Eur. Phys. J. E (2013) 36: 122 DOI 10.1140/epje/i2013-13122-0
THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Phase ordering kinetics in uniaxial nematic liquid crystals with second- and fourth-rank interactions Amrita Singh and Shri Singha Department of Physics, Banaras Hindu University, Varanasi-221005, India Received 5 October 2012 and Received in final form 10 July 2013 c EDP Sciences / Societ` Published online: 28 October 2013 – a Italiana di Fisica / Springer-Verlag 2013 Abstract. We present comprehensive results of the Monte Carlo (MC) simulations of the phase ordering dynamics in d = 2 nematic liquid crystals. We study a system of size N 2 (N = 512) with molecules confined to a 512 × 512 square lattice and report the results for two LC Hamiltonians: generalized Lebwohl-Lasher (GLL) model and r−6 dependent anisotropic dispersion interaction potential. In these Hamiltonians a fourth-rank Legendre polynomial P4 interaction term is added to the usual second-rank P2 term. We find that in both the cases the presence of the P4 interaction term significantly influences the nematic domains morphology. Our numerical data show a diffusive growth law with a logarithmic correction: L(t) ∼ (t/ ln t)1/2 .
1 Introduction The study of phase ordering dynamics in nematic liquid crystals (NLCs), that is, the growth of order via domain coarsening when a system is quenched from a homogeneous disordered (isotropic liquid) phase to an ordered broken symmetry (uniaxial nematic) phase, has attracted considerable attention in experiments [1–10], theories [4, 11–24] and computer simulations [25–36]. In this paper, we expand upon our recent works [37, 38] (referred to as I and II, respectively) of Monte Carlo (MC) simulations for the ordering dynamics in d = 2 nematic liquid crystals. In I [37] we presented the results of 2d MC simulations for a system of size N 2 (N = 4096) of rod-like nematic molecules interacting via the two-components LebwohlLasher (LL) Hamiltonian P2 (cos θij ) HLC {S i } = − ij
= −
1 (S i · S j ) − . 2 2
3 ij
2
(1)
In II [38] a system of size N 2 (N = 512) of the LC Hamiltonian, 1 P2 (cos θij ), HLC {S i } = − r n j>i ij 1 3 1 2 (S (2) · S ) − = − i j r n 2 2 j>i ij a
e-mail: [email protected]
was studied. Here represents the coupling strength, θij = θi − θj and rij = |r i − r j |. In both the equations we have considered a set of two-component spins {S i }, where i denotes the sites of a d = 2 square lattice. Each spin is described by an angle θi , where S i = (cos θi , sin θi ). P2 is a second-degree Legendre polynomial and ij indicates that the sum is restricted to nearest-neighbor (nn) pairs of sites. Equation (2) represents the LL Hamiltonian generalized to include long-ranged interactions with index n characterizing the range of interaction; n = ∞ corresponds to the nearest-neighbor (nn) interactions, and n = 0 corresponds to the mean-field limit. In both I and II, we compared our MC results with an analytic form obtained for the correlation function of the LC order parameter and found that the analytic and numerical results are in excellent agreement. It is worth noting that the presence of inversion symmetry (S i → −S i ) in HLC creates defects which drive the ordering kinetics. The MC simulation naturally includes the effects of thermal fluctuations. Using cell dynamics simulations [39], several groups [26–32] have reported interesting results on domain growth in nematic liquid crystals. Blundell and Bray [26] have simulated a spin (O(n)) model in the equal constant approximation for the phase ordering and shown that the tail in the structure factor decays as S(k) ∼ k ν with ν = −4.0 ± 0.1 in 2d and ν = −5.3 ± 0.1 in 3d simulations and that the 3d simulations has a growth exponent φ = 0.44±0.1 and a power law tail χ(= d+n) = 5.3±0.1. Puri et al. [27] studied the zero temperature ordering kinetics of conserved XY model in both the spatial dimensions d = 2 and d = 3 and found the growth law L(t) ∼ t0.25 in 2d and L(t) ∼ (t ln t)0.25 in 3d. Adopting the same scheme of cell dynamics simulations, Zapotocky et al. [28] have
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computed the correlation function, structure factor, energy density and number densities of topological defects in the uniaxial and biaxial nematic phases of a quasi-2d nematic liquid crystal. In this work a 2d sample refers to a film with thickness that is smaller than the equilibrium correlation length prior to the quench. These authors [28] have reported several important results: the correlation function and the structure factor apparently collapse onto scaling curves over a wide range of time showing a growth law L(t) ∼ t−0.407±0.005 . However, the growth exponent obtained from the energy length (φen ) and the defect number densities (φdef ) differ from the correlation growth law. In case of biaxial nematic systems of the four topologically distinct species of defects only two are present in large numbers at late times obeying the growth law with powers of 0.391 ± 0.007 and 0.366 ± 0.007. Using a cell dynamics simulation scheme Dutta and Roy [29] simulated the phase ordering kinetics in a 2d uniaxial nematics and found that asymptotically the growth laws are the same as that of the 2d XY model quenched from above the Kosterlitz-Thouless transition temperature (TKT ), i.e., L(t) ∼ [t/ ln(t/t0 )]1/2 (with nonuniversal time scale t0 ). A large momentum dependence k −4 of the structure factor was observed which establishes the presence of topologically stable 1/2-disclination points. A discrete model was constructed by Toyoki [31,32] to describe the growth kinetics of a 2d system with P 2 symmetry. He performed the numerical simulations to describe the dynamics of nematics in which the topological defects dominate the spatial pattern and found that the struc2 ture factor obeys the scaling law S(k, t) = kt g(k/kt ) where the first moment kt ∼ t−0.42 and that the asymptotic power law tail shows the behavior g(x) ∼ x−4.5 . The effects of the rate of heating and cooling and aligning fields on the nematic-isotropic (NI) transition has been investigated by Verma [33] using a 3dMC simulation confining 103 molecules to a cubic lattice interacting via nn interactions. It has been observed that the rate of quenching of system influences the final configuration appreciably. Khandkar and Barma [34] have studied orientational ordering in a 2d grand canonical system of thin hard rods (needles) using deposition-evaporation MC moves and observed that the ratio κ of deposition to evaporation rates controls the equilibrium density of the rods. An increase in κ leads to entropy driven transition to a nematic phase, in which the orientational correlation functions (both static and dynamic) decay as power law with exponent varying continuously with κ. The values of the critical exponents and the behavior of the orientational cumulant have been found to be consistent with Kosterlitz-Thouless theory. In an interesting work, Billeter et al. [35] have performed MD simulation for the quench of 65536 nematic molecules interacting via a Gay-Berne potential [40]. They have observed twist disclination lines, type-1 lines and monopoles and of the spatial correlation function and the density of disclination lines. In the structure factor simulation they [35] observed the breakdown of dynamical scaling and the crossover from a defect-dominated regime at small values of the wave vector to a thermal-fluctuation-dominated
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regime at large wave vector. In a lattice Boltzmann simulation of a 2d liquid crystal under a quench from isotropic to nematic phase [17,18], it was interesting to find that local nematic order is quickly established, but the domains with the molecules pointing in different directions are separated by the topological defects. The kinetic pathways for the IN transition in a fluid of colloidal hard rods have been studied by Cuetos and Dijkstra [36]. They developed a cluster criterion that distinguishes nematic clusters from the isotropic phase and based on this criterion performed MC simulation and observed spinodal decomposition as well as nucleation and growth depending upon the supersaturation. The purpose of this paper is to present the results of 2d MC simulations for the coarsening dynamics of nematic liquid crystals subjected to a temperature quench. We have performed the simulations for the two interaction potentials in which a fourth-rank Legendre polynomial interaction term (P4 ) is added to the usual secondrank term (P2 ). The influence of fourth-rank term on the domain morphology has been analyzed by comparing the results with and without P4 term in the potential. The manuscript is organized as follows: In sect. 2, we describe, in brief, the interaction models with a particular emphasis on the role of P4 interaction on the orientational and NI transition properties and simulation details are given in sect. 3. The main results and discussion are presented in sect. 4 and finally the paper ends with the conclusions in the last sect. 5.
