Impact of random-field-type disorder on nematic liquid crystalline structures

Abstract

We study bicomponent systems where one component represents a liquid crystalline (LC) phase, and the other component randomly perturbs the LC order. Such systems can serve as a testbed to systematically analyse the impact of qualitatively different types of random-type sources of perturbation on the orientational and/or translational order. This mini-review presents typical representatives of such systems, where orientational and translational order is probed in nematic and smectic A LCs, respectively. As a source of perturbation, we consider either different porous matrices (control-pore glass, aerogels) or aerosil nanoparticles, which can form in LCs' different fractal-like network organizations. In such complex systems, LC ordering fingerprints the interplay among LC elastic forces, interfacial forces, and randomness. The resulting LC behaviour could be characterised by either long-range, quasi long-range, or short-range order. We demonstrate under which conditions random-field-like phenomena or interfacial effects dominate. However, these effects are relatively strongly entangled in most experimental systems, and individual impacts cannot be precisely identified.

Graphical abstract

Appendices

Appendix A

Mesoscopic bulk LC behaviour

Below we present a simple phenomenological description of the LC ordering using the Landau-de Gennes–Ginzburg [8]-type phenomenological description.

The orientational ordering is presented by the nematic tensor order parameter $\underline{Q}$. The latter allows both uniaxial and biaxial states. In this presentation, we neglect biaxial states and consequently, we use the parametrisation in terms of the nematic director field $\overrightarrow{n}$ and the nematic uniaxial order parameter S as

$$\underline{Q}=S\left(\overrightarrow{n}\otimes \overrightarrow{n}-\underline{I}/3\right)$$

(A1)

Note that states with S > 0 (S < 0) determine to prolate (oblate) uniaxial shapes in the mesoscopic geometric presentation of nematic order. In bulk equilibrium $\underline{Q}$ is spatially homogeneous.

The SmA layering is described by the complex order parameter in terms of the smectic phase $\phi$ and the translational order parameter amplitude $\eta :$

$$\psi =\eta {e}^{i\phi }$$

(A2)

It approximately describes the LC mass density spatial variation $\rho ={\rho }_{0}\left(1+\psi +{\psi }^{*}\right),$ where ${\rho }_{0}$ is a constant. In bulk equilibrium $\eta$ is spatially homogeneous, $\phi ={q}_{0}\overrightarrow{n}.\overrightarrow{r}$ and the periodicity ${q}_{0}=2\pi /{d}_{0}$ determines the smectic layer spacing d₀.

Thermotropic LCs exhibiting either an I-N-SmA phase sequence or a direct I-SmA phase transition are considered below. In addition, we consider cases where LC is either confined by an interface (e.g., in CPG matrices) or encloses NPs (e.g., aerosils).

The free energy F of the system is expressed as

$$F=\iiint \left({f}_{c}^{(n)}+{f}_{e}^{(n)}+{f}_{c}^{(s)}+{f}_{e}^{(s)}+{f}_{cp}\right){d}^{3}r+\iint \left({f}_{i}^{(n)}+{f}_{i}^{(s)}\right){d}^{2}r$$

(A3)

where the first integral runs over the LC body and the second integral over the interfaces in contact with LC. Using minimal model, we express the free energy densities in Eq. (A3) as

$${f}_{c}^{(n)}={a}_{n}\left(T-{T}_{n}^{*}\right)Tr{\underline{Q}}^{2}-{b}_{n}{\underline{Q}}^{3}+{c}_{n}{\left({\underline{Q}}^{2}\right)}^{2}$$

(A4a)

$${f}_{c}^{(s)}={a}_{s}\left(T-{T}_{s}^{*}\right){\left|\psi \right|}^{2}+{b}_{s}{\left|\psi \right|}^{4}$$

(A4b)

$${f}_{e}^{(n)}=L{\left|\nabla \underline{Q}\right|}^{2}$$

(A4c)

$${f}_{e}^{(s)}={C}_{\parallel }{\left|\left(i{q}_{0}\overrightarrow{n}-\nabla \right)\psi \right|}^{2}+{C}_{\perp }{\left|\left(\overrightarrow{n}\times \nabla \right)\psi \right|}^{2}$$

(A4d)

$${f}_{cp}=-D\nabla \psi .\underline{Q}\nabla {\psi }^{*}$$

(A4e)

$${f}_{i}^{(n)}=-{W}_{n}\overrightarrow{e}.\underline{Q}\overrightarrow{e}$$

(A4f)

$${f}_{i}^{(s)}=-\left({W}_{s}^{*}\psi +{W}_{s}{\psi }^{*}\right)$$

(A4g)

