Osmotic repulsion force due to adsorbed surfactants

Colloids and Surfaces B: Biointerfaces 91 (2012) 137–143

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Osmotic repulsion force due to adsorbed surfactants Alexander J. Babchin a , Laurier L. Schramm b,∗ a b

School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel Saskatchewan Research Council, 15 Innovation Blvd., Saskatoon, SK, S7N 3X8, Canada

a r t i c l e

i n f o

Article history: Received 10 June 2011 Received in revised form 4 October 2011 Accepted 27 October 2011 Available online 3 November 2011 Keywords: Colloid stability Surfactant Osmotic force Disjoining pressure

a b s t r a c t When considering interaction forces in surfactant-stabilized colloidal dispersions a factor that has been rarely discussed is the possible effect of osmotic force due to overlapping adsorbed surfactant monolayers. In the present work, the osmotic repulsion force is built-in on the basis of DLVO mechanics and based on Fischer’s consideration of the analogous situation for adsorbed polymer layers on solid surfaces [E.W. Fischer, Kolloid Zeitschrift 160 (1958) 120–141] and on Langmuir’s earlier concept of osmotic pressure excess due to overlapping adsorption layers [I. Langmuir, J. Chem. Phys. 6 (1938) 873–896]. The advanced method for calculation of the net repulsion force in overlapping surfactant monolayers is developed and applied to real adsorbed surfactant systems. We show that the value of disjoining pressure can reach values as high as 8 MPa for the condition of fully overlapping surfactant adsorption layers, based on the calculation of the first virial term of the general expression for osmotic pressure. Thus, we have shown that osmotic forces can be substantial at distances of close interfacial approach, and that they can easily be of the same or greater order of magnitude than the forces that have been more conventionally considered. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Surfactants have a well-known ability to adsorb at interfaces and stabilize, or contribute to the stabilization of aqueous and nonaqueous dispersions of all kinds, including suspensions, emulsions, and foams. In the simplest cases, the charged head groups of adsorbed ionic surfactant molecules contribute to electrostatic repulsions among dispersed particles, droplets or bubbles. This happens for interfaces that, compared with the charge on the surfactant head groups, were originally of like charge, opposite charge, or electrically neutral: • where the interface is originally of like charge, the adsorbed ionic surfactant acts to increase the total interfacial charge (such as when an anionic surfactant is added to a dispersion of air bubbles in aqueous solution to stabilize a common foam), • where the interface is originally of opposite charge, the adsorbed ionic surfactant acts to neutralize and then reverse the charge (such as when an anionic surfactant is added to a suspension of positively charged particles in a selective flotation process), and • where the interface is originally electrically neutral, the adsorbed ionic surfactant creates interfacial charge (such as when an anionic surfactant is added to a dispersion of mineral oil droplets in aqueous solution to stabilize an oil-in-water emulsion).

∗ Corresponding author. Tel.: +1 306 933 5402; fax: +1 306 933 7519. E-mail address: [email protected] (L.L. Schramm). 0927-7765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfb.2011.10.050

In each of the above cases the electrical contributions to the stability contributed by the ionic surfactant to the emulsion, suspension, or foam can be understood and quantitatively described using the stability of lyophobic colloids developed by Derjaguin and Landau [1,2], and Verwey and Overbeek [3], and commonly known as DLVO theory. Using DLVO theory, one can calculate the energy changes that take place when two dispersed species (particles, droplets or bubbles) approach each other by estimating the potential energies of attraction (London-van der Waals dispersion, VA ) and repulsion (electrostatic including Born, VR ) versus interparticle distance. These are then summed to yield the total interaction energy, VT = VR + VA . More advanced approaches also include terms for steric and hydration (solvation) forces. The DLVO theory has been well developed for two special cases, the interaction between parallel plates of infinite area and thickness, and the interaction between two spheres; and although the original calculations of dispersion forces employed a model due to Hamaker, more precise treatments now exist [4]. Less easy to understand and quantitatively describe have been the cases where nonionic surfactants (or an amphoteric surfactant in Zwitterionic form) adsorb at interfaces that are electrically neutral. Nevertheless it is well known that colloid stability can be created or enhanced through the adsorption of surfactants that are neutral (or net neutral) at interfaces that are also electrically neutral. Similarly, ionic surfactants can adsorb at the surfaces of dispersed species having the opposite charge and stabilize those dispersed species, even at the point of zero charge. In the case of emulsions, surfactant adsorption can not only promote stability but

