pubs.acs.org/Langmuir © 2009 American Chemical Society
Nonisothermal Model for the Direct Isotropic/Smectic-A Liquid-Crystalline Transition Nasser Mohieddin Abukhdeir* and Alejandro D. Rey* Department of Chemical Engineering, McGill University, Montr eal, Qu ebec, Canada Received May 4, 2009. Revised Manuscript Received June 24, 2009 An extension to a high-order model for the direct isotropic/smectic-A liquid-crystalline phase transition was derived to take into account thermal effects including anisotropic thermal diffusion and latent heat of phase ordering. Multiscale multitransport simulations of the nonisothermal model were compared to isothermal simulation, showing that the presented model extension corrects the standard Landau-de Gennes prediction from constant growth to diffusionlimited growth under shallow quench/undercooling conditions. Nonisothermal simulations, where metastable nematic preordering precedes smectic-A growth, were also conducted, and novel nonmonotonic phase-transformation kinetics were observed.
Introduction 1
The kinetics of phase transformation is a fundamental subject in material and interfacial science that has a widespread impact on material manufacturing and use. Three fundamental kinetic phenomena associated with phase transformations are microstructure, growth rate, and shape evolution2 (Figure 1). The growth processes associated with phase transformations vary, depending on driving forces such as bulk free-energy minimization and the reduction of interfacial area. Growth laws in diffusive and nondiffusive transformations typically differ in that the former has a conserved order parameter.1,2 The differing kinetics of nondiffusive and diffusive phase transitions are manifested in their growth laws l tn, where l is the characteristic length of the growing domain. Nondiffusive phase transitions exhibit constant growth where l scales linearly (n = 1). Diffusive phase transitions exhibit diffusion-limited growth, where n = 1/2. In the case of liquid crystals, both (mass and thermal) diffusive and (phase-ordering) nondiffusive transformation dynamics are present simultaneously. The study of diffusive transformations in liquid-crystalline materials has mainly focused on mass diffusion. Two main areas of this research involve liquid-crystalline materials where either a small amount of (undesired) impurity is present or where a composite material is desired, such as polymerdispersed liquid crystals3 (PDLCs). In the former case, much work has focused on the study of diffusion-driven growth instabilities4,5 (refs 6 and 7 and chapter B.IX of ref 8) in nematic *E-mail:
[email protected];
[email protected]. (1) Balluffi, R.; Allen, S.; Carter, W. Kinetics of Materials; Wiley-Interscience: Hoboken, NJ, 2005. (2) Sutton, A.; Baluffi, R. Interfaces in Crystalline Materials; Oxford University Press: Oxford, U.K., 1995. (3) Drzaic, P. S. Liquid Crystal Dispersions; World Scientific: Singapore, 1995; Vol. 1. (4) Mullins, W. W.; Sekerka, R. F. J. Appl. Phys. 1963, 34, 323–329. (5) Mullins, W. W.; Sekerka, R. F. J. Appl. Phys. 1964, 35, 444–451. (6) Coriell, S.; McFadden, G.; Sekerka, R. Annu. Rev. Mater. Sci. 1985, 15, 119– 145. (7) Bechhoefer, J. Pattern Formation in Liquid Crystals; Springer: New York, 1996; pp 257-283. (8) Oswald, P.; Pieranski, P. Nematic and Cholesteric Liquid Crystals: Concepts and Physical Properties Illustrated by Experiments; Liquid Crystals Book Series; Taylor & Francis/CRC Press: Boca Raton, FL, 2005. (9) Bechhoefer, J.; Simon, A. J.; Libchaber, A.; Oswald, P. Phys. Rev. A 1989, 40, 2042–2056.
