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Adhesion, entropy and surface forces
Colloids and Surfaces A: Physicochemical and Engineering Aspects 167 (2000) 209 – 214 www.elsevier.nl/locate/colsurfa
Adhesion, entropy and surface forces Ha˚kan Wennerstro¨m * Di6ision of Physical Chemistry 1, Center for Chemistry and Chemical Engineering, PO Box 124, S-221 00 Lund, Sweden Received 30 November 1998; accepted 19 April 1999
Abstract The adhesion energy at contact between two surfaces can be seen as due to a sum of a positive direct interaction term and a term of entropic origin appearing for a free surface exposed to a solvent. When the former dominates we have a lyophobic system while if the entropic term wins the system is lyophilic. These statements are illustrated by three examples (i) a solid ionizing surface; (ii) a liquid-like surface with protrusion modes; (iii) a surface with solvent polarization. It is pointed out that a theory of surface forces also have implications for adhesion energies. In particular it is found that the Marc¸elja theory of hydration forces leaves as unexplained the absence of a strong adhesive minimum at contact. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Adhesion; Entropy; Surface forces
1. Introduction It is a general rule that chemically identical colloidal particles or surfaces adhere at equilibrium in a solvent. However, there are exceptions, particularly common in biological systems, where the adhesion is nonexisting or very weak. Traditionally one made the distinction between lyophobic (solvent hating) and lyophilic (solvent liking) colloids. In textbooks [1 – 3] the discussion of surface forces is based on the DLVO theory, in which a repulsive double layer and an attractive van der Waals’ force combine to generate a global
This article was originally submitted to the Per Stenius Special Issue. * Tel.: + 46-46-2229767; fax: +46-46-2224413. E-mail address: [email protected] (H. Wennerstro¨m)
minimum at contact between surfaces with a possible energy barrier further out. The DLVO theory has formed our expectations about surface forces in general although Verwey and Overbeek made it clear already in the title of their classical treatise [4] that the theory really apply only to lyophobic colloids. It took some time before the scientific interest was switched to the lyophilic colloids and the forces that make them stable. During the last few decades there has, due to advances in experimental techniques, appeared increasingly more detailed observations of strong deviations from the DLVO-description [5–7]. For lyophilic colloids these consists of short range repulsive forces. When observed in aqueous systems they have been called hydration forces [8]. Although experimentally well characterized the possible molecular origin of these forces has given
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rise to a heated debate, and a consensus has not been reached. Prime examples of observations of hydration forces’ are between zwitterionic liquidlike bilayers [5,8], between solid mica surfaces [9] and between solid silica surfaces [10]. Together with Jacob Israelachvili we have argued [11] that repulsive forces between like colloidal objects have in general an entropic origin and this should also apply to the ‘hydration force’ [12,13]. To provide a complementary perspective on the role of entropy contributions to surface forces we will below analyze adhesion free energies between chemically identical surfaces in the presence of a medium.
2. Adhesion, interaction at contact, and surface forces According to the Dupre´ equation [1] the work of adhesion, W121, between two similar materials (1) in a medium (2) is related to the corresponding work of adhesion, W11 and cohesion W12, across a vacuum W121 = W11 + W22 −2W12
(1)
When analyzing the work of adhesion in a microscopic model we thus need to consider the interaction energy at contact between two halfplanes of material (1), the same quantity for the medium (2) and substract twice the interaction energy material-medium. In addition it is essential to include changes in entropy to obtain the free energy. This microscopic picture of the adhesion energy is based on the assumption that the minimum of the free energy occurs at molecular contact between the surfaces. Although this is usually the case there are many exceptions to the rule. In an alternative, and more general, formulation we can calculate W121 as minus the integral of the force per unit area, F(z)/A, between the surfaces in the medium W121 = − 1/A
&
f(z)dz
(2)
dm
Note that by convention an attractive force is considered to be negative the adhesion energy is the difference in the (free) energy at infinite sepa-
ration minus the energy at contact. In Eq. (2) dm is the distance at the free energy minimum, F(dm)= 0. This distance can equal the value at contact, dc, defined by the condition that the medium is absent between the surfaces or the minimum can occur at a larger separation. It follows from Eq. (2) that there is a direct connection between the surface force and the adhesion energy. When making a statement about one of them you necessarily implies something about the other. The work of adhesion is simply related to the interfacial free energy or the interfacial tension, (3)
2g12 = W121
For g12 we can identify three typical cases. The most common is that g12 is of a substantial magnitude, not more than an order of magnitude smaller than the surface tension so that g12 \ 1 mJ m − 2. A second possibility is that g12 is small but positive 0B g12 B 1 mJ m − 2. The low interfacial free energy is typically an indication that the medium is present to some extent between the surfaces at equilibrium, i.e. dm \ dc. A third possibility is that the surfaces repel one another to the extent that infinite separation represents the lowest free energy state. In the latter two cases we can define a contact free energy, W121(contact), by considering the work needed to bring the two surfaces from infinity to molecular contact W121(contact)= − 1/A
&
dm
dc
F(z)dz + 2g12
(4)
Using Eq. (4) we can now analyze all adhesion energies by only considering the direct contact between two surfaces of the material and the surface exposed to an infinite medium. At contact the interaction free energy is dominated by direct electrostatic and dispersion interactions across the interface, while for the free surface in a medium there can also be important contributions to the free energy from an increase in entropy due to the increased flexibility at a free surface. Below we will illustrate this point of view through three examples: (i) a surface with ionizable groups; (ii) a surface with molecular protrusions; (iii) a surface inducing solvent polarization.
