Solitons in a discrete model of chiral liquid crystals with competing interactions
aa r X i v : . [ c ond - m a t . s o f t ] M a y Solitons in a discrete model of chiral liquid crystals with competing interactions
Alain M. Dikand´e ∗ Laboratory of Research on Advanced Materials and Nonlinear Science (LaRAMaNS),Department of Physics, Faculty of Science, University of Buea P. O. Box 63 Buea, Cameroon
Bernard Y. Nyanga
Department of Physics, Faculty of Science, University of Buea PO Box 63 Buea, Cameroon
S. E. Mkam Tchouobiap
Laboratory of Research on Advanced Materials and Nonlinear Science (LaRAMaNS),Department of Physics, Faculty of Science, University of Buea P. O. Box 63 Buea, Cameroon (Dated: May 29, 2018)Chiral liquid crystals exhibit in-plane spontaneous polarizations, however in their smectic phasethe primary order parameter is a tilt vector associated with molecular rotations around the longmolecular axis parallel to the director. The molecular rotations lead to several distinct phasesamong which a domain-wall texture with a periodic-kink soliton profile. In this study the formationof domain walls in smectic chiral liquid crystals is analyzed, with emphasis on the competitionbetween ising-type symmetric and antisymmetric nearest-neighbor interactions, and an in-planeelectric field. It is found that antisymmetric intermolecular interactions, which are of chiral origin,increase the width of kink structures in the domain wall at moderate intensity of the y componentof the electric field. Increasing the x component of the electric field creates unstable condition forsoliton formation irrespective of magnitudes of the symmetric and chiral intermolecular interactions.Stability condition for single-kink domain-wall structures in the discrete molecular chain, is discussedby estimating the Peierls stress experienced by the single-kink soliton. Results suggest that chiralitylowers the Peierls-Nabarro barrier, hence increasing the lifetime of single-kink structures in thediscrete medium. PACS numbers: 42.60.Da, 42.65.Sf, 42.65.Tg, 05.45.Pq
I. INTRODUCTION
Discrete linear (i.e. one-dimensional) chains exhibit abroad range of phase transitions resulting from the com-petition between different interactions among atoms ormolecules [1]. The soliton condensation, one of most in-triguing of these phases, occurs as a disclination due to vi-olation of rotational symmetries in molecular crystals [1–5], as dislocation in crystal lattices due to misalignementof atoms in the crystal structure of a Frenkel-Kontorovalattice [1, 6–8], as a discommensuration associated with asuperlattice structure forming in a charge-density-wave,antiferromagnetic or ferromagnetic lattices [9–14], or asdomain walls in incommensurate systems in general. Inthese systems solitons are topological defects formed fromthe presence of different nonlinear interactions compet-ing with the chain discreteness and, in some physical con-texts, an external field. In mathematical physics they aresolutions to nonlinear partial differential equations, forwhich they represent waves of long lifetime consequentupon the balance of lattice dispersion by nonlinearity.Among physical systems exhibiting the soliton conden-sation phase are a class of soft matters composed ofnon-spherical molecules, known to be prone to structuralorders governed by molecular tilts with respect to the ∗ [email protected] molecular-chain axis [15, 16]. In this class liquid crys-tals [17] have attracted a great deal of attention becauseof the elongated (i.e. rod-like) shape of their molecularconstituants [18–21], this strong anisotropy of moleculesindeed favors phase transitions related to the moleculartilts leading to several distinct phases in liquid crystals.In the smectic (Sm) phase, liquid crystals possess non-zero molecular tilt angles between the director and thenormal to the smectic layers. In general there exists twodistinct tilt directions i.e. a parallel and an antiparalleltilt directions, associated with two possible orders i.e. theferroelectric order, in which molecules are tilted in uni-son with an