## Abstract

Based on the phenomenological model first presented by van der Schoot et al., which predicts the alignment of carbon nanotube (CNT) dispersions in thermotropic nematic liquid crystals, we present the extensive results concerning the phase diagram and the orientational properties of the mixture in this chapter.

### Keywords

- liquid crystal
- carbon nanotube
- phase transition

## 1. Introduction

A method to obtain aligned CNTs (necessary for applications) is to disperse them into liquid crystals (LCs) [with their self-organization (long-range orientational order) and fluidity] [1, 2, 3, 4, 5, 6, 7, 8, 9]. The thermotropic [10, 11, 12, 13] and the lyotropic LCs [14, 15, 16, 17] have been used to align CNTs parallel as well as perpendicular to average direction of alignment of long axes of LC molecules called the director. As a consequence, the orientational order parameter of CNTs could have the values between

The main hypothesis used in the theoretical study of the collective behavior of CNTs dispersed in the isotropic solvents [26, 27, 28] as well as in LCs [29, 30, 31, 32, 33, 34, 35, 36] is that they can be considered as rigid rod polymers [37].

Using the density functional theory, the isotropic-liquid crystal phase transition has been analyzed considering the van der Waals attractive interactions [26]. Onsager theory of rigid rods [38] was used to study the phase behavior of CNTs dispersed into organic and aqueous solutions [27]. Also, the Onsager model including length polydispersity and solvent-mediated interaction was considered to study the dispersions of CNTs in superacids [28]. These theoretical studies lead to the conclusion that to obtain orientational order of CNTs at room temperature it is necessary that the van der Waals interactions must be screened out, i.e., the CNTs must be dispersed in a good solvent. In the case of a non-good solvent, no liquid crystalline phases of CNTs form at room temperature because only dilute solutions are thermodynamically stable.

Matsuyama [29, 30] used a mean field theory to analyze the phase behavior of a mixture composed of low-molecular-weight liquid crystal and a rigid rod-like polymer such as CNTs. The free energy is constructed on the basis of Onsager model for excluded volume interactions, the Maier-Saupe model for orientational-dependent attractive interactions, and the Flory-Huggins theory for binary mixtures.

In the previous papers [31, 32, 34, 35, 36], we have presented a phenomenological theory to describe the alignment of CNT dispersions in thermotropic nematic LCs. We combined the Landau-de Gennes [39, 40] free energy for thermotropic ordering of the LC solvent and the Doi free energy [41, 42, 43] for the lyotropic orientational order of CNTs. Because the CNT is much thinner than the elastic penetration length, the alignment of CNTs in the nematic solvent is caused by the coupling of the LC director field to the anisotropic interfacial tension of the CNTs. This is true only for very dilute solutions without large aggregates [33]. Density functional calculations [44] show that CNT alignment mechanism in LC is associated with a strong interaction due to surface anchoring with a binding energy of

In the present chapter, we extend this model to generalize the anisotropic interaction form between a CNT and the liquid crystal molecules including the possibility of perpendicular alignment.

The remainder of this chapter is organized as follows. In Section 2, we describe the model. In Section 3, we illustrate the results and finally give the main conclusions in Section 4.

## 2. Model

The free energy density of the binary mixture composed of CNTs and thermotropic nematic LC contains three terms:

where

### 2.1 Free energy of carbon nanotubes

At the mesoscopic level of description, the free energy of CNTs can be written in the following form [31]:

where the first two terms represent the entropy of isotropic mixing of CNTs and LC components [45] and the third one is obtained from the Onsager theory [38] using the Smoluchowski Equation [41, 42]. In this form, the van der Waals attractions between CNTs are neglected. The long-ranged intermolecular attractions between CNTs were considered in [36].

Eq. (2) defines a first-order isotropic-nematic phase transition of CNTs with the equilibrium values of the order parameters

### 2.2 Free energy of thermotropic liquid crystal

To characterize the isotropic-nematic phase transition of thermotropic LC, we use the Landau-de Gennes [39] free energy:

where

### 2.3 The coupling free energy

The condition of weak anchoring limit of the interaction between the two components is

The coupling free energy has been explained in [31], and it has the following form:

where

We note that positive values of

### 2.4 The total free energy density of the binary mixture

Finally, the total free energy density of the binary mixture (1) is the sum of the free energy densities of CNTs (2), thermotropic LC (3), and interaction between nematic CNTs and nematic LC (4):

## 3. Results

We present the main results concerning the phase behavior of the CNT-LC binary mixture as a function of temperature, volume fraction

### 3.1 Positive γ : critical point of CNT phase transition

The positive value of the interaction parameter

The dependence of

In the nematic phase of the liquid crystal, the critical interaction parameter is relatively small (because

The dependence of

In Figure 3 the volume fractions of CNTs at the transition as a function of the coupling parameter for a fixed value of the temperature are shown. For negative values of

The CNT order parameters at the transition as a function of the coupling parameter for a fixed value of the temperature are plotted in Figure 4. For negative values of the coupling parameter (the region I in the figure), the first-order phase transition of CNTs takes place between a paranematic phase with a negative order parameter (perpendicular alignment) and a nematic phase with a positive order parameter (parallel alignment). For positive values of

### 3.2 Phase diagram

The equilibrium phase diagram can be obtained by minimizing the free energy with respect to

and equating the chemical potentials of CNTs and LC in the two phases. The chemical potentials are given by

where the free energy density is given by Eq. (5).

#### 3.2.1 Positive coupling constant

A positive coupling parameter corresponds to a paranematic (

For

To see in more detail the orientational order of the two components, in Figure 6, we have plotted the order parameter profiles

The transition between the isotropic and nematic phases of LC is first order (see Figure 6a), and the values of the LC order parameter depend only on temperature and are not influenced by CNT phase transition (because of

#### 3.2.2 Negative coupling constant

The negative coupling parameter corresponds to a paranematic (

For

The order parameter profiles

Again the transition between the isotropic and nematic phases of LC is first order (see Figure 8a), and the values of the LC order parameter depend only on temperature. The behavior of CNT orientational order with temperature is shown in Figure 8b. In the nematic phase of LC (

## 4. Conclusions

In the present chapter, we have extended the previous mesoscopic phenomenological model [31, 32, 34, 35, 36] (used to describe the phase behavior of a binary mixture composed of CNTs and thermotropic nematic LC) to include the possibility of a perpendicular alignment of CNTs to LC molecules. The model contains the CNT free energy density (2), the Landau-de Gennes free energy (3) for thermotropic LC order, and an interaction term between the nematic CNTs and nematic LC (4). This interaction term generates the possibility of existence of two nematic phases of CNTs: (i) the mean direction of orientation of CNTs is parallel with the direction of the director of LC molecules, when

The phase behavior and orientational properties of the mixture are discussed considering different values of temperature

The theoretical model we have presented describes quite well (comparing with experimental results) the phase properties of the CNTs into thermotropic nematic LC. The model could be improved by considering the polydispersity of the CNT component and by considering, in more detail, the form of the interaction energy between the two components.