Semiflexible polymers are ubiquitous in biological systems, e.g., as building blocks of the cytoskeleton, and they also play an important role in various materials due to their ability to form liquid-crystalline order. These rigid macromolecules are characterized by numerous (hierarchical) length-scales that define their static and dynamic properties. Confinement can promote uniform order, e.g., through capillary nematization in narrow slits, but it can also introduce long-ranged disruptions of the nematic ordering field through (unavoidable) topological defects in spherical containers. This Perspective concentrates on the theoretical description and computational modeling of such confined systems, with the focus on spherical containers that play an important role in the injection/ejection of double-stranded DNA from viral capsids and the fabrication of nematic droplets. Basic principles and recent developments are reviewed, followed by a discussion of open questions and potential directions for future research in this field.

The year 2020 marks the 100th anniversary of Staudinger’s seminal paper,1 which introduced the concept of macromolecules consisting of a large number of repeating units. Since then, our basic understanding of polymers and techniques for synthesizing new materials have taken large strides due to the concentrated efforts of science and industry. Polymers have become an integral part of our modern society because of their numerous applications. It was also realized that polymers play a key role in many biological systems, e.g., in the form of polysaccharides, proteins, and DNA. The vast majority of synthetic polymers can be considered as fully flexible, with persistence lengths p on the order of few nanometers, which is much smaller than typical polymer contour lengths L.2 Both in solution and in melt, such flexible macromolecules are characterized by a self-similar fractal structure, which allows for an elegant and simple description of their static and dynamic properties based on scaling arguments.3,4 On the opposite side of the spectrum lie rod-like objects with pL, which can form nematic liquid crystals at low concentrations.5 Prominent examples for these anisotropic particles include the tobacco mosaic virus,6,7 which was the first virus to be discovered and crystallized, and the short molecule 4-cyano-4′-pentylbiphenyl (5CB),8 which established the base for viable liquid-crystal displays.

Semiflexible polymers occupy the intermediate regime pL, which is realized for various biological macromolecules such as double-stranded DNA9 (p ≈ 50 nm in 0.1M aqueous NaCl), the fd virus10 (p ≈ 2.8 μm, L = 880 nm), and filamentous actin11 (p ≈ 10 μm–17 μm). Synthetic semiflexible polymers typically have much smaller persistence lengths in the range of p ≈ 6 nm–25 nm for ortho-phenylated poly(p-phenylene) (PPP),12, p ≈ 50 nm for poly(p-phenylene terephthalamide) (PPTA, Kevlar),13 and p ≈ 120 nm for poly(γ-benzyl-L-glutamate) (PBG).14 These (locally) stiff macromolecules share some of the properties of rod-like objects and fully flexible polymers. For instance, semiflexible polymers can form liquid-crystalline phases in (concentrated) solutions,15–22 resulting in interesting material properties such as good thermal conductivity and high tensile strength along the nematic director.23 Furthermore, (semi)dilute solutions of semiflexible chains exhibit a qualitatively similar response to strong shear as their fully flexible analogs.24–26 Despite these similarities, important key differences arise from the local bending rigidity of semiflexible polymers: Their conformations are not self-similar but are instead characterized by several crossover length-scales,27,28 which, in turn, introduce numerous disparate time scales that are relevant for accurately describing their dynamics.26,29–32

The unique properties of semiflexible polymers become even more apparent in confinement. In general, confined systems are characterized by a large surface to volume ratio, which can lead to strong deviations from the bulk behavior, even at regions far away from the confining interface. While confinement effects are predominantly entropic for fully flexible polymers and rigid rods,33 confinement incurs also energetic penalties for semiflexible polymers. The balance between the two contributions is primarily controlled by the contour length L and the persistence length p of the polymer, as well as by the geometry and characteristic length-scale of the container. Soft materials under confinement are ubiquitous in nature and technology,34 ranging from the crowded and fluctuating interior of biological cells35–38 to nematic droplets.39–44 Confinement can be imposed externally through a specific geometry or an external field.45 Alternatively, confinement can emerge spontaneously through the collective self-organization of the constituents into ordered assemblies, e.g., the lipid bilayer formation.46,47 One crucial aspect of confined systems is the shape of the enclosing geometry, which can be, for example, planar, cylindrical, or spherical. Furthermore, the confining interface can be rigid, flexible, porous, or semipermeable.

In this perspective, we will explore the ordering, phase behavior, and correlations of rigid rods and semiflexible chains in confinement, focusing on theoretical and simulation work of lyotropic solutions and melts. Section II contains a brief summary of some common models for describing these systems. The bulk properties of rigid rods and semiflexible polymers are briefly discussed in Sec. III. The behavior in planar, nanochannel, and spherical confinements is discussed in Secs. IV–VI, respectively. Section VII concludes with a brief summary and an outlook on future directions of this field.

Various theoretical models and simulation techniques have been developed for studying the structure and dynamics of rigid rods and semiflexible chains in confinement. The methods differ in, e.g., the resolved molecular detail, accessible length- and time scales, and computational complexity. A comprehensive list of existing models is much longer than what can be discussed here in sufficient detail, and different representations could be advantageous depending on the considered problem. Therefore, only a brief overview of selected models for lyotropic liquid crystal polymers in solutions and melts will be given. A more comprehensive discussion of the computational modeling of liquid-crystalline polymers can be found in, e.g., Refs. 22 and 48.

Molecular-level descriptions of semiflexible polymers allow for the detailed analysis of the conformation and dynamics of individual polymers as well as the collective behavior on mesoscopic length- and time scales. Depending on the level of coarse-graining, microscopic models can be used to simulate specific liquid-crystalline compounds49–51 or generic semiflexible polymers.22 Any such molecular-level model must include two independent length-scales, i.e., the persistence length p and the contour length L. Figure 1 provides schematic illustrations of some of the models that have been used for the simulation of semiflexible polymers.

FIG. 1.

Schematic representations of typical semiflexible chain models: (a) self-avoiding walk on a square lattice, (b) worm-like chain model with thickness d, and (c) bead-spring model with spherical monomers of diameter d. In (a) and (c), the distance b between discrete repeat units is indicated, while the bending angle θijk between subsequent bond vectors is shown in (c). The tangent vector u(s) of a chain with a continuous coordinate s along its contour is shown in (b).

FIG. 1.

Schematic representations of typical semiflexible chain models: (a) self-avoiding walk on a square lattice, (b) worm-like chain model with thickness d, and (c) bead-spring model with spherical monomers of diameter d. In (a) and (c), the distance b between discrete repeat units is indicated, while the bending angle θijk between subsequent bond vectors is shown in (c). The tangent vector u(s) of a chain with a continuous coordinate s along its contour is shown in (b).

Close modal
Early works used Monte Carlo (MC) simulations of self-avoiding random walks on square or simple cubic lattices [see Fig. 1(a)] due to their simplicity and computational efficiency.27,28,52–55 The bond length b is dictated by the lattice spacing, and the contour length of a polymer with N monomers is simply given by L = (N − 1)b. Subsequent bonds between beads i, j, and k can form angles of θijk = 0 or θijk = ±π/2. Bending stiffness can be introduced through a potential
(1)
where εp controls the interaction strength. The potential Ubend(θijk) vanishes for θijk = 0, i.e., when monomers i, j, and k lie on a straight line, and it is maximized when the three monomers form a sharp bend with θijk = ±π/2. Simple lattice models enable the efficient simulation of single chains with up to L=O(105) segments,27,28,55 which is useful for studying the various crossover regimes of single semiflexible chains in dilute solution. However, lattice models fail to capture the regime of small bending angles and the associated physics. Lattice artifacts are exacerbated in multi-chain systems with liquid-crystalline order since the director of the nematic ordering field is constrained to be aligned with one of the lattice axes.
Long macromolecules can also be regarded as continuous worm-like chains (WLCs),56 where the polymer conformation is described via the tangent vector u(s), with 0 ≤ sL being the coordinate along the contour [see Fig. 1(b)]. The Hamiltonian is given by the integral over the local curvature along the chain
(2)
with Boltzmann’s constant kB and temperature T. From Eq. (2), one can calculate the autocorrelation function of the tangent unit vectors
(3)
Hence, the tangent unit vectors u(s) lose the correlations of their orientation over the persistence length p. It should be noted that the standard continuous WLC model neglects excluded volume interactions among monomers. However, such interactions are crucial in crowded environments and strongly confined systems, where multiple chains are in close proximity to each other. Excluded volume can play a significant role even in dilute solutions when pL because local steric effects can lead to swelling of the chain.27,28,31 Furthermore, semiflexible chains can form hairpins and start to interact with themselves (see Sec. III).21,57,58
In contrast, it is much easier to include steric interactions into particle-based simulations, where polymers are represented by (spherical) particles connected through (rigid) joints with length b. In principle, a hard-sphere potential UHS could be employed for incorporating the excluded volume interactions. However, this choice is problematic in Molecular Dynamics (MD) simulations where the forces F = −∇U are required for propagating the system in time. To avoid this problem, UHS is usually replaced by a continuously differentiable function, e.g., the Weeks–Chandler–Andersen (WCA) pair potential59 
(4)
where r is the radial distance between a pair of beads, ε controls the strength of the repulsion, and d sets the diameter of the spherical beads. When the solvent is treated only implicitly, these interactions correspond to good solvent conditions for dilute polymer solutions. To vary the solvent quality in such an implicit solvent description, one can replace UWCA by the following non-bonded interaction:60 
(5)
with purely attractive pair potential UA(r). The dimensionless parameter λ controls the solvent quality (or the effective temperature of the system), with good solvent conditions at λ = 0, and deteriorating solvent quality with increasing λ. The attractive contribution is typically chosen as
(6)
where ULJ is the standard Lennard-Jones (LJ) potential with cutoff radius rc. Varying λ instead of the thermodynamic temperature T has the advantage that the strength of the excluded volume, bond, and bending interactions (and thus the persistence length p) remain unaffected.
Bonds between neighboring monomers of a chain are usually modeled using (harmonic) springs, which can be treated much more easily than rigid joints in MD simulations. A common choice for the bond potential is
(7)
with spring constant k and equilibrium bond length b. Typically, one chooses kkBT/d2 and bd to impede unphysical bond stretching and bond crossing. Small bond lengths in Eq. (7), e.g., b = 0.5d in Ref. 61, lead to chains of overlapping spheres with a smooth surface similar to the WLC model. In this case, excluded volume interactions between the nearest (and next-nearest) neighbors along the chains are turned off.
Alternatively, bonded interactions can be incorporated through the finitely extensible non-linear elastic (FENE) potential62 
(8)
with spring constant k and maximum extension r0. A typical choice for these parameters is k = 30kBT/d2 and r0 = 1.5d,63 which prevents the unphysical crossing of bonds. Employing UWCA for the excluded volume interactions (with ε = kBT) in combination with UFENE for the bonded interactions leads to an equilibrium bond length of b ≈ 0.97d.
Bending stiffness is typically included through a bending potential Ubend [see Eq. (1)]. For sufficiently small angles θijk, the bending potential in Eq. (1) can be approximated by a harmonic potential
(9)
The resulting persistence length can be estimated through
(10)
For semiflexible chains in dilute solutions and with sufficiently strong interaction strengths κεp/(kBT) ≫ 1, the persistence length can be approximated by p/bκ, as expected from the equipartition theorem.21,31

In particle-based models, confinement can be realized through microscopically resolved walls. This approach allows for arbitrary container shapes, patterned surfaces, and also deformable containers by introducing flexible bonds between neighboring wall particles. However, such a microscopic description of confinement can quickly become computationally taxing due to the potentially large number of required wall particles. Alternatively, confinement can be imposed by applying an external potential that exerts a force along the normal direction of the walls. This approach is computationally efficient, and its implementation is rather straightforward for planar, cylindric, or spherical confinement geometries. However, describing walls with surface roughness or more complex containers with local variations in curvature poses a significant challenge for such field-based descriptions.

