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Conformational Properties of Semiflexible Chains at Nematic Ordering Transitions in Thin Films: A Monte Carlo Simulation Victor A. Ivanov,*,† Alexandra S. Rodionova,† Julia A. Martemyanova,† Mikhail R. Stukan,† Marcus Müller,‡ Wolfgang Paul,§ and Kurt Binder∥ †

Faculty Institut § Institut ∥ Institut ‡

of Physics, Moscow State University, Moscow 119991, Russia für Theoretische Physik, Georg-August-Universität, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany für Physik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany für Physik, Johannes-Gutenberg-Universität, Staudinger Weg 7, 55099 Mainz, Germany

ABSTRACT: Athermal solutions of semiflexible macromolecules with excluded volume interactions and with varying concentration (dilute, semidilute, and concentrated solutions) in a film of thickness D between two hard walls have been studied by means of grand canonical Monte Carlo simulation using the bond fluctuation lattice model. In earlier work, we have reported on the phase diagram of this model system, which exhibits a continuous quasi-twodimensional order−disorder transition at rather small concentration and (for thick enough films) a “capillary nematization”-type first-order phase transition. In the “semi-infinite” case (i.e., macroscopically thick films) the onset of nematic order is triggered by the walls (surface-induced ordering). While the focus of this previous work was on the order parameters of these phase transitions and associated surface effects, in the present paper we focus on the interplay between the conformational statistics of the semiflexible chains and these orientational ordering phenomena. In particular, we study how characteristic lengths of the chains (persistence length, end-to-end distance) depend on the local and global orientational order in the system. We show that there is a strong coupling between single-chain properties and long-range orientational order. The relation between mean-square end-to-end distance of short semiflexible chains and their persistence length predicted by the Kratky−Porod model is found to be applicable (within 10% errors) in the dilute limit, while it fails as soon as nematic short- or long-range order is present.

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INTRODUCTION By increasing concentration and/or chain length and/or chain stiffness, one can introduce a phase transition from an isotropic fluid to a nematically ordered state in semiflexible polymer solutions under good solvent conditions.1−12,14,18−28 Good solvent conditions mean that the interaction between the chains is predominantly repulsive, so that this transition is analogous to the entropically driven transition of standard lyotropic liquid-crystalline systems.13,29,30 By this analogy, the choices of chain stiffness and of chain length combine to a choice of effective aspect ratio of the macromolecule, which is one control parameter in the Onsager theory. However, differences to the small molecule case are expected due to the additional configurational entropy of the semiflexible macromolecules.2,6,28 The effect of confining walls on liquidcrystalline order in general is on the one hand very important from an application point of view and on the other hand a challenging problem of statistical mechanics. For semiflexible macromolecules this has been studied by various computational and analytical approaches.31−38 In ref 38, we presented a detailed analysis of the intricate interactions between the different phase transitions which occur in a film of semiflexible polymers confined by hard walls. A sketch of the qualitative phase diagram for this case is presented in Figure 1. The diagram is presented in the variables ϕ−D © 2014 American Chemical Society

Figure 1. Sketch of the qualitative phase diagram of a solution of semiflexible macromolecules (having the polymer volume fraction ϕ) confined in a thin film of thickness D. For details see the text.

Received: October 16, 2013 Revised: January 9, 2014 Published: January 27, 2014 1206

dx.doi.org/10.1021/ma402138c | Macromolecules 2014, 47, 1206−1220

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where ϕ is the polymer volume fraction and D is the thickness of the film. The entropic enrichment of chains at the hard walls introduces a two-dimensional (2d) isotropic−nematic transition preceding the bulk transition. The 2d transition is shown by the dashed line in Figure 1 separating the surface disordered, SD, from the surface ordered, SO, region; the open circle at the end of this line indicates that this transition is absent for a pure bulk system with periodic boundary conditions along all axes, and for a bulk system in an infinitely thick film (D → ∞) this transition only is a singularity of its surface excess free energy (due to the hard walls); it does not show up in the total (bulk) free energy density. When one considers the limit when the chemical potential μ approaches its value μc at coexistence after the limit D → ∞, one observes a complete wetting transition;38 when one performs the limits in the reversed order, one observes capillary nematization.39 This means that the densities of the isotropic as well as the nematic phase at coexistence are reduced with respect to their bulk values (full lines in Figure 1, where the two filled circles indicate the densities of isotropic and nematic phases at the coexistence in the bulk). However, there is a critical thickness Dc (indicated by the black rhombus) which is about the size of an elongated chain, below which the three-dimensional isotropic−nematic transition disappears; i.e., the rhombus indicates the location of a critical point. For smaller D a line of fluctuation maxima (dotted line) connects this critical point with the 2d ordering transition at the walls. While we have investigated in detail the liquid crystal physics of the orientational ordering transitions in thin films of semiflexible polymers in earlier publications,37,38 we now want to turn to the polymeric character of this system. The different transitions discussed above should all influence the conformational properties of the macromolecules, like their end-to-end distance, the distribution of chain ends with respect to the walls, or the stiffness of the chains. As is well-known, the standard model to describe semiflexible chains is the Kratky− Porod worm-like chain model,40 whose Hamiltonian is /WLC =

1 κ 2

∫0

3

⎛ ∂ 2t (⃗ s) ⎞2 ds ⎜ ⎟ ⎝ ∂s 2 ⎠

related with a typical liquid-crystalline control parameter,13,14 the aspect ratio of the molecules A=

⟨R e 2⟩ dchain

(2)

where dchain is the thickness of a chain (= 2 lattice spacings in our model, which will be discussed in the next section). For semiflexible chains, this control parameter is represented in the theoretical descriptions15−17 by the Kuhn length divided by the thickness of the chains A′ =

lk dchain

=

lp a

(3)

assuming that one can define an equivalent Gaussian freely jointed chain for the considered model in the limit chain length to infinity (a is the unit of length here). The last equality connects the Kuhn length to the persistence length for the worm-like chain model defined in eq 1 for our case of dchain = 2. We will show that, while indeed it is straightforward to define an energy term in the Hamiltonian that describes the energy cost of the chain bending which is independent of the environment in which the chain molecule is embedded (see eq 4), there is no simple environment-independent relation between this energy parameter and the Kuhn or persistence lengths employed in the theoretical descriptions. The remainder of this paper is organized as follows: After presenting our model and simulation technique in the next section, we will turn to the discussion of our results and finally present some conclusions.

