Unmixing of Polymer Blends Confined in Ultrathin Films: Crossover

6544

J. Phys. Chem. B 2005, 109, 6544-6552

Unmixing of Polymer Blends Confined in Ultrathin Films: Crossover between Two-Dimensional and Three-Dimensional Behavior† A. Cavallo, M. Mu1 ller,*,‡ and K. Binder Institut fu¨r Physik, WA 331, Johannes Gutenberg UniVersita¨t, D-55099 Mainz, Germany ReceiVed: September 14, 2004; In Final Form: NoVember 10, 2004

The interplay between chain conformations and phase separation in binary symmetric polymer mixtures confined into thin films by “neutral” hard walls (i.e., walls that do not preferentially attract or repel one of the two components of the mixture) is studied by Monte Carlo simulations. Using the bond fluctuation model on a simple cubic lattice in the semi grand canonical ensemble, we locate the critical temperature of demixing via finite size scaling methods for a wide range of chain lengths (16 e N e 256 effective monomers per chain) and film thicknesses (2 e D e 19 lattice spacings). Simultaneously, we investigate the geometrical structure of the chains, showing that despite using melt densities there are pronounced “correlation hole effects”, in particular for the smaller values of D. Also the components of the radius of gyration and end-to-end distance parallel and perpendicular to the confining walls are analyzed and their scaling behavior is studied. Evidence is presented that for strictly two-dimensional polymers (as occur for D ) 2) the average number of intermolecular contacts scales with chain length N as zc ∝ N-3/8 and therefore the critical temperature scales as Tc ∝ N5/8, whereas for values of D that exceed the excluded volume screening length, zc remains nonzero for N f ∞, and hence Tc ∝ N. However, strong deviations from the Flory-Huggins theory occur as long as the unperturbed chain dimension exceeds D, and the critical behavior falls in the universality class of the two-dimensional Ising model for any finite value of D.

I. Introduction Many polymeric materials are multicomponent systems. By “alloying” chemically different polymers, one wishes to design a material that can combine favorable characteristics of the individual components.1-5 A growing number of applications in materials science use polymer blends in thin film geometry to exploit their adhesive properties, or utilize them as lubricants, protective coatings against corrosion, or insulating layers in microelectronic devices.6-9 The current trend toward nanotechnology also creates a strong interest in thin films, i.e., films whose thickness is comparable to or even smaller than the size of a polymer coil under bulk melt conditions. However, even in the bulk the miscibility of polymers is a complicated problem,1,-5,10-13 and the situation is even more involved for polymer blends confined in thin films.14,15 Note that already a single polymer chain adsorbed on a wall or a homopolymer melt interacting with a wall poses challenging problems to the description in the framework of statistical thermodynamics because of the interplay between enthalpic forces the wall exerts and the entropy changes the polymer coils suffer upon confinement.7-10,16,17 For a polymer blend in thin film geometry, the additional complications due to the finite thickness of the film and the preferential adsorption of one of the components cause an intricate interplay of finite-size and wetting effects with phase separation,14,15,18 which, in general, is only incompletely understood. In the present work, we consider a selected aspect of this general problem, namely the phase separation of a symmetric polymer mixture confined by perfectly flat and structureless †

Part of the special issue “David Chandler Festschrift”. Department of Physics, University of Wisconsin, 1150 University Avenue, Madison, WI 53706. ‡

walls into thin films. We present simulation results of a coarsegrained model and interpret them by a variety of phenomenological arguments. Note that we deliberately use the advantage of computer simulations to choose a model that omits many complicated effects whichsfor our purposesswe consider as being side issues (for instance, the preferential adsorption of one species of the mixture at the surface and the resulting precursor of wetting layers,19 or effects caused by asymmetries in chain length,20 chain stiffness, or monomer chemical architecture,21 and effects due to irregular roughness or periodic corrugation of a real surface on the atomic scale22). Our use of “neutral” (i.e., nonselective), perfectly flat surfaces thus highlights the effects of geometrical confinement only. We employ the bond fluctuation model23,24 that does not capture the chemical structure on the atomistic scale but rather lumps a small number (n ≈ 3-5) of chemical repeat units into one effective monomer. These monomers interact via coarsegrained interactions that are simplified (e.g., no torsional potentials) and softer.25-27 This coarse-graining results in a significant computational speed up, which allows us to investigate large time and length scales. In the next section, we describe this model and the simulation techniques in more detail. Then, we present geometrical properties (density profile across the film, profiles of the chains' center of mass, chain extensions in the directions parallel and perpendicular to the confining surfaces, and pair correlation functions). Due to our use of a square well potential of finite range on the lattice, the coordination number, zc, is computationally both well-defined and easily obtainable in our model and can be related to the Flory-Huggins parameter.20,21,28 We present detailed results on how this coordination number depends on film thickness and chain length. These results are

