Effect of magnetic nanoparticles on the nematic-smectic-A phase transition

TL;DR

This paper develops a phenomenological mean-field model to analyze how ferromagnetic nanoparticles influence the nematic-smectic-A phase transition in liquid crystals, revealing a critical impurity concentration that alters transition order.

Contribution

It introduces a new mean-field model combining Flory-Huggins and Landau-de Gennes theories to quantify nanoparticle effects on phase transitions in liquid crystal mixtures.

Findings

Identification of a critical nanoparticle concentration causing a transition from second to first order.

Model predictions align well with experimental observations.

Analysis of phase diagram topologies under varying impurity levels.

Abstract

Recent experiments on mixed liquid crystals have highlighted the hugely significant role of ferromagnetic nanoparticle impurities in defining the nematic-smectic-A phase transition point. Structured around a Flory-Huggins free energy of isotropic mixing and Landau-de Gennes free energy, this article presents a phenomenological mean-field model that quantifies the role of such impurities in analyzing thermodynamic phases, in a mixture of thermotropic smectic liquid crystal and ferromagnetic nanoparticles. First we discuss the impact of ferromagnetic nanoparticles on the isotropic-ferronematic and ferronematic-ferrosmectic phase transitions and their transition temperatures. This is followed by analysis of various topologies in the phase diagrams. Our model results indicate that there exists a critical concentration of nanoparticle impurities for which the second order N-SmA transition becomes first order at a tricritical point. Calculations based on this model show remarkable agreement with experiment.

Equations

$$f=f_{\text{mix}}+f_{NP}+F_{LC}+f_{\text{int}},$$

$$\frac{f_{\text{mix}}}{k_{B}T}=\phi\ln\phi+(1-\phi)\ln(1-\phi)+\chi\phi(1-\phi)+\frac{\lambda}{2}{\bigg{(}{\bf\nabla}\phi\bigg{)}}^{2},$$

$$f_{NP}=\varphi \left[\frac{1}{2}p{\mathbf{M}}^2+\frac{1}{4}q{\mathbf{M}}^4\right]$$

$$f_{LC}$$

$$f_{\text{int}}=-\varphi (1-\varphi )\left[\frac{1}{2}\gamma M_i{M}_j{Q}_{ij}+\frac{1}{3}{\eta }_1{\mathbf{M}}^2{Q}_{ij}{Q}_{ij}+\frac{1}{2}{\eta }_2{M}_i{M}_k{Q}_{ij}{Q}_{kj}+\frac{1}{2}\omega {\mathbf{M}}^2{\left|\psi \right|}^2\right]$$

$$f$$

$$\frac{{\partial }^{2}f}{\partial S^2}>0,,\frac{{\partial }^{2}f}{\partial M^2}>0,\frac{{\partial }^{2}f}{\partial {\psi }_0^2}>0,$$

$$\frac{{\partial }^{2}f}{\partial u^2}\cdot \frac{{\partial }^{2}f}{\partial v^2}-{\left(\frac{{\partial }^{2}f}{\partial u\partial v}\right)}^2$$

$$a^{\ast }-2b^{\ast }S+3c^{\ast }{S}^2>0,$$

$$p-(1-\varphi )({\gamma }^{\prime }S+{\eta }^{\prime }{S}^2)<0,$$

$$a_1-2b_{1}S+3c_1{S}^2>0,$$

$$f$$

$$M^2=-\frac{1}{q}(p-(1-\varphi )({\gamma }^{\prime }S+{\eta }^{\prime }{S}^2)),$$

$$\varphi \frac{p{\gamma }^{\prime }}{2q}+a^{\ast }S-b^{\ast }{S}^2+c^{\ast }{S}^3=0$$

$$f_{NA}$$

$$f_{FN}(S)=F_0(T),f_{FN}^{\prime }(S)=0,f_{FN}^{\prime \prime }(S)\ge 0$$

$$T<T_{I-FN}^{\ast }+\frac{\varphi (1-\varphi ){\gamma }^{\prime 2}}{2q{a}_0^{\ast }}$$

$$\Delta T_{I-FN}=\frac{\varphi (1-\varphi ){\gamma }^{\prime 2}}{2q{a}_0^{\ast }}$$

$$a^{\ast \ast }-2b^{\ast \ast }S+3c^{\ast \ast }{S}^2>0,$$

$${\alpha }^{\ast }-{\delta }^{\ast }S-\frac{\varphi (1-\varphi )\omega {\eta }^{\prime }}{q}{S}^2<0,$$

