Research ArticleHysteresis in Two-Dimensional Liquid Crystal Models
Slavko BuIek,1 Samo Kralj,2,3,4 and T. J. Sluckin5
1Osnovna sola I Murska Sobota, Stefana Kovaca 32, Sl-9000 Murska Sobota, Slovenia2Faculty of Natural Sciences and Mathematics, University of Maribor, Koroska 160, Sl-2000 Maribor, Slovenia3Condensed Matter Physics Department, Jozef Stefan Institute, Jamova 39, Sl-1000 Ljubljana, Slovenia4Jozef Stefan International Postgraduate School, Jamova 39, Sl-1000 Ljubljana, Slovenia5Division of Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UK
Correspondence should be addressed to Slavko Bucek; [email protected]
Received 20 November 2014; Revised 28 January 2015; Accepted 8 February 2015
Academic Editor: Mark Bowick
Copyright Β© 2015 Slavko Bucek et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We make a numerical study of hysteresis loop shapes within a generalized two-dimensional Random Anisotropy Nematic (RAN)model at zero temperature.Thehysteresis loops appear on cycling a static external ordering field.Ordering in these systems is historydependent and involves interplay between the internal coupling constant J, the anisotropy randomfieldD, and the ordering externalfieldH. Here the external field is represented by a traceless tensor, analogous to extension-type fields in continuummechanics.Thecalculations use both a mean field approach and full lattice simulations. Our analysis suggests the existence of two qualitativelydifferent solutions, which we denote as symmetric and symmetry breaking. For the set of parameters explored, only the symmetricsolutions are stable. Both approaches yield qualitatively similar hysteresis curves, which are manifested either by single or doubleloops. But the quantitative differences indicate that mean field estimates are only of limited predictive value.
1. Introduction
Understanding the effect of disorder on structures reachedvia continuous symmetry breaking is of interest for variousbranches of physics [1, 2]. Such systems exhibit almostunavoidably topological defects [3, 4], which are in generalstabilized by disorder [5β7]. The resulting structures are ingeneral rich in metastable states, which are separated fromeach other by relatively high energy barriers with respectto thermal energies. Transitions between competing statesoften require local topological changes, which are in generalenergetically costly. Consequently, the sample history can besignificant even on the macroscopic properties of the system.
The pioneering investigations of spatially randomly per-turbed system exhibiting continuous symmetry breakingwere carried out inmagnetism [8β10]. Other studies [11] showthat disorder strongly influences phase behavior, as a resultof the presence of Goldstone modes in gauge fields reflectingbroken symmetries [11]. For example, the Larkin-Imry-Matheorem [11, 12] claims that even infinitesimally weak randomfield-type disorder destabilizes long range order (LRO) withrespect to short range order (SRO). However, later studies
[13] demonstrate the occurrence rather of quasi-long-range-order (QLRO), characterized by algebraic decay of spatialcorrelations. A well-known example of such a system is theBragg glass phase in dirty superconductors [14]. Glass-likeproperties are typically observed in these systems, whichas a rule manifest history-dependent macroscopic responseswhen subject to an external ordering field [10]. Becauseof their potential applications in various memory storagedevices, the hysteresis response [15, 16] to cycling externalfields are of particular interest. Even in simple minimalmodels [17], the competition between the inherent orderinginteractions, the disorder strength, and the external orderingfield can lead to a relatively rich set of behaviors.
In recent decades, there have been several studies ofthe influence of disorder on continuous symmetry brokenphases in a number of different liquid crystal (LC) phases[6, 7, 18, 19]. These studies include a rich variety of dif-ferent LC phases and also enable the type and strength ofthe disorder to be relatively well controlled. As such theyrepresent a useful testing ground for the investigation ofuniversal features related with disorder. We note that liquidcrystals are relatively easily experimentally accessible due
Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2015, Article ID 834867, 10 pageshttp://dx.doi.org/10.1155/2015/834867
2 Advances in Condensed Matter Physics
to their liquid character, softness, and optical anisotropy[20]. The simplest representative is the nematic uniaxialLC phase, whose local mesoscopic orientational ordering istypically represented by the nematic director field n, wherethe directions Β±n are physically equivalent [21, 22], and|n| = 1. In the bulk, equilibrium nematic LCs tend toalign parallel along a single symmetry direction, in analogousfashion to local magnetization m in ferromagnets. Randomfield-type disorder in LC structures can be imposed using anumber of different procedures. Well-known examples areLCs confined to porous matrices such as controlled-poreglasses, aerogels or Russian glasses, or LC-aerosil mixtures[6, 18]. The latter systems are of particular interest becauseone can tune disorder strength, and even the type of disorder,by varying aerosil nanoparticle concentration [6, 23, 24].
