the difference ␥Total y ␥ LW , which is also plotted in the figures. Fig. 22a shows that the total surface energy evaluated decreases with the water contact angle and ranges between 20 and 70 mJrm2 when equation of state approach was followed. The marked difference from geometric mean approach is that the total surface energy evaluated, for ␪W ) 66⬚, is less than the Lifshitz᎐van der Waals component of the surface energy and hence gives rise to negative acid᎐base component. The acid᎐base component of surface energy ranges in between ᎐25 and 40 mJrm2 . Fig. 23a shows the total surface energy evaluated by Equation of state approach decreases with the formamide contact angle and ranges in between 20 and 57 mJrm2 . Above the formamide contact angle of 50⬚, the total surface energy evaluated by formamide contact angle is less than the Lifshitz᎐van der Waals component of surface energy evaluated using ␣-bromonapthalene contact angle. Hence, the acid᎐base component is negative after 50⬚, ranging between y20 and 25 mJrm2 . Surprisingly the total surface energy evaluated by Equation of state approach also follows a quadratic relation when ␥ is plotted with cos␪ ŽFigs. 22b and Fig. 23b.. The second degree polynomial fit for the ␥sTotal vs. cos␪W is as follows 2 ␥Total Eq .st s 28.7 q 33.3cos␪ W q 9.4cos ␪ W

with R 2 of 0.998 and S.D. of 0.597.

Ž 127c .

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 431

Fig. 22. The polar, apolar and total surface energy of bacterial cell surfaces as a function of water contact angle Ža. and cosines water contact angle Žb., using the Equation of state approach.

432 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 23. The polar, apolar and total surface energy of bacterial cell surfaces as a function of formarnide contact angle Ža. and cosines formamide Žb. contact angle, using the Equation of state approach.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 433

The second degree polynomial fit for the ␥sTotal vs. cos␪ F is as follows 2 ␥Total Eq .st s 20.6 q 22.9cos␪ F q 14cos ␪ F

Ž 127d.

with R 2 of 0.999 and S.D. of 0.13. The ␥sAB , which is equal to ␥Total y ␥ LW evaluated from equation of state approach using polar and apolar liquid contact angles, also follows a quadratic fit with respect to cos␪. The second degree polynomial fit for the ␥sAB vs. cos␪W is as follows AB 2 ␥ Eq .st s y13.3 q 33.5cos␪ W q 10.2cos ␪ W

Ž 126c .

with R 2 of 0.895 and S.D. of 5.03. The second degree polynomial fit for the ␥sAB vs. cos␪ F is as follows AB 2 ␥ Eq .st s y17.8 q 18.7cos␪ F q 14cos ␪ F

Ž 126d.

with R 2 of 0.621 and S.D. of 4.85. The second-degree polynomial fits for the ␥sTotal vs. cos␪ evaluated by equation of state approach are very good which is clear from the R 2 values above 0.99. Whereas when the Geometric mean approach is used then the second-degree polynomial fits have R 2 values in the range of 0.91᎐0.97. This clearly shows that the quadratic fit explains 99% of the variability of ␥sTotal with respect to cos␪ even when the equation of state approach is used. As mentioned earlier Fig. 24a shows that the total surface energy evaluated by the Geometric mean approach is very similar to the surface energy evaluated by the equation of state approach using polar liquid contact angles. This is evident from Fig. 38a. When the formamide contact angle is used, then, the total surface energy evaluated by the two approaches is the same as the ␥sTotal vs. ␪ and ␥sTotal vs. cos␪ overlap shown in Fig. 24a,b, respectively. This similarity is also evident from Fig. 25a where the correlation between the total surface energy evaluated by the two approaches follow the 45⬚ line. This similarity continues until the formamide contact angle of 55⬚, where the ␥sTotal has reached approximately 40 mJrm2 . Above 55⬚ formamide contact angle the ␥sTotal evaluated by the two approaches start to become different. For the equation of state approach ␥sTotal continues to drop but for the Geometric mean approach it almost becomes constant at 35.6 mJrm2 because after this point total surface energy is composed entirely by the Lifshitz᎐van der Waals component and the acid᎐base component approaches zero. When the water contact angle is used then the Geometric mean approach gives the total surface energy similar to the one given by the Equation of state approach but approximately 2.5 mJrm2 higher. For water contact angle higher than 70⬚ the difference between the surface energies evaluated by the two approaches increases. The surface energy evaluated by the Geometric mean approach is 45 mJrm2 at ␪W f 70⬚ and after that it gradually decreases to a value of 35.6 mJrm2 and stays constant. At higher water contact angle the surface energy is entirely composed by Lifshitz᎐van der Waals component and acid᎐base component contribution ap-

434 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 24. Total surface energy of bacterial cell surfaces evaluated by using the Geometric mean and Equation of state approaches vs. water and formarnide contact angles Ža., cosines of water and formamide contact angles Žb..

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 435

Fig. 25. Comparison of the Ža. total surface energy and Žb. polar component of surface energy when evaluated using either the Geometric mean or Equation of state approaches. The comparison is done when water or formamide contact angles are used.

436 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

proaching zero. However, the total surface energy evaluated by the Equation of state approach decreases continuously after ␪W f 70⬚ hence, giving rise to a negative acid᎐base component. This is also evident from Fig. 25a where the plot for surface energy evaluated by the two approaches is parallel to the 45⬚ line and follow a relation of Eq. Ž128., until the surface energy value of 45 mJrm2 and then it curves and becomes horizontal. Total Total ␥GM ;W s 2.5 q ␥ Eq.st;W

Ž 128.

In the whole range of surface energy the total surface energies evaluated by the two approaches follow a third degree polynomial relation wEq. Ž129.x. Total Total Total ␥GM ;W s 49.7 y 1.41␥ Eq.st;W q 0.04 Ž ␥ Eq.st;W .

y 2.25 = 10y4 Ž ␥Total Eq .st;W .