average zero phase difference between tiltsof neighbor molecules, and the antiferroelectric order inwhich phase differences fluctuate around π [19, 20, 22–30]. When molecules have no internal planes of symme-try, the pitch becomes chiral and molecular tilts cause he-licoidal distorsions in which molecular rotations aroundthe layer axis are strongly biased. This leads to a frustra-tion in interayer interactions, and in-plane polarizationsprecessing around the helical axis from a layer to neigh-bour layers. In the presence of an external field the pre-cessions of molecular polarizations are confined withinthe smectic layers thus triggering a long-range orienta-tion order via a Freedericksz transition [36, 37].In real Sm chiral (SmC) liquid crystals, however, the di-rections of in-plan polarizations on neighboring smecticlayers are neither exactly parallel nor exactly antiparal-lel. Actually the helical superstructure [19] is stabilizedby a competition between the frustration caused by chi-rality and the ising-type coupling promoting ferroelec-tric or antiferroelectric orders along the tilt axis. Twoimportant consequences of such competition are a slowprecession of in-plane electrical polarizations [19, 25–30],and the possibility of an electroclinic effect [31–33] suchas chiral piezoelectricity, a perpendicular alignement ofthe tilt direction with in-plane polarizations or with anapplied external field [31–35].In this work, we consider a discrete model for chiral SmCliquid crystals in which both the symmetric and anti-symmetric intermolecular interactions are taken into ac-count. We also take into consideration electroclinic ef-fects via a cross coupling between the tilt order param-eter and an applied electric field, assumed to describea perpendicular alignement of the tilt direction with anapplied planar electric field [32]. From the proposed dis-crete model we investigate the effects of the competitionbetween the symmetric (i.e. ising-type) and antisymmet-ric (i.e. chiral-type) intermolecular interactions, and theelectric field confined within the smectic layers, on thegeneration and shape profiles of domain-wall structuresin the chiral SmC liquid crystal. Under specific con-ditions the model reduces to the standard sine-Gordonequation without suppression of chiral intermolecular in-teractions, and hence can support sine-Gordon kink soli-tons. Since the discrete equation is not integrable, weresort to a continuum approximation wich requires ananalysis of the effect of lattice discreteness on continuumsoliton profiles. In this respect we carry out numericalsilumations which show that the continuun periodic-kinksoliton has the same profile as the exact numerical so-lution to the discrete problem, but not the continuumsingle-kink solution. Therefore the discreteness effect willbe more effective on the single-kink soliton, and we de-termine the Peierls stress experienced by this structurein the discrete model. We discuss the implictions of thevariations of the Peierls-Nabarro barrier with the chi-ral and ising-type intermolecular interaction coefficients,and magntitudes of the two components of the externalfield, on the single-kink lifetime in the discrete molecuarmedium. II. THE DISCRETE MODEL AND SOLITONSTRUCTURES
Our model is a discrete linear chain of rod-shapedmolecules describing a liquid crystal in the SmC phase.In this phase the tilt of the director from the normal(here the z axis) to the smectic layers (xy plane) in the n t h smectic layer is a two-component vector field u n ,representing the magnitude and direction of the tilt. Weconsider the system in the vicinity of the smectic-A to thetilted (i.e. SmC) phase transition, and assume that in ad-dition to achiral nearest-neighbor interactions molecules are also antisymmetrically coupled, as a result of chiralinteractions between molecules on neihgbor smectic lay-ers. Due to the chirality the tilt direction will tend toalign perpendicular to any applied electric field withinthe smectic layer by virtue of the electroclinic effect.Taking this last effect into consideration, the