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Liquid Crystals

Field: Soft Matter Physics

Matter in a state that has properties between those of conventional liquids and those of solid crystals.

Sequence of Expressions

Definition

Order Parameter

Define the nematic director n⃗\vec{n} as the unit vector quantifying the average local orientation of the molecular long axes: n⃗=⟨u/∣u∣⟩/⟨1/∣u∣⟩\vec{n} = \langle \mathbf{u} / |\mathbf{u}| \rangle / \langle 1 / |\mathbf{u}| \rangle, where u\mathbf{u} is the unit vector along a molecule's axis. The degree of orientational order is quantified by the second-rank tensor QijQ_{ij}, the order parameter tensor: \nQij=⟨uiuj−13δijI⟩Q_{ij} = \langle u_i u_j - \frac{1}{3} \delta_{ij} \mathbf{I} \rangle \nFor uniaxial systems, this simplifies to Qij=S(ninj−13δijI)Q_{ij} = S (n_i n_j - \frac{1}{3} \delta_{ij} \mathbf{I}), where SS is the scalar order parameter (0≤S≤10 \le S \le 1).
Let ψ(x)\psi(\mathbf{x}) be the order parameter field. The two-point correlation function is defined as ⟨ψ(x)ψ(x′)⟩\langle \psi(\mathbf{x}) \psi(\mathbf{x}') \rangle. The correlation length ξ\xi is defined by the exponential decay of this function with separation ∣x−x′∣|\mathbf{x} - \mathbf{x}'|: \nlim⁡∣x−x′∣→∞ln⁡⟨ψ(x)ψ(x′)⟩∣x−x′∣=−1ξ\lim_{|\mathbf{x} - \mathbf{x}'| \to \infty} \frac{\ln \langle \psi(\mathbf{x}) \psi(\mathbf{x}') \rangle}{\left|\mathbf{x} - \mathbf{x}'\right|} = -\frac{1}{\xi} \nAlternatively, in Fourier space, ξ−2\xi^{-2} is related to the coefficient of the quadratic term in the inverse susceptibility χ−1(k)\chi^{-1}(\mathbf{k}):\nχ−1(k)≈1ξ2+c2k2+…\chi^{-1}(\mathbf{k}) \approx \frac{1}{\xi^2} + c_2 k^2 + \dots
Consider a dynamical system defined by dxdt=F(x,t)\frac{d\mathbf{x}}{dt} = \mathbf{F}(\mathbf{x}, t). The evolution of a small perturbation δx(t)\delta\mathbf{x}(t) is governed by the linearized equation: \nddtδx=J(x,t)δx\frac{d}{dt} \delta\mathbf{x} = \mathbf{J}(\mathbf{x}, t) \delta\mathbf{x} \nwhere J\mathbf{J} is the Jacobian matrix of F\mathbf{F}. The Lyapunov exponents λk\lambda_k are defined by the asymptotic growth rate of the magnitude of the perturbation: \nλk=lim⁡t→∞1tln⁡(∣δx(t)∣∣δx(0)∣)\lambda_k = \lim_{t \to \infty} \frac{1}{t} \ln \left( \frac{|\delta\mathbf{x}(t)|}{|\delta\mathbf{x}(0)|} \right) \nPositive exponents indicate exponential divergence and chaotic behavior.
Let ψ(x,t)\psi(\mathbf{x}, t) be the order parameter field, and let α\alpha and β\beta be coefficients. The dynamics are governed by the relaxation equation:\n∂ψ∂t=−ΓδFδψ+ξ(x,t)\frac{\partial \psi}{\partial t} = -\Gamma \frac{\delta F}{\delta \psi} + \xi(\mathbf{x}, t)\nwhere the free energy functional FF is defined as:\nF[ψ]=∫Ω[a(T−Tc)ψ2+bψ4+c2∣∇ψ∣2]d3xF[\psi] = \int_{\Omega} \left[ a(T - T_c) \psi^2 + b \psi^4 + \frac{c}{2} |\nabla \psi|^2 \right] d^3x\nand ξ(x,t)\xi(\mathbf{x}, t) is a thermal noise term.
Let n(x,t)\mathbf{n}(\mathbf{x}, t) be the unit director vector field, and let K1,K2,K3K_1, K_2, K_3 be the splay, twist, and bend elastic constants, respectively. The evolution of n\mathbf{n} is governed by the torque balance equation:\nn×(∂n∂t+csn×∂n∂t)=−δFelδn\mathbf{n} \times \left( \frac{\partial \mathbf{n}}{\partial t} + c_s \mathbf{n} \times \frac{\partial \mathbf{n}}{\partial t} \right) = -\frac{\delta F_{el}}{\delta \mathbf{n}}\nwhere the elastic free energy density felf_{el} is:\nfel=12[K1(nx⋅∇nx)2+K2(ny⋅∇ny)2+K3(nz⋅∇nz)2]f_{el} = \frac{1}{2} [K_1 (n_x \cdot \nabla n_x)^2 + K_2 (n_y \cdot \nabla n_y)^2 + K_3 (n_z \cdot \nabla n_z)^2]\n(Note: The full form involves derivatives of n\mathbf{n} and is simplified here for clarity of the governing principle.)
