Self-consistent field theory of chiral nematic worm-like chains

Abstract

Many macromolecules of biological and technological interest are both chiral and semi-flexible. DNA and collagen are good examples. Such molecules often form chiral nematic (or cholesteric) phases, as is well-documented in collagen and chitin. This work presents a method for studying cholesteric phases in the highly successful self-consistent field theory of worm-like chains, offering a new way of studying many biologically relevant molecules. The method involves an effective Hamiltonian with a chiral term inspired by the Oseen–Frank (OF) model of liquid crystals. This method is then used to examine the formation of cholesteric phases in chiral-nematic worm-like chains as a function of polymer flexibility, as well as the optimal cholesteric pitch and distribution of polymer segment orientations. Our approach not only allows for the determination of the isotropic–cholesteric transition and segment distributions, beyond what the OF model promises, but also explicitly incorporates polymer flexibility into the study of the cholesteric phase, offering a more complete understanding of the behavior of semiflexible chiral-nematic polymers.

I. INTRODUCTION

The tendency of semi-flexible polymers to align is well-known and often leads to liquid-crystalline structures, such as aligned or nematic phases.^1–3 Chiral molecules can form chiral nematic or cholesteric structures, where molecules align, as in the nematic phase, but the direction of alignment changes throughout the sample.^1–3 The alignment direction rotates about an axis normal to the nematic director. This behavior is illustrated in Fig. 1. A chiral-nematic phase is characterized by its chiral pitch, which is the distance over which the director rotates by 360°.

FIG. 1.

Illustration of a cholesteric arrangement of rigid rods. Layers are shaded successively lighter to aid in visualization. Successive layers are rotated relative to one another about an axis normal to the alignment direction.

The cholesteric pitch of chiral-nematic phases is often close to the wavelengths of visible light, making them useful for technological applications in optics.^4–9 Many biological molecules are also chiral and exhibit this behavior.¹⁰ Well-known examples are the structural proteins collagen^11–18 and chitin;¹⁹ a chiral-nematic behavior has also been observed in viruses.^20–23 A thorough understanding of chiral-nematic behavior is thus not only of theoretical interest but also crucial to engineering and bioengineering endeavors.^24–26 For instance, current methods ignore important parameters, such as polymer flexibility as well as the statistics of how polymers are locally arranged.

Studying nematic semi-flexible polymers is no simple task partly because it involves anisotropic (angle-dependent) interactions. For many applications, it is sufficient to treat them as aligned rigid rods and study the alignment behavior in various circumstances. A noteworthy approach, particularly for chiral-nematics, is the Oseen–Frank (OF) model.^1,27 The OF model represents a system of nematic molecules by a vector field, n(r), of unit size (i.e., n · n = 1) and defines a free energy functional of the system. The free energy functional has terms corresponding to different types of deformations; the propensity for the system to form these deformations is controlled through the coefficients of these terms, which are taken as inputs to the model.

The traditional cholesteric phase can be described by a unit vector field^1,27

$$\mathbf{\text{n}}(z)=\widehat{\mathbf{\text{x}}}\cos (kz)+\widehat{\mathbf{\text{y}}}\sin (kz)$$ (1)

specifying the direction of nematic ordering, where $\widehat{\mathbf{\text{x}}}$ and $\widehat{\mathbf{\text{y}}}$ are the unit vectors in the x- and y-directions, respectively. Here, 2π/k is the pitch, i.e., a height along the z axis, over which the director n makes a complete turn (see the right panel in Fig. 1). The free energy density in the OF model, including a term corresponding to chiral interactions, is given by

$$f_{\mathrm{O}\mathrm{F}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{l}}=\frac{1}{2}{K}_{22}{\left(\mathbf{\text{n}}\cdot \nabla \times \mathbf{\text{n}}+\frac{k_2}{K_{22}}\right)}^2.$$ (2)

The parameter K₂₂ gives the strength of cholesteric interactions and k₂ controls the pitch. This free energy term is minimized by the configuration described by Eq. (1) when k = k₂/K₂₂.^1,27 Although derived for the study of fully rigid molecules, the OF model has been successful in predicting the behavior of semi-flexible molecules, such as collagen.^28,29 The full OF free energy contains more terms and is available in Refs. 1 and 27–29. These other terms correspond to splay, bend, splay–bend, and saddle–splay deformations and are not directly relevant to the cholesteric behavior. We have focused on the OF model as it is the most widely used theoretical model to describe the chiral-nematic phase.

