Effect of Confinement on the Structural, Dielectric, and Dynamic Properties of Liquid Crystals in Anopores

Abstract

Based on data from broadband dielectric spectroscopy (BDS) and a molecular model based on the Landau–de Gennes concept, the effect of confinement on the structural, dielectric, and dynamic properties of 4-n-pentyl-4′-cyanobiphenyl (5CB) in the nematic phase is studied. The dielectric permittivity and relaxation times were previously obtained by the BDS technique in a wide frequency range ($1\mathrm{MHz}\le \mathrm{f}\le 1\mathrm{GHz}$) in the nematic phase composed of 5CB molecules confined to Anopore membranes with pore sizes of 0.2 $\mathsf{\mu }$m. The distance-dependent values of the order parameter $P_2\left(r\right)$, the relaxation time $\tau \left(r\right)\equiv {\tau }_{00}^1\left(r\right)$, the rotational diffusion coefficient $D_{\perp }\left(r\right)$, and both rotational viscosity coefficients ${\gamma }_i\left(r\right)$ ($i=1,2$) as functions of the distance r away from the bounding surface are calculated by a combination of existing statistical-mechanical approaches and data obtained by the BDS technique. Reasonable agreement between the calculated and experimental values of ${\gamma }_i(i=1,2)$ for bulk 5CB is obtained.

1. Introduction

The broadband dielectric spectroscopy (BDS) technique provides a powerful tool for investigation both the dielectric and dynamic properties of complex systems such as liquid crystalline (LC) materials confined in randomly interconnected porous media [1,2,3,4,5] or in cylindrical channels of Anopore membranes [5,6,7,8,9]. In BDS, the dynamical behavior of a sample is studied on the basis of the analysis of its frequency-dependent function, such as the complex dielectric permittivity tensor ${ϵ}_{ij}\left(\omega \right)=Re{ϵ}_{ij}\left(\omega \right)-iIm{ϵ}_{ij}\left(\omega \right)$, in a wide frequency $f=\omega /2\pi$ range (${10}^{-2}\mathrm{kHz}\le \mathrm{f}\le 10\mathrm{GHz}$). Data are conventionally interpreted in terms of superimposed Debye, stretched exponential, Havriliak–Negami [10], and other relaxation functions. The characteristics of relaxation frequencies can be associated with dynamic processes at the interfacial regions of such complex systems. From the dielectric response, it is possible to obtain the values of dipole strength, dielectric losses, and correlation times of the relaxation processes present in the LC system confined to microporous cylindrical channels. In particular, the high surface-to-volume ratio makes it convenient to examine LC near solid substrates [11]. The temperature dependence of a particular relaxation process in such confined systems gives information on mobilities [12], phase transition temperatures [13], and thermal activation. On the other hand, the geometrical restrictions can produce interesting changes in the structural and molecular diffusion properties and may influence the thermodynamic properties of the confined phase. For instance, one can observe the shift of phase transition temperatures or the enthalpy jump in the LC phase confined in randomly interconnected porous media, such as aerosils [14,15]. Anopore membranes with well-defined cylindrical pores (see Figure 1) are very often used to study the surface ordering effect of confined LCs, such as pretransitional order [14] and dielectric spectra [1], to take advantage of its large surface-to-volume ratio. We chose broadband spectroscopy as a convenient tool to study the dynamics of 4-n-pentyl-4′-cyanobiphenyl (5CB) molecules being confined to Anopore membranes with pore sizes of 0.2 $\mathsf{\mu }$m for a wide temperature range.

In this work, we revised the broadband spectroscopy results in order to clarify the temperature dependence of the relaxation times corresponding to 5CB molecules in the pore volume and in an interfacial phase close to the pore wall in the frequency range from 1 MHz to 1 GHz [16]. From the relaxation times, which have been interpreted in terms of Debye and Havriliak–Negami relaxation models [10], the calculations of the rotational diffusion coefficient (RDC) $D_{\perp }$ and rotational viscosity coefficients (RVCs) ${\gamma }_i$ ($i=1,2$) are possible, both in the bulk of the pore and close to the pore wall.

The outline of this article is as follows. A brief revision of dielectric spectra in the LC phase confined to Anopore membranes (treated and untreated) [16] is given in Section 2, whereas the temperature dependence of relaxation times is discussed in Section 3. Formulas for rotational diffusion and viscosity coefficients and numerical calculations for these constants are given in Section 4. The conclusions are given in Section 5.