2

3

Ž 129.

Fig. 25b correlates the acid᎐base component of surface energy evaluated by the two approaches. For formamide contact angle, the acid᎐base component evaluated by the two approaches follow Eq. Ž130.. This means that the acid᎐base component evaluated by the two approaches is similar but the one evaluated by the Equation of state approach is approximately 6 mJrm2 lower than the one evaluated by the Geometric mean approach. Because of this, the Equation of state approach gives rise to a negative acid᎐base component of surface energy. Total Total ␥GM ;Fo s 6 q ␥ Eq.st;Fo

Ž 130.

The acid᎐base component of surface energy from water contact angle and evaluated by the two approaches is also correlated in Fig. 25. In the whole range, the points follow a third degree polynomial wEq. Ž131.x. However, a straight line relation of Eq. Ž132. can correlate most of the points in the range of 10᎐37 mJrm2 which means that the acid᎐base component evaluated by the Equation of state is approximately 9 mJrm2 lower to the one evaluated by the Geometric mean approach. 2

AB AB AB y4 AB ␥GM Ž ␥ Eq.st;W . ;W s 9.57 y 0.84␥ Eq.st;W q 0.012 Ž ␥ Eq.st;W . y 2.99 = 10

AB AB ␥GM ;W s 9 q ␥ Eq.st;W

3

Ž 131. Ž 132.

Fig. 24b shows the plot of ␥sTotal vs. cos␪ for water and formamide contact angles. If we consider these plots as reverse Zismann plots and since the ␥sTotal is evaluated by using Equation of state and Geometric mean approach, we can compare the two approaches based on this. When water is used for measuring contact angle on the bacterial surfaces, then according to Equation of state approach water will spread or give zero contact angle for a bacterial cell surface having the surface energy of 71.4 mJrm2 Žand higher., which is very close to the

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 437

water surface tension. According to the geometric mean approach water will spread on the bacterial cell surface with surface energy of 75 mJrm2 Žand higher. which, is close to the water surface tension but higher by 3 mJrm2 . The difference can be due to the fact that Equation of state approach needs only one Žwater. contact angle to evaluate the surface energy but Geometric mean approach needs two contact angle values. Geometric mean approach evaluates the total surface energy of the bacterial cell surface by adding the Lifshitz᎐van der Waals component Ževaluated from apolar liquid contact angle. and acid᎐base component Ževaluated from polar liquid contact angle. hence, the errors in the contact angle measurement propagating to the final surface energy value is much higher. When the formamide contact angle is used, the ␥sTotal vs. cos␪ plots by the two approaches overlap each other. According to Equation of state approach formamide will spread on the bacterial cell surface having a surface energy of 57.5 mJrm2 or more, whereas according to Geometric mean approach formamide will spread on bacterial cells with surface energy of 59.2 mJrm2 and more. Both the values are quite close to each other and are very close to the formamide surface tension of 58 mJrm2 . Fig. 26 shows the correlation between the surface energy evaluated by different liquids but by using the same approach. The total surface energy evaluated by water contact angle does not correlate to the one evaluated by formamide contact angle, this is true for both the approaches. If the approaches were internally consistent then the surface energies evaluated by them should be the same irrespective to the liquid contact angle used. The acid᎐base component of the surface energy also follows the same trend. The acid᎐base component evaluated by Geometric mean approach confines itself in the first quadrant with all positive values but acid᎐base component evaluated by Equation of state approach give negative values for ␪W ) 70⬚ or ␪ F ) 50⬚. This elucidates another shortcoming of the Equation of state approach. 4.5. Polar and dispersion component of surface energy using Least square method to fit the contact angle data with four liquids to Geometric mean approach The matrix in Eq. Ž33. is solved for all the 147 microbial surfaces. This method gives both the polar and dispersion component of the surface energy. Fig. 18 shows the distribution of the dispersion component of surface energy for the 147 microbial isolates. It can be seen clearly the distribution of is similar to the distribution of dispersion energy evaluated by the Fowkes approach with the use of methyleneiodide liquid contact angle. This become more obvious in Fig. 27a where the dispersion energy evaluated by the least square method correlates well with the dispersion energy evaluated by Fowkes approach using methyleneiodide and the data points fall on the 45⬚ line. The dispersion energy evaluated by Fowkes approach using ␣-bromonapthalene contact angles are higher than the ones obtained by the least square method Žalso seen from Fig. 18. and hence the points do not fall on the 45⬚ line, but correlate well.

438 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 26. Correlation of total and polar surface energy evaluated using either water or formarnide contact angles, the correlation is shown for both the Geometric mean and Equation of state approach.

Fig. 27b correlates the acid᎐base component obtained by the least square method to the ones obtained by solving pair-wise equation from waterr␣bromonapthalene and formamider␣-bromonapthalene. The least square method evaluates the acid᎐base component similar to the ones obtained from pair-wise solution using water and ␣-bromonapthalene as the points lie on the 45⬚ line. There is no correlation between the acid᎐base component obtained from least square method and pair-wise solution using formamide and ␣-bromonapthalene. This is also evident from the distribution diagram in Fig. 34b where the distribution of acid᎐base component of surface energy evaluated by least square method is very similar to the one obtained by pair-wise solution using water and ␣-bromonapthalene, but very different from the one obtained by pair-wise solution using formamide and ␣-bromonapthalene. Fig. 27c correlates the total surface energy obtained by the least square method to the ones obtained by solving pair-wise equation from waterr␣-bromonapthalene and formamider␣-bromonapthalene. Total surface energy evaluated by least square method is similar to the ones obtained by pair-wise solution using water and ␣-bromonapthalene, although the data points follow the straight line fit for Eq. Ž133., which crosses the 45⬚ line at 30 mJrm2 but we can see that all the data points lie very close to the 45⬚ line. There is no correlation between the total surface energy obtained from the least square method and pair-wise solution using formamide and ␣-bromonapthalene.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 439

Fig. 27. Correlation of Ža. dispersion component; Žb. polar component; and Žc. total surface energy when either the least square method is used or the pair-wise solution is used for the Geometric mean approach.