discreteLandau-Ginzburg free energy corresponding to our modelwill be [38, 39]: G = N X n =1 [ A u n + B u n + J u n . u n +1 + f ( u n × u n +1 ) z + ( E × u n ) z ] . (1)In Eq. (1) the parameter A ( T ) = A ( T − T c ), where A is positive ensuring a continuous transition from thesmectic A phase to the SmC phase at the mean-fieldcritical temperature T C . B in the second term is pos-itive, the third term is an ising-type symmetric nearest-neighbor interactions while the four term is an antisym-metric nearest-neighbor interaction due to molecular chi-rality. The last term takes into account the electrocliniceffect caused by chirality, which forces the directions ofthe tilt vector and an applied electric field E to be per-pendicular [32]. The supscript z in Eq. (1) indicates aprojection along the smectic layer normal z .Express the local order parameters u n as two-componentverctor fileds i.e. u n ≡ u (cos ϕ n , sin ϕ n ), where thetilt amplitude u is assumed homogeneous and only theangle of helicoidal motion ϕ n varies locally. As forthe applied electric field, we assume that its lies withinthe layer planes and hence is a two-component vector E ≡ ( E x , E y ). In terms of ϕ n Eq. (1) becomes: G = G + N X n =1 [ Ju cos( ϕ n +1 − ϕ n ) + f u sin( ϕ n +1 − ϕ n ) − ( E x sin ϕ n − E y cos ϕ n )] . (2)The spatial configuration of the helicoidal order in thechiral SmC phase is obtained by minimizing (2) with re-spect to ϕ n , and is governed by the discrete equation:0 = sin( ϕ n +1 − ϕ n ) + sin( ϕ n − − ϕ n ) − c [cos( ϕ n +1 − ϕ n )+ cos( ϕ n − − ϕ n )] + ǫ x cos ϕ n + ǫ y sin ϕ n , (3)where: c = fJ , ǫ x = E x Ju , ǫ y = E y Ju . (4)To solve Eq. (3) we will isolate ϕ n +1 from the localvariables ϕ n and ϕ n − . To this end we define: β = ǫ x ǫ y , α = ǫ y s β c , (5)such that Eq. (3) reduces to: ϕ n +1 = ϕ n + arctan c + arcsin [sin( ϕ n − ϕ n − − arctan c ) − α sin( ϕ n + arctan β )] . (6)The equilibrium solutions of Eq. (6) will generally de-pend on the signs and magnitudes of characteristic pa-rameters of the model. The equilibrium states for in-stance, in the absence of applied electric field (i.e. E =0), have been discussed in some past works consideringsecond-neighbor interactions of both achiral and chiraltypes [38, 39]. Thus, it is well established that an antifer-roelectric groundstate is expected mainly when second-nearest neighbor interactions are taken into account[35].From the standpoint of Eq. (6) without the externalfield but with addition of second-nearest neighbor inter-actions, this state will correspond to an equilibrium con-figuration where the phase differences between tilt direc-tions increase nearly by π [35]. In the present context,where there is no second-nearest neihghbor interactionterms in Eq. (6), we can rule out antiferroelectric order-ing. This is anyhow evident given that the only equili-bruim state suggested by Eq. (6), when E = 0, is thezero phase difference between neighbor tilts along thechain axis corresponding to a ferroelectric order.Looking for the general solution of Eq. (6) for arbitraryvalues of the model parameters, it is instructive recall-ing that a similar equation was obtained in the studyof discrete-soliton and soliton-lattice generations duringthe unwinding process of SmC ∗ α phase to SmC phasedriven by an electric field [40]. Eq. (6) is more priciselya perturbed version of the so-called sine-lattice equation[8, 41, 42], and as such is not exactly integrable. How-ever, in some specific contexts approximate solutions canbe found. For instance, when c = β = 0, a familyof approximate solutions have been shown to exist withsome dispersion relation [8, 41, 42]. These solutions are π -kink solitons and are also solutions to the discrete equa-tion (6) with c = 0. Indeed, with the variable change ϕ n = φ n − arctan β Eq. (6) can be transformed to [8]: φ n +1 = φ n + arcsin [sin( φ n − φ n − ) − α sin φ n ] . (7)The single-soliton solution to the sine-lattice equation (7)is a π kink as shown in ref. [42], with the help of Hirotatransformations.The single-kink soliton as a general solution to the