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frequently also determines which of the two liquid phases will form the dispersed phase and which the continuous phase. A number of empirical “rules of thumb” have been developed for some of these situations (including Bancroft’s rule, the oriented wedge theory, the hydrophile–lipophile balance (HLB [5,6]), and the volume balance value [7]). Where surfactant adsorption promotes colloidal stability and the stabilizing force is clearly not electrostatic in nature then one typically considers the possibility of the so-called “steric” effects. Although not the focus of this paper, there is also the non-DLVO, short-range repulsive force due to perturbations of the molecular ordering at hydrated surfaces. This hydration force arises when surfaces contain water molecules adsorbed by hydrogen bonding, and can be either attractive or repulsive depending on the situation [4]. Longer range hydration repulsive forces provide extension of the electric double layer and DLVO theories for the case of high surface potentials and are due to the dielectric saturation of water in the strong electric field near an interface [8–10]. The ability of surfactants to contribute a steric stability force is usually attributed to one or more of the following: lowering the effective Hamaker constant, physically restricting the freedom of motion of surfactant chains that are adsorbed on approaching dispersed species (volume restriction), or mechanically preventing the approach of surfactant chains that are adsorbed on approaching dispersed species (mechanical barrier necessitating desorption) [11,12]. In this paper we consider an osmotic mechanism for the colloidal repulsion force arising from the interaction of non-ionic surfactant-adsorption layers on dispersed species as such species approach each other. While creation of a self-consistent theory, accounting for all aspects of the colloidal stability problem is beyond the scope of the present work, the osmotic component of the repulsion forces should always be present for different surfaces, surfactants and solvents, and can be quantitatively described. This is not a new idea, but it has been largely overlooked in mainstream colloid science. In 1958, Fischer [13] introduced the osmotic pressure as the principal reason for repulsion between two surfaces carrying adsorbed polymer layers. Although, Fischer described the Gibbs free energy equation for overlapping polymer layers in integral form, he stopped-short of calculating the repulsion force explicitly as function of the distance between the planar surfaces. In a 1987 review article on the same subject de Gennes [14] independently refers to the repulsion force due to the osmotic pressure of overlapping polymer layers. The present work is based on Fischer’s initial approach, which we extend for overlapping surfactant adsorption layers, taking a more general approach for osmotic pressure determination and by which one can calculate the mechanical force of repulsion as developed within DLVO theory, although for the case where electric double layers are absent.

2. Osmotic pressure in overlapping surfactant adsorption mono-layers Langmuir was the first to introduce the excess of osmotic pressure in interfacial layers compared to bulk solution [15]. The excess of osmotic pressure he equated with the repulsion force acting on solid plates carrying adsorption layers, for the case where such adsorption layers overlap. Although Langmuir applied this concept for overlapping electric double layers, the concept of osmotic pressure has much broader application, including adsorption layers in non-electrolyte solutions [13,14,16–20]. The present paper undertakes to extend Langmuir’s initial approach to the general case of adsorbed surfactants. In what follows we consider the mechanical repulsive force for the case of two overlapping surfactant adsorption layers carried by approaching flat parallel surfaces.

For a dilute surfactant solution, below the critical micelle concentration (cmc), the osmotic pressure is represented in the standard form: Posm (bulk) = RTC(bulk)

(1)

where R is the gas constant, T is the absolute temperature, and C(bulk) is the bulk surfactant concentration in (mol/L). As the concentration of surfactants within adsorption layers is not expected to be low, the expression for the corresponding osmotic pressure should be presented with virial terms included: Posm = RT {C + A C 2 + A C 3 + · · ·} A ,

(2)