Although the OF model has been highly successful in predicting the particular configurations of aligned molecules, it has limitations. It assumes that locally, molecules are perfectly aligned and only the alignment direction varies in space. This results in |n(r)| = 1. One cannot, in general, expect that molecules would be perfectly aligned on small length scales. Indeed, it should be recognized that these molecules are not always locally aligned. This is made obvious by the very existence of the isotropic phase, where molecules are not aligned at all. The formation of nematic structures in materials processing often involves taking molecules that are not aligned (in the isotropic phase) and varying system parameters so as to cause them to align into desirable structures.^24–26 In addition to bioengineering, the isotropic–nematic transition has been studied in the context of viruses.^20–23

Relaxing the local alignment constraint is theoretically challenging. One could simply allow the norm of the director, |n(r)|, to vary, but this ignores information about the distribution of molecular alignments. Let u represent the alignment of a molecule with its probability distribution given by ψ(r, u). The degree of alignment of molecules in a region is essentially the average of their alignments, u, in the region,

$$⟨\mathbf{\text{u}}⟩(\mathbf{\text{r}})=\int \mathbf{\text{u}}\psi (\mathbf{\text{r}},\mathbf{\text{u}})d\mathbf{\text{u}}.$$ (3)

Simply characterizing the system by n(r) as in the OF model in Eq. (1) ignores the information in ψ(r, u). Luckily, there is a highly successful approach to polymer theory, known as self-consistent field theory (SCFT), which readily incorporates all of this.^30–32 As it is, however, it merely lacks a robust way to describe cholesteric interactions.

SCFT considers the statistical behavior of a single molecule in a field that represents its interactions with all other molecules. This statistical behavior dictates molecular distributions, which are related to the field, since the field represents molecule–molecule interactions. The field and concentrations are then solved for self-consistently. SCFT has been used successfully to calculate the phase behavior of a wide variety of polymer systems, including the liquid-crystalline behavior in semi-flexible polymers.^33,34

The goal of this work is to extend the scope of SCFT by incorporating chiral-nematic interactions in order to study cholesteric phases in collagen-like polymers.^11–18 To this end, we define an effective Hamiltonian, inspired by the OF model, which causes adjacent molecules to align at a non-zero relative angle, so as to mimic the behavior of chiral molecules. We use this model to calculate the equilibrium properties of the chiral-nematic phase, such as the cholesteric pitch, the probability distribution of polymer–segment tangent vectors, and the location of the isotropic–cholesteric transition, as a function of the strength of the cholesteric interactions. In addition to enhancing the understanding already provided by the OF model, we expect that the ability to explicitly incorporate polymer flexibility and the degree of polymer ordering into the study of the cholesteric phase will lead to a more complete understanding of the behavior of semiflexible chiral-nematic polymers beyond what the OF model can offer.

II. THEORY

This work considers an incompressible melt (or uniform concentration solution) of chiral semi-flexible polymers, modeled as worm-like chains (WLCs) with chiral interactions. The melt, of volume V, contains n_c chiral polymer chains, each of which has a contour length L and persistence length ℓ_p. Each polymer is composed of N segments; each segment has a length a and occupies a volume ${\rho }_0^{-1}=V/n_{\text{c}}N$. The position along the chain is parameterized by s in the range 0 < s < 1. The unit tangent vector to a chain is given by

$$\mathbf{\text{u}}\equiv \frac{1}{L}\frac{d\mathbf{\text{r}}(s)}{ds},$$ (4)

where r(s) denotes the spatial position of the segment at location s along the contour. The reduced concentration, ψ(r, u) ≡ ρ(r, u)/ρ₀, is the local concentration of segments with orientation u, i.e., ρ(r, u) scaled by the segment density ρ₀. The quantity ψ(r, u) can be thought of as a probability density of segment orientations. We think of the system of segments, with orientations u, and argue for an effective Hamiltonian that resembles Eq. (2) to describe the chiral behavior of segments.