440 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 Total Total ␥GM ;W ,␣yBr s 1.185␥GM ;LeastSq . y 5.63

Ž 133.

The above analysis brings out few important points, which needs highlighted. 1. Equation of state approach and Geometric mean approaches are not internally consistent. They evaluate different surface energies when contact angle with different liquids are used ŽFig. 26.. 2. When water contact angle is used, the Equation of state approach gives similar surface energy values as Geometric mean approach with a difference of 2᎐3 mJrm2 in the contact angle range of 16᎐70⬚ and only at higher contact angles the difference becomes larger ŽFig. 24aFig. 25a.. 3. When formamide contact angle is used then both the approaches evaluate the same surface energy values in the contact angle range of 7᎐55⬚ and only above this the difference in surface energy values evaluated by the two approaches increase ŽFig. 24aFig. 25a.. 4. When water contact angle is used then the 147 bacterial cell isolates have surface energy values of approximately 75᎐35 mJrm2 when Geometric mean approach is followed. However, when the Equation of state approach is used, then the value ranges from approximately 70 to 20 mJrm2 . 5. When formamide contact angle is used then the bacterial surface energy ranges from approximately 60 to 35 mJrm2 in Geometric mean approach. However, when Equation of state approach is used, it ranges between approximately 60 and 20 mJrm2 . 6. The proponents of equation of state approach have totally refuted the use of the geometric mean for combining the components of solid᎐liquid surface energies. However, the ␥ vs. cos␪ plots follow a quadratic relation of the type Eq. Ž127. which originates from geometric mean and is very different from the proposed Eq. Ž60. or Eq. Ž71. or Eq. Ž76. ŽFig. 19b, Fig. 24b.. Although the coefficients for cos 2 ␪, cos␪ do not correspond to the expected ones ŽFrom Geometric mean approach. but the second degree fit of ␥ vs. cos␪ for Equation of state approach is better than that for Geometric mean approach itself. 7. When the least square method is used to fit the geometric mean to the contact angles of water, formamide, methyleneiodide and ␣-bromonapthalene, then the total and acid᎐base component of the 147 microbial isolates is similar to the ones obtained from waterr␣-bromonapthalene pair and the Lifshitz.van der Waals component is similar to the one obtained by using methyleneiodide ŽFig. 27a᎐c.. 4.6. Total surface energy and its components using Lifshitz᎐¨ an der Waals r acid᎐base (LW᎐AB) approach The Lifshitz᎐van der Waalsracid᎐base ŽLW᎐AB. approach uses Eq. Ž112. to evaluate the surface energy of the bacterial cell surface. This equation has three

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 441

Fig. 28. Correlation of water and formarnide contact angles. Ž⽧. Experimental data Žq. theoretical fit of Equation of state approach the relation is irrespective to the value of ␥ LW , ␪ F s 2.27 = 10y4 Ž ␪W . 3 y0.054 Ž ␪W . 2 q 5.35␪W y 138.11 ŽU .. Theoretical fit of Geometric mean approach using ␥ LW s 35.5 mJrm2 , ␪ F s 3.72 = 10y4 Ž ␪W . 3 y 0.089Ž ␪W . 2 q 8␪W y 207 Ž`.. Theoretical fit of LW᎐AB approach using ␥ LW s 35.5 mJrm2 .

unknowns and hence needs three simultaneous equations to be solved. As discussed before it is preferable that contact angle with one apolar liquid and two polar liquids are to be used. The Lifshitz᎐van der Waals component of surface energy is evaluated by using the Fowkes approach wEq. Ž14. without the spreading pressure termx and contact angle with apolar liquid. The acid᎐base component and the electron᎐donorrelectron᎐acceptor characteristics are evaluated by using the two polar liquid contact angles. The data available were with water and formamide liquids, which is good because this pair gives stable results as analysed in Table 4. Fig. 28 plots the experimentally measured water contact angles vs. the formamide contact angle for 147 bacterial isolates. As can be seen the contact angles are scattered and do not follow a well-defined behaviour. The figure also contains the curves showing expected correlation between ␪W and ␪ F by Equation of state and Geometric mean approach along with the ␪W vs. ␪ F scatter expected by LW᎐AB approach. The expected contact angle of water and formamide are presented in Table 8. An average value of 35.57 mJrm2 is assumed for ␥ LW and then the ␥ AB is assumed from 1 to 26 mJrm2 . Eq. Ž28. is solved using the solid and liquid surface

442 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 Table 8 Expected correlation between ␪ Wa ter and ␪ Formamide according to Geometric mean and Equation of state approach Geometric mean appraoch ␥LW

␥AB

␥Total

cos␪W

cos␪F

␪W

35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57

1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 22 24 26

36.57 37.57 38.57 39.57 40.57 41.57 42.57 43.57 44.57 45.57 47.57 48.57 49.57 50.57 51.57 52.57 53.57 54.57 55.57 57.57 59.57 61.57

y0.04 0.04 0.10 0.16 0.20 0.24 0.28 0.32 0.35 0.38 0.44 0.47 0.50 0.52 0.55 0.57 0.60 0.62 0.64 0.68 0.72 0.76

0.43 0.49 0.54 0.58 0.62 0.65 0.68 0.71 0.73 0.76 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.95 0.99 1.02 1.05

