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\begin{document} |
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\title[Real space electrostatics for multipoles. III. Dielectric Properties] |
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{Supplemental Material for: Real space electrostatics for multipoles. III. Dielectric Properties} |
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\title{Supplemental Material for: Real space electrostatics for |
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multipoles. III. Dielectric Properties} |
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|
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\author{Madan Lamichhane} |
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\affiliation{Department of Physics, University |
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\date{\today}% It is always \today, today, |
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% but any date may be explicitly specified |
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|
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\begin{abstract} |
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This document includes useful relationships for computing the |
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interactions between fields and field gradients and point multipolar |
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representations of molecular electrostatics. We also provide |
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explanatory derivations of a number of relationships used in the |
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main text. This includes the Boltzmann averages of quadrupole |
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orientations, and the interaction of a quadrupole density with the |
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self-generated field gradient. This last relationship is assumed to |
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be zero in the main text but is explicitly shown to be zero here. |
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\end{abstract} |
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|
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\maketitle |
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|
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\newpage |
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\section{Generating Uniform Field Gradients} |
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One important task in carrying out the simulations mentioned in the |
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main text was to generate uniform electric field gradients. To do |
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this, we relied heavily on both the notation and results from Torres |
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del Castillo and Mend\'{e}z Garido.\cite{Torres-del-Castillo:2006uo} |
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In this work, tensors were expressed in Cartesian components, using at |
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times a dyadic notation. This proves quite useful for computer |
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simulations that make use of toroidal boundary conditions. |
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|
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\section{Boltzmann averages for orientational polarization} |
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The dielectric properties of the system is mainly arise from two |
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different ways: i) the applied field distort the charge distributions |
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so it produces an induced multipolar moment in each molecule; and ii) |
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the applied field tends to line up originally randomly oriented |
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molecular moment towards the direction of the applied field. In this |
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study, we basically focus on the orientational contribution in the |
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dielectric properties. If we consider a system of molecules in the |
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presence of external field perturbation, the perturbation experienced |
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by any molecule will not be only due to external field or field |
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gradient but also due to the field or field gradient produced by the |
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all other molecules in the system. In the following subsections |
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\ref{subsec:boltzAverage-Dipole} and \ref{subsec:boltzAverage-Quad}, |
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we will discuss about the molecular polarization only due to external |
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field perturbation. The contribution of the field or field gradient |
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due to all other molecules will be taken into account while |
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calculating correction factor in the section \ref{sec:corrFactor}. |
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An alternative formalism uses the theory of angular momentum and |
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spherical harmonics and is common in standard physics texts such as |
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Jackson,\cite{Jackson98} Morse and Feshbach,\cite{Morse:1946zr} and |
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Stone.\cite{Stone:1997ly} Because this approach has its own |
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advantages, relationships are provided below comparing that |
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terminology to the Cartesian tensor notation. |
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|
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\subsection{Dipoles} |
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\label{subsec:boltzAverage-Dipole} |
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Consider a system of molecules, each with permanent dipole moment |
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$p_o$. In the absense of external field, thermal agitation orients the |
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dipoles randomly, reducing the system moment to zero. External fields |
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will tend to line up the dipoles in the direction of applied field. |
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Here we have considered net field from all other molecules is |
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considered to be zero. Therefore the total Hamiltonian of each |
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molecule is,\cite{Jackson98} |
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The gradient of the electric field, |
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\begin{equation*} |
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\mathsf{G}(\mathbf{r}) = -\nabla \nabla \Phi(\mathbf{r}), |