Here, we propose an effective Hamiltonian for chiral nematic WLCs, $\mathcal{H}=U_{\text{B}}+U_{\text{C}}$, given as a sum of the usual bending energy

$$\frac{U_{\text{B}}}{k_{\text{B}}T}=\frac{{\ell }_{\text{p}}}{2L}\sum _{\alpha =1}^{n_{\text{c}}}{\int }_0^{1}ds{\left|\frac{d{\mathbf{\text{u}}}_{\alpha }}{ds}\right|}^2,$$ (5)

where α refers to different WLCs. The cholesteric energy, U_C, arises as follows: Polymers that exhibit cholesteric interactions orient such that adjacent molecules (or molecular segments) tend to lie almost parallel to each other but not exactly parallel as in the usual nematic phase. This can be explicitly incorporated as in Ref. 43. However, this formulation requires the use of highly non-local interactions. Rather, we posit that each molecule (or molecular segment) has a minimum energy when the nearby molecules on the one side of it are offset by some angle κΔr, while the molecules on the other side are offset by an angle −κ|Δr|, where Δr is the distance to the molecules and κ is some constant defining the preferred angle between adjacent molecules. Summing over all of these interactions with adjacent molecules, this argument tells us that the energy is minimized for a given molecule (or segment) when the curl of the average direction of its surroundings is some finite value, i.e., u · ∇ × ⟨u′⟩(r) = κ, where ⟨u⟩(r) ≡ ∫uψ(r, u)du is the average alignment of molecules surrounding a test molecule (or a segment) at position r and alignment u. This argument relies on the assumption of a slowly varying average orientation (director). Expanding around this minimum and summing over all molecules (integrating over the spatial position and probability distribution of segment orientations) results in the energy

$$\frac{U_{\text{C}}}{k_{\text{B}}T}=\frac{\gamma {\rho }_0}{2}\int d\mathbf{\text{r}}d\mathbf{\text{u}}{\left[\mathbf{\text{u}}\cdot \nabla \times ⟨{\mathbf{\text{u}}}^{\prime }⟩(\mathbf{\text{r}})+\kappa \right]}^2\psi (\mathbf{\text{r}},\mathbf{\text{u}}).$$ (6)

The constants γ and κ give the strength of the interaction and the preferred magnitude of the curl, respectively. Comparing with Eq. (2), we see that it resembles closely the Oseen–Frank model and the parameters γ and κ take the place of K₂₂ and k₂/K₂₂, respectively. Note that Eq. (6) reduces to Eq. (2) when segments are perfectly aligned with each other at a given position, i.e., ψ(r, u) = δ(u − n(r)).

There is, however, a subtle difference between Eqs. (2) and (6). In Eq. (2), |n| = 1, whereas $|⟨\mathbf{\text{u}}⟩|\le 1$. Let us relax the constraint that |n| = 1. In a sense, we are now thinking of n as representing $⟨\mathbf{\text{u}}⟩$. For simplicity, let us define $A\equiv \sqrt{⟨\mathbf{\text{u}}\cdot \mathbf{\text{u}}⟩}$. We then consider a segment orientation field $\mathbf{\text{n}}=A\left(\widehat{\mathbf{\text{x}}}\cos kz+\widehat{\mathbf{\text{y}}}\sin kz\right)$. This essentially describes the WLC system by only its local average orientation and degree of orientation, A. The energy in Eq. (6) is minimized when A²k* = κ. The combination A²k in Eq. (6) plays the same role as k in Eq. (2). Note that we have ignored much of the physics that influences the actual distribution, ψ(r, u), which arises through not just segment interactions but also chain connectivity, and thus the parameters L and ℓ_p.

