Langmuir 2008, 24, 541-546
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Isotropic-to-Nematic Phase Transition in a Liquid-Crystal Droplet Xuemei Chen, Benjamin D. Hamlington, and Amy Q. Shen* Mechanical and Aerospace Engineering, Washington UniVersity in St. Louis, St. Louis, Missouri 63130 ReceiVed June 21, 2007. In Final Form: October 11, 2007
In this paper, we focus on the isotropic-to-nematic phase transition in a liquid-crystal droplet. We present the results of an experiment to measure the growth of the nematic phase within an isotropic phase liquid-crystal droplet. Experimentally, we observe two primary phase transition regimes. At short time scales, our experimental results (R(t) ∼ t0.51) show good agreement with a Stefan-type model of the evolution of the nematic phase within the isotropic phase of a liquid crystal. As time progresses, the growth of the nematic phase is restricted by increased confinement of the droplet boundary. During this stage of growth, the nematic phase grows at a slower rate of R(t) ∼ t0.31. The slower growth at later stages might be due to a variety of factors such as confinement-induced latent heat reduction; a change of defect strength during its evolution; or interactions between the defect and the interface between the liquid crystal and oil or between adjacent defects. The presence of two growth regimes is also consistent with the molecular simulations of Bradac et al. (Bradac, Z.; Kralj, S.; Zumer, S. Phys. ReV. E 2002, 65, 021705) who identify an early stage domain regime and a late stage confinement regime. For the domain and confinement regimes, Bradac et al. (Bradac, Z.; Kralj, S.; Zumer, S. Phys. ReV. E 2002, 65, 021705) obtained growth exponents of 0.49 ( 0.05 and 0.25 ( 0.05. These are remarkably close to the values 0.51 and 0.31 observed in our experiments.
1. Introduction Liquid-crystalline phases have been referred to as a fourth state of matter.1 The range of liquid-crystalline phases represents transitional states between liquids and crystalline solids. Many liquid-crystalline phases exist, with each being distinguished by the spatial arrangement of the constituent molecules.2 The nematic phase is that most commonly used in commercial applications. Indeed, a wide range of liquid-crystal-based applications including, but not limited to, consumer electronics are very important and central to everyday life. The phases of liquid crystals in particular have interesting applications in chemistry and biology. Lyotropic liquid crystals attract particular attention in the field of biomimetic chemistry. More specifically, biological membranes and cell membranes are a form of liquid crystal,3 and studying phase transitions provides greater insight into the workings of membranes. The past few years have seen growing interest in geometrically confined systems and related surface phenomena.4 In the field of liquid crystals, such systems are of particular interest due to their importance in electro-optics.5-8 The study presented in this paper focuses specifically on liquid-crystal dispersions and their phase transitions under confinement. Nematic liquid-crystalline dispersions have unique optical and rheological properties. Examples of this are found in the interactions between water droplets dispersed in nematic liquid crystals,9 liquid-crystal droplets in polymer solutions,10 defect gels in cholesteric liquid * To whom correspondence should be addressed. E-mail: aqshen@ me.wustl.edu. (1) Templer, R.; Attard, G. New Sci. 1991, 1767, 25. (2) de Gennes, P. G.; Prost, J. The Physics of Liquid Crystals; Oxford University Press: Oxford, U.K., 1993. (3) Seddon, J.; Templer, R. New Sci. 1991, 1769, 45. (4) Kralj, S.; Zumer, S.; Allender, D. W. Phys. ReV. A 1991, 43, 2943-2952. (5) Trebin, H. Liq. Cryst. 1998, 24, 127-130. (6) Spergel, D. N.; Turok, N. G. Sci. Am. 1992, (March), 36. (7) Brinkman, W. F.; Cladis, P. E. Phys. Today 1982, (May), 48. (8) Repnik, R.; Mathelitsch, L.; Svetec, M.; Kralj, S. Eur. J. Phys. 2003, 24, 481. (9) Poulin, P.; Stark, H.; Lubensky, T. C.; Weitz, D. A. Science 1997, 275, 1770. (10) Hamlington, B.; Steinhaus, B.; Feng, J. J.; Link, D.; Shelley, M.; Shen, A. Q. Liq. Cryst. 2007, 34, 861.
