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Liquid Crystal - Director distribution
Posted Feb 24, 2011, 7:08 a.m. EST Modeling Tools & Definitions, Parameters, Variables, & Functions Version 4.1, Version 4.3 4 Replies
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I would like to calculate the director distribution of liquid crystals.
The formula
fs=1/2(K11*(div n)^2 + K22*(n * rot n + q0)^2 +K33* |(n x rot n)|^2)
describes the Oseen-Frank free energy density for nemantic crystals.
K11, K22, K33 and q0 are constants. n are unit vectors presenting each director. (n is a vector with nx, ny and nz as elements).
At at least one boundary the director has known value (nx and ny are known).
A second energy density comes from the electric field:
fe=div grad(phi),
where phi is the electric potential.
The potential is given at the boundarys, at least at two edges:
------------------------------ +phi
LC
------------------------------ -phi
The energy has to be minimized:
Fs-Fe -->0
I think I have to use mathematical euqtions --> PDE in Comsol, but I do not know how to fill the template(s).
I have 2/3 variables, if I try to solve a 2D problem:
phi --> scalar at every grid point
n --> vector consiting of two scalars nx and ny
fs and fe are scalar fields.
Is it possible to compute the solution with Comsol and one of the PDE forms? Or do I need a weak form?
How can I fill in the PDE templates for this special task?
I thought about setting one energy as source term, but during my tries I got the problem, that n is a vector and fs, fe are just scalars.
Thanks a lot for any help.
Irina
Hello Irina Stockem
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Since your post is year and half ago, you probably already figure out how to deal with liquid crystal in comsol. But here is what I have in mind.
I am new to comsol, so correct me if anything is wrong.
When we work with liquid crystal we want to minimize the energy of the system (bulk and surface interaction). Since comsol is based on PDE, we may derive the equation of motion in the bulk and the surface boundary conditions in PDE form. And then use these PDEs to simulate.
The problem with liquid crystal is that it is not a scalar field, it is a vector field (for nematic). There might be a way to model a vector field in comsol (I am still learning). However, if your system is not too complicated, you can assume the director to be a function of angle/angles (azimuthal, polar, etc.), then the PDEs would be in terms of scalar and the calculation would also be simpler. In simple word, do it as we usually do for the calculation of liquid crystal cells, but just let comsol solve for the PDEs for you.
Hope that makes some sense.
Quan
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the solution you described is exactly that what I did.
Nevertheless, thanks a lot for your reply.
Irina
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I am doing something like that, but I want first solve the problem without electric field. Could I ask you some things?
1) Did you use the euler lagrange formulation to solve the problem?
2) When I solve the rotational term appear d/dz, what did you do with those if you work in 2D?
3) How I can undo the spurius solutions?
Thanks a lot for any help!!
:)
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Can you please share your model with us?
Or if anyone else has a working Frank-Oseen energy minimization model for computing liquid crystal orientation in absence of electric field please let me know.
Thanks!