10.1021/jp0458506 CCC: $30.25 © 2005 American Chemical Society Published on Web 01/05/2005

Unmixing of Polymer Blends Confined in Ultrathin Films

J. Phys. Chem. B, Vol. 109, No. 14, 2005 6545

used to interpret our findings on the thermodynamics of the blend. We present evidence that in ultrathin films, which are thinner than the screening length of excluded volume interactions, a dramatic enhancement of compatibility occurs in comparison with the corresponding bulk system and that standard concepts of the theory of polymeric systems (such as the Flory-Huggins theory1,11,29,30 of unmixing or the “random phase approximation” RPA10) utterly fail. In the concluding section, we summarize our results and give a brief qualitative discussion of possible extensions if some of the omitted complications are reintroduced into our model. II. Model and Simulation Technique We use the bond fluctuation model of polymers on the simple cubic lattice.23,24,26-28,31,32 Each effective monomer blocks all 8 corners of a unit cell from further occupancy, thus incorporating excluded volume interactions1,10 that prevent atoms from “sitting on top of one another”. Monomers along a polymer are connected by one of 108 bond vectors, B b, which may take the x x x lengths b ) 2, 5, 6, 3, and 10 in units of the lattice spacing. These bond vectors are chosen such that the excluded volume interactions automatically prevent a crossing of bond vectors during “local hopping” motion, i.e., random displacements of an effective monomer by one lattice unit in the x, y, or z direction. This set of bond vectors (omitting the possible bond vectors with b ) x8 or larger than x10) is used because the same model is often used to study dynamical properties of polymer melts33 or polymer blends34,35 as well. The large number of bond vectors allows for 87 different bond angles. For this reason this lattice model approximates the continuum behavior rather well and, in many respects, is a more efficient and versatile model than the ordinary self-avoiding walk model where a monomer occupies a single lattice point and bonds are the elementary links between adjacent sites.25-27,31,32 In particular, chains as short as N ) 10 already exhibit the Gaussian statistics expected for polymers in three-dimensional melts1,10 (monomer number density, F ) 1/16; i.e., half of all lattice sites are occupied); i.e., the end-to-end distance Re scales as Re ∝ xN, and the radius of gyration is given by Rg ≈ Re/x6. Moreover, the single-chain structure factor is rather well approximated by the Debye function.1,10 In addition to the “random hopping” move mentioned above also “slithering snake”-type moves26,27,31,32,36,37 have been implemented for this model:20,38,39 A segment of a chain is removed from one end of the chain and added to the opposite one. This “slithering snake” move cannot be used when a qualitatively realistic description of polymer dynamics in terms of Rouse or reptation models10,40 is desired, but it is advantageous for applications where purely static properties are addressed, as done here, because it relaxes the chain conformations a factor N faster than the “random hopping” algorithm.31,36 Because chain lengths up to N ) 256 will be studied in the following, the availability of this algorithm was crucial for the success of our study. The binary polymer mixture contains two components, which we henceforth denote A and B. Monomeric units of the same type are chosen to attract each other, whereas monomeric units of different types repel each other. We use the “most symmetric” choice of a square well potential,

≡ - AA ) - BB ) AB

(1)

In the following, we choose as our unit of energy, which has to be compared to the thermal energy scale, kBT, where kB