2. Dielectric Spectra of Confined 5CB

A polar LC compound, such as 5CB, with a large positive dielectric anisotropy ${ϵ}_a={ϵ}_{\Vert }-{ϵ}_{\perp }$, where ${ϵ}_{\Vert }$ and ${ϵ}_{\perp }$ are the dielectric constants parallel and perpendicular to the director $\widehat{\mathbf{n}}$, respectively, with an axial dipole moment, $\mu \sim 5D$, directed from the polar head to the hydrophobic tail of the molecule [17], and with a convenient temperature range for nematic LC (NLC) phase between 297 and $309\mathrm{K}$ ($T_{\mathrm{NI}}=308.8\mathrm{K}$), is considered here. The main reason why our research interest is concerned with the dielectric coefficients of well-known materials like 5CB is that their measurements are nontrivial. The dielectric measurements of bulk 5CB have been a subject of interest from the beginning of investigations of LC materials and continue to this day. But, the next step in investigating the dielectric properties of these LC compounds is to extend these measurements to cyanobiphenyls confined in cylindrical channels of Anopore membranes. The dielectric spectra for 5CB embedded in untreated and treated Anopore membranes for three different temperatures are shown in Figure 2 and Figure 3 [17].

Anopore membranes with well-defined cylindrical pores 0.2 $\mathsf{\mu }$m in size, in which the cylindrical axes are directed to be parallel to the probing electric field, are used for the study of modified LC behavior and effects of confinement on LCs in small pores. It can be seen that for 5CB in cylindrical pores, at the temperatures corresponding to the nematic phase, the dielectrically active mode related to the rotation of the molecule around the short axis has one loss peak with a frequency of ∼$10\mathrm{MHz}$. Typical spectra corresponding to the isotropic (circles) and supercooled (up-triangles) phases have also been presented in the Figure 3. One can suppose that in untreated Anopore membranes, with the probing electric field directed parallel to the director, the molecules (at least the majority of them) are oriented along the pore axis. This means that the structure of 5CB in untreated alumina Anopore cylindrical pores can be considered as axial, with the parallel axial alignment at the surface along the cylindrical axis, as depicted in Figure 1a (in the case of planar anchoring). It should be pointed out that the supercooling of the phase by at least $20\mathrm{K}$ below the bulk crystallization temperature $T_{\mathrm{cr}}\sim 296\mathrm{K}$ is also observed for 5CB molecules confined in 0.2 $\mathsf{\mu }$m pores. Figure 3a,b show the loss spectra obtained for 5CB molecules,= confined in the cylindrical pores of the matrix treated with lecithin [6]. In the case of homeotropic boundary conditions at the pore walls (see Figure 1b), dynamics of the librational mode can be easily investigated by BDS methods. Figure 3b shows the loss spectra corresponding to the isotropic, nematic, and supercooled phases at the same temperatures as presented in Figure 2b. It is clear from Figure 3a,b that the relaxation process in the case of treated matrices (see Figure 3b) is much faster than that in the case of untreated matrices (see Figure 2b), and the characteristic loss peak shifts to a higher frequency region. Since the molecules of 5CB do not have a component of the dipole moment directed perpendicular to the long axis, the high-frequency loss peak, in the case of the LC phase confined in treated cylindrical channels of Anopore membranes, can be only explained by associating it with the librational motion [18,19]. It is important to stress that in the case of $\mathbf{E}$‖$\widehat{\mathbf{n}}$, the dielectric spectra corresponding to the supercooled phase (up-triangles) are similar to the dielectric spectra corresponding to the nematic one (see Figure 3b (squares)).

3. Temperature Dependence of Relaxation Times

Figure 4 shows the temperature dependence of the relaxation times corresponding to the hindered rotation of the molecules around their molecular short axes and the librational motion for 5CB, both in untreated and treated Anopore membranes. Both the bulk nematic and isotropic 5CB data are included for comparison. As seen in the figure, these dependencies are very different from the one corresponding to the bulk sample.