sine-lattice equation is interesting, but concerning the specificproblem at hand periodic structures provide a better pic-ture of the topology of the helicoidal superstructure cre-ated in the discrete system. In want of analytical methodenabling the derivation of an exact solution consistentwith this periodic helicoidal superstructure, we shall re-sort to a continuum-limit approximation. In this goalwe assume the phase differences φ n +1 − φ n to remain al-ways small, such that we can expand sin( φ n +1 − φ n ) ≈ φ n +1 − φ n , sin( φ n − φ n − ) ≈ φ n − φ n − . Substitutingthese expansions in Eq. (6) and defining a continuousspatial position x = na , where a (hereafter assumed tobe unity) is the separation between neighbor smectic lay- ers at equilibrium, we can readily rewrite (6) as: φ xx = − α sin φ, (8)where the subscript ” xx ” means a second-order derivativewith respect to x . The periodic-soliton solution to Eq.(8) is obtained as: φ κ ( x ) = ± (cid:20) sn (cid:18) xℓ κ , κ (cid:19)(cid:21) , = ± am (cid:18) xℓ κ , κ (cid:19) , ℓ κ = κ √ α , (9)in which sn and am are Jacobi elliptic functions of modu-lus κ obeying 0 ≤ κ ≤ ℓ κ and equal separation d κ = 2 ℓ κ K ( κ ), where K ( κ ) is thecomplete elliptic integral of the first kind.According to the expression of ℓ κ given in formula (9),an increase of the kink width with increase of the chi-ral interaction strength at fixed value of the symmetricinteraction J , is balanced by an increase of the x com-ponent of the electric field for a fixed value of ǫ y . How-ever, this balance costs a uniform shift of the periodic-kink soliton by a phase factor arctan( β ), as reflected inthe expression of the real solution to our problem i.e. ϕ ( x ) = φ ( x ) − arctan β . Variations of the period d κ with c and β , are the same as the variations of thekink width with these two parameters. In fig. 1, weplotted the amplitude-function solution given by (9) for κ = 0 .
97 (left graph) and κ = 1 (right graph). It is re-markable that profile of the amplitude-function solutionwhen κ = 1, coincides with the analytical expression: φ ( x ) = ± h exp xℓ i ∓ π, (10)while the period d κ =1 → ∞ . Clearly, when κ →
1, thehelicoidal superstructure decays to a single-kink soliton.Fig. 2 summarizes the variations of the periodic-kinkwidth with β and c , for κ = 0 . ε y to 0 . c and β . Curves in the left graph of Fig. 3 are spatial profilesof the periodic-kink soliton when β = 0 and c = 0,0 .
2, 0 .
5, 0 .
75. In the right panel, the periodic-kink soli-ton profiles were generated for a fixed value of c (i.e. c = 0 .
1) while β was varied as β = 0, 0 .
1, 0 .
16, 0 . . FIG. 1. (Color online) Saptial profile of the analytical periodic-kink soliton solution (9), for κ = 0 .
97 (left graph) and κ = 1(right graph). All other parameters are taken to be unity.FIG. 2. (Color online) Width of kinks (in units of the latticespacing) in the periodic-kink solution, plotted versus β and c for a fixed value of ε y (i.e. β = 1). kink tructures are shape preserving and always stable inthe discrete system even for an electric field reduced toonly its y component. As we increase the chiral interac-tion coefficient (which corresponds to increasing c ) kinksin the soliton lattice get sharper while their widths areincreased, consistent with the behavior observed in Fig.2. On the other hand, when the ratio of the chiral tothe symmetric intermolecular interactions is fixed, andthe x component of the electric field is increased from azero, the periodic-kink profile seems to survive only upto some critical value of the ratio E x /E y as seen in theright graph of Fig. 3. Beyond this critical value the soli-ton structure decays into a kink-antikink lattice, whichin turn will survive within some finite range of values of β beyond which soliton structures become unstable inthe system. III. ENERGETIC CONSIDERATIONS ONSOLITON EXISTENCE AND STABILITY