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liquid crystals through both experimental and theoretical approaches.9-16 Higher-order mesophases such as smectic and columnar liquid crystals have been studied, in this context, to a lesser extent.17,18 Much of this interest stems from the fact that liquid-crystal transformations occur on experimentally accessible time scales and are inherently anisotropic. In the latter case, the study of PDLC materials3 has focused on different mechanisms of phase separation19-28 (polymerization, solvent, and thermally induced), where the growth morphology of liquid-crystal-rich domains and their texture are studied in order to control functional physical properties, such as the electro-optical response. Subsequently, PDLCs composed of higher-order liquid crystals have also been studied to a lesser extent.29-31 (10) Simon, A. J.; Libchaber, A. Phys. Rev. A 1990, 41, 7090–7093. (11) Oswald, P. J. Phys. II 1991, 1, 571–581. (12) Bechhoefer, J.; Langer, S. A. Phys. Rev. E 1995, 51, 2356–2362. (13) Misbah, C.; Valance, A. Phys. Rev. E 1995, 51, 1282–1290. (14) Mesquita, O.; Figueiredo, J.; Vidal, A. J. Cryst. Growth 1996, 166, 222–227. (15) Ignes-Mullol, J.; Oswald, P. Phys. Rev. E 2000, 61, 3969–3976. (16) Gomes, O. A.; Falc~ao, R. C.; Mesquita, O. N. Phys. Rev. Lett. 2001, 86, 2577–2580. (17) Gonzalez-Cinca, R.; Ramı´ rez-Piscina, L.; Casademunt, J.; Hernandez-Machado, A.; Toth-Katona, T.; B€orzs€onyi, T.; Buka, A. J. Cryst. Growth 1998, 193, 712–719. (18) Oswald, P.; Pieranski, P. Smectic and Columnar Liquid Crystals; Liquid Crystals Book Series; Taylor & Francis/CRC Press: Boca Raton, FL, 2006. (19) Dorgan, J. R. J. Chem. Phys. 1993, 98, 9094–9106. (20) Amundson, K. R.; Srinivasarao, M. Phys. Rev. E 1998, 58, R1211–R1214. (21) Borrajo, J.; Riccardi, C. C.; Williams, R. J. J.; Siddiqi, H. M.; Dumon, M.; Pascault, J. P. Polymer 1998, 39, 845 –853. (22) Lucchetti, L.; Simoni, F. J. Appl. Phys. 2000, 88, 3934–3940. (23) Harrison, D.; Fisch, M. R. Liquid Crystals 2000, 27, 737 –742. (24) Nakazawa, H.; Fujinami, S.; Motoyama, M.; Ohta, T.; Araki, T.; Tanaka, H.; Fujisawa, T.; Nakada, H.; Hayashi, M.; Aizawa, M. Comput. Theor. Polym. Sci. 2001, 11, 445 –458. (25) Hoppe, C. E.; Galante, M. J.; Oyanguren, P. A.; Williams, R. J. J. Macromolecules 2002, 35, 6324–6331. (26) Hoppe, C.; Galante, M.; Oyanguren, P.; Williams, R. Mater. Sci. Eng., C 2004, 24, 591–594. Selected Papers Presented at Symposia B and F at the Brazilian Society for Materials Research. (27) Hoppe, C. E.; Galante, M. J.; Oyanguren, P. A.; Williams, R. J. J. Macromolecules 2004, 37, 5352–5357. (28) Das, S.; Rey, A. Comput. Mater. Sci. 2006, 38, 325–339. (29) Benmouna, F.; Daoudi, A.; Roussel, F.; Buisine, J.; Coqueret, X.; Maschke, U. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 1841–1848. (30) Benmouna, F.; Daoudi, A.; Roussel, F.; Leclercq, L.; Jean-Marc, B.; Coqueret, X.; Benmouna, M.; Ewen, B.; Maschke, U. Macromolecules 2000, 33, 960–967. (31) Graca, M.; Wieczorek, S.; Holyst, R. Macromolecules 2003, 36, 6903–6913.
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Figure 1. Schematic showing the inter-relationships between the phase-ordering processes. Adapted from Figure 1 of ref 35.
However, thermal diffusion in liquid-crystalline phase transitions has typically been neglected because of the relatively low latent heat contributions of (nematic) liquid-crystalline transformations and the pervasive use of thin film geometries7 (where evolved latent heat escapes in the vertical dimension). Focusing on pure/single-phase liquid crystals, where impurity concentrations are below the saturation limit, evidence has shown that latent heat effects can also result in diffusive dynamics.32 Thus, heat diffusion must be taken into account to reproduce experimentally observed growth laws under these conditions.33,34 As the study of these nonisothermal effects progresses,32,35,36 it is becoming clear that they play a non-negligible role in the complex kinetics and dynamics of liquid-crystalline phase transitions. As previously mentioned, a relatively large amount of effort in the study of liquid crystals has focused on those that exhibit some degree of orientational order, or nematics. Higher-order mesophases that exhibit some degree of positional order, in addition to orientational, have been less studied. The simplest of these higherorder liquid crystals is the smectic-A mesophase, which exhibits lamellar translational ordering in addition to the orientational ordering of nematics. Recently, an increasing amount of interest in this mesophase, in particular, of materials exhibiting a direct isotropic/smectic-A (disordered/ordered) transition, has resulted in many experimental and theoretical results. Nonetheless, the understanding of this mesophase is in a nascent stage. Much of this is due to the time and length scales at which the structures and dynamics occur being on the nanoscale. The additional positional order of smectics, where the underlying primary order is orientational, allows the possibility of dual-order kinetics and dynamics. This has been experimentally observed in a smectic polymer liquid crystal that exhibits extremely slow dynamics.37 The presence of a transient metastable nematic phase preceding the stable smectic phase in this polymer liquid crystal was observed under certain quench conditions.38 The general phenomenon of metastable states in phase transitions involving dual nonconserved order was theoretically shown over a decade ago,39 but this generalized approach is not suitable for liquid-crystal phase transitions (Figure 1). A first approximation of the experimental system,37 taking into account shape kinetics/growth kinetics/texturing (Figure 1), has been modeled,40 shedding light on some of the nanoscale phenomena resulting in these experimental observations. In this context, the kinetics of liquid-crystal phase transitions, in particular, smectics, becomes more complex and diverse than (32) Huisman, B. A. H.; Fasolino, A. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2007, 76, 021706. (33) Dierking, I. Appl. Phys. A: Mater. Sci. Process. 2001, 72, 307–310. (34) Dierking, I.; Russell, C. Physica B 2003, 325, 281–286. (35) Abukhdeir, N.; Soule, E.; Rey, A. Langmuir 2008, 24, 13605–13613. (36) Soule, E. R.; Abukhdeir, N. M.; Rey, A. D. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2009, 79, 021702. (37) Tokita, M.; Kim, K.-W.; Kang, S.; Watanabe, J. Macromolecules 2006, 39, 2021–2023. (38) Tokita, M.; Funaoka, S.-i.; Watanabe, J. Macromolecules 2004, 37, 9916– 9921. (39) Bechhoefer, J.; L€owen, H.; Tuckerman, L. S. Phys. Rev. Lett. 1991, 67, 1266–1269. (40) Abukhdeir, N.; Rey, A. Arxiv preprint 2009, arXiv:0902.1544. Accepted for publication in Liquid Crystals 09MAR2009.