denotes Boltzmann’s constant and T the temperature. The square well potential is extended over a range of x6. It comprises the 54 neighbors of a lattice site that constitute the first peak of the pair correlation function.20,24,26,28,31 Of course, the form of this potential is not at all chemically realistic, but it is computationally convenient, and we do not expect the specific form of the potential to qualitatively alter the properties studied here. We utilize identical chain lengths, NA ) NB ) N. Note that besides their labels, A or B, both types of chains have identical properties. It is possible to generalize this model to asymmetric blends,28 and consider asymmetry of chain lengths,28,20 different chain stiffness,21 or monomer shapes. However, the computational efficiency of our simulation technique deteriorates rather fast if the conformations of the two species substantially differ. Thus, we do not attempt to incorporate any such asymmetry in the present work. For the symmetric model the use of the semi grand canonical ensemble is rather straightforward:31,41,42 An A chain is transformed into a B chain at fixed configuration, or vice versa. As described in more details elsewhere,28,42 one randomly mixes the random hopping attempts, “slithering snake” moves, and identity switches (A T B) in suitable proportions to relax both the chain configurations and the concentrations, FA and FB, of the two species on roughly equal time scales. Note that the order parameter, m, of the symmetric mixture simply is the difference of relative concentrations, m ) (FA - FB)/F, the total monomer number density, F ) FA + FB, being held fixed.41 As usual in Monte Carlo sampling, attempted moves are accepted only if the transition probability, w, exceeds a random number, σ, that is uniformly distributed between zero and one. w is zero if an excluded volume or bond length constraint is violated and w ) min[1, exp(-∆E/kBT)] otherwise with ∆E being the energy change (to be computed using the energies quoted in eq 1) associated with the move. A factor exp[(∆µN/kBT] needs to be included in the transition probability for the identity switches, where ∆µ is the chemical potential difference between A and B monomers. Being interested in the critical temperature and the two-phase coexistence curve of the symmetric model, we set ∆µ ) 0. In the simulation, one gathers a joint histogram of the energy and the composition and, then, one employs histogram extrapolation26,28 and finite size scaling methods28,41,42 to sample and analyze the order parameter distribution, PL(m), in the considered finite size geometry. We consider a rectangular simulation cell of L × L × D geometry, with two hard walls at z ) 0 and z ) D + 1, whereas in lateral x and y directions periodic boundary conditions are utilized. III. Single Chain Conformations in Confined Films In the studies of three-dimensional polymer blends20,28,41-43 it has been found that the structure of the polymers is not independent of the thermodynamic state of the blend. For instance, there is a weak dependence of the radii of gyration, RAg and RBg , of the two species on the composition and temperature of the system although it is rather weak for long chain lengths. For m ) 0 symmetry requires RAg ) RBg ) Rg. We expect that these effects are larger in d ) 2 dimensions as well as in ultrathin films, because the critical temperature of unmixing is then significantly lower as for a d ) 3 bulk system. All the data for two-dimensional films presented in this section are taken at the respective critical temperature of the film. One can recognize from Figure 1 that close to the walls there is a strong layering effect (density oscillations). The density in

6546 J. Phys. Chem. B, Vol. 109, No. 14, 2005

Figure 1. Density profiles along the z-direction, perpendicular to the hard walls confining the film, for thicknesses from D ) 4 to D ) 19, as indicated in the figure. The origin of the z-axis has always been chosen to be in the midplane of the film. The average density is fixed at F ) 0.5 in all cases. Data are for chain length N ) 64 (but the chain length dependence of these density profiles is negligibly small).

Figure 2. Density profiles of the center of mass of the chains for film thickness D ) 19. Chain lengths from N ) 16 to N ) 256 are included, as indicated in the figure.

the layer adjacent to the wallsin the wall itself the density is zero by construction of the modelsattains a maximum, in the next layer there is a minimum, etc. Note that each monomer blocks sites in two neighboring lattice planes, and therefore in a film with D lattice planes D - 1 layers are only accessible; e.g., for D ) 4, monomers can be located in planes 1+2, 2+3, or 3+4, respectively. This layering effect is well-known from previous studies of athermal versions of the bond fluctuation model19,44 and analogous phenomena are known from various off-lattice models.19,31,45-48 Only for the thicker films, D ) 13 and 19, have these density oscillations decayed in the center of the film, such that there the bulk density, Fbulk ) 1/16, is reached. The packing effects at the walls give rise to a small, negative density excess. If we fix the average density in the film to the value, F ) 1/16, the density at the center of the film will be larger by an amount of the order 1/D.19 Because the FloryHuggins parameter, χ, is proportional to the density, this thickness dependence of the density also gives rise to a trivial shift of the critical temperature, Tc, of the order 1/D in addition to a shift of Tc upon confinement.49 For the thickest film, D ) 19, we have also analyzed the density profile of the center of mass of the chains (see Figure 2). Unlike the monomer density, there is no layering effect in this profile, but it exhibits a pronounced chain length dependence. Note that for the short chain lengths, N ) 16 and 32, a maximum of center of mass density occurs a few layers away from the wall, and in the center of the film, z ) 0, there is a shallow minimum. For N ) 64, this minimum is filled up and

Cavallo et al.