Inspired by the anisotropic phases observed in dilute solutions of the tobacco mosaic virus,6,7 Onsager demonstrated that nematic order in systems of strictly hard rods emerges if the gain in positional entropy outweighs the accompanied decrease in orientational entropy.5 The degree of orientational order in the liquid-crystalline phase is typically quantified through the traceless tensor Qαβ with
(11)
where α, β are the Cartesian components x, y, z, and u is the unit vector along the axis of the rigid rods. In the case of semiflexible polymers, u is the unit bond vector between two monomers in discretized models or the tangent vector along the continuous chain contour (see Sec. II). In the nematic phase, the three eigenvalues of Qαβ are λ3 = S and λ2 = λ1 = −S/2, respectively.
For lyotropic solutions of rigid rods, the corresponding Helmholtz free energy F per molecule can be written as the sum of an ideal, Fid, and an excess contribution, Fexc. The ideal term is given by
(12)
where Nmol is the total number of (rod-like) molecules in volume V, f(ω) is the orientational distribution function of the molecules, and ω is a shorthand for the two polar angles θ and ϕ describing the orientation of the molecules. Within the second virial approximation, the excess term can be written as
(13)
with Vexcl(ω, ω′) characterizing the excluded volume between two (isolated) molecules with orientations ω and ω′, respectively. For fully rigid rods, Vexcl can be determined analytically,5 with the following simple solution in the limit Ld:
(14)
where γ is the angle between the long axes of the two rods. The equilibrium orientational distribution function f(ω) can be obtained by minimizing the free energy F, e.g., through a variational method with a suitable trial function. For long thin rods, the isotropic-nematic transition occurs at rather low concentrations5 so that the description in terms of only the second virial coefficient in Eq. (13) becomes self-consistent.

For short rigid rods (L ≈ 5d), the isotropic-nematic transition is weakly first order without a strong density jump,64 and the phase separation process has all the typical features of spinodal decomposition.65 As a consequence, these systems are typically characterized by polydomain nematic structures with disclination defects between them. For longer rods (L = 15d) and weak supersaturation, Cuetos and Dijkstra observed in their MC simulations the formation of monodomain nematic clusters, which gradually grew until all particles became nematically ordered with a (nearly) uniform orientation.65 Alternatively, uniform nematic order can be achieved by applying external stimuli, such as shear flow.66–68 

Khokhlov and Semenov15,16 extended Onsager’s approach to semiflexible chains by dividing, as a first step, the persistent macromolecule into discrete segments of length p such that dpL. Because the resulting elementary units are rigid rods, one can use Eq. (14) to describe their interactions. In Fid, the translational entropy of the molecules is still ∝ ln(Nmol/V), but the rotational entropy is now on the order of L/p due to the orientation of the individual persistent segments. The extension to continuous semiflexible chains is considerably more challenging as the problem involves the solution of an integral-differential equation. Khokhlov and Semenov proposed approximate interpolation expressions,15,16 which were later (qualitatively) confirmed by Chen through exact numerical calculations.18 

One key finding of these Onsager-style models was that the degree of nematic order, S, was consistently smaller in solutions of semiflexible chains compared to strictly rigid rods of the same contour length and concentration. Furthermore, the polymer concentration of the isotropic-nematic transition increased substantially with decreasing chain stiffness [see Fig. 2(a)], which is at odds with the underlying description based on the second virial approximation [cf. Eq. (13)]. The need to include higher-order virial terms has been widely recognized,69–75 but unfortunately there is no unique way of extending Onsager-style theories.19,20,70–74

FIG. 2.

(a) Nematic order parameter S and (b) persistence length p normalized by the value at infinite dilution, p,0, vs monomer number density, ρ, for semiflexible chains with N = 32 and various κ, as indicated. Vertical arrows in (b) indicate estimates for the onset of nematic order in the solution [see panel (a)]. Reproduced with permission from Milchev et al., J. Chem. Phys. 149, 174909 (2018). Copyright 2018 AIP Publishing LLC.

FIG. 2.

(a) Nematic order parameter S and (b) persistence length p normalized by the value at infinite dilution, p,0, vs monomer number density, ρ, for semiflexible chains with N = 32 and various κ, as indicated. Vertical arrows in (b) indicate estimates for the onset of nematic order in the solution [see panel (a)]. Reproduced with permission from Milchev et al., J. Chem. Phys. 149, 174909 (2018). Copyright 2018 AIP Publishing LLC.

Close modal

To characterize the isotropic-nematic transition and chain conformations of (thermotropic) semiflexible chains in solvent-free systems, Warner and co-workers developed a mean-field approach76,77 using a Maier–Saupe-like model.78 Within the assumption that Lp, they self-consistently determined the order parameter S by exploiting that the mean-field potential experienced by a molecule must be of even order due to the symmetry of the nematic ordering field. In the asymptotic case of strong nematic order, their theory predicted that the isotropic-nematic transition is first order. Liu and Fredrickson79 developed an alternative mean-field description for (thermotropic) solutions and blends of semiflexible polymers with isotropic and quadrupolar interactions. They used chain statistics in the isotropic state and performed a Landau–Ginzburg expansion in terms of two order parameters, i.e., the deviation of the local volume fraction of the polymer from its average and the local nematic tensor order parameter. Hence, the phase behavior was only characterized below or near the isotropic–isotropic and isotropic–nematic phase transitions where the order parameters are still sufficiently small.79,80 To overcome this limitation, Spakowitz and Wang81 reprised the field-theoretical formulation of Liu and Fredrickson79 but derived the exact mean-field equations valid for both the isotropic and nematic states. They systematically studied the phase behavior of semiflexible polymers in solution, finding, e.g., spinodal decomposition into a polymer-lean and a polymer-rich isotropic phase followed by subsequent nucleation of the nematic phase from the polymer-rich isotropic phase for systems with sufficiently strong isotropic repulsion. Decreasing the polymer–solvent repulsion shifted the critical point of the isotropic–isotropic transition inside the isotropic–nematic coexistence region.

In general, such field-theoretical descriptions are elegant as they reduce the many-chain problem to a single chain in a self-consistent field, allowing for the efficient exploration of phase diagrams of, e.g., homopolymers,18–21,76,77,79,81,82 polymer blends,79,80,83 and block copolymers.84–86 It should be noted, however, that the predicted mean-field critical behavior is often not quantitatively accurate as one finds instead, e.g., Ising-like behavior in solutions of semiflexible polymers in solvents of variable quality.60 Furthermore, effects due to long wavelength fluctuations in the nematic phase are not properly included in such mean-field models.19,20 Nevertheless, polymer field theories are valuable tools for studying the behavior of semiflexible chains in bulk and confinement (see Ref. 48 for a recent review on these topics).

The differences between the ordering of rigid rods and semiflexible polymers can be traced back to the finite persistence length of the latter, which allows for long wavelength fluctuations of the bond vector orientations. Hence, it is instructive to study the conformation of semiflexible polymers in more detail. In this regard, the WLC model has been examined extensively,3,87–89 as it allows for an (exact) analytical treatment. For a single (ideal) chain at infinite dilution, the mean-squared end-to-end distance is given by
(15)
In the limit pL, the chain effectively behaves like an infinitely thin rigid rod, whereas in the opposite limit pL, the chain conformation becomes essentially an ordinary random walk with Kuhn length K = 2p and mean-squared end-to-end distance Re2=2Lp.3  Figure 3 shows Re2/(2Lp) compared to simulation results of a bead-spring model with and without excluded volume effects (see Sec. II) at infinite dilution.31 For rather stiff chains with p ≳ 0.1L, the results of the bead-spring model agree with Eq. (15). For more flexible chains, however, local exclusion effects become increasingly important, resulting in a distinct swelling compared to the WLC model. These differences disappear when excluded volume interactions are turned off.
FIG. 3.

Reduced end-to-end length, Re2/(2Lp), of semiflexible chains with N = 48 and N = 96 beads and variable stiffness vs reduced contour length L/p. The solid line corresponds to the WLC model [see Eq. (15)], whereas filled and open symbols are the simulation results for bead-spring polymers with (+EV) and without (−EV) excluded volume, respectively. The arrows indicate the crossover from a Gaussian behavior to a self-avoiding walk-type behavior. Reproduced with permission from A. Nikoubashman, A. Milchev, and K. Binder, J. Chem. Phys. 145, 234903 (2016). Copyright 2016 AIP Publishing LLC.

FIG. 3.

Reduced end-to-end length, Re2/(2Lp), of semiflexible chains with N = 48 and N = 96 beads and variable stiffness vs reduced contour length L/p. The solid line corresponds to the WLC model [see Eq. (15)], whereas filled and open symbols are the simulation results for bead-spring polymers with (+EV) and without (−EV) excluded volume, respectively. The arrows indicate the crossover from a Gaussian behavior to a self-avoiding walk-type behavior. Reproduced with permission from A. Nikoubashman, A. Milchev, and K. Binder, J. Chem. Phys. 145, 234903 (2016). Copyright 2016 AIP Publishing LLC.

Close modal
At high concentrations above the isotropic-nematic transition, the semiflexible chains can be considered confined in a fictitious tube created by the nematic ordering field due to the surrounding chains (see also Sec. V).17,20,21,90,91 This view is inspired by the reptation model for entangled polymers,92,93 which assumes that the confining environment remains fixed over the characteristic relaxation time of a selected test chain. In the nematic state, the bond vectors along the chain contour u are aligned almost parallel to the nematic director n so that the bond vector autocorrelation function [Eq. (3)] can be expanded for small angles θ, resulting in
(16)
By restricting the degree of orientational order to θ2=2/α, one effectively introduces a new length-scale, namely, the so-called deflection length,91,
(17)
Deep in the nematic phase, α ≫ 1 and bond orientations deviate from the director only by small angles of order α−1/2. Thus, each semiflexible chain could be regarded as a sequence of L/d segments of length d that undulate inside a narrow virtual tube, with configurations similar to that in a nanochannel (see Sec. V). Using a Gaussian approximation for the orientational distribution function f(θ), Odijk derived an asymptotic expression for the parameter α,91,
(18)
Recent MD simulations and Density Functional Theory (DFT) calculations21 confirmed that Eq. (18) correctly reproduces the qualitative trends, i.e., α increases with increasing p and ρ, but there was neither quantitative matching nor the same qualitative scaling with ρ. Inserting Eq. (18) into Eq. (17) provides an analytic expression for the deflection length d, which was found to be in semi-quantitative agreement with MD simulations using a bead-spring model.21 For concentrated polymer solutions with strong nematic order, the resulting values of d were, however, not much larger than the bond length b,21 rendering the underlying assumption Lpdbd inapplicable. At this point, it should be noted that the employed WLC model assumes a constant persistence length p independent of the (local) polymer concentration [see Eq. (2)]: Although this approximation is reasonably accurate for dilute solutions in the isotropic phase, recent MD simulations of semiflexible chains in solution have revealed enhancements of p by up to 50% in the nematic phase [see Fig. 2(b)].21 Thus, theoretical models relying on chain properties sampled in the dilute limit should be used carefully in more concentrated systems.
Different chain statistics are expected near the isotropic-nematic transition, where S is no longer close to saturation [see Fig. 2(a)] and α ≲ 10.21 For sufficiently long chains Lp, Vroege and Odijk argued that the resulting polymer conformations can be described as anisotropic random walks with steps of length g (g) parallel (perpendicular) to the nematic director n.17 In this view, the parameter g (often referred to as the “global persistence length”) quantifies the average distance between hairpins and can be seen as the one-dimensional analog to the three-dimensional persistence length p in free solution. Assuming gd, one can estimate the energetic cost of forming a hairpin in the WLC model from Eq. (2) as
(19)
which is on the order of few kBT near the isotropic-nematic transition. According to these considerations, a sizable fraction of chains should form one or more hairpins at the onset of the nematic phase, which was indeed confirmed in recent MD simulations.21 
While Onsager-style theories and particle-based simulations describe liquid-crystalline systems from the bottom up, one can alternatively employ a top-down approach based on the macroscopic elastic constants of the material K1, K2, and K3. These three constants correspond to pure splay, pure twist, and pure bend distortions, respectively, and a combination of these terms can be used to represent any arbitrary deformation in a liquid crystal. For a non-chiral nematic liquid crystal, the increase in the free energy due to deviations of the uniformly aligned state can be written as94 
(20)
where n(r) is the slowly varying director field, describing the local molecular orientation. Theoretical descriptions based on Eq. (20) are especially useful for studying confined liquid-crystalline systems, e.g., nematic droplets (see Sec. VI), by choosing the appropriate integration volume and boundary conditions. Here, one important aspect is the connection between the (microscopic) particle properties, e.g., contour length L, thickness d, and persistence length p, and the resulting (macroscopic) elastic constants K1, K2, and K3. Although this relation has been studied extensively for lyotropic solutions of hard rods and semiflexible chains,21,95 there are still many open questions. For example, the theoretical result K1 = 3K2 is sometimes brought up in the literature as an exact solution, but it is a consequence of the (erroneous) assumption that orientational fluctuations are the only relevant degree of freedom. In reality, however, local density fluctuations of the suspended rods (or chain segments) are strongly coupled to the orientational fluctuations and therefore need to be taken into account as well.

