Model 0 Results

Liquid Crystal/Flexible Polymer Mixture for Model 0:

Investigating Domain Growth Kinetics

We study the growth kinetics of domains formed in an incompressible binary mixture of short (rigid) liquid crystals and long (flexible) polymer chains. We simulate temperature quenches and gather data on how the concentration of liquid crystals (phi) and the orientational density (S) evolve over time.

For all quenches, the system begins in the isotropic, homogeneous one-phase regime and is quenched to a temperature T in the unstable (not metastable) two-phase regime. Because our model monitors two order parameters (phi and S), we can calculate two spinodals (assuming the system initially is disordered). Once quenched, the system initially becomes unstable with respect to phase separation (PS), phase ordering (PO), or both. Consult our phase diagram for a better understanding.

To study the domain growth kinetics, we must quantify domains. We can define a domain based on either of our two order parameters, phi and S (where S is the magnitude of the tensor S), and calculate the structure factor,

S(k, t) = 1/N² sum_r [e^ik.r sum_r_o [delta_x(r_o, t) delta_x(r_o+r, t)]]
where x is phi or S, and sum_r = sum over all r values. Spherical averaging then results in S(k, t). (Note that summing over lattice sites rather than integrating over space is OK since we deal with a discrete lattice.)

phi-based S(k, t): domains based on delta_phi(r, t) = phi(r, t) - phi^avg(t)
S-based S(k, t): domains based on delta_S(r, t) = S(r, t) - S^avg(t)
(I have some concerns about using this definition.)

We then calculate the first moment of the structure factor, k₁(t):

"new" k₁(t) = sum_k [k² S(k,t)] / sum_k [k S(k,t)]
where sum_k = sum over all k values up to some cutoff value.
[Explanation for why we believe this to be the correct definition of the first moment of S(k, t).]
"old" k₁(t) = sum_k [k S(k,t)] / sum_k [S(k,t)]
where sum_k = sum over all k values up to some cutoff value.
Note: This "old" k₁ is that defined in S.C. Glotzer's paper (Annual Reviews of Computational Physics 2, 1 (1996)).

For some quenches, we conduct several runs, each with a different initial configuration. For each run of a given quench, we calculate S(k) for each time t. From this, we determine k₁(t). We then calculate the average k₁(t) (= [sum of all k₁(t)'s] / [# initial configurations]) and use this value o fk₁ for that given quench.

Finally, we calculate the average domain size: R(t) = 2 pi / dx k₁(t).
dx is the pixel size and is set to be 0.25 or 0.10. We examine how R evolves over time, making comparisons to known growth laws: R(t) ~ t^1/3 (Model B) and R(t) ~ t^1/2 (Model A) for intermediate/late times. Calculated power-law behavior shown in our plots are stated merely for comparison with models A and B growth laws and are not to imply any scaling behavior (unless stated otherwise).

Definition of headings in the tables below:

phi0 = initial concentration of liquid crystals (LCs); note: 1-phi0 = concentration of polymer chains.
T = temperature to which system is quenched
Suffix Run = datafile suffix
N = linear system size; we examine an N x N system.
# Configurations = maximum number of different initial configurations

I. Particular Quenches.

phi0	T	Suffix Run	N	# Configurations	Comments
0.68	0.95	A4-1	150	10	Initially unstable wrt PS only.
0.85	0.75	C2-1	150	9	Initially unstable wrt PS and PO, but more unstable wrt PS.
0.89	0.75	E7-2	150	10	Initially unstable wrt PS and PO, but more unstable wrt PO.

II. Resolving the "inconsistencies" among our results.

We attempt to show that results obtained by A. Al Sunaidi really may not be inconsistent with ours.

Critical quench: phi0=0.76, T = 0.95.

Off-critical quench: phi0=0.68, T=0.95.