Now that we know how polymer segments interact, i.e., the energy functionals, Eqs. (5) and (6), we can carry out self-consistent field theory (SCFT) calculations in the usual way. Given a field, w(r, u), which represents interactions between segments, we can calculate the propagator, q(r, u, s), of a polymer subject to this field. The propagator is the partition function for a portion of a WLC up to a point on the chain, s, which is fixed at spatial position r and has orientation u. The propagator for a WLC obeys

$$\frac{\partial q}{\partial s}=\left[\frac{L}{2\xi }{\nabla }_u^2-L\mathbf{\text{u}}\cdot \nabla -w(\mathbf{\text{r}},\mathbf{\text{u}})\right]q(\mathbf{\text{r}},\mathbf{\text{u}},s),$$ (7)

which is solved with uniform initial conditions, q(r, u, s = 0) = 1. The back propagator, q^†(r, u, s), is the partition function for the rest of the chain and is solved similarly but with one side of Eq. (7) multiplied by −1. Segment concentrations are then given by

$$\psi (\mathbf{\text{r}},\mathbf{\text{u}})=\frac{V}{Q}{\int }_0^{1}dsq(\mathbf{\text{r}},\mathbf{\text{u}},s)q^{†}(\mathbf{\text{r}},\mathbf{\text{u}},s),$$ (8)

where

$$Q=\int d\mathbf{\text{r}}d\mathbf{\text{u}}q(\mathbf{\text{r}},\mathbf{\text{u}},s=0)$$ (9)

is the partition function for a WLC subject to the field w(r, u).

Once the behavior of the polymer in the field is calculated, the field is adjusted iteratively using Anderson mixing^35,36 until it satisfies the field equations, which are calculated from the internal energy functionals. As the field is conjugate to the concentration, it is given by the derivative of the internal energy with respect to concentration. Applying this to the internal energy defined above results in

$$w(\mathbf{\text{r}},\mathbf{\text{u}})=\gamma \int d{\mathbf{\text{u}}}^{\prime }{\left(\mathbf{\text{u}}\cdot ⟨{\mathbf{\text{u}}}^{\prime }⟩+\kappa \right)}^2\psi (\mathbf{\text{r}},{\mathbf{\text{u}}}^{\prime }).$$ (10)

SCFT calculations are carried out by starting from an initial guess for w(r, u) followed by calculating the ϕ(r, u) as above and adjusting the w(r, u) to satisfy Eq. (10). A root mean-square error of 10⁻⁴ between the left- and right-hand sides of Eq. (10) is sufficient for our calculations. We can then write the free energy per unit chain as

$$\frac{F}{n_{\text{c}}{k}_{\text{B}}T}=-\ln Q-\frac{1}{2V}\int d\mathbf{\text{r}}d\mathbf{\text{u}}w(\mathbf{\text{r}},\mathbf{\text{u}})\psi (\mathbf{\text{r}},\mathbf{\text{u}}).$$ (11)

The first term is the free energy of non-interacting chains, subject to the field, w(r, u), but counts doubly the enthalpic contribution to the free energy from chain–chain interactions. The second term corrects for this by subtracting off half of this chain–chain interaction energy.

The equilibrium isotropic phase does not have any position dependence and its bulk properties can be gleaned by considering a single point. To study the bulk behavior of the cholesteric phase, we need to include a spatial dependence. This can be done using a standard discretization of the grid into m points, each representing a spacing, Δ. Standard boundary conditions would, however, become problematic for studying the bulk behavior. Periodic boundary conditions impose a pitch, as they require an integer number of twists over the length of the box. Reflecting boundaries are similarly problematic. We use a special case of Neumann boundary conditions.