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currently regarded where coupled nondiffusive (phase-ordering), diffusive (mass/thermal), and preordering transformation dynamics are present. The general objective of this work is to contribute to the fundamental understanding of mesophase formation under simultaneous orientational and positional symmetry breaking and under non-negligible latent heat evolution. The specific objectives of this work are (1) to extend an existing high-order isothermal model for the direct isotropic/smectic-A transition to account for thermal effects and (2) to conduct a basic study of the phase-transition landscape: (1) Derive an energy balance to extend an existing highorder model to the isotropic/smectic-A phase transition41,42 to account for nonisothermal effects including latent heat of phase ordering and thermal diffusion. (2) Simulate a 1D growing smectic-A front, using the extended model, to show that the nonisothermal extension corrects the standard Landau-de Gennes prediction of volume-driven growth kinetics (l t) to diffusion-limited growth kinetics (l t1/2) under shallow quench/undercooling conditions, consistent with experimental observations.18,33,34 (3) Simulate a 1D growing smectic-A front using the nonisothermal model under conditions where metastable nematic preordering is observed.43 This work neglects nucleation mechanisms,44 fluctuations,45,46 impurities,3,8 and convective flow46 while taking into account energetically the intercoupling between orientational/translational order and the variation of smectic layer spacing. This work is organized into effectively three sections: background/ model, energy balance derivation, and simulation. The background/model section provides a brief introduction to relevant types of liquid-crystal phase ordering and the Landau-de Gennes-type model of Mukherjee, Pleiner, and Brand41,42 used in this work. The derivation of the thermal energy balance extension to this model is then presented. Finally, simulation results are presented and discussed, followed by a summary of the conclusions.
Background and Model Liquid-Crystal Order. As mentioned above, liquid-crystalline phases or mesophases are materials that exhibit partial orientational and/or translational order. They are composed of anisotropic molecules that can be disklike (discotic) or rodlike (calamitic) in shape. Thermotropic liquid crystals are typically pure-component compounds that exhibit mesophase ordering most greatly in response to temperature changes. Lyotropic liquid crystals are mixtures of mesogens (molecules that exhibit some form of liquid crystallinity), possibly with a solvent, that most greatly exhibit mesophase behavior in response to concentration changes. Effects of pressure and external fields also influence mesophase behavior. This work focuses on the study of calamitic thermotropic liquid crystals that exhibit a first-order mesophase transition. (41) de Gennes, P.; Prost, J. The Physics of Liquid Crystals, 2nd ed.; Oxford University Press: New York, 1995. (42) Mukherjee, P. K.; Pleiner, H.; Brand, H. R. Eur. Phys. J. E: Soft Matter 2001, 4, 293–297. (43) Abukhdeir, N. M.; Rey, A. D. Macromolecules 2009, 42, 3841–3844. (44) Ziabicki, A. Multidimensional Theory of Crystal Nucleation. In Mathematical Modelling for Polymer Processing: Polymerization, Crystallization, Manufacturing; Capasso, V., Ed.; Springer: Berlin, 2003; pp 59 -115. (45) Singh, S. Liquid Crystals: Fundamentals; World Scientific: Singapore, 2002. (46) Pleiner, H.; Brand, H. R. Pattern Formation in Liquid Crystals; Springer: New York, 1996; pp 15-57.
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Figure 2. Schematics of the (a) isotropic, (b) nematic, and (c) smecticA phases.