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3. Surface ionization
knowing the surface charge density s using the Gouy–Chapman theory [14]
Most solid surface acquire a charge when immersed into water. Freshly cleaved mica have potassium ions electrostatically bound on the surface. In pure water these ions diffuse into the solution and are partly replaced by protons from the water giving the surface a net negative charge. This spontaneous dissociation process contributes to reducing the work of cleavage in water relative to dry air. In a similar way there are silanol groups at the surface of silica. For two silica surfaces in contact these silanol groups carry protons, while when exposed to an aqueous solution a dissociation can occur. The dissociation is driven by an increase in entropy, while it is gradually stopped as the surface charge density increases giving a high electrostatic energy. Let us make a quantitative estimate of the free energy charge associated with a surface dissociation. Consider the reaction scheme
s= − enAXA(eq)= 2kTkoro0 sinh(eFs/2kT)/e
AHs X As− +H+ b
(5)
Where subscripts s and b denote surface and bulk, respectively. At a given surface potential Fs, pH, and mole fractions of AH and A− at the surface the free energy change DGH in releasing one proton is DGH =2.3kT(pKa −pH) + kT ln XA/XAH − eFs, (6) where Ka is the intrinsic equilibrium constant at the surface. As the dissociative process continues both the ratio XA/XAH and the surface potential change. Equilibrium is reacted at a degree of dissociation XA(eq) where DGH of Eq. (6) is zero. The total free energy change is simplest obtained as the difference between final and initial state and DGdiss/area=kTnA{XA(eq)2.3(pKa −pH) + XA(eq) ln XA(eq) +XAH(eq) ln XAH(eq)} + Gel(Fs)/area,
(7)
where nA is the surface density of titratable groups A. The surface potential Fs we can estimate
(8)
The electrostatic free energy is in the Gouy– Chapman theory [15] Gel/area= 2nAXA(eq) × kT{ln[s + (s 2 + 1)1/2 + 1/s − (s 2 + 1)1/2/s}
(9)
with the dimensionless surface charge parameter s defined by s= sinh(− eFs/2kT)
(10)
To obtain an estimate of the typical magnitude of the free energy in Eq. (9) take as an example a surface with a density nA of one per 20 A, 2, pKa 5, approximately representing a solid fatty acid, exposed to an electrolyte solution at pH 7. Fig. 1 shows how DGdiss varies with the concentration of the 1:1 electrolyte. The free energy is typically in the range −5 to − 30 mJ m2 increasing in magnitude with increasing electrolyte concentration. This amounts to a substantial, negative, contribution to the work of adhesion which we can write approximately as a sum of a direct interaction term W121(int) and a surface dissociation term W121 : W121(int)+ 2DGdiss
(11)
In practice models of intermolecular interactions yield a positive effective interaction term [16] w= wAB − 1/2(wAA + wBB),
(12)
where wij is the pair interaction between molecules i and j. To what extent this result can be generalized to hold also for surface interaction W121(int) is not uneqivocally established, but a positive sign is expected and explicitly obtained for the van der Waals interaction [17]. Depending on the chemical nature of the surface W121(int) can vary substantially and it is concievable that DGdiss can provide a large enough negative contribution to change the sign of W121 in Eq. (11). However, for perfectly flat charged surfaces this does not seem to happen, while for approximately spherical colloidal particles with small areas of direct contact a redissolu-
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tion, so called repeptization, can occur at higher electrolyte concentrations [18]. To see the connection between the work of adhesion and the surface force we note that the long range double layer repulsion between two charged surfaces is precisely due to a decrease in DGdiss of Eq. (7). In the confined space between the surfaces there is less entropy for the counterions. Thus the molecular mechanism process that leads to a decrease in W121 through surface dissocation is also responsible for creating a repulsive component of the surface force.