crystals that are stabilized by particles11 with enhanced elastic moduli, and electro-optically tunable scattering of light with polymer-dispersed liquid crystals (PDLC).12-15 PDLC materials are dispersions of micrometer-sized liquid-crystalline droplets embedded in a solid polymer matrix. PDLC technology has led to the emergence of a new type of electro-optic display and has stimulated research on liquid-crystalline droplets. The effects of confinement on the phase behavior of liquid crystals is of interest for fundamental physics of phase transitions and for applications involving porous media filled with liquid crystals.16-18 Iannacchione and Finotello16 reported a high-resolution calorimetric study for the nCB liquid-crystal systems confined to the submicrometer cylindrical pores of membranes. The specific heat was observed to deviate from the bulk value when the samples were confined with different orientations. Kutnjak et al.17 also performed a calorimetric study to measure the heat capacity and latent heat response of 8CB liquid crystals during isotropic-tonematic and nematic-to-smectic phase transitions when they are confined to a controlled-pore glass. Wittebrood et al.18 observed that the effective latent heat becomes smaller compared to the bulk value when a thin nematic film is being confined. Theoretical work has been done in the past on the phase transition in liquid-crystalline droplets. A Landau-de Gennes phenomenological model was used16,17 to estimate the confinement effects on the latent heat and specific heat variations, and it showed good agreement with experimental results. Working with a generalization of a model developed by Allen and Cahn19 to model the motion of antiphase boundaries, Bray20 predicted (11) Zapotocky, M.; Ramos, L.; Poulin, P.; Lubensky, T. C.; Weitz, D. A. Science 1999, 283, 209. (12) Drzaic, P. S. Liq. Cryst. 1988, 3, 1543. (13) Doane, J. W.; Vaz, N. A.; Wu, B. G.; Zumer, S. Appl. Phys. Lett. 1986, 48, 269. (14) Fergason, J. SID Int. Symp. Dig. Tech. Pap. 1985, 16, 68. (15) Doane, J. W.; Golemme, A.; West, J. L.; Whitehead, J. B.; Wu, B. G. Mol. Cryst. Liq. Cryst. 1988, 165, 511. (16) Iannacchione, G. S.; Finotello, D. Phys. ReV. E 1994, 50, 4780. (17) Kutnjak, Z.; Kralj, S.; Lahajnar, G.; Zumer, S. Phys. ReV. E 2003, 68, 021705. (18) Wittebrood, M. M.; Luijendijk, D. H.; Stalling, S.; Rasing, Th.; Musevic, I. Phys. ReV. E 1996, 54, 5232. (19) Allen, S. M.; Cahn, J. W. Acta Metall. 1979, 27 (6), 1085. (20) Bray, A. J. AdV. Phys. 2002, 51 (2), 481.
10.1021/la701844s CCC: $40.75 © 2008 American Chemical Society Published on Web 12/15/2007
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that interfaces between the nematic and isotropic phases obey classical diffusive growth. Bhattacharya et al.21 used a molecular dynamics approach to study the isotropic-to-nematic quench. Bradac et al.22 simulated the growth of the nematic phase within the isotropic phase for cylindrical and spherical confinements using molecular dynamics. The results for the spherical confinement show good agreement with the experimental results presented in this paper, in which the liquid-crystalline droplets simulate spherical confinement. Bradac et al.22 define several regimes of nematic growth for both bulk samples and confined samples. For the confined case, Bradac et al.22 define a domain regime in which the growth is only weakly influenced by the confinement and has a scaling law of tγ where γ ∼ 0.49 ( 0.05. As the transition continues, the nematic growth crosses into the confinement regime in which γ ∼ 0.25 ( 0.05. In this regime, the confinement exerts a strong influence and slows the nematic growth greatly. The data obtained in our experiments appear to fall in similar regimes, and we obtained γ values of ∼0.51 for the domain regime and ∼0.31 for the confinement regime, regardless of the length scale difference between our experiment and Bradac’s studies. Wincure and Rey23,24 also employed the Landau-de Gennes model to simulate the growth of the nematic phase in an isotropic-to-nematic transition of 5CB. The first paper23 investigated the growth, shape, and texturing dynamics of single 2D spherulites by using the Landau-de Gennes liquidcrystal model of isotropic-to-nematic phase ordering. The second paper24 described numerical simulations of the defect core. A liquid crystal dynamic shape function was derived to model shape changes, growth kinetics, cusp formation, and defect nucleation. The nematic phase is found to grow linearly with time and thus is distinct from what we observe. Goyal and Denn37 recently investigated orientational multiplicity and transitions in liquidcrystal droplets and discovered that droplet elongation led to surface-induced transitions. Other researchers have investigated a phenomenon called “nematic wetting” or “pretransitional surface