$$p^{\ast }-(1-\varphi ){\gamma }^{\ast }S-(1-\varphi ){\eta }^{\ast }{S}^2<0.$$

$$a_2-2b_{2}S+2c_2{S}^2>0,$$

$$\omega <\sqrt{q\beta /\varphi (1-\varphi )}$$

$${\psi }_0^2$$

$$M^{2}=-\frac{1}{q}\bigg{(}p^{*}-(1-\phi)(\gamma^{*}S+\eta^{*}S^{2})\bigg{)}.$$

$$\frac{\delta {\alpha }^{\ast }}{2{\beta }^{\ast }}+\frac{\varphi {\gamma }^{\prime }{p}^{\ast }}{2q}+a^{\ast \ast }S-b^{\ast \ast }{S}^2+c^{\ast \ast }{S}^3=0.$$

$$f_{FSmA}$$

$$f_{FSmA}(S)=f_{FN}(S),f_{FSmA}^{\prime }(S)=0,f_{FSmA}^{\prime \prime }(S)\ge 0$$

$${\alpha }^{\ast }-{\delta }^{\ast }S-\frac{\varphi (1-\varphi )\omega {\eta }^{\prime }}{q}{S}^2=0,f_{FN}^{\prime }(S)=0,f_{FSmA}^{\prime \prime }(S)\ge 0$$

$$f$$

$$F=F_{FN}^{\ast }(S_0)+\frac{1}{2}{\alpha }^{\ast \ast }{\psi }_0^2+\frac{1}{4}{\beta }^{\ast \ast }{\psi }_0^4$$

Full text

Recent experiments on mixed liquid crystals have highlighted the hugely significant role of ferromagnetic nanoparticle impurities in defining the nematic-smectic-A phase transition point. Structured around a Flory-Huggins free energy of isotropic mixing and Landau-de Gennes free energy, this article presents a phenomenological mean-field model that quantifies the role of such impurities in analyzing thermodynamic phases, in a mixture of thermotropic smectic liquid crystal and ferromagnetic nanoparticles. First we discuss the impact of ferromagnetic nanoparticles on the isotropic-ferronematic and ferronematic-ferrosmectic phase transitions and their transition temperatures. This is followed by plotting and discussing various topologies in the phase diagrams. Our model results indicate that there exists a critical concentration of nanoparticle impurities for which the second order N-SmA transition becomes first order at a tricritical point. Calculations based on this model show remarkable agreement with experiment.

Liquid crystals; nanoparticles; phase transition

I Introduction

In recent years, many experiments have found that liquid crystals doped with dispersive materials e.g. carbon nanotubes, silica microbeads, nanoparticles and various collides exhibit remarkable new physical phenomena. Experiments have shown that nanoparticles and ferroelectric nanoparticles can greatly enhance the physical properties of nematic and smectic liquid crystals reznikov ; li1 ; milkulko ; blach ; cook ; reshetnyak ; kurochkim ; ouskova ; gharde ; basu ; rzoska .

The mixture of nematic and smectic liquid crystals and magnetic nanoparticles is known as a ferronematic and ferrosmectic. A number of experimental studies miranda ; potocova ; cordo3 ; fabre ; ramos ; ponsinet1 ; ponsinet2 ; spoliansky1 ; spoliansky2 are devoted to study of ferrosmectic phase in both in thermotropic and lyotropic liquid crystals. The most essential feature of the ferrosmectics is that their structure aligns parallel to an external magnetic field. Potocova et al. potocova studied the structural instabilities of the ferronematic and ferrosmectic phases prepared from 8CB (CH3(CH2)7(C6H4)2CN). Martínez-Miranda et al. miranda studied how the surface coating interacts with the liquid crystal in conjunction with the ferromagnetic nanoparticles (FNP). They have found out that depending on the surface coating the interaction of the ferromagnetic nanoparticles with the liquid crystal varies. Cordoyiannis et al. cordo3 experimentally studied the impact of magnetic nanoparticles on the isotropic-nematic (I-N) and nematic-Semectic-A (N-SmA) phase transitions of 8CB. This work shows that the I-N transition remains weakly first order even in the presence of FNP. For the N-SmA phase transition the results of this work show a crossover from anisotropic criticality towards tricriticality.