An unordered liquid, where there is neither orientational nor translational order (apart from an average intermolecular separation distance) of the molecules, is referred to as isotropic. Liquidcrystalline order involves partial orientational order (nematics) and, additionally, partial translational order (smectics and columnar mesophases). The simplest of the smectics is the smectic-A mesophase, which exhibits 1D translational order in the direction of the preferred molecular orientational axis. It can be thought of as layers of 2D fluids stacked on each other. A schematic representation of these different types of ordering is shown in Figure 2. Theoretical characterization of orientational and translational order of the smectic-A mesophase (Figure 2) is accomplished using order parameters that adequately capture the physics involved. Partial orientational order of the nematic phase is characterized using a symmetric traceless quadrupolar tensor41 1 1 ð1Þ Q ¼ S nn - I þ Pðmm -llÞ 3 3 where n/m/l are the eigenvectors of Q, which characterize the average molecular orientational axes, and S/P are scalars that characterize the extent to which the molecules conform to the average orientational axes.47-49 Uniaxial order is characterized by S and n, which correspond to the maximum eigenvalue (and its corresponding eigenvector) of Q, S = 3/2μn. Biaxial order is characterized by P and m/l, which correspond to the lesser eigenvalues and eigenvectors, P = 3/2(μm - μl). The 1D translational order of the smectic-A mesophase in addition to the orientational order found in nematics is characterized through the use of primary (orientational) and secondary (translational) order parameters together.50 A complex order parameter can be used to characterize translational order41 Ψ ¼ ψeiφ ¼ A þ iB
ð2Þ
where φ is the phase, ψ is the scalar amplitude of the density modulation, and A/B is the real/imaginary component of the complex order parameter. The density wave vector, which describes the average orientation of the smectic-A density modulation, is defined as a = rφ/|rφ|. The smectic scalar order parameter ψ characterizes the magnitude of the density modulation and is used in a dimensionless form in this work. In the smectic-A mesophase, the preferred orientation of the wave vector is parallel to the average molecular orientational axis, n. Model for the Direct Isotropic/Smectic-A Transition. A two-order-parameter Landau-de Gennes model for the first order isotropic/smectic-A phase transition is used that was (47) Rey, A.; Denn, M. Annu. Rev. Fluid Mech. 2002, 34, 233-266. (48) Yan, J.; Rey, A. D. Phys. Rev. E 2002, 65, 031713. (49) Rey, A. D. Soft Matter 2007, 3, 1349–1368. (50) Toledano, J.-C.; Toledano, P. The Landau Theory of Phase Transitions: Application to Structural, Incommensurate, Magnetic, and Liquid Crystal Systems; World Scientific Lecture Notes in Physics; World Scientific: Singapore, 1987.
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initially presented by Mukherjee, Pleiner, and Brand41,42 and later extended by adding nematic elastic terms51,52 1 1 1 f -f0 ¼ aðQ : QÞ - bðQ 3 QÞ : Q þ cðQ : QÞ2 2 3 4 1 1 1 2 4 2 þ RjΨj þ βjΨj - δjΨj ðQ : QÞ 2 4 2 1 1 1 - eQ : ðrΨÞðrΨ/ Þ þ l1 ðrQÞ2 þ l2 ðr 3 QÞ2 2 2 2 1 1 1 þ l3 Q : ðrQ : rQÞ þ b1 jrΨj2 þ b2 jr2 Ψj2 ð3Þ 2 2 4 a ¼ a0 ðT -TNI Þ; R ¼ R0 ðT -TAI Þ
ð4Þ
where f is the free-energy density, f0 is the free-energy density of the isotropic phase, terms 1-5 are the bulk contributions to the free energy and terms 6 and 7 are couplings of nematic and smectic order;both the bulk order and coupling of the nematic director and smectic density-wave vector, respectively. Terms 8-10/1112 are the nematic/smectic elastic contributions to the free energy. T is temperature, TNI/ TAI are the hypothetical second-order transition temperatures for isotropic/nematic and isotropic/smectic-A mesophase transitions (refer to ref 53 for more detail), and the remaining constants are phenomenological parameters. This free-energy density expression is a real-valued function of the complex order parameter Ψ and its complex conjugate Ψ*, which makes it convenient to reformulate using the real and imaginary parts of the complex order parameter (eq 2). As previously mentioned, the dual-order nature of the isotropic/smectic-A liquid-crystal transition has been shown to result in metastable nematic preordering. Figure 3 shows a schematic of this phenomenon, where a metastable nematic front precedes the stable smectic-A front, growing into an isotropic matrix phase. Under standard conditions, metastable nematic preordering is not observed, but when the dynamic time scales between nematic/ orientational and smectic-A/translational ordering differ greatly, metastable nematic preordering is observed and predicted by the model based on eq 3.37,39,43,67 The previously mentioned dynamic time scales are included in this model via the Landau-Ginzburg time-dependent formulation.55 The general form of the time-dependent formulation is as follows55 10 1 0 1 0 1 δF DQ 0 0 B C B δQ C C B Dt C B μn CB B DA C B B δF C B C ¼B 0 1 0 C B C ð5Þ CB B Dt C B μS C@ δA C @ A @ A DB 1A δF 0 0 Dt μS δB Z ð6Þ F ¼ f dV V
where μn/μs is the rotational/smectic viscosity and V is the control volume. A higher-order functional derivative must be used because of the second-derivative term in the free-energy 0 equation 1 (eq 3) ! δF Df D Df D D @ Df A þ ð7Þ ¼ 2θ Dθ δθ Dθ Dxi D Dx Dx i Dxj D DxD Dx i i
j
(51) Brand, H. R.; Mukherjee, P. K.; Pleiner, H. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2001, 63, 061708/1–061708/6. (52) Mukherjee, P. K.; Pleiner, H.; Brand, H. R. J. Chem. Phys. 2002, 117, 7788– 7792. (53) Coles, H. J.; Strazielle, C. Mol. Cryst. Liq. Cryst. 1979, 55, 237–50. (54) Coles, H. J.; Strazielle, C. Mol. Cryst. Liq. Cryst. 1979, 49, 259–64. (55) Barbero, G.; Evangelista, L. R. An Elementary Course on the Continuum Theory for Nematic Liquid Crystals; Series on Liquid Crystals; World Scientific: Singapore, 2000; Vol. 3.