ordering,” where the nematic phase initiates at the surface of an isotropic droplet when the liquid crystal is cooled to a temperature higher than the bulk transition temperature. Specifically, Beaglehole25 and Kasten and Strobl26 investigated nematic wetting at the free surface. Beaglehole25 reported complete wetting for 5CB, while Kasten and Strobl26 reported partial wetting for 5CB and complete wetting for 6CB, 7CB, and 8CB. Hsiung27 studied the nematic wetting phenomenon between the interface of nCB and a surfactant-coated glass substrate, and they observed partial wetting for 5CB and complete wetting for n ) 6, 7, 8, and 9. In this paper, we aim to understand an isotropic-to-nematic phase transition in a liquid-crystal droplet. We describe an experiment that measures the growth of a nematic phase liquidcrystalline droplet within an isotropic phase. We also compare our experimental data with results from a Stefan-type model. This model shows good agreement only at short time scales. On the other hand, our experiments match the molecular simulation results of the isotropic-to-nematic quench from Bradac et al.22 throughout the growth of the nematic phase. 2. Experimental Section 2.1. Materials. The thermotropic liquid crystal 4′-pentyl-4biphenylcarbonitrile (5CB) was purchased from Merck and used as received. Rai et al.23 found that 5CB undergoes a crystal-to-nematic transition at approximately 21 °C and a nematic-to-isotropic transition (21) Bhattacharya, A.; Rao, M.; Chakrabarti, A. Phys. ReV. E 1996, 53, 4899. (22) Bradac, Z.; Kralj, S.; Zumer, S. Phys. ReV. E 2002, 65, 021705. (23) Wincure, B. M.; Rey, A. D. Discrete Contin. Dyn. Syst.: Ser. B 2007, 8 (3), 623. (24) Wincure, B. M.; Rey, A. D. Continuum Mech. Thermodyn. 2007, 19, 37. (25) Beaglehole, D. Mol. Cryst. Liq. Cryst. 1982, 89, 319. (26) Kasten, H.; Strobl, G. J. Chem. Phys. 1995, 103, 6768. (27) Hsiung, H.; Rasing, Th.; Shen, Y. R. Phys. ReV. Lett. 1986, 57, 1149.
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Figure 1. Rheology characterization of 5CB under temperature sweep by using an AR2000 rheometer. The shear viscosity is the dissipative part of the complex viscosity in small-amplitude oscillation. at approximately 35 °C. Using an AR2000 stress-controlled rheometer with a cone and plate geometry (diameter ) 60 mm, cone angle ) 1°, truncation ) 29 µm), we verified these transitions using a temperature sweep with a shear rate of 100 s-1, as illustrated in Figure 1. This plot shows the dissipative component of the complex viscosity η′ as a function of temperature. Since the storage modulus of 5CB is negligible, the dissipative of the complex viscosity is essentially the shear viscosity. A large jump in the shear viscosity occurs at 33.2 °C due to the phase transition from the nematic to the isotropic phase of the liquid crystal. 2.2. Experimental Setup. A 5CB liquid-crystalline (abbreviated as LC hereafter) droplet with a diameter ranging from 50 to 300 µm was formed using a syringe and injected in a 7 cm well filled with 20-cSt silicone oil. The well and droplet were then placed on a Leica DM-IRB inverted microscope and viewed using crossed polarizers. The LC droplet was initially in the nematic phase at room temperature. The temperature of the system was increased to 40 °C using the Instec STC200 temperature stage, at which point the LC droplet had fully undergone a transition from the nematic phase to the isotropic phase (see Figure 4). After the transition, the temperature stage was turned off and the setup was allowed to cool to room temperature. Using a high-speed camera, the transition from the isotropic phase to the nematic phase was recorded. In our experiments, the nematic phase grows from a single-point defect arising from the center of the isotropic phase LC droplet. The temperature distribution within the setup is important when considering the phase transition. It is difficult to definitively determine the exact temperature profile with the imaging and measurement techniques used in this experiment, as any type of probe inserted directly into the liquid crystal would disturb the droplet and affect the phase-transition dynamics. To model our experiment, we hypothesize that the temperature of the entire system (both LC drop and the silicone oil ocean) is below the isotropic-to-nematic transition temperature when we first observe the single-point defect appearing in the center of the LC drop. In other words, the system is initially supercooled. Specifically, even though the temperature in the center of the LC drop is higher than that at the drop/oil interface, we assume that it is lower than the isotropic-to-nematic transition temperature. Geometrical constraints then induce a symmetry that requires that the defect nucleates at the center of the drop (see movie in the Supporting Information). To