Figure 3. Profiles of the normalized parallel component R||(z)/Rbulk | and of the normalized perpendicular component R⊥(z)/Rbulk of the ⊥ gyration radius plotted vs distance z of the center of mass of the chains. All data refer to film thickness D ) 19 and three chain lengths are shown as indicated in the figure.

the density profile is essentially flat throughout the bulk region of the film. For N g 128; however, the center of mass density profile has a pronounced peak in the center of the film. For these chain lengths, the chain extension is comparable to or larger than the film thickness, and the chains pack such that their conformations are least restricted by the hard wall boundary condition. The instantaneous shape of a polymer coil in the bulk melt looks like a soap-shaped object;50 i.e., all three eigenvalues of the gyration tensor differ from each other. In the bulk, no direction in space is singled out for the orientation of this object, and there is no physical effect attributable to this anisotropic shape of single coils. Near the wall confining a semi-infinite melt, however, it is clearly preferable for simple geometric reasons to orient the coil such that the axis corresponding to the smallest eigenvalue is oriented perpendicular to the wall. In this way it becomes possible for the center of mass to be close to the wall without deforming the coil much and to minimize the “elastic” free energy of chain deformation. It is instructive to consider profiles of R⊥(z)/Rbulk and R|(z)/ ⊥ bulk R| as a function of the distance, z, of the molecule’s center of mass from the wall. In a semi-infinite system, one finds ) R|(zf∞), Rbulk ) R⊥(zf∞), and Rbulk ) x2Rbulk Rbulk | ⊥ | ⊥ because there are two parallel but only one perpendicular direction. Due to the alignment effect that a hard wall exerts on the polymer coils, R| (z)/Rbulk adopts values larger than | unity near the wall and decreases to unity as one approaches the center of the film. Similarly, R⊥(z)/Rbulk takes value much ⊥ less than unity in the vicinity of the surface, and increases toward unity on the scale of Rgbulk. Such a behavior (see Figure 3) has been noted for various lattice19,44,51 and off-lattice31,45,48 models. In Figure 3, it is seen that for N ) 16 one indeed observes this behavior as anticipated above, and in the central region of the film (i.e., for 5 e z e 12) R|(z)/Rbulk ) R⊥(z)/Rbulk ) 1, | ⊥ bulklike behavior is reached. However, already for N ) 64 no such bulk behavior shows up any longer and there remains some alignment effect of the wall on the chains even for chains that have their center of mass in the middle of the film. This alignment becomes very pronounced for N ) 256. In this case it is also quite clear that for chains with small z a simple orientation of the coils without deformation cannot suffice to bring them that close to the walls, simply because their smallest eigenvalue of the gyration tensor would already be too large. Note that the center of mass density is very low in the vicinity of the surface (cf. Figure 2) and these profiles overemphasize the orientation and distortion of the polymer conformations. Many experimental techniques cannot resolve the center of mass position and yield only average properties across the entire film. In Figure 4 we present the average molecular dimensions parallel and perpendicular to the film surfaces on various length scales.



Figure 4. Normalized, parallel, and perpendicular chain extension for different film thicknesses, D, and chain lengths, 16 e N e 256. Data are averaged over all chains in the film irrespective of the distance of their center of mass from the walls. Panels a and b present the behavior of the square root of the mean-squared end-to-end vector, panels c and d show the behavior of the radius of gyration, and panels e and f depict the behavior of bond vectors. The inset of panel f presents the data for the perpendicular components of the bond vectors as a function of the film thickness. The parallel extensions are plotted vs film thickness, D, and the perpendicular components are shown as a function of D/Rbulk e .

The figure includes a wide range of film thicknesses and chain lengths. As expected from the profiles, the thinner the film is the larger are the chain extensions parallel to the surface and the smaller are the chain extensions perpendicular to the surface. There is, however, a remarkable difference between the thickness dependence of the parallel and perpendicular chain extension.52 The data for the perpendicular extension collapse onto a common curve if plotted vs the normalized film thickness measured in units of the bulk chain extension, D/Rbulk e , whereas the Monte Carlo results for the parallel chain extension exhibit a data collapse when plotted vs the film thickness, D. This behavior is in accord with Silverberg’s argument,53 which simply conceives the chain conformations at a surface of a dense melt as random walks reflected at the boundary. Parallel and perpendicular components of a random walk are independent and, consequently, the reflection at the surface only affects the perpendicular components but not the parallel chain extension. If the film thickness becomes on the order of the unperturbed chain extension, i.e., D/Rbulk ) O(1), a sizable fraction of the e chains configurations become reflected and the perpendicular chain extension decreases. This is exactly what we observe in panels (c) and (d) of Figure 4.