92.32 87.67 84.08 81.05 78.35 75.89 73.61 71.45 69.41 67.45 63.72 61.94 60.19 58.47 56.78 55.11 53.46 51.82 50.18 46.91 43.61 40.24

␪F 64.37 60.34 57.14 54.35 51.80 49.43 47.16 44.98 42.85 40.75 36.59 34.49 32.37 30.19 27.94 25.58 23.08 20.38 17.36 9.16 NA NA

Equation of state approach ␥Total 57.00 56.00 55.00 54.00 50.00 45.00 40.00 35.00 30.00 25.00 20.00

␪W 44 46 47 49 56 64 73 81 89 96.5 104.5

␪F 11 15.5 19 22 32 43 52 61.5 71 80 90

energy parameters and, cos␪ and then ␪ are determined for Geometric mean approach. For Equation of state approach the total surface energy is assumed in the region of 20᎐57 mJrm2 and the water and formamide contact angle are read from the conversion table w130x. The ␪W vs. ␪ F scatter for LW᎐AB approach is generated by first assuming ␥ LW to be 35.57 mJrm2 . The electron accepting and electron-donating parameters are

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 443

then allotted values between 1.8 = 10y5 and 11 and 0.01 and 80, respectively. By combining ␥q and ␥y the acid᎐base component is evaluated wEq. Ž107.x and then Eq. Ž112. is solved using the liquid surface energy parameter for water and formamide to obtain water and formamide contact angles. The values obtained are presented in Table 9. A similar approach has been used by de Meijer w132x to correlate the water and formamide contact angles. It is clear from Fig. 28 that the contact angle of the two liquids does not follow the expected curve that is expected by the Equation of state approach and the Geometric mean approach. However, the scatter predicted by the LW᎐AB approach almost covers the scatter of the two contact angles. The incomplete coverage may be due to the fact that the scatter of contact angle values using the LW᎐AB approach are generated using the ␥ LW of only 35.57 mJrm2 . This shows that the LW᎐AB approach can explain the natural behaviour of the water and formamide contact angles. The LW᎐AB approach evaluates the electron-donor, electron-acceptor and acid᎐base component of the surface energy of the bacterial cells using water and formamide contact angle. Fig. 29 correlates the aforementioned parameters to the water contact angle. There is no straightforward correlation between the parameters and water contact angle but the random scatter of the different parameters vs. ␪W is overlapped by the theoretically estimated behaviour for the parameters vs. ␪W ŽTable 9.. Similarly Fig. 30 correlates the formamide contact angle to the evaluated parameters, in this case also the theoretically generated scatter at ␥ LW s 35.57 mJrm2 ŽTable 9. overlaps the scatter in the plots. The LW᎐AB approach needs contact angles with two polar liquids and one with apolar liquid. For the 147 bacterial isolates, contact angle data are available with two polar liquids and two apolar liquids therefore the internal consistency of LW᎐AB approach can be checked only by changing the apolar liquid contact angle. Fig. 31 shows the correlation of surface energy parametersᎏ␥y, ␥q, ␥ AB and ␥Total when they are evaluated by LW᎐AB approach using water, formamide and either methyleneiodide or ␣-bromonapthalene contact angles. Fig. 31a shows the electron-acceptor Ž ␥q. characteristics of microbial cells. The value of ␥qranges from 1.8 = 10y5 to 15 mJrm2 , more than 90% of microbial cells have the value within 1.8 = 10y5 to 5 mJrm2 Žinset figure.. The correlation of the ␥q evaluated by using methyleneiodide or ␣-bromonapthalene contact angles along with water and formamide does not follow the 45⬚ line but is a scatter with a straight line fit with a slope of 0.74. Fig. 31b shows the electron-donor Ž ␥y. characteristics of the microbial cells. Values of ␥y range from 0.03 to 105 mJrm2 . For a microbial cell surface the value of ␥y does not depend on the apolar liquid used in the calculation, which is evident from the fact that all the data points lie on the 45⬚ line when ␥y-values evaluated using methyleneiodide are plotted against the ones evaluated using ␣-bromonapthalene. For the 147 microbial isolates under study, the ␥y parameter lie in two different groups. For approximately 38% of the microbial cells the ␥y parameter lie between 0.03 and 30 mJrm2 and for the other 57% it lies between 40

␥LW

␥q

␥y

␥AB

␥Total

␪W

␪F

␥LW

␥q

35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57

1.8 = 10y5 1.8 = 10y5 1.8 = 10y5 1.8 = 10y5 1.8 = 10y5 1.8 = 10y5 1.8 = 10y5 1.8 = 10y5 1.8 = 10y5 1.8 = 10y5 1.8 = 10y5 1.8 = 10y5 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.5 0.5 0.5 0.5 0.5

0.01 0.1 1 10 20 30 40 50 60 70 72 79.1 0.01 0.1 1 10 20 30 40 50 60 70 72 73 73.5 0.01 0.1 1 10 20

0.001 0.003 0.009 0.027 0.038 0.047 0.054 0.060 0.066 0.071 0.072 0.076 0.064 0.2 0.64 2 2.83 3.47 4 4.48 4.9 5.3 5.37 5.41 5.43 0.15 0.45 1.42 4.48 6.33

35.58 35.58 35.58 35.6 35.61 35.62 35.63 35.63 35.64 35.65 35.65 35.65 35.64 35.77 36.21 37.57 38.4 39.04 39.57 40.05 40.47 40.87 40.94 40.98 41 35.72 36.02 36.99 40.05 41.9

102.74 100.99 95.49 78.22 67.3 58.31 49.99 41.71 32.84 22.14 19.48 1.07 100.21 98.47 93.01 75.67 64.58 55.34 46.67 37.83 27.94 14.17 9.59 6.19 3.4 97.07 95.34 89.9 72.44 61.09

