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Smectic C and nematic phases in strongly adsorbed layers of semiflexible polymers Andrey Ivanov Milchev, and Kurt Binder Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b01948 • Publication Date (Web): 05 Jul 2017 Downloaded from http://pubs.acs.org on July 7, 2017
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Smectic C and nematic phases in strongly adsorbed layers of semiflexible polymers Andrey Milchev1,2∗ , and Kurt Binder2 1
Institute for Physical Chemistry, Bulgarian Academia of Sciences, 1113 Sofia, Bulgaria Institut f¨ ur Physik, Johannes Gutenberg Universit¨ at Mainz, 55099 Mainz, Germany (Dated: July 5, 2017)
2
Molecular Dynamics simulations of semiflexible polymers in a good solvent reveal a dense adsorbed layer when the solution is exposed to an attractive planar wall. This layer exhibits both a nematic and a smectic phase (smA for short and smC for longer chains) with bond vectors aligned strictly parallel to the wall. The tilt angle of the smC phase increases strongly with the contour length of the polymers. The isotropic - nematic transition is a Kosterlitz - Thouless transition, and also the nematic - smectic transition is continuous. Our finding demonstrates thus a two-dimensional realization of different liquid crystalline phases, ubiquitous in three dimensions, that occurs in a single monomolecular layer ordered at least over mesoscopic scales. Keywords: semiflexible polymers, smectic, nematic, phase transition
Phase transitions in thin surface layers or confined films of liquid crystalline systems have found a broad and longstanding interest1–13 . These systems exem-
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Figure 1: (a) Log-log plot of the density ρ(z) vs the distance z from the attractive wall, for the case N = 16, ǫb = 128, Lx = 192, Ly = 199, Lz = 64, and Nmono = 235936 effective monomeric segments in the system. Various temperatures T (in units of ǫ/kB T ) are included, as indicated. The peak near z ≈ 1.3 is due to the adsorbed Rlayer. The density z ρ1 (T ) in this layer is defined as ρ1 (T ) = 0 min ρ(z)dz, with zmin ≈ 1.9. Insert shows ρ1 (T ) vs T . (b) The same as in (a) but for N = 24, κ = ǫb /kR T = 144 at T = 0.6 for different attraction strength ǫM ie .
∗ Corresponding
author’s email:
[email protected] plify fundamental physical phenomena as the Kosterlitz - Thouless transition14,15 , and have potential application in various devices16,17 . Moreover, semiflexible biopolymers like double-stranded (ds) DNA or actin adsorbed at substrates may exhibit related phenomena, given the broad evidence for liquid-crystalline order in many types of semiflexible polymers in bulk solutions18–20 . While most of the existing experimental work on surface behavior of small molecule liquid crystals has considered free surfaces against air (or even free-standing films1,2,6,8 ), much less work has addressed liquid crystals at strongly adsorbing substrates where the molecules align parallel to the substrate. Such systems can be studied with the surface force apparatus, for instance22,23 . By calorimetric and birefringence measurements, a surface-induced isotropic-nematic transition of the molecule 7CB was reported13 . For sufficiently stiff and not too short semiflexible polymers, a rather dense quasi-two-dimensional layer of macromolecules adsorbed at the wall and oriented parallel to it can be expected with moderately strong wall - monomer attraction already. Indeed, there are numerous experimental reports on DNA adsorbed on lipid membranes in an essentially two-dimensional (2d) conformation24,25 or confined between 2d lipid bilayers26–29 . For short rod-like DNA strands or related objects25 under such conditions, various nematic, smectic, or columnar orderings were reported24,29 . The present work addresses such systems performing Molecular Dynamics (MD) simulations of a coarsegrained model of semiflexible polymers interacting with a planar structureless attractive substrate surface in solution under good solvent conditions. The polymers in solution are still in the isotropic phase, and just serve as a “reservoir” from which the adsorption takes place. In a coarse-grained view, semiflexible polymers are characterized by three lengths30–32 , namely, the contour length L, the persistence length ℓp , and the chain molecule diameter D. In a MD context, these properties are realized by using a bead-spring model33 , amended by a bending potential Ubend (θijk )34,35 , which