However, the symmetries ofn andmdiffer. Consequently,some features observed in magnetic and LC systems couldalso differ [19]. For example, magnetic systems exhibit linedefects only with integer values of the correspondingwindingnumberπ [3, 20]. But the head-to-tail invariance of π permitsnematic LCs also to exhibit line defects characterized by half-integer winding numbers.
In this paper we consider the hysteresis properties of arandomly perturbed two-dimensional nematic LC describedby theRandomAnisotropyNematic (RAN) [25] latticemodelat zero temperature. The orientational order of the LC ismeasured by a tensor nematic order parameter. The externalextensional ordering field is a traceless tensor and possessesthe symmetry of an extensional stress of a type shown oftenin continuum mechanics [26]. We then calculate the globalnematic response to the cycling of this field. Our particularinterest lies in the existence of qualitatively different typesof hysteresis loop. The calculations use both lattice-typesimulations and also a mean field-type approach.
The plan of the paper is as follows. In Section 2 weformulate the problem. We introduce our two-dimensionallattice model in terms of tensor nematic order parameter atzero temperature. In Section 3 we derive mean field solutionfor average nematic ordering for a given external orderingfield. Results are presented in Section 4 where we focuson possible shapes of hysteresis loops on cycling the staticexternal field. We analyze both mean field behaviour andlattice model simulation results. In the final section wesummarize the results of the calculations.
2. Formulation of Problem
2.1. Model. We consider a π-dimensional lattice model ofπ dimensional nematic liquid crystal molecules. These aresubject to external fields as well as local disordering randomfields.Themodel is a generalization of the random anisotropynematic (RAN) model discussed elsewhere by some ofthe present writers [25], which in turn is a random fieldgeneralization of the classic Lebwohl-Lasher lattice model[27] of a nematic liquid crystal. A rotor, whose direction isdefined by n β‘ βn, is placed at each site π. In this paper,π = π = 2.
Orientational properties of the rotors at site π are deter-mined by the traceless tensor quantityQπ:
ππ
πΌπ½= ππ
πΌππ
π½β1
π
πΏπΌπ½. (1)
Greek indices label components of the Cartesian coordinatesystem, while Latin indices determine lattice sites. We willrefer to the quantitiesππ
πΌπ½as tensor spins, to distinguish them
from vector spins, whose hysteresis properties have been wellstudied in the literature [17].
The Hamiltonian of the system reads
H = β1
2
π½β
ππ,ππ
ππ
πΌπ½ππ
πΌπ½β π»πΌπ½β
π
ππ
πΌπ½β π·β
π
ππ
πΌπ½ππ
πΌπ½, (2)
where repeated Greek indices in a product indicate theapplication of a summation convention.
The meaning of the terms in (2) is as follows.(a) The first term involves interactions between nearest
neighbor sites and favours parallel orientation ofneighboring LC molecules. π½ > 0 is a measure of theinteraction strength between the nearest neighbors(ππ). The factor 1/2 is introduced to avoid doublecounting. Later we will suppose π§ nearest neighbors,with π§ = 4 in the square lattice, and π§ = 6 in thehexagonal lattice. We refer to this term as the internalfield term.
(b) The second term describes the coupling with anexternal tensor ordering field π»
πΌπ½. We define this in
terms of a single parameter π», and a set of principalaxes, which in this case coincide with π₯ and π¦ axes:
π»πΌπ½
= 2π»(ππ₯,πΌππ₯,π½
β1
2
πΏπΌπ½) = β2π»(π
π¦,πΌππ¦,π½
β1
2
πΏπΌπ½) ,
(3)
where eπ₯is the unit vector in the π₯ direction and
eπ¦is the unit vector in the π¦ direction. This tensor
field takes the form of an extension field, of the typethat frequently occurs in continuum mechanics. Weaddress elsewhere in the paper the difference betweenthis tensor form and the uniaxial random tensor fieldintroduced by Cleaver et al. [25], but note at this stageonly that for π = 2 the two forms are completelyequivalent. We will refer to this term as the externalfield term.
(c) The final term represents the coupling between thelocal tensor spin and a time-independent quenchedlocal field of uniform strengthπ· > 0, where the tensor
ππ
πΌπ½= ππ
πΌππ
π½β1
2
πΏπΌπ½
(4)
is expressed in terms of unit vectors eπ β‘ βeπ. Thedirections {eπ} are drawn randomly from an isotropicorientational distribution, and the directions at differ-ent sites are uncorrelated.As in the case of the externalfield term, this tensor is an extensional field. We referto this term as the random field term.