A

where etc. are the virial coefficients, and C is the surfactant concentration within an adsorption layer. By introducing the classical equations for the osmotic phenomena it is implied that macroscopic osmotic pressure determination is still applicable within an interfacial layer of adsorbed surfactants, and the surfactant molecules are much larger than the solvent molecules. According to Langmuir the excess of the osmotic pressure within the interfacial layer over its bulk value should be considered for the mechanical force, thus it is represented as a difference between Eqs. (2) and (1). However, for the case of dilute bulk solution C(bulk)  C, so that the small term can be neglected and Eq. (2) provides the value for the osmotic pressure within the interfacial layer. This assumption simplifies the problem, as we avoid the application of Eq. (2) to the bulk solution with another set of unknown virial coefficients, as would be the case for high bulk surfactant concentration (above the cmc) [21]. At this point the following clarification should be made. In the case of non-overlapping adsorption layers, the Posm from Eq. (2), acting on the plate, is fully compensated by the molecular forces exerted by adsorbed surfactant on the same plate. Otherwise, there would be a non-zero net force available for work without any physical or chemical changes in the system. In fact, only an excess of osmotic pressure in overlapping layers over the equilibrium pressure from Eq. (2) is expected to provide a net mechanical force acting on the plates. The surface excess concentration of non-interacting adsorbed surfactant, Cs (mol/cm2 ), adsorption layer (non-interacting) thickness, Ho , and corresponding volume concentration are inter-related through: Cio =

Cs Ho

(3)

where Cio is the volume concentration of surfactant within a nonoverlapping adsorption layer. When the gap between two approaching surfaces is 2Ho the surfaces are located in close proximity with each other, but do not yet overlap. At this point the volume concentration of surfactants in interlayer of 2Ho thickness is still determined by Eq. (3), and the system is still in thermodynamic equilibrium with the bulk solution. As soon as the surfaces move any closer, however, the surfactant concentration within the gap will increase as well, along with the appearance of the resulting repulsion force. Allowing the gap to vary and considering H as half thickness of the variable gap, the osmotic pressure can be calculated for the corresponding volume concentration within the gap, Ci = Cs /H as:



Posm(i) = RTCs

A Cs A Cs2 1 + + + ··· H H2 H3



(4)

In order to sustain this new osmotic pressure, Eq. (4), a mechanical force must be applied from the opposite side of the plate. This mechanical force, F, related to unit area is obtained as the difference between osmotic pressure, described by Eq. (4) and the initial osmotic pressure in the gap determined by Eqs. (2) and (3). We

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stress here again that equilibrium osmotic pressure, Eqs. (2) and (3) is fully compensated by the mechanical action of the solute on the plate, so this value must be subtracted from the total osmotic pressure, represented by Eq. (4). This subtraction allows for determination and calculation of the net mechanical force induced by the system on the plates for the case of overlapping surfactant adsorption layers. Thus, the net force is given by the following formula: F = Posm = RTCs

 1

H

 +A Cs

−

1 Ho



1 1 − 2 H2 Ho



 + A Cs2

1 1 − 3 H3 Ho



 − ···

(5)

The case under consideration assumes a constant value of surface concentration Cs , independent of the gap width 2H. The final value of half gap thickness at the condition of full overlap is H = Ho /2, so the first term of the virial expansion for the repulsion force can be expressed as:  = F = Posm

RTCs Ho

(6)

 where Posm denotes the part of osmotic pressure attributed to the first term of the virial expansion. The first term of osmotic pressure, as represented by Eq. (6), represents the lower limit for the osmotic pressure estimate. Some information is lost due to omitting the second and third terms in Eq. (5), however Eq. (6) contains no unknown coefficients and is reliable for making an order of magnitude, lower limit estimate. The algebraic formulas presented above, including Eq. (5), outline an approach to the problem’s solution, or zero approximation solution. For this approach to be valid, the surfactant layer should not be too dense, but allow the entry and residence of solvent molecules between surfactant molecules in the adsorption layer. Otherwise the osmotic pressure concept is not applicable to the present problem. Ideally, the full virial expansion would be used in any calculations, but the virial coefficients are difficult to access. We expect that the three first terms in Eq. (5) will have the same order of magnitude in the condition of high surfactant concentration as in the adsorption layer. As the second and third virial coefficients are not determined, we employ only the first term of the virial expansion, Eq. (6), for the estimation of the osmotic pressure and the repulsion force orders of magnitude. Additional explanation is provided in Appendix A. The thickness of the surfactant layer Ho may exceed the thickness of surfactant monolayer, allowing for a diffusion part of the adsorption layer. Unfortunately it is difficult to indicate the basic scale of Ho , as was done in DLVO theory on the basis of the Debye–Hukkel length of the electrical double layer. Thus, determining the actual scale of Ho remains an experimental task. This is discussed further in Section 4.