At some film thickness D, however, the description of the polymer conformations as reflected, random walks will fail.52 When the film thickness becomes very small, the chain folds back many times into its own volume and the density inside of the Gaussian coil increases. The fractal structure of the segments of a single chain gradually becomes compact and it crosses over to two-dimensional behavior. When the density inside of the coil becomes comparable to the average density of the melt, the parallel chain extension begins to grow such that the density of the film remains laterally homogeneous. The stretching 2 parallel to the surface is only negligible when FD(Rbulk e ) , N 52 or

1 D . bulk Re xN

with

N)

[

]

3 F(Rbulk e ) N

2

(2)

where N measures the degree of interdigitation of different polymers. Because N ∝ N, the breakdown of Silverberg’s argument occurs at film thickness, Dbd ∝Rbulk e /xN, which is independent of the chain length. This film thickness scales as a function of density and chain length like the screening length,


Cavallo et al. parallel and perpendicular components of the bond vectors, B b, are shown in Figure 4e and f, respectively. In both cases the data for different chain length collapse onto a common curve when plotted against the film thickness, D. Only bond vectors that are in the ultimate vicinity of the surface are aligned, but bonds that belong to blobs away from the surface are unperturbed. IV. Intermolecular Correlations in Thin Films

Figure 5. (a) Schematic representation of two polymer chains in a melt in terms of blobs. The melt is confined in a film with thickness D larger than the blob size, ξ. (b) Snapshot of three selected chains of the melt. The chain length is N ) 128 and the density F ) 1/16. Different chains are drawn in different colors. The thickness of the film is D ) 13 ≈ 2ξ. Blobs on the order of the screening length ξ ≈ 7 can be distinguished.

ξ, of excluded volume interactions. Indeed, we observe in panels (a) and (c) of Figure 4 a steep increase of the parallel chain extensions around film thickness D ) 7 independent of chain length. In this context, we recall that on small length scales the excluded volume interactions are still effective, only on distances exceeding the screening length, ξ, of the excluded volume interactions the latter are negligible. The analysis of the crossover scaling between swollen coils and Gaussian coils as a function of density, F, has yielded the estimate ξ ≈ 7 for F ) 1/ , which is only slightly larger than the average bond length.33 16 Thus, our Monte Carlo results strongly support that the breakdown of Silverberg’s argument occurs for Dbd ∝ ξ and the numerical value of the proportionality constant is close to unity. These results suggest a simple picture: In analogy to the bulk behavior, the chain conformations in a thin film can be described by a random walk of blobs of size ξ. The inside of a blob is filled to a large extent with monomers that belong to the same polymer, but blobs can almost freely interdigitate (see Figure 5). On length scales larger than ξ, the chains adopt Gaussian conformations. For film thicknesses D . ξ, the Silverberg argument holds and only the perpendicular chain extensions are distorted by the confining walls. For D < ξ, however, the chain can be conceived as two-dimensional (d ) 2). Although the large scale chain extensions obey Gaussian statistics, Re ∝ xN, the lateral chain dimensions depend on the film thickness. The confinement “squeezes” the blob size. The occurrence of two length scalessRe and ξsalso explains why previous simulations observed a minimum of the total mean squared chain extension as a function of the film thickness.54 First, for Re > D > ξ, the lateral chain dimension remains unaffected but the perpendicular one is squeezed (back-folding), resulting in a decrease of the total mean squared chain extension in the film upon decreasing D. However, at extremely small film thicknesses, D < ξ, the lateral chain dimensions increase and give rise to an increase of the total mean squared chain extension. It is interesting to see that the orientation of the smallest length scale, the bond vectors, differs from the behavior of the endto-end vector or the radius of gyration. The results for the

Despite this random walk-like behavior of the chains on large length scales, excluded volume interactions do have some remaining effects on the structure of the coils, particularly in ultrathin films, D < ξ. This fact is most clearly observed in the form of correlation hole effects. We consider the intermolecular pair correlation function, g(r|), without distinction of the types (AA, AB, BB) in Figure 6. It is seen that g(r|) for r| ) 0 always has a minimum and g(r|f∞) approaches unity as it must be for any fluid system. The scale for this approach is set by Rg| ∝ xN. The physical interpretation of this correlation hole effect is that in the volume taken by a chain the density of other chains is reduced due to the excluded volume interactions. While in the bulk three-dimensional melt, g(r) deviates from unity only by a very small amount,10 namely 1/xN; this effect is much stronger in thin films.52 Note that in d ) 2 dimensions the “fractal dimension”,55 df ) 2, of a simple random walk agrees with the dimension d of the embedding space, and therefore there cannot be much interdigitation of chains in d ) 2, when the excluded volume constraint is obeyed. In d ) 3 the density of a coil inside the volume, Re3, it occupies is of the order Fself/F ) N/(FRe3) ) 1/xN and this quantity sets the depth of the correlation hole. This is in marked contrast to the behavior in a thin film, where the same argument yields Fself/F ) N/(FRe2D) ) Re/(DxN ) ∝ ξ/D in agreement with eq 2. Thus, we can expect that the scale for 1 - g(r|) on short and intermediate distances, r| , Rg|, is of the order ξ/D. In Figure 6c we see for D ) 13 at the smallest possible values of r| still a rather significant depression, and for D ) 4, a value that is less than ξ, g(r|) is almost zero for r| ) 0, i.e., monomers that have the same lateral coordinates. The very small nonzero value of g(r|)0) for values of D < ξ is due to the residual overlap, that still occurs between the excluded volume blobs of different chains. Of course, for D ) 2, the strictly twodimensional case, r| ) 0 can no longer occur at all, i.e., g(r|)0) ≡ 0. In this limit it is a well defined question to ask how this limit is approached as r| f 0. We assume that there is a power law