Note that the only difference between the terms in (3) and (4)is that the field in (3) aligns the spins in the same directionat each site, whereas in (4) the direction of the aligning fieldvaries from site to site.
Advances in Condensed Matter Physics 3
2.2. Angular Representation. In the (π₯, π¦) Cartesian frame,the dynamical state of the system can be parameterized ateach site in terms of a local spin direction nπ = (cos π
π, sin ππ)
and a locally favored orientation eπ = (cosπΌπ, sinπΌ
π). Then it
is readily shown that
Qπ = 1
2
(
cos 2ππ
sin 2ππ
sin 2ππβ cos 2π
π
) ,
H = π»(
1 0
0 β1
) ,
Nπ = 1
2
(
cos 2πΌπsin 2πΌ
π
sin 2πΌπcos 2πΌ
π
) .
(5)
The energy can now be expressed in terms of the quantities{ππ, πΌπ}. Substituting (5) into (2) we obtain
H = βπ½
4
β
ππ,ππ
cos 2 (ππβ ππ) β π»β
π
cos 2ππ
βπ·
2
β
π
cos 2 (ππβ πΌπ) .
(6)
Although the model is originally expressed in terms of localtensor quantities, we note the similarity of this form to theclassical random fieldππ vector spin model Hamiltonian
Hβ= β
π½
2
β
ππ,ππ
cos (ππβ ππ) β οΏ½οΏ½β
π
cos ππβ π·β
π
cos (ππβ πΌπ) ,
(7)
where now π½ is the local spin interaction energy, οΏ½οΏ½ is amagnetic field, and π· is the coupling with a random fielddirection. Apart from the factors of 2 in the angles and dif-ferences in convention concerning the values of parameters,(6) and (7) are extremely similar. We expect the results ofstatistical mechanism calculations to mirror this.
We further observe that the field energy at a site isβπ»πΌπ½ππ
πΌπ½= βπ» cos 2π
π. Thus positive π» prefers alignment
in the π₯ direction, while negativeπ» prefers alignment in theπ¦ direction. Likewise the random energy at site π is givenby βπ·ππ
πΌπ½ππ
πΌπ½= β(1/2)π· cos 2(π
πβ πΌπ), implying that for
sufficiently largeπ·, ππ= πΌπ.
2.3. Order Parameters. Our study concerns the averagebehaviour of the whole lattice. We can define a mean tensororder parameter
S = β¨Qπβ© = (
ππ₯π₯
ππ₯π¦
ππ¦π₯
ππ¦π¦
) = (
π π
π βπ
) , (8)
where β¨β β β β© indicates spatial averaging. Thus the key orderparameters are
π = β¨cos 2ππβ© =
1
π
β
π
cos 2ππ, (9a)
π = β¨sin 2ππβ© =
1
π
β
π
sin 2ππ, (9b)
where the averaging is over the allπ sites. In the limitsπ» β
Β±β, the external field dominates the internal coupling andthe random field terms, and hence π(π» β +β) β 1,whereas π(π» β ββ) β β1. This corresponds to a rotationfrom the π₯ to the π¦ directions asπ» goes from +β to ββ.
Thequantity S responds toH, and thuswe expect it to takethe same form. The order parameter π tends to adjust to theimposed fieldπ». In this case the principal axes of S lie in thesame directions as those of the forcing fieldπ»
πΌπ½. By contrast,
the quantity π is an off-diagonal term in the principal frameof reference. In general, the symmetry of the system mightsuggest π = 0, if we may suppose π¦ and βπ¦ to be equivalent. Ifπ = 0, the symmetry of the system is broken, and we would ingeneral expect a pair of solutions with positive and negativevalues. In this case, the principal axes of S will be rotated atan (plus or minus) oblique with respect toH. We henceforthrefer to solutions with π = 0 and π = 0 as symmetric solutionsand symmetry breaking solutions, respectively.
2.4. Procedure. The dynamics of the system at π = 0 isgoverned by the instantaneous local field at site π, given by
πΊπ
πΌπ½= β
πΏH
πΏππ
πΌπ½
= π»πΌπ½+ π·π
π
πΌπ½+ π½ β
ππ,π =π
ππ
πΌπ½, (10)
where the local fields πΊππΌπ½
are necessarily symmetric andtraceless. Each individual tensor spin seeks to minimize thelocal energy
Hπ= βπΊπ
πΌπ½ππ
πΌπ½, (11)
where we bear in mind that βπHπ
= H. The sum ofthe local energies does not equal the total Hamiltonian,because of the need to include a compensating factor of+(1/2)π½β
ππππ
πΌπ½ππ
πΌπ½, to avoid double counting of nearest
neighbor interaction terms.The dynamics of the system proceeds in discrete time,
such thatπππΌπ½(π‘+1)minimizes the quantityβπΊπ
πΌπ½(π‘)ππ
πΌπ½(π‘+1).