3. Comparison of the osmotic pressure and DLVO approaches It is useful to introduce the disjoining pressure concept, which served the application of the DLVO theory so well, to the problem under consideration. Indeed, by subtracting Eqs. (4) from Eq. (5) we could follow an analogy to DLVO theory, by determining disjoining pressure at any virtual plane between two plates carrying overlapping layers. The disjoining pressure in the virtual plane within the gap, following a DLVO theory analogy, would be given as the difference between osmotic pressure and the Maxwell stress component of the electric field at the location of the same plane [22]. This value is constant at any location within the gap and determines the disjoining pressure, as introduced by Derjaguin prior to the original

139

DLVO development. At the plane of symmetry, with the Maxwell tensor component equal to zero, osmotic pressure alone, as introduced by Langmuir, determines the repulsion force and thus the disjoining pressure. In the problem under consideration, the overlapping of non-ionic adsorption layers of surfactants, especially in non-polar solvents or in aqueous solutions having electrolyte concentrations of 1 M or higher, the effective electric field is assumed to be zero. Osmotic pressure in this case is caused by the interplay of molecular forces between solid plates, adsorbed surfactants, and solvent molecules, and is the macroscopic manifestation of hydration (solvation) effects quantified in Eq. (2). Compared with the application of DLVO theory for adsorbed ionic surfactant molecules, in the osmotic theory case for adsorbed non-charged adsorbed surfactant molecules, the role of the electric field in DLVO theory is taken over here by the field of molecular forces, acting on the surfactant molecules by the molecules of the solid plate and by solvent molecules. In the case of unperturbed surfactant adsorption layers (the non-overlapping condition) all molecular forces and osmotic pressure are balanced and the solid plate does not experience any resulting force. This equilibrium state is characterized by following equation: Posm + Fm = 0

(7)

where Fm is the force, induced by surfactant and solvent molecules on the plate and is compensated by osmotic pressure. This sum (with corresponding signs of terms) becomes non-zero in the overlapping case and determines the resultant repulsive force between the plates, Fr as: Posm (H) + Fm = Fr

(8)

Following the DLVO analogy, Eq. (7) states that: Fm = −Posm (Ho )

(9)

so the resultant repulsive force acting on the plate is finally determined as: Fr (H) = Posm (H) − Posm (Ho )

(10)

The derivation of Eq. (5) has accounted for this procedure. This argument can be supported in a less heuristic and more pictorial way. Derjaguin, in his first publication, which started the DLVO theory development, provided the formula for the electrostatic repulsion force arising due to overlapping electric double layers in the case of small and constant surface potentials [23]. He based his calculation on taking the difference between Maxwell stress components on the outside and the inside of the plate immersed into electrolyte solution, interacting with an identical plate inside the gap. His result was supported by Langmuir [15], who provided another way of arguing and equated the repulsion force to the excess osmotic pressure in the plane of symmetry, but obtaining the equivalent result for the same problem. In short, both Derjaguin and Langmuir subtracted the initial state of nonoverlapping double layers from the result of their overlap. Eqs. (5) and (10) are derived on the same basis. The initial state in the problem under consideration is described by Eqs. (2) and (9). Not doing the subtraction as it was done in Eq. (5), would stop at Eq. (4) as a result. But by doing so, we would have to invoke a step function behavior of repulsive force at separation distance 2Ho . The repulsive force is zero at 2Ho and is a finite, large value at any distance smaller than 2Ho . The transition from Eqs. (4) to (5) removes this contradiction. Thus, the DLVO and Langmuir approaches are advanced by including in the net repulsive force the difference between the osmotic pressures in overlapping surfactant monolayers and the osmotic pressure in non-overlapping surfactant monolayer. Finally, as the maximum thickness of interacting adsorption layers is restricted by the length of two surfactant molecules (for

