Ind. Eng. Chem. Res. 1998, 37, 427-434
427
Osmotic Stresses and Wetting by Polymer Solutions R. M. Ybarra and P. Neogi* Chemical Engineering Department, University of MissourisRolla, Rolla, Missouri 65409
J. M. D. MacElroy Chemical Engineering Department, University College Dublin, Dublin, Ireland
Polymer solutions do not wet the substrate, even when the solvent is wetting. It is suggested here that it becomes difficult to squeeze polymer molecules into thin films found near the contact line due to adverse changes in the polymer entropy. Models are used to show that as the film thins a potential barrier is encountered due to the entropy changes in the polymer. This is observed in the calculated values of disjoining pressure as a function of film thickness. Equations arising from force balances are used to predict the contact angles in terms of a model disjoining pressure. The results can vary with the nature of constraints on the system, and care has been exercised to be faithful to the experiments reported earlier. Qualitatively, the dependence of the contact angles on various parameters are correctly predicted. Introduction In a previous work by Nieh et al. (1996) we have reported that dilute solutions of dibutyl phthalate (DBP) containing polystyrene did not wet glass and showed small contact angles, even though pure DBP wet glass. This was unexpected because the surface tension of DBP is ∼33 mN/m and the critical surface tension of polystyrene is ∼33-35 mN/m, and we suggested possible mechanisms for this lack of wetting. We questioned if adsorption of the polymer could account for this behavior. As polymer adsorption is insensitive to molecular weights (Hervet, 1990), we tried polystyrene with a larger molecular weight and obtained lower contact angles and even saw some restoration of the expected wetting behavior. We also changed the substrate from the high-energy glass to low-energy poly(methyl methacrylate) (critical surface tension ∼ 39 mN/m) and found that the contact angles decreased but did not disappear. Finally, although the polymer adsorption isotherms reach saturation at very low polymer concentrations (Hervet, 1990), the contact angles continued to increase with concentration even up to 40 wt %. The Young-Dupre equation predicts that as adsorption decreases the solid-liquid interfacial tension, it would also decrease the wettablity. This is true unless material moves ahead of the drop onto the solid surface to decrease the solid-vapor interfacial tension by an adequate amount. This may not happen, and we show in our model that the system can support a precursor film and still not be wetting. Although adsorption, and some effects of adsorption may exist, we are left with the distinct possibility that the main reason for the lack of wetting is the unadsorbed polymer in the solution. In Figure 1 we depict schematically the difficulty of squeezing a polymer molecule into the corner of a drop, a process strongly dependent on the molecular weight of the polymer. This aspect can also be couched as a thin film problem. The situation resembles the contact angles on foam films reported among others by Princen and * To whom all correspondence must be addressed.
Figure 1. A schematic view of a polymer molecule near a contact line is shown. Obviously it is difficult for the polymer molecule to squeeze into a narrow corner, a difficulty which grows with increasing molecular weight. The same is also shown through thin films. The entropy of the polymer is lowered on film thinning, and thus film thinning is neither entropically nor energetically favored.
Frankel (1971), Dimitrov et al. (1990), and Huisman and Mysels (1969), although the contact angles in foams are a lot smaller. When the film is thick, the polymer molecule can assume many configurations, but when the film is thin, both the number of configurations and the accompanying entropy decrease, the free energy increases, and the film thinning is hindered. The condition of equilibrium inside a drop is
p+φ)C
(1)
where p represents the pressure, φ the potential due to molecular forces, and C a constant. Since condition applies on the surface, we set p ) 2MγL, the Laplace pressure, and φ ) Π, the disjoining pressure:
2MγL + Π ) C
(2)
If we consider a wedge as shown in Figure 2, we find that to the left the curvature 2M ) 0 and the film is sufficiently thick such that the disjoining pressure is zero, leading to C ) 0. Equation 2 becomes
-γL
d2h +Π)0 dx2
(3)
where h represents the local film thickness and x the coordinate in the tangential direction. Since we calcu-
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428 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998
balance the pressure in the drop at the contact line
Π(h0) )
Figure 2. A wedge showing how the contact angle λ, film thickness h(x), and the thin precursor film of constant thickness h0 are defined. The fact that the contact angle is taken to be the extrapolation of the macroscopic shape to the horizontal is to be noted.