First of all, in the case of an untreated matrix, with pore sizes of 0.2 $\mathsf{\mu }$m, the experimental value of the $\mathrm{NI}$ transition temperature shift $\Delta T_{\mathrm{NI}}=T_{\mathrm{NI}}(\mathrm{untr}.)-T_{\mathrm{NI}}$ is $-0.3\mathrm{K}$, whereas in a treated matrix, with the same pore size, it is $-0.7\mathrm{K}$, respectively. Secondly, for the process that describes the hindered rotation of the molecules around their molecular short axes in the confined geometry, ${log}_{10}\tau$ shows that both magnitudes decrease with growth in the temperature and are smaller than in the bulk phase. It is important to stress that in the case of untreated Anopore membranes, the relaxation process in the full temperature range ($275\mathrm{K}\le T\le 323\mathrm{K}$) is faster than the same process in the bulk phase. The temperature dependence of $\tau \sim exp\left(U/RT\right)$ in the Arrhenius form in the broad range of temperatures ($275\mathrm{K}\le T\le 307\mathrm{K}$) has an activation energy of ∼$61.0\mathrm{kJ}/\mathrm{mol}$. However, the temperature dependence of relaxation times will be better suited to the Arrhenius form if one subdivides the temperature interval into two parts: $\Delta T_{\mathrm{A}}$ ($275\mathrm{K}\le T<296.4\mathrm{K}$) with the activation energy $U_{\mathrm{A}}$ of ∼$68.9\mathrm{kJ}/\mathrm{mol}$ and $\Delta T_{\mathrm{B}}$ ($296.4\mathrm{K}\le T\le 307\mathrm{K}$) with the activation energy $U_{\mathrm{B}}$ of ∼$60.0\mathrm{kJ}/\mathrm{mol}$. Thus, the value of the activation energy corresponding to the temperature range $\Delta T_A$ is closer to the value of the energy corresponding to the bulk nematic phase. The relaxation process observed in the lecithin-treated matrix is very different from that observed in the untreated sample. In the temperature range of 302.7 K–308.1 K, where the nematic phase of 5CB exists, the relaxation time ${log}_{10}\tau$ shows magnitudes that slightly increase with growth in the temperature (see Figure 4 (diamonds)), whereas below $302.7\mathrm{K}$, the relaxation time again increases continuously with decreasing temperature.

The temperature-dependent dielectric strength $\Delta {ϵ}_j={ϵ}_j-{ϵ}_{\infty }$ ($j=\Vert ,\perp$) for 5CB molecules confined in untreated ($\mathbf{E}$‖$\widehat{\mathbf{n}}$) and treated ($\mathbf{E}$⊥$\widehat{\mathbf{n}}$) Anopores membranes with pore sizes of 0.2 $\mathsf{\mu }$m are shown in Figure 5.

Here, ${ϵ}_{\infty }$ is the high-frequency limit of the permittivity. The dielectric strength $\Delta {ϵ}_{\Vert }$ ($\mathbf{E}$‖$\widehat{\mathbf{n}}$), both for bulk and LC phases confined in untreated Anopore membranes, shows magnitudes that decrease with growth in the temperature over the temperature range ($275\mathrm{K}\le T\le 309\mathrm{K}$), whereas the magnitudes of the dielectric strength $\Delta {ϵ}_{\perp }$ ($\mathbf{E}$⊥$\widehat{\mathbf{n}}$), on the contrary, increase with increasing the temperature, at least in the nematic phase ($297\mathrm{K}\le T\le 309\mathrm{K}$). Measurements of the dielectric strength $\Delta {ϵ}_{\perp }$ in the temperature range of the bulk nematic phase of 5CB can be treated as temperature-independent, whereas in the case of $\mathbf{E}$‖$\widehat{\mathbf{n}}$, one deals with the strong, approximately three times larger, temperature dependence of $\Delta {ϵ}_{\Vert }$. Looking for some more detail at the experimental data relating to Figure 5, we can see that the magnitudes of $\Delta {ϵ}_{\perp }(\mathrm{treat}.)$ in the temperature range corresponding to the nematic phase are approximately one-fourth that in the untreated case $\Delta {ϵ}_{\perp }(\mathrm{untreat}.)$. Taking into account that the dielectric strength $\Delta {ϵ}_j\sim {\displaystyle \frac{{\mu }^2}{k_{B}T}}{g}_j$ ($j=\Vert ,\perp )$, one can estimate the Kirkwood correlation factor $g_j=1+z\langle cos{\theta }_{ij}\rangle$, where z is the number of nearest neighbors, ${\theta }_{ij}$ is the angle between the dipole moments of the central and nearest neighboring molecules, and $\langle ....\rangle$ denotes the statistical-mechanical average. As seen in Figure 5, the magnitudes of the Kirkwood correlation factor ($g_{\Vert }$) for planar alignment ($\mathbf{E}$‖$\widehat{\mathbf{n}}$) of 5CB molecules at the solid wall are greater, approximately four times, than that ($g_{\perp }$) for homeotropic alignment ($\mathbf{E}$⊥$\widehat{\mathbf{n}}$). As a result, the correlations among the molecules of 5CB confined in untreated Anopores membranes are much stronger than those in the treated case. This result seems important, particularly in the point of view of the dynamical process in the vicinity of the bounding surface, but it has not been widely investigated.