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MODEL AND SIMULATION TECHNIQUE We perform Monte Carlo simulations using the bond fluctuation lattice model.43 On the simple cubic lattice, each effective monomeric unit is represented by an elementary cube. The eight corners of this cube are blocked from further occupation, realizing thus the excluded volume interaction. The lattice spacing a ≡ 1 is taken as our unit of length. The bond vectors connecting two consecutive monomeric units along the chain are taken from the set {(±2,0,0), (±2,±1,0), (±2,±1,±1), (±2,±2,±1), (±3,0,0), (±3,±1,0)}, including also all permutations between these coordinates. Such a set automatically forbids any crossings of bond vectors in the course of local movements of monomeric units (i.e., their displacements on one lattice spacing along a randomly chosen coordinate axis). Altogether 108 different bond vectors occur, leading to 87 different angles between successive bonds, and in this sense, the bond fluctuation model can be considered as a quasi-continuum model. To describe variable chain stiffness, an intramolecular bending potential depending on the angle ϑ between two successive bond vectors along the chain is used:

(1)

where t ⃗ (s) is a local tangent vector along the chain contour and s is a coordinate running along the chain contour (3 is the contour length of the chain). It normally is assumed that the bending stiffness κ can be related to the persistence length lp of the chain as κ = kBTlp (where kB is the Boltzmann constant and T is the temperature). The most naive assumption would be that both the bending stiffness κ and the persistence length lp are “intrinsic” properties of the chain (i.e., determined already by specifying its chemical structure, independent of its environment). Concerning the latter property, it has recently been worked out that the concept of a persistence length as a local measure of the intrinsic stiffness of a polymer chain has to be used with care.41 This concept relies on properties of ideal random walk chains, and it has been shown that, e.g., bond orientation correlations in polymer systems, on which the concept of the persistence length is based, never decay in the exponential form valid for ideal chains. However, two measures of intrinsic stiffness remain meaningful.42 One is the slope of the bond-orientation correlation function at zero chemical distance (a local measure which agrees for a discrete model with the average cosine between adjacent bonds), and the other is the scaling behavior of the end-to-end distance, Re. When the chain conformations are close to rod-like, the latter one can be

Ubend = −f cos ϑ(1 + c cos ϑ)

(4)

The parameter f in eq 4 is the main parameter for the intramolecular stiffness. In this study the value f = 8.0 (note the convention putting kBT equal to unity) was used for which the isotropic-to-nematic transition has already been carefully studied in the bulk28 and under thin film confinement.37,38 The value of the parameter c in eq 4 was quite small, c = 0.03, and it is only used to be compatible with previous work.14 The chain length was equal to N = 20 monomeric units, as in our previous studies.24,28,37,38 For this choice of stiffness parameter 1207


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and chain length we are working in the desirable regime lb < lp < 3 , where lb is the bond length (more precisely, the average bond length in our model). Reducing the bending energy parameter (e.g., f = 2.68 and f = 4 have been studied in the bulk before24,28) stabilizes the isotropic phase, i.e., shifts the isotropic coexistence density to larger values, increasing the computational effort for the study of different phase transitions without changing the qualitative picture. Note that with this model the local measure of stiffness, i.e., the average cosine of the bond angle, is basically given by the average energy in the system. Monte Carlo simulations were carried out in the grand canonical (μVT) ensemble using the configurational bias method.44,45 In the configurational bias moves, one needs to utilize a biased chain insertion method to let a polymer “grow” successively into the system without violating the excluded volume constraint. At each step all possible 108 bond vectors having their origin at the current effective monomeric unit are examined, and a position for inserting the next monomeric unit along the chain is chosen, respecting the excluded volume condition and using the Boltzmann weight calculated from the intramolecular energy, eq 4. The statistical weight of the generated polymer configuration hence is easily calculated recursively, and thus the bias can be accounted for in the acceptance probability for the move. For more details on the algorithm see our previous papers.24,28,37,38 One Monte Carlo step (MCS) in our simulations then includes one configurational bias move and, additionally, either one attempt per effective monomeric unit in the system to perform a local random hopping move or one attempt per chain of a slitheringsnake move. The simulation box had geometry L × L × D, with hard repulsive (impenetrable) walls at z = 0 and z = D + 1, while periodic boundary conditions were applied in both x and y directions. Because our bending potential, eq 4 does not depend on the bond length the equilibrium length of a free single chain totally elongated along one of the coordinate axes lies between 40 and 60. We have considered film thickness D from 10 to 500, so that the smallest box width used here (D = 10) is much smaller than a totally elongated chain, and this case corresponds to a quasi-two-dimensional system. The linear dimension L along the walls was varied from L = 60 to 500. The chemical potential per chain μ was chosen in the range between −195 and −150 (in kBT units), leading to values of polymer volume fraction ϕ in the range from 0.0 to 0.55, and it is important to distinguish between average volume fraction in the whole box and the value of the “bulk” volume fraction in the center of the box (provided a plateau is formed on the density profile in the middle of the box) which then can be compared to the corresponding bulk density. The total simulation time was typically between 107 and 2 × 107 MCS. The relaxation time in the most extreme cases (in largest boxes in the vicinity of isotropic−nematic transition) was about 5 × 106 MCS; however, this is only the nonequilibrium relaxation time in the transition region (i.e., the time needed for the systems to switch from a nonequilibrium state to an equilibrium one in a situation when the free energy difference between the orientationally ordered and disordered states is small). The initial configurations of the system were either an isotropic dilute solution or a dense nematic state. We observed a hysteresis of calculated quantities in the vicinity of the isotropic−nematic transition for simulations started from these two different initial config-

urations. However, we do not discuss this hysteresis in the present paper because we have previously studied it24,28,37,38 and because it is not an essential issue for polymeric properties studied here. Therefore, most data below are presented for simulation runs started from a nematic initial configuration. We have recorded and analyzed the following quantities in our simulations. We have calculated profiles of all collective variables (polymer volume fraction ϕ, eigenvalues S1, S2, S3 of the tensor characterizing the nematic order) in the z-direction across the film; i.e., we sample ϕ(z), S1(z), S2(z), and S3(z). The tensor characterizing the nematic order is defined in terms of unit vectors ei⃗ along the bonds connecting monomeric units i and i + 1 (eαi is the α-th Cartesian component). In the bulk, for a system of 5 chains each having N effective monomeric units and hence N − 1 bonds, this order parameter tensor is Q αβ =

1 5(N − 1)

5(N − 1)

∑ i=1

1 α β (3ei ei − δαβ) 2

(5)

The largest eigenvalue of this tensor S1 is a good order parameter for characterizing the isotropic−nematic transition, while the combination of the second and the third eigenvalues P = S2 − S3 is known to be a good order parameter for transitions between uniaxial and biaxial phases.46 In our simulations, we have recorded also standard singlechain characteristics such as the mean-squared end-to-end distance ⟨Re2⟩ of the chains and the components of ⟨Re2⟩ along the z-axis, ⟨Re,z2⟩, and in the xy-plane, ⟨Re,xy2⟩ = (⟨Re,x2⟩ + ⟨Re,y2⟩)/2, i.e., perpendicular and parallel to the walls, as well as their profiles along the z-axis. From the measurements of the average end-to-end distance, we calculate the aspect ratio A, eq 2, as our global stiffness measure. For ideal chains, the orientational correlation function, Ck, between bond vectors separated by k monomer units along the chain Ck = ⟨ ei⃗ · ei⃗ + k⟩ = ⟨cos θi , i + k⟩