A large number of theoretical works lopatina1 ; lopatina2 ; soule ; popa1 ; kralj ; muk1 ; muk2 have been carried out to describe the effect of anisotropic nanoparticles, carbon nanotubes and ferroelectric nanoparticles in nematic and smectic liquid crystals. However, very few or practically no theoretical work has been attempted to explain the phase behavior of the N-SmA phase transition in thermotropic liquid crystals (TLC) in the mixture of FNP. Only one theoretical work muk3 on ferrosmectic phase in LLC is available in the literature but that is not quantitatively explicit either. Thus it is interesting to see how the ferromagnetic nanoparticles influence the character of the N-SmA phase transition in TLC. Based on this core question, here we to develop a phenomenological model structured around the Flory-Huggins theory flory to discuss the I-N and N-SmA phase transitions in the in the mixture of FNP.

II Model

In this section, we use the combination of Flory-Huggins theory and Landau-de Gennes theory for the binary mixture of calamitic SmA liquid crystal and ferromagnetic nanoparticles. First we describe the order parameters necessary in the model free energy. The smectic-A phase has both the orientational and translational ordering. The nematic order parameter, originally proposed by de Gennes degennes , is a symmetric, traceless tensor described by $Q_{ij}=\frac{S}{2}(3n_{i}n_{j}-\delta_{ij})$, where $n_{i}$ are unit vectors specifying the preferred orientation of the primary molecular axes, also called directors. The quantity $S$ defines the strength of the nematic ordering . The layering in the SmA phase is characterized degennes by the order parameter $\psi(\bf{r})=\psi_{0}\exp(-i\Psi)$, which is a complex scalar quantity whose modulus $\psi_{0}$ is defined as the amplitude of a one dimensional density wave characterized by the phase $\Psi$. The magnetic order is described by the magnetization ${\bf{M}}=M\hat{\bf{m}}$ such that $|M|=0$ in paramagnetic state and $|M|\neq 0$ in the ferromagnetic state. Thus we use $Q_{ij}$, $\psi$, and $\bf{M}$ as order parameters necessary for the description of the I-N and N-SmA phase transitions in the mixture of FNP.

The total free energy per unit volume of the mixture can be written as

[TABLE]

where $f_{mix}$ is the free energy mixing of isotropic liquids; $f_{NP}$ describes the contribution of the FNP dispersed in liquid crystal, $f_{LC}$ represents the free energy of SmA ordering of liquid crystals and $f_{int}$ describes the coupling between FNP and SmA ordering, respectively.

The isotropic mixing free energy density may be approximated in terms of the Flory-Huggins theory flory

[TABLE]

where $k_{B}$ is the Boltzmann constant and $T$ is absolute temperature. $\phi$ and $(1-\phi)$ describe the volume fractions of FNP and SmA liquid crystals. \frac{\lambda}{2}{\bigg{(}{\bf\nabla}\phi\bigg{)}}^{2} is the Ginzburg-Landau term. $\chi$ is a temperature-dependent function which can be defined as $\chi=A+\frac{B}{T}$ is known as the Flory-Huggins interaction parameter with $A$ and $B$ are constants.

The contribution of the FNP free energy density can be expressed as

[TABLE]

The material parameter $p$ can be assumed as $p=p_{0}(T-T_{3}^{*})$. $p_{0}$ is a positive constant and $T_{3}^{*}$ is the virtual transition temperature. We assume $q>0$ for the stability of the free energy.

The SmA free energy density can be expressed as

[TABLE]

$a$ and $\alpha$ can be assumed as $a=a_{0}(T-T_{1}^{*})$ and $\alpha=\alpha_{0}(T-T_{2}^{*})$, $a_{0}>0$ and $\alpha_{0}>0$. $T_{1}^{*}$ and $T_{2}^{*}$ are the virtual transition temperatures. We choose $c>0$, $b>0$ and $\beta>0$ for the stability of the free energy density (4). Equations (2.2)-(2.5) are all tacitly structured around the standard symmetricity argument.

The contribution to free energy density due the interactions is written as

[TABLE]

The parameters $\delta$, $\gamma$, $\eta_{1,2}$, and $\omega$ are coupling constants. $\delta$ is chosen positive to favor the smectic-A phase over the nematic phases The positive values of $\gamma$, $\eta_{1,2}$, and $\omega$ ensures the ferromagnetic order induced by the nematic order and translational order.

