Nematic ordering of worm-like polymers near an interface

Abstract

The phase behavior of semi-flexible polymers is integral to various contexts, from materials science to biophysics, many of which utilize or require specific confinement geometries as well as the orientational behavior of the polymers. Inspired by collagen assembly, we study the orientational ordering of semi-flexible polymers, modeled as Maier–Saupe worm-like chains, using self-consistent field theory. We first examine the bulk behavior of these polymers, locating the isotropic–nematic transition and delineating the limit of stability of the isotropic and nematic phases. This effort explains how nematic ordering emerges from the isotropic phase and offers insight into how different (e.g., mono-domain vs multi-domain) nematic phases form. We then clarify the influence of planar confinement on the nematic ordering of the polymers. We find that while the presence of a single confining wall does not shift the location of nematic transition, it aligns the polymers in parallel to the wall; above the onset of nematic transition, this preference tends to propagate into the bulk phase. Introducing a second, perpendicular, wall leads to a nematic phase that is parallel to both walls, allowing the ordering direction to be uniquely set by the geometry of the experimental setup. The advantage of wall-confinement is that it can be used to propagate mono-domain nematic phases into the bulk phase.

I. INTRODUCTION

The organization of semi-flexible polymers is relevant in a variety of contexts and technologies, ranging from materials science to biology, biotechnology, collagen assembly,^1,2 and DNA packing in a viral capsid.³ A distinguishing feature of these polymers is their anisotropic interactions, which are often responsible for the formation of ordered structures, such as nematic or smectic phases, with novel and useful mechanical⁴ or optical⁵ properties. Utilizing these polymers often requires confining them, either fully by encapsulating or partially by restricting their degrees of freedom in one or more dimensions, as dictated by the experimental or biological setting. Confining semi-flexible polymers to small spaces is a field of active research.^6–9 However, there also remain open questions about how partial confinement, i.e., the presence of a plate adjacent to bulk samples, affects their bulk behavior.

The worm-like chain (WLC) is a standard coarse-grained representation of semi-flexible polymers.^10–12 Although much simplified compared to more microscopic (e.g., atomistic) approaches, WLCs store many degrees of freedom (e.g., conformational and orientational). This poses a significant barrier to both theoretical and computational modeling of WLCs. Accordingly, a number of approximate methods have been developed,^13–16 most notably self-consistent field theory (SCFT) methods^12,17–19 (see Refs. 20 and 21 for recent applications of these methods to nematic polymers). These methods have been applied to a wide variety of systems, including DNA,¹³ bottle-brush polymers,²² and nematic polymers.^19,23,24

Although the isotropic–nematic coexistence line has been located,²⁵ other factors play a role in the transition. These include the limits of stability of each phase, free energy barriers, the mechanism of the transition, and critical nucleus sizes. Although the majority of these are beyond the scope of this paper, we examine the limits of stability.

Furthermore, open questions remain regarding how confinement will affect the equilibrium behavior of WLCs. Confinement often brings about changes in polymer behavior that are of both fundamental and practical interest.^26–29 For instance, it has non-trivial effects on the self-assembly or phase behavior of heteropolymers (e.g., diblock copolymers)²⁶ or biomolecules,^28,29 as well as on the glassy behavior of polymers (e.g., enhanced mobility).²⁷ In a polymer melt, planar confinement induces segregation of short chains toward the confining boundary;^30–32 in a rigid-rod system, it aligns the rods parallel to walls.^9,33 Clarifying how polymers will behave next to surfaces or under confinement can be challenging. Nonetheless, the problem can be approached in a few different ways (see Ref. 9 and relevant references therein).

Of particular interest is the possibility of long-range effects caused by single-wall confinement (e.g., a biological interface). The effect of a single repulsive flat wall on (mono-disperse) flexible polymers is nominally expected to be local. The wall effect will not propagate much beyond a characteristic length scale set by chain statistics.³⁴ What remains to be explored is the interplay between wall-confinement and anisotropic interactions in semi-flexible polymers.

Earlier effort along this line has been focused on short, rigid molecules,^33,35 which are of obvious interest for applications such as liquid-crystal displays.³⁶ For WLCs, another length scale, i.e., their persistence length, becomes relevant; locally, they behave as rigid rods, but at large scales, they resemble flexible polymers. As a result, they combine both features of flexible chains and rods. It is thus unclear how the local rigidity of WLCs and wall-confinement are intertwined in determining their phase behavior and orientational ordering.

Here, we study the orientational organization of WLCs near an interface using SCFT and choosing parameters that mimic those of collagen. The orientation-dependent interactions in WLCs can be characterized by Maier–Saupe (MS) or Onsager theory.^12,37,38 If Onsager interactions arise from purely repulsive, excluded-volume interactions that discourage misalignment, MS interactions encourage alignment;^12,37,38 orientation-dependent van der Waals interactions are the quintessential Maier–Saupe interactions. Despite the apparent difference, both interactions tend to align WLCs. We choose the MS model primarily because of its easy implementation and historical success³⁹ (also see Ref. 25 for relevant discussions).

We first examine the bulk behavior of collagen-like WLCs, locating the isotropic–nematic transition, delineating the limit of stability of the isotropic and nematic phases. This gives insight into the formation of the nematic phase, as it allows one to predict whether the nematic transition will proceed via nucleation or spinodal decomposition.

We then clarify the role of confinement on the ordering of WLCs. Above the onset of nematic ordering in the corresponding bulk (unconfined) system, the planar-wall confinement tends to align WLCs parallel to the wall at z = 0 all the way up into the bulk phase far from the wall. This naturally sets the direction of nematic ordering, i.e., parallel with the x–y plane. Nevertheless, the WLCs near the confining wall remain randomly oriented in its plane. Below the onset of nematic ordering, however, the wall effect does not propagate into the bulk phase and is local. In the presence of two perpendicular plates, the direction of nematic ordering is uniquely determined. Our results suggest that wall-confinement can be used to produce mono-domain nematic phases.

II. THEORY

This section describes self-consistent field theory (SCFT) for examining the equilibrium behavior of a solution of semi-flexible polymers in the presence or absence of a planar surface, as illustrated in Fig. 1. Each polymer is approximated as a worm-like chain (WLC): a continuous space curve of contour length L and persistence length ℓ_p. Each WLC consists of N segments, or monomers, of length b = L/N each.

FIG. 1.

Worm-like chains adjacent to a flat surface (left) and a field representation (right). The field representation focuses on one chain and replaces the influence of others by a field w(r, u) it is subject to. In self-consistent field theory (SCFT), w(r, u) and the spatial distribution of “particles” (e.g., chain segments), denoted as ρ(r, u), are determined self-consistently. First, note that w(r, u) and ρ(r, u) are interrelated. We adjust w(r, u) and ρ(r, u) iteratively until they are “correctly” related—this is the essence of SCFT.

Our calculations reflect n WLCs in some volume V. Let ρ₀ be the average segment concentration: ρ₀ = nN/V. The volume explored by each WLC is N/ρ₀ = V/n. Note that these segments are not closely packed, except in a melt; the remaining space in V is filled with solvent molecules of volume v_s each.