(6)

shows an exponential behavior Ck ∼ exp( −k /lp′)

(7)

and the persistence length lp′ can be defined from this equation. For nonideal systems, where excluded volume interactions play a role, and independent of the solvent quality or solution density, such an analysis is not applicable.41,42 Nevertheless, the average cosine between two subsequent bond vectors along the chain, ⟨cos θi,i+1⟩, still gives a reasonable estimation of the chain stiffness41,42 lp′ = −⟨lb⟩/ln⟨cos θi , i + 1⟩

(8)

where ⟨lb⟩ is the average bond length which we monitor in the simulation. As an alternative to the average cosine of the angle between subsequent bond vectors, for our model Hamiltonian the energy of the system can be used as well to quantify this local stiffness. For both the global and the local stiffness measure, their profiles A(z) and lp(z) have been determined as well. All profiles have been determined by locating the center of mass of a chain first, and using its z-coordinate to accumulate the statistics for A(z), Re,xy2(z), Re,z2(z), lp(z), or the distribution of positions of chain ends along the z-axis (cf. ref 47). All these quantities depend of course on the average polymer volume fraction ϕ = 85 Na3/V in the system which is 1208


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controlled by the value of the chemical potential μ in our grand canonical simulations. All quantities were averaged over all chains and all generated system configurations.

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RESULTS AND DISCUSSION Dependencies of Chain Linear Dimensions and Nematic Order on μ at Different z. The dependence of the aspect ratio A = (Re2)1/2/dchain, and the orientational order parameter S1 on μ in different z-layers for a box with dimensions D = 150, L = 100 is shown in Figures 2a and 2b. The dependence of the components Re,xy2(z) and Re,z2 of the squared end-to-end distance on μ in different layers z for this box is shown in Figure 3. Both the isotropic−nematic transition and the surface-induced 2d ordering transition are well visible in these plots and are indicated by arrows. The curves for layers in the center of the film show a pronounced jump around μ ≈ −167.5 in both A and S1 indicating the isotropic−nematic transition, while the curves for layers at the walls show either a large jump-like increase (in S1) or a smaller sharp but continuous increase (in A) around μ ≈ −172.5, indicating the 2d ordering transition at the walls. The plots of both A and S1 vs the polymer volume fraction ϕ are very similar to these plots, but instead of a vertical jump at μiso−nem there occur two kinks with a straight line rising steeply under an angle somewhat below 90° in between the kinks. This happens because the densities of isotropic ϕiso and nematic ϕnem phases at coexistence differ from each other although those values are very close to each other. The 2d transition in the layers at the wall occurs around ϕ ≈ 0.2 while the three-dimensional (3d) transition in the center of the film takes place around ϕ ≈ 0.3, which is the value of the transition in the bulk.24,28 Intermediate layers show a shift from 2d to 3d behavior. The layers at z ≈ 20 show both the 3d and 2d features because there is both the jump at the 3d transition (both in A and S1) and a small increase at the 2d transition (only in S1) visible. It is also well visible in Figure 3b that the value of Re,z2 at small μ-values for z = 20 is only 2/3 of the corresponding value for z = 75. Actually, only the layers at z ≥ 40 are already completely bulk-like. This distance from the wall is comparable with the length of totally elongated chain which can be estimated as the product of the average bond length ⟨lb⟩ ≈ 2.65 by the number of bonds between monomeric units Nbonds = N − 1 = 19, giving the value 2.65 × 19 ≈ 50. It is interesting to observe that in the layers at the walls the variation of both the aspect ratio A and Re,xy2 with increasing chemical potential is nonmonotonic: a maximum value is reached at about μ ≈ −166. We interpret this observation by noting that with increasing density more bonds of length 2 rather than 3 in x and y directions occur, to allow a better filling of space. This is confirmed by the dependence of the average bond length on μ in Figure 9b (see the discussion in the subsection on intramolecular stiffness below). Arguably, reducing the set of available directions of bond vectors in the layers closer to the walls in comparison to the layers in the center of the film is responsible for the nonmonotonic trend in the aspect ratio in different z-layers in dilute solution as well. At μ-values below the surface ordering transition, e.g., μ < −175, the aspect ratio decreases with increasing z-coordinate (chains are essentially two-dimensional when their center of mass is forced to be in the layers adjacent to the walls), but in the center of the film the value of the aspect ratio is slightly larger than the lowest value reached at z between 10 and 20 (see Figure 2b) because the population of the set of bond vectors in the center of the box is unperturbed

Figure 2. Dependence of the orientational order parameter, S1, (a) and of the aspect ratio, A = ((Re2)1/2)/2, (b) on the chemical potential μ for box size D = 150, L = 100 and different z-values shown in the legend. Runs were started from a nematic initial conformation. Arrows indicate the locations of the 2d transition (μ = −172.5) and the 3d transition (μ = −167.3), as estimated in our previous papers.37,38 In (c) the profile of average orientation of bond vectors is shown for μ = −168.0 (for details see the text).

by presence of the walls. Another possible explanation of this observation is as follows. At very small μ, corresponding to a dilute solution, and at small distances z from the wall, flexible or long semiflexible chains adopt swollen conformations of 2d selfavoiding walks with Re ∼ N3/4. Farther away from the surface, however, 3d behavior is observed with Re = bN= . Thus, the chain dimensions strongly decrease with z. In our case, which rather corresponds to a semidilute solution, this crossover from 2d to 3d scaling in dilute solution gives rise to a decreasing blob 1209


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from Figure 2b we see that chains adjacent to the walls have a distinctly larger aspect ratio (near 22) than chains in the bulk (near 19), for chemical potentials in the range −185 ≤ μ ≤ −175, where no long-range orientational order exists anywhere in the system. When the SD → SO orientational transition at the surface occurs, for the chains in the vicinity of the walls the aspect ratios are still increasing, up to a maximum value of 23.5, reached after the onset of bulk order. Since this onset of bulk order is accompanied by the thickening of a nematic precursor layer (“surface-induced ordering”, as discussed in ref 38), there is a closely related precursor effect also seen in the aspect ratio of inner layers in the film; of course, the further away from the walls the considered layers are located, the closer to the transition point of the capillary nematization transition (highlighted by the rightmost arrow in Figures 2 and 3) this enhancement of the aspect ratio occurs. It is also worth mentioning that deeply in the well-ordered region (S1 > 0.9 irrespective of z for μ > −150, see Figure 2a) the aspect ratio in all layers roughly is identical. The nematic surface-induced ordering can also be detected rather clearly from the behavior of the individual components of the squared end-to-end distance (see Figure 3): layers affected by surface-induced order show both an increase of Re,xy2 and a decrease of Re,z2; Figure 3 does not distinguish between orientation of the long axis of the stiff chain and true conformational changes, while the latter show up clearly in the aspect ratio (Figure 2b). Dependence of Average Values of Chain Linear Dimensions and Nematic Order on μ for Several Film Thicknesses. The dependence of the average aspect ratio on the chemical potential μ for different D-values is shown in Figure 4. The jump in the aspect ratio at μ ≈ −167.5 for D =