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which simplifies to dW Df DrQ Df DrA ¼ l þ dt DrQ Dt DrA 3 Dt þ
! Df DrrB Df DQ : þ r3 : þ DrrB Dt DrQ Dt
Figure 3. Schematic of 1D direct isotropic/smectic-A growth under conditions where metastable nematic preordering is present. The light/dark-gray shading indications the magnitude of the nematic/smectic-A scalar order parameter S/ψ growing in an unstable isotropic phase. Arrows indicate the growth direction, where the metastable nematic front grows independently from the trailing smectic-A front. The increase in the nematic scalar order parameter is due to coupled enhancement from the presence of smectic order.42,53,54
where θ corresponds to the order parameter. Thermal Energy Balance Derivation. The energy balance for a differential volume, without flow, is du dW ¼ -r 3 q ð8Þ dt dt where u is the internal energy density, q is the heat flux, and dW/dt is the rate of mechanical work on the differential control volume. According to eq 4.46 of ref 56, the rate of mechanical work on the system can be written as the sum of elastic energy stored and the rate of dissipation. The rate of elastic energy storage is the change in nematic/smectic-A energy due to changes in the order parameters, so it is ∂f/∂t taken at constant temperature. The rate of dissipation in the absence of flow is equal to57 DQ DA DB þ HA þ HB ð9Þ HQ : Dt Dt Dt where A/B is the real/imaginary part of the complex order parameter (Ψ), the molecular fields HQ, HA, and HB are the negative of the variational derivatives of the free-energy functional with respect to Q, A, and B . With no flow, the Landau-de Gennes model gives (symmetric-traceless contributions are implied throughout this work)
! Df Df DA þ r3 -rr : DrA DrrA Dt
! Df Df DB -rr : þ r3 DrB DrrB Dt
DQ Df Df ¼ þ r3 Dt DQ DrQ DA Df Df Df HA ¼ μs ¼ þ r3 -rr : Dt DA DrA DrrA DB Df Df Df HB ¼ μs ¼ þ r3 -rr : ð10Þ Dt DB DrB DrrB
Expanding the time derivatives yields Df Df DT Df DQ Df DrQ ¼ þ : þ l Dt DT Dt DQ Dt DrQ Dt Df DA Df DrA Df DrrA þ þ : DA Dt DrA 3 Dt DrrA Dt Df DB Df DrB Df DrrB þ þ : þ DB Dt DrB 3 Dt DrrB Dt
þ
-T
þ þ
D2 f DQ D2 f DA D2 f DB -T : þ þ DQDT Dt DADT Dt DBDT Dt
ð15Þ
where it has been taken into account in eq 15 that all gradient terms in the free energy f are assumed to be temperatureindependent. Additionally, the first term in eq 15 includes the specific heat (per unit volume): Cp ¼ -T
D2 f DT 2
du DT D2 f DQ D2 f DA D2 f DB ¼ Cp -T : þ þ dt Dt DQDT Dt DADT Dt DBDT Dt ð11Þ
(56) Stewart, I. The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction; Taylor & Francis: London, 2004. (57) Qian, T.; Sheng, P. Phys. Rev. E 1998, 58, 7475–7485.
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!
ð16Þ
Substituting eqs 14-16 into eq 13 yields the total change in internal energy:
Df DrrA Df DB Df DrB : þ þ DrrA Dt DB Dt DrB 3 Dt
Df DrrB DQ DA DB : þ HQ : þ HA þ HB DrrB Dt Dt Dt Dt
ð14Þ
D2 f D2 f DT ¼ -T 2 DtDT DT Dt
The rate of mechanical work is thus dW Df DQ Df DrQ Df DA Df DrA ¼ : þ l þ þ dt DQ Dt DrQ Dt DA Dt DrA 3 Dt
ð12Þ
The left-hand side of eq 8 can be written in terms of the Helmholtz free energy: du D Df Df Df DT D2 f ¼ f -T ¼ -T ð13Þ dt Dt DT Dt DT Dt DtDT
H Q ¼ μn
Df DrrA Df DrB : þ DrrA Dt DrB 3 Dt
þ þ
!