2 The interaction models The Lebwohl-Lasher (LL) model [41], a lattice version of the Maier-Saupe potential, has proven to be simple yet a powerful potential for the study of orientational properties of liquid crystals and of the characteristic features of NI transition. Each site of the lattice is occupied by a molecule which interacts with the molecules located at its nn sites. The total potential energy takes the form of eq. (1). A large number of studies has been performed on the LL model by the groups in liquid crystals field and also by theoretical physicists interested in the phase transitions of RPN spin model relevant in elementary particle physics [42, 43]. The computer simulation studies carried out on a planar Lebwohl-Lasher (PLL) model, defined on a square lattice, suggest that the PLL model presents a topological defect driven continuous transition of the Berezinskii-Kosterlitz-Thouless (BKT) type [44,45]. Some differences between the transition of the PLL model and that of the 2d XY model have been reported recently [46]. It has been agreed [47, 48] that a complete knowledge of order in a nematic liquid crystal can be obtained by formally expanding the orientational distribution function (ODF) in terms of a complete basis set. Employing the Legendre polynomial PL (cos θ) the orientational pseudopotential was written [47, 48] as u(θ) =
L
u¯L P¯L PL (cos θ),
(3)
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where the prime restricts the summation to the even values of L, θ is the angle between the unique molecular symmetry axis and the director and P¯L represents the Lth rank orientational order parameter. So for a complete knowledge of order of a liquid crystal one requires all rank order parameters. However, for nematics, the P¯L do decrease rapidly with increasing L. For this reason, retaining only the first two terms (L = 2, 4), the pseudo-potential can be approximated as (4) u(θ) = u¯2 [P¯2 P2 (cos θ) + λP¯4 P4 (cos θ)], where λ = u¯4 /u¯2 , is a measure of relative strength of the fourth-rank term. The second-rank order parameter P¯2 is relatively easy to determine. The P¯4 is usually determined by Raman scattering and fluorescence depolarization techniques. Order parameters of higher rank than four are extremely difficult to determine experimentally [49]. So one restricts the discussion to only P¯2 and P¯4 . The NI transition properties have been calculated [48] for various values of λ. It has been observed that the addition of the P4 term (λ > 0) clearly enhances the anisotropy of the pseudo-potential whereas its subtraction (λ < 0) reduces this anisotropy. However, it was very interesting to find that the enhancement of properties by a positive P4 term is greater than the reduction due to a comparable negative P4 term. Assuming a simple generalized Lebwohl-Lasher (GLL) model, incorporating the first two terms P2 and P4 in the interaction potential, the orientational and NI transition properties have been investigated [50–60] and it has been found that the non-negligible P4 term significantly influences the behavior of the system. Hamley et al. [49] studied the orientational ordering in the nematic phase of PAA using small angle neutron scattering method and have shown that only the P2 and P4 terms in the expansion of the single molecule scattering in a basis of Legendre functions are statistically significant. These authors [49] further analyzed that the higher rank term in the expansion were not found to make significant contributions to the scattering patterns. For a GLL model Chiccoli et al. [53] performed an extensive MC simulations for a range of the fourth- to the second-rank relative strengths and have shown that the presence of P4 term in the interaction potential significantly influences the temperature dependence of the order parameters. More importantly, it has been shown that the fourth-rank contributions larger than 20% worsen the agreement of the model with the typical temperature behavior of nematics order. Based on a MD simulations Sandstrom et al. [52] have shown that the knowledge of both P2 and P4 terms improves the ODF estimates. It has also been shown [54, 55] that the presence of P4 term in addition to P2 term in potential makes otherwise weak first-order NI transition stronger first order. The simulations studies [58, 59] for the NI transition in two dimensions using an O(2) vector model characterized by nonlinear nn spin interactions governed by the P4 term show very interesting consequences on the behavior of system. In this work, we perform, in a NVT ensemble, 2d MC simulations for the domain coarsening in a nematic liquid
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crystal of molecules interacting via pair potentials of the form uij = − rij −n [P2 (cos θij ) + λP4 (cos θij )].
(5)
We report the results for the two pair potentials: n = ∞ and n = 6. For n = ∞, eq. (5) represents the generalized Lebwohl-Lasher (GLL) model with nn interactions and n = 6 corresponds to the r−6 dependent anisotropic dispersion interaction. For both the potentials, we have performed 2d MC simulations for a set of three values of λ = 0, 0.1, 0.3. This helps us in analyzing the role of P4 interaction term relative to P2 term on the evolution of the domain morphology, the scaling properties of correlation function, structure factor and the length scale L(t).