4. Effect of a Solid Wall on Relaxation Processes

The rotational dynamics of molecules in an anisotropic phase can be described by the small step diffusion model [20,21]. In general, however, the rotational motion of an uniaxial molecule in a nematic phase is conveniently characterized using the orientational time correlation functions (TCFs) ${\Phi }_{mn}^L\left(t\right)=\langle D_{mn}^{L*}(\Omega \left(0\right))D_{mn}^L(\Omega \left(t\right))\rangle$, where $D_{mn}^L(\Omega )$ is the Wigner rotational matrix element of rank L and $\Omega =(\alpha ,\beta ,\gamma )$ is a set of time-dependent Euler angles, which define the orientation of the molecular axis system relative the director frame. The projection index m is related to the director coordinate system, whereas molecular properties are dictated by the projection index n. Different spectroscopic methods provide correlation functions with different rank values of L. First-rank $(L=1)$ TCFs are relevant for infrared and dielectric spectroscopies, while TCFs with $L=2$ appear in the expressions for nuclear spin relaxation rates and Raman band shapes. The initial values of the TCFs $\langle D_{mn}^{L*}(\Omega \left(0\right))D_{mn}^L(\Omega \left(t\right))\rangle$ can be expressed in terms of the orientational order parameters (OPs) $P_{2L}$ ($L=1,2$) [22]. In general, the components of the tensor dipole autocorrelation function $C_i\left(t\right)$ may be represented by [20,21]

$$\begin{array}{c}\hfill C_{\Vert }\left(t\right)=\langle e_Z\left(0\right)e_Z\left(t\right)\rangle =\\ \hfill {\Phi }_{00}^1\left(t\right){\Phi }_{00}^1\left(0\right)exp\left(-{\displaystyle \frac{t}{{\tau }_{00}^1}}\right)={\displaystyle \frac{1+2P_2}{3}}exp\left(-{\displaystyle \frac{t}{{\tau }_{00}^1}}\right),\end{array}$$

(1)

$$\begin{array}{c}\hfill C_{\perp }\left(t\right)=\langle e_X\left(0\right)e_X\left(t\right)\rangle =\langle e_Y\left(0\right)e_Y\left(t\right)\rangle =\\ \hfill {\Phi }_{10}^1\left(t\right)={\Phi }_{10}^1\left(0\right)exp\left(-{\displaystyle \frac{t}{{\tau }_{10}^1}}\right)={\displaystyle \frac{1-P_2}{3}}exp\left(-{\displaystyle \frac{t}{{\tau }_{10}^1}}\right),\end{array}$$

(2)

where ${\Phi }_{i0}^1\left(t\right)$ $(i=0,1)$ represents first-rank time correlation functions and $e_{\alpha }$ represent the projections of the unit vector $\widehat{\mathrm{e}}$ along the dipole moment onto the laboratory axis $\alpha$ $(\alpha =X,Y,Z)$. Notice that the first-rank TCFs at $t=0$ depend on $P_2$ only. Based on the short time expansion of the TCFs, expressions for the correlation times ${\tau }_{00}^1$ and ${\tau }_{10}^1$ are solely determined by the tumbling motion of the molecule [21,23]:

$$\begin{array}{c}\hfill {\tau }_{00}^1={\left[D_{\perp }{\displaystyle \frac{2-2P_2}{1+2P_2}}\right]}^{-1},\end{array}$$