As sketched in Fig. 1, a field theoretical representation reduces the n-chain system to one chain in an external field, $w\left(\mathbf{r},\mathbf{u}\right)$, which captures the influence of others. The field w(r, u), which the chain is subject to, determines its segment distribution ρ(r, u). In addition, the interactions between molecules determine a relationship between ρ(r, u) and w(r, u), a field condition. The former statement allows us to calculate ρ(r, u) from w(r, u), while the latter enables us to determine w(r, u) from ρ(r, u). Combining these relationships produces self-consistent field theory (SCFT), whereby ρ(r, u) and w(r, u) are determined self-consistently. We start with an initial guess for the field, w(r, u), which we then adjust iteratively until ρ(r, u) and w(r, u) are “correctly” related, i.e., related by the field condition detailed below.

Each polymer is identified with a space curve, r_α(s), where s is the normalized (0 < s < 1) position along the chain. We can thus define a dimensionless, scaled, polymer-segment or monomer concentration (i.e., the number of polymer segments per volume),

$$\widehat{\rho }\left(\mathbf{r},\mathbf{u}\right)=\frac{N}{{\rho }_0}\sum _{\alpha =1}^n{\int }_0^{1}ds\delta \left(\mathbf{r}-{\mathbf{r}}_{\alpha }\left(s\right)\right)\delta \left(\mathbf{u}-{\mathbf{u}}_{\alpha }\right),$$ (1)

where

$${\mathbf{u}}_{\alpha }\left(s\right)\equiv \frac{1}{L}\frac{d{\mathbf{r}}_{\alpha }}{ds}$$ (2)

is the unit tangent vector to the αth chain at monomer s. The prefactor N/ρ₀ ensures that the segment concentration is normalized such that $\int \int \widehat{\rho }d\mathbf{r}d\mathbf{u}=V$.

Each segment interacts not only with other segments but also with solvent molecules. In the much-celebrated Flory–Huggins approach, monomer–solvent interactions as well as the entropy of mixing are subsumed into the Flory–Huggins parameter, χ. If the former favors segregation between the polymer and the solvent, the latter encourages mixing. In this implicit-solvent picture, the effect of solvent is viewed as renormalizing the monomer–monomer interaction. The strength of the resulting isotropic interaction is set by the excluded-volume parameter ${\nu }_0={\left(v_s{\rho }_0\right)}^{-1}-2\chi$, as discussed in Appendix A.

In the MS model employed in this work, nematic ordering of rod-like molecules arises from attractive interactions (e.g., van der Waals interactions) between them. Imagine considering a rod in such a rod system. What kind of effective interaction will it experience? This interaction should remain invariant under u → −u, unless the rods have polarity. The simplest choice is −νP₂(u ⋅ u′)^12,37,38 (or −ν(u ⋅ u′)² up to an additive constant), where ν is the nematic strength and P₂(x) = (3x² − 1)/2 is the Legendre polynomial of degree 2. The parameter ν is expected to increase approximately linearly with the rod concentration; it can thus be, loosely, thought of as a proxy for the rod concentration (see below for a similar issue in the Onsager model).

In the alternative Onsager approach, the excluded-volume repulsion between rods is responsible for their alignment. If D is the diameter and L is the length of each rod, this interaction is approximately proportional to ρ₀DL²|u × u′|; rods can pack more effectively at higher concentrations.

The resulting Hamiltonian for the system or WLCs with the MS interactions can then be written as

$$\begin{array}{ccc}\hfill \frac{\mathcal{H}}{k_{\mathrm{B}}T}& =\hfill & \frac{{\ell }_{\mathrm{p}}}{2L}\sum _{\alpha =1}^n{\int }_0^{1}ds{\left|\frac{d{\mathbf{u}}_{\alpha }}{ds}\right|}^2+\frac{{\rho }_0}{2}\int \int \int d\mathbf{r}d\mathbf{u}d{\mathbf{u}}^{\prime }\widehat{\rho }\left(\mathbf{r},\mathbf{u}\right)\hfill \\ \hfill & \hfill & \times \left[{\nu }_0-\nu P_2\left(\mathbf{u}\cdot {\mathbf{u}}^{\prime }\right)\right]\widehat{\rho }\left(\mathbf{r},{\mathbf{u}}^{\prime }\right).\hfill \end{array}$$ (3)

Here, k_B is the Boltzmann constant and T is the temperature. The first term in Eq. (3) represents the bending energy. The second term combines the isotropic excluded-volume interactions with the excluded-volume parameter ν₀ and anisotropic liquid-crystalline interactions with strength ν.

SCFT involves calculating the statistics of a WLC in a field, w(r, u), to which a polymer segment at r with an orientation specified by the tangent vector u is subject. The first step is to calculate the propagator, q(r, u, s), which is the (restricted) partition function of a WLC fragment of sN segments, with one free end and another fixed at r with orientation u. The propagator satisfies

$$\frac{\partial q}{\partial s}=\left[\frac{L}{2{\ell }_{\mathrm{p}}}{\nabla }_{\mathbf{u}}^2-L\mathbf{u}\cdot \nabla -w\left(\mathbf{r},\mathbf{u}\right)\right]q,$$ (4)

where s now parameterizes the position along a space curve of length L. Equation (4) is to be solved subject to the uniform initial condition, q(r, u, 0) = 1.

For computational ease and mathematical simplicity, it is convenient to expand the propagator and other quantities in terms of real spherical harmonics ${\tilde{Y}}_l^m\left(\mathbf{u}\right)$,⁴⁰

$$q\left(\mathbf{r},\mathbf{u},s\right)=\sum _{l=0}^{\infty }\sum _{m=-l}^l{q}_l^m\left(\mathbf{r},s\right){\tilde{Y}}_l^m\left(\mathbf{u}\right).$$ (5)

Our implementation is similar to the one described in Refs. 21, 32, 41, and 42, but it relies on the expansion of q(r, u) in real spherical harmonics⁴⁰ rather than in Legendre polynomials.^21,32,41,42 Details of the real spherical harmonic representation are presented in Appendix B, and further details of the implementation can be found in Refs. 21, 32, 41, and 42. Real spherical harmonics constitute a set of real (as opposed to complex) orthonormal basis functions that are linear combinations of the regular (complex) spherical harmonics and as such are also eigenfunctions of Laplace’s equation. For our purposes, they are functionally the same as the regular spherical harmonics, except that their real-valued nature makes them computationally simpler.

From the propagator, we can calculate the ensemble-averaged scaled polymer segment concentration, $\rho \left(\mathbf{r},\mathbf{u}\right)\equiv ⟨\widehat{\rho }\left(\mathbf{r},\mathbf{u}\right)⟩$, which is given by

$$\rho \left(\mathbf{r},\mathbf{u}\right)=\frac{V}{Q}{\int }_0^{L}dsq\left(\mathbf{r},\mathbf{u},s\right)q^{†}\left(\mathbf{r},\mathbf{u},s\right),$$ (6)

where the “back” propagator q^†(r, u, s) is defined as the s-reversed q, thus satisfying Eq. (4) with s replaced by $L$ − s. The partition function, Q, is given by

$$Q=\int d\mathbf{r}d\mathbf{u}q\left(\mathbf{r},\mathbf{u},L\right).$$ (7)

The scaling prefactor V/Q in Eq. (6) ensures that ∫drduρ(r, u) = V, effectively setting to 1 the average scaled segment concentration in Eq. (6). This is equivalent to saying that the segment concentration is scaled by the average ρ₀.