3 Details of simulations Adopting Metropolis prescription with periodic boundary conditions, we perform, in a (N, V, T ) ensemble, 2d MC simulations to study the phase ordering dynamics in a system of 262144 molecules that are confined to a 512 × 512 square lattice with a preferred axis and interact via pair potentials (eq. (5)). At t = 0, first we quench the homogeneous isotropic phase to T = 0.2 and perform the simulations for = 1, and two values of n = ∞ and 6 and in each case for three values of fourth-rank interaction parameter λ = 0, 0.1, 0.3. We start with an arbitrary initial configuration and employing the usual Metropolis algorithm the system is allowed to move towards the equilibrium. The energy of the initial configuration is calculated. Now we select randomly a molecule and keeping its position fixed on the lattice point, its orientation is changed randomly by a small amount. We accept or reject the move according to standard Metropolis algorithm. We carry out the above acceptance/rejection step N 2 number of times and this completes one Monte Carlo Sweep (MCS). The periodic boundary conditions are applied in both the X- and Y -directions. The results are averaged over five independent initial configurations and after each 1000 MCS the order parameter is obtained. The simulation for n = 6 has been carried out by taking the cut off distance rc = 2.5 and for n = ∞ accounting the nn interactions. We have studied the domain morphology of the system quenched to nematic order at t = 0 from the isotropic liquid. We simulate several statistical quantities to characterize the domains coarsening. These are described below. Far from equilibrium the evolution of ordering system is usually characterized [61] by the domain growth law and the correlation function or its Fourier transform, the structure factor, which is a measure of domain morphology. The coarsening domains have a single characteristic length scale L(t) which grows as a power law in time, usually as L(t) = tα . The exponent α depends strongly on the nature of defects driving the ordering process, the nature of conservation laws governing the coarsening process and the relevance of hydrodynamic flow fields. This unique length can be obtained as the rescaling factor needed to collapse the correlation function and the structure factor.
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Thus the correlation function C(r, t) exhibits a dynamical scaling property [62] and has the scaling form C(r, t) ≡ φ(R, t)φ(R + r, t) − φ(R, t)φ(R + r, t) r =g , (6) L(t) where φ(r, t) is the nematic order parameter at point r and time t. The angular brackets in eq. (6) represent averaging over independent initial conditions and thermal fluctuation. g(x) is a scaling function which is independent of time. This dynamical scaling property reflects the fact that the morphology of coarsening system is statistically self-similar in time and only changes by a scale factor. Actually, most of the experiments, e.g. neutron or light scattering, probe the time-dependent structure factor, which is the Fourier transform of the real-space correlation function S(k, t) = dr eik·r C(r, t). The corresponding dynamical scaling form for the structure factor is S(k, t) = L(t)d f (k L(t)),
(7)
where d is the dimensionality and the scaling function f (y), given as f (y) = dx eiy·x g(x), is also independent of time. Most of the works on the phase ordering are concerned with the study of the behavior of L(t) and the scaling functions g(x) and f (y).
4 Numerical results In the present paper, we have performed MC simulations of ordering kinetics in d = 2 nematic liquid crystals for the interaction Hamiltonian (eq. (5)) for n = ∞ (nn interactions) and n = 6 (dispersion interaction) and three values of λ (0, 0.1 and 0.3), which measures the strength of the P4 interaction term relative to the P2 interaction. In presenting these results our purpose is twofold: We analyze systematically the effects of the P4 term on the domain coarsening and how its strength influences the characteristic features of the domains growth and scaling behavior. Secondly, we intend to study how the nematic phase ordering dynamics is affected in case of dispersion interaction (n = 6) in comparison with the nn interactions (n = ∞). Here it is worth noting, as mentioned in I and II also, that the Mermin-Wagner theorem states that a system with continuous order parameter and short-ranged interactions does not exhibit long-range order (LRO) at a temperature T > 0 in d = 2. Therefore, as in case of I and II, our initially disordered system (with short-ranged interactions) will “order” towards a final state without LRO when thermal fluctuations are present.
In fig. 1, we show the evolution of domains pattern of the configurations obtained for our MC simulations for the case of n = ∞ at various times, t = 10000 MCS, 50000 MCS and 100000 MCS, following a temperature quench to T = 0.2 at t = 0. In these snapshots panels (a), (b) and (c) correspond to λ = 0 (LL model) and panels (d), (e) and (f) to λ = 0.1 (GLL model). It can be seen that during the same time span in case of GLL model, due to the presence of the P4 term in the interaction potential, domains grow bigger in size in comparison with their growth for the LL model. In both the cases as time advances the nematic domains grow in size and defects regions disappear, but more slowly in case of λ = 0. Comparing the evolution of the domains morphology corresponding to λ = 0.1 and λ = 0.3 (not shown here), we find that the growth of domains is much faster in case of λ = 0.3. This shows the role of the P4 interaction term on the domains coarsening dynamics. Figure 2 shows the evolution snapshots of nematic domains coarsening at various times (10000 MCS, 50000 MCS and 100000 MCS) corresponding to n = 6 (dispersion interaction potential) for λ = 0 (panels (a), (b) and (c)) and λ = 0.1 (panels (d), (e) and (f)). It can be seen that the domains coarsening dynamics is significantly affected by the presence of the P4 interaction term in the Hamiltonian. Next, we simulate the correlation function, structure factor and length scale for both the potentials n = ∞ and 6 with λ = 0.1 and 0.3. The length scale L(t) is defined as the distance over which C(r, t) decays to half its maximum value (C(L(t), t) = 0.5). In fig. 3, we plot C(r, t) vs. r/L for λ = 0.1 at three different times. Figure 3(a) corresponds to the case of n = ∞ and fig. 3(b) to n = 6. In these figures the solid line denotes the results corresponding to λ = 0 [38]. For both the interaction potentials, we find a neat scaling collapse over an extended time, confirming the validity of dynamical scaling. We plot in fig. 4 the scaled structure factor, S(k, t)L−2 vs. kL, for n = 6 with λ = 0.1, corresponding to data set in fig. 3(b). The structure factor has been obtained as the Fourier transform of the correlation function. Here also we find a neat data collapse, confirming the dynamical scaling in the LC ordering. Figure 4 also shows that the structure factor tail obeys the generalized Porod law, S(k) ∼ k −4 , which is obtained from scattering off the cores of the vortex/antivortex defects [13, 63]. Finally, in figs. 5(a) and 6(a) we show the time dependence of L(t) for n = ∞ and 6, respectively. In both the figures, the solid line denotes the Allen-Cahn growth law, L(t) ∼ t1/2 . Our data show a little slower growth than this law. In figs. 5(b) and 6(b), we show the universality of growth law by collapsing the length scale data to a single curve by using 1/2 t , (8) L(t) ∼ L0 τ (n) where τ (n) is defined as the ordering time scale [64] and assumes values: τ (n) = 0.686 for λ = 0.1 and it is 1 for λ = 0 in case of n = 6 (dispersion interaction) and
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(a)
(d)
(b)
(e)
(c)
(f)
Fig. 1. Evolution snapshots of nematic domain coarsening in 2d MC simulations. These snapshots correspond to nn interactions (GLL model) with λ = 0 (panels (a), (b) and (c)) and λ = 0.1 (panels (d), (e) and (f)). Panels (a) and (d) correspond to time 10000 MCS, Panels (b) and (e) to time 50000 MCS and panels (c) and (f) to 100000 MCS.
