Configurations of confined nematic polymers

SEMINAR 1 1 st YEAR, SECOND CYCLE DEGREE Configurations of confined nematic polymers Author: Danijel Vidaković Mentor: Daniel Svenšek Ljubljana, March 2018 Abstract In this seminar, I present the construction of free energy of main-chain nematic polymers contained in a confining sphere. I begin with a quick description of liquid crystalline phases in liquid crystals and I introduce concepts that will be useful later (e.g. director field, order parameter, etc.). After that I define a complete nonunit nematic field a. Polymer density and field a are used as variables for describing our system. I take into account the connectivity of nematic monomers and acquire the continuity condition for polymer current density. After that, I describe free energy terms (i.e., energy of the nematic-isotropic phase transition, elastic energy, density constraints and the penalty potential for continuity requirement). The presented free energy expression is capable of describing the transition between an isotropic and a nematic polymer solution. I end this seminar with a short discussion of research finding (i.e., calculated equilibrium states). 1

Contents 1. Introduction... 2 2. Basics of liquid-crystalline phases... 3 3. The complete nematic director field aa... 5 4. Continuity equation... 5 5. Free energy density for nematic polymers... 6 5.1. Energy of nematic-isotropic phase transition... 6 5.2. Elastic energy... 7 5.3. Density constrains... 8 5.4. Penalty potencial for continuity requirement... 8 6. Results... 8 7. Conclusion... 10 References:... 10 1. Introduction: Polymers are long chain-like molecules composed of monomers. Even though chemical composition is important, most of physical properties are independent of chemical properties. Polymers that form liquid crystalline phases can be divided into two groups: main and side-chain nematic polymers (Fig.1). In this paper, we focus solely on main-chain nematic polymers. [1] Liquid crystalline molecule Side-chain Main-chain Fig. 1: The sketch of main and side-chain nematic polymers. In main-chain polymers liquid crystalline molecules (i.e elongated molecules) are included as a part of main-chain; in side-chain polymers liquid crystalline molecules are part of side-chain components. Adapted from [8]. An interesting representative of main-chain polymers is DNA molecule. It carries genetic information of growth, development, functioning, and reproduction. DNA molecules are usually entrapped in cell nucleus, or in a capsid (in simple viruses). Therefore, studying confined main-chain nematic polymers is crucial for understanding the packing of DNA molecules in the capsid. It is important to point out that almost all processes governing DNA packing are biologically unspecific physical mechanisms. Description of entrapped nematic polymers presented in this seminar is not limited only to the DNA molecule, but it can be applied to any confined main-chain polymer. [2][3] 2

2. Basics of liquid-crystalline phases Liquid-crystalline phases are types of mesophases, in which matter s mechanical properties are the same as those of a liquid, but its optical properties are those of a solid. The simplest liquid-crystalline phases are: isotropic, nematic, smectic, and columnar phases. Our focus is mainly on the first two phases. Isotropic phase has neither positional nor orientational order. In this phase matter behaves like ordinary liquid. The nematic phase has an orientational order but no positional order (Fig.2). We can define a director nn for it, which is a headless unit vector pointing in the direction of the average molecular orientation. [4] nn Fig. 2: Schematic representation of isotropic and nematic phases. Adapted from [9]. The simplest theory that can describe the existence of the nematic phase and the required conditions is the Onsager theory [5]. It is based on matter, consisting of hard rods of length L and of diameter D. If we want to write an equation for free energy for such a system, it would contain two terms that compete with each other. The first can be identified as a result of translation entropy, and the second as a