III. Other quenches.

We present overall reviews, showing the results of the runs listed below. Choose a temperature (T) value to view a comparison of systems quenched to that temperature. Choose an initial LC concentration (phi0) to view a comparison of systems with that same initial LC concentration. With these quenches we hope to gain a better understanding of our "focussed" quenches (listed above in I.).

phi0 T Suffix Run N # Configurations Comments
0.68 0.75 F3-2 150 1 Initially unstable wrt PS only.
0.75 0.75 H21-2 150 1 Initially unstable wrt PS only.
0.80 0.75 H41-2 150 1 Initially unstable wrt PS only.
0.83 0.75 F6-2 150 1 Initially unstable wrt PS only, but quite near the PO spinodal.
0.84 0.75 F2-2 150 1 Initially unstable wrt PS and PO, but more unstable wrt PS.
0.85 0.75 C2-1 150 9 Initially unstable wrt PS and PO, but more unstable wrt PS.
0.86 0.75 F5-2all 150 1 Initially unstable wrt PS and PO, but more unstable wrt PO.
0.87 0.75 F1-2 150 1 Initially unstable wrt PS and PO, but more unstable wrt PO.
0.88 0.75 F4-2 150 1 Initially unstable wrt PS and PO, but more unstable wrt PO.
0.89 0.75 E7-2 150 10 Initially unstable wrt PS and PO, but more unstable wrt PO.

0.68 0.85 G11-2 150 1 Initially unstable wrt PS only.
0.75 0.85 H11-2 150 1 Initially unstable wrt PS only.
0.80 0.85 H31-2 150 1 Initially unstable wrt PS only.
0.83 0.85 G22-2 150 1 Initially unstable wrt PS only.
0.84 0.85 G31-2 150 1 Initially unstable wrt PS only.
0.85 0.85 G41-2 150 1 Initially unstable wrt PS only.
0.86 0.85 G51-2 150 1 Initially unstable wrt PS only.
0.87 0.85 G61-2 150 1 Initially unstable wrt PS only.

phi0	T	Suffix Run	N	# Configurations	Comments
0.68	0.75	F3-2	150	1	Initially unstable wrt PS only.
0.75	0.75	H21-2	150	1	Initially unstable wrt PS only.
0.80	0.75	H41-2	150	1	Initially unstable wrt PS only.
0.83	0.75	F6-2	150	1	Initially unstable wrt PS only, but quite near the PO spinodal.
0.84	0.75	F2-2	150	1	Initially unstable wrt PS and PO, but more unstable wrt PS.
0.85	0.75	C2-1	150	9	Initially unstable wrt PS and PO, but more unstable wrt PS.
0.86	0.75	F5-2all	150	1	Initially unstable wrt PS and PO, but more unstable wrt PO.
0.87	0.75	F1-2	150	1	Initially unstable wrt PS and PO, but more unstable wrt PO.
0.88	0.75	F4-2	150	1	Initially unstable wrt PS and PO, but more unstable wrt PO.
0.89	0.75	E7-2	150	10	Initially unstable wrt PS and PO, but more unstable wrt PO.
0.68	0.85	G11-2	150	1	Initially unstable wrt PS only.
0.75	0.85	H11-2	150	1	Initially unstable wrt PS only.
0.80	0.85	H31-2	150	1	Initially unstable wrt PS only.
0.83	0.85	G22-2	150	1	Initially unstable wrt PS only.
0.84	0.85	G31-2	150	1	Initially unstable wrt PS only.
0.85	0.85	G41-2	150	1	Initially unstable wrt PS only.
0.86	0.85	G51-2	150	1	Initially unstable wrt PS only.
0.87	0.85	G61-2	150	1	Initially unstable wrt PS only.