late the disjoining pressure assuming plane parallel films, the slope of the wedge must be small, an assumption we also made in writing the expression for the curvature. Multiplying with dh/dx and integrating the result, we find
dh ) -xE dx E ) λ2 -
2 γL
(4a)
∫h∞Π dh′
(4b)
where we have used the boundary condition that dh/dx ) -tanλ = -λ far to the left where the film is thick. Rearranging eq 4a we find that
dh ) -dx xE
(4c)
Equation 4c clearly shows that integration is difficult if for some value of h ) h0, E ) 0. If E ) 0 at some h ) h0, then the integration results in a film profile reaching h ) h0 only as x goes to infinity (far right in Figure 2). If this is the only admissible view of the structure of the contact line, then the condition for a nonwetting liquid is E ) 0 or
λ2 )
2 γL
∫h∞Π dh 0
(5)
For small angles λ we note that cos λ = 1 - λ2/2 + ..., and one can express eq 5 as
cos λ = 1 -
1 γL
∫h∞Π dh 0
(6)
Equation 6 is the Frumkin-Derjaguin equation (Frumkin, 1938; Derjaguin, 1940; Derjaguin et al., 1987) discussed by Churaev (1994) and Hirasaki (1990), which we have derived for the first time using disjoining pressure as a body force. Our derivation also shows clearly the constraints that exist behind the result and emphasizes the conditions under which the system turns wetting. Equation 6 provides us with an equation relating λ to h0, where h0 must be g0. It does not provide a means to calculate either one independently. The present formulation also clarifies the definition of the contact angle λ as the angle obtained by extrapolating the macroscopic profile to the solid surface. Experimentally, the contact angles are just these as photomicrography, conventionally used in these measurements, cannot see anything in the thin film region. In the experiments the drop shapes are seen as segments of spheres, and the force in the thin film must
2γL R0
(7)
where R0 represents the radius of curvature of the bulk of the drop. R0 depends on the drop volume V and the contact angle λ. Thus, like eq 6 we again have an equation relating h0 to λ, with V added on this time. Equation 7, which is similar to the condition of equilibrium in foam films near Plateau borders discussed by Toshev and Ivanov (1975) and Bergeron and Radke (1992), also says that such contact angles are nonlocal properties. Since thin films containing polymers have seen considerable investigations, we now look at the other cases studied and how they are related to this system. One important area has been in the study of polymer stabilized colloids discussed in detail by Napper (1983) and Scheutjens and Fleer (1982). When two colloidal particles approach each other, a thin film forms between them. If the approach occurs slowly, the film can equilibrate and the polymer molecules in the film can stay within the film or move into the bulk solution as the system attempts to reach a minimum energy configuration. (The first case leads to depletion stabilization and the second to depletion flocculation.) We will call such a system an open system. In contrast, if the two particles approach each other rapidly, then the polymer molecules may not have enough time to diffuse to redistribute between the film and bulk following the conditions of equilibrium. That is, this case is a constant concentration case. Such a system is generally not investigated in this area and will be called a closed system. A proper thin film is without ends, such as those investigated by Yerushalmi-Rozen and Klein (1995a,b,c, 1996) and Martin et al. (1996), for the particular case of polymer solutions where the solid substrate is covered by grafted polymer chains. Other films are finite and when they undergo thinning one can assume that the total moles of solvent and polymer remain fixed. In determining the properties of such films, Halperin et al. (1985) and Boudoussier (1987) use a constant volume constraint. These films are not Klein’s films. Boudoussier (1987) showed that constant volume polymer films could exist in different thicknesses and polymer concentrations and more than one type can coexist with one another. The latter case implies a lack of wetting. Fondecave and Borchard-Wyart (1997) observed experimentally that at higher polymer concentrations the polymer solution was nonwetting (even though the pure solvent was wetting) and the equilibrium contact angle increased with an increase in polymer concentration, a trend also observed by Nieh et al. (1996). At lower polymer concentrations Fondecave and Bouchard-Wyart (1997) observed a thin film (∼1.5 nm) around the base of the drop. The film itself was nonwetting and of finite extent. Their system was grafted on a solid substrate, the solution was a highmolecular-weight polymer dissolved in a low-molecularweight polymer, and both were poly(dimethylsiloxane). These are constant volume systems and not Klein’s thin films. The present day theory and the experiments of Klein and co-workers, show that a great many effects take place when one grafts a polymer to a substrate and
Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 429
the solution contains a low-molecular-weight oligomer besides the polymer. These complications are much beyond the scope of this work. We concentrate here on relating the thin film properties to wettability and contact angle. We have already shown how the contact angle is related to the thin film properties. In the next section we examine the features of a polymer containing thin films necessary to obtain a robust model to work with. We use these features in the last section to obtain expressions for the contact angle in terms of known or observable quantities. We note that since wetting plays an important role in two-phase fluid flow and distribution in porous media and in coating processes, the ability to affect the wettability by choosing polymeric additives is significant.