g(r|) ) c

( ) r|

N1/2

y

for

r| f 0

(d ) 2)

(3)

where c is a constant, and y is an exponent that has been predicted to be55 y ) 3/4. The inset of Figure 6a shows that the data are indeed roughly compatible with this prediction. In a brief preliminary study56 of the d ) 2 case that preceded the work of Semenov and Johner,55 we analyzed the chain conformations, the single chain structure factor, and the intermolecular pair correlation function and clearly demonstrated that the twodimensional chains do not adopt segregated, disklike conformations but are much more elongated and irregular. This observation has also been supported by off-lattice simulations.57,58 Lacking an analytical prediction, we erroneously assumed an analytic behavior of g(r| f 0) for r| f 0.56 However, the nonanalytic behavior, eq 3, which results from the fractal boundary of a self-avoiding chain in a two-dimensional melt, is also compatible with our simulation data. Semenov and



Figure 7. (a) Average number zc of contacts per monomer with monomers from other chains plotted as function of 1/xN. The thick straight line shows the bulk behavior zc ) 2.1 + 2.8/xN taken from previous work.34 All film thicknesses studied are included, as indicated in the figure. (b) Extrapolated value a ) z(D,∞) when a straight line is fitted as z(D,N) ) a + b/xN to the data zc shown in (a). The constant b is of the order of b ≈ 4. (c) log-log plot of zc vs N for d ) 2. Broken straight line shows the best fit, yielding an exponent x ≈ 0.43 in the power law zc ∝ Nx. Full straight line indicates the theoretical55 value x ) 3/8.

behavior of zc simply is

c(D) zc ) zc(D,∞) + xN

Figure 6. Scaling plot of the intermolecular pair correlation function g(r|) vs r|/N1/2 ∝ r/Rg|, for the case D ) 2, i.e., the two-dimensional case (a), D ) 4 (b), and D ) 13 (c). Chain lengths from N ) 16 to N ) 256 are included, as indicated in the key. For short chain length oscillations on the length scale of a few lattice units are visible and arise from the packing effects that also lead to the density oscillations in the density profile, Figure 1. The inset in (a) shows g(r|) vs r|/N1/2 on a log-log plot, including data for N g 32 only. The theoretical exponent y ) 3/4 is indicated by a straight line.

Johner55 also predict that the length of the fractal boundary scales such as N5/8. A geometrical quantity that is of key interest for the statistical thermodynamics of the blend is the number of contacts, zc, between monomers of different chains, defined from the condition that the distance r ) |r b| is smaller than the interaction range de ) x6 of the interaction chosen in eq 1. Thus we write

zc ) F

r ∑ g(b)

(4)

|b|ed r e

Figure 7a shows a plot of zc vs 1/xN. In bulk, threedimensional melts one knows that for the present model the

(5)

the 1/xN correction being due to the correlation hole effect. For the bulk one finds, zc(bulk,∞) ) 2.1 and c(bulk) ) 2.8.28,34 Therefore, we also choose to plot the results for thin films in this way. Extrapolating the simulation results for thin films toward N f ∞ we obtain the data shown in Figure 7b. Because there is some curvature in the plots shown in Figure 7a, the extrapolation in Figure 7b may suffer from some systematic errors. In particular, the value for zc(2,∞) which is slightly positive is likely not to be trusted, and rather one expects55,56 that zc vanishes as N f ∞ in d ) 2 according to a power law, zc ∝ N-x. Figure 7c shows that the numerical data are consistent with such an interpretation as well. The best fit to the data yields an exponent x ) 0.43 as noted in ref 56. This simulation result is between the simple geometrical estimate, x ) 1/2, and the new theoretical prediction55 for the exponent, x ) 3/8. If we would disregard the last data point, the data would agree much better with the theoretical prediction,55 which might indicate sampling problems at the largest chain length. The theoretical prediction of the exponent has also been confirmed by recent off-lattice calculations58 using the efficient recoil growth algorithm.59 In Figure 8 we test various phenomenological hypotheses to study the possible behavior of the numerical data for zc(D,N). It is seen that a plot of zcf 0 for D/N1/2 does not exhibit a good data collapse, but the data are compatible with zc f 0 for D/N1/2 f 0. The absence of a data collapse indicates the presence of two distinct length scales: On one hand, the unperturbed chain extension, Rbulk ∝ xN, describes the dee