obtain the approximate temperature at which the transition occurred, a resistance thermometer detector (RTD) was placed in the silicone oil well, in close proximity to the liquid-crystalline droplet. A schematic of the setup is shown in Figure 2. Our model predicts the temperature profile in the LC drop. Future experiments should be done to confirm the validity of this prediction. Using video microscopy to study the isotropic-to-nematic transition, the growth over time of the nematic phase LC within the isotropic phase LC was measured for different initial liquid-crystal droplet sizes (ranging from 50 to 300 µm). In the initial stages of the transition, the nematic phase arises as a single-point defect. This point defect grows radially over time, allowing for easy measurement of the diameter. For different initial droplet sizes, we observed similar growth dynamics. The nondimensional plot of nematic and isotropic boundary positions versus time is shown in Figure 3 in which the
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Figure 2. Schematic showing the transition from isotropic liquid crystal to nematic liquid crystal. The temperature of the system is initially above the isotropic-to-nematic transition temperature. The temperature is then lowered until the nematic phase begins to appear in the isotropic phase. Three actual images of the growth of the nematic phase are given below the schematic. transition, the nematic phase would disappear at several different locations, making it difficult to pinpoint one location to take measurements. A sequence of images for the nematic-to-isotropic phase transition is shown in Figure 4, illustrating the difficulty in quantifying the rate of the nematic phase shrinking in the isotropic phase.
3. Theory Several papers have presented simulations or theoretical models for the isotropic-to-nematic phase transition in liquid crystals.22,17,21 We use a continuum-based model to provide a comparison with our experimental data. For simplicity, we assume radial symmetry. The energy balance for the isotropic portion of the drop is
Figure 3. Normalized plot of isotropic-to-nematic phase transition for liquid-crystal droplets with different initial sizes. normalized time t ) t/t0 is plotted against the nematic/isotropic phase interface normalized by d/d0 , with d0 being the initial liquidcrystal droplet diameter and t0 being the time at which the isotropicto-nematic transition is completed (on the order of seconds). The experimental data with different initial LC droplet sizes follow a similar trend, which can be fit by a power-law relationship with an exponent of ≈1/2 at the initial growth stage and an exponent of ≈1/3 at the later growth stage when d/d0 goes beyond ∼0.75. This observation suggests that the nematic growth can be hindered due to edge effects. It is important to note that we only focused on the isotropic-tonematic transition in this study. The transition from nematic-toisotropic occurred in a manner that is difficult to quantify and measure. When transitioning from isotropic to nematic, the nematic phase of the liquid crystal begins as a single-point defect that grows in the radial direction over time. Thus, it is easy to measure the change in the diameter of the nematic phase (see Figure 2). During the nematic-to-isotropic transition, we noticed little consistency in the shrinking of the nematic phase. At the initial stages of the transition, the interface between the isotropic and nematic phases would not propagate radially but would instead travel from one side of the droplet to the other. Furthermore, at the final stages of the
Fcθ˙ )
k ∂ 2∂θ r , r > R(t) r2 ∂r ∂r
( )
(3.1)
where F, c, and k denote the density, specific heat, and thermal conductivity of the isotropic phase, respectively, and θ is a dimensionless temperature deviation defined in terms of the absolute temperature T and the isotropic-to-nematic transition temperature T0 by θ ) (T - T0)/T0. We assume that thermal diffusion occurs in a time scale that is much shorter than the time scale of the phase transformation so that a steady-state can be assumed and eq 3.1 reduces to a Laplace equation for θ. We assume that the temperature in the nematic phase is uniform and equals to that of the interfacial limit of the isotropic phase. In addition, we assume that the director field in the nematic phase is radial, with a single-point defect (hedgehog) at r ) 0. At the phase interface r ) R(t), the energy balance at the interface gives
l dR ∂θ ) ∂r T0 dt
-k
(3.2)
where l is the latent heat of transformation. In addition, we use a generalization of the Gibbs-Thomson equation which accounts
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Figure 4. Images showing stages of nematic-to-isotropic phase transition. The temperature is increasing in the images from left to right.