73.12 72.44 70.28 63.22 58.73 55.14 51.98 49.08 46.35 43.72 43.2 41.37 69.02 68.32 66.1 58.79 54.07 50.26 46.87 43.72 40.71 37.77 37.19 36.9 36.75 63.71 62.98 60.66 52.91 47.81

35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57

1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 5

␥y 1 10 20 30 40 50 53 55 57 58.9 0.1 1 10 20 30 40 50 53 55 56 0.1 1 10 20 30 40 42.2 0.1 0.1 0.1

␥AB

␥Total

␪W

␪F

2.45 7.75 10.96 13.42 15.5 17.33 17.84 18.17 18.5 18.8 0.9 2.83 8.95 12.65 15.5 17.89 20 20.6 20.98 21.17 1.1 3.47 10.96 15.5 18.98 21.91 22.51 1.1 1.27 1.42

38.02 43.32 46.53 48.99 51.07 52.9 53.41 53.74 54.07 54.37 36.47 38.4 44.52 48.22 51.07 53.46 55.57 56.17 56.55 56.74 36.67 39.04 46.53 51.07 54.55 57.48 58.08 36.67 36.84 36.99

85.78 68.07 56.27 46 35.68 23.67 19.12 15.48 10.8 1.59 89.71 84.27 66.43 54.44 43.86 33.01 19.59 13.8 8.09 2.05 87.19 81.72 63.65 51.27 40.08 28.04 24.98 87.19 85.05 83.17

52.96 44.32 38.38 33.23 28.26 23.1 21.44 20.3 19.11 17.93 52.58 49.95 40.84 34.41 28.65 22.79 16.06 13.61 11.76 10.72 47.42 44.57 34.36 26.62 18.81 7.73 0.83 47.42 42.71 38.18

444 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Table 9 Expected correlation between ␪ Wa ter and ␪ Formamide according to LW᎐AB approach

Table 9 Ž Continued. ␥q

35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57

0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 1 1 1 1 1 1 1.5 10.7 3 4 5 6 6.5 6.7 7 8 9 9.1 9.3

␥y

␥AB

␥Total

␪W

␪F

␥LW

␥q

␥y

␥AB

␥Total

␪W

␪F

30 40 50 60 65 66 67 0.1 1 10 20 30 40 50 55 60 61 62 0.1 0.01 1 1 1 1 1 1 1 1 1 1 1

7.75 8.95 10 10.96 11.41 11.49 11.58 0.64 2 6.33 8.95 10.96 12.65 14.15 14.84 15.5 15.63 15.75 0.78 0.66 3.47 4 4.48 4.9 5.1 5.18 5.3 5.66 6 6.04 6.1

43.32 44.52 45.57 46.53 46.98 47.06 47.15 36.21 37.57 41.9 44.52 46.53 48.22 49.72 50.41 51.07 51.2 51.32 36.35 36.23 39.04 39.57 40.05 40.47 40.67 40.75 40.87 41.23 41.57 41.61 41.67

51.47 42.23 32.43 20.34 11.02 8.05 2.97 93.01 87.57 69.98 58.39 48.43 38.64 27.79 21.16 11.95 9.16 5.07 91.22 76.54 81.72 79.57 77.65 75.91 75.09 74.77 74.29 72.78 71.35 71.21 70.93

43.6 39.78 36.15 32.56 30.76 30.39 30.03 58.82 56.39 48.19 42.68 38.03 33.69 29.42 27.24 24.99 24.53 24.07 55.49 1.41 44.57 39.6 34.74 29.76 27.15 26.07 24.4 18.13 9.03 7.6 3.28

35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57

6 7 8 9 10 10.1 10.2 10.3 3.5 4 5 6 7 8 9 10 10.5 10.6 10.7 6 7 7.9 4 5 6 6.1 6.2 6.3 6.4 4 4.5

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 4 4 4 10 10 10 10 10 10 10 20 20

1.55 1.68 1.79 1.9 2 2.01 2.02 2.04 0.38 0.4 0.45 0.49 0.53 0.57 0.6 0.64 0.65 0.66 0.66 9.8 10.59 11.25 12.65 14.15 15.5 15.63 15.75 15.88 16 17.89 18.98

37.12 37.25 37.36 37.47 37.57 37.58 37.59 37.61 35.95 35.97 36.02 36.06 36.1 36.14 36.17 36.21 36.22 36.23 36.23 45.37 46.16 46.82 48.22 49.72 51.07 51.2 51.32 51.45 51.57 53.46 54.55

81.45 79.87 78.39 77 75.67 75.54 75.41 75.22 87.8 86.77 84.89 83.19 81.61 80.14 78.76 77.44 76.8 76.68 76.55 67.53 65.83 64.39 61.24 59.08 57.08 56.89 56.7 56.51 56.32 48.48 47.18

33.66 28.96 23.85 17.86 9.23 7.91 6.34 2.61 45.94 43.66 39.22 34.81 30.27 25.41 19.86 12.65 6.93 5.1 2.04 23.01 15.63 2.78 27.91 20.79 11.12 9.73 8.11 6.1 2.97 17.79 11.9

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 445

␥LW

␥LW

␥q

␥y

␥AB

␥Total

␪W

␪F

␥LW

␥q

␥y

␥AB

␥Total

␪W

␪F

35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57

9.32 4 5 6 7 8 9 9.5 9.6 9.7 9.75 9.75 9.75 9.75 9.75 9.75 9.75 1 2 3 4 5

1 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 4 4 4 4 4

6.11 2.83 3.17 3.47 3.75 4 4.25 4.36 4.39 4.41 4.42 4.42 4.42 4.42 4.42 4.42 4.42 4 5.66 6.93 8 8.95

