A novel Landau-de Gennes model with quartic elastic terms
A novel Landau-de Gennes model with quartic elastic terms
复制标题
DOI：
10.1017/s095679252000008x
复制
发表时间：
2021-02-01
影响因子：
1.9
通讯作者：
Sternberg, Peter
中科院分区：
文献类型：
--
作者：
Golovaty, Dmitry;Novack, Michael;Sternberg, Peter
在广义Landau-de Gennes理论的框架内，我们确定了一个基于q张量的能量，当它被认为是可定向的单轴向列态时，它可以减少到四常数osee - frank能量。虽然通常认为的朗多-德-热纳理论的弹性贡献在q张量及其导数的分量中最多是三次的，但这里提供的替代方案是这些变量的四次。与三次理论相比，我们的方法的一个明显优势是，相关的最小化问题适用于更广泛的弹性常数选择。特别是，这种四次能量可以用来模拟高度不同弹性常数的向列到各向同性相变。除了证明Landau-de Gennes理论的拟合性外，我们还通过在消失向列相关长度极限下的伽马收敛论证，建立了该理论与Oseen-Frank对应理论之间的严格联系。我们还证明了相关最小值的强收敛性。
Within the framework of the generalised Landau-de Gennes theory, we identify a Q-tensor-based energy that reduces to the four-constant Oseen-Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the Q-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimisation problem is well-posed for a significantly wider choice of elastic constants. In particular, this quartic energy can be used to model nematic-to-isotropic phase transitions for highly disparate elastic constants. In addition to proving well-posedness of the proposed version of the Landau-de Gennes theory, we establish a rigorous connection between this theory and its Oseen-Frank counterpart via a Gamma-convergence argument in the limit of vanishing nematic correlation length. We also prove strong convergence of the associated minimisers.