In the π = 2, π = 2 case discussed here, this only involvesensuring that
(a) the principal axes of πππΌπ½(π‘ + 1) coincide with those
πΊπ
πΌπ½(π‘);
(b) the normalization ofπππΌπ½(π‘+1), given by (1), is correct.
We thus reach the result that
Qπ (π‘ + 1) = 1
2
(
cos 2ππ (π‘ + 1) sin 2π
π (π‘ + 1)
sin 2ππ (π‘ + 1) β cos 2π
π (π‘ + 1)
)
=1
tr (Gπ (π‘))2πΊπ
πΌπ½(π‘)
=1
2 (πΊπ
11(π‘)2+ πΊπ
12(π‘)2)
(
πΊπ
11(π‘) πΊ
π
12(π‘)
πΊπ
12(π‘) βπΊ
π
11(π‘)
) .
(12)
4 Advances in Condensed Matter Physics
Given a lattice configuration of random tensor fieldsπππΌπ½
anda starting configuration of spinsQπ(0), the discrete dynamicsis iterated until a stable configuration, or equivalently a fixedpoint of the dynamics is reached. The condition for stabilityat site π, from (12), is then that
1
2
(
cos 2ππ
sin 2ππ
sin 2ππβ cos 2π
π
)
=1
2 [(πΊπ
11)2+ (πΊπ
12)2]
(
πΊπ
11πΊπ
12
πΊπ
12βπΊπ
11
) .
(13)
This corresponds to a local minimum of the energy, butnot necessarily, of course, a global minimum. The local fieldcombines the randomfield, the internal field, and the externalfield. Which of these dominates depends on a number offactors. If the magnitude of the external field is sufficientlylarge, other terms must be a small perturbation. Likewise, ifthe magnitude of the internal field is small compared to therandom field, it can never play more than a minor role, evenat low external fields. Finally, the history of the sample alsoplays a major role. Under some circumstances it is hard todestroy previously well-established order.
We study the hysteretic behavior of ensembles of spins atzero temperature. In each numerical experiment we keep thesame lattice with fixed value of π½ > 0 and π· > 0 and fixedvalues ofπΌ
πat each site.The ordering field strengthπ» is cycled
between very large positive (π» = π»max > 0) and negative(π» = βπ»max) values. The energy is first minimised at largepositiveπ», corresponding to the vector e
πaligning along the
π₯ axis at all sites. The field is progressively reduced in smallsteps (the βdownβ part of the cycle). At each step, a new energyminimum is sought, starting from the previous equilibriumconfiguration, and following an energy-downhill procedure.Eventually, after many steps, π» is large and negative, and all{eπ} essentially lie along the π¦ direction, that is, rotated by π/2
with respect to their original direction. The process is thenreversed, and π» is taken from ββ to +β (the βupβ part ofthe cycle).
We also develop mean field equations to describe thisprocess and compare these results with the results of oursimulations.Hysteresis corresponds to unambiguously differ-ent values of the order parameters during the up and downparts of the cycle. We also examine in detail the process ofrelaxation and tensor spin rotation.
3. Mean Field Approximation
3.1. The Local Field. In the following we calculate orderingproperties of an ensemble driven by Hamiltonian defined in(2), using a mean field approximation valid in the limit thatall sites interact equally with all others.This approach followsprevious work by Strogatz et al. [28], who were interestedin conduction in charge density waves, and, by Shukla andKharwanlang [17], who focused on the much more closelyrelated problem of random vector spins.
A mean field theory requires the determination of self-consistent local field. This involves ignoring correlations
between sites in (10), leading to a mean local field at site πgiven by
πΊπ
πΌπ½= π»πΌπ½+ π·π
π
πΌπ½+ π½π§ππΌπ½. (14)
3.2. Symmetric Solution. Assuming no symmetries are bro-ken and taking into account π
π₯π₯+ ππ¦π¦
= 0 we can define Susing a single variable. From the definition (9a) it follows that
S = 1
2
(
π 0
0 βπ
) ,
πΊ
= (
π» +1
2
π· cos 2πΌπ+1
2
π½π§π1
2
π· sin 2πΌπ
1
2
π· sin 2πΌπ
βπ» β1
2
π· cos 2πΌπβ1
2
π½π§π
) .