The condensation nematic ${(f}_{c}^{(n)})$, condensation smectic ${(f}_{c}^{(s)})$, and the coupling $\left({f}_{cp}\right)$ terms determine the equilibrium value of the nematic ($S={S}_{eq}$) and smectic ($\eta ={\eta }_{eq}$) order parameter in bulk elastically undistorted samples. The quantities ${a}_{n},{b}_{n},{c}_{n},{T}_{n}^{*},$ ${a}_{s},{b}_{s},{T}_{s}^{*},D$ are positive material constants. In particular, $D$ determines the coupling strength between the nematic and the smectic order parameters. The nematic and smectic elastic properties are determined by the nematic elastic constant L (we use the approximation of equal nematic elastic constants) and the smectic compressibility ${(C}_{\parallel })$ and smectic bend ${(C}_{\perp })$ elastic constant. The positive nematic elastic constant enforces spatially homogeneous nematic order. Furthermore, the positive smectic elastic constants enforce smectic layer spacing ${d}_{0}=2\pi /{q}_{0}$ and orientation of $\overrightarrow{n}$ along the smectic layer normal. The interfacial terms are characterised by positive nematic (${W}_{n}$) and smectic ${(W}_{s})$ surface interaction strengths, and $\overrightarrow{e}$ stands for the easy (i.e., local interface preferred) orientation. Such surface interaction terms tend to support LC ordering.

If the nematic and smectic order parameters are decoupled (D = 0) the equilibrium phase behaviour is as follows. The isotropic supercooling temperature is equal to ${T}_{n}^{*}$, and the first order I-N phase transition takes place at ${T}_{IN}={T}_{n}^{*}+\frac{{b}_{n}^{2}}{24{a}_{n}{c}_{n}}$. Above and below ${T}_{IN}$ it holds ${S}_{eq}\left[T>{T}_{IN}\right]=0$ and ${S}_{eq}\left[T\le {T}_{IN}\right]={S}_{0}\frac{3+\sqrt{9-8\left(T-{T}_{n}^{*}\right)/({T}_{IN}-{T}_{n}^{*})}}{4}$, where ${S}_{0}={S}_{eq}\left[T={T}_{IN}\right]=\frac{{b}_{n}}{4{c}_{n}}.$ The second order N-SmA phase transition is realised at ${T}_{NA}={T}_{s}^{*}<{T}_{IN}.$ It holds ${\eta }_{eq}\left[T>{T}_{NA}\right]=0$ and ${\eta }_{eq}\left[T\le {T}_{NA}\right]={\eta }_{0}\sqrt{{(T}_{NA}-T)/{T}_{NA}}$, ${\eta }_{0}=\sqrt{{T}_{NA}{a}_{s}/(2{b}_{s})}$.

In the case of coupled order parameters it is convenient to introduce scaled order parameter amplitudes: $\widetilde{S}=\frac{S}{{S}_{0}}, \widetilde{\eta }=\frac{\eta }{{\eta }_{0}},$ dimensionless temperatures ${r}_{n}=\frac{\left(T-{T}_{n}^{*}\right)}{{T}_{IN}-{T}_{n}^{*}}$ and ${r}_{s}=\frac{\left(T-{T}_{s}^{*}\right)}{{T}_{s}^{*}}$, the dimensionless free energy density $\widetilde{f}=\frac{3f}{{a}_{n}{(T}_{IN}-{T}_{n}^{*}){s}_{0}^{2}}$, where $f={f}_{c}^{(n)}+{f}_{e}^{(n)}+{f}_{c}^{(s)}+{f}_{e}^{(s)}+{f}_{cp},$ and the dimensionless coupling strength $\widetilde{D}=\frac{3D{a}_{s}^{2}{T}_{s}^{*}{\eta }_{0}^{2}}{{2a}_{n}{(T}_{IN}-{T}_{n}^{*}){s}_{0}^{2}}.$ In the absence of elastic distortions it follows

$$\widetilde{f}={r}_{n}{\widetilde{S}}^{2}-2{\widetilde{S}}^{3}+{\widetilde{S}}^{4}+\Omega \left({r}_{s}{\widetilde{\eta }}^{2}+\frac{{\widetilde{\eta }}^{4}}{2}\right)-\widetilde{D}{\widetilde{\eta }}^{2}\widetilde{S}$$

(A5)

where $\Omega =\frac{3{a}_{s}^{2}{T}_{s}^{*}{\eta }_{0}^{2}}{{2a}_{n}{(T}_{IN}-{T}_{n}^{*}){S}_{0}^{2}}$. The equilibrium LC order is obtained by minimizing $\widetilde{f}$ with respect to $\widetilde{S}$ and $\widetilde{\eta }$. On increasing $\widetilde{D}$ two qualitative changes in LC behaviour appear, which are determined by the critical ${\widetilde{D}}_{c}^{(1)}$ and ${\widetilde{D}}_{c}^{(2)}$. Below ${\widetilde{D}}_{c}^{(1)}$ the N-SmA phase transition is continuous and discontinuous in the regime ${\widetilde{D}}_{c}^{(1)}<\widetilde{D}<{\widetilde{D}}_{c}^{(2)}$. For $\widetilde{D}>{\widetilde{D}}_{c}^{(2)}$ the system exhibits a first order I-SmA phase transition.