Langmuir 1997, 13, 5751-5755
5751
Critical Point Wetting for Binary Two-Phase Polymer-Solvent Mixtures on Solid Interfaces F. A. M. Leermakers,* L. J. M. Schlangen,† and L. K. Koopal Department of Physical and Colloid Chemistry, Wageningen Agricultural University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands Received March 10, 1997. In Final Form: July 22, 1997X Cahn argued that the wetting temperature, Tw, is always below the critical temperature, Tc, of a binary solvent mixture. A self-consistent field theory is used to show that this phenomenon is expected to be best observable for low molecular weight compounds. In polymer-solvent mixtures in contact with a rigid surface made of the same material as the polymer units, the difference Tw - Tc becomes minimal for an intermediate degree of polymerization N ) N** (∼65). The polymer wets the surface for N < N**, whereas for larger N the solvent is at the wall. Critical wetting is the rule; first-order wetting is only found for relatively short chains, 2 e N e 16.
Introduction Wetting processes are of large importance in many industrial processes and for numerous every day applications. Frequently, the wetting characteristics of a solvent mixture near interfaces are the key properties that need to be controlled, e.g., in printing technology, painting applications, material sciences (surface modification), and cleaning. Much is known about wetting. The scientific roots go back to Young1 and Laplace2 who already indicated that wetting is an interplay of the surface tensions of the interfaces involved: γRβ (fluid R-fluid β), γSR (solid S-fluid R), and γSβ (solid S-fluid β). For a partial wetting case we can find a contact angle ϑ of the three-phase contact region by a free energy balance argument, viz.
γRβ cos ϑ ) γSR - γSβ
(1)
The universal scaling dependence of the solid-liquid and liquid-vapor (or liquid R-liquid β) surface tensions on the temperature near the critical temperature, has prompted Cahn to predict that a wetting transition from partial to complete wetting should occur before the bulk passes through the two-phase to one-phase transition at the critical temperature.3 Critical point wetting is an important phenomenon as it is a universal mechanism to achieve complete wetting. The scaling arguments used by Cahn do not allow a precise estimate on how far from the critical temperature, Tc, one should expect to find the wetting temperature, Tw. A key observation is that the fluid-fluid interfacial tension γRβ approaches zero at the Rβ critical point as γRβ ∝ (Tc T)µ, where µ is known4 from experiments and theory to be about 1.3. As the R and β phases approach their critical point, the two phases become identical, and therefore γSR and γSβ should approach a common critical surface tension γc. Just below Tc the difference between γSR and γSβ will be due to the difference in the densities of the R phase and the β phase at the wall. It is found4 that near Tc, both the differences γSR - γc and γSβ - γc should vanish with a power law (Tc - T)ν, where ν ≈ 0.8. As ν < µ, it is clear that the contact angle ϑ approaches zero at a temperature * To whom all correspondence should be addressed. † Present address: Philips National Laboratory, Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands. X Abstract published in Advance ACS Abstracts, September 15, 1997. (1) Young, T. Phil. Trans. R. Soc. London 1805, 95, 65. (2) de Laplace, P. A. Me´ canique Ce´ leste, suppl. au X. Livere; Courcier: Paris, 1805. (3) Cahn, J. W. J. Chem. Phys. 1977, 66, 3667.