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depends on the angle between two subsequent bond vectors ~ai = ~rj − ~ri and ~aj = ~rk − ~rj (with j = i + 1, k = j + 1) along the chain. Each chain contains N monomers at positions ~ri , i = 1, . . . , N . The energy parameters ǫb of the bending potential {Ubend (θijk ) = ǫb [1 − cos(θijk )]}, and ǫM ie of the adsorption potential, UM ie (z) = ǫM ie (σ/z)10 − (σ/z)4 , are two basic control parameters36 of the considered system; here z is the distance from the substrate, and the size of the monomeric subunit of the chains is the unit of length σ ≡ 1. Note that thesesubunits repel each other with a potential Urep (r) = 4ǫ (σ/r)12 − (σ/r)6 + 1/4 , for distances r < rc = 21/6 σ, where ǫ = 1 sets the energy scale. Neighboring beads along the chain are bonded by the spring
potential33 U F EN E (r) = −0.5kr02 ln 1 − (r/r0 )2 , r < r0 , with r0 = 1.5σ, k = 30ǫ/σ 2 . With these choices an effective bond length lb ≈ 0.97σ results, while L = (N −1)lb and ℓp ≈ ǫb lb /kB T (for chains in dilute 3d solutions; in dilute 2d solution ℓp is twice as large). In previous work the bulk behavior of solutions of semiflexible polymers described by this model (solvent molecules not being considered explicitly) has been extensively studied as function of bead density ρ, N , and ǫb (choosing temperature kB T ≡ 1)37–39 . However, we shall consider here only densities ρ in the bulk solution far below the onset of nematic order in the bulk as well as strong adsorption40 .
Figure 2: Snapshot pictures of well equilibrated configurations of semiflexible polymers confined in the adsorbed layer projected onto the x − y plane. Only the bond vectors but not the monomers are shown so each polymer is represented by a continuous line. Cases refer the choice (top, from left to right): N = 16, ǫb = 128, Lx = 192, Ly = 199, and T = 3.2 (a), T = 2.0 (b), and T = 0.86 (c). Also shown are the cases (bottom): N = 24, ǫb = 144, T = 0.80 (d), N = 32, ǫb = 128, T = 1.0 (e), and N = 64, ǫb = 128, T = 1.0 (f).
For the MD work a large simulation box with linear dimensions such as Lx = 192, Ly = 199, Lz = 64 is chosen with one attractive wall at z = 0, and a repulsive wall at z = Lz (described by the potential Urep (z ′ ) where z ′ = Lz −z). In x and y directions periodic boundary conditions (PBC) are applied. For total density ρ of the or-
der of ρ = 0.1 and parameters such as N = 16, ǫb = 128, the bulk solution is in the disordered isotropic phase, but in the adsorbed layer at the attractive wall the density ρ1 is much higher (Fig. 1). It increases from about ρ1 (T ) = 0.3 to ρ1 (T ) = 0.9 as T is lowered from 3.2 to 0.8 (in units of ǫ/kB T ). Increasing the density of this
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quasi-two-dimensional adsorbed layer causes a transition from nematic short-range order, Fig. 2 (a), to almost long-range order, Fig. 2 (b), and ultimately smectic order, Fig. 2 (c). Equilibration of the systems has to be done with great care [see Ref.41 and the Supplementary material42 ]. In simulations of chiral rigid rods in 2d43 also a Kosterlitz - Thouless type14 isotropic-nematic transition and a smectic-C phase was observed. However, while the occurence of smectic-C type order for chiral molecules is expected16 , for the nonchiral semiflexible molecules considered here smectic C order comes as a surprise. Of course, in d = 2 dimensions long-range order (LRO) breaking a continuous symmetry is a delicate problem44–46 since long wavelength fluctuations destroy both the strict orientational long range order of the d = 2 nematic phase and the positional long range order of crystals and smectics. Rather, a “quasi - LRO” is expected, i.e., a power-law decay of the appropriate orientational or positional correlation functions46 . In fact, in a finite system with periodic boundary conditions, order in the system will be stabilized to some extent by the finite size and boundary effects. We emphasize, however, that our linear dimensions Lx , Ly are large enough so that a visible bias on the orientation of the director is avoided, at least for N = 16 (Figs. 2, top row). In the case of smectic order, however, we observe just 12 layers for N = 16, and less for the larger choices of N . Thus, for smectic order finite size clearly matters. We also note that for N > 32 the director coincides more or less with a lattice axis in the observed smectic C phases. It should be emphasized that the structures in Fig. 2 are not related to the smectic layering