The crux of SCFT is the field condition that relates the monomer concentration and the field,^12,37,38

$$w\left(\mathbf{r},\mathbf{u}\right)=\int d{\mathbf{u}}^{\prime }\left[{\nu }_0-\nu P_2\left(\mathbf{u}\cdot {\mathbf{u}}^{\prime }\right)\right]\rho \left(\mathbf{r},{\mathbf{u}}^{\prime }\right).$$ (8)

Recall that ν₀ and ν characterize the isotropic and anisotropic interactions, respectively. It proves useful to decompose the field w(r, u) into angle-independent (isotropic) and angle-dependent (anisotropic) parts as w(r, u) = w_i(r) + w_a(r, u). In addition to being useful, this separation is a natural consequence of the spherical harmonic decomposition (see Appendix B for details).

The isotropic field condition thus becomes

$$w_i\left(\mathbf{r}\right)={\nu }_0\int d\mathbf{u}\rho \left(\mathbf{r},\mathbf{u}\right).$$ (9)

In our numerical implementation, we adjust the field iteratively until

$$\frac{1}{V}\int d\mathbf{r}{\left(w_i\left(\mathbf{r}\right)-{\nu }_0\int \rho \left(\mathbf{r},\mathbf{u}\right)d\mathbf{u}\right)}^2<\epsilon ,$$ (10)

where the error tolerance ε should be chosen to be sufficiently small. We find ε = 10⁻⁴ suffices for our numerical calculations.

If we know a priori the expected concentration profile, ρ_e(r), then this procedure can be simplified; it amounts to adjusting the field so that

$$\int \rho \left(\mathbf{r},\mathbf{u}\right)d\mathbf{u}={\rho }_e\left(\mathbf{r}\right)$$ (11)

within a tolerance of ε = 10⁻⁴, as discussed above. For the bulk behavior, we expect a uniform profile and set ρ_e(r) = 1. When ν is varied, the width of the interfacial profile changes somewhat, but the changes are small and quantitative (unless ν is far above the transition). The essential features of the confined WLCs (e.g., the location of the nematic transition and spinodals) will not be sensitive to the shape of the interfacial profile, as discussed in Appendix C. Indeed, the interfacial profile is not a universal feature but depends on the boundary condition imposed: absorbing vs reflecting, as well as the details of how excluded-volume interactions are implemented. To ease the computational demand and to enhance stability of calculations, we use Eq. (10) to calculate the interfacial profile for a given value of ν and reuse it for other values. For further discussion on the variation of the profile with ν and the historical context of imposing a profile, see Appendix C. We also find that the behavior does not change qualitatively with ν₀, and for convenience, we set ν₀ = 20, where relevant in this paper.

The anisotropic part of the field equation in Eq. (8) reads

$$w_a\left(\mathbf{r},\mathbf{u}\right)=-\nu \int d{\mathbf{u}}^{\prime }{P}_2\left(\mathbf{u}\cdot {\mathbf{u}}^{\prime }\right)\rho \left(\mathbf{r},{\mathbf{u}}^{\prime }\right).$$ (12)

As before, we solve for w_a(r, u) iteratively by adjusting it until

$$\frac{1}{V}\iint d\mathbf{r}d\mathbf{u}{\left(w_a\left(\mathbf{r},\mathbf{u}\right)+\nu \int d{\mathbf{u}}^{\prime }{P}_2\left(\mathbf{u}\cdot {\mathbf{u}}^{\prime }\right)\rho \left(\mathbf{r},{\mathbf{u}}^{\prime }\right)\right)}^2<\epsilon ,$$ (13)

where ε = 10⁻⁴ is sufficient to achieve our desired precision. Recall that this is computationally simpler when expanded in real spherical harmonics. Also note that in order to satisfy Eq. (8), we must simultaneously satisfy Eqs. (12) and (9) [or (11)]. We thus iterate until both are satisfied within the given tolerance.

In order to investigate the phase behavior of WLCs, we calculate the free energy per polymer, given by

$$\frac{F}{n{k}_{\mathrm{B}}T}=-\ln Q-\frac{1}{2V}\int \int d\mathbf{r}d\mathbf{u}w\left(\mathbf{r},\mathbf{u}\right)\rho \left(\mathbf{r},\mathbf{u}\right).$$ (14)

The isotropic–nematic transition can be predicted by numerically determining the value of ν, where the free energies of the two phase are equal.

A central quantity in characterizing the distribution of polymer orientations and nematic ordering is the nematic order parameter matrix,¹²

$$S_{\alpha \beta }\left(\mathbf{r}\right)=\frac{V}{nL}\int d\mathbf{u}\rho \left(\mathbf{r},\mathbf{u}\right)\left(u_{\alpha }{u}_{\beta }-\frac{1}{3}{\delta }_{\alpha \beta }\right),$$ (15)

where α and β are coordinate directions. This matrix is 2/3 times the so-called Q tensor, which is also used to study nematic order.⁴³ The meaning of this matrix is most obvious when we consider its eigenvalue decomposition. Each eigenvector of this matrix corresponds to an alignment direction, and the corresponding eigenvalue sets the strength of alignment in that direction. It can thus be used to study molecular alignment. For instance, the nematic phase is simple mono-axial order. It is characterized by a single large eigenvalue, which sets the degree of nematic order, and the corresponding eigenvector, which determines the alignment direction (director). The two other eigenvalues are small, degenerate, and correspond to eigenvectors that are orthogonal to the director. In the results below, we focus on the largest eigenvalue, labeled S, which is identified with the degree of order.

The presence of a confining surface tends to suppress orientations perpendicular to the surface. This results in two large, degenerate eigenvalues corresponding to eigenvectors parallel with the plane and a third, small eigenvalue corresponding to an eigenvector perpendicular to the plane. This is a special case of biaxiality; however, in general, biaxial order occurs when the two smaller eigenvalues are different in magnitude.⁴⁴

In our theoretical consideration, the wall-confinement is mimicked by the absorbing boundary condition (impenetrable wall). If there is a wall at z = 0, the system is infinite in the directions parallel with the wall, i.e., in the x and y directions. For the SCFT computation, the fields, i.e., w(r, u) and ρ(r, u), are represented discretely on a square lattice of lattice constant chosen to be ℓ_p/40. We can exploit symmetries to effectively reduce the dimensionality of the problem. In the case of a single wall at z = 0, the system does not change in the x and y directions and can be represented in a one-dimensional lattice.

Similarly, the propagator [Eq. (4)] is solved in a discretized s space with a step size Δs = 20ℓ_p/500 = ℓ_p/25. In the real spherical harmonic expansion in Eq. (5), we include terms up to l = 5, for our typical value of L/ℓ_p = 20. More terms are required for more rigid polymers.

III. RESULTS

A. Equilibrium phase behavior in a free space

1. Isotropic vs nematic

We first examine the equilibrium phase behavior of worm-like chains in the bulk. Indeed, SCFT can be used to determine the free energy of the nematic and isotropic phases [see Eq. (14)]. This allows us to locate the transition in terms of the Maier–Saupe interactions strength ν. For this, we have primarily chosen the parameters that are representative of collagen: the persistence length ℓ_p = 15 nm (thus, a Kuhn length of 30 nm) and contour length L = 300 nm.^45–47 The thermal energy k_BT is scaled out of the interactions and only appears in the scale of the free energy in Eq. (14).