(3)

$$\begin{array}{c}\hfill {\tau }_{10}^1={\left[D_{\perp }{\displaystyle \frac{2+P_2}{1-P_2}}\right]}^{-1},\end{array}$$

(4)

where $D_{\perp }$ is the rotational diffusion coefficient (RDC), which corresponds to molecular tumbling in the nematic phase. Having obtained both the relaxation time ${\tau }_{00}^1$ and OP $P_2$, one can now calculate the RDC $D_{\perp }$. Taking into account that the relaxation time ${\tau }_{00}^1\left(R\right)\equiv \langle \tau \left(R\right)\rangle$, both in untreated and treated Anopore membranes, can be obtained by BDS methods, one may reconstruct the distance-dependent function $\tau \left(r\right)\equiv {\tau }_{00}^1\left(r\right)$, which describes the relaxation process in the vicinity of the solid substrate. Indeed, for cylindrical symmetry, the average relaxation time $\langle \tau \left(R\right)\rangle$ can be presented in the form

$$\begin{array}{c}\hfill \langle \tau \left(R\right)\rangle ={\displaystyle \frac{{\int }_0^R\tau \left(r\right)rdr}{{\int }_0^{R}rdr}}=2{\int }_0^1\tau \left(Ry\right)ydy,\end{array}$$

(5)

where $y=r/R$, r is a distance away from the wall, and R is the radius of the cylindrical pores, whereas for the calculation of the surface OP $P_2\left(r\right)$ in the vicinity of the solid surface, we employ the Landau–de Gennes formalism [24]. The liquid crystal material is assumed to be confined in a cylindrical cavity of radius R, with both rotational and translational symmetries along the cylindrical axis. For a system with a uniform director orientation, the free energy density can be phenomenologically described in terms of the Landau–de Gennes theory as follows [24]:

$$\begin{array}{c}\hfill f_n=f_0+{\displaystyle \frac{1}{2}}{a}_0[T-T_{\mathrm{NI}}]P_2^2-{\displaystyle \frac{1}{3}}b{P}_2^3+{\displaystyle \frac{1}{4}}c{P}_2^4+{\displaystyle \frac{L}{2}}{\left[\nabla P_2\right]}^2+GP\delta \left(r\right),\end{array}$$

(6)

where $f_0$ is the "isotropic" free energy density; $P_2\equiv P_2\left(r\right)$ is the order parameter at $T_{\mathrm{NI}}$; $a_0$, b, c, and L are positive constants; and G is the strength of the anchoring energy. ²H-NMR spectra were measured for different radii of cavities and for two different surface conditions, which provide perpendicular anchoring at the untreated and lecithin-treated Anapore cavities filled with 5CB [25]. Based on these measurements, the anchoring strength of G was estimated as $3\times {10}^{-5}\mathrm{J}/{\mathrm{m}}^2$. The solution for the OP $P_2\left(r\right)$ in the bulk as a function of distance r is expressed as [24]

$$\begin{array}{c}\hfill P_2\left(r\right)=P_2^s{\displaystyle \frac{cosh\left(\frac{r}{\xi }\right)}{cosh\left(\frac{R}{\xi }\right)}},\end{array}$$

(7)

where $\xi =\sqrt{{\displaystyle \frac{L}{a(T_{\mathrm{NI}}-T)}}}={\xi }_0\sqrt{{\displaystyle \frac{T_{\mathrm{NI}}}{T_{\mathrm{NI}}-T}}}$. Here, ${\xi }_0$ is the order of the characteristic length of molecules, and $P_2^s\equiv P_2(r=0)$. If the size of the cavity is much larger than the correlation length ($\xi \ll R$), Equation (7) can be rewritten in the following form:

$$\begin{array}{c}\hfill P_2\left(r\right)=P_2^{s}exp\left({\displaystyle \frac{(r-R)}{\xi }}\right).\end{array}$$

(8)

In order to obtain the distance dependence of the relaxation time $\tau \left(r\right)$, let us assume a power functional form

$$\tau \left(r\right)=\left\{\begin{array}{c}{\tau }_{\mathrm{surf}}{\left({\displaystyle \frac{r}{R}}+1\right)}^x,r\le R,\hfill \\ {\tau }_{00}^1,r>R,\hfill \end{array}\right.$$

where the unknown associated exponent x can be obtained by solving the transcendent equation

$$\begin{array}{c}\hfill \mathcal{G}\left(x\right)=2^x-\mathcal{H}\left(x\right)=0,\end{array}$$