When examining the bulk cholesteric phase on a grid, we consider a single point and calculate spatial derivatives by assuming uniformity in a plane (say, the x–y plane), and in the direction normal to this plane, we calculate derivatives assuming that w(z ± Δ, θ, ϕ) = w(z, θ, ϕ ± kΔ). This imposes a helical twist around the z axis. As we will show, we can find the equilibrium pitch for the cholesteric phase by minimizing the free energy with respect to k.

SCFT calculations are conducted with a spatial resolution of Δ = 0.01L and an angular resolution of 20 points in the θ direction and 32 points in ϕ. Equation (7) is solved with 5 × 10⁴ steps in s using an Euler step. The large number of contour steps is to retain stability. This difficulty can likely be avoided by using a more sophisticated numerical algorithm, based on non-uniform spacing in ϕ, to eliminate bunching of points at the poles or conducting the SCFT in the spherical harmonic space. All of these are standard methods³⁷ but were not necessary for our purposes here.

III. RESULTS

This section describes the equilibrium phase behavior of a system of chiral worm-like chains (WLCs) described above. Here, we define lengths relative to the contour length of each chiral WLC, L, and fix κ = L⁻¹. Our choice of κ is arbitrary as it can be scaled out of the problem. This will be discussed further later. Unless otherwise stated, the WLCs described below have a persistence length of ℓ_p = L/20. This flexibility was chosen as a compromise between the limits of flexible polymers and rigid rods and also to mimic collagen.^24,38,39

We start by carrying out SCFT calculations with boundary conditions that impose a cholesteric pitch 2π/k. Figure 2 displays the free energy per chain of the cholesteric phase relative to the isotropic phase as a function of k. As shown in the figure, the free energy changes non-monotonically with k and develops a local minimum at a special value of k, denoted as k*. For γ = 18L², k* ≈ 2.030 and at this value, the free energy is negative. This indicates that the optimally twisted cholesteric phase is more stable than the isotropic phase. For γ < 17.314, the isotropic phase is always more stable. For γ > 17.314, however, the optimally twisted cholesteric phase is more stable than the isotropic. At γ = 17.314, the optimal pitch is k* = 2.000 74. We will expand on the isotropic–cholesteric transition later.

FIG. 2.

Free energy, per chain, of the cholesteric phase relative to the isotropic phase, $\left(F_{\text{C}}-F_{\text{I}}\right)/n{k}_{\text{B}}T$, as a function of the magnitude of the twist wave vector, k. We have used three choices of γ: γ = 17L², 17.314L², 18L². Note that $\left(F_{\text{C}}-F_{\text{I}}\right)/n_{\text{c}}{k}_{\text{B}}T$ changes non-monotonically with kL. There exists an optimal value of k $(\approx 2/L)$ at which $\left(F_{\text{C}}-F_{\text{I}}\right)/n_{\text{c}}{k}_{\text{B}}T$ is minimized. For the largest value of γ used, $\left(F_{\text{C}}-F_{\text{I}}\right)/n_{\text{c}}{k}_{\text{B}}T<0$ in an intermediate range of kL. In this case, the system tends to minimize its free energy at the optimal value of k by forming a cholesteric phase. We do not collect free energies below kL ≈ 1 because the cholesteric phase becomes unstable, i.e., it ceases to be metastable and there is no solution to the SCFT equations.

At low k, the cholesteric phase destabilizes, giving way to the isotropic phase. The benefit of forming the cholesteric phase is counterbalanced by the entropic penalty incurred. Note that the minimum is not at k = κ = L⁻¹, as it is in the OF model, i.e., if we minimized Eq. (2) with respect to k using n given by Eq. (1). To see why, consider that if n(r) is replaced by a vector field that is not of unit magnitude, the minimum in Eq. (1) shifts away from k = κ. For γ = 17.314, at the free energy minimum, |⟨u⟩| = 0.531, not unity. Furthermore, what is specified at a location r is not a single u but a distribution of segment orientations. In short, when we relax the assumption of perfect alignment and the degree of alignment can vary, κ no longer represents the equilibrium cholesteric pitch [see the relevant discussion in Eq. (6)].