Figure 2 shows our results for the nematic order parameter S (left axis) and the free energy difference between the nematic and isotropic phases F_n − F_i (right axis), as a function of ν. Recall that S is the largest eigenvalue of S_αβ in Eq. (15); the free energy is given in Eq. (14). There is a big jump in the S curve at ν ≡ ν_tr ≈ 135.423. This is correlated with the free energy curve, which indicates F_n − F_i = 0 at this value of ν. The discontinuities in both the order parameter and the derivative of the free energy at the transition indicate that the transition between the two phases is first order.⁴⁸

FIG. 2.

Equilibrium nematic order parameter S (left axis) and free energy (right axis), as a function of MS interaction strength ν for collagen-like WLCs in a free space; recall S is the largest eigenvalue of S_αβ in Eq. (15). We have chosen the persistence length ℓ_p = 15 nm and the contour length L = 300 nm: L/ℓ_p = 20. At ν ≈ 135.423, there is a sharp change in the nematic order parameter S (blue curve). This is a signature of a first-order transition. The transition between isotropic and nematic phases occurs when their free energies (F_i and F_n, respectively) cross, i.e., F_n − F_i = 0, indicated by the dotted magenta curve.

We have repeated the calculation for different ratios of contour and persistence lengths, L/ℓ_p, and obtained a phase diagram, as shown in Fig. 3. As the rest of this work only considers the isotropic and nematic phases, we have restricted the phase diagram to these phases, ignoring more complex ones such as smectic or cholesteric phases.^4,38 The location of the nematic–isotropic transition (where F_n = F_i) or simply the phase boundary shown in Fig. 3 is consistent with the one previously calculated.²⁵ As the chains become more flexible, the transition occurs at higher anisotropic interaction strength. The entropic penalty for straightening and aligning chains will be higher if they are more flexible, since they have more degrees of freedom to loose.

FIG. 3.

(Top) Phase diagram for (Maier–Saupe) worm-like chains. The isotropic–nematic transition and spinodal lines are shown in the ν − L/ℓ_p space; also included are metastable regions (see the free energy curves on the right). The isotropic phase is stable below the “transition” line and metastable in the blue region. Similarly, the nematic phase is stable (above the transition line) and metastable (magenta) regimes are identified. (Bottom) This shows variable organization of WLCs as the polymer or segment concentration increases; the gray arrow at the bottom points toward higher densities. At low MS interaction strength ν (equivalent to low polymer or segment concentration), rods orient randomly (isotropic phase) as shown on the left; as ν increases, they orient parallel to each other, eventually entering the nematic phase, as shown on the right. In the isotropic phase, random motion leads to small, short-lived aligned regions (dashed circles). If quenched into an isotropic-metastable region, these alignments can form nuclei of the nematic phase. For visual simplicity, chain flexibility is ignored.

2. Stability limits: Spinodal decomposition vs nucleation

In addition to locating the transition, we have considered the stability limit or spinodal of the isotropic and nematic phases, as shown in Fig. 3. In our approach, this can be done by examining the “shape” of the free energy as a function of the degree of nematic order, S. When the phase adopted by the system is stable or metastable, the system is at a local free energy minimum in the S-space: δ²F[S]/δS² > 0⁴⁹ [recall the free energy F is defined in Eq. (14)]. Small changes in S thus increase the free energy. The phase becomes unstable when these small changes begin to decrease the free energy, as is the case for δ²F[S]/δS² < 0. Thus, the boundary between stable and unstable regions, known as the stability limit or spinodal, is set by the condition δ²F[S]/δS² = 0.^12,18,48

In the nematic stable region, for instance, the free energy curve has a single minimum at a non-zero value of S. In the isotropic-metastable region, it develops a second local minimum at S = 0, which is higher than the global minimum at a non-zero S value. The upper spinodal line in Fig. 3 sets the boundary between the two regions. The lower spinodal line can be understood similarly. In practice, in SCFT, this is done by finding where the system is unstable to small changes in w(r, u): we locate the spinodal by considering the functional derivative δ²F[w]/δw² = 0, rather than δ²F[S]/δS² = 0.

In principle, small changes in S, which would by themselves increase the free energy, may decrease the free energy when coupled with small changes in concentration. Including this consideration is beyond the scope of this work. It would involve calculating a matrix of second order derivatives δ²F[ρ, S]/δX_idX_j, where X_i or X_j can be either ρ(r) or S(r).⁵⁰ An instability develops when the smallest eigenvalue of this matrix becomes negative, representing a negative curvature in the free energy along the direction in ρ(r)–S(r) space specified by the corresponding eigenvector. The spinodal is thus found by locating where the smallest eigenvalue of the matrix vanishes.

We expect that ignoring these density variations will be a good approximation when the isotropic and nematic phases for given ν are close in concentration and density variations are unlikely to stabilize alignment fluctuations. Our analysis assumes a fixed bulk concentration, ρ₀, regardless of the phase. Ignoring concentration variation in the stability analysis likely yields an upper limit in the sense that concentration fluctuations should shrink the region of metastability. The degree to which the metastable region would shrink should depend on the strength of the excluded-volume interactions.

As shown in Fig. 3, isotropic metastability occurs over a wider ν range for larger L/ℓ_p. This is correlated with the observation that the transition occurs at larger ν for more flexible polymers. In the regions of metastability (Fig. 3), transitions to the stable phase proceed via nucleation. Small fluctuations in the metastable phase, such as locally aligned regions of the isotropic phase (dashed circles in Fig. 3), increase the free energy, and when they occur, these aligned regions tend to shrink and disappear. In other words, there is a free energy barrier to the transition. Only large fluctuations, such as supercritical nuclei of the stable (e.g., nematic) phase, will grow and cause a phase transition via nucleation (e.g., from an isotropic to a nematic state).⁵¹

Beyond the stability limit (i.e., the spinodal), the transition proceeds via spinodal decomposition. The barrier to the transition disappears, and the formerly metastable phase no longer occupies a local free energy minimum. Small fluctuations decrease the free energy and thus grow. If an isotropic phase is brought into the nematic stable region of the phase diagram, even tiny (infinitesimal) variations in S will spontaneously increase and grow, leading to an isotopic to nematic phase transition. Since this happens throughout the sample simultaneously, spinodal decomposition of the isotropic phase can be expected to simply amplify local directional fluctuations and result in a multi-domain sample with many nematic grains pointing in different directions.

In our attempt to go beyond the bulk phase behavior, we constrain our calculations to a particular ratio of contour to persistence length of L/ℓ_p = 20. This ratio was chosen to avoid the extremes of the rigid or flexible limits, making it a suitable representative of WLCs. This particular ratio is also of technological and biological relevance because it corresponds to collagen used in bioengineering.⁴⁷ Below we describe our results for the phase behavior of collagen-like polymers near an interface.

B. Nematic ordering under confinement

Many applications involve the presence of boundaries or confinement of WLCs. We thus turn our attention to the effects of confinement. For this, we first introduce one absorbing boundary at z = 0, to mimic the presence of a flat wall, and one reflecting boundary condition at z = z_rfl. (Two walls will also be considered below.) Technically, this procedure simulates a system confined between two parallel plates a distance h = 2z_rfl. In the results presented below, however, the separation is large enough that the middle of the system (z ≈ z_rfl) would not feel the “direct” effects of this confinement and is expected to be bulk-like.