the model is that at the surface of a monolayer formed by lipids in the liquid state individual molecules can protrude out from the surface. When the monolayer is exposed to a solvent the motion is mainly restricted by the free energy increase due to alkyl chain-solvent interaction. In the simple case of laterally independent protrusions the free energy for this degree of freedom is [12] G(z)/area = nkT{2 ln(kT/a) −ln[1− (1+ za/kT) exp(− za/kT)}
4. The protrusion model When the material is a solid we do not expect any major entropy increase when a surface is created, except for the ionization process described above. For a liquid material, on the other hand, there are major entropic contributions to the free energy and these could increase when an interface is created. One simple illustration of effect is found in the protrusion model [12] for the force between lipid bilayers. The basic feature of
(13)
where z is the separation between the surfaces, and a is the coefficient in the protrusion potential V(z)= az. For typical values of density of molecules n = 1/(50 A, 2), kT/a=1.5 A, and a separation at contact dc = 2 A, the free energy gain in going from contact to infinite separation is 9 mJ m − 2 per surface. In this model we have for simplicity, focused on only one particular molecular degree of freedom. In practice one expects several degrees of freedom per molecule to be affected by the presence of a second surface.
Fig. 1. The free energy change on dissociating a surface consisting of a solid fatty acid at a packing density of one molecule per 20 nm2 as a function of the bulk 1:1 electrolyte concentration c0. The system is at pH 7 and the intrinsic surface pKa 5. The curve is calculated using Eq. (7) with the electrostatic free energy calculated from the Gouy – Chapman theory.
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The simple model demonstrates that there can be substantial negative contributions to the work of adhesion due to the greater molecular freedom at a free surface. These entropic contributions to W121 are balanced by positive energy terms and the final outcome for the sign of W121 depend on the molecular details affecting both the entropic and the energetic contributions. The protrusions and other surface excitations contribute to both the lowering of the work of adhesion and to the surface force. We see that in analogy with the ionizing surfaces that it is the same mechanism that lowers the adhesion and generates the repulsive force. This coupling might appear self-evident but let us now analyze a model where such a coupling is actually absent.
5. The surface polarization model Shortly after the first publication of accurate measurements of repulsive forces between phospholipid bilayers by Parsesian et al. [5], Marc¸elja and Radic [19] published a theoretical derivation accounting for the short range repulsion in terms of a solvent structure-dipole polarization effect. The idea was subsequently clarified by Gruen and Marc¸elja [20], who were able to construct a microscopic model, based on ice with defects, and elegantly evaluate its thermodynamic properties. The essential physical idea in the model is that at short range surface solvent interaction induces a polarization of the solvent layer in direct contact with the surface. This polarization is propagated by short range solvent – solvent interactions to subsequent layers of the solvent. Since the bulk is unpolarized the polarization decays to zero for a free surface, while for two opposing surfaces a frustrated situation arises, when two polarized layers meet. This gives rise to a repulsive interaction. For a solvent that behaves as an ideal dielectric there is no orientational coupling between successive solvent layers and the effect comes from having some coupling between layers that are not dipolar in nature. In the Gruen – Marc¸elja model this coupling is introduced through the Bernal– Fowler rules of an ice lattice.
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For the free surface the existence of a polarization profile is a result of a compromise between two conflicting demands. On one hand the solvent molecules want to be randomly oriented to maximise entropy as in the bulk. On the other hand they have to adjust to the surface and by assumption this would tend to orient the molecules. Meeting these demands lead to an increase in free energy; the solvent is not as flexible as an ideal dielectric. Using eq. (3) of Ref. [20] for the free energy of the polarized liquid we find that the free energy to set up the polarization is Gpol/area= P 20orj/{2(or − o )o o0}.
(14)
Here P0 is the given surface polarization, j the decay length of the polarization profile, or is the relative dielectric permitivity at zero frequency, while o is that at high frequencies, vkT/&, and o0 is the permitivity of vacuum. In the limit or o Eq. (14) can be interpreted as the free energy/area of a capacitor of charge density s=Po and thickness j in a medium with permitivity o o0. To account for the observed forces P0 has to be of order 0.02 C m − 2 and the decay length j :2 A, with the estimated value of o= 6. With these values of the parameters Gpol/area: 1 mJ m − 2.