Df DQ Df DrQ Df DA Df DrA : þ l þ þ DQ Dt DrQ Dt DA Dt DrA 3 Dt
Df DrrA Df DB Df DrB Df DrrB : þ þ þ : DrrA Dt DB Dt DrB 3 Dt DrrB Dt ð17Þ Langmuir 2009, 25(19), 11923–11929
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Finally, substituting the final expression for the total change in internal energy (eq 17) and rate of mechanical work (eq 12) into the energy balance (eq 8) yields DT DQ DQ ¼ μn : þ μs Cp Dt Dt Dt
( 2 ) DA 2 DB þ Dt Dt
! D2 f DQ D2 f DA D2 f DB : þ þ -r 3 q þT DQDT Dt DADT Dt DBDT Dt
ð18Þ
This can be further simplified by taking the derivatives of the freeenergy density (eq 3) DT DQ DQ ¼ μn : þ μs Cp Dt Dt Dt
(
DA Dt
2
2 ) DB þ Dt
! 1 DQ 1 DA DB þ R0 A þB -r 3 q þ T a0 Q : 2 Dt 2 Dt Dt
ð19Þ
where terms 1 and 2 correspond to orientational/translational dissipation, respectively, term 3 corresponds to the energy of phase ordering, and term 4 is the heat flux. A constitutive relationship for the heat flux that can be used is the anisotropic Fourier law q ¼ -K 3 rT
ð20Þ
where the thermal conductivity tensor K is used because of the anisotropy of the smectic phase. The thermal conductivity tensor can be written as the sum of isotropic and anisotropic contributions k þ 2k^ δ þ ðk -k^ ÞQ K ¼ kiso δ þ kan Q ¼ 3 )
)
ð21Þ
)
where kiso and kan are the isotropic and anisotropic contributions to the thermal conductivity and k and k^ are the conductivities in the directions parallel and perpendicular to the average orientational axis (nematic director), respectively. Whereas eq 21 takes into account the dependence of thermal conductivity on the degree of orientational order,58,59 the effect of the translation ordering of the smectic-A phase on the thermal conductivity is neglected on the basis of past experimental work. This work determined that the shape anisotropy of the mesogens (liquid-crystal molecules) is the main contribution to the thermal transport anisotropy compared to smectic layering.60 Equations 3, 5, 18, and 21 constitute the model system of equations. Simulation Conditions. One-dimensional simulation of the model was performed using the Galerkin finite element method (Comsol Multiphysics). Quadratic Lagrange basis functions were used for the Q-tensor variables/temperature, and quartic Hermite basis functions were used for the complex order parameter components. Standard numerical techniques were utilized to ensure the convergence and stability of the solution including an adaptive backward-difference formula implicit time-integration method. A uniform mesh was used such that there was a density of 6 nodes/ (58) Zammit, U.; Marinelli, M.; Pizzoferrato, R.; Scudieri, F.; Martellucci, S. Phys. Rev. A 1990, 41, 1153–1155. (59) Marinelli, M.; Mercuri, F.; Foglietta, S.; Zammit, U.; Scudieri, F. Phys. Rev. E 1996, 54, 1604–1609. (60) Rondelez, F.; Urbach, W.; Hervet, H. Phys. Rev. Lett. 1978, 41, 1058–1062.
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Figure 4. Schematic of the 1D simulation domain where the initial smectic-A nucleus grows (from left to right) into the unstable isotropic phase. Both the density modulation (Re(Ψ), solid line) and scalar nematic order parameter (S, dotted line) are shown. The scalar smectic-A order parameter ψ can be calculated √ using both components of the complex order parameter ψ = (A2 + B2).
equilibrium smectic-A layer in the liquid-crystal computational domain of 500 layers (approximately 2 μm). A coupled thermal domain, where only transient thermal diffusion was solved for, leveraged the fact that the characteristic thermal length scale (on the order of μm) is much greater than that of the characteristic smecticA/metastable nematic length scale61 (on the order of nanometers). Neumann boundary conditions were used at both boundaries, and the initial condition consisted of uniform temperature with a single smectic-A layer nucleus. Additionally, a correction factor of Θ = 0.96 was used to normalize the slight difference between the latent heat value predicted by the phenomenological model parameters used (term 3 of eq 18) and an experimentally determined approximate reference value of ΔHexp = 485 kJ/mol.62 The latent heat predicted by the model and phenomenological parameters (determined previously61) is ΔH ¼ Tb
Df ðTb Þ 1 1 ¼ Tb a0 Sb 2 þ R0 ψb 2 DT 3 2
ð22Þ
where Tb is the bulk isotropic/smectic-A transition temperature based on an approximate reference value experimentally determined53 to be 331.35 K. Computational limitations constrain simulations to 1D in order to resolve the multiple length scales of smectic-A ordering (nanometers) and thermal diffusion (micrometers). Because of the use of the full complex order parameter (eq 2), an additional constraint imposed on 1D simulation requires that the smectic-A layers are parallel to the interface (imposed via initial/boundary conditions). A growing front where the smectic layers are perpendicular to the interface is inherently 2D and cannot be captured using 1D simulation.