(15)
Within a mean field philosophy, we may suppose that πππΌπ½
isparallel to πΊπ
πΌπ½at each site. Hence
ππ
πΌπ½=1
2
(
cos 2ππ
sin 2ππ
sin 2ππβ cos 2π
π
)
= π(
π» +1
2
π· cos 2πΌπ+1
2
π½π§π1
2
π· sin 2πΌπ
1
2
π· sin 2πΌπ
βπ» β1
2
π· cos 2πΌπβ1
2
π½π§π
) ,
(16)
and hence the equation for ππis
cos 2ππ= (π» +
1
2
π· cos 2πΌπ+1
2
π½π§π)
β {[π» +1
2
π· cos 2πΌπ+1
2
π½π§π]
2
+1
4
π·2sin22πΌ
π}
β1/2
.
(17)The reader should note the strong analogy between procedureoutlined above and other work in the mean field theoryof disordered systems at zero temperature (see, e.g., [17,28]) The philosophy is similar to that of mean field theoryin temperature-dependent nonglassy materials. However,unlike in the case of temperature-dependent nonrandomsystems, as far as we are aware, there is no detailed theorydiscussing the accuracy and dimensional dependence ofmean field theories in these disordered systems. There is, ofcourse, an expectation, based on the applicability of meanfield theory in temperature-dependent nonglassy materials,that mean field theory becomes more accurate in the limitsof long-ranged interactions and high dimensionality. At thisstage, we merely present the mean field results because thecalculation is heuristically appealing and at this stage the onlyapproximation available.
The self-consistent equation for π involves taking theaverage, which we do by integration:
π =1
π
β«
π
0
ππΌ β¨cos 2ππ (πΌ)β© , (18)
Advances in Condensed Matter Physics 5
leading to a final self-consistent equation of
π =1
π
β«
π
0
ππΌ((π» +1
2
π· cos 2πΌ + 1
2
π½π§π)
β {[π» +1
2
π· cos 2πΌ + 1
2
π½π§π]
2
+1
4
π·2sin22πΌ}
β1/2
) .
(19)
This equation is exactly of the same formas that in the Shukla-Kharwanlangππmodel solution [17], apart from scaling.
3.3. Symmetry Breaking Solution. Note that we might expectthe liquid crystal direction to swing around, rather than justto change fromπ₯ toπ¦, so that thewhole system swings aroundcoherently. Then we need to take into account also the orderparameter π; see (9b). Now (8) becomes
S = (
π π
π βπ
) ,
πΊ
= (
π» +1
2
π· cos 2πΌπ+1
2
π½π§π1
2
π· sin 2πΌπ+1
2
π½π§π
1
2
π· sin 2πΌπ+1
2
π½π§π βπ» β1
2
π· cos 2πΌπβ1
2
π½π§π
) .
(20)
Requesting parallel orientation of ππ and πΊ it follows
ππ
πΌπ½=1
2
(
cos 2ππ
sin 2ππ
sin 2ππβ cos 2π
π
)
= π(
π» +1
2
π· cos 2πΌπ+1
2
π½π§π1
2
π· sin 2πΌπ+1
2
π½π§π
1
2
π· sin 2πΌπ+1
2
π½π§π βπ» β1
2
π· cos 2πΌπβ1
2
π½π§π
) .
(21)
The (two) self-consistent equation is now
π =1
π
β«
π
0
ππΌ((π» +1
2
π· cos 2πΌ + 1
2
π½π§π)
β {[π» +1
2
π· cos 2πΌ + 1
2
π½π§π]
2
+ [1
2
π· sin 2πΌ + 1
2
π½π§π]
2
}
β1/2
) ,
π =1
π
β«
π
0
ππΌ((π» +1
2
π· sin 2πΌ + 1
2
π½π§π)
β {[π» +1
2
π· cos 2πΌ + 1
2
π½π§π]
2
+ [1
2
π· sin 2πΌ + 1
2
π½π§π]
2
}
β1/2
) .
(22)
3.4. Effective Energy. The self-consistent equations can bederived from an effective energy per site which we define as
π (π, π) =1
4
π½π§ (π2+π2) β πΈdis (π, π) ,
πΈdis =1
π
β β«
π
0
ππΌ [(π» +1
2
π· cos 2πΌ + 1
2
π½π§π)
2
+(1
2
π· sin 2πΌ + 1
2
π½π§π)
2
]
1/2
.
(23)
With this definition of the effective energy, the self-consistentequations become
ππ
ππ
= 0;ππ
ππ
= 0. (24)
In the restricted theory (π = 0), corresponding to symmetricsolutions, the coercive field occurs when
π2π
ππ2= 0. (25)
4. Numerical Results
We fix the value of π½ > 0 and π· > 0 at zero temperatureand focus on the hysteretic behavior of ensembles on cyclingthe ordering field strengthπ». We first analyze behavior usingthemean field approximation. Afterwards we perform amoreaccurate lattice model simulation.