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monolayer adsorption), the variation of osmotic pressure within this length cannot be provided by the macroscopic approach applied. Thus, the averaged value as described by Eq. (5) determines the overall disjoining pressure, which should be only gap dependent. 4. Comparison with the Derjaguin adsorption component The disjoining pressure induced by overlapping surfactant monolayers is so large, that the diffuse part of the adsorption layer is usually not considered. However, the adsorption component of disjoining pressure as introduced by Derjaguin et al. actually ignores the first layer of adsorbed solute molecules and instead focuses on the solute molecules in the diffuse layer, but in the range of dispersion forces near the interface (see Chapter 5 in [2], Chapter 6 in [24], and [25]). In these original papers by Derjaguin et al. they deal with the effect of the diffuse layer solute molecules by applying a correction to the initial Hamaker constant, due to the identical functional dependence on the gap which separates the approaching interfaces. The mechanism is described in detail by Boinovich [26]. Consider the gap between non-overlapping adsorbed surfactant monolayers. The surfactant molecules in the bulk solution will interact with the surfactant molecules that are adsorbed at the solid surfaces. The gap thickness, d, in this case can be defined as the distance between the surfactant tails or heads facing the solution from the two approaching surfaces. Following Boinovich [26], the concentration of surfactant molecules at different locations within the gap can be expressed by a Boltzmann distribution as follows:



C(x) = C(bulk) exp

−U(x)



kT

(11)

where x is the distance coordinate within the gap, k is the Boltzmann constant, and U is the interaction energy. Assuming that the energy of molecular interactions at the symmetry plane within the gap U* < kT, the osmotic pressure given by an excess or a deficit of concentration can be written as in [20] as: ∗ = RTC(bulk) Posm

 −U ∗  kT

(12)

If we take, for the purposes of illustration, a bulk surfactant concentration near the CMC point as C(bulk) = 8 × 10−3 mol/L, and U* = −0.5 kT, the osmotic pressure in the symmetry plane, which ∗ determines the repulsion force, will be Posm = 10 kPa. Such a value is at least two orders of magnitude lower than the repulsion force induced by overlapping adsorbed surfactant monolayers (see Table 3). Nevertheless the effect is important, as repulsion acts at larger distances and can appear as a negative Hamaker constant, contributing to overall colloid stability. At this point we should stress that the estimation of repulsion forces by calculating osmotic pressure in the symmetry plane, as suggested by Langmuir [15], is applicable to adsorption layers of different natures. At U* = −0.1 kT, the expected disjoining pressure drops to 2 kPa. By no means are these values smaller than disjoining pressure for the overlapping electric double layers in the case of small potentials in the symmetry plane. For univalent, symmetric electrolytes the corresponding osmotic pressure is determined as in [15] as: Posm = RTC(bulk)

 −U ∗ 2 kT

(13)

where U* is determined as a product of the electric field potential in the symmetry plane within the gap and the electric charge of the ion. At field potential values below 25 mV the value of U* is also small with respect to kT.

However, there is an essential difference between electrolytes and non-electrolytes, the latter including non-ionic surfactants. For electrolytes, Eq. (13), osmotic pressure is always positive due to the square power of the non-dimensional energy (U*/kT). For nonelectrolytes, osmotic pressure may have either sign, as determined by Eq. (12). In the case where surfactant molecules are attracted to the first dense monolayer, the variable value of energy U < 0. However, if the surfactant molecules are repelled from the dense monolayer, U > 0, and osmotic pressure within the diffuse part of the adsorption layer becomes negative. Corresponding to the work of Boinovich [26], this effect is equivalent to an increase in the value of the Hamaker constant, contributing to destabilization of a colloidal system. As dense monolayers cause greater repulsion when overlapped, any flocculation structure will be mechanically weak in strength. In which case, using surfactant to cause the flocculation, if desired, may not be the most practical choice. The distance of separation for the adsorption component of disjoining pressure is assumed to be much larger than the thickness of a dense surfactant monolayer, so the gap can be estimated to be between 15 and 50 nm. At these separation distances the potential U* in the plane of symmetry can definitely be assumed to be smaller than the molecular (or ionic) thermal energy kT. 5. Calculations Information on the area per adsorbed surfactant molecule can be obtained from the Gibbs adsorption equation, which describes the lowering of surface free energy due to monolayer surfactant adsorption [11]:  =−