S0743-7463(97)00270-9 CCC: $14.00
Tw < Tc. It is important to note here that the wetting arguments of Cahn survive in a mean-field treatment of the system,4 where the predictions for the governing exponents are4 µ ) 1.5 and ν ) 0.5. In other words, in a mean-field approach it is expected that the Cahn wetting rule should apply. It is believed that usually the Cahn wetting rule applies but that there are scenarios to escape from the rule.5,6 These scenarios can take place under circumstances of competing long-range and short-range interactions. In the present analysis we apply a mean-field treatment to the problem in which we only include short-range interactions (the full problem is left for future work), so a full examination of the Cahn wetting rule is not within reach of the present study. It remains important, however, to predict which systems in a mean-field theory with only short-range interactions are likely to undergo a wetting transition that is well separated from the critical point and which ones are only wet near the critical point. These last systems are then potential candidates to escape from the Cahn wetting rule. Experimentally, wetting near the critical point has been found for several systems with compounds of low molecular weight; see, e.g., ref 7. No experiments on wetting near a critical point for polymeric systems are known to us. Systems in which there are one or more molecular components (polymeric systems) are potentially advantageous, because in these systems the solvent condition can be chosen such that the critical temperature is near room temperature. This will facilitate the experimental observations. To know the relative ease to induce a wetting transition for polymeric systems is certainly a relevant issue, because for many wetting applications chain molecules are an essential part of the formulation. If the Cahn wetting rule can be utilized in intermediate or high molecular weight systems, useful applications may exist. In considering the characteristics of the wetting transition,4 in general, a distinction has been made between critical wetting and first-order wetting. In the case of critical wetting the film thickness of the adsorbed layer at coexistence increases continuously with temperature and becomes infinite at the wetting temperature, Tw. In the case of first-order wetting, the film thickness at coexistence jumps discontinuously from a finite value (4) Schick, M. In Liquids at interfaces; Les Houches Session 48 NATO ASI; Charvolin Joanny, J. F., Zinn-Justin, J., Eds.; Elsevier Science Publishers b.v.: Amsterdam, 1990. (5) Nightingale, M. P.; Indekeu, J. O. Phys. Rev. B 1985, 32, 3364. (6) Ebner, C.; Saam, W. F. Phys. Rev. B 1987, 35, 1822. (7) Moldover, M. R.; Cahn, J. W. Science 1980, 207, 1073.
© 1997 American Chemical Society
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below the wetting temperature to an infinite value at Tw. Here we mean with “infinite” a macroscopically large value and that the thickness of this film is governed by the amount available rather than by some thermodynamic barrier. First-order wetting is linked to a prewetting step occurring off coexistence at temperatures below Tw. At the prewetting step, the film thickness changes from thin to thick but the film remains microscopic. In this paper a self-consistent-field theory applicable for complex inhomogeneous systems containing many different molecules is used to predict the wetting of a solid in the presence of a binary polymer solvent mixture. All (molecular) parameters in this model are measurable, which makes the method a powerful new tool in wetting studies. The main goal is to calculate the wetting temperature and compare that to the bulk critical temperature. The temperature difference is examined as a function of the molecular weight of the polymer in the binary liquid. Emphasis is given to chain length effects, and some attention is paid to the type of wetting. In our theory we use (local) mean-field and lattice approximations based upon the work of Scheutjens and co-workers.8,9 Other mean-field treatments of the wetting problem are available. Of particular interest are theories based upon the Landau-type phenomenological free energy functional. This approach can be used near the critical point where the interfaces are wide compared to the polymer size. The wetting of a polymer blend near a solid interface has been studied already more than a decade ago by Schmidt and Binder.10 Our theory is also applicable for systems further away from the critical point and is in many aspects similar to the off-lattice theory of Hong and Noolandi.11 Relatively recently, this work has been applied to wetting problems in polymer blends.12 One should realize, however, that in the numerical SCF theories one always has to discretize at some stage of the process. We choose to do this from the start, which helps us to solve easily any problems with the boundary conditions (i.e., what happens near the solid boundary), which is of significant importance in wetting studies. We further mention that in polymer blends both first-order and critical wetting behavior have been found by Monte Carlo simulation methods.13 The remainder of this paper is composed of two parts. In the following, we briefly mention the main characteristics of the theory. Then we will discuss typical results for a specially chosen system with a minimum number of parameters. SCF Theory A homogeneous mixture of a monomeric solvent with segment type V and a polymeric component with segment type C and chain length N, can be modeled, to a first approximation, with the Flory-Huggins theory.14 In this theory a regular spaced lattice with cell-size, l, is used. On the lattice there are sites with characteristic length such that they fit a polymer segment or a monomeric solvent molecule. Within this coordinate system chain conformations are analyzed with a Markov approximation, where chain back-folding is allowed. This implies a freely (8) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619-1635. (9) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980, 84, 178-190. (10) Schmidt, I.; Binder, K. J. Phys. 1985, 46, 1631-1644. (11) Hong, K. M.; Noolandi, J. Macromolecules 1981, 14, 747. (12) Yeung, C. Y.; Desai, R. C.; Noolandi, J. Macromolecules 1994, 27, 55-62. (13) Wang, J.-S.; Binder, K. J. Chem. Phys. 1990, 94, 8537-8541. (14) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.