41.68 38.4 38.74 39.04 39.32 39.57 39.82 39.93 39.96 39.98 39.99 39.99 39.99 39.99 39.99 39.99 39.99 39.57 41.23 42.5 43.57 44.52

70.9 81.92 80.02 78.3 76.7 75.2 73.79 73.1 72.97 72.84 72.77 72.77 72.77 72.77 72.77 72.77 72.77 79.57 76.2 73.58 71.35 69.35

2.46 40.95 36.25 31.48 26.44 20.77 13.52 8.06 6.47 4.34 2.72 2.72 2.72 2.72 2.72 2.72 2.72 52.71 45.91 40.12 34.63 29.06

35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57 35.57

4.6 4.7 4.8 4.9 3.5 3.6 3.7 3.8 3.9 4 4.5 4.8 4.9 4.92 3.5 3.6 3.7 3.9 3.1 3.11 3.12 3.13

20 20 20 20 20 20 20 20 20 20 20 20 20 20 30 30 30 30 40 40 40 40

19.19 19.4 19.6 19.8 20.5 16.98 17.21 17.44 17.67 17.89 18.98 19.6 19.8 19.84 20.5 20.79 21.08 21.64 22.28 22.31 22.35 22.38

54.76 54.97 55.17 55.37 56.07 52.55 52.78 53.01 53.24 53.46 54.55 55.17 55.37 55.41 56.07 56.36 56.65 57.21 57.85 57.88 57.92 57.95

46.93 46.68 46.42 46.18 38.34 49.57 49.29 49.02 48.75 48.48 47.18 46.42 46.18 46.13 38.34 38 37.66 36.99 27.55 27.5 27.45 27.4

10.39 8.65 6.48 3.1 12.39 21.58 20.68 19.75 18.79 17.79 11.9 6.48 3.1 1.78 12.39 10.74 8.83 2.16 4.34 3.85 3.28 2.59

Surface energies in ŽmJrm2 . and angles in Ž⬚..

446 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Table 9 Ž Continued.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 447

Fig. 29. Correlation of experimentally obtained and the theoretically evaluated Ža. total, acid᎐base, Žb. electron acceptor and Žc. electron donor surface energy parameters of bacterial cells with respect to water contact angle using the LW᎐AB approach.

448 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 30. Correlation of experimentally obtained and the theoretically evaluated Ža. total, acid᎐base, Žb. electron acceptor and Žc. electron donor surface energy parameters of bacterial cells with respect to formamide contact angle using the LW᎐AB approach.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 449

Fig. 31. Correlation of Ža. electron-donor, Žb. electron-acceptor, Žc. acid-base component and Žd. total surface energy parameters for 147 different bacterial isolates evaluated using LW᎐AB approach, when the liquid for the apolar contact angle is changed keeping the two polar liquids the sameᎏwater and formamide.

450 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 31. Ž Continued..

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 451

and 70 mJrm2 with a very less number of microbial isolates having the ␥y parameter between 30 and 40 mJrm2 Žinset figure.. This clearly shows that the 147 isolated have two groups with different types of surfaces. Fig. 31c shows the polar component, i.e. acid᎐base component of the surface energy of the microbial cells. The values of ␥ AB range from 0.06 to 70 mJrm2 but more than 97% of the microbial cells have the ␥ AB between 0.06 and 30 mJrm2 Žinset figure.. The correlation of ␥ AB values evaluated using methyleneiodide and ␣-bromonapthalene show that the data points are scattered and follow a straight line fit with a slope of 0.82. Fig. 31d shows the total surface energy of the microbial isolates evaluated by the LW᎐AB approach. The total surface energy for the 147 microbial isolates range between 30 and 100 mJrm2 but more than 98% of the microbial cells have the surface energy between 30 and 60 mJrm2 Žinset figure.. The correlation of ␥Total evaluated by using methyleneiodide and ␣-bromonapthalene is a scatter, which follows a straight-line fit of Eq. Ž134.. Total ␥Total LW yAB ; B r s 7.43 q 0.85␥ LWyAB ; M

Ž 134.

The presence of two peaks in the distribution of ␥y parameter for 140 bacterial isolates Žinset of Fig. 31b. is further investigated in Fig. 32, where the distribution of ␥y parameter is plotted against frequency in form of numbers. Out of the 140 bacterial isolates 29 are gram-negative and the other 111 are gram-positive. The distribution of ␥y parameter for gram-positive, gram-negative and overall 140 bacterial isolates is plotted separately in Fig. 32. If we consider the overall distribution then 35.7% of the bacterial isolates have the ␥y parameter between 0 and 25 mJrm2 and 57.9% have ␥y parameter between 35 and 65 mJrm2 . Gram-positive bacterial cells have the same behaviour as the overall distribution, i.e. having two separate groups, with 38.7% in the range of 0᎐25 mJrm2 and 54.9% in between 35 and 65 mJrm2 . However, for the gram-negative bacterial cells 69% have the ␥y parameter between 35 and 65 mJrm2 and only 24.1% in between 0 and 25 mJrm2 . This shows that most of the gram-negative bacterial isolates have higher characteristics. The X-ray photoelectron spectroscopic ŽXPS. surface composition for 116 out of the 140 bacterial isolates is available from van der Mei w23x. Fig. 32 shows the distribution diagrams for NrC, OrC and PrC ratios for 116 bacterial isolates. Out of these 116, 26 are gram-negative and 90 are gram-positive bacterial strains. Fig. 33a shows the distribution of nitrogen to carbon ratio ŽNrC. for 90 gram-positive, 26 gram-negative and overall 116 bacterial isolates. The NrC ratio for the 116 bacterial isolates lie in between 0.026 and 0.199 and the distribution is similar to the distribution of their ␥y parameter in Fig. 32 with two groups of bacterial isolates having different nitrogen to carbon ratio on their surface. Of the bacterial cells Ž67.3%. have 0.026 - NrC F 0.105 and 32.7% have 0.105 - NrC - 0.199. Approximately 59% of gram-positive bacterial cells have 0.026 - NrC F 0.105 and 41% have 0.105 - NrC - 0.199. However, 96% of gram-negative bacte-