IV. Individual Results.

Jump to the individual results of the quench with (phi0, T, N) of:

A2: 0.76, 0.95, 100 A21-2: 0.76, 0.95, 150 A4: 0.68, 0.95, 100 A4-1: 0.68, 0.95, 150 A41-250-2: 0.68, 0.95, 250
F31-2: 0.68, 0.75, 150 H21-2: 0.75, 0.75, 150 H41-2: 0.80, 0.75, 150 F61-2: 0.83, 0.75, 150 F21-2: 0.84, 0.75, 150
C2-1: 0.85, 0.75, 150 C21-250-2: 0.85, 0.75, 250 F5-2: 0.86, 0.75, 150 F11-2: 0.87, 0.75, 150 F4a-2: 0.88, 0.75, 150
E7-2: 0.89, 0.75, 150 E71-250-2: 0.89, 0.75, 250 G11-2: 0.68, 0.85, 150 H11-2: 0.75, 0.85, 150 H31-2: 0.80, 0.85, 150
G21-2: 0.83, 0.85, 150 G31-2: 0.84, 0.85, 150 G41-2: 0.85, 0.85, 150 G51-2: 0.86, 0.85, 150 G61-2: 0.87, 0.85, 150

V. Individual Profiles.

Jump to the individual DynamicLattice profiles (phi, S, theta) of the quench with (phi0, T, N) of:

A2: 0.76, 0.95, 100 A21-2: 0.76, 0.95, 150 A4: 0.68, 0.95, 100 A4-2: 0.68, 0.95, 150 A41-250-2: 0.68, 0.95, 250
F31-2: 0.68, 0.75, 150 H21-2: 0.75, 0.75, 150 H41-2: 0.80, 0.75, 150 F61-2: 0.83, 0.75, 150 F21-2: 0.84, 0.75, 150
C2-1: 0.85, 0.75, 150 C21-250-2: 0.85, 0.75, 250 F5-2: 0.86, 0.75, 150 F11-2: 0.87, 0.75, 150 F4a-2: 0.88, 0.75, 150
E7-2: 0.89, 0.75, 150 E71-250-2: 0.89, 0.75, 250 G11-2: 0.68, 0.85, 150 H11-2: 0.75, 0.85, 150 H31-2: 0.80, 0.85, 150
G21-2: 0.83, 0.85, 150 G31-2: 0.84, 0.85, 150 G41-2: 0.85, 0.85, 150 G51-2: 0.86, 0.85, 150 G61-2: 0.87, 0.85, 150

VI. Summarizing Results.

Ordering does appear to have a nontrivial effect on the spinodal decomposition of a LC/polymer mixture. We observe not only morphologies different from mixtures of two istropic components, but also domain growth kinetics different from that of pure blends (Model B).

Overall, the presence of ordering tends to slow down domain growth. We observe "growth laws" slower than the well-established one for Model B: R ~ t^1/3. (Note that this growth law is valid for critical and off-critical quenches, particularly deep ones. The growth law for shallow, off-critical quenches may be slightly slower (t^1/4)).

www.chem.ucla.edu/~aml/research.html
aml@chem.ucla.edu

Last updated August 26, 1999.

A2: 0.76, 0.95, 100	A21-2: 0.76, 0.95, 150	A4: 0.68, 0.95, 100	A4-1: 0.68, 0.95, 150	A41-250-2: 0.68, 0.95, 250
F31-2: 0.68, 0.75, 150	H21-2: 0.75, 0.75, 150	H41-2: 0.80, 0.75, 150	F61-2: 0.83, 0.75, 150	F21-2: 0.84, 0.75, 150
C2-1: 0.85, 0.75, 150	C21-250-2: 0.85, 0.75, 250	F5-2: 0.86, 0.75, 150	F11-2: 0.87, 0.75, 150	F4a-2: 0.88, 0.75, 150
E7-2: 0.89, 0.75, 150	E71-250-2: 0.89, 0.75, 250	G11-2: 0.68, 0.85, 150	H11-2: 0.75, 0.85, 150	H31-2: 0.80, 0.85, 150
G21-2: 0.83, 0.85, 150	G31-2: 0.84, 0.85, 150	G41-2: 0.85, 0.85, 150	G51-2: 0.86, 0.85, 150	G61-2: 0.87, 0.85, 150