result of orientational entropy. The most important result of this theory is the condition for existence of the nematic phase. Large density is not enough to describe the existence of the nematic phase (Fig. 3). Molecules must be elongated, which means L>>D. When this condition is met, phase is governed solely by density, regardless of the system being in the isotropic or the nematic phase. [5] Fig. 3: The left picture represents the dependence of free energy on order parameter S for two distinct φφφφ/dd. The right picture shows a phase diagram in respect to parameters φφ and LL/DD. φφ stands for volume packing fraction, φφ = NNVV 0. Where V is VV volume of system, N is number of liquid crystalline molecules and VV 0 volume of one liquid crystalline molecule. Both graphs are calculated from Onsager free energy. Reproduced from [10] At low densities we observe the isotropic phase, and at high enough density we observe the nematic phase. That means there must be a transition between these phases, and the theory that describes this transition is called Landau-de Gennes theory. The first step to describe the N-I transition is achieved by finding a parameter that can distinguish these phases. The tensor of magnetic 3

susceptibility is one of those parameters (Magnetic anisotropies are usually small. So even better parameter would be electric susceptibility). Magnetic susceptibility for isotropic and nematic phases respectively is shown below [6]: χχ 0 0 χχ II = 0 χχ 0 0 0 χχ χχ 0 0 χχ NN = 0 χχ 0 (1) 0 0 χχ Let us define a traceless tensor QQ iiii χχ iiii 11 χχ 33 kkkkδδ iiii, called the tensor order parameter. It describes the nematic order and distinguishes between liquid-crystalline phases. Below we can see the tensor order parameter for the isotropic and the nematic phase for a uniaxial nematic liquid crystal: 0 0 0 QQ II = 0 0 0 0 0 0 ss/2 0 0 QQ NN = 0 ss/2 0 (2) 0 0 ss The parameter s above is the scalar order parameter, ss = PP 22 (cccccc(θθ)), where θθ represents the deviation of the molecular main direction from the director nn. Free energy per unit volume for the nematic-isotropic transition must be expanded in terms of the scalar invariants of QQ iiii. In isotropic phase the minimum of free energy is located at s=0, so there cannot be a first-order invariant; therefore we use the second-order QQ iiii QQ jjjj and the third-order invariant QQ iiii QQ jjjj QQ kkkk. The Landau-de Gennes expansion reads: ff LLLLLL = ff 00 + 11 22 aa(tt TT )QQ iiii QQ jjjj 11 33 bbqq iiiiqq jjjj QQ kkkk + 11 44 cc QQ iiiiqq jjjj 22, (3) where a, b and c are phenomenological constants, and T* is the super-cooling temperature. If we calculate this for a uniaxial nematic liquid crystal, we get: ff LLLLLL = ff 00 + 33 44 aa(tt TT )ss 22 11 44 bbss33 + 99 1111 ccss44 (4) In the nematic phase (s 1), we usually do not have the uniform director field nn. Mostly we have spatially modulated director field nn, which is usually caused by the external forces (e.g., Electric and magnetic external fields or surface anchoring). To describe this deformation of director we must introduce the elastic theory, in our case Frank elasticity [6]. Frank free energy density has three terms: ff eeee FFFFFFFFFF = KK 11 22 ( nn )22 + KK 22 22 (nn nn )22 + KK 33 22 (nn nn )22. (5) The first term corresponds to splay deformation of director field, the second term corresponds to twist deformation and the third term corresponds to the bend deformation of the director field (Fig. 4). KK 1, KK 2 and KK 3 are elastic constants. [6] Fig. 4: Different types of director field deformations. Reproduced from [11] 4

