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van 't Hoff Institute for Molecular Sciences
Kinetics of liquid crystal phase transitions:A phase field approach
Bastiaan A.H. Huisman1,*, Annalisa Fasolino1,2
1 Van 't Hoff Institute for Molecular Sciences, University of Amsterdam. Nieuwe Achtergracht 166, 1018 WV Amsterdam.
2 Theoretical Physics, Institute for Molecules and Materials, Radboud University Nijmegen. Toernooiveld 1, 6525 ED Nijmegen.
Asymptotic behavior
Growth exponent versus time for (exponentially) decreasing p. Dotted lines are fitted to
101
102
103
104
10-4
10-3
10-2
10-1
100
decreasing p
t101
103
104
-0.5
-0.4
-0.3
-0.2
-0.1
0
102
decreasing p
tGrowth exponent versus time for (exponentially) decreasing p. Asymptotic behavior reached at later times for lower p
Exponent fitted to
t0 found to be linear in 1/p
Motivation
. Liquid Crystals (LC's) isotropic to nematic phase transition: Nematic domains experimentally1,2 found to grow as R ∝ t 1/2 into undercooled isotropic phase.. Diamagnetic LC's found1,2 to grow faster in high magnetic fields. Possibly with different, higher power.. Reason for faster growth unknown.
. Isotropic to nematic phase transition is weakly first order. . Time scales of thermal diffusion and order parameter kinetics are 5 orders apart3.
Liquid crystals
BoundariesThe boundary conditions act as a thermostat, keeping the systems boundaries at a fixed temperature.
Once the u-profile widenes beyond the system size, the growth exponent changes.
Temperature profile for a system with nearby boundaries kept at the undercooling temperature. Note the difference before and after the field widened to the system size.
0
-0.10
-0.08
-0.06
-0.04
-0.02
x
u
t=200t=4200t=8200t=12200
0.00
101
102
103
104
10-3
10-̆2
10-̆1
100
decreasing p&increasing system size
tGrowth exponent for increasing p and increasing system size. When temperature profile width exceeds the system size, the exponent grows.
Temperature field width fitted for systems with far away boundaries. Extrapolated for parameters describing 8CB, found to be approx. 25 cm: much larger than experimental set up.
Phase field approachOrder parameter m evolves to minimize free energy:
where F is given by a local (see figure) and a surface energy part3
functional derivation gives:
Phase change enters as heat source in heat equation:
Such that the latent heat is:
Solving the equationsEquations discretized by finite differences. Time integration by series expansion.
m
f
T<T0
T=T0
T>T0
0 1
Determining ExponentsAssume: then:
At low undercooling:Asymptotic behavior unreachable for p=10-5, typical for the liquid crystal 8CB.We estimate onset of powerlaw from simulations at higher p.
Liquid crystals
p: ratio thermal and order parameter diffusion constants. p extremely low in LC's -> Time and length scales of u and m very different at low undercooling.
Order parameter and temperature profiles for p=5.10-3, low undercooling. Order parameter interface width 21/2 in adimensional units. Thermal profile much wider.
Two regimesLow undercooling or slow thermal diffusion: latent heat increases interface temperature towards T. The dynamics of the order parameter front R is dominated by thermal diffusion: R ∝ t 1/2
High undercooling or fast thermal diffusion: order parameter front propagates at constant velocity: R ∝ t . Interface temperature approximately constant.
0.0
0.2
0.4
0.6
0.8
1.0
m
-2000 -1000 0 1000 2000
-0.08
-0.06
-0.04
-0.02
0.00
x
u
t=200
t=4200
t=8200
t=12000
-0.10
Conclusions. Nematic domains experimentally found to grow into isotropic undercooled liquid as powerlaw where .. Phase field equations difficult to integrate in range of parameters suitable for liquid crystals. Extrapolation necessary.. In the estimated time t0 for the powerlaw behavior to set in, the thermal profile of liquid crystals is estimated to be around 25cm wide.. Experimental set up much smaller: therefore exponent expected to be higher:
Remaining.... Neighbouring domains. Can latent heat stall growth?. Can we get without temperature field?. The effect of dimensionality on the time t0 (spherical coords.).. Coupling to magnetic field.
References1 G. Tordini, P.C.M. Christianen, J.C. Maan. cond-mat/0408208 (2004).2 G. Tordini, private communication.3 H. Lowen, J. Bechhoefer, L.S. Tuckerman. Phys. Rev. Lett. 45, 2399 (1992).