Figure 3 shows the distribution ψ(r, u) obtained with γ = 1000 (large peak) and 17.314 (small peak); it shows a slice through θ = π/2. The distribution is more strongly peaked for larger γ. The magnitudes of the average local alignment vector are |⟨u⟩| ≈ 0.531 and 0.853 for γ = 17.314 and 1000, respectively. By contrast, the distribution for the isotropic phase is uniform and given by ψ(r, u) = 1/4π.

FIG. 3.

Distributions, ψ(r, u), are shown for γ = 1000 (solid line with squares peak) and 17.314 (solid line with circles) as well as for γ = 15.5 (dashed line with inverted triangles). The plot shows the distribution in the ϕ space—a slice through θ = π/2. The distribution is more strongly peaked for larger γ. The magnitudes of the average local alignment vector are |⟨u⟩| ≈ 0.531 and 0.853 for γ = 17.314 and 1000, respectively.

When the distribution ψ(r, u) is sharply peaked at a single u value, we expect to recover the behavior seen in the OF model. This is indeed the case for a large value of the cholesteric interaction strength γ. In this case, the field acting on segments is strong and tends to effectively suppress fluctuations in segment orientation. The resulting ψ(r, u) is more sharply peaked, as shown in Fig. 3.

This sharp peak leads to an optimal k, denoted as k*, that is closer to what the OF model suggests, as shown in Fig. 4(a): as γ increases, k tends to be k* = L⁻¹, at which the free energy in Eq. (2) is minimized. Except in this case, the values of k* in Fig. 4 (top) are larger than this limiting value. This is, however, a seeming discrepancy. When |u| = 1, the helical pitch is 2π/k*. However, this is no longer the case for finite γ.

FIG. 4.

Optimal twist wave vector, k* = |k*| and rescaled k*, i.e., k*A² (left axis) (a) and the helical pitch, 2π/k*A²L (b), as functions of cholesteric interaction strength, γ. (a) The optimal twist wave vector k is the value of k at which the free energy in Fig. 2 is minimized. Note that k* changes non-monotonically with γ. Also shown is $A=|⟨\mathbf{u}⟩|$ (right axis). Unlike k*, k*A² increases monotonically with γ and approaches 1 (dashed line), as γ → ∞. (b) The helical pitch is shown as a function of γ. It decreases monotonically and tends to π (dashed line) with increasing γ. Also shown is the rescaling factor A² (right axis).

Also shown in Fig. 4(a) are k*A², which describes the “effective” helical pitch, and A (right axis); recall that A²k in Eq. (6) plays the same role as k in Eq. (2). Indeed, k*A² increases monotonically and approaches 1 as γ increases.

Figure 4(b) displays our results for the (rescaled) helical pitch, 2π/k*A²L, as a function of γ. Also shown is the rescaling factor A² used to convert 2π/k*L into 2π/k*A²L (right axis). The helical pitch shown decreases monotonically and tends to 2π (dashed line) as γ increases. Only in the limit γ → ∞ will the calculated pitch reduce to the one based on minimization of f_{OF chol} in Eq. (2). Entropy tends to diminish the cholesteric ordering, more so for smaller γ. This tends to increase the helical pitch, as shown in Fig. 4(b).

We now turn to the free energy of the cholesteric phase obtained at k = k* relative to the isotropic phase, as shown in Fig. 5(a). The free energy of the cholesteric phase is calculated at the optimal pitch, k = k*, which is determined as illustrated in Fig. 2. The free energy of the isotropic phase, which has a uniform field, w(r, u), a uniform distribution, ψ(r, u) = 1/4π, and no molecular alignment (i.e., $A\equiv \sqrt{⟨\mathbf{\text{u}}\cdot \mathbf{\text{u}}⟩}=0$) is subtracted to give us the free energy difference between the phases. When γ = 17.314L², the free energy curve crosses the dashed line: the two phases have the same free energy. In addition, the slopes of the isotropic and cholesteric free energies are different at the transition, indicating that the transition between the two phases is first order.