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(a)
(d)
(b)
(e)
(c)
(f)
Fig. 2. Evolution snapshots of nematic domain coarsening at various times. These snapshots correspond to anisotropic dispersion interaction potential (n = 6 in eq. (5)) with λ = 0 (panels (a), (b) and (c)) and λ = 0.1 (panels (d), (e) and (f)). Panels (a) and (d) correspond to time 10000 MCS, panels (b) and (e) to time 50000 MCS and panels (c) and (f) to 100000 MCS. Simulation details are same as those in fig. 1.
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n=
1
C(r,t)
1
t=10000 MCS 20000 50000
0.8
0.6
0.4
0.4
0.2
0.2 (a) 0
t=10000 MCS 20000 50000
0.8
0.6
0
(b)
0 1
2 r/L
n=6
3
4
0
1
2 r/L
3
4
t=10000(MCS) 20000 50000
1
S(k,t) L
-2
Fig. 3. (a) Scaling plot of correlation function, C(r, t) vs. r/L, for λ = 0.1 and (a) n = ∞; (b) n = 6. We superpose data sets for three different times, as shown in the figure. The solid line denotes the results corresponding to λ = 0.
-4
0.01
1
10 kL
Fig. 4. Scaling plot of the structure factor, S(k, t)L−2 vs. kL, for n = 6 with λ = 0.1. The structure factors are obtained as the Fourier transform of the correlation functions shown in fig. 3(b). The solid line of slope −4 denotes the generalized Porod law: S(k, t) ∼ L2 (kL)−4 in d = 2 [63].
τ (n) = 0.349, 0.643 and 1.0 corresponding to λ = 0.3, 0.1 and 0 respectively for n = ∞ (nn) case. In fig. 7, we plot t/(L2 τ (n)) vs. ln t for the data set of fig. 6(b) (n = 6, λ = 0, 0.1). This plot should be linear in case of diffusive growth law with logarithmic correction. We observe (fig. 7) an approximately linear plot, confirming the
presence of logarithmic correction in the growth law, i.e., L(t) ∼ (t/ ln t)1/2 . From a close inspection of figs. 5, 6 and 7, we observe the following: The strength of the P4 interaction term influences significantly the domains length scale. The average domain size is much larger in case of λ = 0.1 and 0.3 as compared to its value, corresponding to λ = 0. This is obvious from the values which τ (n) assumes corresponding to values of n and λ. As compared to nn interactions significant coarsening of nematic domains occurs in case of dispersion interaction. During the same time span the length scale grows much larger in case of dispersion interaction, which is long-ranged as compared to nn interactions, in both the cases λ = 0 and 0.1. For both the interaction potentials, the inclusion of the P4 term affects the domains coarsening; even the ordering time scale τ (n) is changed due to this term.
5 Conclusions To summarize we would like to focus on the main findings of the present work. Adopting the usual Metropolis prescription with periodic boundary conditions, in this work we have undertaken comprehensive MC simulations for the phase ordering kinetics of nematic liquid crystals in d = 2 of a system of size 512 × 512 molecules confined to a square lattice with preferred axes. We report the results for the two pair potentials: the generalized LebwohlLasher (GLL) model (nn interactions) and the rij −6 dependent anisotropic dispersion interaction potential. The dispersion interaction is long-ranged as compared to nn interactions. These models contain two even order Legendre
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1/2
1
1
(a)
1000
L(t) ()
1/2
10
L(t)
10
10000 t
(b)
1000
10000 t
Fig. 5. (a) Plot of the characteristic length scale, L(t) vs. t, for n = ∞ (nn interactions case) with λ = 0, 0.1 and 0.3. The straight line corresponds to the Allen-Cahn growth law, L(t) ∼ t1/2 . (b) Scaling plot of data in (a).
10
10
L(t)
L(t)()1/2
1/2
(a)
1 1000
10000 t
1 1000
(b)
10000 t
Fig. 6. (a) Plot of the characteristic length scale, L(t) vs. t, for n = 6 (dispersion interaction case) with λ = 0 and 0.1. The straight line corresponds to the Allen-Cahn growth law, L(t) ∼ t1/2 . (b) Scaling plot of data in (a).
polynomial interaction terms P2 (cos θij ) and P4 (cos θij ). We have analyzed the role of the P4 interaction term as well as the range of interaction on the evolution of domains morphology and the scaling properties of correlation function, structure factor and the domains length scale L(t). We have found that in case of both the interaction potentials the domains morphology and its scaling behavior are significantly influenced by the presence of the P4 interaction term in eq. (5) and that the growth law is consistent with the Allen-Cahn law with logarithmic corrections: L(t) ∼ (t/ ln t)1/2 . Such a growth law was first proposed
for the d = 2 XY model and we understand that it applies here also because of the similarity of defects which drive the ordering kinetics.
We gratefully acknowledge useful discussions with Sanjay Puri and Sriram Ramaswamy. We also would like to thank the referees for their comments and suggestions which have been very useful in revising the manuscript. AS is grateful to CSIR, New Delhi and SS to DST, New Delhi for financial support.
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t/(L
2
)
960
920
880
9
lnt
10
11
2
Fig. 7. Plot of t/(L τ ) vs. ln t for n = 6 with λ = 0 and 0.1. We get a similar behavior for n = ∞ case also.