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jointed chain approximation. Interactions of polymer units with solvent molecules are parameterized with the well-known Flory-Huggins (FH) dimensionless exchange interaction parameter χ. The χ parameter is defined for all unlike contacts and expresses the difference between the interaction between the unlike segments (e.g., A and B) and the like contacts, or mathematically χAB ) (Z/kT)[UAB - 1/2(UAA + UBB)]. In this equation, Z is the lattice coordination number, kT is the thermal energy UAA and UBB are the interaction energy for the like contacts, and UAB is the interaction energy of the unlike contacts. A FH parameter is usually a decreasing function of the temperature T. In this paper we will assume that χ ∝ T-1, implying that we assume that the interaction energy U is not a function of T. Below, typically the interaction parameter χCV has been varied, which implies a concomitant variation in the temperature in the system. Incompressibility constraints are used, which means that, on average, all lattice sites are occupied exactly once. In this mean-field treatment it is well-known that the critical value of χ for demixing is a function of the chain length N:14
χc )
(
)
1 1 1+ 2 xN
2
(2)
Thus for N ) 1 the critical FH parameter is 2. For polymers with infinite N, χc ) 0.5. This point is better known as the Θ temperature and has been tabulated for many polymer-solvent systems.15 For χ > χc a polymer rich phase can coexist with a solvent rich phase. To investigate the wetting behavior, these two coexisting phases are brought in contact with a solid surface. Due to the presence of the surface, an inhomogeneous situation occurs and the ordinary FH theory is no longer adequate. The model applied in this paper is a straightforward generalization of the FH theory for inhomogeneous polymer systems, originally proposed by Scheutjens and Fleer. Only the main characteristics of the theory will be presented here. For full details, we refer to the literature.16 We consider a 3D lattice system, similarly as in the FH treatment, with lattice layers perpendicular to the surface, numbered z ) 1, ..., M. In these layers L indistinguishable sites are present, which allows the use of volume fractions φ(z) in each layer. The computation of the density gradient information in the z-direction is the main objective of the model. Boundary conditions near z ) 1 and z ) M need to be specified. The solid (S) interface is modeled by fixing an S profile to be unity for z < 1 and zero elsewhere. In the system there are three segment types x, y ) S, C, and V and, consequently, three FH interaction parameters. Each segment type is conjectured to a dimensionless segment potential of the following form:
ux(z) ) u′(z) +
∑y χxy(〈φy(z)〉 - φby) ≈ u′(z) +
(
∑y χxy
)
2 1 2∂ φy(zl) φy(z) + l - φby (3) 2 3 ∂(zl)
where the superindex b indicates the value of the segment density in the bulk solution. The angular brackets indicate the averaging of the density over three layers z - 1, z, z + 1, and mathematically, this averaging means that the (15) Polymer Handbook, 3rd ed.; Brandrup, I., Ed.; John Wiley & Sons: New York, 1989. (16) Fleer, G. J.; Scheutjens, J. M. H. M.; Cohen Stuart, M. A.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman & Hall: London, 1993.