FIG. 5.

Free energy (a) and phase diagram (b). (a) Free energy as functions of cholesteric interaction strength, γ. Free energies are calculated at the optimal pitch. As the cholesteric strength increases, the cholesteric phase becomes more stable with reference to the isotropic state. We did not extend to higher γ as calculations become unstable. (b) Phase diagram specifying the boundary between the cholesteric and isotropic phases as a function of polymer flexibility, L/ℓ_p. While the cholesteric phase is more stabilized for larger γ (a), the effect of chain entropy becomes more important with increasing L/ℓ_p. This competition determines the shape of the boundary.

Repeating the calculations above at different polymer flexibilities allows us to plot a phase diagram in the γ-L/ℓ_p space. This is shown in Fig. 5(b). As polymers become more flexible, a stronger interaction parameter is needed to bring about the cholesteric phase. In the limit of a rigid rod, the transition occurs at γ = 1.370 24L².

The strength of cholesteric interactions, γ, in this work has been treated simply as an input parameter. In experiments, the tendency for molecules to align into a cholesteric phase can be controlled using the concentration of molecules and crowding agents, such as proteoglycans (PGs).^26,42 These effects can, in principal, be mapped onto changes in γ, allowing one to predict the location of the transition in terms of the polymer flexibility and the control parameters.

IV. DISCUSSION

Self-consistent field theory has a long history of success in modeling the self-assembly of macromolecules (see Refs. 31 and 32 and references therein). It has grown to incorporate Flory–Huggins interactions for inter- and intra-molecular interactions, local chain rigidity, and nematic interactions among others.^33,38 This work expands its repertoire to include chiral-nematic interactions.

The use of SCFT to study chiral-nematic polymers, as opposed to phenomenological models, such as the Oseen–Frank (OF) model, yields a number of advantages and new pieces of information. One advantage is the explicit incorporation of chain flexibility. Previous work on collagen organization was able to obtain a qualitative behavior by tuning the parameters of the OF model.^28,29 Using SCFT allows one to explicitly study the organization of cholesteric molecules with their parameters set by experiments, including chain flexibility^24,39,40 as well as parameters describing the cholesteric phase itself, such as γ and κ.^11–18

In addition, this work enabled us to study systematically the isotropic–cholesteric transition. Indeed, this transition is realized in our work because it allows the degree of alignment to vary. In contrast, it is suppressed if the alignment is assumed to be simply a unit vector field, thereby ignoring the emergence of the isotropic phase, as is the case for the original OF model in Eq. (2). Although this is a step forward, there remain questions about how this transition will proceed. These include the limits of stability for the phases and the dynamics of the transition.

Throughout the results shown, we have fixed κ = L⁻¹. This choice is convenient but arbitrary. Repeating the calculations at a different value of κ merely shifts the free energies shown in Fig. 2 horizontally such that when plotted in terms of k/κ, the curves lie on top of each other. Similarly, the phase diagram, Fig. 5, is stretched vertically such that when plotted in terms of γκ², the transition is independent of κ. This follows from the fact that the internal energy and parameters in the model can be rescaled, as in Ref. 29; the parameter κ can be scaled out of the problem.

As the theory presented is a mean-field theory, there are also open questions about the effects of fluctuations. It is well-known that fluctuations play a large role in the nature of the order-disorder transition in block copolymers, for example.⁴¹ Addressing fluctuation effects typically requires sophisticated and computationally costly tools that are so far computationally out of reach for worm-like chains. In block copolymers, fluctuation effects on transitions are not significant if the transition is strongly first order.⁴¹ If this is the case for chiral molecules, we suspect that fluctuation corrections will be small based on Fig. 2, which suggests that the transition is somewhat strongly first order. Further clarifications are, however, left for future work.