We determine the ordering properties of the system oncycling π» from +π»max > 0 to βπ»max and back to +π»max.We solve equations iteratively for a given set of externalparameters. A simulation is stopped when the differencebetween the new and old configuration is sufficiently small.In this context the criterion of smallness is that at each sitethe absolute difference between the previous and the updatedconfiguration is below 10β5.
In practice we begin simulations at a high enough valueof π» for which π βΌ 1 and π = 0. Next we decrease the fieldfor a small enough step Ξπ» and calculate the new solution byusing the previous solution as an initial guess. In such a waywe obtain {π(π»), π(π»)} curves by carrying out one cycle inπ» values. We choose a large enough step Ξπ» which does notaffect {π(π»), π(π»)} profiles.
6 Advances in Condensed Matter Physics
0.0 0.5 1.0
0.0
0.5
1.0
β1.0
β1.0
β0.5
β0.5
J = 1
H
π
(a)
0.0
0.5
1.0
β1.0
β0.5
π
0.0 0.5 1.0β1.0 β0.5
H
D = 1
(b)
Figure 1: Mean field approximation. Dependence of π(π») on external fieldπ» through a complete hysteresis cycle. Either no hysteresis, singleloop hysteresis, or double loop hysteresis is observed. (a) π½ = 1; dashed line: π· = 0.3, full line: π· = 0.8, and dotted line: π· = 1.2. (b) π· = 1;full line: π½ = 0.5, dashed line: π½ = 1.2, and dotted line: π½ = 2.
4.1. Mean Field Results. We solve (22) iteratively using theMathematica software [29]. For a specific set of parameterswe stop iterations when the differences |π(π+1) β π
(π)| and
|π(π+1)
β π(π)| between the πth and π + 1th iteration steps are
below 10β5.Qualitatively different solutions on cycling π» are shown
in Figure 1. In Figure 1(a) we varyπ· for π½ = 1 and converselyin Figure 1(b) we vary π½ for π· = 1. In all cases studied weobtain only symmetric solutions; that is, we find that π =
0 within numerical accuracy for all set of parameters andsystem histories. In dependence of the ratio π = π·/π½ weobserve either (i) no hysteresis, (ii) a single hysteresis loop,or (iii) a double hysteresis loop, in π(π»). For relatively lowvalues π we obtain hysteretic behavior, where values of πswitch at a critical value ofπ» = π»
π(π) between π βΌ Β±1. With
increasing values of πdepartures froma rectangularly shapedhysteretic profile become increasingly more pronounced.In addition the width of the hysteresis loop decreases. Onfurther increasing π, a double hysteretic loop begins to occurat π = π
(1)
π(e.g., π(1)
π= π·(1)
π/π½ βΌ 0.7 for π½ = 1). This persists
till π = π(2)
π(e.g., π(2)
π= π·(2)
π/π½ βΌ 1.2 for π½ = 1), beyond which
we observe gradual evolution in π(π») without hysteresis.In Figure 2 we plot the effective energy per site π versus π
for specific values of π· and π»; π½ = 1. The energy iscalculated using (23). In Figure 2(a) we demonstrate the casecorresponding to the coercive fieldπ»
πdefined by (25). In the
case shownπ»π= 0.22 (π»
π= β0.22) for the βupβ (down) part
of the cycle for π· = 0.3. The corresponding hysteresis loopis plotted in Figure 1(a) with the dashed line. For |π»| < |π»
π|
the π(π) dependence exhibits double minima and at π» = π»π
one minimum ceases to exist. In Figure 2(b) we show typicalfree energy landscape in the regime where double hysteresis
loop exists. In the case shown π· = 0.8 and π» = Β±0.09
corresponding to the double hysteresis loop depictedwith thefull line in Figure 1(a).
The observed behavior can be qualitatively understood inthe following way. In the limit π β« 1 the disorder dominantlyinfluences structural behavior if π» < π·. Therefore, forπ» = 0 it is always the case that π = 0 for a large enoughsystem because the disorder enforces isotropic symmetry. Afinite value of π» breaks the symmetry of the system andconsequently π = 0. Because π½ βͺ π· some degree of localdisorder field preference prevails in the regime π· < π».Consequently the collective ordering tendency favored by theinteraction constant π½ is always overwhelmed, either by theexternal field π» or by the local random nematic disorderπ·. Conversely, in the limit π βΌ 0 the collective behaviortendency favored by the constant π½ is large if π½ < π».Therefore,for a relatively weak ordering field π» the system tends tobe aligned along a symmetry breaking direction, which ishistory dependent, giving rise to a pronounced single loophysteresis behavior.