452 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 32. Distribution of the electron-donating characteristics for 140 bacterial species, 111 Gram-positive and 29 Gram-negative bacterial cell surfaces.

rial isolates have the 0.026 - NrC F 0.105 and only 4% have 0.105 - NrC 0.199. This shows that most of the gram-negative bacteria have a lower nitrogen content on their surface, whereas equal proportion of gram-positive bacterial cells which have high and low nitrogen concentration on their surface. This delineates a clear difference in the surface composition of gram-positive and gram-negative bacterial isolates under investigation. Out of all the surface polymers nitrogen is present in the peptidoglycan and proteins w23,133x. In general, gram-negative cell surface lacks peptidoglycan which could have got reflected for the gram-negative cells under investigation. However, the other reason can be that they have lower protein content on their surface whereas gram-positive cells have both high and low protein or peptidoglycan content on its surface. Fig. 33b shows the distribution of oxygen to carbon ratio ŽOrC. for gram-positive, gram-negative and overall 116 bacterial isolates. The OrC ratio for the 116 bacterial isolates lie in between 0.203 and 0.655 and the distribution of the oxygen to carbon ratio shows the presence of two different groups but the demarcation is not so clear. For the 116 bacteria along with the 90 gram-positive bacteria have the OrC ratio distributed in between 0.203 and 0.705, but most of the gram-negative bacteria Ž) 96%. have the OrC between 0.203 and 0.55. Here also we can see that most of the gram-negative bacteria have lower oxygen content on their surface as compared to gram-positive bacteria. Fig. 33c shows the distribution of phosphorus to carbon ratio ŽPrC. for the

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 453

Fig. 33. Distribution of the chemical composition of bacterial cells surface in the form of NrC, OrC and PrC obtained by XPS spectroscopy Žvan der Mei et al. w23x.. The distribution is presented for Gram-positive Žq with dashed line. and Gram-negative Žy with dotted line. bacterial cells along with the distribution for them together ŽSolid square with whole line.. Inset figures show correlation between the ␥y parameter and the surface composition by XPS.

454 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

gram-positive, gram-negative and overall bacterial isolates. The PrC ratio for the 116 bacterial isolate lie in between 0 and 0.05 and the distribution shows a presence of three groups with different phosphorus contents on their surfaces. Gram-positive bacterial isolates also have the PrC ratio in the range of 0᎐0.05, but 26 gram-negative bacterial cells have the PrC ratio in the range of 0.002᎐0.027. Out of all the surface polymers phosphorus is present in teichoic acid ŽGram q. and lipopolysaccharides ŽGram y. w23,133x. The plots inset in Fig. 33a᎐c show the correlation between the ␥y parameter of gram-negative bacterial cells to NrC, OrC and PrC ratios respectively, as obtained by XPS and taken from van der Mei w23x. From the inset plots we can see that there is no direct correlation between the surface composition of gram-negative bacterial cells Žin form of NrC, OrC and PrC ratios. to the ␥y parameter of the surface energy. From the above analysis of LW᎐AB approach we can conclude that: 1. The 98% of the 147 microbial isolates under investigation have total surface energy in between 30 and 60 mJrm2 . The acid᎐base component varying between 0.06 and 30 mJrm2 ŽFig. 31c,d.. 2. Most of the microbial cells Ž99%. have low electron-accepting characteristics between 1.85 = 10y5 and 10 mJrm2 . However, the microbial cells have from very low Ž0.03. to very high Ž105. electron-donating characteristics ŽFig. 31a,b.. 3. Unlike Equation of state and Geometric mean approach, the LW᎐AB approach with water and formamide contact angles on bacterial cell surface are expected to form a scatter and not to follow a well-defined polynomial fit. This is actually observed in the experimentally measured water and formamide contact angle on 147 microbial cell surfaces ŽFig. 28.. 4. When the electron-accepting, electron-donating, acid᎐base component and total surface energy of bacterial cells are correlated to either water or formamide contact angle, then they give rise to a scatters which are overlapped by the theoretically generated scatters for LW᎐AB approach ŽFigs. 29 and 30.. 5. The electron-donating parameter of bacterial cells remains unchanged when the apolar liquid for contact angle is changed for evaluation of surface energy by LW᎐AB approach ŽFig. 31b.. 6. Electron-accepting parameter, acid᎐base component and the total surface energy are different when the apolar liquid for contact angle is changed ŽFig. 31a,c,d.. 7. The 140 bacterial isolates under investigation can be divided in two groups having different surface properties, in form of their electron-donating characteristics Žinset Fig. 31b.. 8. The gram-negative bacterial cells have predominantly higher electron-donating parameter between 35 and 65 mJrm2 , whereas some gram-positive bacterial cells have high Ž35᎐65 mJrm2 . and some low Ž0᎐25 mJrm2 . electron-donating parameter of surface energy ŽFig. 32..