Following Pleiner et al. pleiner we consider the ordering directions between $\hat{\bf{n}}$ and $\hat{\bf{m}}$ make an angle $\theta_{f}$ i.e. $\hat{\bf{n}}\cdot\hat{\bf{m}}=\cos\theta_{f}$. Then the total free energy density (1) leads to

[TABLE]

Minimization of Eq. (6) with respect to $S$, $\psi_{0}$, $M$ and $\theta_{f}$ yields the following five stable solutions excluding the ferromagnetic state:

1. (I)

Isotropic phase (I): $S=0$, $\psi_{0}=0$, $M=0$, $\theta_{f}=0$. This phase exists for $a>0$, $\alpha>0$, and $p>0$. 2. (II)

Nematic phase (N): $S\neq 0$, $M=0$, $\psi_{0}=0$, $\theta_{f}=0$. This phase exists for $a<0$, $\alpha-\delta S>0$, $p-(1-\phi)\gamma^{\prime}S-(1-\phi)\eta^{\prime}S^{2}>0$. 3. (III)

Smectic-A phase (SmA): $S\neq 0$, $\psi_{0}\neq 0$, $M=0$, $\theta_{f}=0$.

This phase exists for $a<0$, $\alpha-\delta S<0$, and $p-(1-\phi)\gamma^{\prime}S-(1-\phi)\eta^{\prime}S^{2}+\omega\psi_{0}^{2}>0$. 4. (IV)

Ferronematic phase (FN): $S\neq 0$, $\psi_{0}=0$, $M\neq 0$, $\theta_{f}=0$ or $\theta_{f}=\frac{\pi}{2}$. This phase exists for $a<0$, $p-(1-\phi)\gamma^{\prime}S-(1-\phi)\eta^{\prime}S^{2}<0$, $\alpha-\delta S>0$. 5. (V)

Ferrosmectic phase (FSmA): $S\neq 0$, $\psi_{0}\neq 0$, $M\neq 0$, $\theta_{f}=0$ or $\theta_{f}=\frac{\pi}{2}$. This phase exists for $a<0$, $p-(1-\phi)\gamma^{\prime}S-(1-\phi)\eta^{\prime}S^{2}+\omega\psi_{0}^{2}<0$, $\alpha-\delta S<0$.

In the description above, we have used $\eta^{\prime}=\eta_{1}+\frac{\eta_{2}}{4}(3\cos^{2}\theta_{f}+1)$ and $\gamma^{\prime}=\frac{1}{2}(3\cos^{2}\theta_{f}-1)\gamma$. For the specific cases in hand, $\gamma^{\prime}=\gamma(or-\frac{\gamma}{2})$ for $\theta_{f}=0$ (or $\frac{\pi}{2}$) and $\eta^{\prime}=(\eta_{1}+\eta_{2})$ (or $(\eta_{1}+\eta_{2}/2)$) for $\theta_{f}=0$ (or $\frac{\pi}{2})$.

The necessary conditions for the four different phases to be stable (Hessian determinant) are given below:

[TABLE]

where $u,v\in\{S,\psi_{0},M\}$.

Now it is clear from the above solutions that I-N, I-SmA, I-FN, I-FSmA, N-SmA, N-FN, FN-FSmA, FN-SmA, SmA-FSmA phase transitions are possible. I-N, I-SmA, I-FN, I-FSmA phase transitions must always be first order because of the cubic invariant $b$ in the free energy expansion (6). Other phase transitions can be first or second order depending on the concentration of FNP. In the following we will discuss only the I-FN and FN-FSmA phase transitions which is observed experimentally.

II.1 I-FN phase transition

In order to ensure the stability of the FN phase, we require

[TABLE]

[TABLE]

[TABLE]

where

$a^{*}=a-\phi(1-\phi)\frac{\gamma^{\prime 2}}{2q}+\phi\frac{p\eta^{\prime}}{q}$, $b^{*}=b+\phi(1-\phi)\frac{3\gamma^{\prime}\eta^{\prime}}{2q}$, $c^{*}=c-\phi(1-\phi)\frac{\eta^{\prime 2}}{q}$, $a_{1}=a^{*}-\phi(1-\phi)\frac{\gamma^{\prime 2}}{2q}$, $b_{1}=b^{*}+\phi(1-\phi)\frac{\gamma^{\prime}\eta^{\prime}}{q}$, $c_{1}=c^{*}-\phi(1-\phi)\frac{2\eta^{\prime 2}}{3q}$.