In contrast to the bulk behavior discussed in Fig. 2, where the concentration tends to be uniform [see Eq. (11)], the presence of a wall affects the concentration, requiring us to solve for the concentration profile using Eq. (10). For weakly nematic polymers (ν ≲ 150 for L/ℓ_p = 20), we find that ν only has a small effect on the profile. Equation (10) is far more computationally costly to solve than Eq. (11). We therefore simplify the problem by calculating the profile ρ(r) at ν = 0 and imposing this profile for finite ν. In other words, we set the expected concentration ρ_e(r) to that determined by solving Eq. (10) at ν = 0. Sample calculations regarding the interfacial profile, as well as the historical context of imposing a profile, are shown in Appendix C. The advantage of this method is that it greatly reduces the computational cost. This is particularly useful for more computationally demanding calculations, e.g., two-dimensional calculations as required when a second wall is introduced (see below).

Adjusting ν₀ changes the length-scale over which the polymer concentration diminishes to 0 at the interface but does not significantly alter other behavior. We choose ν₀ = 20 for all results presented with walls. We also scale the amount of material to set the bulk density to 1. If the bulk is in the isotropic phase, this is equivalent to ignoring the V/Q factor in Eq. (6), since Q = V for the bulk isotropic phase. This scaling allows for a more meaningful comparison with the bulk behavior.

We first examine the effects of a plate wall placed at z = 0. Figure 4 shows the resulting nematic order parameters S (left axis) and segment densities ρ(z) (right axis). For this, we have chosen L = 20ℓ_p and h = 2z_rfl = 30ℓ_p and used two choices of the anisotropic interaction parameter ν [see Eqs. (3) and (12)]: ν = 0 (dashed lines) and ν = 140 (solid lines). Note that the presence of the wall does not shift the location of the isotropic–nematic transition (data not shown).

FIG. 4.

Nematic order parameter S [the largest eigenvalue of S_αβ in Eq. (15)] and monomer density ρ of collagen-like WLCs vs z/ℓ_p with L = 20ℓ_p confined between two plates a distance h apart h = 30ℓ_p; ρ(z/ℓ_p) is normalized so that ρ(z/ℓ_p) → 1 when z = h/2. The MS interaction [see Eqs. (3) and (12)] is set to ν = 0 (dashed lines) or ν = 140 just above the nematic transition (solid lines). (Left axis) For ν = 0, the presence of the plate wall tends to align the confined WLCs parallel with the plate over a short distance ≈2ℓ_p. This is not to be taken as nematic ordering. For ν = 140, above the onset of nematic transition in the corresponding unconfined case in Fig. 2, the confined WLCs form a nematic phase, except for z < 2ℓ_p (the Kuhn length). As shown in Fig. 5, the confining wall leads to biaxial order, where the alignment perpendicular to the wall is suppressed, and molecules are equally likely to align in any direction parallel to the wall, i.e., the alignment distribution is azimuthally symmetric. In the bulk phase, the azimuthal symmetry is broken and the biaxial order gives way to simple mono-axial order. (Right axis) The effect of the plate wall at z = 0 is to diminish ρ for z < 2ℓ_p for both ν = 0 and 140; for ν = 140, this reduces S near the wall by effectively diminishing ν.

As described by the blue dashed line in Fig. 4, for ν = 0, the effect of the wall-confinement is local. It tends to align the confined WLCs parallel with the plate over a short distance ≈2ℓ_p. This is not an indication of nematic ordering, but it simply reflects the tendency of the chains to align in the plane of the wall, i.e., biaxial alignment.

For ν = 140, slightly above the onset of the isotropic–nematic transition in the corresponding unconfined case in Fig. 2, the confined WLCs form a nematic phase, except in a few persistence lengths from the wall (the blue solid line). Based on this observation, the wall effect may be mistakenly interpreted as local. As evidenced below, the confining wall limits the orientation of nematic ordering to being parallel to the wall. The trend of the chains to be arranged parallel to the wall persists in the bulk phase, i.e., far away from the wall, as shown in Fig. 5. The symmetry-breaking transition into the nematic phase only needs to break azimuthal symmetry, i.e., symmetry in the direction of ϕ in Fig. 5, rather than full directional symmetry. Some features of the wall-confinement effects on S can be understood in parallel with the wall effect on ρ (right axis, Fig. 4).

FIG. 5.

Orientation distributions of collagen-like molecules (worm-like chains with the choices of L/ℓ_p = 20 and ν = 140) close to the wall of a plate normal to the z-axis, shown in (a), i.e., at z = 0.4ℓ_p, and far from the wall, shown in (b), i.e., at z = 15ℓ_p (half way between the plate and another plate at z = 30ℓ_p), for the MS interaction parameter [see Eqs. (3) and (12)] set to ν = 140, as in Fig. 4 (right axis). The angle θ is relative to the z-direction, and ϕ is the angle relative to the x-direction in the plane normal to z, as shown in the top panel. (a) Close to the wall, the molecules are parallel to the wall (θ = π/2) but are uniformly distributed in ϕ. (b) At the midplane, they remain parallel with the x–y plane (θ = π/2) but align in the ±y-direction (ϕ = π/2 and 3π/2), thus displaying nematic ordering. If combined with Fig. 4, this suggests that the wall-confinement restricts the orientation of nematic ordering in the entire z-range, except for z < 2ℓ_p. The top panel illustrates what is displayed in the bottom panel [(a) and (b)]; the farthermost plane on the right corresponds to the (imaginary) midplane at z = 15ℓ_p. For visual clarity, the bending of WLCs is ignored in the illustration.

As indicated by the magenta curves in Fig. 4, the effect of the plate wall at z = 0 is to diminish ρ(z) for z ≲ 2ℓ_p. This is consistent with what we would expect from the absorbing boundary condition imposed at z = 0. For ν = 140, this reduces S near the wall by effectively diminishing ν. For z/ℓ_p ≳ 2, ρ(z) recovers the bulk behavior.

To further clarify the wall effects, we examine the ordering direction of collagen-like polymers adjacent to a planar wall at z = 0, with a reflecting wall at z = z_rfl = 15ℓ_p, as discussed in this section and Fig. 4, focusing on ν = 140. Figure 5 displays our results for the orientation distribution of the polymers: ρ(θ, ϕ), where θ is the polar angle and ϕ is the azimuthal angle, as illustrated on the top left corner. Close to the wall (a), i.e., z = 0.4ℓ_p, polymers align parallel to the wall (θ = π/2) but show no preferred direction in this plane (uniform in ϕ) normal to the z axis. The corresponding distribution obtained with ν = 0 (not shown) is essentially the same. This corresponds to the second ordering scenario discussed in Sec. II (a special case of biaxial order), where the two degenerate eigenvalues of S_αβ are now larger than the third. Although larger than the third eigenvalue, the degenerate ones remain small, reflecting a weak tendency to align in the plane of the wall, since no true nematic ordering is present here.

The presence of a wall excludes trajectories/orientations that would have crossed the wall, leaving only those that do not. This manifests as a tendency for polymers to align with the confining wall. In terms of the eigenvector decomposition, this suppression of order manifests as a small eigenvalue corresponding to the eigenvector that is in the direction perpendicular to the wall. The azimuthal symmetry [see Fig. 5(a)] results in two degenerate eigenvalues corresponding to eigenvectors parallel to the wall. Even when ν is large, the polymers close to the wall do not enter the nematic phase. This can be attributed to the reduced segment concentration next to the wall, which effectively weakens the anisotropic interactions (see Fig. 4).