 1  RT

d d ln Cbulk

 (14)

where  is the Gibbs surface excess of surfactant (mol/cm2 ), Cbulk is the solution concentration of surfactant, and  the surface free energy (mN/m). According to Eq. (14) the surface excess at surface saturation in a filled monolayer ( m ) is related to the slope of the linear portion of a plot of surface free energy versus the logarithm of solution concentration. The value of  is close to its maximum value when the solution concentration of surfactant is high enough that the surface tension of water has been reduced by 20 mN/m. As a result, a common reference point for comparing the efficiency of surfactants is to take the concentrations at which the surface tension has been reduced by 20 mN/m; usually denoted C20 . This is almost always less than the critical micelle concentration. The corresponding value of  is denoted  20 . The values of  20 and  m are so close that they are often used interchangeably. This is of practical importance because experimental results for  20 are more readily available than those for  m . Ideally, one would like to have specific experimental  m results for surfactant adsorption at the air/aqueous interface to relate to the colloidal stability of bubbles in a foam, at the oil/aqueous interface for the colloidal stability of emulsions, and at the solid/aqueous interface for the colloidal stability of suspensions. Unfortunately, of the limited amount of experimental information on  20 and  m , much of it comes from measurements of surfactant adsorption at the air/aqueous interface (principally because it is for these interfaces that the appropriate laboratory equipment is the most readily available and the surface free energies are the easiest to measure). Table 1 shows some representative values for  20 for polyoxyethylenated alchohol surfactants in air/aqueous systems drawn from Refs. [27,28]. The results of Miller [29], recalculated in terms of surface excess concentration, suggest that the situation for anionic surfactants at the air/aqueous interface, whether in

A.J. Babchin, L.L. Schramm / Colloids and Surfaces B: Biointerfaces 91 (2012) 137–143

141

Table 1  20 values for polyoxyethylenated alchohol surfactants in air/aqueous systems. Surfactant

Interface

Temperature (◦ C)

 20 (×1010 mol/cm2 )

References

C6 H13 (OC2 H4 )6 OH C10 H21 (OC2 H4 )6 OH C12 H25 (OC2 H4 )6 OH C16 H33 (OC2 H4 )6 OH C12 H25 (OC2 H4 )7 OH C16 H33 (OC2 H4 )7 OH C12 H25 (OC2 H4 )9 OH C16 H33 (OC2 H4 )9 OH C12 H25 (OC2 H4 )12 OH C16 H33 (OC2 H4 )12 OH C16 H33 (OC2 H4 )15 OH C16 H33 (OC2 H4 )21 OH

Air/aqueous Air/aqueous Air/aqueous Air/aqueous Air/aqueous Air/aqueous Air/aqueous Air/aqueous Air/aqueous Air/aqueous Air/aqueous Air/aqueous

25 25 25 25 23 25 23 25 23 25 25 25

2.7 3.0 3.7 4.4 2.6 3.8 2.3 3.1 1.9 2.3 2.1 1.4

[27] [27,28] [27,28] [27,28] [27] [27] [27,28] [27,28] [27,28] [27,28] [27,28] [27]

*(OC2 H4 ) = POE = polyoxyethylene. Table 2  m values calculated from literature results for adsorbed surfactants at surface saturation in gas/liquid, liquid/liquid, and solid/liquid systems. Surfactant

Interface

Temperature (◦ C)

 m (×1010 mol/cm2 )

References

Sucrose monolaurate Sucrose monolaurate Dodecyl hexaethylene oxide Dodecyl hexaethylene oxide Sodium dodecyl sulfate Glyceryl-1-monostearate n-octanol Dodecyl amine Dodecylammonium Sodium dodecyl sulfonate Triton X-100 Dodecylammonium acetate

Air/water Air/1.0 M NaCl Air/water Air/1.0 M NaCl Air/oil Oil/aqueous Oil/aqueous Mercury/aqueous Mercury/aqueous Solid/aqueous Solid/aqueous Quartz/aqueous

25 25 25 25 50 70 20

2.8 3.0 2.5 3.8 2.7 4.4 6.9 0.5–0.9 2.1–2.7 1.4 3.5 2.5

[29] [29] [29] [29] [30] [30] [31] [32] [32] [33] [34] [35]