(15)
In contrast to the two previous examples there is actually a free energy penalty associated with setting up the solvent structure that is responsible for the repulsion. We can now estimate the work of adhesion at contact using Eq. (4), the experimentally measured hydration force, and the adhesion energy at equilibrium: 5×10 − 2 mJ m − 2 [22], and
&
W121(contact) =−
dm
4× 108 exp(− z/1.7)dz + 5
dc
× 10 − 5 J m − 2 = − 21 mJ m − 2
(16)
which is strongly negative. We could also try to estimate W121(contact) using the Dupre´ equation. In this approach the W121(contact) is given by the direct energetic interaction term plus possible entropic contributions from the medium solvent interface. The latter provides, in contrast to the first
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two examples, a positive contribution of 2 mJ m − 2 according to Eqs. (4) and (15). To match the value of − 21 mJ m − 2 in Eq. (16) we would require that the direct interaction contribution as W121(int) in Eq. (11) equals: − 23 mJ m − 2. As discussed in Section 3 the general expectation is that W121(int) is positive. A large negative value can’t a priori be ruled out but it certainly requires a non-trivial explanation. Additionally this unknown energetic contribution would by itself generate a repulsive force possibly leaving additional explanations of a repulsion unwarranted. We have previously pointed out [13] that an unsatisfactory feature of the Marc¸eljas theory of ‘hydration forces’ is that it is based on the concept that the solvent (water) can’t easily adjust to the conditions at the surface. Here we have made a quantitative analysis of this effect and demonstrated that there is a major problem in accounting for the observed highly negative value of the work of adhesion at contact between two liquid bilayers within this conceptual framework.
6. Conclusions We have analysed the relation between the work of adhesion of two similar surfaces and the force between the surfaces in a medium. By introducing the novel concept of a contact adhesion energy even when this is not the equilibrium state we have been able to demonstrate the mechanistic connection between forces and adhesion for specific models. This point of view is conceptually useful and it provides a criterion for mechanistic models of surface forces. In particular for repulsive forces it is not only sufficient to account for the distance dependence of the force, but one should also be able to relate this to the adhesive energy at contact. We have found that the Marc¸elja–Gruen theory of hydration forces lacks in this respect. Systems with adsorption from solution is an area where the general aspect developed in the paper can be of considerable help for the conceptual understanding of surface forces [21]. Clearly .
adsorption from bulk solution lowers the free energy of a surface. When now two surfaces are brought together under equilibrium with the bulk the integral of the force up to contact between surfaces has a higher value, less negative or more positive, than without the adsorbed species. For example, it is sometimes stated for surfaces with adsorbed polymers that the force is monotonicly attractive when polymers are allowed to equilibrate with the solution. This is an erroneous conclusion and our argument would give that since the polymers adsorbed to lower the free energy there is a free energy cost associated with forcing them away.
References [1] J. Israelachvili, Intermolecular and Surface Forces, 2nd edition, Academic Press, New York, 1991. [2] D.F. Evans, H. Wennerstro¨m, The Colloidal Domain, 2nd edition, Wiley, New York, 1999. [3] P.C. Hiemenz, R. Rajagopalan, Principles of Colloid and Surface Chemistry, 3rd edition, Marcel Dekker, New York, 1997. [4] E.J.W. Verwey, J.Th.G. Overbeek, Theory of stability of lyophobic colloids, Elsevier, Amsterdam, 1948. [5] D.M. LeNeveu, R.P. Rand, V.A. Parsegian, Nature 259 (1976) 601. [6] See Ref. [1], chapter 13. [7] See Ref. [2], sections 5.3 – 5.6. [8] R.P. Rand, V.A. Parsegian, Biochem. Biophys. Acta 988 (1989) 351. [9] R.M. Pashley, Adv. Colloid Interface Sci. 16 (1982) 57. [10] R.G. Horn, D.T. Smith, W. Haller, Chem. Phys. Lett. 162 (1989) 404. [11] J.N. Israelachvili, H. Wennerstro¨m, J. Phys. Chem. 96 (1992) 520. [12] J.N. Israelachvili, H. Wennerstro¨m, Langmuir 6 (1990) 874. [13] J.N. Israelachvili, H. Wennerstro¨m, Nature 397 (1996) 219. [14] See, for example, Ref. [2], eq. (3.8.15). [15] See Ref. [2]. eq. (3.9.8). [16] See, for example, Ref. [2], eq. (1.4.16). [17] See Ref. [1], chapter 11. [18] G. Frens, Faraday Discuss. 90 (1990) 143. [19] S. Marc¸elja, N. Radic, Chem. Phys. Lett. 42 (1976) 129. [20] D.W. Gruen, S. Marc¸elja, J. Chem. Soc. Faraday Trans. II 79 (1983) 225. [21] See Ref. [2], section 5.4 [22] J. Marra, J. Israelachvili, Biochemistry 24 (1985) 4608.