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local second derivative of the density profile is included in the potential as expressed in eq 3. In eq 3 u′(z) is a potential coupled to the incompressibility constraint. It takes a value such that the resulting density profiles have the property that the volume fractions in each layer add up to unity. In eq 3 we have no explicit contributions to long range Van der Waals forces. If necessary, Van der Waals interactions, e.g., due to the surface, can be included, but we choose here not to do this. Next, for each segment type a free segment distribution function Gx(z) ) exp(-ux(z)) is defined. For chain segments, these Boltzmann weights are generalized. If segment s of molecule i is of type x then Gi(z,s) ) Gx(z). Using these quantities, one can compute so-called chain end distribution functions (cedf), Gi(z,s|1) and Gi(z,s|N), by a recursive scheme:
Gi(z,s|1) ) Gi(z,s)〈Gi(z,s-1|1)〉 Gi(z,s|N) ) Gi(z,s)〈Gi(z,s+1|N)〉
(4)
These equations are started by Gi(z,0|1) ) Gi(z,N+1|N) ) 1 for all z > 0. The eqs (4) imply a first-order Markov process for generating and weighting the various chain conformations in the external potential fields. The cedf’s are Green’s functions that obey the appropriate diffusion equation.17 On a lattice, this diffusive process is given by eq 4. The angular brackets represent a second derivative with respect to the space coordinate similar to that in eq 3. In this scheme, the segment densities follow from the so-called composition law:
φi(z,s) ) Ci
Gi(z,s|1) Gi(z,s|N)
(5)
Gi(z,s)
where the normalization constant Ci fixes the total amount θi ) ∑sφi(z,s) of molecule i in the system. The excess amount of component i follows θσi ) θi - Mφbi . Equations 3-5 are such that usually they only can be solved numerically. From the ranking number dependent segment densities one can easily compute the overall densities per molecule φi(z) and per segment type φx(z) in the system. For an acceptable solution the densities and the potentials are consistent with each other. Moreover, the volume constraint is obeyed. Using the equilibrium densities and potentials, one can calculate various thermodynamic quantities such as the interfacial tensions. The chemical potentials are found by applying the Flory-Huggins theory to the bulk solution. With respect to the bulk situation, the presence of an interface creates an excess free energy. This excess free energy per unit area, Aexc, is a function of the densities φx(z) and φi(z) and free segment distribution functions Gx(z):
Aexcl2 ) kT 1
∑x ∑y
2
′
[
∑z ∑i
φi(z) - φbi
-
+ Ni
∑x φx(z) ln Gx(z) +
]
′χxy{φx(z)(〈φy(z)〉 - φby) - φbx(φy(z) - φby)} (6)
In eq 6 the primes indicate that the segment type of the surface component S is not included in the summations. For liquid-liquid interfaces Aexc is calculated by taking on both lattice boundaries reflecting conditions and fixing the amount of one of the components to such a value that approximately half the system is filled with that component. Then the interface between the two liquids is (17) Edwards, S. F. Proc. Phys. Soc. 1965, 85, 613.
positioned approximately near z ) M/2. In the case of a solid surface we have an absorbing (near z ) 0) and a reflecting (near z ) M) boundary condition. Polymers near interfaces have been studied extensively with this theory.16 It is well-known that a solid boundary limits the number of conformations that a polymer chain can locally adopt. The possible number of ways to put a segment-segment bond near the boundary is less than in solution. This results in an entropy loss for each bond that is in the first layer next to the wall. This entropy loss needs to be compensated for by interaction energy, or else the polymer molecules will not accumulate at the interface. This critical adsorption energy depends slightly on the lattice type used. Results and Discussion To simplify the parameter choice, we consider in this paper a special case of a two-component solution composed of a monomeric solvent (V) and a polymeric component (CN). This binary solution is always considered below the bulk critical temperature. It is brought in contact with a rigid solid surface made of segments of type C. This means that the chemistry of the surface is identical to that of the chains. Thus, only one short range interaction parameter is present in the system, namely the χ parameter between the C and V units. We explicitly assume that the interface between the surface and the solution remains infinitely sharp (the surface is not critical). As the CN chain is chemically similar to the surface (matched surface), one might be inclined to expect that the polymer molecules will preferentially adsorb onto the interface and that the solvent is displaced from it. However, this is not necessarily true. As told above, polymers only adsorb (accumulate at the surface) when the energy gain upon adsorption is more than a critical value.16 For the present model, the adsorption energy is only a function of the interaction parameter between C cr (ads) ) (3/2) ln(3/2) ≈ 0.61. This and V and is given by χCV result is, in principle, true for infinitely long chains.16 For finite chain lengths this value is only slightly larger. Therefore, near a matched surface, the polymer may not necessarily be preferentially at the surface, nor is complete wetting automatically the case. In the case of a finite contact angle there are two stable layer thicknesses at the coexistence point (saturation point): a thin polymer film at the surface in contact with the solvent rich phase and a macroscopically thick polymer layer. For both layers we compute the excess free energy per unit area and label these by “thin” and “thick”, respectively. The excess free energy per unit area between exc ) γs the fluid phases is also calculated. Notice that Athin exc exc - γSV, Athick ) γs + γCV - γSC, and ACV ) γCV where C indicates the polymer rich phase and V the solvent rich phase. The contact angle, ϑ, can be computed by18,19
cos ϑ )
exc exc - Athick Athin
Aexc CV
+1
(7)
To show the effect of chain length on the wetting behavior, we first investigate the case when the χ parameter is high (χCV ) 2; temperature is constant) and the chains accumulate near the interface. Results shown (18) Schlangen, L. J. M. Adsorption and wetting; experiments, thermodynamics and molecular aspects. Ph.D. Thesis, Wageningen Agricultural University, 1995. (19) Schlangen, L. J. M.; Leermakers, F. A. M.; Koopal, L. K. J. Chem. Soc., Faraday Trans. 1996, 92, 579-587.
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Figure 1. Contact angle ϑ as a function of the chain length N. Interaction parameter χCV ) 2. The line is drawn to guide the eye. For other parameters, see text.
Figure 2. Comparison of the FH interaction parameters for the bulk critical point (dashed curve) and for the wetting transition (solid line) as a function the indicated function f(N) ) (1/2)(1 + 1/xN)2. Note that f(1) ) 2, f(2) ) 1.46, f(3) ) 1.24, ..., f(∞) ) 0.5. In panel b an expanded scale is used for the region 0.5 e f(N) e 1. The lines are drawn to guide the eye. For further parameters, see text.
in Figure 1 indicate that the contact angle is an increasing function of the chain length N. From extrapolation of the curve in Figure 1, we further notice that for N < 4 complete wetting occurs at this temperature. For N ) 3 the critical FH parameter equals 1.24, which corresponds with a larger temperature than χCV ) 2. This result is thus in line with Cahn’s wetting argument. From eq 2 we know that the critical temperature is a decreasing function of the chain length and thus for fixed interaction parameters the shorter chains are closer to the critical point than the longer ones. Also in line with the Cahn argument, our model shows that for χCV < 2 the critical chain length for which wetting is found becomes larger18 (not shown). As N can change only with integer numbers, it is more convenient to compute the critical FH parameter for wetting as a function of the chain length N. Results of this type of calculation are collected in Figure 2. In this figure we give the χ parameter for which the wetting transition from partial to complete wetting occurs (solid line) and the critical bulk χ parameter (dashed line) as a function of c (N)). By realizing that χ ∼ T-1, we (1/2)(1 + 1/xN)2 ()χCV can conclude from Figure 2 that the difference between the wetting temperature and the critical bulk temperature is a nonmonotonic function of the chain length. For low molecular weight the χCV for the wetting transition is high and the (oligomeric) component wets the surface. For intermediate and high molecular weight the wetting χCV is close to the bulk critical point. The decrease in χCV for wetting also means that for a certain N the adsorption energy becomes below the value needed for the polymer to overcome the conformational entropy loss. Consequently, the polymer is depleted from the surface and the solvent is enriched near the interface (adsorption transition). The adsorption transition leads to a wetting transition: at sufficiently high N, at the wetting temperature, the surface is wet by solvent and not by the
Leermakers et al.