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THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Phase ordering kinetics in uniaxial nematic liquid crystals with second- and fourth-rank interactions Amrita Singh and Shri Singha Department of Physics, Banaras Hindu University, Varanasi-221005, India Received 5 October 2012 and Received in final form 10 July 2013 c EDP Sciences / Societ` Published online: 28 October 2013 – a Italiana di Fisica / Springer-Verlag 2013 Abstract. We present comprehensive results of the Monte Carlo (MC) simulations of the phase ordering dynamics in d = 2 nematic liquid crystals. We study a system of size N 2 (N = 512) with molecules confined to a 512 × 512 square lattice and report the results for two LC Hamiltonians: generalized Lebwohl-Lasher (GLL) model and r−6 dependent anisotropic dispersion interaction potential. In these Hamiltonians a fourth-rank Legendre polynomial P4 interaction term is added to the usual second-rank P2 term. We find that in both the cases the presence of the P4 interaction term significantly influences the nematic domains morphology. Our numerical data show a diffusive growth law with a logarithmic correction: L(t) ∼ (t/ ln t)1/2 .
1 Introduction The study of phase ordering dynamics in nematic liquid crystals (NLCs), that is, the growth of order via domain coarsening when a system is quenched from a homogeneous disordered (isotropic liquid) phase to an ordered broken symmetry (uniaxial nematic) phase, has attracted considerable attention in experiments [1–10], theories [4, 11–24] and computer simulations [25–36]. In this paper, we expand upon our recent works [37, 38] (referred to as I and II, respectively) of Monte Carlo (MC) simulations for the ordering dynamics in d = 2 nematic liquid crystals. In I [37] we presented the results of 2d MC simulations for a system of size N 2 (N = 4096) of rod-like nematic molecules interacting via the two-components LebwohlLasher (LL) Hamiltonian P2 (cos θij ) HLC {S i } = − ij
= −
1 (S i · S j ) − . 2 2
3 ij
2
(1)
In II [38] a system of size N 2 (N = 512) of the LC Hamiltonian, 1 P2 (cos θij ), HLC {S i } = − r n j>i ij 1 3 1 2 (S (2) · S ) − = − i j r n 2 2 j>i ij a
e-mail: [email protected]
was studied. Here represents the coupling strength, θij = θi − θj and rij = |r i − r j |. In both the equations we have considered a set of two-component spins {S i }, where i denotes the sites of a d = 2 square lattice. Each spin is described by an angle θi , where S i = (cos θi , sin θi ). P2 is a second-degree Legendre polynomial and ij indicates that the sum is restricted to nearest-neighbor (nn) pairs of sites. Equation (2) represents the LL Hamiltonian generalized to include long-ranged interactions with index n characterizing the range of interaction; n = ∞ corresponds to the nearest-neighbor (nn) interactions, and n = 0 corresponds to the mean-field limit. In both I and II, we compared our MC results with an analytic form obtained for the correlation function of the LC order parameter and found that the analytic and numerical results are in excellent agreement. It is worth noting that the presence of inversion symmetry (S i → −S i ) in HLC creates defects which drive the ordering kinetics. The MC simulation naturally includes the effects of thermal fluctuations. Using cell dynamics simulations [39], several groups [26–32] have reported interesting results on domain growth in nematic liquid crystals. Blundell and Bray [26] have simulated a spin (O(n)) model in the equal constant approximation for the phase ordering and shown that the tail in the structure factor decays as S(k) ∼ k ν with ν = −4.0 ± 0.1 in 2d and ν = −5.3 ± 0.1 in 3d simulations and that the 3d simulations has a growth exponent φ = 0.44±0.1 and a power law tail χ(= d+n) = 5.3±0.1. Puri et al. [27] studied the zero temperature ordering kinetics of conserved XY model in both the spatial dimensions d = 2 and d = 3 and found the growth law L(t) ∼ t0.25 in 2d and L(t) ∼ (t ln t)0.25 in 3d. Adopting the same scheme of cell dynamics simulations, Zapotocky et al. [28] have
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computed the correlation function, structure factor, energy density and number densities of topological defects in the uniaxial and biaxial nematic phases of a quasi-2d nematic liquid crystal. In this work a 2d sample refers to a film with thickness that is smaller than the equilibrium correlation length prior to the quench. These authors [28] have reported several important results: the correlation function and the structure factor apparently collapse onto scaling curves over a wide range of time showing a growth law L(t) ∼ t−0.407±0.005 . However, the growth exponent obtained from the energy length (φen ) and the defect number densities (φdef ) differ from the correlation growth law. In case of biaxial nematic systems of the four topologically distinct species of defects only two are present in large numbers at late times obeying the growth law with powers of 0.391 ± 0.007 and 0.366 ± 0.007. Using a cell dynamics simulation scheme Dutta and Roy [29] simulated the phase ordering kinetics in a 2d uniaxial nematics and found that asymptotically the growth laws are the same as that of the 2d XY model quenched from above the Kosterlitz-Thouless transition temperature (TKT ), i.e., L(t) ∼ [t/ ln(t/t0 )]1/2 (with nonuniversal time scale t0 ). A large momentum dependence k −4 of the structure factor was observed which establishes the presence of topologically stable 1/2-disclination points. A discrete model was constructed by Toyoki [31,32] to describe the growth kinetics of a 2d system with P 2 symmetry. He performed the numerical simulations to describe the dynamics of nematics in which the topological defects dominate the spatial pattern and found that the struc2 ture factor obeys the scaling law S(k, t) = kt g(k/kt ) where the first moment kt ∼ t−0.42 and that the asymptotic power law tail shows the behavior g(x) ∼ x−4.5 . The effects of the rate of heating and cooling and aligning fields on the nematic-isotropic (NI) transition has been investigated by Verma [33] using a 3dMC simulation confining 103 molecules to a cubic lattice interacting via nn interactions. It has been observed that the rate of quenching of system influences the final configuration appreciably. Khandkar and Barma [34] have studied orientational ordering in a 2d grand canonical system of thin hard rods (needles) using deposition-evaporation MC moves and observed that the ratio κ of deposition to evaporation rates controls the equilibrium density of the rods. An increase in κ leads to entropy driven transition to a nematic phase, in which the orientational correlation functions (both static and dynamic) decay as power law with exponent varying continuously with κ. The values of the critical exponents and the behavior of the orientational cumulant have been found to be consistent with Kosterlitz-Thouless theory. In an interesting work, Billeter et al. [35] have performed MD simulation for the quench of 65536 nematic molecules interacting via a Gay-Berne potential [40]. They have observed twist disclination lines, type-1 lines and monopoles and of the spatial correlation function and the density of disclination lines. In the structure factor simulation they [35] observed the breakdown of dynamical scaling and the crossover from a defect-dominated regime at small values of the wave vector to a thermal-fluctuation-dominated
Eur. Phys. J. E (2013) 36: 122
regime at large wave vector. In a lattice Boltzmann simulation of a 2d liquid crystal under a quench from isotropic to nematic phase [17,18], it was interesting to find that local nematic order is quickly established, but the domains with the molecules pointing in different directions are separated by the topological defects. The kinetic pathways for the IN transition in a fluid of colloidal hard rods have been studied by Cuetos and Dijkstra [36]. They developed a cluster criterion that distinguishes nematic clusters from the isotropic phase and based on this criterion performed MC simulation and observed spinodal decomposition as well as nucleation and growth depending upon the supersaturation. The purpose of this paper is to present the results of 2d MC simulations for the coarsening dynamics of nematic liquid crystals subjected to a temperature quench. We have performed the simulations for the two interaction potentials in which a fourth-rank Legendre polynomial interaction term (P4 ) is added to the usual secondrank term (P2 ). The influence of fourth-rank term on the domain morphology has been analyzed by comparing the results with and without P4 term in the potential. The manuscript is organized as follows: In sect. 2, we describe, in brief, the interaction models with a particular emphasis on the role of P4 interaction on the orientational and NI transition properties and simulation details are given in sect. 3. The main results and discussion are presented in sect. 4 and finally the paper ends with the conclusions in the last sect. 5.