Clearly, there are numerous theoretical approaches and open questions for describing the bulk behavior of liquid-crystalline molecules and polymers. However, describing all facets of this topic is beyond the scope of this review, and the interested reader is referred to Refs. 22, 91, and 95–99 for a more comprehensive overview of this topic.

Materials with uniform nematic order are needed for numerous applications, e.g., liquid crystal displays,100,101 ferroelectric liquid crystals,102 and liquid crystal elastomers.103 Furthermore, many (experimental) techniques for measuring the elastic constants in liquid crystals require the preparation of monodomain samples.95,104 However, creating systems with long-range nematic order is challenging because the nematic phase typically develops either through spinodal decomposition or nucleation and growth at various locations in the system,65 resulting in polydomain samples with disclination defects. One successful strategy for biasing the orientation of the director field is to confine the particles in (narrow) slits. There are various alignment techniques, including unidirectional rubbing,105 oblique evaporation,100 and optical etching,101,106 which result in an ordered layer at the solid surface with either parallel or perpendicular alignment of the anisotropic particles. This order then propagates to the bulk region of the system through elastic forces.

Using a Landau–de Gennes description for the confined liquid-crystalline material, Sheng determined the variation of the nematic order parameter in films with finite thickness H.107 His model predicted that the nematic–isotropic transition shifts to higher temperatures due to the ordering field exerted by the planar walls. Furthermore, the transition should be continuous for thin films below a critical thickness Hc, while it should become second order for thicker films, H > Hc. However, an important feature missing from this early model is the critical surface field, which should end the nematic–isotropic coexistence at a fixed film thickness H. Furthermore, note that the employed Landau theory is based on a power series expansion of the order parameters, in this case S [see Eq. (11)]. Therefore, this approach is strictly applicable only when S ≪ 1, which is not the case in the confined systems considered by Sheng, where S = 1 at the walls. Finally, such mean-field theories often provide inaccurate descriptions of order–disorder phenomena, in particular at low dimensionality.108 Despite these limitations, Sheng’s early theoretical work sets an important foundation for following efforts studying the surface-induced alignment of rod-like nematogens in thin films.109–113 For example, Escobedo and de Pablo studied the nematic ordering of athermal rigid rods between two parallel plates in more detail by performing expanded grand-canonical MC simulations.111 For a fixed chemical potential, the average nematic order parameter S was a non-monotonic function of the film thickness H, with a maximum at thicknesses commensurate with the correlation length of the effective intermolecular interactions. Furthermore, they found that the nematic order parameter was not uniform; it was largest at the walls and gradually decreased toward the center of the film.

Polymers in such one-dimensional confinement exhibit an even more complex behavior due to their internal degrees of freedom:114–116 On the one hand, a single polymer experiences a loss in configurational entropy when it is placed close to a surface, resulting in a local depletion of polymers compared to the bulk region. On the other hand, in concentrated solutions and melts, the macromolecules can pack against the surface to optimize the available free volume in the entire system, leading to an excess of polymers near the surfaces. With increasing chain stiffness, the typical polymer extension increases so that depletion effects become more prominent. At the same time, the bond angle distribution becomes more narrow, which affects the packing of the macromolecules against the surface. To better understand the interplay between these effects, Yethiraj performed MC simulations of a tangent-hard-sphere model in planar confinement,117 focusing on systems that are still in the isotropic regime in the bulk. At low densities, a distinct depletion of chains was observed near the walls, while the chains increasingly packed against the surface at higher densities. Furthermore, the degree of ordering increased with increasing chain stiffness. In all cases, there were more chain ends at the surface than in the bulk since placing an end monomer at the walls incurs a smaller entropic loss compared to placing a middle monomer there.

The phase behavior of athermal semiflexible chains in planar confinement was further characterized by Ivanov et al. using a bond fluctuation lattice model.118–120 They identified a bulk isotropic–nematic first-order transition ending in a critical point upon decreasing H (see Fig. 4). The chemical potential at this transition decreased with decreasing H, indicating that planar confinement can promote uniform order through capillary nematization. Another continuous transition was observed in the layers near the hard walls from a disordered to a quasi-two-dimensional nematic phase, while the bulk region of the film was still disordered. Finally, a strong coupling between single chain properties and long-range orientational order was found. In particular, they found that the WLC model accurately describes the mean-square end-to-end distance and persistence length of (short) semiflexible chains in dilute solutions but fails in systems with short- or long-range nematic order.

FIG. 4.

Qualitative phase diagram of semiflexible polymers confined between two hard walls as functions of the inverse film thickness H−1 and polymer volume fraction φ. The regions of isotropic (ISO) and nematic (NEM) phases both in three-dimensional (3D) and two-dimensional (2D) space are indicated. The transition between disordered (SD) and ordered (SO) surface regions is indicated by the dashed line, while the solid line marks the isotropic–nematic transition. Reproduced with permission from Ivanov et al., Macromolecules 47, 1206 (2014). Copyright 2014 American Chemical Society.

FIG. 4.

Qualitative phase diagram of semiflexible polymers confined between two hard walls as functions of the inverse film thickness H−1 and polymer volume fraction φ. The regions of isotropic (ISO) and nematic (NEM) phases both in three-dimensional (3D) and two-dimensional (2D) space are indicated. The transition between disordered (SD) and ordered (SO) surface regions is indicated by the dashed line, while the solid line marks the isotropic–nematic transition. Reproduced with permission from Ivanov et al., Macromolecules 47, 1206 (2014). Copyright 2014 American Chemical Society.

Close modal

The lattice model employed in Refs. 118–120 allows, however, only for two discrete director orientations in the wall-attached layers so that the observed order–disorder transition becomes an Ising model-like transition, which is not a realistic representation of the nematic order of semiflexible polymers. Due to this inherent shortcoming of lattice models, Binder and co-workers revisited the problem using off-lattice MD simulations and DFT calculations.82,121–124 When semiflexible polymers in a good solvent were confined between two purely repulsive and featureless plates, some nematic order was already stabilized in the isotropic phase.82,121,122 The average nematic order in the entire system increased with decreasing film thickness H, while the local nematic order was always most pronounced near the walls and then decayed monotonically toward the film center. Hence, there was capillary nematization, that is, the transition to uniform nematic order occurred at a smaller polymer concentration compared to the bulk (see Fig. 5).82,121,122 It should be noted, however, that the quasi-two-dimensional long-range nematic order observed in the wall-attached layers was likely just a finite size effect;82 rather, a power law decay of the orientational correlation is expected for sufficiently large systems, analogous to the Kosterlitz–Thouless transition125 of two-dimensional XY-ferromagnets.

FIG. 5.

Average nematic order parameter S (from MD simulations) in slits with width H vs monomer number density ρ for semiflexible chains with N = 32 and κ = 32. Corresponding data for bulk systems with periodic boundary conditions are shown by open symbols. Reproduced with permission from S. A. Egorov, A. Milchev, and K. Binder, Macromol. Theory Simul. 26, 1600036 (2017). Copyright 2017 John Wiley and Sons.

FIG. 5.

Average nematic order parameter S (from MD simulations) in slits with width H vs monomer number density ρ for semiflexible chains with N = 32 and κ = 32. Corresponding data for bulk systems with periodic boundary conditions are shown by open symbols. Reproduced with permission from S. A. Egorov, A. Milchev, and K. Binder, Macromol. Theory Simul. 26, 1600036 (2017). Copyright 2017 John Wiley and Sons.

Close modal

When the semiflexible chains were confined between two attractive walls, no capillary nematization was found anymore.123 This effect was attributed to the reduction in the polymer density in the center of the film because of the strong adsorption of polymers at the walls. The excess free energy due to the walls, i.e., the surface tension, was strongly negative since the adsorbed polymers lowered the potential energy of the system. In contrast, for purely repulsive walls, the surface tension was much smaller and positive, and it originated from the various entropic contributions of the confined chains.123 In Ref. 124, Milchev and Binder studied in more detail the ordering in the polymer layers adsorbed on attractive walls, finding both nematic and smectic phases. Furthermore, by carefully studying the orientational correlation function between all bond vectors, they demonstrated that the isotropic–nematic transition is indeed a Kosterlitz–Thouless transition.

The issue of semiflexible polymers in nanochannel confinement has generated significant attraction over the past decades,126–128 especially due to its relevance for sequencing single DNA molecules. The underlying idea is to linearly array the genome sequence by confining and stretching very long single DNA molecules in nanochannels. This approach keeps the sequence ordering intact and allows for single molecule analysis, as opposed to traditional sequencing methods such as gel electrophoresis129,130 that use large ensemble averages of short DNA fragments. Furthermore, because the extended state is the equilibrium conformation in the nanochannel, there is no need for complex flows to maintain chain stretching.131,132 Nanochannel confinement of DNA molecules can also be employed for other applications, including DNA sorting,133 barcoding DNA,134 and studying DNA–protein interactions.135 In this section, we focus on the conformation of a single semiflexible chain in a (cylindrical) nanochannel with diameter D, using a generic description of the polymer in terms of its thickness d, persistence length p, and contour length L.

The orientation of rigid rods in such two-dimensional confinement is restricted to angles that prevent collisions with the surrounding walls, e.g., sin(γ) ≤ D/L for thin rods (dDL). In contrast, semiflexible polymers can have a much wider range of configurations due to their finite persistence length (see Fig. 6). In the regime LD ≳ 2p, the conformation of confined semiflexible chains can be described accurately by the de Gennes blob model,33 with a one-dimensional arrangement of isometric compression blobs for channel diameters down to Dp2/d and compressed anisometric blobs for p2/dD2p.136–138 The limit LpD corresponds to the classic Odijk regime90 of undulating deflection segments (see also Sec. III). The intermediate regime Dp is perhaps the most challenging case since the confined polymer could potentially fold back and interact with itself; several theories136,139,140 have been proposed to predict and rationalize the chain conformations, but recent simulations141–143 have provided convincing evidence in favor of Odijk’s backfolding model.136 In the following, we will briefly discuss the salient features of these four confinement regimes and the corresponding theories.

FIG. 6.

Schematic representation of the various confinement regimes for a semiflexible chain with persistence length p and thickness d in a nanochannel with diameter D [the parameter ξ is defined in Eq. (32)]: (a) classic Odijk, (b) backfolded Odijk, (c) extended de Gennes, and (d) classic de Gennes regimes.

FIG. 6.

Schematic representation of the various confinement regimes for a semiflexible chain with persistence length p and thickness d in a nanochannel with diameter D [the parameter ξ is defined in Eq. (32)]: (a) classic Odijk, (b) backfolded Odijk, (c) extended de Gennes, and (d) classic de Gennes regimes.