The renormalized coefficients show that the Landau coefficients $a^{*}$, $b^{*}$ and $c^{*}$ change with change of the concentration of FNP.

Now the free energy density near the I-FN phase transition can be expressed as

[TABLE]

The value of the magnetization in the FN phase can be expressed as

[TABLE]

where the value of $S$ in the FN phase can be calculated from the equations

[TABLE]

The temperature variation of the order parameter ($S$) for pure sample and a fixed concentration of FNP in the FN phase is shown in Fig.1. This is done for a set of phenomenological parameters for which the direct I-N and I-FN phase transitions are possible. Figure 1 shows that I-FN transition temperature and the jump of the order parameter $S$ decrease with the increase of the concentration of FNP. For a fixed set of parameter values, we find for pure sample $S_{I-N}=0.4$ and $T_{I-N}=313.36K$. For the volume fraction $\phi=0.02$, we find $T_{I-FN}=312.51$ are $S_{I-FN}=0.31$. The low value of $S_{I-FN}$ indicates the weakly first order character of the I-FN phase transition. Thus the I-FN transition is still a weakly first order transition even in the mixture of FNP. The present analysis completely agree with experimental results of Cordoyiannis et al. cordo3 .

The substitution of $M$ from Eq. (11) into Eq. (10), we get

[TABLE]

The free energy density (13) describes the I-FN phase transition. At the dimensional level, the Ginzburg-Landau free energy term [\frac{\lambda}{2}{\big{(}{\bf\nabla}\phi\big{)}}^{2}] effectively renormalizes the value of the parameter $\chi$ by rescaling the quadratic $\phi^{2}$-term with a $\frac{\lambda}{L^{2}}$ type term, where $L$ is the typical scaling length of the system. Beyond the mean-field level, the $\frac{1}{L^{2}}$-dependence will further renormalize the spatial correlation function. The cubic coefficient $b^{*}$ in the free energy density (13) shows that the I-FN phase transition must always be first order in mean field approximation. Lower the value of $b^{*}$, weakly the first order character of the I-FN phase transition.

The conditions for the first order I-FN phase transition can be obtained as

[TABLE]

The conditions for phase equilibrium require that the chemical potentials in the isotropic and FN phases are equivalent i.e $\mu_{iso}=\mu_{FN}$.

The FN phase appears only for $a^{*}<0$ $i.e.$

[TABLE]

where $T_{I-FN}^{*}=\frac{a_{0}T_{1}^{*}+\frac{\phi\eta^{\prime}p_{0}T_{f}}{q}}{a_{0}^{*}}$,

$a_{0}^{*}=a_{0}+\frac{\phi\eta^{\prime}p_{0}}{q}$.

From Eq. (15) we observe the decrease of the I-FN transition temperature with the increase of the concentration of FNP as

[TABLE]

II.2 FN-FSmA phase transition

We now discuss the FN-FSmA phase transition. In order to ensure the stability of the FSmA phase we require

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

$a^{**}=a-\frac{\delta\delta^{*}}{2\beta^{*}}-\frac{\phi(1-\phi)\gamma^{*}\gamma^{\prime}}{2q}+\frac{\phi\eta^{\prime}p^{*}}{q}$, $b^{**}=b+\frac{\phi(1-\phi)\delta\omega\eta^{\prime}}{2q\beta^{*}}+\frac{\phi(1-\phi)\gamma^{*}\eta^{\prime}}{q}+\frac{\phi(1-\phi)\gamma^{\prime}\eta^{*}}{q}$, $c^{**}=c-\frac{\phi(1-\phi)\eta^{\prime}\eta^{*}}{q}$, $a_{2}=a^{**}-\frac{\phi(1-\phi)\gamma^{\prime 2}}{2q}$, $b_{2}=b^{**}+\frac{\phi(1-\phi)\gamma^{\prime}\eta^{\prime}}{q}$, $c_{2}=c^{**}-\frac{\phi(1-\phi)2\eta^{\prime 2}}{3q}$, $\alpha^{*}=\alpha+\frac{\phi\omega p}{q}$, $\delta^{*}=\delta+\frac{\phi(1-\phi)\omega\gamma^{\prime}}{q}$, $\beta^{*}=\beta-\frac{\phi(1-\phi)\omega^{2}}{q}$, $p^{*}=p+\frac{(1-\phi)\omega\alpha^{*}}{\beta^{*}}$, $\eta^{*}=\eta^{\prime}+\frac{\phi(1-\phi)\omega\eta^{\prime}}{q\beta^{*}}$, $\gamma^{*}=\gamma^{\prime}+\frac{\omega\delta^{*}}{\beta^{*}}$.