(9)

where ${\mathcal{H}}^{-1}\left(x\right)=\alpha \left(x^2+(3-{\displaystyle \frac{2}{\alpha }})x+2\right)$ and $\alpha ={\displaystyle \frac{\langle \tau \left(R\right)\rangle }{2{\tau }_{00}^1\left(\mathrm{bulk}\right)}}$ with ${\tau }_{\mathrm{surf}}={\tau }_{00}^1\left(\mathrm{bulk}\right)/2^x$. There is only one root $x\approx 3.651$ that satisfies the condition ${\tau }_{\mathrm{r}=\mathrm{R}}=\tau \left(\mathrm{bulk}\right)={\tau }_{00}^1\left(\mathrm{bulk}\right)$. Since the values of the relaxation time $\langle \tau (R,T)\rangle$ corresponding to the restricted geometry are proportionally less than the values of ${\tau }_{\mathrm{bulk}}\left(T\right)$ (see Figure 4, solid circles and open triangles), one can use that analytical expression for $\tau \left(r\right)$ in the full temperature range where the nematic phase of 5CB exists. It should be noted here that the exponential form of the distance dependence of the relaxation time $\tau \left(r\right)={\tau }_{\mathrm{surf}}exp\left({\displaystyle \frac{xr}{R}}\right)$ for $\alpha <1$ cannot exist because the corresponding equation for x does not have real roots. Having obtained both the relaxation time $\tau \left(r\right)$ and the data for $P_2\left(r\right)$, one can calculate, using Equation (3), the distance-dependent values of the RDC $D_{\perp }\left(r\right)$, which correspond to molecular tumbling in the vicinity of the solid wall and the orientational relaxation time ${\tau }_{00}^1\left(r\right)$. In order to calculate the absolute values of the $P_2\left(r\right)$, in the case of untreated cylindrical pores (case of the planar alignment) with sizes of 0.2 $\mathsf{\mu }$m, $P_2(r=0)$ was fixed to be $P_{\mathrm{surf}}=1.0$, whereas ${\xi }_0$ was fixed to the value ∼$0.67\mathrm{nm}$ [26].

The distance dependencies of the relaxation time $\tau \left(r\right)$ and RDC $D_{\perp }\left(r\right)$ are shown in Figure 6a,b.

Recently, the relaxation times ${\tau }_{i0}^1$ $(i=0,1)$ for bulk 5CB at one temperature $(T=300\mathrm{K})$ were calculated using dynamical parameters obtained from the MD simulation based on realistic atom–atom interaction potential [27] and the NMR technique based on the decoupling model for the correlated internal motions [28]. For completeness, we tabulate the measured and calculated ${\tau }_{00}^1$ and ${\tau }_{01}^1$ values of the bulk 5CB nematic phase at a temperature of $300\mathrm{K}$ ($P_2(r=R)=P_2\left(\mathrm{bulk}\right)=0.51$) in

Table 1. The orientational relaxation times ${\tau }_{0i}^1$ ($i=0,1$) measured by the BDS technique for 5CB molecules in the bulk nematic phase at $T=300\mathrm{K}$. The last four columns were obtained from the MD simulation [27] (second), by NMR technique [28] (third), calculations using Equation (4) (forth), and the MD simulation [27] (fifth), respectively.

${\mathit{\tau }}_{00}^1\left[\mathbf{ns}\right]$	${\mathit{\tau }}_{00}^1\left[\mathbf{ns}\right]$	${\mathit{\tau }}_{00}^1\left[\mathbf{ns}\right]$	${\mathit{\tau }}_{01}^1\left[\mathbf{ns}\right]$	${\mathit{\tau }}_{01}^1\left[\mathbf{ns}\right]$
$34.7$	$28.9$	$38.9$	$3.28$	$2.83$

.