Adhesion, entropy and surface forces Ha˚kan Wennerstro¨m * Di6ision of Physical Chemistry 1, Center for Chemistry and Chemical Engineering, PO Box 124, S-221 00 Lund, Sweden Received 30 November 1998; accepted 19 April 1999
Abstract The adhesion energy at contact between two surfaces can be seen as due to a sum of a positive direct interaction term and a term of entropic origin appearing for a free surface exposed to a solvent. When the former dominates we have a lyophobic system while if the entropic term wins the system is lyophilic. These statements are illustrated by three examples (i) a solid ionizing surface; (ii) a liquid-like surface with protrusion modes; (iii) a surface with solvent polarization. It is pointed out that a theory of surface forces also have implications for adhesion energies. In particular it is found that the Marc¸elja theory of hydration forces leaves as unexplained the absence of a strong adhesive minimum at contact. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Adhesion; Entropy; Surface forces
1. Introduction It is a general rule that chemically identical colloidal particles or surfaces adhere at equilibrium in a solvent. However, there are exceptions, particularly common in biological systems, where the adhesion is nonexisting or very weak. Traditionally one made the distinction between lyophobic (solvent hating) and lyophilic (solvent liking) colloids. In textbooks [1 – 3] the discussion of surface forces is based on the DLVO theory, in which a repulsive double layer and an attractive van der Waals’ force combine to generate a global
This article was originally submitted to the Per Stenius Special Issue. * Tel.: + 46-46-2229767; fax: +46-46-2224413. E-mail address: [email protected] (H. Wennerstro¨m)
minimum at contact between surfaces with a possible energy barrier further out. The DLVO theory has formed our expectations about surface forces in general although Verwey and Overbeek made it clear already in the title of their classical treatise [4] that the theory really apply only to lyophobic colloids. It took some time before the scientific interest was switched to the lyophilic colloids and the forces that make them stable. During the last few decades there has, due to advances in experimental techniques, appeared increasingly more detailed observations of strong deviations from the DLVO-description [5–7]. For lyophilic colloids these consists of short range repulsive forces. When observed in aqueous systems they have been called hydration forces [8]. Although experimentally well characterized the possible molecular origin of these forces has given
0927-7757/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 9 9 ) 0 0 2 2 1 - 6
H. Wennerstro¨m / Colloids and Surfaces A: Physicochem. Eng. Aspects 167 (2000) 209–214
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rise to a heated debate, and a consensus has not been reached. Prime examples of observations of hydration forces’ are between zwitterionic liquidlike bilayers [5,8], between solid mica surfaces [9] and between solid silica surfaces [10]. Together with Jacob Israelachvili we have argued [11] that repulsive forces between like colloidal objects have in general an entropic origin and this should also apply to the ‘hydration force’ [12,13]. To provide a complementary perspective on the role of entropy contributions to surface forces we will below analyze adhesion free energies between chemically identical surfaces in the presence of a medium.
2. Adhesion, interaction at contact, and surface forces According to the Dupre´ equation [1] the work of adhesion, W121, between two similar materials (1) in a medium (2) is related to the corresponding work of adhesion, W11 and cohesion W12, across a vacuum W121 = W11 + W22 −2W12
(1)
When analyzing the work of adhesion in a microscopic model we thus need to consider the interaction energy at contact between two halfplanes of material (1), the same quantity for the medium (2) and substract twice the interaction energy material-medium. In addition it is essential to include changes in entropy to obtain the free energy. This microscopic picture of the adhesion energy is based on the assumption that the minimum of the free energy occurs at molecular contact between the surfaces. Although this is usually the case there are many exceptions to the rule. In an alternative, and more general, formulation we can calculate W121 as minus the integral of the force per unit area, F(z)/A, between the surfaces in the medium W121 = − 1/A
&
f(z)dz
(2)
dm
Note that by convention an attractive force is considered to be negative the adhesion energy is the difference in the (free) energy at infinite sepa-
ration minus the energy at contact. In Eq. (2) dm is the distance at the free energy minimum, F(dm)= 0. This distance can equal the value at contact, dc, defined by the condition that the medium is absent between the surfaces or the minimum can occur at a larger separation. It follows from Eq. (2) that there is a direct connection between the surface force and the adhesion energy. When making a statement about one of them you necessarily implies something about the other. The work of adhesion is simply related to the interfacial free energy or the interfacial tension, (3)
2g12 = W121
For g12 we can identify three typical cases. The most common is that g12 is of a substantial magnitude, not more than an order of magnitude smaller than the surface tension so that g12 \ 1 mJ m − 2. A second possibility is that g12 is small but positive 0B g12 B 1 mJ m − 2. The low interfacial free energy is typically an indication that the medium is present to some extent between the surfaces at equilibrium, i.e. dm \ dc. A third possibility is that the surfaces repel one another to the extent that infinite separation represents the lowest free energy state. In the latter two cases we can define a contact free energy, W121(contact), by considering the work needed to bring the two surfaces from infinity to molecular contact W121(contact)= − 1/A
&
dm
dc
F(z)dz + 2g12
(4)
Using Eq. (4) we can now analyze all adhesion energies by only considering the direct contact between two surfaces of the material and the surface exposed to an infinite medium. At contact the interaction free energy is dominated by direct electrostatic and dispersion interactions across the interface, while for the free surface in a medium there can also be important contributions to the free energy from an increase in entropy due to the increased flexibility at a free surface. Below we will illustrate this point of view through three examples: (i) a surface with ionizable groups; (ii) a surface with molecular protrusions; (iii) a surface inducing solvent polarization.