The above renormalized coefficients show that the interaction parameters $\delta$, $\eta^{\prime}$ and $\gamma^{\prime}$ change with the change of the concentration of FNP.

The values of the smectic ordering and the magnetization in the FSmA phase can be expressed as

[TABLE]

[TABLE]

where the values of $S$ in the FSmA phase can be calculated from the equation

[TABLE]

Equation (24) is a cubic equation that admits of an exact solution (Cardan’s method) involving the parameters involved.

It is clear from Eq. (22) and Eq. (23) that a nonzero real value of $\psi_{0}$ and $M$ exist only when $\alpha^{*}-\delta^{*}S-\frac{\phi(1-\phi)\omega\eta^{\prime}}{q}S^{2}<0$ and $p^{*}-(1-\phi)(\gamma^{*}S+\eta^{*}S^{2})<0$. Since there is a small temperature range where $\alpha>0$, $\delta>0$, $\gamma^{\prime}>0$ and $\eta^{\prime}<0$ in this region.

The substitution of $\psi_{0}$ and $M$ from Eqs. (22) and (23) into Eq. (6) gives

[TABLE]

The free energy density (25) describes the FN-FSmA phase transition. The cubic coefficient $b^{**}$ in the free energy density (5) shows that the FN-FSmA phase transition must be first order in mean field approximation in the mixture of FNP.

The temperature variations of the order parameters ($S$ and $\psi_{0}$ ) for $\phi=0$ in the FSmA phase are shown in Fig. 1 and Fig. 2, respectively for $\theta_{f}=0,\pi/2$. Figs. 3 and 4 show the temperature variations of the same order parameters at the first order FN-FSmA transition point with the change of the concentration of FNP for $\phi\neq 0$ (= 0.02). Figs 3 and 4 show discontinuous phase transitions, as expected.

For threshold temperatures $T_{1}=T_{2}=10$ (in non-dimensionalized units), we find continuous (second order) phase transition for the parameter values $\theta_{f}=0.0$ and $\phi=0.02$ (Fig. 1). However, changing the parameter values to $\theta_{f}=\pi/2$ and $\phi=0$ (Fig. 2) or $\phi=0.02$ (Fig. 3), we find discontinuous jumps, confirming our prediction of a first order phase transition.

The figures above have been drawn using the parameter values $a_{0}$= 0.012, $b=0.44$, $c=1.56$, $\alpha_{0}$=0.012, $p=0.1$, $q=2.0$, $\beta$ =2.07, $\gamma^{\prime}$= 2.98, $\eta^{\prime}$=-0.82, $\delta=1.01$ and $\omega=0.72$. The coefficients (parameters) entering the Landau free energy are arbitrary and can at best be fitted to the experimental data so as to give the observed physical behavior. Since enough experimental data are not available in the literature, we cannot provide precise quantitative comparison of the values of the coefficients used here against known experimental benchmark. Hence, we have restricted ourselves to generic parametric windows that portray the physically relevant behavior, focusing on the phase transition aspect. Our choice of parameter values have been guided by the constraint of eliciting continuous variations of $S$ and $\psi_{0}^{2}$ against $T$ for $\phi=0$ (second order) while showing a discontinuous variation for $\phi\neq 0$ (first order). Generally, all we can say here is that the values of $\gamma^{\prime},\omega>0$ while $\eta^{\prime}<0$.

The conditions for the first order FN-FSmA phase transition are given by

[TABLE]

The conditions for the second order phase N-SmA transition read

[TABLE]

Again the conditions for phase equilibrium require that the chemical potentials in the FN and FSmA phases are equivalent i.e $\mu_{FN}=\mu_{FSmA}$.

The $\phi$ versus temperature $T$ variation in the FSmA and FN phases, obtained from a self-consistent solution of Eq. (27), is shown in Fig. 5. The results are shown over a range of values of $S$, including the phase transition value $S\sim 0.3$, as previously demonstrated in Figs. 1-4. Apart from the generic decaying trend, the results remain largely unaffected by changes in $\theta_{f}$ values.