Figure 3. Dependence of Re,xy2 (a) and Re,z2 (b) on μ for different z (shown in the legend) for a box of dimensions D = 150 and L = 100. Runs were started from a nematic initial conformation. Arrows indicate the locations of the 2d transition (μ = −172.5) and the 3d transition (μ = −167.3), as estimated in our previous papers.37,38

size as a function of z ≪ Re and a concomitant decrease of Re. In the regime z ≤ Re, however, the lateral chain dimensions are independent from z, but the chain extension perpendicular to the walls is constraint and smaller than in a lateral direction. As the chains center of mass moves farther away from the wall, the average chain dimensions in all three directions become equal, and consequently, the asymptotic 3d bulk Re is reached from below. These two mechanisms give rise to a nonmonotonous dependence of the chain extension from the distance to the wall. Such an effect is also well visible in the z-profiles of the aspect ratio in the isotropic films (see Figure 12). To provide a better understanding of how the preferred orientation changes on moving away from the walls, we present in Figure 2c a snapshot of orientations of bond vectors in the box of the size D = 150, L = 100 at μ = −168 which is between the μ-values for the 2d and 3d transitions. A projection of the box on the xz-plane is shown (the z-axis is horizontal). The colors indicate a direction of the average bond orientation along x- (red), y- (green), or z-axis (blue). Averaging is performed along the y-axis at each (x,z)-point. The chains are nematically ordered at both walls (because the system is already above the 2d transition) while the solution is still isotropic in the center of the box (because the system is still below the isotropic− nematic transition). Thus, Figures 2 and 3 already give clear indications for an interesting interplay between the tendency of the walls to orient the rather stiff chains along the x- or y-axes parallel to the walls and their effect on the internal structure of the chains. Thus,

Figure 4. Dependence of the average aspect ratio, A = ((Re2)1/2/2), on the chemical potential μ for different D-values shown in the legend. Vertical dashed lines indicate the locations of transitions for these films, as estimated in our previous papers.37,38

150 and D = 500 indicates the isotropic−nematic transition in the whole film, while this jump disappears for D ≤ 50. For the thin films there is a steep but continuous decrease in the aspect ratio shifting to the smaller μ-values, i.e. smaller densities, and the position of the inflection point (where the slope is maximal) represents the maximum of fluctuations in the aspect ratio which coincides with the maximum of fluctuations in the orientational order parameter.38 This increase of the size of the worm-like chain along the director in the nematic state was first 1210


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theoretically predicted in ref 48 and called the “stiffening” of the persistent macromolecule induced by the increase in the degree of anisotropy of the environment. The μ-dependence for ⟨Re,xy2⟩ and ⟨Re,z2⟩ are shown for different box sizes D in Figure 5. Again, the capillary

Figure 6. (a) Variation of the total energy per chain with the chemical potential for different box sizes (indicated in the legend). Runs were started from a nematic initial conformation. (b) Analysis of the latent heat estimated as the jump in the energy in part (a). Squares show the estimations of ΔG from part (a) for D = 70, 150, and 500; the solid line is a linear fit.

reveals that transitions in our system are not of purely entropic nature due to the presence of the intramolecular energy. Remember that our model exhibits intramolecular bending energy only, and we do not vary the temperature in our system. So the energy changes seen in Figure 6a result indirectly from the interplay with entropic effects. In Figure 6b, we present our analysis of the latent heat, ΔG, estimated as the energy jump at the isotropic−nematic transition. We have performed an attempt to check whether the latent heat shows a power law dependence on D − Dc. According to the analysis of ref 38, the critical point at which the capillary nematization transition ends can be interpreted as belonging to the 3-state Potts model in a field that favors two of the states, i.e., the Ising model. We have tried to fit the dependence of ΔG on D − Dc by a power law ΔG ∼ (D − Dc)? using the value of the critical film thickness Dc (where the 3d transition disappears) as the adjustable parameter to reach a mostly linear plot in double-logarithmic coordinates. The value Dc ≈ 57 corresponds quite well to the maximal elongation of a chain along a coordinate axis (3 ×19) and is also close to the average length of a totally elongated chain (2.65 × 19 ≈ 50). As regards the exponent χ, we have obtained an estimation χ ≈ 0.14. Since already the accuracy of our estimate for Dc is somewhat uncertain, and it is possible that the critical region where the correct asymptotic exponent can be seen could be very narrow, we do not speculate about the interpretation of this empirical exponent estimate.

Figure 5. Dependence of the components of end-to-end distance on the chemical potential. Box sizes are indicated in the legend. Runs were started from a nematic initial conformation. Vertical dashed lines indicate the locations of transitions for these boxes, as estimated in our previous papers.37,38

nematization transitions in the thick films, which shows up by a jumpwise increase of the order parameter S1 measuring long-range nematic order, is reflected in a jump-wise increase of Re,xy2 and a corresponding jumpwise decrease of Re,z2. On the other hand, the continuous transition of the surface layer is seen in these quantities that are averaged over the whole films as a mild inflection point in Re,xy2 only for the very thin films. Also, the gradual, nonsingular onset of orientational order in the film, which is reflected in the fluctuation maxima in Figure 1 (dotted curve), shows up as a smooth inflection point in Figure 5 for the somewhat thicker films (D = 30 to D = 50). Such gradual changes in the local order cannot be distinguished from true continuous phase transitions, from a consideration of local chain conformations alone. The variation of the total energy per chain with the chemical potential for different box sizes D and L is shown in Figure 6a for runs from the nematic to the isotropic state. The figure 1211


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Intramolecular Stiffness. The energy shown in Figure 6a is closely related to the measure of the local stiffness of the chains, the average bond angle, as discussed above. A histogram of the cosine of the bond angles below and above the isotropic−nematic transition in a thin box is shown in Figure 7.