It should be noted that other potentials have been applied to study the cholesteric phase, such as⁴³

$$U_{\text{C}2}(\mathbf{\text{r}},\mathbf{\text{u}},{\mathbf{\text{r}}}^{\prime },{\mathbf{\text{u}}}^{\prime })=V(\mathbf{\text{r}},{\mathbf{\text{r}}}^{\prime })(\mathbf{\text{u}}\cdot {\mathbf{\text{u}}}^{\prime })\frac{(\mathbf{\text{r}}-{\mathbf{\text{r}}}^{\prime })}{|\mathbf{\text{r}}-{\mathbf{\text{r}}}^{\prime }|}\cdot (\mathbf{\text{u}}\times {\mathbf{\text{u}}}^{\prime }),$$ (12)

giving interaction energy between segments at positions r and r′ with orientations u and u′, where V(r, r′) is a non-local potential giving the strength of the interaction. We chose the OF-inspired method over this one primarily due to the popularity of the OF model;^1,27 in addition, it will not involve complications associated with implementing non-local interactions as in Eq. (12).

Our choice of modeling cholesteric interactions after the OF model leaves open the possibility of including other interaction terms used therein.^1,27 These terms correspond to splay, bend, splay–bend, and saddle–splay deformations and would add another layer of complexity and predictive power to this model. Including them involves a treatment similar to what we have presented: imagine a polymer segment with orientation u interacting with an ensemble of other segments, {u′}, with an average orientation given by their ensemble average, ⟨u′⟩(r), and replace vector derivatives of the field n(r) in the OF model with vector derivatives of the vector field given by ⟨u′⟩(r).

Importantly, this approach can be extended to the practically relevant case of cholesteric molecules, possibly mixed with crowding molecules (e.g., PGs), near an interface.^24–26,38 In a cellular environment or in an experimental setting, these molecules experience inevitably the effects of confinement (e.g., plate-wall confinement). In experiments, these effects can be used to control the structure cholesteric molecules form as desired for their function.^{24–26,38,42} This is most obvious in corneal collagen for which the collagenous structure dictates their mechanical and optical properties.

Self-consistent field theory (SCFT) is well-suited to the study of confined polymer mixtures. Earlier, we clarified the effect of plate-wall confinement on the organization of nematic polymers using SCFT.³⁸ This can be extended to take into account cholesteric interactions proposed in this work [Eq. (6)] in the presence or absence of crowding agents. Indeed, there has been much effort to understand how confinement or crowding can be used to control the structure of the collagen assembly, inspired by the desire to engineer tissue constructs/substitutes (e.g., cornea and tendon).^24–26 We believe that our work constitutes a first step toward offering a physical model of the collagen assembly in this direction.

This work effectively combines two historically successful models, the OF model for liquid crystals, which describes the chiral-nematic behavior, and the self-consistent field theory of polymers, which describes precisely the behavior of polymers. This union allows for precise descriptions of chiral-nematic polymers. Although prevalent in a wide variety of biological and technological areas of research, semi-flexible polymers with chiral-nematic interactions were generally either treated crudely or ignored: approximated by the OF model, with parameters adjusted to approximate the qualitative behavior, and largely ignored by self-consistent field theory of polymers.

V. SUMMARY

We have presented a method for using self-consistent field theory to study cholesteric phases in chiral nematic semi-flexible polymers. The method employs a term in the effective Hamiltonian that was inspired by the historically successful, phenomenological liquid crystal model: the Oseen–Frank model. It allowed us to study the transition between the isotropic and cholesteric phases with varying polymer flexibility.

Including chirality in SCFT allows for the description of many types of behavior exhibited by semi-flexible molecules of biological and technological interest, such as collagen. Previous descriptions of cholesteric phases did not explicitly include polymer flexibility, a key parameter of semi-flexible polymers. As a result, they ignore much of the important physics in semi-flexible molecules. It is our hope that this attempt to incorporate molecular chirality into SCFT will shed light on the important aspects of the phase behavior of chiral nematic semi-flexible molecules.

AUTHOR DECLARATIONS

Conflict of Interest

The authors declare no conflict of interest.

DATA AVAILABILITY

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