Phase Diagrams: Liquid Crystal/Flexible Polymer Mixture

To model the dynamics of any multicomponent mixture, we must know its behavior at equilibrium. We use the catastrophe method¹ to construct phase diagrams for a mixture of short liquid crystals (Na = 10) and long flexible polymers (Nb = 100).

For the following critical values²:

   Tc = 1.14  chi_c = 0.0867  phi_c= 0.760
           where chi_c = chi0 + chi1/Tc.


   in 2D:  Tni = 1.30
   in 3D:  Tni = 1.72
           where Wni = W0 + W1/Tni.

Values of the parameters in the catastrophe program:

  set Na = 10
  set Nb = 100
  set W0 = 5.500000e-01
  set W1 = 2.600000e-01
  set chi0 = 5.500000e-02
  set chi1 = 3.600000e-02
  set mustep = 1.000000e-03
  set phi0 = 1.000000e-03
  set phi1 = 1.000000e-08
  set dphimax = 1.000000e-03
  set maxnpts = 100000
  set hashbins = 500
  set orderN = true
  set lowE = true
  set labels = false
  set verbose = false

In two dimensions the phase diagram is:

In three dimensions the phase diagram is:

To compare two-dimensional behavior to three-dimensional behavior, we present the two coexistence curves on the same axes.

Realize that both coexistence curves should go down to T = 0 for the pure liquid crystal and the pure flexible polymer mixtures.

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²Tc is the temperature at which the system begins to phase separate in the absence of orientational order. chi_c and phi_c are the chi and phi values, respectively, at Tc. Tni is the temperature at which the pure liquid crystal system undergoes the isotropic-nematic transition. This temperature in 2D is different from that in 3D, because in 2D this transition is second-order, while in 3D it is first-order.

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Last updated September 5, 1997.

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