2 The interaction models The Lebwohl-Lasher (LL) model [41], a lattice version of the Maier-Saupe potential, has proven to be simple yet a powerful potential for the study of orientational properties of liquid crystals and of the characteristic features of NI transition. Each site of the lattice is occupied by a molecule which interacts with the molecules located at its nn sites. The total potential energy takes the form of eq. (1). A large number of studies has been performed on the LL model by the groups in liquid crystals field and also by theoretical physicists interested in the phase transitions of RPN spin model relevant in elementary particle physics [42, 43]. The computer simulation studies carried out on a planar Lebwohl-Lasher (PLL) model, defined on a square lattice, suggest that the PLL model presents a topological defect driven continuous transition of the Berezinskii-Kosterlitz-Thouless (BKT) type [44,45]. Some differences between the transition of the PLL model and that of the 2d XY model have been reported recently [46]. It has been agreed [47, 48] that a complete knowledge of order in a nematic liquid crystal can be obtained by formally expanding the orientational distribution function (ODF) in terms of a complete basis set. Employing the Legendre polynomial PL (cos θ) the orientational pseudopotential was written [47, 48] as u(θ) =
L
u¯L P¯L PL (cos θ),
(3)
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where the prime restricts the summation to the even values of L, θ is the angle between the unique molecular symmetry axis and the director and P¯L represents the Lth rank orientational order parameter. So for a complete knowledge of order of a liquid crystal one requires all rank order parameters. However, for nematics, the P¯L do decrease rapidly with increasing L. For this reason, retaining only the first two terms (L = 2, 4), the pseudo-potential can be approximated as (4) u(θ) = u¯2 [P¯2 P2 (cos θ) + λP¯4 P4 (cos θ)], where λ = u¯4 /u¯2 , is a measure of relative strength of the fourth-rank term. The second-rank order parameter P¯2 is relatively easy to determine. The P¯4 is usually determined by Raman scattering and fluorescence depolarization techniques. Order parameters of higher rank than four are extremely difficult to determine experimentally [49]. So one restricts the discussion to only P¯2 and P¯4 . The NI transition properties have been calculated [48] for various values of λ. It has been observed that the addition of the P4 term (λ > 0) clearly enhances the anisotropy of the pseudo-potential whereas its subtraction (λ < 0) reduces this anisotropy. However, it was very interesting to find that the enhancement of properties by a positive P4 term is greater than the reduction due to a comparable negative P4 term. Assuming a simple generalized Lebwohl-Lasher (GLL) model, incorporating the first two terms P2 and P4 in the interaction potential, the orientational and NI transition properties have been investigated [50–60] and it has been found that the non-negligible P4 term significantly influences the behavior of the system. Hamley et al. [49] studied the orientational ordering in the nematic phase of PAA using small angle neutron scattering method and have shown that only the P2 and P4 terms in the expansion of the single molecule scattering in a basis of Legendre functions are statistically significant. These authors [49] further analyzed that the higher rank term in the expansion were not found to make significant contributions to the scattering patterns. For a GLL model Chiccoli et al. [53] performed an extensive MC simulations for a range of the fourth- to the second-rank relative strengths and have shown that the presence of P4 term in the interaction potential significantly influences the temperature dependence of the order parameters. More importantly, it has been shown that the fourth-rank contributions larger than 20% worsen the agreement of the model with the typical temperature behavior of nematics order. Based on a MD simulations Sandstrom et al. [52] have shown that the knowledge of both P2 and P4 terms improves the ODF estimates. It has also been shown [54, 55] that the presence of P4 term in addition to P2 term in potential makes otherwise weak first-order NI transition stronger first order. The simulations studies [58, 59] for the NI transition in two dimensions using an O(2) vector model characterized by nonlinear nn spin interactions governed by the P4 term show very interesting consequences on the behavior of system. In this work, we perform, in a NVT ensemble, 2d MC simulations for the domain coarsening in a nematic liquid
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crystal of molecules interacting via pair potentials of the form uij = − rij −n [P2 (cos θij ) + λP4 (cos θij )].
(5)
We report the results for the two pair potentials: n = ∞ and n = 6. For n = ∞, eq. (5) represents the generalized Lebwohl-Lasher (GLL) model with nn interactions and n = 6 corresponds to the r−6 dependent anisotropic dispersion interaction. For both the potentials, we have performed 2d MC simulations for a set of three values of λ = 0, 0.1, 0.3. This helps us in analyzing the role of P4 interaction term relative to P2 term on the evolution of the domain morphology, the scaling properties of correlation function, structure factor and the length scale L(t).