Hamaker forces V(-AH/h3), where V is the total volume of the thin film and is independent of the film thickness h. Differentiating with respect to the number of moles of polymer np and the number of moles of the solvent ns, we get the two chemical potentials
µop + RT ln φ∞ ) µop - RT ln K* +
µos + RT ln (1 - φ∞) ) µos +
K)
φ φ∞
(8)
where φ and φ∞ are the polymer volume fractions in the film and in a reservoir. It is also possible to calculate a change in energy when the polymer is brought from the bulk into the thin film. This is expressed as a decrease in configuration entropy ∆S ( 0 as appropriate for intrinsically wetting liquids described here. In an open system, which eventually evolves, the excess energy is compensated by changes in the polymer concentrations and entropy that enable the chemical potentials of the polymer to balance. If we calculate the partition coefficient of the polymer, eq 8, it unfortunately forces the solvent partition coefficient at (1 - φ)/ (1 - φ∞), and does not allow it to be independent. Thus, the system will not equilibrate without an additional degree of freedom, which means that the pressure must change inside the film. The total free energy of a thin film in an open system is made up of the entropy of dispersion, the changes in the configurational entropy, the effects of the excess pressure in the thin film Vp2, and the effect of the
ln
1 - φ∞ K*φ∞ ) N ln φ 1-φ
(12)
where vp/vs, the ratio between the molar volumes, is approximated as N, the number of repeat units, and for p2
p2 -
AH h3
)
FpRT K*φ∞ FsRT 1 - φ∞ ln ln ) (13) M φ Ms 1-φ
Equations 8 and 12 show that K* is not equal to the partition coefficient K, except at infinite dilution or when K tends to 1. The net force on the film, therefore, becomes
Π ) -p2
(14)
The pressure p2 is negative and the difference between pressures in the film and the reservoir is balanced by the osmotic pressure difference. But the difference between the pressure in the film against that in the atmosphere is not balanced unless a force of -p2 is applied at the interface, which amounts to an increased potential as seen in eq 14. Equation 14 has been derived by using the basic definitions introduced by Everett (1972) and is equivalent to the lattice theory model of Scheutjens and Fleer (1982) if a geometric mean of the solvent volume fractions in the vertical direction (lattices) is replaced with the arithmetic mean. Simple simulations with representative numbers show that such an approximation is reasonable if the polymer solution is dilute, which holds in the present case. Details are given in the Appendix. Further, when K tends to 1 and φ tends to φ∞, eq 14 reduces to eq 10. If K ) 0 and φ ) 0, one has the osmotic stress model of LeNeveu et al. (1976) and Leikin et al. (1993) for the case where the steric restriction excludes all polymers. It is important to note that in the thin film literature the disjoining pressure is defined by the pressure in the bulk liquid in equilibrium with the thin film. In the open system this leads to zero excess pressure in the thin film but a pressure of -p2 in the bulk, leading eventually to the same results using the same procedure. Finally, if we substitute eq 13 into eq 14 and compare with the results of the closed system in eq 10, it is easy to see that the two differ because the energy in the open
430 Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998
Figure 3. Equations 10 and 15 (closed system) are shown for a single concentration, φ ) 0.05, and two different Hamaker constants. The bold and dashed lines are for molecular weight of 45 000 and the other two are for a molecular weight of 280 000. The effects of the polymer dominate at large values of h and the effects of Hamaker forces dominate at small values of h. In general, the effects of polymer molecular weights are low. (HamakerCassasa closed.)
