Abstract
We consider a Landau-de Gennes model for a connected cubic lattice scaffold in a nematic host, in a dilute regime. We analyse the homogenised limit for both cases in which the lattice of embedded particles presents or not cubic symmetry and then we compute the free effective energy of the composite material. In the cubic symmetry case, we impose different types of surface anchoring energy densities, such as quartic, Rapini-Papoular or more general versions, and, in this case, we show that we can tune any coefficient from the corresponding bulk potential, especially the phase transition temperature. In the case with loss of cubic symmetry, we prove similar results in which the effective free energy functional has now an additional term, which describes a change in the preferred alignment of the liquid crystal particles inside the domain. Moreover, we compute the rate of convergence for how fast the surface energies converge to the homogenised one and also for how fast the minimisers of the free energies tend to the minimiser of the homogenised free energy.
Highlights
We consider a cubic microlattice scaffold constructed of connected particles of micrometer scale, within a nematic liquid crystal
We treat the particles of the cubic microlattice as being inclusions from the mathematical point of view, while they might be interpreted as colloids from the physical point of view, even though they do not possess all of their properties
If (a, b, c) are the parameters from (1.2) of the nematic liquid crystal used in the homogenisation process and (a, b, c ) are the desired parameters for the homogenised material, our goal is to choose the lenghts of the model particle used for constructing the scaffold and a surface energy density fs such that if, for example, the bulk energy chosen is the one from (1.2), in the limit ε → 0, we want to obtain a fhom with the following property: fb(Q) + fhom(Q) = a tr(Q2) − b tr(Q3) + c tr(Q2)2
Summary
We consider a cubic microlattice scaffold constructed of connected particles of micrometer scale, within a nematic liquid crystal. – in the case where the cubic symmetry is lost, we obtain a new term into the homogenised limit that can be seen as a change in the preferred alignment of the liquid crystal particles inside the domain (see Thm. 3.7);. If (a, b, c) are the parameters from (1.2) of the nematic liquid crystal used in the homogenisation process and (a , b , c ) are the desired parameters for the homogenised material, our goal is to choose the lenghts of the model particle used for constructing the scaffold and a surface energy density fs such that if, for example, the bulk energy chosen is the one from (1.2), in the limit ε → 0, we want to obtain a fhom with the following property: fb(Q) + fhom(Q) = a tr(Q2) − b tr(Q3) + c tr(Q2).
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