Let n0\mathbf{n}_0 be the uniform, constant director vector. The Kelvin state is characterized by the minimization of the free energy functional F[n]F[\mathbf{n}] subject to the constraint of uniform orientation, implying that the order parameter ψ\psi is constant in space and time:\n∂ψ∂t=0,∇ψ=0\frac{\partial \psi}{\partial t} = 0, \quad \nabla \psi = \mathbf{0} \nFurthermore, the free energy density ff must achieve its minimum value fminf_{min} corresponding to the maximum entropy state, such that the elastic energy contribution vanishes:\nF[n]=∫Ωfmind3xF[\mathbf{n}] = \int_{\Omega} f_{min} d^3x
Let qj(t)\mathbf{q}_{j}(t) be the displacement of the jj-th molecule from its equilibrium position, and ωj\omega_{j} be its natural frequency. The collective dynamics are modeled by the coupled equations of motion for the generalized coordinates q=(q1,…,qN)\mathbf{q} = (\mathbf{q}_1, \dots, \mathbf{q}_N): \nd2qdt2+Γdqdt+Kq=Fext(t)\frac{d^2 \mathbf{q}}{dt^2} + \Gamma \frac{d \mathbf{q}}{dt} + \mathbf{K} \mathbf{q} = \mathbf{F}_{ext}(t) \nwhere Γ\Gamma is the damping matrix, and K\mathbf{K} is the coupling matrix derived from the harmonic potential energy V(q)=12qTKqV(\mathbf{q}) = \frac{1}{2} \mathbf{q}^T \mathbf{K} \mathbf{q}. The low-temperature dynamics are analyzed by solving this system in the frequency domain.
Let GG be the point group symmetry of the liquid crystal phase. The physical properties are invariant under the action of GG. The order parameters ηi\eta_{i} must transform according to the irreducible representations (irreps) Γk\Gamma_{k} of GG. The free energy density ff must be a scalar invariant under the group action: \nf(ηi)=f(g⋅ηi)∀g∈Gf(\eta_{i}) = f(g \cdot \eta_{i}) \quad \forall g \in G \nThis requires that ff can be expressed as a linear combination of the basis invariants of the group.
Consider the free energy density ff of the system, which depends on the local density ρ\rho and the order parameter SS. The transition is modeled by minimizing the free energy functional F\mathcal{F}: \nF[ρ,S]=∫[fliquid(ρ,S)+fsolid(ρ,S)]d3r\mathcal{F}[\rho, S] = \int \left[ f_{liquid}(\rho, S) + f_{solid}(\rho, S) \right] d^3r \nThe melting transition is characterized by the condition that the free energy difference between the liquid and solid phases vanishes at the transition temperature TmT_m: \nΔf=fsolid(ρm,Sm,Tm)−fliquid(ρm,Sm,Tm)=0\Delta f = f_{solid}(\rho_m, S_m, T_m) - f_{liquid}(\rho_m, S_m, T_m) = 0 \nThis requires the system to satisfy the equilibrium condition δFδρ=0\frac{\delta \mathcal{F}}{\delta \rho} = 0 and δFδS=0\frac{\delta \mathcal{F}}{\delta S} = 0 simultaneously.
Define the total free energy functional F[r]F[\mathbf{r}] for the system, where r\mathbf{r} represents the molecular configuration. The contribution from weaker bonds, FweakF_{weak}, is modeled by a directional potential VdirV_{dir} acting on the bond vectors rij\mathbf{r}_{ij}. The effective Hamiltonian is then modified such that:\nHeff=Hstandard+∑i<jVdir(rij)⋅cos⁡(θij−θpref)H_{eff} = H_{standard} + \sum_{i<j} V_{dir}(\mathbf{r}_{ij}) \cdot \cos(\theta_{ij} - \theta_{pref})\nwhere θij\theta_{ij} is the angle between the bond rij\mathbf{r}_{ij} and the preferred alignment direction θpref\theta_{pref}, and VdirV_{dir} quantifies the strength of the weaker bond interaction.