Close modal
In the original de Gennes model, a (confined) polymer is described as a sequence of spherical blobs with diameter 2Rbl = D, each consisting of polymer segments with conformations following Flory statistics [see Fig. 6(d)].3 The mean-squared end-to-end distance of an unconfined random walk with cylindrical monomers scales as3,
(21)
with Flory exponent ν = 3/5 for a self-avoiding random walk in an athermal solvent and ν = 1/2 for an ideal chain. Hence, the contour length Lbl of a self-avoiding chain segment contained in a blob with diameter D scales as
(22)
The extension of the polymer parallel to the channel axis is then
(23)
which was confirmed by lattice MC144,145 and off-lattice MD simulations.146 
When the channel becomes narrower, the polymer contour per spherical blob Lbl could become so small that the comprised segments do not follow Flory statistics anymore. One can estimate the minimum channel diameter for which the classic de Gennes regime is still valid by setting the repulsion energy between blobs to kBT so that neighboring blobs cannot overlap,136 
(24)
Thus, for channel diameters smaller than Dp2/d, the confined semiflexible polymer cannot be considered as a sequence of spherical blobs anymore but instead should be regarded as an array of cylindrical (or ellipsoidal) blobs with dimensions Rbl,⊥ = D/2 and Rbl,∥ perpendicular and parallel to the channel axis [see Fig. 6(c)]. In this extended de Gennes regime, excluded volume interactions are sufficiently strong to avoid overlap of the blobs, but not strong enough to reach Flory statistics of the polymer segments inside the blobs. Assuming ideal polymer scaling along the channel axis, i.e., ν = 1/2 in Eq. (21), and replacing the blob volume D3 by D2Rbl,∥ in the denominator of Eq. (24) results in
(25)
and
(26)
The resulting chain extension along the channel axis, R = (L/Lbl)Rbl,∥, still scales as Eq. (23). However, the scaling of the free energy differs between the classic and extended de Gennes regimes due to the compression of the blobs in the latter case.
In the limit LpD, the channel boundary starts to have a strong influence on the local polymer conformation, resulting in rather stretched out configurations with large bending energies. Odijk argued that such confinement effects become increasingly relevant when the fluctuations of the chain contour in the radial direction become comparable to the pore diameter, i.e., ΔR2(d)(D/2)2, with deflection length d [see Fig. 6(a)].90,147 Using the asymptotic expression ΔR2(s)=2s3/(3p), derived by Yamakawa and Fujii for the WLC model near the rod limit,87 Odijk determined d as
(27)
By analyzing all accessible configurations, the increase in free energy due to confinement was estimated to scale as
(28)
for Lpd. It should be noted that Eq. (28) is not identical to the expression of Fc for a freely jointed coil of L/P rigid links in a pore90 because the latter always has Ubend = 0 and thus would permit the formation of hairpins. For semiflexible chains, by contrast, the formation of a single hairpin incurs an energetic penalty on the order of Ubend/(kBT) ∼ p/D [cf. Eq. (19)], which is much larger than unity and therefore unlikely. Hence, the mean extension of a confined semiflexible chain in the deflection regime is given by
(29)
with average deflection angle cos(γ) and amplitude αc = 0.1701, which was determined numerically for nanochannels with a cylindrical cross section.148 
The theoretical description of confined semiflexible chains with Dp is rather challenging because there are no dominant length-scales. Yet, this narrow range between the extended de Gennes and the classic Odijk regime is of great importance for practical applications such as genome mapping.126 Odijk proposed a mean-field Flory type expression for the free energy of a confined polymer consisting of L/d deflection segments,136,141
(30)
with global persistence length g (see Sec. III) and excluded volume between a pair of deflection segments vdd2d(D/p)1/3. The first term in Eq. (30) is due to the confinement of the deflection segments,90 the second term describes the entropic elasticity of an ideal chain, and the last term arises from the excluded volume interactions between the deflection segments. Minimizing Eq. (30) with respect to Re results in a (deceivingly) simple scaling law for the average chain extension along the channel axis,
(31)
where the dimensionless parameter
(32)
describes the ratio between the required and available volume to form a hairpin. For ξ > 1, the chain cannot fold back and instead follows the classic Odijk regime [see Fig. 6(a)]. The main challenge of this model is to accurately determine the global persistence length g that appears in Eq. (32). Odijk developed a mechanical approximation for g,149 and Muralidhar et al. performed extensive simulations to compute g and test Odijk’s theory.141 Although the simulation data showed the same qualitative trends as the theory, i.e., g/p increased rapidly with decreasing D/p, the theory overestimated g by about two orders of magnitude.141 A similar discrepancy between the (asymptotic) analytical expression for g and simulation results was also observed in bulk systems of nematically ordered chains.17,21

Figure 7 shows a summary of the relative chain extension R/L, highlighting the scaling behavior in the different confinement regimes.

FIG. 7.

Relative extension X/LR/L of a semiflexible chain with thickness wd vs reduced channel diameter Deff/p ≡ (Dd)/p. The blue shaded region corresponds to the classic and extended de Gennes regimes, which exhibit the same scaling [see Eq. (23)]. The beige and green shaded regions correspond to the backfolded and classic Odijk regimes, respectively. Note that the lower bound in ξ for the backfolded Odijk regime is a function of d/p, and the value of 10−3 was only used to illustrate that this regime can span many decades in ξ. Reproduced with permission from A. Muralidhar, D. R. Tree, and K. D. Dorfman, Macromolecules 47, 8446–8458 (2014). Copyright 2014 American Chemical Society.

FIG. 7.

Relative extension X/LR/L of a semiflexible chain with thickness wd vs reduced channel diameter Deff/p ≡ (Dd)/p. The blue shaded region corresponds to the classic and extended de Gennes regimes, which exhibit the same scaling [see Eq. (23)]. The beige and green shaded regions correspond to the backfolded and classic Odijk regimes, respectively. Note that the lower bound in ξ for the backfolded Odijk regime is a function of d/p, and the value of 10−3 was only used to illustrate that this regime can span many decades in ξ. Reproduced with permission from A. Muralidhar, D. R. Tree, and K. D. Dorfman, Macromolecules 47, 8446–8458 (2014). Copyright 2014 American Chemical Society.

Close modal

The static and dynamic properties of semiflexible chains in spherical confinement are largely dictated by the size ratios between the radius R of the container, the contour length of the macromolecules L, and their persistence length p. In the following, we will discuss the regimes of weak (RpL), intermediate (RLp), and strong confinement (LR).

The size regime RL is typical for self-assembled membranes150 and droplets containing small rod-like molecules.39–44 The natural tendency of these anisotropic particles to align parallel to each other is suppressed when they are confined to a spherical surface. In this case, the emergence of topological defects in the order parameter field is unavoidable according to the hedgehog theorem.151,152 Lubensky and Prost studied the ground states of order parameters with p-fold symmetry on a sphere using the one Frank constant approximation in the distortion free energy Fd.153 In this case, the order parameter rotates by 2π/p on a circuit that encloses the minimum energy defect. The winding angle is then specified as 2πp/nj, where nj is an integer characterizing the jth defect, e.g., nj = 6 for the hexatic order, where rotations by 2πp/6 lead to physically equivalent states. The Poincaré–Hopf theorem then implies that for any ordered texture on a surface with the topology of a sphere (Euler characteristic 2),
(33)
For instance, the ground state of a two-dimensional nematic texture on a sphere (p = 2) consists of four nj = +1 disclinations at the vertices of a tetrahedron,153 whereas aligned states of vectorial order parameters (p = 1) will be disrupted by two nj = +1 vortices located on opposite sides of the sphere (see Fig. 8).
FIG. 8.

(a) Splay vector order parameter configuration on a sphere. There are +1 vortices at the north and south pole. (b) Two views of a splay configuration of headless vectors on the sphere. There are four disclinations at the vertices of a tetrahedron. Reproduced with permission from D. R. Nelson, Nano Lett. 2, 1125 (2002). Copyright 2002 American Chemical Society.

FIG. 8.

(a) Splay vector order parameter configuration on a sphere. There are +1 vortices at the north and south pole. (b) Two views of a splay configuration of headless vectors on the sphere. There are four disclinations at the vertices of a tetrahedron. Reproduced with permission from D. R. Nelson, Nano Lett. 2, 1125 (2002). Copyright 2002 American Chemical Society.

Close modal
Nelson extended these ideas and studied in more detail the free energy of liquid-crystalline order with p-fold symmetry on a sphere surface.154 In such a quasi-two-dimensional system, twist elastic distortions are absent (K2 = 0), and using the approximation K = K1 = K3, the distortion free energy in Eq. (20) can be written as
(34)
with angle βij = dij/R between two defects i and j at geodesic distance dij. The term E(R) ∝ πK/p2 ln(R/a) is a defect self-energy, which depends on the details of the particle interactions near the defect. From Eq. (34), one sees that the interaction between defects on a sphere surface is purely repulsive and long-ranged. To assess the thermal stability of the ground state, Fd was studied for small disruptions, finding that the distortion from perfect alignment is proportional to kBT/K for both nematic and vectorial order parameters. Hence, the position of topological defects should be insensitive to thermal fluctuations in the high density limit where KkBT. However, one expects KkBT at low densities close to the isotropic–nematic transition154 and thus significant fluctuations of the defect locations. Considering that typical nematogens are strongly anisotropic objects with Ld, the elastic constant for bend distortions K3 can become much larger than for splay distortions K1.21,155 In such a case, the disruptions of the nematic ordering field become asymmetric, favoring textures that follow the lines of longitude.

Following these initial efforts, several numerical studies have been conducted to explore in more detail the organization of topological defects on the sphere surface. Vitelli and Nelson derived an exact solution for the nematic ground state in spherical shells of varying thickness.156 For very thin shells, the nematic ground state consisted again of four disclinations at the vertices of a tetrahedron, but a competing three-dimensional defect texture emerged as the shell thickness increased, which was characterized by two pairs of half-hedgehogs at the north and south pole. Dhakal et al. performed MC simulations of rigid rods confined to the surface of a sphere for various surface densities, temperatures, and size ratios 0.23 ≲ L/R ≲ 0.31.157 They observed a continuous transition from a tetrahedral defect arrangement at high temperatures (K1K3) to a configuration where the defects lied on a great circle as the temperature was lowered (K1 < K3).

So far, it was always assumed that the long axis of the anisotropic particles was oriented tangentially to the droplet surface (parallel anchoring). However, tilt angles of π/2 are also possible (homeotropic anchoring) by manipulating the effective surface interactions, e.g., through the addition of surfactants.158,159 The most common configuration observed for nematogens with K1 < K3 is a radial hedgehog located at the center of the drop. For nematogens favoring bend over splay distortions (K3 < K1), an equatorial disclination ring forms on the surface of the drop.42 For inhomogeneous boundary conditions, intermediate defect configurations between tetrahedral and radial were observed in experiments and simulations.160 Of course, this was just a short overview on the rich ordering behavior of short anisotropic particles in spherical confinement, and we refer the reader to, e.g., Refs. 42 and 44 for a more comprehensive review on this topic.

More recently, several simulation studies have focused on the case where all relevant length-scales are comparable, i.e., RLp. Milchev et al. employed both MD simulations and DFT calculations to investigate the ordering and conformation of semiflexible chains in spherical cavities with purely repulsive interactions.161–163 At low concentrations below the isotropic–nematic transition of the bulk, the centers-of-mass of the polymers were primarily located in the central region of the sphere to minimize the chain bending due to the curved container walls.161 The size of the resulting depletion zone near the walls increased with increasing contour length L (at fixed pL), noting that the monomer density remained almost constant everywhere. However, the polymers still explored a substantially larger portion of the sphere compared to (infinitely thin) rigid rods of the same length, where geometric constraints restrict center-of-mass positions with rR2(L/2)2. Furthermore, the chains in the sphere center were oriented randomly, while nematic bundles of chains with more or less parallel orientation of the bonds occurred close to the walls.