3 Details of simulations Adopting Metropolis prescription with periodic boundary conditions, we perform, in a (N, V, T ) ensemble, 2d MC simulations to study the phase ordering dynamics in a system of 262144 molecules that are confined to a 512 × 512 square lattice with a preferred axis and interact via pair potentials (eq. (5)). At t = 0, first we quench the homogeneous isotropic phase to T = 0.2 and perform the simulations for = 1, and two values of n = ∞ and 6 and in each case for three values of fourth-rank interaction parameter λ = 0, 0.1, 0.3. We start with an arbitrary initial configuration and employing the usual Metropolis algorithm the system is allowed to move towards the equilibrium. The energy of the initial configuration is calculated. Now we select randomly a molecule and keeping its position fixed on the lattice point, its orientation is changed randomly by a small amount. We accept or reject the move according to standard Metropolis algorithm. We carry out the above acceptance/rejection step N 2 number of times and this completes one Monte Carlo Sweep (MCS). The periodic boundary conditions are applied in both the X- and Y -directions. The results are averaged over five independent initial configurations and after each 1000 MCS the order parameter is obtained. The simulation for n = 6 has been carried out by taking the cut off distance rc = 2.5 and for n = ∞ accounting the nn interactions. We have studied the domain morphology of the system quenched to nematic order at t = 0 from the isotropic liquid. We simulate several statistical quantities to characterize the domains coarsening. These are described below. Far from equilibrium the evolution of ordering system is usually characterized [61] by the domain growth law and the correlation function or its Fourier transform, the structure factor, which is a measure of domain morphology. The coarsening domains have a single characteristic length scale L(t) which grows as a power law in time, usually as L(t) = tα . The exponent α depends strongly on the nature of defects driving the ordering process, the nature of conservation laws governing the coarsening process and the relevance of hydrodynamic flow fields. This unique length can be obtained as the rescaling factor needed to collapse the correlation function and the structure factor.
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Thus the correlation function C(r, t) exhibits a dynamical scaling property [62] and has the scaling form C(r, t) ≡ φ(R, t)φ(R + r, t) − φ(R, t)φ(R + r, t) r =g , (6) L(t) where φ(r, t) is the nematic order parameter at point r and time t. The angular brackets in eq. (6) represent averaging over independent initial conditions and thermal fluctuation. g(x) is a scaling function which is independent of time. This dynamical scaling property reflects the fact that the morphology of coarsening system is statistically self-similar in time and only changes by a scale factor. Actually, most of the experiments, e.g. neutron or light scattering, probe the time-dependent structure factor, which is the Fourier transform of the real-space correlation function S(k, t) = dr eik·r C(r, t). The corresponding dynamical scaling form for the structure factor is S(k, t) = L(t)d f (k L(t)),
(7)
where d is the dimensionality and the scaling function f (y), given as f (y) = dx eiy·x g(x), is also independent of time. Most of the works on the phase ordering are concerned with the study of the behavior of L(t) and the scaling functions g(x) and f (y).
4 Numerical results In the present paper, we have performed MC simulations of ordering kinetics in d = 2 nematic liquid crystals for the interaction Hamiltonian (eq. (5)) for n = ∞ (nn interactions) and n = 6 (dispersion interaction) and three values of λ (0, 0.1 and 0.3), which measures the strength of the P4 interaction term relative to the P2 interaction. In presenting these results our purpose is twofold: We analyze systematically the effects of the P4 term on the domain coarsening and how its strength influences the characteristic features of the domains growth and scaling behavior. Secondly, we intend to study how the nematic phase ordering dynamics is affected in case of dispersion interaction (n = 6) in comparison with the nn interactions (n = ∞). Here it is worth noting, as mentioned in I and II also, that the Mermin-Wagner theorem states that a system with continuous order parameter and short-ranged interactions does not exhibit long-range order (LRO) at a temperature T > 0 in d = 2. Therefore, as in case of I and II, our initially disordered system (with short-ranged interactions) will “order” towards a final state without LRO when thermal fluctuations are present.
In fig. 1, we show the evolution of domains pattern of the configurations obtained for our MC simulations for the case of n = ∞ at various times, t = 10000 MCS, 50000 MCS and 100000 MCS, following a temperature quench to T = 0.2 at t = 0. In these snapshots panels (a), (b) and (c) correspond to λ = 0 (LL model) and panels (d), (e) and (f) to λ = 0.1 (GLL model). It can be seen that during the same time span in case of GLL model, due to the presence of the P4 term in the interaction potential, domains grow bigger in size in comparison with their growth for the LL model. In both the cases as time advances the nematic domains grow in size and defects regions disappear, but more slowly in case of λ = 0. Comparing the evolution of the domains morphology corresponding to λ = 0.1 and λ = 0.3 (not shown here), we find that the growth of domains is much faster in case of λ = 0.3. This shows the role of the P4 interaction term on the domains coarsening dynamics. Figure 2 shows the evolution snapshots of nematic domains coarsening at various times (10000 MCS, 50000 MCS and 100000 MCS) corresponding to n = 6 (dispersion interaction potential) for λ = 0 (panels (a), (b) and (c)) and λ = 0.1 (panels (d), (e) and (f)). It can be seen that the domains coarsening dynamics is significantly affected by the presence of the P4 interaction term in the Hamiltonian. Next, we simulate the correlation function, structure factor and length scale for both the potentials n = ∞ and 6 with λ = 0.1 and 0.3. The length scale L(t) is defined as the distance over which C(r, t) decays to half its maximum value (C(L(t), t) = 0.5). In fig. 3, we plot C(r, t) vs. r/L for λ = 0.1 at three different times. Figure 3(a) corresponds to the case of n = ∞ and fig. 3(b) to n = 6. In these figures the solid line denotes the results corresponding to λ = 0 [38]. For both the interaction potentials, we find a neat scaling collapse over an extended time, confirming the validity of dynamical scaling. We plot in fig. 4 the scaled structure factor, S(k, t)L−2 vs. kL, for n = 6 with λ = 0.1, corresponding to data set in fig. 3(b). The structure factor has been obtained as the Fourier transform of the correlation function. Here also we find a neat data collapse, confirming the dynamical scaling in the LC ordering. Figure 4 also shows that the structure factor tail obeys the generalized Porod law, S(k) ∼ k −4 , which is obtained from scattering off the cores of the vortex/antivortex defects [13, 63]. Finally, in figs. 5(a) and 6(a) we show the time dependence of L(t) for n = ∞ and 6, respectively. In both the figures, the solid line denotes the Allen-Cahn growth law, L(t) ∼ t1/2 . Our data show a little slower growth than this law. In figs. 5(b) and 6(b), we show the universality of growth law by collapsing the length scale data to a single curve by using 1/2 t , (8) L(t) ∼ L0 τ (n) where τ (n) is defined as the ordering time scale [64] and assumes values: τ (n) = 0.686 for λ = 0.1 and it is 1 for λ = 0 in case of n = 6 (dispersion interaction) and
Eur. Phys. J. E (2013) 36: 122
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(a)
(d)
(b)
(e)
(c)
(f)
Fig. 1. Evolution snapshots of nematic domain coarsening in 2d MC simulations. These snapshots correspond to nn interactions (GLL model) with λ = 0 (panels (a), (b) and (c)) and λ = 0.1 (panels (d), (e) and (f)). Panels (a) and (d) correspond to time 10000 MCS, Panels (b) and (e) to time 50000 MCS and panels (c) and (f) to 100000 MCS.