Results and Discussion Simulations were performed for three different cases: a shallow isothermal quench (neglecting the thermal energy balance, T = 331 K), a shallow nonisothermal quench (T = 331 K), and a deep nonisothermal quench (T = 330 K) where metastable nematic preordering is predicted to be present (the time scales of nematic and smectic-A ordering differ greatly43). Shallow quench conditions describe the situation where the domain is quenched below the bulk (61) Abukhdeir, N.; Rey, A. Arxiv preprint arXiv 2008, arXiv:0807.4525. Accepted for publication in Communications in Computational Physics 12MAR2009. (62) Oweimreen, G. A.; Morsy, M. A. Thermochim. Acta 2000, 346, 37–47.
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Figure 5. Plots of the (a) power law fit exponent (n in l tn) and (b) front velocity versus interface position for both the isothermal (solid line, T = 331 K and μs/μn = 25) and nonisothermal (dashed line, T = 331 K and μs/μn= 25) simulations. The material parameters and phenomenological coefficients, based upon 12CB,61 are TNI = 322.85 K, TAI = 330.5 K, a0 = 2 105 J/m3 K, b = 2.823 107 J/m3, c = 1.972 107 J/m3, R0 = 1.903 106 J/m3 K, β = 3.956 108 J/m3, δ = 9.792 106 J/m3, e = 1.938 10-11 pN, l1 = 1 10-12 pN, l2 = 1.033 10-12 pN, b1 = 1 10-12 pN, b2 = 3.334 10-30 J m, and μn = 8.4 10-2 N s/m2. The thermal material parameters used are based on values for 9CB and 10CB66 (12CB data was not available, Tref = 330.0 K for thermal conductivity values): k = 0.3100 W/mK, k^ = 0.1300 W/mK, Cp = 2600 J/kg K, and F = 1000 kg/m3.
isotropic/smectic-A transition temperature Tb but above the lower stability limit of the isotropic phase TAI. Subsequently, deep quench conditions describe a quench temperature below both Tb and TAI. As previously mentioned, the vast majority of studies of diffusive dynamics of liquid-crystalline materials are focused on mass transport. However, past experimental work was conducted by Dierking et al. on growth laws of different types of purecomponent liquid-crystal mesophases,33,34,63,64 including a study of the direct isotropic/smectic-A transition.34 The culmination of this work, following the findings of Huisman and Fassolino,32 includes a generalized nonisothermal model65 showing the growth law evolution as a function of quench depth. This evolution transitions from purely nonconserved growth dynamics L t at deep quenches/undercooling to thermal diffusion-limited growth L t1/2 under shallow quench/undercooling conditions. The presented extension (eq 18) of the high-order model of Mukherjee, Pleiner, and Brand41,42 to account for latent heat effects allows for the resolution of the minimum physics (Figure 1) to describe the direct isotropic/smectic-A transition of a purecomponent liquid crystal. Figure 5a,b shows the computed evolution of the power law exponent fit (l tn) and interface velocities for the first two simulation cases (isothermal and nonisothermal) versus the position of the growing interface. Figure 5a shows that the inclusion of latent heat effects corrects the standard Landau-de Gennes model prediction of constant growth to that consistent with experimental observations of this specific system34 and other liquid-crystal mesophases.33,34,63,64 Figure 5b shows that in addition to the correction of the growth law, the trend and magnitude of the interface velocity are modified. The initial relative increase in the front velocity is due to the fact that in this work nucleation effects are neglected and the initial nucleus size and degree of liquid-crystalline ordering are assumed to be at bulk equilibrium values. As the stable smectic-A nucleus grows into the unstable isotropic phase, there is an initial decrease in the degree of smecticA order as predicted by the high-order model. Due to the inclusion of latent heat effects, this decrease in smectic-A ordering requires thermal energy, effectively resulting in local cooling. (63) Bronnikov, S.; Dierking, I. Phys. Chem. Chem. Phys. 2004, 6, 1745–1749. (64) Bronnikov, S.; Dierking, I. Physica B 2005, 358, 339–347. (65) Chan, H.; Dierking, I. Phys. Rev. E 2008, 77, 31610.