For more concentrated systems with LR, a gradual onset of nematic order was also seen in the entire spherical cavity, with distinct topological defects on the sphere surface.162,163 At high concentrations and intermediate stiffness 1/3 ≲ p/L ≲ 3/4, the semiflexible chains formed a latitudinal texture on the sphere surface [see Figs. 9(a-i) and 9(b-i)] with a pair of adjacent topological defects near each pole. At intermediate concentrations and high stiffness p ≈ 3L, a longitudinal bipolar phase was observed, as shown in Figs. 9(a-ii) and 9(b-ii). Densely filled spheres with stiffer semiflexible chains, 3/4 ≲ p/L ≲ 3, exhibited four topological defects in a tetrahedral arrangement, closely resembling a “tennis-ball” or “baseball” structure [cf. Figs. 8(b), 9(a-iii), and 9(b-iii)]. The topological defects were characterized using the eigenvalues of the tensor
(35)
which was computed from all unit vectors u(r) lying in the shell r > 0.85R. Here, M denotes the total number of bonds in this shell, and ui × ui+1 characterize the orientation of the plane formed by two successive bonds. The dyadic product in Eq. (35) is taken to symmetrize the chains (since they do not have “heads” or “tails”).
FIG. 9.

(a) Schematic representation of four defect structures on a spherical surface and the corresponding eigenvalues and eigenvectors of the order parameter Ω [see Eq. (35)]: (i) latitudinal bipolar phase, (ii) longitudinal bipolar phase, (iii) quadrupolar “tennis-ball” phase with defects arranged on the vertices of a tetrahedron, and (iv) quadrupolar phase with coplanar defects. (b) Simulation snapshots of semiflexible polymers confined in a spherical cavity with LR (adapted from Refs. 162 and 163), showing (i) latitudinal bipolar phase, (ii) longitudinal bipolar phase, and (iii) quadrupolar “tennis-ball” phase. (iv) Cross-sectional view on the equatorial plane in a quadrupolar “tennis-ball” system. The right inset highlights the smectic ordering of the chains, while the left inset shows the separation in “bulk” (blue) and “surface-attached” (red) chains.

FIG. 9.

(a) Schematic representation of four defect structures on a spherical surface and the corresponding eigenvalues and eigenvectors of the order parameter Ω [see Eq. (35)]: (i) latitudinal bipolar phase, (ii) longitudinal bipolar phase, (iii) quadrupolar “tennis-ball” phase with defects arranged on the vertices of a tetrahedron, and (iv) quadrupolar phase with coplanar defects. (b) Simulation snapshots of semiflexible polymers confined in a spherical cavity with LR (adapted from Refs. 162 and 163), showing (i) latitudinal bipolar phase, (ii) longitudinal bipolar phase, and (iii) quadrupolar “tennis-ball” phase. (iv) Cross-sectional view on the equatorial plane in a quadrupolar “tennis-ball” system. The right inset highlights the smectic ordering of the chains, while the left inset shows the separation in “bulk” (blue) and “surface-attached” (red) chains.

Close modal

The sphere interior was characterized by a distorted smectic structure with L/(2R) layers [see Fig. 9(biv)].162,163 When the chain contour length was comparable to the sphere radius, LR, only chains with one end close to the sphere center could fit along the axis connecting the poles, whereas chains further away from the center were strongly bent. Semiflexible chains with LR/2 formed four smectic layers, while no such simple ordering could be identified for chains with L ≈ 3R/2. In all cases, the nematic order parameter S of the confined chains was much smaller compared to the corresponding bulk systems [see Figs. 10(a) and 10(b)]. Furthermore, the chains developed a distinct biaxiality due to the confinement, which was most pronounced for the semiflexible chains near the curved surfaces [see Fig. 10(c)]. To better understand the coupling between the ordering in the interior of the cavity and the topological defects on the sphere surface, we conducted additional simulations where the polymers were confined to thin spherical shells.164,165 While fully flexible chains were entirely disordered, orientational order coupled with topological defects developed as the chain stiffness was increased. However, bipolar phases (either longitudinal or latitudinal) were unstable in shell confinement, and only quadrupolar defect structures were found, which changed their relative orientation with increasing chain rigidity.165 These findings underline the close relationship between the ordering behavior of short rod-like objects and long semiflexible chains.

FIG. 10.

Eigenvalues λ3 > λ2 > λ1 of the average nematic order tensor [Eq. (11)] vs stiffness parameter κ at fixed monomer density ρ = 0.7d−3 and chain length N = 32 in (a) the bulk and (b) a sphere with radius R = 35d. Solid lines were computed for all bonds in the system, while dashed and dotted curves show λ3 of bonds with r > 0.85R (i.e., from the surface region) and r < 0.85R, respectively. (c) Biaxiality parameter B = (λ2λ1)/2 plotted vs chain stiffness κ for the same case as in (b). The blue curve shows the data averaged over all bonds in the system, while the red and green curves show the bonds in the surface and bulk regions, respectively. Reproduced with permission from Milchev et al., Macromolecules 51, 2002 (2018). Copyright 2018 American Chemical Society.

FIG. 10.

Eigenvalues λ3 > λ2 > λ1 of the average nematic order tensor [Eq. (11)] vs stiffness parameter κ at fixed monomer density ρ = 0.7d−3 and chain length N = 32 in (a) the bulk and (b) a sphere with radius R = 35d. Solid lines were computed for all bonds in the system, while dashed and dotted curves show λ3 of bonds with r > 0.85R (i.e., from the surface region) and r < 0.85R, respectively. (c) Biaxiality parameter B = (λ2λ1)/2 plotted vs chain stiffness κ for the same case as in (b). The blue curve shows the data averaged over all bonds in the system, while the red and green curves show the bonds in the surface and bulk regions, respectively. Reproduced with permission from Milchev et al., Macromolecules 51, 2002 (2018). Copyright 2018 American Chemical Society.

Close modal

The aforementioned studies focused on the structure formation of monodisperse solutions of semiflexible chains, but many (biological) macromolecules are circular166 or have other topologies. Furthermore, the inside of biological cells is typically crowded and consists of numerous (semiflexible) components.35–38 As a first step toward understanding such multicomponent systems, Zhou et al. simulated bidisperse mixtures of unknotted and unlinked ring polymers.167 The spherical container was predominantly filled with short stiff rings (p ≈ 5L, LR/2) and few long semiflexible rings with varying stiffness (p ≲ 2L, L ≈ 5R/2). Fully flexible long rings were randomly immersed in a matrix of short rings, but with increasing stiffness, the long polymers adopted a rigid ring conformation and attached to the surface of the sphere. A similar mixed state was observed in mixtures of short stiff rings and semiflexible chains (L ≈ 5R/4) for persistence lengths p ≲ 2L. However, as the chain stiffness was increased further to p ≈ 4L, the semiflexible chains accumulated in the sphere center to maximize their orientational entropy and to minimize the energetic penalty due to wall-induced bending. In a recent simulation study,168 Zhou et al. extended these simulations to mixtures of semiflexible chains and rings inside oblate ellipsoids. At low concentrations, both species were distributed homogeneously, irrespective of the chain stiffness (the persistence length of the rings was kept fixed at p ≈ 5L/2). At high polymer concentrations, moderately stiff chains (pL) occupied the central region of the ellipsoid and pushed the rigid rings to the walls. When the chain stiffness was increased further to p ≈ 6L, the chains developed a (distorted) liquid-crystalline order, while the rings formed a bent stack on the equator of the ellipsoid.

A considerable amount of research has been devoted to the regime LR, which is of particular interest for a range of biological systems, such as DNA-histone complexation in the eukaryotic nucleus169–171 or the packaging and injection of DNA in bacteriophage.172–176 The former system can be (approximately) described by a semiflexible polymer confined onto a sphere surface.177–181 Spakowitz and Wang derived analytic expressions for the polymer statistics in such effectively two-dimensional systems by neglecting excluded volume interactions.179 They found that in the limit Rp, the chains essentially behaved like semiflexible polymers confined to a plane, with their maximum end-to-end distance bounded by the sphere radius. A qualitatively different behavior was observed, however, for pR, where orientational correlations caused the chains to lie on the equator of the sphere, and conformation fluctuations spread the segment density from the equator toward the poles. At high surface densities, excluded volume interactions play an important role since the chain can wrap around the sphere multiple times and interact with itself. Indeed, Zhang and Chen have shown in MC simulations of a bead-spring model that excluded volume effects were responsible for the formation of an ordered anisotropic state with a “tennis-ball” texture.181 

The conformation and packaging of semiflexible polymers change completely when the chains are not confined to the sphere surface anymore but are allowed to explore the inside of the container.180,182–193 Structured configurations were found for pR, even in the absence of excluded volume interactions between chain segments due to the interplay of bending correlations and confinement.180 (The case Rp is less interesting as the chain essentially behaves like an ideal polymer with a bounded end-to-end length.) A qualitatively similar behavior was also observed when excluded volume interactions were included [see Fig. 11(c)].182,188–193 For semiflexible chains with self-attraction (e.g., cholesteric interaction in DNA), a toroidal structure formed spontaneously when the chain was inserted into a spherical cavity.182,189 As more and more segments filled the container, the toroidal structure expanded parallel to its central axis into a spool-like structure with a nearly empty internal cylindrical core. Once the sphere shell was fully covered with chain segments, the core of the spool was filled with strands parallel to this axis. In Figs. 11(a) and 11(b), this process is shown for the ejection of a DNA strand from a capsid (read from right to left for DNA injection). Curk et al. investigated the ground states of spherically confined semiflexible chains in more detail.193 Their theoretical calculations and simulations identified up to three nested concentric spools as the dominant structure at low concentrations, followed by more complex morphologies resembling Hopf fibrations at higher concentrations. Liang et al. determined the phase diagram of a single semiflexible chain confined in a sphere (Fig. 12) through self-consistent field theory calculations and mean-field MC simulations,194 demonstrating that the polymers can undergo several phase transitions during injection/ejection.

FIG. 11.

Snapshots from three ejection runs. The DNA cholesteric interactions were considered for runs shown in (a) and (b), as well as for generating the initial fully packaged state. The two runs differ for the (a) absence and (b) presence of an initial lag phase. The cholesteric interaction was neglected in (c). The initial arrangements of the packaged genome are shown (left) and followed by snapshots taken at various time intervals and percentage of packaged genome, as indicated. For visual clarity, beads are colored with a rainbow scheme (red → yellow → green → blue), and their rendered size is decreased systematically going from the red to the blue end. Reproduced with permission from Marenduzzo et al., Proc. Natl. Acad. Sci. U. S. A. 110, 20081 (2013). Copyright 2013 National Academy of Sciences.

FIG. 11.

Snapshots from three ejection runs. The DNA cholesteric interactions were considered for runs shown in (a) and (b), as well as for generating the initial fully packaged state. The two runs differ for the (a) absence and (b) presence of an initial lag phase. The cholesteric interaction was neglected in (c). The initial arrangements of the packaged genome are shown (left) and followed by snapshots taken at various time intervals and percentage of packaged genome, as indicated. For visual clarity, beads are colored with a rainbow scheme (red → yellow → green → blue), and their rendered size is decreased systematically going from the red to the blue end. Reproduced with permission from Marenduzzo et al., Proc. Natl. Acad. Sci. U. S. A. 110, 20081 (2013). Copyright 2013 National Academy of Sciences.

Close modal
FIG. 12.

Left: phase diagram of a single semiflexible chain confined in a sphere with radius R and volume V = 4πR3/3, shown in the reduced variables ρ0 = 2Lℓpd/V vs p/R. The morphologies are isotropic (ISO), coaxial spools (CSs), Hopf fibrations (HFs), and condensed Hopf fibrations (CHFs). The first-order phase transition is represented by a black solid line, while second-order ones are represented by blue dashed lines. The orange square indicates the critical point where the first-order line terminates. Right: examples produced from typical DNA parameters according to the theory by Liang et al. Reproduced with permission from Q. Liang, Y. Jiang, and J. Z. Y. Chen, Phys. Rev. E 100, 032502 (2019). Copyright 2019 American Physical Society.

FIG. 12.

Left: phase diagram of a single semiflexible chain confined in a sphere with radius R and volume V = 4πR3/3, shown in the reduced variables ρ0 = 2Lℓpd/V vs p/R. The morphologies are isotropic (ISO), coaxial spools (CSs), Hopf fibrations (HFs), and condensed Hopf fibrations (CHFs). The first-order phase transition is represented by a black solid line, while second-order ones are represented by blue dashed lines. The orange square indicates the critical point where the first-order line terminates. Right: examples produced from typical DNA parameters according to the theory by Liang et al. Reproduced with permission from Q. Liang, Y. Jiang, and J. Z. Y. Chen, Phys. Rev. E 100, 032502 (2019). Copyright 2019 American Physical Society.