3. The complete nematic director field aa Nematic liquid crystals are described by the director field nn. As the density is homogenous, the above variable is enough to describe such a system. But with nematic polymers this isn t the case. Since our description must be suitable for both phases (i.e. nematic and isotropic), the director field is replaced with a complete director field aa, as it covers not only the orientation of polymer chains, but also the degree of order. It is important to notice, that field aa is no longer headless vector (i.e., it is a true vector). The reason for that is the vector nature of continuity equation (i.e., to write continuity equation in tonsor form is not an easy task). The complete director field aa is defined as follows: aa =< cccccccc > nn = ssnn. (6) Where vector nn is a unit vector pointing in the direction of the average molecular orientation (i.e., vector nn is very similar to director nn. Being the only difference, that vector nn is a true vector and director nn is not.). The above definition makes sense only in hydrodynamic limit (i.e. continuous regime). That means aa is defined in a given hydrodynamic volume V by the following expression: aa = 11 ii ddll ll ii ii ii ddrr ii = ll ii 11 ii ddll ll ii ii ii ddll ii ccccccθθ ii nn, (7) ll ii Where integrals go over the included parts of the i-th chain in hydrodynamic volume V. θθ ii is the angle between the director nn (i.e. average orientation) and the local tangent of i-th chain. Triangular brackets represent a thermodynamic average. The usefulness of the defined field aa can be observed when we compare nn and aa at s 0. The director field nn cannot be zero (as it is a unit vector), which means the direction at s=0 is not well defined. The complete director field aa, on the other hand, vanishes at s=0, which is in coherence with the vanishing orientation order (i.e. isotropic phase). Therefore, the complete director field aa is a good variable for our system. In nematic polymers, density is not necessary spatially homogeneous (in particular since we have polymer solutions). Therefore, in order to fully describe nematic polymers, beside director field aa, we need local density ρρ. [3] 4. Continuity equation In nematic polymers, density and director field are coupled. This coupling is taken into account through an equation similar to continuity equation. In the limit of very long polymer chains, this equation is defined as: ( ρρ ss nn ) = 0, (8) where ρρ ss is the surface number density of chains crossing the plane perpendicular to the director nn. In simple words, if we have an infinitesimal volume dv and two of the boundary planes perpendicular to nn, the number of polymers that cross the plane perpendicular to nn from the left side is the same as the number of polymers that cross the plane from the right side. It means that the polymers do not disappear, neither do they originate. Here the basic assumption is that hairpins (i.e. sharp U turns of the chain) are absent. We define current density as ȷ = ρρ ss nn. However, this is not a current in the usual sense, since it does not describe a rate. An average number of subunits of length ll 0 perforating a plane perpendicular to the director nn is: NN = ρρρρ ddll iiccccccθθ ii, ii NN (9) 5

where ρρ is a number (volume) density of subunits, NN = ii 1 is the average number of subunits in hydrodynamic volume; integrals go over the included parts of the i-th chain in hydrodynamic volume V. We take into account that a subunit penetrates the plane only if the distance between its center and the plane is smaller than 1 2 ddll iiccccccθθ ii. Thus we can express the surface density as an equation (10) and the current density can be expressed by equation (11). ρρ ss = ρρ ii ddll iiccccccθθ ii ii ddll ii ii ddll ii = ρρll NN 00 cccccccc, ȷ = ρρll 0 aa. (10) (11) If the polymers are of a finite length, we must take into account the position of chain beginnings or ends. So the volume number density of beginnings (ρρ + ) and ends (ρρ ) appear in continuous equation as the source terms: jj (rr ) = ρρ + (rr ) ρρ (rr ) = ρρ ± (rr ) (12) It is important to notice that all identities above are express in the hydrodynamic limit. All quantities are expressed as averaged microscopic quantities over the hydrodynamic volume V. [3] 5. Free energy density for the nematic polymers Free energy for nematic main-chain polymers mainly consists of four parts: energy of nematic-isotropic phase transition, elastic energy, penalty potencial for continuity requirement, and other penalty potentials (i.e. density restrictions). We will take a look at every part separately. 