Cavallo et al.

Figure 9. Critical temperature kBTc/ plotted vs chain length, for various film thicknesses D, as indicated in the key.

Figure 8. (a) Scaling plot of zc(D,N) as a function of the scaling variable D/N1/2. (b) Scaling plot of the coordination number zpoly of a whole polymer chain, i.e., the number of different other chains it interacts with; zpoly/D shown as function of D/N1/2. The inset of (b) shows the unscaled data for zpoly plotted vs 1/xN.

pendence of zc in the bulk (cf. eq 5) and controls the “filling up” of the correlation hole due to backfolding of the reflected chains in films, D . ξ, according to Silverberg’s argument. On the other hand, the N-independent screening length of excluded volume interactions, ξ, sets the film thickness below which Silverberg’s argument breaks down and the chains adopt two-dimensional conformations. These two length scales also control the perpendicular and lateral chain extensions (cf. section III). Another important structural quantity is the number of other molecules a reference chain interacts with. In the bulk, this quantity is described by the invariant degree of polymerization, N. In the limit, N f ∞, long-wavelength fluctuations are strongly suppressed and, thus, three-dimensional melts of long polymers are accurately described by mean-field theory. In the bulk, one finds zpoly ) FRe3/N ≡ xN ∝ xN. In a thin film, Re > D . ξ, one obtains by the same token zpoly ) FRe2D/N ) DxN/Rbulk ∝ DN0. In ultrathin films, ξ > D, a reference chain e interacts only with a small, finite number of other molecules. This expectation is confirmed by the data in Figure 8b. V. Critical Temperature of Demixing in Ultrathin Polymer Films Utilizing the techniques summarized in section II,19-21,28,31,41,42 we obtained the critical temperature, Tc, of demixing for our model and present Tc as a function of N in Figure 9. Apart from the strictly two-dimensional case (D ) 2), all data are compatible with a linear increase of Tc with N, which already results from the Flory-Huggins theory1,10-14,30 and has been confirmed in

-1/2. Various values of Figure 10. Plot of Tc/TFH c ) kBTc/zcN versus N film thickness are shown, as indicated in the figure. Bulk threedimensional data42 shown as open circles are included for comparison.

d ) 3 by experiment,60 simulation,28,42 and more sophisticated theories such as P-RISM.61 Because the Flory-Huggins theory simply relates Tc and zc according to

kB TFH c / ) Nzc

(6)

it is of interest to plot Tc in units of TFH c (see Figure 10). One can see that for all bulk data,42 which are included for comparison, Tc/TFH c f 1 as N f ∞, as expected, because zpoly f ∞ in this limit, and mean field theory then should become exact. However, for any finite film thickness this is no longer the case and zpoly ∝ D remains finite even in the limit N f ∞ (cf. Figure 8b inset). As a result, mean field theory cannot become exact, and fluctuation effects tend to be nonnegligible and will lead to a reduction of Tc. In addition, one needs to consider that near the critical temperature the correlation length of order parameter fluctuations can grow to infinity only in two rather than three directions. Therefore, the critical behavior ultimately belongs to the universality class62 of the twodimensional Ising model rather than the three-dimensional Ising model. When one extends56 the standard Ginzburg criterion63 from the case of d ) 3 polymer mixtures64-66 to mixtures in d ) 2, no mean field critical behavior is predicted even for N f ∞. However, the situation becomes more complicated for thin films67 where zpoly is large but finite and intermediate regimes where mean field critical behavior applies may again occur. The crossover scaling67 that then may occur is beyond the scope of the present study. We only note that in Figure 10 for the thicker films, D ) 13 and 19, the apparent change of trends Tc/zN is almost independent of N for N e 128, but then starts to distinctly increase for larger Nscould be related to this interplay with crossover effects.