Close modal

The slow dynamics of semiflexible chains26,29–32 become even more exacerbated in densely filled containers, implying that the resulting chain configurations of the non-equilibrium polymer injection/ejection strongly depend on the applied protocol.195 To better understand the relevance of polymer dynamics, the packing of a single long semiflexible chain into icosahedral and spherical cells at constant insertion force was studied via MD simulations.182,184,185,187 At early stages of packing, when the capsid was still mostly empty, the packing curves were highly regular with roughly linearly decaying insertion velocities.184,185 During the final stages of packing, however, the insertion process was characterized by random stalling and bursts due to the reorganization of the confined polymer segments.185 Ali et al. investigated through hybrid MD simulations the impact of the container shape on the insertion and ejection dynamics,187 finding that flexible polymers were released more quickly from elongated capsids than from spherical ones. For semiflexible polymers, however, the opposite trend was found, which was partly attributed to the larger loss in bending energy incurred by spherical confinement. Irrespective of the container shape, packing and ejection times were considerably faster for flexible polymers,184,187 which is consistent with their faster relaxation in unconfined solutions.29–31 Furthermore, significant hysteresis was found,187 that is, the force required for injecting a polymer was considerably larger than the one experienced during ejection. More recently, Marenduzzo et al. investigated the impact of knotting on the release kinetics,189 revealing that ordered DNA spools experience a much lower effective friction compared to disordered entangled chains.

Understanding and controlling the properties of semiflexible polymers in confinement is of peculiar interest for many biological systems and for applications that leverage their liquid-crystalline order. In bulk solutions of lyotropic semiflexible polymers, increasing the polymer concentration can induce a transition from an isotropic to a nematic state, solely driven by entropic effects. This (weakly) first-order transition is typically accompanied by a marked change in the material’s elastic and optical properties, which distinctly depend on the microscopic polymer characteristics, e.g., their length and bending stiffness. Studying the ordering of semiflexible polymers in confinement is, however, much more challenging due to the numerous (competing) length-scales involved. Theoretical modeling and computer simulations are valuable tools for shedding light on these problems as they allow for systematic control over the relevant parameters and for a (microscopically) detailed analysis of the resulting trajectories. For example, it was found that confinement can enhance the inherent tendency to form ordered structures, e.g., through capillary nematization in thin films, whereas it can also suppress the long-range order through the emergence of topological defects in spherical containers. Despite the rapid progress in the past decades, there are still many open questions and directions for future research.

The vast majority of previous work has focused on the equilibrium structures in confined systems but largely neglected the associated dynamic properties. These effects are of particular importance for non-equilibrium situations, e.g., the injection/release of semiflexible polymers into/from capsules,172–175 which depends on the specific process pathway.187,195 Already in bulk solutions, solvent-mediated hydrodynamic interactions can have a significant effect on the resulting dynamics of semiflexible chains;26,29,31 for example, the diffusion of the chains slows down and becomes increasingly anisotropic with increasing chain stiffness,26,31 features that are absent in the Rouse model, where the noise force is assumed to act independently on each bead and to be uncorrelated in time.4 Various techniques have been developed for efficiently treating hydrodynamics in computer simulations,196–198 but extending these methods to confined systems is often challenging due to many-body effects in concentrated solutions and the inherently long-ranged nature of hydrodynamic interactions.199,200

Another promising avenue for future research is the development of multiscale simulation techniques,201–203 which combine the fast convergence of coarse-grained representations with the accuracy of (chemically accurate) fine-grained models. Such hybrid approaches have been successfully applied for studying the phase behavior of small liquid-crystalline molecules with atomistic details,204,205 and similar procedures should also be applicable for simulating long semiflexible macromolecules. However, the parameterization of such multiscale models requires great care as the chain properties are highly state dependent [see, e.g., Fig. 2(b)]. One could also use fine-grained simulations to compute the elastic constants in the bulk21,51,206 and then use these parameters in a continuum model to determine the ground states in confined systems. In principle, one could also infer the elastic constants of liquid-crystalline molecules and polymers by systematically varying K1, K2, and K3 in continuum level descriptions of confined systems until the number and distribution of topological defects match the fine-grained simulations.

Furthermore, a particular restriction shared by the majority of previous studies is the complete rigidity of the cavity. This constraint is, however, rarely satisfied in biological systems, where, e.g., microtubules confined in lipid bilayer membranes207 as well as actin networks in vesicles208 are able to significantly deform their weak confinements. A similar deformation of the enclosing capsule was also observed in experiments of (oxidized) carbon nanotubes in oil-in-water and water-in-oil mixtures.209 Only few theoretical studies have addressed this issue to date. Notably, Vetter et al. investigated the packaging of elastic wires confined by elastic shells210,211 using the Kirchhoff rod model212 with dry stick-slip friction. They observed the emergence of two distinct packing patterns depending on the friction between segments of the filament and the wall (see Fig. 13). Without friction, the filament bundled into a tight toroidal curve, whereas it packed into a highly disordered hierarchic structure at high friction. Thus, understanding the interplay between polymer stiffness and shell elasticity is a crucial next step for establishing a more comprehensive picture of the packing and ordering in confinement.

FIG. 13.

Simulation snapshots of single elastic filaments of reduced length l = L/R in elastic confinement without (top) and with (bottom) friction. Simulations performed at fixed filament diameter d = R/10, shell thickness ds = R/200, and ratio between Young’s modulus of the filament and shell Ef/Es = 100. Reproduced with permission from R. Vetter, F. K. Wittel, and H. J. Herrmann, Europhys. Lett. 112, 44003 (2015). Copyright 2015 IOP Publishing.

FIG. 13.

Simulation snapshots of single elastic filaments of reduced length l = L/R in elastic confinement without (top) and with (bottom) friction. Simulations performed at fixed filament diameter d = R/10, shell thickness ds = R/200, and ratio between Young’s modulus of the filament and shell Ef/Es = 100. Reproduced with permission from R. Vetter, F. K. Wittel, and H. J. Herrmann, Europhys. Lett. 112, 44003 (2015). Copyright 2015 IOP Publishing.

Close modal

To date, most theoretical and computational studies considered passive systems and thermodynamic equilibrium arguments for studying the ordering of semiflexible chains in confinement. Biological systems are, however, inherently out of equilibrium as external sources of energy are constantly consumed for sensing, motion, or reproduction. For example, microtubules and filamentous actin seemingly move through the cytoplasmic matrix via the continuous addition and removal of subunits.213 In a recent numerical study,214 Das and Cacciuto reevaluated the packaging of a single long semiflexible filament inside a spherical cavity (LpR) with an additional active force acting along the chain contour. They found that small amounts of activity were enough to unwrap the filament from a multispool configuration to a single spool with a latitudinal bipolar order [cf. Fig. 9(ai)]. At higher activity, the filament could cross more easily the energy barrier between different states, and it switched almost periodically between (multi)spool, “tennis-ball,” and disordered configurations. Manna and Kumar studied the ordering and collective dynamics of active filaments in the size regime RLp,215 finding that contractile chains self-knotted into entangled melts already at low concentrations and weak activity, whereas extensile polymers formed an entangled coherently moving state only above a threshold concentration and activity (see Fig. 14). Contractile chains confined to the sphere surface developed isotropic, orientationally ordered, and micro-phase separated states, while extensile chains showed a transition between isotropic and nematic states with increasing density. The models considered in Refs. 214 and 215 assumed that the filaments have constant activity and that the nutrients are homogeneously distributed within the cavity. In reality, however, the consumed nutrients are not instantaneously replenished, resulting in gradients or even the complete exhaustion of energy sources in the cell, which should impact the activity of the particles. Furthermore, it would be interesting to simulate mixtures of active and passive or of contractile and extensile filaments.

FIG. 14.

(a) and (b) Simulation snapshots of activity-induced knotting of two contractile filaments recorded at two different times. (c) Contractile and (d) extensile filaments at volume fraction φ = 0.23. Chains are colored from green to red with increasing normalized magnitude of their local curvature. Selected chains are shown in blue as guides to the eye. Reproduced with permission from R. K. Manna and P. B. S. Kumar, Soft Matter 15, 477 (2019). Copyright 2019 The Royal Society of Chemistry.

FIG. 14.

(a) and (b) Simulation snapshots of activity-induced knotting of two contractile filaments recorded at two different times. (c) Contractile and (d) extensile filaments at volume fraction φ = 0.23. Chains are colored from green to red with increasing normalized magnitude of their local curvature. Selected chains are shown in blue as guides to the eye. Reproduced with permission from R. K. Manna and P. B. S. Kumar, Soft Matter 15, 477 (2019). Copyright 2019 The Royal Society of Chemistry.

Close modal

I gratefully acknowledge financial support from the German Research Foundation (DFG) under Grant Nos. NI 1487/2-1, NI 1487/2-2, and NI 1487/4-2. I also thank K. Binder, A. Milchev, and S. Egorov for many fruitful discussions.