Energies of solitons are relevant parameters when con-sidering their formation as well as their stability in agiven medium. For the problem at hand there are twodifferent energies that are relevant for the existence of thekink solitons obtained in the previoeus section, they aretheir creation energy and the energy related to the dis-creteness of the molecular chain. Numerical simulationsof the discrete equation (6) carried out in sec. II, haveshown that profiles of the exact periodic-kink solutions tothis equation were identical with the continuum periodic-kink soliton solution obtained in formula (9). This meansEq. (9) can be readily regarded as an exact solution tothe discrete equation, and too is its continuum energy.On the contrary, the single-kink solution Eq. (10) is notreproduced by numerical simulations and hence can by nomeans be exact to the discrete problem. Reason why thesingle-kink soliton Eq. (10) is expected to suffer the lat-tice discreteness, resulting in energy dispersion in effortsto overcome the discrete relief of the molecular chain. Tostart we calculate the creation energy of the periodic-kinksoliton, and next determine the potential barrier erectedby the lattice discreteness and to which the single-kinksoliton can be trapped.The periodic-kink soliton solution (9) was obtained by in-tegration of Eq. (8) with periodic boundary conditions.The corresponding energy integral, in the specific case ofJacobi-elliptic function solutions, is given by:12 φ x = ω κ (cid:2) − κ sin ( φ/ (cid:3) , = ω V ( φ ) , ω = α . (11)This relation provides the right condition of energy con-servation for the periodic-kink soliton, and hence canbe used to define the express of the total energy of the T il t ang l e n -10 0 10 20 30 40 50 60 70 0 50 100 150 200 T il t ang l e n FIG. 3. (Color online) Spatial profiles of the periodic-kink soliton from numerical simulations ofthe discrete equation (6), for ε = 0 .
2. Left panel: β = 0 (fixed) and c = 0, 0 .
2, 0 . .
75 (from bottom to top curves). Right panel: c = 0 . β = 0, 0 .
1, 0 .
16, 0 .
17 and 0 .
176 (from top to bottom curves). periodic-kink soliton i.e.: E sol ( κ ) = Ju Z d κ dx (cid:20) φ x + ω V ( φ ) (cid:21) . (12)Substituting φ ( x ) given by (9) in the last integral we find: E sol ( κ ) = Ju √ ǫ y κ (cid:18) β c (cid:19) / E ( κ ) , (13)where E ( κ ) is the complete elliptic integral of the secondkind [43]. Formula (13) indicates that an enhancement ofchirality will increase the periodic-kink creation energy,while an increase of the x component of the electric fieldwill be detrimental to the creation of periodic-kink soli-ton. This behavior, once again, is consistent with resultsof numerical simulations of the discret equation (6) dis-cussed in the previous section. In the single-kink limitformula (13) reduces to: E sol = 4 Ju ω . (14)As emerged in our previous discussions, strickly formula(14) is valid only in the continuum medium given thatthe single-kink structure is not exact for the discrete sys-tem, to find the actual energy of the single-kink solutionplaced in the discrete molecular chain, we must use theanalytical solution (10) in the discrete total energy givenby (2). To this aim we must explicely introduce a pin-ning coordinate for the single-kink soliton, here denoted X and coinciding with the soliton centre-of-mass position[47] in the discrete discrete system. Thus the argumentof (10) is shifted from x = n to x − X . Next keepingthe ”ferroelectric-ordering” argument i.e. φ n +1 − phi n is always very small, using formula (10) and groupingall constant terms in an homogeneous function F , the discrete energy (2) can be written: F = F − Ju q c α N X n =1 sech (cid:18) n − Xℓ (cid:19) (15)+ 4 Ju q α (1 + c ) arctan c N X n =1 sech (cid:18) n − Xℓ (cid:19) , where F = G − G . The discrete sum over n in (15) isexact when N → ∞ , yielding: F = U + 4 Ju q c [ dn (2 XK ( ν )) arctan c − (cid:0) K ( ν ) dn (2 XK ( ν )) − E ( ν ) (cid:1) ] K ( ν ) ,U = F − Ju q α (1 + c ) . (16) dn () is one of Jacobi elliptic functions [43] while K ( ν )and E ( ν ) are complete elliptic integrals of the first andsecond kinds respectively, here given in terms of a newmodulus ν obeying the transcendental relation [48]: πℓ = K ( ν ′ ) K ( ν ) , ν ′ = p − ν , ≤ ν ≤ . (17)To easily