J. Phys. Chem. B, Vol. 109, No. 14, 2005 6551 of the critical temperature, Tc ∝ N, holds. Gratifyingly, this too simple-minded scaling assumption does not find much support from the data in Figure 12. The failure of this scaling ansatz corroborates that the demixing behavior in thin films is controlled by two length scales, Re and ξ. VI. Conclusions

Figure 11. log-log plot of the two-dimensional and the bulk critical temperatures kBTc/ versus N. Broken straight lines through the data points indicate best fits with exponent 1 - x ≈ 0.65 (2d) and 1 (bulk).

Figure 12. Test of crossover scaling behavior for the critical temperature, plotting Tc/N versus D/N1/2. The different film thicknesses are indicated in the key of the figure. Note that the data systematically deviate from collapsing on a master curve.

A very naive scaling would assume that the relative depression of the critical temperature is ruled by a comparison of lengths, namely, the film thickness, D, and the chain extension, Re ∝ xN, which also sets the length scale of the correlation length of composition fluctuations in the bulk. Then, one would write

( )

kBTc D )f N xN

(7)

With f (σ) f const for σ f ∞ one recovers the linear scaling of the critical temperature for the bulk. Such a scaling behavior also is appropriate for not too thin films, D ≈ O(Re) . ξ, because then the unperturbed chain extension is the only relevant length scale. To recover two-dimensional behavior within this simple scaling ansatz, we require f (σ) ∝ σ2x for σ f 0. This scaling behavior of two-dimensional polymer mixtures is confirmed in Figure 11. The best fit to the data yields x ) 0.35, which is compatible with the theoretical prediction, x ) 3/8.55 However, eq 7 cannot be correct in this limit, because it would imply that Tc ∝ N1-x for any finite film thickness in the limit N f ∞. We rather expect this scaling to be restricted to ultrathin film, D < ξ, whereas for larger thicknesses, D > ξ, a linear N dependence

In this investigation extensive Monte Carlo simulation results for the structure of dense symmetric binary polymer blends confined between neutral repulsive walls confining the system in ultrathin films were presented and discussed. Our main findings are summarized in Table 1. For the strictly twodimensional case, i.e., film thickness D ) 2 < ξ in our model, a breakdown of Flory-Huggins theory occurs, because Tc ∝ N1-x ()N5/8) is found in agreement with theoretical predictions,55 rather than the standard linear scaling in N. This gives rise to a substantial enhancement of compatibility (or, equivalently, a pronounced depression of the critical temperature) in comparison with the bulk three-dimensional blend. The failure of the FloryHuggins theory invalidates also its many extensions (such as the random phase approximation, which considers extensions to wavevector-dependent properties or the self-consistent field theory, etc.) and whether theories that include the correlation hole effect, such as P-RISM theory,61 can account for this behavior, remains to be seen. We expect this two-dimensional behavior to describe not only strictly two-dimensional polymer films (i.e., D ) 2 in our model) but also ultrathin film with a thickness, D, smaller than the screening length, ξ, of excluded volume interactions. Of course, it would be interesting to consider other phase transitions of polymers in this ultrathin film geometry (mesophase ordering of block copolymers, etc.). Such extensions are left to future work. We also note that anomalous chain length dependence of the liquid-vapor critical point has been observed in recent two-dimensional simulations.68 When the film thickness is of the order of a few times the excluded volume screening length, ξ, the coordination number starts to converge toward its bulk value (cf. Figure 7), and then the depression of the critical temperature in comparison with the bulk is only about a factor of 2, or even less. But even then the critical behavior is still that of the two-dimensional Ising model, at least close enough to the critical point. A systematic investigation of this critical behavior is beyond the scope of this study. We think that our use of a lattice model is not a serious restriction, and similar results could be obtained for off-lattice models as well,57,58 although with considerably more effort. In fact, the result zc ∝ N-3/8 has already been verified for an offlattice model.58 We have only considered strictly symmetric blends for the sake of computational convenience. However, it it likely that most of our qualitative findings carry over to systems with asymmetry of chain lengths, chain stiffness, different shapes and sizes of effective monomers, as well as asymmetries in the interaction parameters. These asymmetries might give rise to an entropic contribution to the Flory-Huggins parameter21 but the salient features of the miscibility behavior still depend on the number of intermolecular contacts.

TABLE 1: Summary of Thin Film Properties film thickness, D

Silverberg’s argument valid R|/Rbulk |

R⊥/Rbulk ⊥

scaling of Tc

mean field theory valid

universality class

bulk Re ∼ D . ξ ∝ Re/xN ξ.D

yes, 1 yes, ≈1 no, >1

1

Unmixing of Polymer Blends Confined in Ultrathin Films: Crossover

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