Figure 8. Orientational correlation function between bond vectors along the chain. Box size and μ-values are indicated in the legend. The slope at k → 0, which is equal to the average cosine between subsequent bond vectors, ⟨cos θi,i+1⟩, is used for estimation of persistence length. Figure 7. Histogram of the cosine of the bond angles below and above the isotropic−nematic transition (chemical potential μ = −177.0 and −170.0, respectively). The box dimensions are D = 10 and L = 500.

curved, and the curvature for small k is concave (as expected, when a power law in k is plotted on a semilog plot) that crosses over to convex curvature for large k. This is borne out by most of our data for the present problem as well: so even if there occur some regimes of k where a straight line fit works, in most cases the result will depend strongly on our choice of chain length N = 20, which by no means has a special physical significance, of course. It makes more sense to look for a measure of persistence length which describes the local intrinsic stiffness of a chain and is independent of its chain length N, of course, and that is achieved by eq 8, which relates to the initial slope of Ck (from k = 0 to k = 1) in Figure 8. This quantity clearly is related to a description of a semiflexible chain (without excluded volume interaction) in terms of the Kratky−Porod model for worm-like chains, but eq 8 makes perfect sense in the presence of excluded volume interactions as well. At this point we recall that experimental work (see, e.g., ref 50 for an application to synthetic polymers and ref 51 for an application to biopolymers) on semiflexible polymers broadly relies on the prediction of the Kratky−Porod worm-like chain model, eq 1, for the end-to-end distance of the chain

Below the transition (in the isotropic phase) all possible angles between subsequent bond vectors are present in the solution, including those larger than π/2, while in the nematic phase only a few angles larger than π/2 survive in the histogram. This is an evidence of an effective stiffening of the chains after the transition to the nematic state (for the film thickness D = 10 shown here, the transition actually is a continuous transition to an Ising-type order where the chains are predominantly aligned along either the x-axis or y-axis, respectively). Figure 7 also illustrates one advantage of the bond fluctuation model, in comparison to the simple self-avoiding walk−type lattice model used by Hsu et al.:41 while the latter allows only for two discrete values cos θ = 0, 1, Figure 7 shows that cos θi,i+1 is almost continuously distributed between −1 and +1, in the isotropic phase. When one analyzes the orientation of individual bonds in more detail and asks what fraction of the bond vectors is oriented along the x- and y-axes in comparison with all other bond vectors for different values of the chemical potential, one finds that the fraction of the bonds along x- and y-axes (there are only 8 such bond vectors) is very large (about 30%) in comparison to the rest (100 bond vectors), and it increases with increasing density (e.g., for the film thickness D = 10 this fraction is about 25% at μ = −177.8 below the ordering transition and about 35% at μ = −170.0 above the transition). As has been already mentioned in the section on our model above, the traditional way of describing chain stiffness uses the concept of the persistence length lp′ , defined as the exponential decay length of the orientational correlation function Ck (eqs 6 and 7) between subsequent bonds. However, it is well-known that this concept makes sense only for strictly ideal chains, and in reality (including chains in the melt,49 and solutions of good or marginal quality) Ck exhibits a power-law decay41,49 with k, for chain length N → ∞. For chains of finite length, this decay is cut off by a rapid loss of correlation when k becomes of the order of N/2. Consequently, a semilog plot of Ck versus k (Figure 8) typically is not simply a straight line (with slope −1/ lp′ ), as eq 7 would imply, but rather the function ln Ck versus k is

2

⟨Re ⟩KP

⎛ ⎛ 3 ⎞⎤⎞ lp ⎡ ⎜ ⎢ = 2lp 3⎜1 − 1 − exp⎜⎜ − ⎟⎟⎥⎟⎟ ⎢ 3 ⎝ lp ⎠⎥⎦⎠ ⎣ ⎝

(9)

In most experiments, the bending stiffness κ cannot be directly measured, so it is inferred from estimates of lp using eq 9: knowing 3 and ⟨Re2⟩, one gets an estimate for lp if the above equation is accurate. For example, for semiflexible polymers adsorbed on substrates, both 3 and ⟨Re2⟩ can be inferred from AFM pictures of the chain configurations, ignoring, however, the problem that the statistical properties of adsorbed chains differ significantly from those of free chains in solutions. Figure 9 shows the dependence of l′p estimated from ⟨cos θi,i+1⟩ via eq 8 on μ for several choices of D and additionally the corresponding variation of the average bond length lb. One sees that the latter smoothly decreases from about 2.66 for dilute solutions to 2.60 for dense nematic melts. Within statistical errors, no significant dependence on film thickness D can be detected. Such a decreasing of average bond length in the bond 1212


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Figure 10. Profiles of the persistence length, l′p, (a) and the average bond length, ⟨lb⟩, (b) for different μ-values (indicated in the legend) for a box width D = 30.

Figure 9. The μ-dependence of the persistence length, l′p, (a) and the average bond length, ⟨lb⟩, (b) for different values of the film thickness (indicated in the legend).

Noting the fact that in our case in the dilute limit the contour length 3 = (N − 1)lb ≈ 50 exceeds the persistence length l′p only by a factor of 2, we can use our estimate lp′ ≈ 22 for D = 150 in the disordered phase to obtain from the above equation the Kratky−Porod estimate ⟨Re2⟩KP ≈ 1353 while the actual observation is ⟨Re2⟩ ≈ 1444. This comparison shows that in the dilute limit as well as in the semidilute disordered phase (for thick films where surface effects are small) the Kratky−Porod model leads only to errors of the order of 5−10%. This is in accord with the results from ref 41, where it was shown that the worm-like chain model reasonably describes the extension of thin semiflexible chains in 3d for 3 /lp′ ≤ 5. On the other hand, we see that in the nematic phase a strong increase of l′p has occurred (Figure 9), and so l′p is not an intrinsic property of the chain but strongly dependent on the conditions of the environment. This conclusion holds in the same way for the thin films, where surface effects lead to additional changes of l′p (see Figure 9). Thus, confinement of semiflexible chains between hard repulsive walls causes an interesting interplay of conformational changes and wall-induced orientational order. This means that one can no longer rely on the relation between end-to-end distance and persistence length predicted by the Kratky−Porod model (eq 9) when a significant amount of orientational order is present in the system. This is demonstrated in Figure 11. In Figure 11a, the prediction for the ⟨Re2⟩ as a function of μ, resulting when l′p is used in eq 9, is compared with the actually observed values. We see that the increase of ⟨Re2⟩ due to the onset of nematic ordering is