1.
H.
Staudinger
,
Ber. Dtsch. Chem. Ges. A/B
53
,
1073
(
1920
).
2.
Polymer Handbook
, 4th ed., edited by
J.
Brandrup
,
E. H.
Immergut
, and
E. A.
Grulke
(
Wiley
,
2003
).
3.
M.
Rubinstein
and
R. H.
Colby
,
Polymer Physics
(
Oxford University Press
,
Oxford
,
2003
).
4.
M.
Doi
and
S. F.
Edwards
,
The Theory of Polymer Dynamics
(
Oxford University Press
,
1988
), Vol. 73.
6.
M. W.
Beijerinck
,
Verh. K. Akad. Wet. Amsterdam, Afd. Natuurkd.
5
,
3
21
(
1898
).
7.
F. C.
Bawden
,
N. W.
Pirie
,
J. D.
Bernal
, and
I.
Fankuchen
,
Nature
138
,
1051
(
1936
).
8.
G. W.
Gray
,
K. J.
Harrison
, and
J. A.
Nash
,
Electron. Lett.
9
,
130
(
1973
).
10.
E.
Barry
,
D.
Beller
, and
Z.
Dogic
,
Soft Matter
5
,
2563
(
2009
).
11.
R.
Zhang
,
N.
Kumar
,
J. L.
Ross
,
M. L.
Gardel
, and
J. J.
de Pablo
,
Proc. Natl. Acad. Sci. U. S. A.
115
,
E124
(
2018
).
12.
A.
Baun
,
Z.
Wang
,
S.
Morsbach
,
Z.
Qiu
,
A.
Narita
,
G.
Fytas
, and
K.
Müllen
,
Macromolecules
53
,
5756
(
2020
).
13.
P.
Metzger Cotts
and
G. C.
Berry
,
J. Polym. Sci., Polym. Phys. Ed.
21
,
1255
(
1983
).
14.
V. N.
Tsvetkov
,
I. N.
Shtennikova
,
V. S.
Skazka
, and
E. I.
Rjumtsev
,
J. Polym. Sci., Part C: Polym. Symp.
16
,
3205
(
1967
).
15.
A. R.
Khokhlov
and
A. N.
Semenov
,
Physica A
108
,
546
(
1981
).
16.
A. R.
Khokhlov
and
A. N.
Semenov
,
Physica A
112
,
605
(
1982
).
17.
G. J.
Vroege
and
T.
Odijk
,
Macromolecules
21
,
2848
(
1988
).
18.
Z. Y.
Chen
,
Macromolecules
26
,
3419
(
1993
).
19.
S. A.
Egorov
,
A.
Milchev
, and
K.
Binder
,
Phys. Rev. Lett.
116
,
187801
(
2016
).
20.
S. A.
Egorov
,
A.
Milchev
,
P.
Virnau
, and
K.
Binder
,
Soft Matter
12
,
4944
(
2016
).
21.
A.
Milchev
,
S. A.
Egorov
,
K.
Binder
, and
A.
Nikoubashman
,
J. Chem. Phys.
149
,
174909
(
2018
).
22.
K.
Binder
,
S. A.
Egorov
,
A.
Milchev
, and
A.
Nikoubashman
,
J. Phys.: Mater.
3
,
032008
(
2020
).
23.
A. M.
Donald
,
A. H.
Windle
, and
S.
Hanna
,
Liquid Crystalline Polymers
(
Cambridge University Press
,
2006
).
24.
R. G.
Winkler
,
Phys. Rev. Lett.
97
,
128302
(
2006
).
25.
R. G.
Winkler
,
J. Chem. Phys.
133
,
164905
(
2010
).
26.
A.
Nikoubashman
and
M. P.
Howard
,
Macromolecules
50
,
8279
(
2017
).
27.
H.-P.
Hsu
,
W.
Paul
, and
K.
Binder
,
Europhys. Lett.
92
,
28003
(
2010
).
28.
H.-P.
Hsu
,
W.
Paul
, and
K.
Binder
,
Europhys. Lett.
95
,
68004
(
2011
).
30.
G. T.
Barkema
,
D.
Panja
, and
J. M. J.
van Leeuwen
,
J. Stat. Mech.: Theory Exp.
2014
,
P11008
.
31.
A.
Nikoubashman
,
A.
Milchev
, and
K.
Binder
,
J. Chem. Phys.
145
,
234903
(
2016
).
32.
N.-K.
Lee
,
J. Korean Phys. Soc.
73
,
488
(
2018
).
33.
P.-G.
de Gennes
,
Scaling Concepts in Polymer Physics
(
Cornell University Press
,
1979
).
34.
S.
Perkin
and
J.
Klein
,
Soft Matter
9
,
10438
(
2013
).
35.
S. B.
Zimmerman
and
S. O.
Trach
,
J. Mol. Biol.
222
,
599
(
1991
).
36.
M.
Otten
,
A.
Nandi
,
D.
Arcizet
,
M.
Gorelashvili
,
B.
Lindner
, and
D.
Heinrich
,
Biophys. J.
102
,
758
(
2012
).
37.
J.
Alvarado
,
B. M.
Mulder
, and
G. H.
Koenderink
,
Soft Matter
10
,
2354
(
2014
).
38.
M. E.
Grady
,
E.
Parrish
,
M. A.
Caporizzo
,
S. C.
Seeger
,
R. J.
Composto
, and
D. M.
Eckmann
,
Soft Matter
13
,
1873
(
2017
).
39.
A.
Fernández-Nieves
,
V.
Vitelli
,
A. S.
Utada
,
D. R.
Link
,
M.
Márquez
,
D. R.
Nelson
, and
D. A.
Weitz
,
Phys. Rev. Lett.
99
,
157801
(
2007
).
40.
T.
Lopez-Leon
,
A.
Fernández-Nieves
,
M.
Nobili
, and
C.
Blanc
,
Phys. Rev. Lett.
106
,
247802
(
2011
).
41.
T.
Lopez-Leon
,
V.
Koning
,
K. B. S.
Devaiah
,
V.
Vitelli
, and
A.
Fernández-Nieves
,
Nat. Phys.
7
,
391
(
2011
).
42.
T.
Lopez-Leon
and
A.
Fernández-Nieves
,
Colloid Polym. Sci.
289
,
345
(
2011
).
43.
I.
Gârlea
,
P.
Mulder
,
J.
Alvarado
,
O.
Dammone
,
D.
Aarts
,
M.
Lettinga
,
G.
Koenderink
, and
B.
Mulder
,
Nat. Commun.
7
,
12112
(
2016
).
44.
I.
Muševicč
,
Liquid Crystal Colloids
(
Springer International Publishing
,
2017
).
45.
H.
Löwen
,
J. Phys.: Condens. Matter
13
,
R415
(
2001
).
46.
S. J.
Singer
and
G. L.
Nicolson
,
Science
175
,
720
(
1972
).
47.
K.
Jacobson
,
O. G.
Mouritsen
, and
R. G. W.
Anderson
,
Nat. Cell Biol.
9
,
7
(
2007
).
49.
G.
Tiberio
,
L.
Muccioli
,
R.
Berardi
, and
C.
Zannoni
,
ChemPhysChem
10
,
125
(
2009
).
50.
N. J.
Boyd
and
M. R.
Wilson
,
Phys. Chem. Chem. Phys.
17
,
24851
(
2015
).
51.
H.
Sidky
,
J. J.
de Pablo
, and
J. K.
Whitmer
,
Phys. Rev. Lett.
120
,
107801
(
2018
).
52.
S.
Doniach
,
T.
Garel
, and
H.
Orland
,
J. Chem. Phys.
105
,
1601
(
1996
).
53.
Y.
Li
,
Q.
Huang
,
T.
Shi
, and
L.
An
,
J. Phys. Chem. B
110
,
23502
(
2006
).
54.
H.
Zhou
,
J.
Zhou
,
Z.-C.
Ou-Yang
, and
S.
Kumar
,
Phys. Rev. Lett.
97
,
158302
(
2006
).
55.
H.-P.
Hsu
and
K.
Binder
,
Soft Matter
9
,
10512
(
2013
).
56.

Sometimes also called the Kratky-Porod model.