at walls observed experimentally21–23 and studied theoretically47,48 , where the director is not parallel to the walls and the vector along the axis of smectic periodicity points along the z−axis. Here, the simulated system exhibits this vector in the xy−plane, and it makes a nontrivial angle with the director (decreasing from about 90o , i.e., a smectic A-phase, for N = 16, to roughly 45o for N = 64. This latter snapshot shows a multi-domain structure rather than the simple smC domains seen for smaller N . With increasing N , and keeping the box linear dimensions unchanged, the distortion of quasi-long range order due to the periodic boundary conditions gets more pronounced, and we interpret the domain structure of Fig. 2c as a metastable state. We have found that the sequence of phases in the adsorbed layer from isotropic to quasi-LRO nematic and smC phases can be observed through the variation of different control parameters. Keeping then T constant, of course, one simply can vary ǫM ie , or the range σ of Urep , or the density ρ of the bulk solution. While in the latter case for large enough ρ the wall-induced order competes with the nematic order in the bulk41 , the behavior displayed in Fig. 2 occurs when one studies the range of rather dilute solutions whereby a strong decrease of ρ1 with ρ takes place only when nearly all the polymers are adsorbed.
To characterize the system more precisely, we have studied the bond-orientation correlation function g2 (r) = hcos 2 [φ(r) − φ(0)]i. Here φ(rlCM ) is the azimutal angle that a bond vector at position ~rlCM = (~rl+1 + ~rl ) /2 makes with the x−axis, and an average is taken over in the system, with r = all bond vectors ~rlCM , ~rlCM ′ |. If there were a nonzero nematic order |~rlCM − ~rlCM ′ parameter S in the thermodynamic limit, we would have g2 (r → ∞) = S 2 , while in the isotropic phase an exponential decay is expected, g2 (r) ∝ exp(−r/ξn ), with a correlation length ξn describing the range of d = 2 nematic short-range order. According to the theory of the Kosterlitz - Thouless (KT) transition14,15,49 , ξn diverges √ as ln ξn ∝ 1/ T − TKT as the transition temperature TKT is approached, while for T < TKT the algebraic decay is described by g2 (r) ∝ r−2kB T /(πκ(T )) , κ(TKT ) = 8kB TKT /π,
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with κ(T ) an effective Frank constant16 . This scenario has been verified in simulations of various models for 2d liquid crystals50–53 . As expected, it applies to the present model as well, Fig. 3. However, the new feature of the model studied in the present work is the appearance of the smC phase at high densities ρ1 (T ), see Fig. 2 and Fig. 3b. For hard rods at high densities a solid (crystal) phase was proposed52 , which perhaps needs to be re-interpreted as a smA-phase, however. So far, smCphases in d = 2 have only been suggested for molecules resembling bent hard needles with a “zigzag” shape54 . While such special molecular shapes are rather uncommon, our results imply that for adsorbed semiflexible polymers smC could be rather common. The periodicity with a period of about L + 1 ≈ 16 in Fig. 3b shows that the smectic phase is very well ordered. To characterize the tilt angle between domain boundaries and the director in the smC-ordered phase, we sample the components ∆x, ∆y of the distance vector between the centers of mass of neighboring molecules (up to a maximum distance that includes next nearest neighbors in the same domain). We define the tilt angle α for the considered pair p of molecules as α = θ + θD , where θ = arccos(|∆x|/ ∆x2 + ∆y 2 ), and θD is determined as orientation of the average molecular director of the system. Thus, for example, for the system with N = 16, shown in Fig. 2 (top row right), one finds a mean value of cos θ ≈ 0.17 = 80o ± 1o , and also θD ≈ 12o , so that the resulting tilt angle α ≈ 92o indicates domain boundaries that are roughly perpendicular to the director, that is, a smA-phase. The probability function, W (cos θ), for N = 24, κ = 144 at T = 0.6, is shown in Fig. 4 (upper panel) for ǫM ie , varied within a broad interval, whereby the smC-phase goes first into nematic, and then into a disordered phase. Neither the KT phase transition, nor the N − smC transition lead to visible singularities in the equation of state, Fig. 4 (lower panel), where the 2d spreading pressure, Pxx , is plotted against density ρ1 of beads in the adsorbed layer. For N = 24 and ǫM ie > 1.8, cf. Fig. 4 (upper panel),