for the curvature elasticity of the nematic phase (Cermelli et al.29). Specifically, following Anderson et al.,30 we linearize the generalized Gibbs-Thomson of Cermelli et al.29 about a state in which the interface is flat and the alignment in the nematic phase is uniform and parallel to the orientation of the interface. This procedure yields
2σ κ ) -lθ - 2 R R
(3.3)
where σ is the surface tension between the isotropic and nematic phases of 5CB and κ ) 2k1 - (k2 + k4) > 0 , with k1, k2, and k4 being the splay, twist, and saddle-splay moduli, respectively, of the nematic phase. Ericksen’s31 inequalities require that κ g 0. In addition, we have the far-field boundary condition
lim θ(r) ) θ∞ ) constant
rf∞
(3.4)
Solving eq 3.1 at steady-state and making use of the far-field boundary condition (eq 3.4) gives
Next, introducing the characteristic radius Rc and time scale tc, and defining the dimensionless drop radius R h and the dimensionless time τ via
R h)
R t , τ) Rc tc
eq 3.7 can be recast in the nondimensional form
C B A R h˙ ) - - 2 2 R h R h R h3
θ(R(t), t) ) θ∞ +
lR(t) dR(t) kT0 dt
(3.6)
Using eq 3.6 in the Gibbs-Thomson relation (eq 3.3), we obtain a single evolution equation for R in the form
kT0θ∞ kT0σ kT0κ dR )-2 2 2 - 2 3 dt lR lR lR
(3.7)
The first and third terms on the right-hand side of eq 3.3 are associated with the thermal and curvature-elastic contributions of the free-energy density of the nematic phase (as measured relative to the free-energy density of the isotropic phase). The second term on the right-hand side of eq 3.7 is associated with the surface tension of the nematic/isotropic phase interface. Equation 3.7 does not determine a solution for the equilibrium radius. The nematic phase will grow indefinitely until it reaches a boundary. (28) Rai, P. K.; Denn, M. M.; Maldarelli, C. Langmuir 2003, 19, 7370. (29) Cermelli, P.; Fried, E.; Gurtin, M. E. Arch. Ration. Mech. Anal. 2004, 174, 151. (30) Anderson, D. M.; Cermelli, P.; Fried, E.; Gurtin, M. E.; McFadden, G. B. J. Fluid Mech. 2007, 581, 323. (31) Ericksen, J. L. Phys. Fluids 1966, 9, 1205.
(3.9)
where R h ) dR h /dτ and the dimensionless parameters A, B, and C are defined as
A)
kT0θ∞tc
B)
(θ(R(t), t) - θ∞) R(t) θ(r, t) ) + θ∞, r g R(t) (3.5) r Differentiating eq 3.5 with respect to r, evaluating the result at r ) R(t), and using the interfacial energy balance (eq 3.2), we find that
(3.8)
C)
lR2c kT0σtc l2R3c kT0κtc l2R4c
(3.10)
The values of the material parameters can be found in the literature. In particular, for 5CB, we have the following reported values:32,34,35
σ ) 2.10 × 10-5 J/m2 κ ) 4.50 × 10-12 J/m l ) 1.59 × 106 J/m3 k ) 0.15 J/m s K For the experiment performed here, the isotropic-to-nematic transition temperature is T0 ) 306.15 K (obtained from Figure 1). The liquid crystal is supercooled so that the ambient temperature is below T0. In particular, we have T∞ ) 306.07 K (measured by a thermometer detector), which gives θ∞ ) T∞/T0 - 1 ) -2.6131 × 10-4. Choosing the characteristic drop radius Rc and time scale tc based on the intrinsic length and time of the problem, we have (32) Ahlers, G.; Cannell, D. S.; Berge, L. I.; Sakurai, S. Phys. ReV. E 1994, 49, 545. (33) Iannacchione, G. S.; Crawford, G. P.; Zˇ umer, S.; Doane, J. W.; Finotello, D. Phys. ReV. Lett. 1993, 71, 2595. (34) Kim, J. W.; Kim, H.; Lee, M.; Magda, J. J. Langmuir 2004, 20, 8110. (35) Yokoyama, H.; Kobayashi, S.; Kamei, H. Mol. Cryst. Liq. Cryst. 1985, 129, 109.