Polymers far from the wall, of course, enter the stable bulk phase, determined by the value of ν. Without the wall (or other symmetry-breaker), there would be no preferred direction for nematic order. When the wall is present, however, the polymers prefer to orient parallel to the wall in order to reduce their unfavorable MS interactions with the polymers next to the wall. In our SCFT calculations, this preference extends arbitrarily far into the bulk. Indeed, the graph in Fig. 5(b) represents our results at z = 15ℓ_p, i.e., halfway between the plate at z = 0 and another one at z = 30ℓ_p. In real systems, however, it competes with fluctuations. As a result, the influence of the wall may not extend indefinitely but might decay some distance into the bulk. Clarifying this is beyond the scope of these calculations and will be left for future work.

The results in Fig. 5 suggest that adding a wall influences the preferred directions for nematic order, causing the director to point in a direction parallel to the wall. This idea can be extended by adding a second (e.g., perpendicular) wall, which provides another constraint.

Figure 6 illustrates a scenario where two perpendicular walls are placed normal to the y- and z-directions. The illustration on the left shows WLCs on an imaginary plane a few persistence lengths away from the wall. The concentration in Fig. 6(a) and orientation distribution in Figs. 6(b)–6(d) are shown for worm-like chains with L/ℓ_p = 20 at ν = 140, with absorbing walls at y = 0 and z = 0, and reflecting walls at y = 10ℓ_p and z = 10ℓ_p.

FIG. 6.

Segment concentrations and orientation distributions of collagen-like molecules confined between two walls, one normal to the y-axis and the other normal to the z-axis. This illustration on the left shows nematic ordering of WLCs several persistence lengths away from the walls; for simplicity, the bending of the WLCs is ignored. (a) The segment concentrations are shown in the rescaled y–z planes; they are represented by varying darkness levels. Solid lines, resembling an inverted L-shape, are included to guide the eye; from the leftmost one, they represent ρ(y, z) = 0.1, 0.5, and 0.9 with 1 representing the bulk-like concentration. Consistent with the results in Fig. 4, the segment distribution vanishes on the walls, i.e., y = 0 and z = 0, and reaches the bulk behavior at the bottom right corner. [(b)–(d)] Orientation distributions ρ(θ, ϕ) are calculated approximately at the positions marked by black dots in the concentration plot in (a) and are shown for small z in (b), small y in (c), and large y–z (bulk-like) in (d); the corresponding dots are labeled as “(B),” “(C),” and “(D),” respectively. [The dots labeled “(B)” and “(C)” in graph (a) are symmetric in the sense that the WLCs there will experience the same wall effect. This is reflected in the distributions in (b) and (c), even though they look seemingly different. Recall that the angle θ is defined with respect to the z axis. This explains the apparent difference.] As in Fig. 5, the polymers near any of the walls preferentially orient parallel to the wall and retain this parallel orientation in the bulk. This sets the bulk orientation, i.e., in the x-direction in order for the polymers to be parallel to both walls. The results in this figure suggest that the presence of two walls uniquely defines the director of the nematic phase to be in the x-direction, rather than simply confining it to a plane as in the case of a single wall (see Fig. 5).

In the concentration graph in Fig. 6(a), concentrations are represented by varying darkness levels. As shown previously (see Fig. 4), the concentration goes to 0 (white) at the walls and increases to 1 (red) in the bulk (i.e., at the right bottom corner). The solid lines, resembling an inverted L-shape, are to guide the eye; from the leftmost one, they represent ρ = 0.1, 0.5, and 0.9.

The graphs in Figs. 6(b)–6(d) show the orientation distribution ρ(θ, ϕ), collected at the points indicated in the concentration plot in Fig. 6(a): small z in Fig. 6(b), small y in Fig. 6(c), and large y–z (bulk-like) in Fig. 6(d); the corresponding dots are labeled “(B),” “(C),” and “(D),” respectively, in Fig. 6(a). At the points labeled as “(B)” and “(C)” in the graph in Fig. 6(a), WLCs will experience the same wall effects. This symmetry is reflected in the distributions plotted in Figs. 6(b) and 6(c), despite the apparent difference between the two. As illustrated on the left, the angle θ is defined with respect to the z axis; θ = 0 in Fig. 6(b) is analogous to θ = π/2 in Fig. 6(c). This asymmetric coordinate system explains the apparent difference.

As in Fig. 5, each wall causes the adjacent polymers to orient with the wall. In order for both of these tendencies to extend into the bulk, i.e., for the nematic phase to be parallel to both walls, the director must be in the x-direction. This provides a mechanism by which partial confinement can be used to uniquely specify an ordering direction. Previous calculations regarding anisotropically interacting confined rigid rods⁹ also concluded that rods align with the wall. However, their results are not conclusive in three dimensions: their rods were unable to simultaneously align parallel to two perpendicular planes, since the calculations were done in two dimensions.

IV. DISCUSSION

In this work, we have first examined the equilibrium phase behavior and stability limits of nematic worm-like chains (WLCs) in the bulk, as shown in Fig. 3. We have then unraveled the effects of planar confinement on this behavior and the propagation of these effects into the bulk.

The diagram in Fig. 3 helps us understand the structures WLCs form and the underlying mechanism. If quenched from an isotropic phase into a region above the isotropic spinodal (labeled “nematic stable” in Fig. 3), a WLC system becomes unstable to arbitrarily small fluctuations in the order parameter (i.e., the nematic order parameter S) and even small orientational fluctuations are amplified.^48,52 This naturally produces a sample with many domains pointing in different directions. We also expect this to be the case close to the spinodal line in Fig. 3, where the barrier to nucleation becomes small and many nuclei form.

The isotropic phase remains metastable well into the region of nematic stability, i.e., through an ∼10% change in the MS interaction parameter ν (Fig. 3) or concentration ρ₀. In this metastable regime (the blue region in Fig. 3), the nucleation mechanism for phase transitions dominates.^48,51,52 This has important implications for materials processing (see below for additional discussions).

Nucleation of the nematic phase from the isotropic phase occurs when the isotropic phase is metastable and starts with a nucleus of the nematic phase,^48,51 i.e., a region of aligned polymers as illustrated by the dashed circles in Fig. 3. The transition proceeds by growth of the nucleus, i.e., nearby polymers align with those in the nucleus, due to nematic interactions. When the system is highly metastable, as is the case if the system is close to the transition line in Fig. 3, nucleation events are rare and single nuclei may grow large. This process should produce a small number of large nematic domains if the nucleation barrier is high. Even if it creates multiple nuclei that intersect,⁵³ the resulting domains will be larger than those created from the system erupting in aligned regions when the barrier disappears. Furthermore, the number of domains that initially nucleate can be reduced by bringing the system closer to the transition line when the transition occurs. In the isotropic metastable regime, the closer the system is to the transition line (further from the spinodal), the higher the barrier to nucleation is (a general feature of first-order transitions). A high barrier to nucleation results in fewer nuclei, thus fewer intersecting domains. In addition, a physical surface, i.e., a plate wall, can be introduced to align polymers. Combining these effects, it may be possible to form a mono-domain sample.