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The interface velocity can be approximated by2 βw ¼ ΔFðTs -T / Þ -C
ð23Þ
where β is the interfacial viscosity, ΔF(TS - T*) is the free-energy driving force (free-energy difference between the ordered/disordered phases), and C is the capillary force. Ts and T* are the interface and transition temperatures, respectively. Thus, any local cooling/heating has a subsequent effect of increasing/decreasing the interface velocity. Figure 6a,b shows the evolution of the power law exponent fit (l tn) and interface velocities for the third simulation case (nonisothermal with metastable nematic preordering) versus the position of the growing interface. Under conditions where metastable nematic preordering occurs, the effect of latent heat results in complex phase-ordering dynamics observed in other material systems.68 In the present case, under deep quench/undercooling conditions, volume-driven growth is expected for the smectic-A phase. Conversely, any metastable nematic preordering is under effectively shallow quench conditions. (See Figure 2a of ref 43 for the free-energy landscape.) As is seen in Figure 6a, the smectic-A front initially undergoes constant volume-driven growth preceded by a metastable nematic front that rapidly exhibits diffusion-limited dynamics. As the metastable nematic front velocity decreases (Figure 6b), it is overtaken by the smectic-A front. This nonmonotonic acceleration/deceleration/acceleration behavior is similar to dual phase-ordering/chemical-demixing growth kinetics recently simulated in nematic PDLC systems.68 A unique differentiation in the growth dynamics as compared to those observed in the phase-ordering/chemical-demixing system is that the evolution of latent heat from the trailing smectic-A front contributes to the isotropic/metastable nematic interface temperature, which results in a small time period where the metastable nematic front begins to melt/recede (shown in Figure 6b). Figure 7 shows the interface temperature versus position of the growing interface for all three simulation cases. Inclusion of the latent heat contribution from phase ordering causes the interfacial (66) Pestov, S. In Landolt-Bernstein - Group VIII Advanced Materials and Technologies; Springer: Berlin, 2003; Chapter 2.1.1 Two ring systems without a bridge. (67) Tuckerman, L. S.; Bechhoefer, J. Phys. Rev. A 1992, 46, 3178–3192. (68) Soule, E. R.; Rey, A. D. Europhys Lett. 2009, 86, 46006.
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Figure 6. Plots of the (a) power law fit exponent (n in l tn) and (b) front velocity versus interface position for the nonisothermal simulation where metastable nematic preordering is present (solid line for the smectic-A front and dashed line for the nematic front, T = 330 K and μs/μn = 250).The inset of plot b shows the full scale of the nematic front velocity.
front velocity is not accessible because of computational limitations. The interface velocity is predicted to converge to a constant value as as result of the deep quench conditions and assumption of infinite volume. These interesting dynamics suggest the possibility of oscillatory splitting/merging behavior of systems exhibiting metastable nematic preordering under realistic experimental conditions where ideal undercooling is not feasible.
Conclusions
Figure 7. Plot of the interfacial temperature versus interface position of the three different simulations: nonisothermal (solid line), isothermal (dashed line), and nonisothermal with metastable nematic preordering (stippled line for the smectic-A front and dashed-dotted line for the nematic front).
temperature to approach the bulk transition temperature Tb, which subsequently causes the free-energy driving force of the isotropic/ smectic-A transition to approach zero. Thus, under these conditions, thermal diffusion limits growth, resulting in dynamics converging to a power law exponent of n = 1/2 as observed experimentally33,34,63,64 and predicted theoretically32,35,65 for liquid-crystal growth under shallow quench/undercooling conditions. In the case of metastable nematic preordering, the evolution of the isotropic/metastable nematic front temperature reveals the influence of the latent heat contribution of the trailing smectic-A front. The thermal energy from the additional heat source causes nonmonotonic behavior of the meta-nematic interface temperature evolution, favoring the eventual merging of the two fronts. Following this merging event, the convergence of the isotropic/smectic-A
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A thermal-energy balance was derived to extend a high-order Landau-de Gennes-type model of the isotropic/smectic-A liquidcrystalline transition to take into account the latent heat of phase ordering, anisotropic thermal diffusion, and dissipation (eq 18). One-dimensional simulations were performed, showing that the inclusion of the thermal-energy balance corrects the standard Landau-de Gennes predictions of volume-driven growth under shallow quench conditions (Figure 5a,b) to experimentally observed33,34 diffusion-limited growth. Additionally, the effect of incorporating thermal effects in the presence of metastable nematic preordering was found to result in nonmonotonic acceleration/deceleration/acceleration growth dynamics (Figure 6a,b), as observed experimentally.37 This work sets the basis for the further study of nonisothermal effects of the direct isotropic/ smectic-A transition where an extended subset of liquid-crystal physics present in the transformation is included (Figure 1). The presented extension to the high-order model of Mukherjee, Pleiner, and Brand41,42 allows for full 3D multiscale, multitransport simulation incorporating the diverse defect and texturing dynamics of lamellar smectic-A ordering coupled with nonnegligible thermal and dissipative effects. Acknowledgment. This work was supported by a grant from the Natural Science and Engineering Research Council of Canada.
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