polymer. In Figure 2b the arrow with label N** indicates the crossover from the condition that the polymer wets the surface to the case that the solvent is doing this. Let us now consider the type of wetting transition that is found. As mentioned above, one distinguishes critical wetting and first-order wetting. In critical (second-order) wetting the amount of wetting component adsorbed at coexistence at the interfaces diverges continuously with temperature at the wetting temperature. For first-order wetting the adsorption at coexistence jumps discontinuously from a finite value below the wetting temperature to an infinite amount at and above this temperature. To obtain the order of the wetting transition, adsorption isotherms are calculated as a function of chain length and χCV values just above the values at which the wetting transition occurs. The calculations show that for our type of systems critical wetting is found more frequently than first-order wetting. With the present choice of parameters first-order wetting is only found in a small range of chain lengths 2 e N e N*, where N* ) 16 is a tricritical point connecting the first-order and second-order regimes. For N ) 1 (C1) our model reduces to a lattice gas model.20 For the system of monomers wetting only occurs at high values of the interaction parameter χCV ) 8. In these cases the numerical solution of the equations indicates layering transitions that show up as a Van der Waals loop with the addition of exactly one equivalent lattice layer to the adsorbed amount. On top of these steps there is a clear indication that the wetting is second order (critical wetting). For an intermediate chain length range N** > N > N* the polymer wets the surface critically, meaning that as the wetting temperature is reached, the adsorbed amount at coexistence increases continuously. Also, obviously, in this case the adsorbed amount at the wetting temperature increases continuously when the chemical potential is increased from a value below the binodal value to this value (no prewetting). For N > N**, however, the solvent wets the surface, again critically. In the literature it is known that a change in the surface field can bring about a change from wetting by one phase to the other.20 In our case the surface field varies with the temperature by way of the FH parameter. For the present conditions, N** ∼ 65. For N > N** we find that the wetting temperature (χCV) is a less strong function of the molecular weight of the polymeric component than of the critical FH parameter. Therefore, the gap between the bulk critical temperature and the wetting temperature increases again with increasing chain length. Near N** the difference between the wetting conditions and the critical conditions is very small indeed. In our numerical procedure we cannot determine whether these conditions become identical as suggested in the literature.20 As N can only change in a discrete manner, we believe that in our study there always will remain a small difference between the wetting and the critical temperatures. Analysis of Figure 2b shows that in the range 6 < N < 16, to a good approximation, χCV(wetting) ) -0.73 + (1 + 1/xN)2. The main conclusion of this paper is that if one looks for systems that have a wetting temperature well below the bulk critical temperature, one should preferably use low molecular weight ones. In the oligomer region the difference Tw - Tc increases with the decreasing molecular weight. Polymers will experience an interface as sharp as when the intrinsic width of the interface is much smaller than the radius of gyration of the molecules. Not before the width of the interface becomes larger than the size of the molecules can we expect that the surface tension (20) Nakanishi, H.; Fisher, M. E. Phys. Rev. Lett. 1982, 49, 1565.
Critical Point Wetting for Polymer-Solvent Mixtures
follows the universal scaling laws. This implies that polymeric systems must be much closer to the critical temperature before Cahn’s argument starts to be operative, as compared to chain molecules with a low molecular weight. Another aspect of using polymers to study wetting is that polymers have a loss of conformational entropy upon adsorption and they only adsorb if the adsorption energy is larger than a critical value. By increasing the temperature toward the bulk critical temperature, it is possible that the adsorption energy falls below this critical value and the solvent displaces the polymers from the surface, so that at the wetting temperature the solvent wets the surface. In this regime the difference Tw - Tc increases with increasing N. This indicates that the depletion of the polymer near the surface assists in reducing the difference between the interfacial tensions of the polymer rich and solvent rich phases with the solid S. For the polymers that have been studied, the wetting near the critical point can occur by either a critical wetting
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or a first-order transition. In our systems we observed for oligomers first-order wetting, whereas for longer chains critical wetting occurs. Apparently, there must be a sizable difference between the wetting and critical temperature (the driving force), as was also noticed by Schick.4 In our systems, this sets an upper limit to the chain lengths for which first-order wetting is found. For other systems with different surface properties, or when, e.g., Van der Waals forces are included, first-order wetting transitions may be more easily observable, especially when complete wetting is induced far away from Tc. Above, a very simple system has been considered. A more systematic study is, however, possible. Moreover, it is feasible to include long range interaction parameters in the modeling. It would be of interest to study within our model the influence of, e.g., Van der Waals and/or electrostatic forces on the wetting characteristics of a system. Work along these lines is in progress. LA9702704