fluctuation model with increasing density has been first reported in ref 52. The story for the persistence length is completely different: for thin films (D = 20) there is a monotonic almost linear increase with μ from lp′ ≈ 23.4 to about lp′ ≈ 30, although lb decreases in the same range (Figure 9a,b). Note that l′p is still smaller than the contour length 3 = (N − 1)lb which varies from 3 ≈ 53.2 to 3 ≈ 52 in this range, but it is also clear that 3 does exceed lp′ only by about a factor of 2. Thus, we work neither in the regime of semiflexible coils (3 ≫ lp′ ), which would have less tendency for nematic order, nor in the regime of almost rigid rods (3 ≪ l′p), but rather in an intermediate regime where 3 and l′p are comparable. Of course, this choice has been deliberately made because only in this intermediate regime can we expect to see an interesting interplay of conformational degrees of freedom with orientational order. For thick films (such as D = 150) we see that l′p stays approximately constant for the whole regime of the disordered phase, from μ = −178 to μ = −168, and at the capillary nematization transition a jumpwise increase of l′p from l′p ≈ 23 to l′p ≈ 26 is seen. Conversely, for films of thickness D = 30 or D = 40 a rather gradual increase is found, as expected from the phase diagram (Figure 1). It is also interesting to refine this analysis and consider the profile of l′p(z) across the film (Figure 10a). Indeed there is a significant enhancement of l′p for layers adjacent to the hard wall. Conversely, no such effect is seen for the bond length (Figure 10b), at least within statistical errors. 1213


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density on the persistence length is negligible (smaller than 20%) in the isotropic phase. Profiles of Aspect Ratio, Chain Linear Dimensions, and Monomer Densities. Profiles of the aspect ratio, A, with dchain = 2, for D = 150, 50, and 30 and different μ-values are shown in Figure 12. The profiles in the films of different thickness are rather similar to each other at the largest density (μ = −166, nematic state) as well as at the smallest density (μ = −174, isotropic phase with disordered surface). For intermediate values of μ, however, there are marked differences between the profiles. First, the isotropic−nematic transition

Figure 11. (a) Comparison of the end-to-end distance (open symbols show “measured” data while solid symbols show data calculated from the intrinsic chain stiffness). (b) Comparison of the persistence length obtained by means of two different procedures.

strongly underestimated. This failure is easy to interpret: according to the Kratky−Porod model, the successive deflections of bond orientations (of order ⟨cos θi,i+1⟩) add up in an uncorrelated fashion (consistent with the exponential decrease of ⟨cos θi,i+k⟩ with k for very long chain). However, for a semidilute solution with non-negligible nematic order (as is present here; cf. Figure 2a), the order requires a much stronger correlation of distant bonds along the chain, and hence ⟨Re2⟩ is much larger than predicted. Alternatively, if we apply the experimental procedure to infer the persistence length from ⟨Re2⟩ and the known contour length in our “computer experiment” (we will denote this estimate for the persistence length by lp in the following) (Figure 11b), the resulting value is dramatically larger than l′p determined from the average bond angle. Again the conclusion is that the formula, eq 9, applicable for relatively stiff and not very long isolated chains (i.e., dilute solutions) cannot be used to estimate local chain stiffness in cases where interchain correlations become important and cause nematic short-range order or even long-range order. We also note that in the theoretical descriptions of the isotropic−nematic transition of worm-like chains15−17 the aspect ratio of the chains, which is given by A′ = lp or A′ = lp′ depending on which observable is employed to determine the persistence length, is assumed to be the second control parameter for this transition, independent of the density. This assumption seems to be quite reasonable for the lyotropic isotropic−nematic transition of semiflexible polymers in solutions because the effect of the increasing

Figure 12. Profiles of the aspect ratio, A = ((Re2)1/2/2), for D = 150 (a), 50 (b), and 30 (c) and different μ-values shown in the legend. 1214


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Figure 13. Profiles of xy- and z-components of the squared end-to-end distance, Re,xy2 and Re,z2, for box 100 × 100 × 150 and different μ-values shown in the legend.




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takes place at different μ-values and is either a sharp transition (for D = 150) or a continuous transition (for D = 50 and 30). In the thick box there is a discontinuous change in the shape of profiles in a narrow interval of μ while the profiles change almost continuously in thin films and thus distinctly reflect the change in the phase diagram with film thickness. Second, the profiles A(z) in thick films (D = 150) in the isotropic state have the maximum at the walls, then show a minimum (at z ≈ 17), and finally reach some constant value in the center of the box (at z ≥ 25). Upon increasing density, the minimum becomes less deep, its position is shifted to the larger z-values (but its right border stays at z ≈ 25), and the plateau value of the aspect ratio in the center of the box stays basically constant, until it jumps to much larger values at the isotropic−nematic transition. In thin films (D = 50 and 30) no plateau in the center of the film is reached, and even the minimum is not visible for D = 30 while it is still visible for D = 50, almost reaching a flat maximum in the center of the film (at z ≈ 25). We can conclude that the position of the minimum and its depth are the same in different boxes, but in a very thin box D = 30 there is no space inside the box for this minimum to be developed. Moreover, the “width” Δz of the minimum (Δz ≈ 25) is comparable with the value of the aspect ratio itself (which is between 19 and 23). From the profiles of the aspect ratio it is obvious that irrespective of μ there is a characteristic range ξ ≈ 20 lattice spacings over which the effect of the walls is felt. Only when D is clearly larger than 2ξ can we have a capillary nematization transition. Profiles of the xy- and z-components of the squared end-toend distance are shown for different box sizes in Figures 13, 14, 15, 16, and 17 (the values of the chemical potential μ are indicated in the legends). In the profiles of the components of the mean-squared end-to-end distance parallel, Re,xy2, and perpendicular, Re,z2, to the walls we now see a superposition of two effects, namely (i) the surface effect on the aspect ratio (as shown in Figure 12) and (ii) the tendency of the rather stiff chains to align the long axis of their gyration tensor in the xyplane in the region where the nematic precursor layer of the surface-induced ordering occurs. In ref 38, we have presented evidence that for the semi-infinite limit (D → ∞) the thickness, S , of this surface-induced layer diverges logarithmically when the bulk transition from the isotropic to the nematic phase is approached, S ∝ −ln|μc − μ|. Thus, for very large D, this length S can become much larger than the length ξ identified for the surface effect of the aspect ratio, which is of the order of 20 lattice spacings both in the isotropic and the nematic phase of the system. In contrast, for D = 150 (and similarly for D = 500; in the interest of saving space the latter data are not shown) we see that it takes about 40 lattice spacings to reach bulk behavior in the isotropic phase (Figure 13). Note that for any finite D the logarithmic divergence is cut off, of course, by the capillary nematization transition (even if one assumes pronounced metastability of the isotropic phase, the growth of the wetting layer cannot exceed S ≈ D/4 to D/3, since there must be a welldefined region of isotropic phase in the center of the film at the transition point μc(D) of this transition). For D = 50 (Figure 14), as expected from the phase diagram38 (no well-developed jumps in any quantity occur any longer, so the first-order capillary nematization transition is already lost), the behavior of Re,xy2 and Re,z2 with μ is completely gradual, and there is no flat plateau in the center of the film in the transition region (of course, due to the symmetry of the profiles with respect to z = D/2, all profiles must show horizontal minima or maxima there,