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Eur. Phys. J. E (2013) 36: 122
(a)
(d)
(b)
(e)
(c)
(f)
Fig. 2. Evolution snapshots of nematic domain coarsening at various times. These snapshots correspond to anisotropic dispersion interaction potential (n = 6 in eq. (5)) with λ = 0 (panels (a), (b) and (c)) and λ = 0.1 (panels (d), (e) and (f)). Panels (a) and (d) correspond to time 10000 MCS, panels (b) and (e) to time 50000 MCS and panels (c) and (f) to 100000 MCS. Simulation details are same as those in fig. 1.
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n=
1
C(r,t)
1
t=10000 MCS 20000 50000
0.8
0.6
0.4
0.4
0.2
0.2 (a) 0
t=10000 MCS 20000 50000
0.8
0.6
0
(b)
0 1
2 r/L
n=6
3
4
0
1
2 r/L
3
4
t=10000(MCS) 20000 50000
1
S(k,t) L
-2
Fig. 3. (a) Scaling plot of correlation function, C(r, t) vs. r/L, for λ = 0.1 and (a) n = ∞; (b) n = 6. We superpose data sets for three different times, as shown in the figure. The solid line denotes the results corresponding to λ = 0.
-4
0.01
1
10 kL
Fig. 4. Scaling plot of the structure factor, S(k, t)L−2 vs. kL, for n = 6 with λ = 0.1. The structure factors are obtained as the Fourier transform of the correlation functions shown in fig. 3(b). The solid line of slope −4 denotes the generalized Porod law: S(k, t) ∼ L2 (kL)−4 in d = 2 [63].
τ (n) = 0.349, 0.643 and 1.0 corresponding to λ = 0.3, 0.1 and 0 respectively for n = ∞ (nn) case. In fig. 7, we plot t/(L2 τ (n)) vs. ln t for the data set of fig. 6(b) (n = 6, λ = 0, 0.1). This plot should be linear in case of diffusive growth law with logarithmic correction. We observe (fig. 7) an approximately linear plot, confirming the
presence of logarithmic correction in the growth law, i.e., L(t) ∼ (t/ ln t)1/2 . From a close inspection of figs. 5, 6 and 7, we observe the following: The strength of the P4 interaction term influences significantly the domains length scale. The average domain size is much larger in case of λ = 0.1 and 0.3 as compared to its value, corresponding to λ = 0. This is obvious from the values which τ (n) assumes corresponding to values of n and λ. As compared to nn interactions significant coarsening of nematic domains occurs in case of dispersion interaction. During the same time span the length scale grows much larger in case of dispersion interaction, which is long-ranged as compared to nn interactions, in both the cases λ = 0 and 0.1. For both the interaction potentials, the inclusion of the P4 term affects the domains coarsening; even the ordering time scale τ (n) is changed due to this term.
5 Conclusions To summarize we would like to focus on the main findings of the present work. Adopting the usual Metropolis prescription with periodic boundary conditions, in this work we have undertaken comprehensive MC simulations for the phase ordering kinetics of nematic liquid crystals in d = 2 of a system of size 512 × 512 molecules confined to a square lattice with preferred axes. We report the results for the two pair potentials: the generalized LebwohlLasher (GLL) model (nn interactions) and the rij −6 dependent anisotropic dispersion interaction potential. The dispersion interaction is long-ranged as compared to nn interactions. These models contain two even order Legendre
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1/2
1
1
(a)
1000
L(t) ()
1/2
10
L(t)
10
10000 t
(b)
1000
10000 t
Fig. 5. (a) Plot of the characteristic length scale, L(t) vs. t, for n = ∞ (nn interactions case) with λ = 0, 0.1 and 0.3. The straight line corresponds to the Allen-Cahn growth law, L(t) ∼ t1/2 . (b) Scaling plot of data in (a).
10
10
L(t)
L(t)()1/2
1/2
(a)
1 1000
10000 t
1 1000
(b)
10000 t
Fig. 6. (a) Plot of the characteristic length scale, L(t) vs. t, for n = 6 (dispersion interaction case) with λ = 0 and 0.1. The straight line corresponds to the Allen-Cahn growth law, L(t) ∼ t1/2 . (b) Scaling plot of data in (a).
polynomial interaction terms P2 (cos θij ) and P4 (cos θij ). We have analyzed the role of the P4 interaction term as well as the range of interaction on the evolution of domains morphology and the scaling properties of correlation function, structure factor and the domains length scale L(t). We have found that in case of both the interaction potentials the domains morphology and its scaling behavior are significantly influenced by the presence of the P4 interaction term in eq. (5) and that the growth law is consistent with the Allen-Cahn law with logarithmic corrections: L(t) ∼ (t/ ln t)1/2 . Such a growth law was first proposed
for the d = 2 XY model and we understand that it applies here also because of the similarity of defects which drive the ordering kinetics.
We gratefully acknowledge useful discussions with Sanjay Puri and Sriram Ramaswamy. We also would like to thank the referees for their comments and suggestions which have been very useful in revising the manuscript. AS is grateful to CSIR, New Delhi and SS to DST, New Delhi for financial support.
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t/(L
2
)
960
920
880
9
lnt
10
11
2
Fig. 7. Plot of t/(L τ ) vs. ln t for n = 6 with λ = 0 and 0.1. We get a similar behavior for n = ∞ case also.
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