Indeed, the presence of a planar wall tends to orient the polymers parallel with the wall, as shown in Fig. 5. Below the onset of bulk nematic alignment, the wall effect is local, i.e., within a few persistence lengths from the wall. Above the onset of nematic alignment, in the corresponding unconfined case, this orientation tends to propagate into the bulk phase.

The presence of a wall helps to choose a particular direction of the nematic phase, which could also be useful to create mono-domain samples. A single wall only decreases the possible alignments from any direction in three dimensions to any direction in two dimensions. Adding a second, however, uniquely specifies an ordering direction, as demonstrated in Fig. 6. The results in this work were calculated using perpendicular walls, which intersect. On physics grounds, however, we suspect that any two non-parallel planar walls should lead to a specific bulk ordering direction, set by the intersection of planes parallel to the walls as this would be the logical result of a tendency to be parallel to both.

In our SCFT calculations, the influence of the wall extends arbitrarily far into the bulk. It is unclear if this would be the case in real systems. Fluctuations may decrease the range of this influence from infinity to some finite value. Existing techniques to include fluctuation effects, such as field-theoretic simulations (FTS), are currently prohibitively computationally expensive for WLCs;^54,55 particle-based simulations are even more so. Sample preparation is another consideration, as it may prevent long-range alignments from forming. An adequate consideration of sample preparation would require dynamical simulations.

What are the implications of our findings for materials processing as well as for natural WLC systems?⁵⁶ If a sample is in the isotropic-metastable region, i.e., the blue region in Fig. 3, close to the transition line, the presence of a wall will not only lower the barrier to nucleation of a nematic domain but also align WLCs parallel to the wall. As a result, a more uniform nematic domain will propagate into the bulk in a parallel orientation from the wall. However, a more complete understanding of nematic ordering via nucleation is beyond the scope of this work.

The walls that we have investigated do not need to represent solid walls that are imposed on the polymer melts or solutions. The behavior that we have found near boundaries should also be present at, for example, polymer–air³⁰ or polymer–polymer⁵⁷ interfaces. The tendency for a disordered region to form on the surface of an ordered material is also well-documented in solids.^58–61

This work has simplified the problem by considering only isotropic and nematic phases, ignoring more complicated configurations such as the smectic or cholesteric phases observed with collagen,^28,29 for instance. Both configurations require modifications of our method. Smectic phases emerge in field-theories of rod-coil polymers, where the combination of polymer types leads to alignment.⁶² Cholesteric phases require a modification of polymer–polymer interactions. This has been done in more coarse-grained analyses by adopting the Oseen–Frank model to incorporate helicity,⁶³ most notably to study fibril formation in collagen.⁶⁴ Interaction energies utilized by this model may be possible to incorporate into SCFT.

V. SUMMARY

This work studied the mean-field behavior of nematic polymers using standard self-consistent field theory (SCFT) techniques. The polymers were represented as worm-like chains (WLCs) with anisotropic interactions represented by Maier–Saupe theory. We studied the mean-field behavior of these nematic polymers, the limits of stability of the nematic and isotropic phases, and importantly the effects of partial confinement on this phase behavior. Partial confinement can be used to influence the nematic ordering direction, which can be set to a single unique direction by imposing two walls parallel to the desired director.

Increasing polymer flexibility increased the stability of the isotropic phase, consistent with the previous predictions.²⁵ It also increased the regions of metastability of the isotropic and nematic phases. The isotropic phase remains metastable well into the region of nematic stability, suggesting a large barrier to the transition and that nucleation may be a viable method for producing mono-domain nematic samples. Confinement or the presence of walls can be used to guide the ordering direction as WLCs tend to align with the walls, and this tendency extends into the nematic phase.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

APPENDIX A: EXCLUDED-VOLUME PARAMETER

Here, we drive the expression for the excluded-volume parameter used in the main text,

$${\nu }_0={\left(v_s{\rho }_0\right)}^{-1}-2\chi ,$$ (A1)

where v_s is the volume of each solvent molecule and ρ₀ is the average segment concentration. The same expression was previously used in a theoretical study of polymer brushes.⁶⁵ Our derivation is essentially identical to the one employed in standard references (see, for instance, Ref. 34); the main difference is that we allow the size of monomers to differ from that of solvent molecules.

The entropy of mixing of solvent molecules can be written as

$$\begin{array}{ccc}\hfill -\frac{S_s}{k_{B}T}& =\hfill & \frac{1}{v_s}\int {\rho }_s\left(\mathbf{r}\right)\ln {\rho }_s\left(\mathbf{r}\right)d\mathbf{r}\hfill \\ \hfill & \approx \hfill & \frac{1}{v_s}\int \left[-\rho \left(\mathbf{r}\right)+\frac{{\rho }^2\left(\mathbf{r}\right)}{2}+\cdots \right]d\mathbf{r},\hfill \end{array}$$ (A2)

where v_s is the volume of each solvent molecule and ρ_s(r) is the concentration of solvent molecules scaled by ρ₀. Note that we have substituted polymer concentration for solvent concentration using ρ_s(r) + ρ(r) = 1, where ρ(r) ≡ ∫duρ(r, u). The log term was expanded in powers of the monomer concentration ρ, assumed to be low.

The enthalpy of solvent–polymer interactions can be expressed as

$$U_{sp}=k_{\mathrm{B}}T{\rho }_0\chi \int \rho \left(\mathbf{r}\right){\rho }_s\left(\mathbf{r}\right)d\mathbf{r}$$ (A3)

or, equivalently,

$$U_{sp}=-k_{\mathrm{B}}T{\rho }_0\chi \int {\rho }^2\left(\mathbf{r}\right)d\mathbf{r},$$ (A4)

where an additive constant is ignored.

By combining the entropic and enthalpic contributions given in Eqs. (A2) and (A4), respectively, we arrive at the isotropic energy term

$$U_i=k_{\mathrm{B}}T\frac{{\nu }_0{\rho }_0}{2}\int {\rho }_s^2\left(\mathbf{r}\right)d\mathbf{r},$$ (A5)

where ν₀ is the (dimensionless) excluded-volume parameter given in Eq. (A1). If we set v_sρ₀ = 1, the expression for ν₀ in Eq. (A1) reduces to the conventional one in Ref. 34: ν₀ = 1 − 2χ, i.e., the excluded-volume parameter in units of the volume of each monomer.

APPENDIX B: SPHERICAL HARMONIC REPRESENTATION

As evidenced below, in our approach, it proves useful to expand u-dependent functions, f(u), in terms of real spherical harmonics, denoted as ${\tilde{Y}}_l^m\left(\mathbf{u}\right)$. These functions are related to the regular, complex, spherical harmonics, $Y_l^m\left(\mathbf{u}\right)$, by

$${\tilde{Y}}_l^m\left(\mathbf{u}\right)=\left\{\begin{array}{cc}\hfill \sqrt{2}{\left(-1\right)}^m\mathfrak{I}\left[Y_l^{\left|m\right|}\right]\hfill & \hfill \mathrm{i}\mathrm{f}m<0,\hfill \\ \hfill Y_l^0\hfill & \hfill \mathrm{i}\mathrm{f}m=0,\hfill \\ \hfill \sqrt{2}{\left(-1\right)}^m\mathfrak{R}\left[Y_l^{\left|m\right|}\right]\hfill & \hfill \mathrm{i}\mathrm{f}m>0,\hfill \end{array}\right.$$ (B1)

where $\mathfrak{R}$ and $\mathfrak{I}$ denote the real and imaginary parts, respectively. Like the regular spherical harmonics, they are orthonormal eigenfunctions of Laplace’s equation.⁴⁰

The real spherical harmonic expansion of f(u) is given by

$$f\left(\mathbf{u}\right)=\sum _{lm}{f}_l^m{\tilde{Y}}_l^m\left(\mathbf{u}\right).$$ (B2)

We apply this expansion to q(r, u, s), w(r, u), and ρ(r, u).