25 25 25

the presence or absence of high electrolyte concentrations, is very similar (Table 2). In terms of liquid/liquid system examples, Lucassen-Reynders and van den Tempel [30], and Aveyard and Briscoe [31], reported some areas per molecule for adsorbed surfactants at surface saturation in oil/aqueous systems. Their results, recalculated in terms of surface excess concentration are shown in Table 2. Leja [32] has reported some surface excess concentrations of ionized and nonionized surfactants adsorbing from aqueous solution to a negatively charged mercury interface, which are also included in Table 2. In terms of solid/liquid system examples, Somasundaran et al. [33], Levitz et al. [34], and Fuerstenau [35] reported some adsorption results for anionic and non-ionic surfactants at surface saturation in solid/aqueous systems. The values of  m calculated from their results are also shown in Table 2. According to Rosen [27] the effectiveness of surfactant adsorption (amount adsorbed at surface saturation) for oil/aqueous systems is generally very similar to that for air/aqueous systems under the same conditions, and the typical range of values for  20 ≈  m for surfactants is 1–5 × 10−10 mol/cm2 . The results shown here, although not necessarily representative of all surfactant classes, interface types, or system conditions, suggest that the effectiveness of surfactant adsorption at air/aqueous, oil/aqueous, and solid/aqueous systems under the same conditions may all be generally similar and within the generalized range reported by Rosen.

Equating Cs =  m , the excess volume concentration of surfactant within the interfacial region, in mol/L, is expressed as: C=

1000 m Ho

The corresponding repulsion force as determined by Eq. (6) is then: F=

1000 RT m Ho

6. Discussion For the system of two identical surfaces separated by a gap of width Ho and containing an overlapping surfactant layer within the gap, the colloidal stability condition can be written in the form of the inequality: Posm − A

 m (× 10 1 1 5 5

mol/cm )

10

Ho (×10 50 15 50 15

m)

>0

(17)

or

Table 3 Posm values calculated from Eq. (6) for different values of  m and Ho . 2

(16)

Based on the ranges suggested from the literature, the practical range of values to be expected in practice for Posm can be calculated. For this purpose we will take the practical range for  m to be 1–5 × 10−10 mol/cm2 . A typical value for the thickness of an adsorbed surfactant monolayer is 50 × 10−10 m or less [27], so we will take the practical range for Ho to be 15–50 × 10−10 m. Using these values and the approximation described by Eq. (6) produces the Posm results shown in Table 3.

6()Ho3

10

(15)

A < 6() Ho3 Posm Posm (MPa) 0.51 1.72 2.53 8.41

(18)

where A is the Hamaker constant related to the molecular interaction between solid surfaces through the surfactant overlapped interlayer. From the scenarios represented in Table 3 consider the example of the lower concentration and smaller separation gap (line 2 in Table 3), which indicates an osmotic pressure of 1.72 MPa.