III Tricritical behavior of the FN-FSmA phase transition

In this section we discuss the tricritical behavior of the FN-FSmA phase transition under the influence of FNP. Assume $S_{0}$ is the order parameter of the FN phase at the FN-FSmA transition point and $f_{FN}(S_{0})$ is the corresponding free energy density at the FN phase. Then the free energy density for a mixture of liquid crystal and FNP near the FN-FSmA phase transition can be written as

[TABLE]

where $F_{FN}(S_{0})$ is the corresponding free energy density of the FN phase and $u=1/\chi_{1}$, $\chi_{1}$ is the response function of the FN phase.

After eliminating the values of $S$ and $M$ from Eq. (28), we get the free energy density as

[TABLE]

The renormalized coefficients are

$F_{FN}^{*}(S_{0})=F_{FN}(S_{0})-\frac{p^{**2}}{4q^{*}}$,

$\alpha^{**}=\alpha-\delta S_{0}-\phi\frac{p^{**}\omega^{*}}{q^{*}}$,

$\beta^{**}=\beta-\frac{\delta^{2}}{2u}-\phi(1-\phi)\frac{\omega^{*2}}{q^{*}}$,

$p^{**}=p+\phi(1-\phi)\frac{\gamma^{\prime 2}}{4u}-(1-\phi)\gamma^{\prime}S_{0}-(1-\phi)\eta^{\prime}S_{0}^{2}$,

$q^{*}=q-\phi(1-\phi)\frac{\gamma^{\prime 2}}{u}-\phi(1-\phi)\frac{2\eta^{\prime 2}}{u}S_{0}^{2}-\phi(1-\phi)\frac{\eta^{\prime}\gamma^{\prime}}{u}S_{0}$,

$\omega^{*}=\omega+\frac{\delta}{u}(\eta^{\prime}S_{0}+\gamma^{\prime})$.

It is clear from the renormalized coefficients that the parameters $\alpha^{**}$ and $\beta^{**}$ change with the change of concentration $\phi$ which indicates change of the order of the FN-FSmA phase transition. For pure 8CB or low value of the concentration $\phi$ of FNP, $\beta^{**}>0$, then a second order transition occurs.

Then renormalization of the second order FN-FSmA transition temperature can be written as

[TABLE]

where

where $T_{FN-FSmA}^{*}=\frac{\alpha_{0}T_{2}^{*}+\frac{p_{0}\omega^{*}T_{f}\phi}{q^{*}}}{\alpha_{0}^{*}}$,

$\alpha_{0}^{*}=\alpha_{0}+\frac{\phi\omega^{*}p_{0}}{q^{*}}$.

For the higher value of concentration $\phi$ of the FNP, $\beta^{**}<0$, the FN-FSmA phase transition is a first order transition. In this case both the N and SmA phases can coexist i.e. a two phase region appears. In this case sixth order term $\frac{e}{6}\phi_{0}^{6}$ should be added into the free energy density (29). Then for the first order FN-FSmA phase transition, the FN-FSmA transition temperature is

[TABLE]

Equations (30) and (31) show that the FN-FSmA transition temperature decreases with increase of the concentration of FNP. This prediction confirms the experimental results cordo3 .

For a tricritical value of the concentration $\phi_{tcp}$, $\beta^{**}=0$, then a tricritical point is obtained. Hence a TCP is achieved with the change of concentration of the FNP.

IV Conclusions

We have developed a phenomenological model combining with Flory-Huggins theory to describe the effect of ferromagnetic nanoparticles on the I-N and N-SmA phase transitions. The I-FN transition is still a weakly first order transition even in the mixture of FNP. The FN-FSmA transition may be first order. In a binary mixture, the N-SmA transition which is second order only in one of the pure forms, becomes first order with the change of concentration of the FNP. This leads to a crossover from second to first order transition via TCP. Furthermore, both the I-FN and FN-FSmA transition temperatures decrease with the increase of the concentration of the FNP. We discuss our analysis by plotting various topology of the phase diagram under different conditions. Our results are qualitative agreement with the experimental results. Our results are expected to encourage further experiments on the impact of FNP on other liquid crystalline phase transitions to verify the validity of the present theory.

In the present work we have discarded spatial variations in the order parameter. The inclusion of these derivative terms will give additional physics into these phase transitions.