capture the physics in the expression (16) ofthe discete energy, we adopt the Fourier series represen-tations of the Jacobi elliptic functions dn () and dn () [43]and find: F = U + ∞ X p =1 U p ( ℓ ) cos(2 πpX ) , (18) U p ( ℓ ) = 4 πJu q c (cid:20) pπ sinh ( pπ ℓ ) − arctan c cosh ( pπ ℓ ) (cid:21) , where the transcendental relation (17) was used to elim-inate the complete elliptic integrals K ( ν ) and E ( ν ). Ac-cording to formula (18), in the discrete regime the single-kink soliton energy is a periodic function of the soli-ton centre-of-mass position X with an energy amplitude U p ( ℓ . When the kink width ℓ is large enough, the lead-ing term U ( ℓ ) in the sum (18) will dominate and theamplitude of the periodic energy reduces to: U P N = 4 πJu q c (cid:20) π sinh ( π ℓ ) − arctan c cosh ( π ℓ ) (cid:21) . (19) U P N , which we refer to as the Peierls-Nabarro barrier,is the amplitude of the periodic potential experiencedby the single-kink soliton due to the discreteness of themolecular chain. Instructively formula (19) reveals thatthe Peierls-Nabarro barrier will be lowered by the chi-rality, while the contribution from the electric field is adecrease of the kink width and hence an increase of thePeierls-Nabarro barrier. However the dependence of U in both the chirality and the electric field, reflected byformula (19, is such that the chirality will have the dom-inant effect for a decrease of ℓ with an increase of β , willbe balanced by the increase of c . IV. CONCLUSION
We investigated the effects of the competition betweenan ising-type nearest-neighbor interaction and an anti-symmetric nearest-neighbor interaction (of chiral origin)between molecules on one hand, and a two-componentelectric field on the other hand, on the formation andstability of domain walls in chiral smectic liquid crystalsin the ferroelectric phase. We found that the equilibriumconfiguration of the discrete liquid-crystal system, result-ing from molecular tilts with respect to the long molec-ular chain axis, is described by a sine-lattice type equa-tion. In the continuum limit this equation can be reducedto the classic sine-Gordon equation, thus admitting twodistinct soliton solutions namely a single-kink and kink-lattice (i.e. periodic-kink) soliton solutions. While nu- merical simulations of the full sine-lattice equation sug-gest that the continuum periodic-kink solution can bea good approximation of the exact solution to the dis-crete problem, the single-kink solution can by no meanbe obtained from the discrete equation and therefore re-mains exact only in the continuum limit. Nevertheless,given that a long-range domain-wall order forms by nu-cleations of single-kink soliton structures, we consideredthe survival of such structures in the discrete molecularchain. In this respect we obtained the amplitude of thePeierls-Nabarro potential, which inverse is proportionalto kink lifetime in the presence of lattice discreteness,and obtained that chirality lowers the Peierls stress andconsequently favors the single-kink stability in the dis-crete system.In this study we were concerned mainly with the com-peting effects of the ising-type and antisymmetric inter-molecular interactions, as well as the electric field, onthe formation of solitonic structures in chiral smectic liq-uid crystals. Although the in-plane polarizations are sec-ondary order parameters, and hence were not consideredin this work, it is well established [35] that because of thechirality in-plane polarizations of molecules are not par-allel with the primary order paramater (i.e. the tilt vec-tor u ). Therefore the interaction of in-plane polarizationvectors and the tilt vectors will introduce Lifshitz termsin the total energy accounting for a chiral piezoelectriceffect [35, 38]). A study of the formation of domain wallstaking into account this chiral piezoelectric effect is arelevant open problem, which will certainly provide richinsight onto the physics of discrete smectic chiral liquidcrystals with competing interactions. ACKNOWLEDGMENTS
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