system has been minimized through the considerations of the chemical potentials and pressure. Models Specific models for K* or ∆S are now considered. We have used below a random walk model by Cassasa (1967), a scaling theory model by Daoud and de Gennes (1977), and a hard spheres model by Glandt (1981) to capture concentration effects. These are described in more detailed later and it suffices to state here that the effect of confinement to be expected in thin films is the main feature sought in any of these models. The parameters represent closely polystyrene in dibutyl phthalate investigated by Nieh et al. (1996). (a) Hamaker-Cassasa. The disjoining pressure is given by eq 10 (closed system) or eqs 13 and 14 (open system) and from Cassasa (1967):
K* )
8
∞
∑
1
π2m)0(2m + 1)2
{
exp -
}
(2m + 1)2π2 2Nb2 4
3h2
(15)
The following values have been used in the calculations below: AH ) Hamaker constant 10-20 and 10-21 J φ ) volume fraction of the polymer in the film φ∞ ) volume fraction of the polymer in the reservoir RT ) universal gas constant times absolute temperature (25 °C) 2478 J/g‚mol Fp ) density of the solid polymer 0.8 g/cm3 M ) molecular weight 45 000 or 280 000 N ) number of steps = M/molecular weight of the monomer, )M/104 for polystyrene m ) M/N ) 104 b ) step size 5 × 10-8 cm Fs ) density of the solvent 1.043 g/cm3 Ms ) molecular weight of the solvent 278.4 The volumes in the solution have been assumed as additive and the moles of polymer negligible compared to the moles of the solvent. Plots have been drawn in Figure 3, for φ ) φ∞ ) 0.05 using eqs 10 and 15 (closed system) to show that positive values of the potential are
Figure 4. Equations 13-15 (open system) are shown for a single concentration, φ∞ ) 0.05, and two different Hamaker constants. All observations are same as before, except that the heights of the barriers increase. (Hamaker-Cassasa open.)
indeed obtained. The peak height increases with the polymer concentration. When φ ) 0.1 (not shown), the peaks increase by about an order of magnitude. At large film thicknesses only the effects of the polymer are seen, and at small thicknesses only the Hamaker interaction exists. The differences between the two cases of two molecular weights are minimal. Figure 4 has been drawn for an open system by combining eqs 13-15 and holding φ∞ constant at 0.05. The extreme left-hand side in the figures is free of polymers and the peak heights are increased and shifted to the left, because it becomes more and more difficult to sustain a film against the suction from the bulk produced by the osmotic pressure. One difficulty with this model is that Cassasa’s calculation of polymer entropies is based on the assumption that the polymer configurations can be approximated by the statistics of random walk, which is invalid particularly when the gap widths are small. (b) Hamaker-Daoud-de Gennes. In the random walk a polymer is allowed to penetrate itself, a feature which occurs more frequently in a thin film. The appropriate form has been given approximately at small separations by Daoud and de Gennes (1977). Equation 10 and
[ (hb) ]
K* ) exp -N
5/3
(16)
have been used to plot in Figure 5 (closed system) for φ ) 0.05. This response is independent of molecular weight. The peaks are a little higher than in Figure 3, which is the corresponding case from Cassasa’s model. For φ∞ ) 0.1 (not shown) the peak heights increase by a factor of about 3. Figure 6 has been drawn for the open system for φ∞ ) 0.05. The plots are equally insensitive to molecular weights. For φ∞ ) 0.1 (not shown) the peak heights increase by a factor of 2. In both cases of Cassasa and Daoud-de Gennes, the open system displays much higher peaks. (c) Hamaker-Glandt. A flaw in the above models for our purposes is that only the intramolecular entropies are considered and not the intermolecular (communal) ones. Thus, the results are valid only in dilute systems. (We do not need the intermolecular or the
Ind. Eng. Chem. Res., Vol. 37, No. 2, 1998 431
Figure 5. Equations 10 and 16 (closed system) are shown for a single concentration, φ ) 0.05, and two different Hamaker constants. The equations are rigorously independent of molecular weights of the polymer. (Hamaker-Daoud-de Gennes closed.)
communal entropy but the effect of confinement on this entropy.) Glandt (1981) has presented an analytic result for hard sphere molecules which is
K1 ) )
π 2µ
(17)
K0 ) 1 - λ
(18)
µ>2
13 π 3 µ - 4µ2 + 16 6 µ
[
K2 ) )
K ) K0 + K1C∞ + K2C∞2 + ...
]
113 π2 840 µ
1