4.2. Lattice Simulation. In the lattice simulation we calculatethe local field tensor defined by (10) and align π
π
πΌπ½parallel
to it. In one sweep we update in such a way all the latticesites of a system. The sweeps are repeated until at each sitethe difference | tr(ππ,new
πΌπ½βππ,oldπΌπ½
)| between the previous (ππ,oldπΌπ½
)
and new (ππ,newπΌπ½
) configuration is below 10β5, signalling thata fixed point structure is reached. From it we calculateorder parameters π and π using (9a) and (9b). We carriedsimulations for the system size 316 Γ 316 (β105) for whichthe finite size effects were negligible.
Advances in Condensed Matter Physics 7
0 1 2
0.0
0.1
0.2
β0.2
β0.3
β0.4
β0.5
β1β2
β1.0
π
π
(a)
0.0 0.5 1.0β1.0 β0.5
β0.407
β0.408
β0.409
β0.410
β0.411
β0.412
β0.413
β0.414
π
π
(b)
Figure 2: The mean field effective energy per site π versus π for different values ofπ» andπ·, π½ = 1, symmetric solutions. (a)π· = 0.3; full line:π» = π»
π= 0.22 obtained for the βupβ part of the single loop hysteresis cycle; dashed line:π»
π= β0.22 obtained for the βdownβ part of the cycle.
Note that the physically sensible regime of π is restricted to the interval [β1, 1]. (b) π· = 0.8; full line:π» = 0.09 obtained for the βupβ part ofthe double loop hysteresis cycle; dashed line:π» = β0.09 obtained for the βdownβ part of the cycle.
0.0 0.5 1.0
0.0
0.5
1.0
β1.0
β0.5
β1.0 β0.5
D = 1
π
H
Figure 3: Lattice simulationπ· = 1: dependence of π(π») on externalfield π» through a complete hysteresis cycle. We observe either (a)π = 4 (dashed line), no hysteresis; (b) π = 0.05 (rectangularlyshaped), single loop hysteresis; (c) π = 0.15 (dash-dotted line),double loop hysteresis; or (d) π = 2, S-shaped single loop hysteresis.
Simulations confirmed our MFA results regarding theabsence of the symmetry breaking solutions. Furthermore, alsoin this case three topologically different responses on cyclingπ» were obtained (i.e., no hysteresis, single hysteresis loop,
or double hysteresis loops). However, hysteresis behavior onvarying the ratio π = π·/π½ exhibits quantitatively and in someregimes also qualitatively different behavior.
Typical behavior on varying π = π·/π½ is shown inFigure 3, where field cycles are calculated for π· = 1. Onincreasing π½ (i.e., decreasing π) we observe the followingbehavior. For π βͺ 1 we obtain roughly rectangularly shapedhysteresis loop, where configurations switch between π =
Β±1. Initially on decreasing π the hysteresis loop width Ξπ» isgradually shrinking towards zero.Within numerical accuracywe obtain Ξπ» = 0 in the interval between π βΌ 0.01 andπ = π
(2)
πβΌ 0.1. For π > π
(2)
πa double hysteresis loop appears
which persists till π(1)π
βΌ 0.22. Above this value we obtain S-shaped hysteresis loopwhereΞπ» gradually decreases. Finally,above π(0)
πβΌ 3.3 no hysteresis is observed.
5. Conclusion
In our numerical study we consider two-dimensional lat-tice model of nematic LC ordering in presence of anexternal ordering field and local disordering random field.We use a generalized Random Anisotropy Nematic modelwhere we introduce an external tensor ordering field ofthe type frequently encountered in continuum mechanics.Our approach resembles work by Shukla and Kharwanlang[17], who examined the shape of hysteresis loops on cyclinga statical external ordering field using random field ππ
model at zero temperature. Despite different symmetries oforder parameters the latter approach and our model yieldsimilar Hamiltonians in angular presentation of orientationalordering.
8 Advances in Condensed Matter Physics
We first calculated ordering properties of generalizedRAN model using the mean field approach. We derived aself-consistent expression for average scalar nematic orderparameters. In general we obtain two qualitatively differentsolutions to which we refer as symmetric solutions andbroken symmetry solutions. In the first type of solutions theprincipal axes of mean nematic tensor order parameter andof the external tensor field coincide. On contrary in thelatter type of solutions the systemβs nematic order parameterpossesses off diagonal terms within the principal frame ofreference imposed by H. For the set of parameters exploredin our simulations we obtained only symmetric solutions, wellrepresented by single scalar order parameter π.