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3. Surface ionization
knowing the surface charge density s using the Gouy–Chapman theory [14]
Most solid surface acquire a charge when immersed into water. Freshly cleaved mica have potassium ions electrostatically bound on the surface. In pure water these ions diffuse into the solution and are partly replaced by protons from the water giving the surface a net negative charge. This spontaneous dissociation process contributes to reducing the work of cleavage in water relative to dry air. In a similar way there are silanol groups at the surface of silica. For two silica surfaces in contact these silanol groups carry protons, while when exposed to an aqueous solution a dissociation can occur. The dissociation is driven by an increase in entropy, while it is gradually stopped as the surface charge density increases giving a high electrostatic energy. Let us make a quantitative estimate of the free energy charge associated with a surface dissociation. Consider the reaction scheme
s= − enAXA(eq)= 2kTkoro0 sinh(eFs/2kT)/e
AHs X As− +H+ b
(5)
Where subscripts s and b denote surface and bulk, respectively. At a given surface potential Fs, pH, and mole fractions of AH and A− at the surface the free energy change DGH in releasing one proton is DGH =2.3kT(pKa −pH) + kT ln XA/XAH − eFs, (6) where Ka is the intrinsic equilibrium constant at the surface. As the dissociative process continues both the ratio XA/XAH and the surface potential change. Equilibrium is reacted at a degree of dissociation XA(eq) where DGH of Eq. (6) is zero. The total free energy change is simplest obtained as the difference between final and initial state and DGdiss/area=kTnA{XA(eq)2.3(pKa −pH) + XA(eq) ln XA(eq) +XAH(eq) ln XAH(eq)} + Gel(Fs)/area,
(7)
where nA is the surface density of titratable groups A. The surface potential Fs we can estimate
(8)
The electrostatic free energy is in the Gouy– Chapman theory [15] Gel/area= 2nAXA(eq) × kT{ln[s + (s 2 + 1)1/2 + 1/s − (s 2 + 1)1/2/s}
(9)
with the dimensionless surface charge parameter s defined by s= sinh(− eFs/2kT)
(10)
To obtain an estimate of the typical magnitude of the free energy in Eq. (9) take as an example a surface with a density nA of one per 20 A, 2, pKa 5, approximately representing a solid fatty acid, exposed to an electrolyte solution at pH 7. Fig. 1 shows how DGdiss varies with the concentration of the 1:1 electrolyte. The free energy is typically in the range −5 to − 30 mJ m2 increasing in magnitude with increasing electrolyte concentration. This amounts to a substantial, negative, contribution to the work of adhesion which we can write approximately as a sum of a direct interaction term W121(int) and a surface dissociation term W121 : W121(int)+ 2DGdiss
(11)
In practice models of intermolecular interactions yield a positive effective interaction term [16] w= wAB − 1/2(wAA + wBB),
(12)
where wij is the pair interaction between molecules i and j. To what extent this result can be generalized to hold also for surface interaction W121(int) is not uneqivocally established, but a positive sign is expected and explicitly obtained for the van der Waals interaction [17]. Depending on the chemical nature of the surface W121(int) can vary substantially and it is concievable that DGdiss can provide a large enough negative contribution to change the sign of W121 in Eq. (11). However, for perfectly flat charged surfaces this does not seem to happen, while for approximately spherical colloidal particles with small areas of direct contact a redissolu-
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tion, so called repeptization, can occur at higher electrolyte concentrations [18]. To see the connection between the work of adhesion and the surface force we note that the long range double layer repulsion between two charged surfaces is precisely due to a decrease in DGdiss of Eq. (7). In the confined space between the surfaces there is less entropy for the counterions. Thus the molecular mechanism process that leads to a decrease in W121 through surface dissocation is also responsible for creating a repulsive component of the surface force.