5.1. Energy of the nematic-isotropic phase transition Similarly as for liquid crystals, we conduct energy of the nematic-isotropic transition in the main-chain liquid crystal polymers. In the case of ordinary liquid crystals, the important parameter that defines the phase of a system is temperature. But in the case of polymers, density takes this role. As the polymers are dissolved (in a solvent) and the temperature is usually constant, or does not change much (e.g. polymers in cell liquid), we control the phase transition with adjusting density of polymers in solution. Also, in the main-chain polymers the scalar order parameter s cannot be lower than zero (this isn t the case in ordinary liquid crystals). A fact that aa is equivalent to - aa, means that the term of the third order cannot exists in the Landau-de Genes energy. Furthermore, the terms must be proportional to the number of molecules, i.e. to the polymer density. By taking into account the upper conditions, we get energy density for N-I transition: ff NN II = 11 22 CCCC ρρ ρρ ρρ +ρρ aa22 + 11 44 ρρccaa44, (13) where constants are adjusted in a way that ensures s<1 (see Fig. 5), and ρρ represents transition density. [3][7] 6

S max=1 Fig. 5: The graph above shows the Landau-de Genes energy with respect to a. The expression ρρ ρρ is larger than -1 and lower ρρ +ρρ than 1. On the picture above we can see that the equilibrium state (df/ds=0) is to be found between s=0 (isotropic) and s=1 (nematic). The above potential is an even function, which can be justified with the symmetry of aa (aa and - aa represent the equivalent state). It is enough to take into account only the right side of the graph (for a>0). 5.2. Elastic energy Frank energy is a good expression for elastic energy of liquid crystals in the nematic phase. The same expression in the isotropic phase cannot be used, because deformations of the director field for low s are cheaper (i.e. easier to deform) than deformations for a higher order parameter. This means that the elastic constants K 1, K 2 and K 3 depend on the order parameter s. Therefore, we define a new expression for elastic energy by replacing director nn with nematic order vector aa in a way that the received expression at a=1 (i.e. s=1) simplifies into the Frank energy density. We also take into account proportionality of energy terms to the square density, as it is the case for any interaction energy density. ff eeee = 11 22 ρρ22 LL 11 ii aa jj 22 + 11 22 ρρ22 LL 22 ( ii aa ii ) 22 + 11 22 ρρ22 LL 33 aa ii aa jj ( ii aa kk ) jj aa kk + 11 22 ρρ22 LL 44 εε iiiiii ii aa jj 22, (14) where LL 1, LL 2, LL 3 and LL 4 are elastic constants, which do not depend on the degree of order. The relations between KK ii and LL jj (where LL jj = ρρ 2 LL ii ) are represented below. SSSSSSSSSS eeeeeeeeeeeeee cccccccccccccccc: KK 11 = aa 22 LL 11 + aa 22 LL 22, (15) TTTTTTTTTT eeeeeeeeeeeeee cccccccccccccccc: KK 22 = aa 22 LL 11 + aa 44 LL 44, (16) BBBBBBBB eeeeeeeeeeeeee cccccccccccccccc: KK 33 = aa 22 LL 11 + aa 44 LL 33. (17) In the case of cholesteric liquid crystal, the fourth term changes into 11 22 ρρ22 LL 44 εε iiiiii ii aa jj + qq 00 22, where qq 0 is the wave vector of the bulk cholesteric phase. The property of cholesteric liquid crystal is the finite spontaneous twist. This occurs because molecules of cholesteric liquid crystal are chiral [5]. Example of such a molecule is DNA molecule. As the Frank energy has only three elastic constants and as the above presented elastic energy has to describe those three matching types of deformation (i.e. a splay, a twist and a bend), we can omit one of the terms in the equation (14) (usually 3. or 4.). It is important to note, that we cannot distinguish different deformation modes (a splay, a twist and a bend) in the limit of small order parameter, as the leading order dependence for all of them is the same (i.e. KK ii aa 2 ). The beauty of the above composed elastic energy density is the ability to describe the nematic liquid crystals for order parameter from 0 to 1. [3][7] 7