Taking into account that the bulk relaxation process for 5CB molecules, measured by the BDS technique in the wide frequency range $1 \mathrm{MHz}\le f\le 1 \mathrm{GHz}$ with high accuracy, can be interpreted in terms of the Debye relaxation function, we may also determine, using the dipole autocorrelation function $C_j\left(t\right)$, represented by Equations (1) and (2), both the real and imaginary parts of the dielectric permittivity ${ϵ}_j\left(\omega \right)$ in the form [21,27,28]

$$\begin{array}{c}\hfill {\displaystyle \frac{{ϵ}_j\left(\omega \right)-{ϵ}_{\infty }}{{ϵ}_j\left(0\right)-{ϵ}_{\infty }}}=1-i\omega {\int }_0^{\infty }{C}_j\left(t\right)exp\left(-i\omega t\right)dt.\end{array}$$

(10)

Figure 7 shows the loss spectra for the bulk 5CB nematic phase at $T=300\mathrm{K}$, calculated using Equation (10) with the relaxation time ${\tau }_{00}^1\left(T\right)=28.9$ $\mathrm{ns}$ obtained by MD simulations [27].

The results of the measurements by the BDS technique of $Im{ϵ}_{\perp }\left(\omega \right)$ for 5CB molecules at a number of temperatures ($T=300.3,302.2,304\mathrm{K}$) are also shown in Figure 7. The longitudinal dielectric loss spectra $Im{ϵ}_{\Vert }\left(\omega \right)$ show magnitudes that increase with the growth in the temperature and have one loss peak that shifts to a higher frequency region (∼$75\mathrm{MHz}$) upon decreasing the temperature.

Recently, a statistical-mechanical approach for the theoretical treatment of rotational viscosity has been proposed [21,27,29]. This theory is based on the rotational diffusion model [21,27,29] and the concept of treating the phenomenological stress tensor $\overline{\sigma }$ as an average of its microscopic equivalent $\sigma$. The averaging can be accomplished using the nonequilibrium statistical operator method [30], and the expression for the rotational viscosity coefficients (RVCs) ${\gamma }_1$ and ${\gamma }_2$ can be written as [21,27]

$$\begin{array}{c}\hfill {\gamma }_1={\displaystyle \frac{k_{B}T\rho }{D_{\perp }}}\mathcal{F}\left(P_2\right),\end{array}$$

(11)

and [29]

$$\begin{array}{c}\hfill {\gamma }_2=-{\displaystyle \frac{k_{B}T\rho }{D_{\perp }}}s{P}_2,\end{array}$$

(12)

where $\mathcal{F}\left(P_2\right)=P_2^2\left(9.54+2.77P_2\right)/\left(2.88+P_2+12.56P_2^2+4.69P_2^3-0.74P_2^4\right)$, $k_B$ is the Boltzmann constant, T is the temperature, $\rho =N/V$ is the number density of molecules, s is a particle geometric factor taken as $s=(a^2-1)/(a^2+1)$, and a is the length-to-breadth ratio of the molecule. Thus, according to Equations (11) and (12), ${\gamma }_i$ $(i=1,2)$ is found to be inversely proportional to the RDC $D_{\perp }$, which corresponds to molecular tumbling in the nematic phase. Having obtained the relaxation times ${\tau }_{00}^1\left(T\right)$ and $\langle \tau (R,T\rangle$, both in treated and untreated Anopore membrances, one can calculate, using Equations (3), (8), (9), (11), and (12), the distance dependent RVCs ${\gamma }_1\left(r\right)$ and ${\gamma }_2\left(r\right)$, respectively.

In order to calculate the absolute values of the RVCs ${\gamma }_1\left(r\right)$ and ${\gamma }_2\left(r\right)$, the length and width of the 5CB molecule were fixed to the values 1.8 nm and 0.6 nm, respectively, whereas the number density $\rho$ of 5CB (at $297\mathrm{K}\le T\le 307\mathrm{K}$) was fixed to the value $1.8\times {10}^{27}$ m⁻³. Calculations of the distance dependence of ${\gamma }_{1,2}\left(r\right)$, as well as comparisons of the calculated values of the ${\gamma }_1\left(\mathrm{bulk}\right)$ with the experimental data, at two temperatures $300\mathrm{K}$ and $304\mathrm{K}$ are given in Figure 8a,b.