the model is that at the surface of a monolayer formed by lipids in the liquid state individual molecules can protrude out from the surface. When the monolayer is exposed to a solvent the motion is mainly restricted by the free energy increase due to alkyl chain-solvent interaction. In the simple case of laterally independent protrusions the free energy for this degree of freedom is [12] G(z)/area = nkT{2 ln(kT/a) −ln[1− (1+ za/kT) exp(− za/kT)}
4. The protrusion model When the material is a solid we do not expect any major entropy increase when a surface is created, except for the ionization process described above. For a liquid material, on the other hand, there are major entropic contributions to the free energy and these could increase when an interface is created. One simple illustration of effect is found in the protrusion model [12] for the force between lipid bilayers. The basic feature of
(13)
where z is the separation between the surfaces, and a is the coefficient in the protrusion potential V(z)= az. For typical values of density of molecules n = 1/(50 A, 2), kT/a=1.5 A, and a separation at contact dc = 2 A, the free energy gain in going from contact to infinite separation is 9 mJ m − 2 per surface. In this model we have for simplicity, focused on only one particular molecular degree of freedom. In practice one expects several degrees of freedom per molecule to be affected by the presence of a second surface.
Fig. 1. The free energy change on dissociating a surface consisting of a solid fatty acid at a packing density of one molecule per 20 nm2 as a function of the bulk 1:1 electrolyte concentration c0. The system is at pH 7 and the intrinsic surface pKa 5. The curve is calculated using Eq. (7) with the electrostatic free energy calculated from the Gouy – Chapman theory.
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The simple model demonstrates that there can be substantial negative contributions to the work of adhesion due to the greater molecular freedom at a free surface. These entropic contributions to W121 are balanced by positive energy terms and the final outcome for the sign of W121 depend on the molecular details affecting both the entropic and the energetic contributions. The protrusions and other surface excitations contribute to both the lowering of the work of adhesion and to the surface force. We see that in analogy with the ionizing surfaces that it is the same mechanism that lowers the adhesion and generates the repulsive force. This coupling might appear self-evident but let us now analyze a model where such a coupling is actually absent.
5. The surface polarization model Shortly after the first publication of accurate measurements of repulsive forces between phospholipid bilayers by Parsesian et al. [5], Marc¸elja and Radic [19] published a theoretical derivation accounting for the short range repulsion in terms of a solvent structure-dipole polarization effect. The idea was subsequently clarified by Gruen and Marc¸elja [20], who were able to construct a microscopic model, based on ice with defects, and elegantly evaluate its thermodynamic properties. The essential physical idea in the model is that at short range surface solvent interaction induces a polarization of the solvent layer in direct contact with the surface. This polarization is propagated by short range solvent – solvent interactions to subsequent layers of the solvent. Since the bulk is unpolarized the polarization decays to zero for a free surface, while for two opposing surfaces a frustrated situation arises, when two polarized layers meet. This gives rise to a repulsive interaction. For a solvent that behaves as an ideal dielectric there is no orientational coupling between successive solvent layers and the effect comes from having some coupling between layers that are not dipolar in nature. In the Gruen – Marc¸elja model this coupling is introduced through the Bernal– Fowler rules of an ice lattice.
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For the free surface the existence of a polarization profile is a result of a compromise between two conflicting demands. On one hand the solvent molecules want to be randomly oriented to maximise entropy as in the bulk. On the other hand they have to adjust to the surface and by assumption this would tend to orient the molecules. Meeting these demands lead to an increase in free energy; the solvent is not as flexible as an ideal dielectric. Using eq. (3) of Ref. [20] for the free energy of the polarized liquid we find that the free energy to set up the polarization is Gpol/area= P 20orj/{2(or − o )o o0}.
(14)
Here P0 is the given surface polarization, j the decay length of the polarization profile, or is the relative dielectric permitivity at zero frequency, while o is that at high frequencies, vkT/&, and o0 is the permitivity of vacuum. In the limit or o Eq. (14) can be interpreted as the free energy/area of a capacitor of charge density s=Po and thickness j in a medium with permitivity o o0. To account for the observed forces P0 has to be of order 0.02 C m − 2 and the decay length j :2 A, with the estimated value of o= 6. With these values of the parameters Gpol/area: 1 mJ m − 2.