57.
58.
T.
Odijk
,
J. Chem. Phys.
105
,
1270
(
1996
).
59.
J. D.
Weeks
,
D.
Chandler
, and
H. C.
Andersen
,
J. Chem. Phys.
54
,
5237
(
1971
).
60.
J.
Midya
,
S. A.
Egorov
,
K.
Binder
, and
A.
Nikoubashman
,
J. Chem. Phys.
151
,
034902
(
2019
).
61.
S.
Naderi
and
P.
van der Schoot
,
J. Chem. Phys.
141
,
124901
(
2014
).
62.
M.
Bishop
,
M. H.
Kalos
, and
H. L.
Frisch
,
J. Chem. Phys.
70
,
1299
(
1979
).
63.
G. S.
Grest
and
K.
Kremer
,
Phys. Rev. A
33
,
3628
(
1986
).
64.
P.
Bolhuis
and
D.
Frenkel
,
J. Chem. Phys.
106
,
666
(
1997
).
65.
A.
Cuetos
and
M.
Dijkstra
,
Phys. Rev. Lett.
98
,
095701
(
2007
).
66.
W.
Helfrich
,
J. Chem. Phys.
56
,
3187
(
1972
).
67.
68.
M. P.
Lettinga
,
Z.
Dogic
,
H.
Wang
, and
J.
Vermant
,
Langmuir
21
,
8048
(
2005
).
69.
M. P.
Allen
and
M. R.
Wilson
,
J. Comput.-Aided Mol. Des.
3
,
335
(
1989
).
70.
B.
Tjipto-Margo
and
G. T.
Evans
,
J. Chem. Phys.
93
,
4254
(
1990
).
71.
R.
Hentschke
and
J.
Herzfeld
,
Phys. Rev. A
44
,
1148
(
1991
).
72.
D. B.
DuPré
and
S.-J.
Yang
,
J. Chem. Phys.
94
,
7466
(
1991
).
73.
S. C.
McGrother
,
D. C.
Williamson
, and
G.
Jackson
,
J. Chem. Phys.
104
,
6755
(
1996
).
74.
M.
Franco-Melgar
,
A. J.
Haslam
, and
G.
Jackson
,
Mol. Phys.
106
,
649
(
2008
).
75.
M. M. C.
Tortora
and
J. P. K.
Doye
,
Mol. Phys.
116
,
2773
(
2018
).
76.
M.
Warner
,
J. M. F.
Gunn
, and
A. B.
Baumgärtner
,
J. Phys. A: Math. Gen.
18
,
3007
(
1985
).
77.
X.-J.
Wang
and
M.
Warner
,
Phys. Lett. A
119
,
181
(
1986
).
78.
W.
Maier
and
A.
Saupe
,
Z. Naturforsch. A
13
,
564
(
1958
).
79.
A. J.
Liu
and
G. H.
Fredrickson
,
Macromolecules
26
,
2817
(
1993
).
80.
S.
Lee
,
A. G.
Oertli
,
M. A.
Gannon
,
A. J.
Liu
,
D. S.
Pearson
,
H.-W.
Schmidt
, and
G. H.
Fredrickson
,
Macromolecules
27
,
3955
3962
(
1994
).
81.
A. J.
Spakowitz
and
Z.-G.
Wang
,
J. Chem. Phys.
119
,
13113
(
2003
).
82.
S. A.
Egorov
,
A.
Milchev
, and
K.
Binder
,
Polymers
8
,
296
(
2016
).
83.
A.
Milchev
,
S. A.
Egorov
,
J.
Midya
,
K.
Binder
, and
A.
Nikoubashman
,
ACS Macro Lett.
,
9
,
1779
(
2020
).
84.
C.
Singh
,
M.
Goulian
,
A. J.
Liu
, and
G. H.
Fredrickson
,
Macromolecules
27
,
2974
(
1994
).
85.
M. W.
Matsen
,
J. Chem. Phys.
104
,
7758
(
1996
).
86.
Y.
Jiang
and
J. Z. Y.
Chen
,
Phys. Rev. Lett.
110
,
138305
(
2013
).
87.
H.
Yamakawa
and
M.
Fujii
,
J. Chem. Phys.
59
,
6641
(
1973
).
88.
J.
Wilhelm
and
E.
Frey
,
Phys. Rev. Lett.
77
,
2581
(
1996
).
89.
S.
Mehraeen
,
B.
Sudhanshu
,
E. F.
Koslover
, and
A. J.
Spakowitz
,
Phys. Rev. E
77
,
061803
(
2008
).
90.
91.
92.
P.-G.
de Gennes
,
J. Chem. Phys.
55
,
572
(
1971
).
93.
M.
Doi
and
S. F.
Edwards
,
J. Chem. Soc., Faraday Trans. 2
74
,
1789
(
1978
).
94.
P.-G.
de Gennes
and
J.
Prost
,
The Physics of Liquid Crystals
, 2nd ed. (
Clarendon Press
,
1995
).
96.
G. J.
Vroege
and
H. N. W.
Lekkerkerker
,
Rep. Prog. Phys.
55
,
1241
(
1992
).
100.
J. L.
Janning
,
Appl. Phys. Lett.
21
,
173
(
1972
).
101.
M.
O’Neill
and
S. M.
Kelly
,
J. Phys. D: Appl. Phys.
33
,
R67
(
2000
).
102.
A.
Mertelj
,
D.
Lisjak
,
M.
Drofenik
, and
M.
Čopič
,
Nature
504
,
237
(
2013
).
103.
R. S.
Kularatne
,
H.
Kim
,
J. M.
Boothby
, and
T. H.
Ware
,
J. Polym. Sci., Part B: Polym. Phys.
55
,
395
(
2017
).
104.
S.-D.
Lee
and
R. B.
Meyer
,
Phys. Rev. Lett.
61
,
2217
(
1988
).
105.
D. W.
Berreman
,
Phys. Rev. Lett.
28
,
1683
(
1972
).
106.
C. J.
Newsome
,
M.
O’Neill
,
R. J.
Farley
, and
G. P.
Bryan-Brown
,
Appl. Phys. Lett.
72
,
2078
(
1998
).
108.
K.
Binder
,
Cohesion and Structure of Surfaces
(
Elsevier Science
,
Amsterdam
,
1995
), Chap. 3, p.
121
.
109.
T. J.
Sluckin
and
A.
Poniewierski
,
Phys. Rev. Lett.
55
,
2907
(
1985
).
110.
111.
F. A.
Escobedo
and
J. J.
de Pablo
,
J. Chem. Phys.
106
,
9858
(
1997
).
112.
I.
Rodríguez-Ponce
,
J. M.
Romero-Enrique
,
E.
Velasco
,
L.
Mederos
, and
L. F.
Rull
,
Phys. Rev. Lett.
82
,
2697
(
1999
).
113.
R.
van Roij
,
M.
Dijkstra
, and
R.
Evans
,
Europhys. Lett.
49
,
350
(
2000
).
114.
J.
Baschnagel
and
F.
Varnik
,
J. Phys.: Condens. Matter
17
,
R851
(
2005
).
115.
M.
Alcoutlabi
and
G. B.
McKenna
,
J. Phys.: Condens. Matter
17
,
R461
(
2005
).
116.
R.
von Klitzing
,
E.
Thormann
,
T.
Nylander
,
D.
Langevin
, and
C.
Stubenrauch
,
Adv. Colloid Interface Sci.
155
,
19
(
2010
).
117.
A.
Yethiraj
,
J. Chem. Phys.
101
,
2489
(
1994
).
118.
V. A.
Ivanov
,
A. S.
Rodionova
,
E. A.
An
,
J. A.
Martemyanova
,
M. R.
Stukan
,
M.
Müller
,
W.
Paul
, and
K.
Binder
,
Phys. Rev. E
84
,
041810
(
2011
).
119.
V. A.
Ivanov
,
A. S.
Rodionova
,
J. A.
Martemyanova
,
M. R.
Stukan
,
M.
Müller
,
W.
Paul
, and
K.
Binder
,
J. Chem. Phys.
138
,
234903
(
2013
).
120.
V. A.
Ivanov
,
A. S.
Rodionova
,
J. A.
Martemyanova
,
M. R.
Stukan
,
M.
Müller
,
W.
Paul
, and
K.
Binder
,
Macromolecules
47
,
1206
(
2014
).
121.
S. A.
Egorov
,
A.
Milchev
,
P.
Virnau
, and
K.
Binder
,
J. Chem. Phys.
144
,
174902
(
2016
).
122.
S. A.
Egorov
,
A.
Milchev
, and
K.
Binder
,
Macromol. Theory Simul.
26
,
1600036
(
2017
).
123.
A.
Milchev
,
S. A.
Egorov
, and
K.
Binder
,
Soft Matter
13
,
1888
(
2017
).
124.
A.
Milchev
and
K.
Binder
,
Nano Lett.
17
,
4924
(
2017
).
125.
J. M.
Kosterlitz
and
D. J.
Thouless
,
J. Phys. C: Solid State Phys.
6
,
1181
(
1973
).
126.
W.
Reisner
,
J. N.
Pedersen
, and
R. H.
Austin
,
Rep. Prog. Phys.
75
,
106601
(
2012
).
127.
L.
Dai
,
C. B.
Renner
, and
P. S.
Doyle
,
Adv. Colloid Interface Sci.
232
,
80
(
2016
).
128.
L.
Rems
,
D.
Kawale
,
J.
Lee
, and
P. E.
Boukany
,
Biomicrofluidics
10
,
043403
(
2016
).
129.
C.
Aaij
and
P.
Borst
,
Biochim. Biophys. Acta
269
,
192
(
1972
).
130.
131.
T. T.
Perkins
,
D. E.
Smith
,
R. G.
Larson
, and
S.
Chu
,
Science
268
,
83
(
1995
).
132.
T. T.
Perkins
,
D. E.
Smith
, and
S.
Chu
,
Science
276
,
2016
(
1997
).
133.
K. D.
Dorfman
,
S. B.
King
,
D. W.
Olson
,
J. D. P.
Thomas
, and
D. R.
Tree
,
Chem. Rev.
113
,
2584
2667
(
2013
).
134.
K.
Jo
,
D. M.
Dhingra
,
T.
Odijk
,
J. J.
de Pablo
,
M. D.
Graham
,
R.
Runnheim
,
D.
Forrest
, and
D. C.
Schwartz
,
Proc. Natl. Acad. Sci. U. S. A.
104
,
2673
(
2007
).
135.
M.
Roushan
,
P.
Kaur
,
A.
Karpusenko
,
P. J.
Countryman
,
C. P.
Ortiz
,
S. F.
Lim
,
H.
Wang
, and
R.
Riehn
,
Biomicrofluidics
8
,
034113
(
2014
).
137.
Y.
Wang
,
D. R.
Tree
, and
K. D.
Dorfman
,
Macromolecules
44
,
6594
(
2011
).
138.
L.
Dai
,
J. R. C.
van der Maarel
, and
P. S.
Doyle
,
Macromolecules
47
,
2445
(
2014
).
139.
C.
Zhang
,
F.
Zhang
,
J. A.
van Kan
, and
J. R. C.
van der Maarel
,
J. Chem. Phys.
128
,
225109
(
2008
).
140.
L.
Dai
,
S. Y.
Ng
,
P. S.
Doyle
, and
J. R. C.
van der Maarel
,
ACS Macro Lett.
1
,
1046
(
2012
).
141.
A.
Muralidhar
,
D. R.
Tree
, and
K. D.
Dorfman
,
Macromolecules
47
,
8446
8458
(
2014
).
142.
A.
Muralidhar
and
K. D.
Dorfman
,
Macromolecules
49
,
1120
1126
(
2016
).
143.
144.
F. T.
Wall
,
W. A.
Seitz
,
J. C.
Chin
, and
P. G.
de Gennes
,
Proc. Natl. Acad. Sci. U. S. A.
75
,
2069
(
1978
).
145.
K.
Kremer
and
K.
Binder
,
J. Chem. Phys.
81
,
6381
(
1984
).
146.
R. M.
Jendrejack
,
D. C.
Schwartz
,
M. D.
Graham
, and
J. J.
de Pablo
,
J. Chem. Phys.
119
,
1165
(
2003
).
147.
148.
Y.
Yang
,
T. W.
Burkhardt
, and
G.
Gompper
,
Phys. Rev. E
76
,
011804
(
2007
).
149.
150.
A. S.
Rudolph
,
B. R.
Ratna
, and
B.
Kahn
,
Nature
352
,
52
(
1991
).
151.
H.
Poincaré
,
J. Math. Pure Appl.
4
,
167
(
1885
).
152.
L. E. J.
Brouwer
,
Math. Ann.
71
,
97
(
1912
).
153.
T. C.
Lubensky
and
J.
Prost
,
J. Phys. II
2
,
371
(
1992
).
154.
155.
S.-D.
Lee
and
R. B.
Meyer
,
J. Chem. Phys.
84
,
3443
(
1986
).
156.
V.
Vitelli
and
D. R.
Nelson
,
Phys. Rev. E
74
,
021711
(
2006
).
157.
S.
Dhakal
,
F. J.
Solis
, and
M.
Olvera de la Cruz
,
Phys. Rev. E
86
,
011709
(
2012
).
158.
K.
Hiltrop
and
H.
Stegemeyer
,
Ber. Bunsenges. Phys. Chem.
82
,
884
(
1978
).
159.
M. J.
Uline
,
S.
Meng
, and
I.
Szleifer
,
Soft Matter
6
,
5482
(
2010
).
160.
O. O.
Prishchepa
,
A. V.
Shabanov
, and
V. Y.
Zyryanov
,
Phys. Rev. E
72
,
031712
(
2005
).
161.
A.
Milchev
,
S. A.
Egorov
,
A.
Nikoubashman
, and
K.
Binder
,
J. Chem. Phys.
146
,
194907
(
2017
).
162.
A.
Nikoubashman
,
D. A.
Vega
,
K.
Binder
, and
A.
Milchev
,
Phys. Rev. Lett.
118
,
217803
(
2017
).
163.
A.
Milchev
,
S. A.
Egorov
,
D. A.
Vega
,
K.
Binder
, and
A.
Nikoubashman
,
Macromolecules
51
,
2002
(
2018
).
164.
A.
Milchev
,
S. A.
Egorov
,
A.
Nikoubashman
, and
K.
Binder
,
Polymer
145
,
463
(
2018
).
165.
M. R.
Khadilkar
and
A.
Nikoubashman
,
Soft Matter
14
,
6903
(
2018
).
166.
M.
Trabi
and
D. J.
Craik
,
Trends Biochem. Sci.
27
,
132
(
2002
).
167.
X.
Zhou
,
F.
Guo
,
K.
Li
,
L.
He
, and
L.
Zhang
,
Polymers
11
,
1992
(
2019
).
168.
X.
Zhou
,
J.
Wu
, and
L.
Zhang
,
Polymer
197
,
122494
(
2020
).
170.
J.
Bednar
,
R. A.
Horowitz
,
S. A.
Grigoryev
,
L. M.
Carruthers
,
J. C.
Hansen
,
A. J.
Koster
, and
C. L.
Woodcock
,
Proc. Natl. Acad. Sci. U. S. A.
95
,
14173
(
1998
).
171.
D. E.
Olins
and
A. L.
Olins
,
Nat. Rev. Mol. Cell Biol.
4
,
809
(
2003
).
172.
W. C.
Earnshaw
and
S. C.
Harrison
,
Nature
268
,
598
(
1977
).
173.
M. E.
Cerritelli
,
N.
Cheng
,
A. H.
Rosenberg
,
C. E.
McPherson
, and
A. C.
Steven
,
Cell
91
,
271
(
1997
).
174.
D. E.
Smith
,
S. J.
Tans
,
S. B.
Smith
,
S.
Grimes
,
D. L.
Anderson
, and
C.
Bustamante
,
Nature
413
,
748
(
2001
).
175.
W.
Jiang
,
J.
Chang
,
J.
Jakana
,
P.
Weigele
,
J.
King
, and
W.
Chiu
,
Nature
439
,
612
(
2006
).
176.
D.
Marenduzzo
,
C.
Micheletti
, and
E.
Orlandini
,
J. Phys.: Condens. Matter
22
,
283102
(
2010
).
177.
K.-K.
Kunze
and
R. R.
Netz
,
Phys. Rev. Lett.
85
,
4389
(
2000
).
178.
S.
Stoll
and
P.
Chodanowski
,
Macromolecules
35
,
9556
(
2002
).
179.
A. J.
Spakowitz
and
Z.-G.
Wang
,
Phys. Rev. Lett.
91
,
166102
(
2003
).
180.
G.
Morrison
and
D.
Thirumalai
,
Phys. Rev. E
79
,
011924
(
2009
).
181.
W.-Y.
Zhang
and
J. Z. Y.
Chen
,
Europhys. Lett.
94
,
43001
(
2011
).
182.
J.
Kindt
,
S.
Tzlil
,
A.
Ben-Shaul
, and
W. M.
Gelbart
,
Proc. Natl. Acad. Sci. U. S. A.
98
,
13671
(
2001
).
183.
P. K.
Purohit
,
J.
Kondev
, and
R.
Phillips
,
Proc. Natl. Acad. Sci. U. S. A.
100
,
3173
(
2003
).
184.
I.
Ali
,
D.
Marenduzzo
, and
J. M.
Yeomans
,
J. Chem. Phys.
121
,
8635
(
2004
).
185.
C.
Forrey
and
M.
Muthukumar
,
Biophys. J.
91
,
25
(
2006
).
186.
E.
Katzav
,
M.
Adda-Bedia
, and
A.
Boudaoud
,
Proc. Natl. Acad. Sci. U. S. A.
103
,
18900
(
2006
).
187.
I.
Ali
,
D.
Marenduzzo
, and
J. M.
Yeomans
,
Phys. Rev. Lett.
96
,
208102
(
2006
).
188.
N.
Stoop
,
J.
Najafi
,
F.
Wittel
,
M.
Habibi
, and
H.
Herrmann
,
Phys. Rev. Lett.
106
,
214102
(
2011
).
189.
D.
Marenduzzo
,
C.
Micheletti
,
E.
Orlandini
, and
D. W.
Sumners
,
Proc. Natl. Acad. Sci. U. S. A.
110
,
20081
(
2013
).
190.
J.
Gao
,
P.
Tang
,
Y.
Yang
, and
J. Z. Y.
Chen
,
Soft Matter
10
,
4674
(
2014
).
191.
192.
Q.
Cao
and
M.
Bachmann
,
Soft Matter
13
,
600
(
2017
).
193.
T.
Curk
,
J. D.
Farr
,
J.
Dobnikar
, and
R.
Podgornik
,
Phys. Rev. Lett.
123
,
047801
(
2019
).