5.3. Density constrains Constraints for the density field can be expressed in many ways (in terms of penalty potentials); they can be sorted into two groups: the terms that limit density and the terms that limit density gradients. We will consider the following terms: 11 22 χχ 00(ρρ ρρ 00 ) 22, χχ 11 [ρρ(ρρ ρρ cc )] 44 and 11 22 LL ρρ( ii ρρ) 22, (19) where the first two refer to the first group, and the third to the second group. χχ 0 and χχ 1 correspond to density compressibility, and LL ρρ is density variation correlation length. The first term prefers bulk equilibrium density ρρ 0 ; the second term prohibits negative densities and densities larger than maximal packing density ρρ cc ; and the last term prefers homogenous density. [3][7] 5.4. Penalty potential for continuity requirement The continuity requirement (Eq. (11) and (12)) will be built in free energy as a penalty potential, rather than as an additional requirement. One of the reasons for this is that finding the equilibrium state with an additional requirement is hard. So instead we will use a penalty potential that will represent continuity requirement: 11 GG 22 ii(ρρaa ii ) ρρ± 22. ll 00 (21) G is a coupling constant; it defines rigidity of continuity constraint. In our discussion ρρ ± is either neglected or considered as a fixed external parameter and not as a variable. At ρρ ± = 0 we can define splay relaxation length ll ρρ that measures the importance of continuity constraint; when we compare density and continuity constraint, it can be calculated: 1 χχ 2 0 = 1 GG 1 2 ; ll 2 ll ρρ = GG. (22) ρρ χχ 0 If the length scales of director deformations are larger than ll ρρ, the director field and density are decoupled; if the length scales are smaller than ll pp, coupling of the orientational field and density is strong. This is expected, as the splay deformation on long scales is extremely expensive. Deviations of ȷȷ on long length scales are compensated by rearranged chain ends (even though we will neglect this effect). [3] 6. Results In previous paragraphs the free energy terms were described. At this point, it is time to write down an equation for free energy density: ff = 11 CCCC ρρ ρρ 22 ρρ +ρρ aa22 + 11 44 ρρccaa44 + 11 22 ρρ22 LL 11 ii aa jj 22 + 11 22 ρρ22 LL 22 ( ii aa ii ) 22 + 11 22 ρρ22 LL 44 εε iiiiii ii aa jj + qq 00 22 + 11 22 GG ii(ρρaa ii ) ρρ± ll 00 22 + χχ 00 [ρρ(ρρ ρρ cc )] 44 + 11 22 LL ρρ( ii ρρ) 22 (23) We get the equilibrium state by minimizing the free energy (Eq. (24)) at the constraint of global mass conservation (Eq. (25)). FF = (ff λλλλ)dddd (λλ is the Lagrange multiplier) (24) ρρρρρρ = mm 00 = cccccccccc. (25) 8

For minimization of the free energy we use quasi-dynamic evolution and write down the Euler- Lagrange equations for director and density field: γγ aa ii = jj and γγ = ( jj aa ii ) aa ii jj + λλ, (26)(27) ( jj ρρ) where γγ represents the time scale (i.e. the numerical time step). Mass conversation constraint can be achieved (at every time step) by density correction. We change density ρρ into ρρ + 1 VV (mm 0 ρρρρρρ) (i.e. a homogeneous field is subtracted from density). The equilibrium state is achieved when γγ aa ii = 0 and γγ = 0. The results below (Fig. 6 and 7) are acquired by solving the above quasi-dynamic Euler-Lagrange equations (Eq. (26) and (27)) in the open source finite volume solver OpenFOAM [14]. The initial conditions are ρρ(rr ) = ρρ 00 and aa (rr ) = 00 + ww (rr ), (28)(29) where ww (rr ) is a small, random perturbation of the director field. For the director field aa on the confining sphere we have tangential boundary condition, and for density ρρ = 0. With the help of nematic correlation length, defined as ξξ = LL 0 ρρ 0 /CC, where LL 0 and ρρ 0 are fixed reference elastic constant and polymer density, we can define dimensionless quantities. In simulation, those dimensionless quantities are used. Correlation length ξξ, corresponds to the distance of two points in or system on which values of scalar order parameter are no longer correlated. The results from simulations can be compared to results from experiments through this correlation length. For DNA molecule ξξ is around 8 nm [13] and diameter of virus capsid is from 20-1000 nm [15]. We set CC = 1, LL 1 = 1, LL 2 = LL 4 = 0, GG = 1, χχ 0 = 1, LL ρρ = 1, ρρ cc = 5, ρρ ± = 0, ρρ = 0.5; radius is set to 32ξξ. [3][7] We do not need any