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Rc )
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κ ) 2.143 × 10-7m σ
lR2c tc ) ) 1.590 × 10-9s kT0 Using these values to calculate the dimensionless parameters in eq 3.10 gives
A ) -2.6131 × 10-4 B)
σ2 ) 6.16 × 10-5 lκ C)B
Note that A is negative and that this supercooling effect ensures the growth of the nematic droplet. Equation 3.9 is solved by MATLAB solver ODE45, which is based on an explicit Runge-Kutta formula, the Dormand-Prince pair. The initial condition is taken from the experimental data at the starting point, and the time span is set to be the same as the experimental time span. From the experimental data, we see that the dimensionless drop radius is of order 100 and higher, so that the dominant term in eq 3.9 is the first term, which gives the Stefan problem. Figure 5 shows that model 3.9 coincides with the Stefan solution and they both match the experimental data very well during the early stages of growth. Comparison with the experimental data also indicates that the Stefan solution overpredicts the rate of the isotropic-to-nematic transition during the late stages of growth. In fact, a curve fit of the raw data (see Figure 5b) indicates that during the late stages the radius of the nematic phase grows at R(t) ∼ t0.31. This deviation may be due to the fact that, at sufficiently long times, confinement of the droplet leads to a reduced latent heat of transformation. Kutnjak et al.17 conducted a calorimetric study of nCB liquid crystals confined to a controlled-pore glass and observed the progressive disappearance of the latent heat l with decreasing pore sizes. Wittebrood et al.18 also observed that the latent heat l was lowered for a nematic film under confinement. In accord with these observations, during the late stage of the nematic phase growth, we use the same model (eq 3.9) but choose the latent heat l* as an empirical fitting parameter to compare with the experimental data. The best fit with the experimental data yields a latent heat l* to be half of the bulk value l at the late nematic growth stage (see Figure 5a). Moreover, referring to eqs 3.9 and 3.10, we see that a decrease in the latent heat l increases the importance of the terms accounting for the effects of surface tension and curvature elasticity. Individually, the surface tension and curvature elasticity terms lead to radial growth rates of R(t) ∼ t1/3 and R(t) ∼ t1/4, respectively. Hence, with reduced latent heat l, these terms would lead to some reduction in the nematic growth rate.
4. Results and Discussion When observing the isotropic-to-nematic transition in the droplet, we notice several stages or regimes. Bradac et al.22 identified three regimes for a bulk sample of liquid crystal: the early regime, the domain regime, confinement regime, and the late stage regime. As stated in their paper, the early regime is dominated by the growth of a long-wave fluctuation of the order parameter of the nematic phase. In this regime, the order parameter exhibits exponential growth and lasts until what was known as the Zurek36 time. This regime is not visible in our experiments. The early regime gives way to the domain regime, which is characterized by a multidomain structure. In this regime, the
Figure 5. (a) Comparison of the experimental data, the solution to the complete evolution equation, and the Stefan solution. Note that the solution to eq 3.9 overlaps the Stefan solution. (b) Scaling laws based on best curve fitting from the experimental data.