Figure 17. Profiles of xy-components of the squared end-to-end distance, Re,xy2, for μ = −166, −170, and −176 (from top to bottom) and different box sizes shown in the legend.

irrespective of μ). It is also interesting to note that profiles of Re,z2 in the disordered phase still exhibit the change of curvature from concave to convex near the walls, even for D = 30 (Figure 15). Only for D = 10, the behavior of this profile is uniformly convex. It is also interesting to note that Re,z2 in the disordered state of the film reaches its maximum value only near the center (z = 25) for D = 50, while for D = 30 the saturation already is reached for z = 12. We interpret this difference by the fact that D = 50 is rather close to the end point of the capillary nematization transition line. This end point is a critical point (falling presumably in the universality class of the twodimensional Ising model) and hence associated with longrange critical fluctuations. The effect of such fluctuations at D = 50 is still somewhat felt, but less so for D = 30. The fact that for D = 50 in the disordered phase Re,xy2 decreases rapidly with increasing distance z from the wall, while the component Re,z2 is increasing less rapidly to reach a 1216


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maximum value in the center of the film, is responsible for the effect that Re2 (and hence A, Figure 4) for μ = −172 and μ = −170 exhibits a shallow minimum near z = 18 or z = 20, while there is a small maximum again at z = 25. All these effects are due to a strong coupling between the chain linear dimension Re,xy2 (and Re2) and the onset of orientational order. It is important to note that a seemingly similar phenomenon occurs in melts of flexible chains near hard walls, but there the origin of the minimum in Re2(z) is completely different:53 in a melt of flexible long chains, Re,z2 decreases for z < Re already, while for z < Re the lateral linear dimension Re,xy2 increases only much closer to the wall, namely, when z becomes comparable to the screening length of the excluded volume interactions (which in a melt is only of the order of a few bond lengths). In our case, the density is low, and the screening length is of the same order as Re, but since the chains are very short, a regime of Gaussian chain statistics is not developed. In Figure 16, we demonstrate that even for very thin films (D = 10) the components of the squared end-to-end distance, Re,xy2 and Re,z2, still exhibit nontrivial profiles across the film. In contrast, the profiles of the (Ising-type) order parameter are close to a trivial horizontal variation.38 Figure 16 shows that the constraint of packing stiff chains between two parallel hard walls a small distance D apart still gives rise to a nontrivial zdependence, the system is not yet completely equivalent to a strictly two-dimensional system. This conclusion is not evident if one studies only orientational order parameters. Finally Figure 17 shows that away from the region where capillary nematization occurs the initial decay of the profile of Re,xy2 vs z does not depend on film thickness D: this region is controlled by the wall effects already identified from the aspect ratio. Let us compare the values of the end-to-end distance in different layers in a thick film D = 150 for different average densities with those values in the bulk which we have obtained in our previous studies.24,28 As has been already mentioned in our previous papers on the nematic ordering in thin films38 and is well visible in Figures 4 and 12, the average end-to-end distance Re ≡ (Re2)1/2 is approximately equal to Re ∼ 35 in the isotropic state and to Re ∼ 45 in the nematic state; i.e., it increases upon increasing average density. In the layers at the walls of the film it is always Re ∼ 45. In the pure bulk the behavior was slightly different:28 there was a very small decrease in Re upon increasing the polymer density from Re ∼ 38 at ϕ ∼ 0.1 to Re ∼ 37.4 at ϕ ∼ 0.3. Profiles of the monomer and end monomer densities (volume fractions) are shown in Figure 18 for D = 150 and L = 100, and Figure 19 shows the ratio of Ne(z)/N(z) = ϕe(z)/ ϕ(z). From Figure 18 we see that in the dilute limit (realized for μ=−180), where the system is in the isotropic phase even close to the walls, monomers are depleted near the walls, while chain ends are already slightly enriched near the walls even then. For μ = −174, still in the disordered phase of the surface but close to the SD → SO transition (Figure 1), this enhancement of chain end density is more pronounced followed by a shallow minimum, and also the density of all monomers is slightly enhanced there. We note that the enhancement of chain ends extends on both sides of the SD → SO transition over a range of about 10−20 lattice spacings. We interpret this fact by the observation (cf. Figure 13) that in the region, where R e,xy 2 is significantly enhanced and R e,z 2 depressed, both chain ends typically are relatively close to the wall, different from a bulk isotropic phase where ϕend = 2ϕ/N =

Figure 18. Density (volume fraction) profiles for all monomers (a) and for chain ends (b) for a box of dimensions 100 × 100 × 150 and different μ-values shown in the legend.

0.1ϕ (for N = 20, as used here). In the dense region of the wellordered nematic phase (e.g., μ = −160), both ϕ(z) and ϕe(z) show pronounced density oscillations near the wall, i.e., the “layering effect” familiar from the packing of hard spheres near hard walls. To get a clearer picture on the chain end enrichment effects, it is useful to normalize ϕe(z) by ϕ(z); this is shown in Figure 19a for D = 150, again varying μ, and (b) for μ = −170 and in (c) for μ = −176, comparing now all the different film thicknesses studied. It is seen that data for D = 150 and D = 500 agree within error, displaying a clear maximum close to the wall (with a superimposed layering effect), followed by a shallow minimum near z = 25 to z = 40 (the precise location of this minimum depends on μ, as is clearly seen in (a)). Figures 19b and 19c also imply clearly that D = 70 is not thick enough to obtain the behavior characteristic for the semi-infinite system, near the minimum, and for D = 50 even the shape of the maximum near the wall gets affected by too small thickness. The enhancement of chain ends at the narrow interface at the solid substrate in the SD state has been observed for flexible chains in self-consistent field theory and simulations.54,55 Chain ends segregate to the narrow interface because only one (instead of two) bonds emerge from them. The surface restricts the possible bond orientations, and therefore chain ends suffer a smaller loss of conformational entropy than middle monomers. This enrichment of chain ends at the narrow interface is compensated by a depletion in wider interphase, which extends a distance Re,z farther away from the surface. On even larger length scales, chain connectivity prevents deviations 1217


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the concomitant reduction in the wider interface is difficult to observe.