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 455

9. Although there is no direct correlation of the electron-donating parameter of gram-negative bacterial cells surface composition ŽXPS., the gram-negative bacterial cells have lower nitrogen, oxygen and phosphorous content on their surface ŽFig. 33.. 4.7. Comparison of LW᎐AB approach to Equation of state and Geometric mean approach Fig. 34a,b show the distribution of the total surface energy and acid᎐base component respectively, evaluated using LW᎐AB approach with water, formamide and ␣-bromonapthalene contact angles, while using equation of state and geometric mean approaches with water and formamide contact angles. For the 147 microbial isolates when water contact angle is used, the total surface energy evaluated by both equation of state and geometric mean approach is distributed over a wide range Ž; 10᎐80 mJrm2 ., but when formamide contact angle is used then the surface energy is distributed only between y30 and 60 mJrm2 ŽFig. 34a.. Total surface energy evaluated using LW᎐AB approach varies in between y30 and 60 mJrm2 and the distribution overlaps to the distribution for equation of state and geometric mean approach using formamide contact angle. The distributions again highlight the internal inconsistency of equation of state and geometric mean approaches but shows the similarity between the two approaches when the same polar liquid contact angle is used. The distribution diagrams of acid᎐base component ŽFig. 34b. show that when LW᎐AB and geometric mean approaches are used then minimum value obtained is zero and there are no negative values, whereas with equation of state approach negative values are obtained. When water contact angle is used with equation of state and geometric mean approaches, the ␥ AB is spread over a wide range Ž0᎐48.9 for geometric mean and y24.2 to 44.2 for equation of state approach.. However, when the formamide contact angle is used for the two approaches then the ␥ AB is spread in narrow range Ž0᎐25 for geometric mean and y20 to 22 for equation of state approach.. The distribution of ␥ AB for the 147 microbial isolates obtained by LW᎐AB approach is similar to the distribution of ␥ AB obtained by using formamide contact angle and other two approaches. Fig. 35 correlates the total surface energies for the 147 microbial isolates obtained by the LW᎐AB approach to the ones obtained by geometric mean and equation of state approaches when water contact angle ŽFig. 35a. and formamide contact angle ŽFig. 35b. are used for the later. We can see that the surface energies obtained by LW᎐AB approach does not correlate with the surface energies obtained by the other two approaches upon using water contact angle. However, there is a definite correlation of surface energy by LW᎐AB to the surface energy by other two approaches upon the use of formamide contact angle, and the data points fall on the 45⬚ line Ž135. for ␥Total ) 50 mJrm2 . For ␥Total - 50 mJrm2 the points fall near the 45⬚ line. Similarly Fig. 35 correlates the ␥ AB for the two cases.

456 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 34. Distribution of total and acid᎐base component of surface energy when evaluated using different approaches and contact angles.

There is no correlation between the ␥ AB obtained by LW᎐AB approach and ␥ AB obtained by the other two approaches upon the use of water contact angle ŽFig. 36a.. However, there is a definite correlation between the ␥ AB obtained by LW᎐AB and other two approaches upon the use of formamide contact angle ŽFig.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 457

Fig. 35. Correlation of the total surface energy evaluated by using either the Geometric mean approach or Equation of state approach with the one evaluated by the LW᎐AB approach Ža. by using the water contact angle Žb. using the formamide contact angle along with ␣-bromonapthalene contact angle for the Geometric mean approach.

458 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 36. Correlation of the acid᎐base component of surface energy evaluated by using either the Geometric mean approach or Equation of state approach with the one evaluated by the LW᎐AB approach Ža. by using the water contact angle Žb. using the formamide contact angle along with the ␣-bromonapthalene contact angle for the Geometric mean approach.

36b.. The ␥ AB values obtained from geometric mean approach using formamide contact angle is the same as the ones obtained by LW᎐AB Ž136. approach but equation of state approach gives the ␥ AB values lower to the ones given by LW᎐AB approach by ; 7 mJrm2 as the data points follow the straight line wEq. Ž137.x.

P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463 459

Fig. 37. Correlation of Ža. the acid᎐base component Žb. total surface energy when evaluated using either the LW᎐AB approach or least square fit of the Geometric mean approach to the four available contact angle values for each microbial surface.

Total Total ␥GM ,Eq.st;Fo s ␥ LWyAB

Ž 135.

AB AB ␥GM ;Fo s ␥ LWyAB

Ž 136.

AB AB ␥ Eq .st;Fo s ␥ LWyAB y 7

Ž 137.

460 P.K. Sharma, K. Hanumantha Rao r Ad¨ ances in Colloid and Interface Science 98 (2002) 341᎐463

Fig. 37a,b shows that when least square fit of Geometric mean approach is used to determine the surface tension then neither the total surface energy nor the acid᎐base component correlate with the total surface energy and acid᎐base component evaluated by the LW᎐AB approach.

5. Summary Using literature and measured contact angle data on 147 different microbial isolates, the theoretical approaches to evaluate solid surface energy are assessed. It has been found that the dispersion component of surface energy for most of the microbial cells is close to 40 mJrm2 " 10%. Though Equation of state approach strictly opposes the division of the total surface energy, it evaluates surface energies similar to the dispersion component of surface energy, as evaluated by Fowkes approach, and total surface energy, as evaluated by Geometric mean approach, when contact angles with apolar and polar liquids are used, respectively. Both Equation of state and Geometric mean approaches lack internal consistency, because the surface energy evaluated depends upon the contact angle used. Though the Lifshitz᎐van der WaalsrAcid᎐base ŽLW᎐AB. approach is effected by mathematical instability, but it has been shown that when water and formamide contact angles along with either methyleneiodide or ␣-bromonapthalene contact angle are used then it gives consistent and non-negative results. On an average the Gram-negative bacteria have higher electron-donating parameter evaluated by the LW᎐AB approach, hence this parameter can differentiate between Gram-positive and Gram-negative bacteria. When water contact angle is used to evaluate the surface energy by Equation of state and Geometric mean approaches the values does not have any correlation to the surface energy evaluated by the LW᎐AB approach whereas, when formamide contact is used then the surface energy values correlate well and are similar to the ones by LW᎐AB approach.

Acknowledgements Financial support from Swedish Foundation for International Cooperation in Research and Higher Education ŽSTINT. and Swedish Foundation for knowledge and Competence Development ŽKKS. is gratefully acknowledged.

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