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where the smC phase is stable and θD ≈ 0, one finds a maximum of W (cos θ) at θmax ≈ 63o which yields α ≈ 63o . In contrast, in the nematic phase for ǫM ie < 1.8, Fig. 4, suggests θmax → 0, whereby the chains are aligned parallel yet with many neighboring molecules showing large displacements along the director. An even larger tilt of domain boundaries is found for N = 32, compatible with the direct observation of the tilt angle from the snapshots, Fig. 2. While for hard rods a “crystal” phase seems to exist for arbitrary aspect ratios of the rods52 , we did not find a smectic phase for short semiflexible polymers with N = 8 (which still have a standard nematic phase in the d = 3 bulk38 ). Rather, we find at low T a disordered variant of the columnar arrangement, Fig. 5 (upper panel), somewhat reminiscent of the ’tetratic’ phase, suggested to occur in non-equilibrium steady state of vibrated rod monolayers55 . With diminishing temperature, T , the distribution W (cos θ), Fig. 5 (lower panel), shows two growing peaks at cos θ ≈ 0.05, and 0.95, reflecting the exis-
Figure 4: (above) Distribution W (cos θ) with θ being the complementary to tilt angle, α, of the smC-phase, measured with regard to the x-axis for N = 24, κ = 144, T = 0.6, and different strength of attraction ǫM ie . The smaller peaks on both sides of the principal maximum arise from distortions in the domain boundaries, cf. Fig. 2. (below) Variation of the 2d spreading pressure, Pxx , with ρ1 , induced by changing attraction of the polymers to the confining plane, ǫM ie . Here N = 16, κ = 128, T = 0.8. The inset displays the tilt angle distribution function W (cos θ) for 1.0 ≤ ǫM ie /kB T ≤ 3.0.
tence of predominantly horizontal and vertical domain boundaries in this system. In summary, we have studied dense layers of strongly adsorbed semiflexible polymers, and demonstrated that various liquid crystal orderings occur, nematic (with quasi-LRO), smA (relatively short chains such as N = 16), and smC. Analyzing bond orientational correlations, we could locate both the I − N KT-transition and the (apparently continuous) N − smC transition, at least roughly. However, the sequence of transitions isotropic nematic - smA - smC in the same system, familiar from some systems in the bulk, has not been found. The physical origin of the smC tilt remains to be clarified. The chain length dependence of the tilt angle must be due to the subtle interplay of the positional excess entropy related to the chain ends and the entropy of chain bending fluctuations. Most probably, the observed rather regular
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Returning to the experimental observations of various liquid-crystalline orderings of short DNAs26–28 , we mention that our model lacks the end-to-end stacking interactions that these rod-like molecules exhibit, the only attractive interaction in our model is the binding of the monomers on the substrate surface. Of course, an interesting extension of the work could be the addition of specific end-to-end interactions to our model, since this clearly is a useful ingredient for chemical design of optimally packed liquid crystalline phases of short anisotropic molecules28 . Special smectic phases are also observed with gapped DNA duplexes (i.e., short pieces of ds DNA linked by a flexible spacer)56 . Thus, we hope that our model studies provide motivations to detect experimentally related behavior in such systems.
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Acknowledgements A. M. acknowledges partial support from the Deutsche Forschungsgemeinschaft (DFG), grant No BI 314/24. We thank the J.-G. University Mainz for computing time on the Mogon cluster (www.hpc.unimainz.de).
Figure 5: (above) A snapshot of a system with N = 8, κ = 32 in a box with Lx = 96, Ly = 100, Lz = 64 at T = 0.4. (below) Probability distribution function W (cos θ) for semiflexible polymers with N = 8, κ = 32 at different temperature T , as indicated.
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The authors declare no competing financial interests.
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Figure 6: TOC graph: Transition from nematic (left) to smectic-C phase (right) in a strongly adsorbed layer of semiflexible polymers of length N = 32, persistence length ǫb = 128 at T = 1.0 upon increase of the monomer number density from ρ = 0.0086 (left) to 0.018 (right). The container size is 192 × 197 × 64.
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