(15)
In contrast to the two previous examples there is actually a free energy penalty associated with setting up the solvent structure that is responsible for the repulsion. We can now estimate the work of adhesion at contact using Eq. (4), the experimentally measured hydration force, and the adhesion energy at equilibrium: 5×10 − 2 mJ m − 2 [22], and
&
W121(contact) =−
dm
4× 108 exp(− z/1.7)dz + 5
dc
× 10 − 5 J m − 2 = − 21 mJ m − 2
(16)
which is strongly negative. We could also try to estimate W121(contact) using the Dupre´ equation. In this approach the W121(contact) is given by the direct energetic interaction term plus possible entropic contributions from the medium solvent interface. The latter provides, in contrast to the first
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two examples, a positive contribution of 2 mJ m − 2 according to Eqs. (4) and (15). To match the value of − 21 mJ m − 2 in Eq. (16) we would require that the direct interaction contribution as W121(int) in Eq. (11) equals: − 23 mJ m − 2. As discussed in Section 3 the general expectation is that W121(int) is positive. A large negative value can’t a priori be ruled out but it certainly requires a non-trivial explanation. Additionally this unknown energetic contribution would by itself generate a repulsive force possibly leaving additional explanations of a repulsion unwarranted. We have previously pointed out [13] that an unsatisfactory feature of the Marc¸eljas theory of ‘hydration forces’ is that it is based on the concept that the solvent (water) can’t easily adjust to the conditions at the surface. Here we have made a quantitative analysis of this effect and demonstrated that there is a major problem in accounting for the observed highly negative value of the work of adhesion at contact between two liquid bilayers within this conceptual framework.
6. Conclusions We have analysed the relation between the work of adhesion of two similar surfaces and the force between the surfaces in a medium. By introducing the novel concept of a contact adhesion energy even when this is not the equilibrium state we have been able to demonstrate the mechanistic connection between forces and adhesion for specific models. This point of view is conceptually useful and it provides a criterion for mechanistic models of surface forces. In particular for repulsive forces it is not only sufficient to account for the distance dependence of the force, but one should also be able to relate this to the adhesive energy at contact. We have found that the Marc¸elja–Gruen theory of hydration forces lacks in this respect. Systems with adsorption from solution is an area where the general aspect developed in the paper can be of considerable help for the conceptual understanding of surface forces [21]. Clearly .
adsorption from bulk solution lowers the free energy of a surface. When now two surfaces are brought together under equilibrium with the bulk the integral of the force up to contact between surfaces has a higher value, less negative or more positive, than without the adsorbed species. For example, it is sometimes stated for surfaces with adsorbed polymers that the force is monotonicly attractive when polymers are allowed to equilibrate with the solution. This is an erroneous conclusion and our argument would give that since the polymers adsorbed to lower the free energy there is a free energy cost associated with forcing them away.
References [1] J. Israelachvili, Intermolecular and Surface Forces, 2nd edition, Academic Press, New York, 1991. [2] D.F. Evans, H. Wennerstro¨m, The Colloidal Domain, 2nd edition, Wiley, New York, 1999. [3] P.C. Hiemenz, R. Rajagopalan, Principles of Colloid and Surface Chemistry, 3rd edition, Marcel Dekker, New York, 1997. [4] E.J.W. Verwey, J.Th.G. Overbeek, Theory of stability of lyophobic colloids, Elsevier, Amsterdam, 1948. [5] D.M. LeNeveu, R.P. Rand, V.A. Parsegian, Nature 259 (1976) 601. [6] See Ref. [1], chapter 13. [7] See Ref. [2], sections 5.3 – 5.6. [8] R.P. Rand, V.A. Parsegian, Biochem. Biophys. Acta 988 (1989) 351. [9] R.M. Pashley, Adv. Colloid Interface Sci. 16 (1982) 57. [10] R.G. Horn, D.T. Smith, W. Haller, Chem. Phys. Lett. 162 (1989) 404. [11] J.N. Israelachvili, H. Wennerstro¨m, J. Phys. Chem. 96 (1992) 520. [12] J.N. Israelachvili, H. Wennerstro¨m, Langmuir 6 (1990) 874. [13] J.N. Israelachvili, H. Wennerstro¨m, Nature 397 (1996) 219. [14] See, for example, Ref. [2], eq. (3.8.15). [15] See Ref. [2]. eq. (3.9.8). [16] See, for example, Ref. [2], eq. (1.4.16). [17] See Ref. [1], chapter 11. [18] G. Frens, Faraday Discuss. 90 (1990) 143. [19] S. Marc¸elja, N. Radic, Chem. Phys. Lett. 42 (1976) 129. [20] D.W. Gruen, S. Marc¸elja, J. Chem. Soc. Faraday Trans. II 79 (1983) 225. [21] See Ref. [2], section 5.4 [22] J. Marra, J. Israelachvili, Biochemistry 24 (1985) 4608.