growth of individual point defects can be measured, and, with increased time, larger domains grow and dominate at the expense of smaller domains. The domain regime can be seen in Figure 6A. The scaling law R ∼ tγ is well obeyed in the domain regime, and our studies show a γ value of about 0.51 in this regime (Figure 5b). When comparing our experiments with the scaling laws for the domain regime of Bhattacharya et al.21 and Bradac et al.22 based on molecular simulations, we see very similar values of γ for all three studies despite different approaches. Bhattacharya et al.21 obtained γ ∼ 0.45, while Bradac et al.22 obtained γ ∼ 0.47, both of which show good correlation with our value for γ ∼ 0.51 in the domain regime. It is worth noting that the domain regime in our experiments can reach up to several hundred micrometers (see Figure 3); this dimension is significantly higher than the domain regime considered by Bradac et al.22 in their molecular simulation results and experimental observations reported in the past.17,18 Following the domain regime, the phase transition enters the confinement regime. From Figure 4, we observe that the growth enters the confinement regime when the nematic growth reaches ∼70% of the initial drop size. In this regime, the growth of the nematic phase is strongly restricted by the confinement, and the elastic and surface interactions can control the growth dynamics. In the case of the spherical confinement, the domain growth is restricted in all directions. The growth restriction is reflected in the scaling law and in particular the value of γ for this regime. Our experiments yield γ values of approximately 0.31 for when the phase transition is in the confinement regime (see Figure 5b). Bradac et al.22 obtained γ ∼ 0.25 ( 0.05 for defects with a planar polar structure in the confinement regime. The confinement regime can be seen in Figure 6B. The experiments of Kutnjak et al.17 and Witterbrood et al.18 report that the latent heat of confined nematic liquid crystals is reduced to half of the measured bulk value when the confinement length scale is in the sub(36) Zurek, W. H. Phys. Rep. 1996, 276, 178. (37) Goyal, R. K.; Denn, M. M. Phys. ReV. E 2007, 75, 021704.
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Figure 6. Images showing different regimes of isotropic-to-nematic phase transition: (A) the domain regime, (B) the confinement regime, and (C and D) the late stage regime.
micrometer range. Based on our experimental observations (Figure 5), our confinement regime ranges from 10 to 30 µm from the liquid crystal and silicone oil interface. Further investigation is required to probe the length scale effect on the confinement regime, possibly incorporating differential scanning calorimetry or NMR measurements into our experimental design. Finally, Bradac et al.22 defined the last regime as the late stage regime. In this regime, the structures of individual defects dominate the nematic pattern. As this regime approaches its end, we begin to see the defect-free nematic structure, as seen in Figure 6C and D. We find that the evolution (eq 3.9) yields excellent predictions during the early stage of growth. At longer time scales, as isotropic-to-nematic phase transition reaches the confinement regime, the growth is restricted and tends to overpredict the rate of growth. The slower growth at later stages might be due to a variety of factors: (a) confinement-induced latent heat reduction might increase the surface tension and curvature elasticity contributions during nematic phase growth; (b) a change of the defect strength in the nematic phase during its evolution; and (c) interactions between the defect and the interface between the liquid crystal and oil or between adjacent defects. It is worth noting that the continuum model leading to eq 3.9 assumes that the defect in the nematic phase is a hedgehog. This differs substantially from the model of Bradac et al.,22 where the defects are the planar polar structure with line defects (PPLD) and the droplet size is on the order of submicrometers, compared to several hundred micrometers of our drop sizes. It would therefore appear that, during the early stages of nematic growth, the kinetics are somewhat insensitive to the type of the defect.
5. Conclusion In this paper, we present an experiment for quantifying the growth of a nematic phase liquid crystal in an isotropic liquid-
crystal droplet. Our experiment facilitates an easy realization of liquid-crystal phase transition in both the bulk and confined conditions. We also present a continuum-based theory that accounts for thermal contribution to the free energy variation between nematic and isotropic phases and also for contributions due to curvature elasticity and surface tension. The model is able to match our experimental data well at short time scales. At the late growth stage, as the isotropic-to-nematic phase transition reaches the confinement regime, the nematic growth slows down. Using the same model but choosing the latent heat as an empirical fitting parameter to compare the model and experimental data, we find that the best fit yields an effective latent heat to be half of the bulk value. This indicates that the latent heat can decrease significantly due to the confinement, and, as a consequence, surface tension and curvature elasticity effects become more important and lead to some reduction in the nematic growth. Furthermore, a change of defect strength or interactions between the defect and the interface between the liquid crystal and oil or between adjacent defects can hinder the nematic growth as well. Finally, the semi-microscopic model used by Bradac et al.22 is able to capture the kinetics of isotropic-to-nematic phase transition observed in our experiments, regardless of the length scale and defect type differences. Acknowledgment. We gratefully acknowledge support from National Science Foundation CBET 0645062 to make this work possible. We also thank the reviewers for many helpful comments. Finally, we thank Professor Eliot Fried for helpful discussions. Supporting Information Available: Movie showing the sequence of isotropic-to-nematic phase transition of a liquid-crystal droplet submerged in silicone oil. The movie was taken by using crossed polarizers. This material is available free of charge via the Internet at http://pubs.acs.org. LA701844S