It is found that the absolute values of the RVCs ${\gamma }_i$ ($i=1,2$) both decrease with increasing the distance away from the surface, and finally, the values of ${\gamma }_1$ and ${\gamma }_2$ in the bulk phase are one-half the values of the RVCs near the wall. It should be noted here that $|{\gamma }_2|>|{\gamma }_1|$ for all distances away from the solid wall, and the director $\widehat{\mathbf{n}}$, at least in the high shear flow, aligns at an angle ${\theta }_{\mathrm{eq}}={\displaystyle \frac{1}{2}}{cos}^{-1}\left(-{\gamma }_1/{\gamma }_2\right)$ with respect to the direction of flow velocity $\mathrm{v}$. For completeness, we reproduce the calculated and measured ${\gamma }_1$ and ${\gamma }_2$ values at a temperature of $300\mathrm{K}$ in

Table 2. The values of the rotational viscosity coefficients ${\gamma }_1$ and ${\gamma }_2$ in the bulk of the nematic 5CB phase at $T=300\mathrm{K}$. All data for RVCs are given in $[\mathrm{kg}/\mathrm{ms}]$.

${\mathit{\gamma }}_1(\mathbf{theor}.)$	${\mathit{\gamma }}_1\left(\mathbf{MD}\right)$ [27]	${\mathit{\gamma }}_1(\mathbf{expt}.)$ [31]	${\mathit{\gamma }}_2(\mathbf{theor}.)$	${\mathit{\gamma }}_2\left(\mathbf{MD}\right)$ [27]
$0.05$	$0.045$	$0.053$	$-0.052$	$-0.049$

.

Note that the second and last columns are values that have been obtained from the MD simulations [27], whereas the experimental value was obtained by direct measurement [31].

5. Conclusions

In this paper, based on data on the real $Re{ϵ}_j\left(\omega \right)$ and imaginary $Im{ϵ}_j\left(\omega \right)$ parts of the dielectric permittivity ${ϵ}_j\left(\omega \right)=Re{ϵ}_j\left(\omega \right)-iIm{ϵ}_j\left(\omega \right)$ and relaxation times ${\tau }_{0i}^1\left(T\right)$ ($i=0,1$) obtained by broadband spectroscopy, the rotational diffusion constant $D_{\perp }\left(T\right)$ and rotational viscosity coefficients (RVCs) ${\gamma }_i\left(T\right)$ ($i=1,2$) in the liquid crystal phase consisting of 4-n-pentyl-4′-cyanobiphenyl (5CB) molecules confined in untreated and treated Anopore membranes were investigated. It was found that in untreated Anopore membranes with pore sizes of 0.2 $\mathsf{\mu }$m and with the probing electric field directed parallel to the director, the molecules (at least the majority of them) are oriented along the pore axis. This means that the structure of 5CB in untreated alumina Anopore cylindrical pores can be considered axial, with the parallel axial alignment at the surface along the cylindrical axis. In turn, the relaxation process in the case of treated matrices is much faster than that in the case of untreated matrices, and the characteristic loss peak shifts to a higher frequency region. Since the molecules of 5CB do not have a component of the dipole moment directed perpendicular to the long axis, the high-frequency loss peak, in the case of the LC phase confined in treated cylindrical channels of Anopore membranes, can only be explained by associating it with the librational motion. Based on measurements by broadband spectroscopy in the confined LC phase, the temperature dependence of relaxation times was obtained corresponding to the hindered rotation of molecules around their molecular short axes and the librational motion for 5CB, both in untreated and treated Anopore membranes.

The distance-dependent values of the order parameter $P_2\left(r\right)$, the relaxation time $\tau \left(r\right)\equiv {\tau }_{00}^1\left(r\right)$, the rotational diffusion coefficient $D_{\perp }\left(r\right)$, and both RVCs ${\gamma }_i\left(r\right)$ ($i=1,2$) as functions of the distance r away from the bounding surface were calculated in the framework of the molecular model based on the concept of the Landau–de Gennes theory.

It was found that the absolute values of the RVCs ${\gamma }_i$ ($i=1,2$) both decrease with increasing the distance away from the surface, and finally, the values of ${\gamma }_1$ and ${\gamma }_2$ in the bulk phase are one-half the values of the RVCs near the wall. Our analysis also shows that surface-induced molecular ordering can play a key role in the interpretation of the dielectric and dynamic properties of complex systems such as liquid crystalline materials confined to porous media.

We believe that this study shows some useful routes not only for understanding the nature of static confining but also for analyzing the rheological behavior of liquid crystals in restricted volumes.