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\end{equation} |
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where $\epsilon _{fs}$ is the permittivity of free space. |
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|
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\section{\label{liquidCrystalSection:methods}Methods} |
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\section{Computational Methodology} |
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|
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\section{\label{liquidCrystalSection:resultDiscussion}Results and Discussion} |
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A series of molecular dynamics simulations were perform to study the |
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phase behavior of banana shaped liquid crystals. In each simulation, |
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every banana shaped molecule has been represented three GB particles |
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which is characterized by $\mu = 1,~ \nu = 2, |
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~\epsilon_{e}/\epsilon_{s} = 1/5$ and $\sigma_{e}/\sigma_{s} = 3$. |
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All of the simulations begin with same equilibrated isotropic |
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configuration where 1024 molecules without dipoles were confined in |
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a $160\times 160 \times 120$ box. After the dipolar interactions are |
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switched on, 2~ns NPTi cooling run with themostat of 2~ps and |
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barostat of 50~ps were used to equilibrate the system to desired |
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temperature and pressure. |
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|
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\subsection{Order Parameters} |
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|
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To investigate the phase structure of the model liquid crystal, we |
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calculated various order parameters and correlation functions. |
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Particulary, the $P_2$ order parameter allows us to estimate average |
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alignment along the director axis $Z$ which can be identified from |
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the largest eigen value obtained by diagonalizing the order |
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parameter tensor |
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\begin{equation} |
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\overleftrightarrow{\mathsf{Q}} = \frac{1}{N}\sum_i^N % |
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\begin{pmatrix} % |
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u_{ix}u_{ix}-\frac{1}{3} & u_{ix}u_{iy} & u_{ix}u_{iz} \\ |
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u_{iy}u_{ix} & u_{iy}u_{iy}-\frac{1}{3} & u_{iy}u_{iz} \\ |
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u_{iz}u_{ix} & u_{iz}u_{iy} & u_{iz}u_{iz}-\frac{1}{3} % |
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\end{pmatrix}, |
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\label{lipidEq:po1} |
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\end{equation} |
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where the $u_{i\alpha}$ is the $\alpha$ element of the unit vector |
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$\mathbf{\hat{u}}_i$, and the sum over $i$ averages over the whole |
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collection of unit vectors. The $P_2$ order parameter for uniaxial |
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phase is then simply given by |
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\begin{equation} |
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\langle P_2 \rangle = \frac{3}{2}\lambda_{\text{max}}. |
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\label{lipidEq:po3} |
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\end{equation} |
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In addition to the $P_2$ order parameter, $ R_{2,2}^2$ order |
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parameter for biaxial phase is introduced to describe the ordering |
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in the plane orthogonal to the director by |
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\begin{equation} |
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R_{2,2}^2 = \frac{1}{4}\left\langle {(x_i \cdot X)^2 - (x_i \cdot |
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Y)^2 - (y_i \cdot X)^2 + (y_i \cdot Y)^2 } \right\rangle |
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\end{equation} |
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where $X$, $Y$ and $Z$ are axis of the director frame. |
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|
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\subsection{Structure Properties} |
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|
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It is more important to show the density correlation along the |
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director |
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\begin{equation} |
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g(z) =< \delta (z-z_{ij})>_{ij} / \pi R^{2} \rho |
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\end{equation}, |
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where $z_{ij} = r_{ij} \dot Z$ was measured in the director frame |
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and $R$ is the radius of the cylindrical sampling region. |
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|
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\subsection{Rotational Invariants} |
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|
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As a useful set of correlation functions to describe |
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position-orientation correlation, rotation invariants were first |
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applied in a spherical symmetric system to study x-ray and light |
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scatting\cite{Blum1971}. Latterly, expansion of the orientation pair |
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correlation in terms of rotation invariant for molecules of |
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arbitrary shape was introduce by Stone\cite{Stone1978} and adopted |
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by other researchers in liquid crystal studies\cite{Berardi2000}. |
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|
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\begin{equation} |
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S_{22}^{220} (r) = \frac{1}{{4\sqrt 5 }}\left\langle {\delta (r - |
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r_{ij} )((\hat x_i \cdot \hat x_j )^2 - (\hat x_i \cdot \hat y_j |
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> |
)^2 - (\hat y_i \cdot \hat x_j )^2 + (\hat y_i \cdot \hat y_j |
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> |
)^2 ) - 2(\hat x_i \cdot \hat y_j )(\hat y_i \cdot \hat x_j ) - |
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2(\hat x_i \cdot \hat x_j )(\hat y_i \cdot \hat y_j ))} |
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\right\rangle |
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\end{equation} |
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|
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\begin{equation} |
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S_{00}^{221} (r) = - \frac{{\sqrt 3 }}{{\sqrt {10} }}\left\langle |
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{\delta (r - r_{ij} )((\hat z_i \cdot \hat z_j )(\hat z_i \cdot |
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> |
\hat z_j \times \hat r_{ij} ))} \right\rangle |
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\end{equation} |
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|
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\section{Results and Conclusion} |
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\label{sec:results and conclusion} |
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|
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To investigate the molecular organization behavior due to different |
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dipolar orientation and position with respect to the center of the |
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molecule, |