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Substitution of the corresponding parameters into Eq. (18) leads to the following requirement for Hamaker constant, in order to have colloidal stability: A < 10−19 J. This condition is easily satisfied in most known cases, except for noble metals [25], even without having surfactants to lower surface energies. For all the other cases represented in Table 1, colloidal stability is even more pronounced, sometimes by orders of magnitude, as for the cases described by lines 3 and 4 of Table 3. If anything, the values calculated above (Table 3) provide a conservative estimate, for practical purposes, of the osmotic stabilization force. It can be seen that the achievement of repulsive forces, per unit area, of the order of several MPa or more will have a huge and normally dominating effect of the stability of particles, droplets, or bubbles containing monolayer surfactant adsorption layers since in most systems for which disjoining pressures have been either measured directly, or calculated for electrostatic stabilization, even the attainment of several hundred kPa is normally taken as a sufficient criterion for “stability” of a dispersion. In support of this statement, we cite here only a few of the many examples that could be drawn from the literature. Takamura et al. [36–38] used electrokinetic measurements and DLVO theory to calculate the disjoining pressures between sand and bitumen surfaces in aqueous solutions with anionic surfactant adsorption at both surfaces. Their maximum values were in the range 120–140 kPa for thicknesses of about Ho = 4–7 nm. Such calculations are necessarily limited by the forces being recognized by the theory, which have not previously included osmotic forces due to adsorbed surfactant. More appropriate comparisons are with direct physical measurement, such as with the disjoining pressures measured for foam films. Radke et al. [39,40] measured disjoining pressures for foam films stabilized with anionic surfactants. Their maximum values were in the range 12–30 kPa for thicknesses of about Ho = 5 nm. Similarly, Bergeron et al. have measured disjoining pressures for foam films stabilized with surfactants and have found values ranging up to about 3 kPa at about Ho = 10 nm for nonionic surfactants [41] and values ranging up to about 30 kPa at about Ho = 15 nm for cationic surfactants [42]. Other investigations have found lower values. For example, Exerowa and Kruglyakov report disjoining pressures for foam films stabilized with saponin of up to about 100 Pa at about Ho = 30 nm [43]. Having shown that the osmotic force can be of comparable, if not greater magnitude than electrostatic and dispersion forces it would be interesting to determine whether osmotic forces may provide the key to understanding the stability of Newton black films, which are to date not well understood and not explained by electrostatic or dispersion considerations [44]. The present results may also have implications beyond that of colloidal stability. In rock mechanics, for example, it is known that a solution of surfactants deposited on a solid (rock or other) surface can reduce the solid’s mechanical strength because the solution can enter fractures [45]. This effect, known as the Rhebinder effect is generally understood to result from the adsorption of the surfactants at the solid surfaces within the fractures, thereby altering the wettability of the surfaces. The change in wettability then causes additional water to invade the fractures, enabling stress corrosions and/or a capillary pressure force that can break-open the solid. The present results suggest that there could be an osmotic component in addition to the forces caused by the capillary imbibition component in the Rhebinder effect. Finally, the interactions among surfactant monolayers (and bilayers) can be very complex, including any or all of electrostatic, van der Waals, hydrophobic, steric, osmotic, and other interactions. In future work it would be valuable to make a quantitative com-

parison of these interaction forces for specific surfactant systems of differing structures and packing densities. 7. Conclusion When considering interaction forces in surfactant-stabilized colloidal dispersions, much attention has been paid to the effects of electrical, dispersion, steric, and hydration (solvation) forces. Typically ignored or overlooked, has been the possible effect of osmotic force. We have shown that osmotic forces can be substantial at distances of close interfacial approach, and that they can easily be of the same or greater order of magnitude than the forces that have been more conventionally considered. Acknowledgements The authors are grateful to Prof. Ludmila Boinovich, Corresponding Member of the Russian Academy of Sciences, for her advice and help during our work on this paper. The authors also thank the journal’s anonymous reviewers of this paper for their helpful perspectives and suggestions. Appendix A. For the convenience of readers we provide a mini-review on the application of van der Waals equation of state for the quantification of osmotic pressure in non-ideal chemical solutions. We were able to trace this problem to as far back as the paper by Porter, published in 1917 [46]. Newer publications [47,48] support the same approach. The equation of state for the osmotic pressure can be represented as:



an2 Posm + 2 V



(V − b) = nRT

(A1)

where V is the volume of the solution, n is the number of moles, a is the constant that introduces the effect of the attraction between solute molecules, and b is the constant associated with the volume excluded by the solute molecules, and where V cannot be smaller than b. Since (n/V) = C and can be expressed in (mol/L), Eq. (A1) can be expended into virial form as follows:





Posm = RT C + b −

a RT



C 2 + b2 C 3 + · · ·



(A2)

Comparing Eq. (A2) with Eq. (2), the second and third virial coefficients are determined as follows: a A = b − (A3) RT A = b2

(A4)

In earlier work [46] it was assumed that A” is always positive, but there is no solid proof of this. Fischer [13] also assumed this coefficient to be positive and picked the second virial term, as in our Eq. (2), for estimating the disjoining pressure in overlapping adsorbed polymer layers on solid surfaces. While considering a problem identical to Fischer’s, de Gennes preferred to rely for the third virial term as an estimate of the disjoining pressure. Indeed, the third virial coefficient is always positive, Eq. (A4), by analogy between the van der Waals and osmotic equations of state. This analogy also indicates that for the high concentration of molecules in the adsorption layer all three virial terms should have the same order of magnitude. So, both Fischer and de Gennes are correct in their estimates. The problem is that these coefficients are largely unknown and would have to be guessed-at. Only the first virial term is well defined and does not contain unknown coefficients.

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