When calculating q(r, u, s), we multiply Eq. (4) by ${\tilde{Y}}_l^m\left({\mathbf{u}}^{\prime }\right)$ and integrate it over u′. We then arrive at

$$\frac{\partial q_l^m}{\partial s}=\frac{L}{a}l\left(l+1\right)q_l^m-L{G}_{1l^{\prime \prime }l}^{j{m}^{\prime \prime }m}\frac{\partial q_{l^{\prime \prime }}^{m^{\prime \prime }}}{\partial x_j}-G_{l^{\prime \prime }{l}^{\prime }l}^{m^{\prime \prime }{m}^{\prime }m}{w}_{l^{\prime }}^{m^{\prime }}{q}_{l^{\prime \prime }}^{m^{\prime \prime }},$$ (B3)

where we employed the Einstein summation convention. The upper index m runs from −l to l; the spatial positions x₋₁, x₀, and x₁ correspond to the y-, z-, and x-positions, respectively; $G_{l{l}^{\prime }{l}^{\prime \prime }}^{m{m}^{\prime }{m}^{\prime \prime }}$ are Gaunt coefficients, which are given as integrals of the product of three real spherical harmonics over u,⁴⁰

$$G_{l^{\prime \prime }{l}^{\prime }l}^{m^{\prime \prime }{m}^{\prime }m}=\int {\tilde{Y}}_{l^{\prime \prime }}^{m^{\prime \prime }}\left(\mathbf{u}\right){\tilde{Y}}_{l^{\prime }}^{m^{\prime }}\left(\mathbf{u}\right){\tilde{Y}}_l^m\left(\mathbf{u}\right)d\mathbf{u}.$$ (B4)

Equation (B3) can be solved using any of many standard numerical integration schemes. We choose that of Ref. 41, which was originally written for the Legendre polynomial expansion but is adapted in Refs. 21 and 42 for use with the real spherical harmonic expansion.

In addition to greatly decreasing the computational cost of the calculations, this expansion reduces the number of field conditions from 1 for every possible (or represented) orientation u, as in Eq. (12), to just 6. One of them corresponds to the isotropic field condition,

$$w_0^0\left(\mathbf{r}\right)={\nu }_0{\rho }_0^0\left(\mathbf{r}\right),$$ (B5)

and the rest correspond to anisotropic interactions,

$$w_2^m\left(\mathbf{r}\right)=-\frac{4\pi \nu }{5}{\rho }_2^m\left(\mathbf{r}\right),$$ (B6)

where m = 0, ±1, ±2. Our separation of the field into isotropic and anisotropic parts thus results naturally from the real spherical harmonic representation. The nematic order parameter in Eq. (15) can also be calculated in real spherical harmonics as

$$S_{\alpha \beta }\left(\mathbf{r}\right)=\frac{V}{nL}\left(\sum _{lm}{\rho }_l^m\left(\mathbf{r}\right)G_{l11}^{m\alpha \beta }-\frac{{\delta }_{\alpha \beta }}{3}\rho \left(\mathbf{r}\right)\right).$$ (B7)

APPENDIX C: ENFORCED PROFILE

Calculating the behavior of polymers at an interface is neither a new nor a simple problem and has been approached with a number of methods. Some studies have chosen to simply impose a sigmoidal profile for the polymer concentration, where the width is set as an input parameter,^30,32,66,67 whereas others used more sophisticated calculations to determine the interfacial profile.^68–70

The way we determine the interfacial profile is similar to the one employed in Refs. 71 and 72. This work is focused on clarifying the effect of walls on the alignment behavior and the propagation of these alignment effects into the bulk; we are not concerned with the precise details of the interfacial profile. It is worth recalling that the wall effect on the segment distribution ρ(z) is short ranged, typically on the order of ℓ_p. Even though it sets the nematic orientation in its plane, its effect should be local otherwise. For instance, the value of ν_tr, at which the nematic transition occurs, will not be influenced by the interfacial profile, neither will stability limits of the bulk region. To ease the computational demand, we approximate the interface by imposing its profile for ν = 0 (see below).

Figure 7 shows the interfacial profiles calculated for the excluded-volume parameter ν₀ = 20 and for several values of the nematic strength ν. As ν increases, the wall effect decays faster and the bulk behavior is reached more quickly, as if the wall repulsion is shorter ranged. To understand this, imagine a hypothetical case where all WLCs are perfectly rigid and aligned with each other (ν = ∞). Since the wall acts as a nematic ordering director, the WLCs will also be aligned with the wall. The repulsion between the wall and the WLCs will be short ranged on the order of the diameter of each WLC. This explains the general trend shown in this figure.

FIG. 7.

Comparison of segment-density distributions of WLCs, ρ(z), near a planar wall at z = 0 for the nematic strength ν = 0, 80, 140, 150, and 160. For a larger value of ν, the distribution converges faster onto the bulk behavior. This can be attributed to the effect of the wall that tends to align the WLCs in parallel. Imagine a hypothetical case where all WLCs are perfectly rigid and aligned with each other. Since the wall acts as a nematic ordering director, the WLCs will also be aligned with the wall. The repulsion between the wall and the WLCs will be short ranged on the order of the diameter of each WLC. This explains the general trend shown in this figure. For ν = 160, the distribution develops a peak at z ≈ ℓ_p/2. This arises from similar wall effects. Near the wall, the WLCs are better aligned, and their repulsion is effectively shorter-ranged than in the bulk phase (z ≫ ℓ_p). This raises ρ(z) slightly beyond the bulk value at z ≈ ℓ_p/2, as seen in the graph.

As nematic interactions strengthen further, the distribution develops a peak close to the interface (z ≈ ℓ_p/2), surpassing the bulk concentration. This can be attributed to the wall-layering effects. At such a large value of ν, WLCs tend to be aligned and packed in parallel, more so near the wall at z = 0. Because of the absorbing boundary condition at z = 0, however, the peak occurs some distance from the wall. There are, however, no qualitative differences in the segment distribution for ν ≲ 150. Furthermore, the wall-layering effect is only local and will not influence the essential features of the confined WLCs such as ν_tr and spinodals.

Even with our simple model, calculating the interfacial profile can be a computationally intensive task and becomes more so when the bulk orders into the nematic phase. The calculations can become slow and numerically unstable but, as seen, are doable for the one-dimensional calculations where there is a single planar wall present. Extending it to two dimensions, however, becomes practically infeasible, as the computational demands are too high. We thus approximate the interface by imposing its profile at ν = 0. Although this will produce a quantitatively inaccurate interfacial profile, it retains the behavior relevant to the questions that we wish to address.