We were particularly interested in hysteresis curvesresulting when the forcing static external field was cycled inthe interval {βπ»max, π»max > 0}, such that the value of π»maxis sufficient to impose a relatively large average alignmentalong e
π₯(eπ¦) for π» = π»max (π» = βπ»max), corresponding
to π βΌ 1 (π βΌ β1). We calculated hysteresis loops at zerotemperature for a fixed value of the ordering interaction anddisorder strength, represented by positive constants π½ andπ·.Both mean field and lattice simulation approaches yieldedqualitatively similar responses on cycling the forcing field.The hysteresis loop was either absent or can consist of singleor double loops. Single loops were obtained for relatively lowratios π·/π½. For π·/π½ βΌ 1, double loops could emerge. Inthis case no hysteresis is observed at small values of π». Forlarge values ofπ» two loops appear in a narrow window ofπ»values for positive and negative values of π», symmetricallyplaced with respect to π» = 0. However, both approachesyield quantitatively different results suggesting that meanfield results well predict only qualitative shape of hysteresisloops.
It is of some interest to pose the question as to why themean field theory is notmore accurate.Themean field theoryhas been constructed, following the work of Shukla andKharwanlang [17], following earlier work on time-dependentsystems by Strogatz et al. [28]. The heuristic philosophyof this disorder-based mean field theory resembles that ofstandard mean field theories in the temperature domain(e.g., Curie-Weiss theory [30], or the well-known Maier-Saupe theory [31] in liquid crystals). There have of coursebeen extensive studies of the circumstances in which thermalmean-field theories might be supposed to be reliable, bothin terms of critical exponents and numerical predictions ofphase transition temperatures. In general this is true forlong-ranged potentials and sufficiently high dimensions, andpresumably this is the case here also, butmore detailed studiesare required for these disorder-based mean field theories. Atthis stage all we can do is to state that the mean field theoriesare heuristically appealing andmight thus be expected to givesome insight into what might be expected.
We note the relationship between our work and a recentpaper involving some of the present authors [32]. This paperconsidered the finite temperature zero field behavior of theπ = 3, π = 3 version of the model under discussionhere, subject to a number of different cooling regimes. Inthis case too, hysteresis and history-dependent properties arefound. There is surely a relation between field-dependent
and temperature-dependent hysteresis, but further studiesare required to elucidate it properly. In [32], the zero-fieldproperties are intimately connected to the nature of longdistance correlations. But here, at zero temperature in afield, we are only concerned with the properties of the totalsystem. It seems likely that having a strong-field history(essentially a field-cooled system), that long-range order ismaintained until the system is reoriented. However, to besure, further size-dependent simulations are necessary, andthe finite temperature hysteresis properties could in principlebe different from the zero-temperature properties even atinfinitesimally small temperatures. A detailed comparisonwith [32] is not currently possible, as the dependence of thevarious properties on π and π is not known at this stage.
The problem posed in this paper is a pilot study ofa three-dimensional liquid crystal in a random field. Theaddition of a third spin and a third spatial dimension presentssome computational challenges. It may change the qualitativeproperties of the hysteresis cycle, in the same way that itdoes change the qualitative properties of the low temperaturephase. The physical problem is that of a liquid crystal ina porous system, in which the volume of the enclosingporous material is a small proportion of the total volume butnevertheless is sufficiently strong to localize the liquid crystal.The hypothesis is that such systems are well described byrandom field systems of some sort. However, much detailedstudy is still required to validate this hypothesis, both interms of comparison between models and comparison withexperiment. The present paper merely indicates that somefeatures of the hysteresis behaviour of systems of liquidcrystals in pores seem to occur in even quite simple randomfield models.
We note from the points above that the primary motiva-tion of this paper is theoretical and concerns the propertiesof random field models. This general problem has attracteda considerable degree of interest. The rather peculiar trace-free maximally biaxial imposed field has been chosen so thatone can cycle reversibly from a positive to a negative fieldand back, in such a way that But one might also ask thequestion as to whether the question as posed finds someexperimental application. In fact random anisotropy modelshave been applied, particularly by Terentjev and coworkers[33β37], and more recently by Lopatina and Selinger [38],to nematic elastomers and related systems. For these cases, abiaxial stress field with the symmetry discussed in this paperis a natural external field, while βshape memoryβ is a naturalconcomitant of the order parameter memory discussed here.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
T. J. Sluckin is grateful to Professor P. Shukla (North-easternHill University, Shillong, India) for continuing discussionand correspondence, to Professor J. V. Selinger (Kent State
Advances in Condensed Matter Physics 9
University) for drawing his attention to [38], and to ProfessorEugene Terentjev (Cambridge) for some remarks and discus-sions on this subject in the dim and distant past. The authorsalso acknowledge discussions with A. Ranjkesh.
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