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CONCLUSIONS AND OUTLOOK In this paper we have presented extensive Monte Carlo simulations of a lattice model for semiflexible polymers confined by hard walls in a slit geometry, in order to elucidate how the orientational order and in particular the associated surface effects due to the repulsive walls affect the structure of the polymer chains. We have varied both the thickness D of the thin film and the concentration ϕ over a wide range so that all points of the phase diagram are explored (Figure 1). As in our previous work where this phase diagram of the model was constructed, we focus on a single chain length (N = 20 effective monomers) and a single energy parameter that controls the chain stiffness ( f = 8.0 in eq 4), which is chosen such that in dilute solution in the bulk the persistence length lp′ comprises about eight effective bond lengths (lb). Of course, in order to exhibit phase transitions from isotropic to nematic phases in a polymer solution, it is essential that the persistence length lp′ of the semiflexible chains is sufficiently larger than the bond length lb; on the other hand, if lp′ ≫ 3 the polymer character of the macromolecules would no longer have much effects, we would have reached essentially the Onsager limit of a solution of hard rods. Our simulations confirm the formation of the surfaceinduced nematic ordering in semidilute solutions of semiflexible chains of moderate stiffness in a thick film (where there is still enough place in the center of the film to represent the bulk behavior). For the chosen values of parameters there exists an entropically driven adsorption of polymersthere is only a small increase of the polymer concentration at the wall in comparison to the bulk in the center of the boxbut a wellpronounced surface-induced nematic ordering (strong increase of the orientational order parameter) is observed. The chain conformations are also different in the center of the film and at the wallsthe aspect ratio has maximal values at the walls; i.e., chains are more elongated at the walls in comparison to the center of the film. Moreover, even in the center of a thick enough film the chains are more elongated than in the bulk. In our study we have investigated various length scales of interest (beyond the size of the effective monomer, the lattice spacing), paying attention to the behavior of these length scales with our control parameters D and ϕ and relating this behavior to the onset of orientational order in the system, including also the variation of this behavior as a function of distance z from the walls. The smallest length scale of interest is the length lb of the effective bonds: we find that its z-dependence is negligible, and it also does not depend on D significantly, while there is a slight decrease of lb with increasing chemical potential μ (and hence increasing polymer density in the system). Recall that physically an effective bond in our model represents something like 3−5 chemical monomers of an atomistically detailed model of a macromolecule, and hence it is this length scale which reflects how densely the monomeric units in the system are packed. As expected, it is this length scale on which in dilute solutions we see some depression of the monomer density at the wall, and in semidilute solution, in contrast, an enhancement, which is linked to the two-dimensional ordering of chains which occur close to the walls (SD → SO transition); in the dense nematic phase, lb is the length scale over which the monomer density exhibits layering, while the chain end density

Figure 19. (a) Profiles of the ratio of the number of end monomers to all monomers Ne(z)/N(z) (equivalent to the ratio of densities of end monomers to all monomers ϕe(z)/ϕ(z)) for a box of dimensions 100 × 100 × 150 and different μ-values shown in the legend. These profiles for D = 150 are very similar to those for D = 500. (b) Same as (a) for μ = −170 and different box sizes shown in the legend. (c) Same as (b) for μ = −176.

of the relative chain end density. Although our chains are semiflexible, we observe a similar behavior in the SD state of our system. In the SO state, however, the driving force for chain-end segregation is significantly reduced because the chains are aligned parallel to the confining wall and, therefore, the preferred bond orientation is compatible with the geometric constraint imposed by the solid substrate. Thus, the entropy loss of middle segments compared to that of end segments in the narrow interface is not as large and the end segregation is reduced. Additionally, for highly aligned chains, the distinction between interface and interphase breaks down. This rational argument is compatible with the simulation data in Figure 19. Since the chain-end enrichment is very small in the SO state, 1218


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exhibits in the semidilute and concentrated solutions a sharp peak in this narrow region adjacent to the wall. The next length scale of interest is the persistence length of the chain. We find that this length reflects the degree of nematic order in the system rather distinctly and cannot hence be taken as an intrinsic property of a single isolated chain; rather, the effective stiffness of the chains that this quantity measures is a collective property already, contrary to naive expectations. There is also a clear enhancement of lp′ for chains that have their center of mass close to the wall; this enhancement (from 22 to about 30 in the semidilute regime and from 28 to about 38 in the concentrated regime, for the example D = 30 shown in this paper) again occurs on the scale of the bond length lb only. Thus, chains that are stretched out adjacent to the walls are effectively somewhat stiffer. A wallinduced stiffening of semiflexible chains has also been observed for chains that undergo an adsorption transition albeit for a different model.56 Despite the fact that lp′ distinctly depends on both chemical potential μ (or density, respectively) and film thickness D (because the latter controls the onset of orientational order), we find that l′p (determined from the local bending; see eq 8) is roughly compatible with the Kratky− Porod model, when one uses the relation between Re2 and 3 to the estimate lp for the persistence length (eq 9). The latter procedure, used widely by experimentalists, hence is found to be valid for dilute solutions of semiflexible chains (apart from possible errors less than 10%), despite the fact that it is not an “intrinsic” property that is determined in this way, but one measures a property that sensitively depends on the environment of the chain (concentration of the solution, degree of orientational order, possible position of a chain close to the wall, etc.). In semidilute solutions, however, these “environment effects” become very strong, and one can no longer rely on eq 9. We emphasize this fact here because we feel that it is not widely appreciated in the literature. On the length scale of single chain extensions (or the persistence length, respectively, since these lengths are of the same order), we also find surface effects on quantities like the aspect ratio, but also local properties such as the density of effective monomers and the density of chain ends show significant variations. These effects are related to the phenomenon that the hard wall has an orienting effect on the end-to-end vector of our rather stiff chains, when the chain is close to the wall, and this effect is present irrespective of the amount of global orientational order in the system. When one looks on individual components such as ⟨Re,xy2⟩, however, one finds a stronger coupling to the orientational order: both when (in a thick film) a surface-induced nematic layer forms and when (in a thin film) gradually nematic order induced by the walls is established in the whole film, it is reflected in this single chain property that ⟨Re,xy2⟩ becomes much larger than ⟨Re,z2⟩. The thickness of the surface-induced nematic ordering layer in very thick films close to the capillary nematization transition actually is a collective length scale that can be much larger than any of the length scales associated with single chains. Thus, studying the profiles of single chain properties in the direction perpendicular to the walls still yield a complete information on the phase behavior of the system as well. We hope that the present work will help to understand better possible experiments on such systems.

Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (V.A.I.). Present Address

M.R.S.: Schlumberger Dhahran Carbonate Research Center, P.O. Box 39011, Al-Khobar 31942, Saudi Arabia. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge the financial support from DFG and RFBR (joint project grants DFG 436 RUS 113/791, RFBR 09-0391339 and 12-03-91334) and from SFB TRR 102. This work was partially supported by the Ministry of Education and Science of the Russian Federation (Contract Nos. 16.523.12.3001 and 8023). Very useful discussions with E. An are kindly appreciated. Simulations have been performed at the Supercomputer Center of MSU.

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REFERENCES

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Macromolecules

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