ansatz in order to get equilibrium density and the director field aa. Therefore, there is no need to make extra assumptions about how the equilibrium state should look and that is the beauty of the above presented solution. Fig. 6:The density and orientation order for different dimensionless average density ρρ = 1 ρρ 0 mm 0 VV 0, where ρρ 0 is a fixed reference polymer density and VV 0 volume of confining sphere; polymers have no chirality. Orientation order is denoted on the color scale from red to green, while density is denoted on the scale from blue to yellow. As expected only for large enough densities we observe orientational order (i.e. nematic phase). Also it is important to notice that we have spontaneous condensation. So in this systems we do not observe orientation defects. Reproduced from [12] Fig. 7: The density and orientation order at dimensionless average density ρρ = 1 for different chiral strengths qq 0. Orientation order is denoted on the color scale from red to green, while density is denoted on the scale from blue to yellow color. The average density is larger ten transition density ρρ, so the polymer is orderd (i.e, in nematic phase). For low chirality we have ordinary toroidal condensates. As the chirality increases further the torus is deformed (twisted) and becomes more globule-like, Elements from (a)-(g) show the representative part of the director. Reproduced from [12] 9

In Fig. 7 we saw for low chirality ordinary toroidal condensates. The same condensates are observed in simple viruses for elevated densities with X-ray spectroscopy and cryomicroscopy [3] (Fig. 8). So the presented results correspond well with the experimental data for DNA molecule. 7. Conclusion The seminar presents the free energy functional that can effectively describe the packing of DNA within a spherical enclosure. The expression is constructed with the help of continuity equation and wellknown concepts from ordinary nematics. With numerically solving quasi-dinamic Euler- Lagrange equations, we would be able to acquire equilibrium state solutions without an ansatz for density or vector field aa. In the seminar, the presented results are in coherence with experimental observations in simple viruses. Even though free energy was based on physical properties of a DNA molecule, it can be used for any main-chain polymer. Hence there is a variety of possible uses of this free energy expression. 8. References: [1] P. Ziherl, Physics of Soft Matter (2014), view 15.3.2018,http://www-f1.ijs.si/~ziherl/smt.pdf, p. 65. [2] DNA, view 4.4.2018 https://en.wikipedia.org/wiki/dna. [3] D. Svenšek, G. Veble and R. Podgornik, Confined nematic polymers: Order and packing in a nematic drop, Phy. Rev. E, 82 011708, (2010). [4] P. Ziherl, Physics of Soft Matter (2014), view 15.3.2018, http://www-f1.ijs.si/~ziherl/smt.pdf p. 39-40. [5] P. Ziherl, Physics of Soft Matter (2014), view 15.3.2018, http://www-f1.ijs.si/~ziherl/smt.pdf p. 41-48. [6] P. Ziherl, Physics of Soft Matter (2014), view 15.3.2018, http://www-f1.ijs.si/~ziherl/smt.pdf p. 53-57. [7] Svenšek D. and Podgornik R., Confined chiral polymer nematics: ordering and spontaneous condensation, Europhys. Lett. 100, 66005 (2012). [8] Polymers, view 4.4.2018, https://patentimages.storage.googleapis.com/us6312770b1/us06312770-20011106-c00001.png [9] Phases, view 4.4.2018, https://www.theiet.org/resources/journals/eletters/4921/images/640_liquidcrystal-phases-640x426.jpg [10] P. Ziherl, Physics of Soft Matter (2014), view 4.4.2018, http://www-f1.ijs.si/~ziherl/smt.pdf str. 44, 45, fig. 3.3 and 3.4 [11] Deformations, view 4.4.2018, http://lcp.elis.ugent.be/sites/lcp.elis.ugent.be/files/twistbendsplay.jpg [12] Svenšek D. and Podgornik R., Confined chiral polymer nematics: ordering and spontaneous condensation, Europhys. Lett. 100, 66005 (2012). Fig. 1 and 2. [13] Svenšek D. and Podgornik R., Confined chiral polymer nematics: ordering and spontaneous condensation, Europhys. Lett. 100, 66005 (2012). [14] OpenFOAM package, view 15.5.2018, https://www.openfoam.com/ [15] Ron Milo and Rob Phillips, Diameter of virus capsids, view 16.5.2018, http://book.bionumbers.org/howbig-are-viruses/. 10