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\end{equation*} |
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where $\Phi(\mathbf{r})$ is the electrostatic potential. In a |
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charge-free region of space, $\nabla \cdot \mathbf{E}=0$, and |
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$\mathsf{G}$ is a symmetric traceless tensor. From symmetry |
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arguments, we know that this tensor can be written in terms of just |
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five independent components. |
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|
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Following Torres del Castillo and Mend\'{e}z Garido's notation, the |
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gradient of the electric field may also be written in terms of two |
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vectors $\mathbf{a}$ and $\mathbf{b}$, |
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\begin{equation*} |
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G_{ij}=\frac{1}{2} (a_i b_j + a_j b_i) - \frac{1}{3}(\mathbf a \cdot \mathbf b) \delta_{ij} . |
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\end{equation*} |
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If the vectors $\mathbf{a}$ and $\mathbf{b}$ are unit vectors, the |
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electrostatic potential that generates a uniform gradient may be |
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written: |
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\begin{align} |
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\Phi(x, y, z) =\; -\frac{g_o}{2} & \left(\left(a_1b_1 - |
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\frac{cos\psi}{3}\right)\;x^2+\left(a_2b_2 |
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- \frac{cos\psi}{3}\right)\;y^2 + |
| 123 |
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\left(a_3b_3 - |
| 124 |
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\frac{cos\psi}{3}\right)\;z^2 \right. \nonumber \\ |
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& + (a_1b_2 + a_2b_1)\; xy + (a_1b_3 + a_3b_1)\; xz + (a_2b_3 + a_3b_2)\; yz \bigg) . |
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\label{eq:appliedPotential} |
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\end{align} |
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Note $\mathbf{a}\cdot\mathbf{a} = \mathbf{b} \cdot \mathbf{b} = 1$, |
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$\mathbf{a} \cdot \mathbf{b}=\cos \psi$, and $g_0$ is the overall |
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strength of the potential. |
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|
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Taking the gradient of Eq. (\ref{eq:appliedPotential}), we find the |
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field due to this potential, |
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\begin{equation} |
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H = H_o - \bf{p_o} .\bf{E}, |
| 135 |
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\mathbf{E} = -\nabla \Phi |
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=\frac{g_o}{2} \left(\begin{array}{ccc} |
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2(a_1 b_1 - \frac{cos\psi}{3})\; x & +\; (a_1 b_2 + a_2 b_1)\; y & +\; (a_1 b_3 + a_3 b_1)\; z \\ |
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(a_2 b_1 + a_1 b_2)\; x & +\; 2(a_2 b_2 - \frac{cos\psi}{3})\; y & +\; (a_2 b_3 + a_3 b_3)\; z \\ |
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(a_3 b_1 + a_3 b_2)\; x & +\; (a_3 b_2 + a_2 b_3)\; y & +\; 2(a_3 b_3 - \frac{cos\psi}{3})\; z |
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\end{array} \right), |
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\label{eq:CE} |
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\end{equation} |
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while the gradient of the electric field in this form, |
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\begin{equation} |
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\mathsf{G} = \nabla\mathbf{E} |
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= \frac{g_o}{2}\left(\begin{array}{ccc} |
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2(a_1\; b_1 - \frac{cos\psi}{3}) & (a_1\; b_2 \;+ a_2\; b_1) & (a_1\; b_3 \;+ a_3\; b_1) \\ |
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(a_2\; b_1 \;+ a_1\; b_2) & 2(a_2\; b_2 \;- \frac{cos\psi}{3}) & (a_2\; b_3 \;+ a_3\; b_3) \\ |
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(a_3\; b_1 \;+ a_3\; b_2) & (a_3\; b_2 \;+ a_2\; b_3) & 2(a_3\; b_3 \;- \frac{cos\psi}{3}) |
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\end{array} \right), |
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\label{eq:GC} |
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\end{equation} |
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is uniform over the entire space. Therefore, to describe a uniform |
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gradient in this notation, two unit vectors ($\mathbf{a}$ and |
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$\mathbf{b}$) as well as a potential strength, $g_0$, must be |
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specified. As expected, this requires five independent parameters. |
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|
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The common alternative to the Cartesian notation expresses the |
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electrostatic potential using the notation of Morse and |
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Feshbach,\cite{Morse:1946zr} |
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\begin{equation} \label{eq:quad_phi} |
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\Phi(x,y,z) = -\left[ a_{20} \frac{2 z^2 -x^2 - y^2}{2} |
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+ 3 a_{21}^e \,xz + 3 a_{21}^o \,yz |
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+ 6a_{22}^e \,xy + 3 a_{22}^o (x^2 - y^2) \right]. |
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\end{equation} |
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where $H_o$ is a function of the internal coordinates of the molecule. |
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The Boltzmann average of the dipole moment is given by, |
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Here we use the standard $(l,m)$ form for the $a_{lm}$ coefficients, |
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with superscript $e$ and $o$ denoting even and odd, respectively. |
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This form makes the functional analogy to ``d'' atomic states |
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apparent. |
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|
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Applying the gradient operator to Eq. (\ref{eq:quad_phi}) the electric |
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field due to this potential, |
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\begin{equation} |
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\braket{p_{mol}} = \frac{\displaystyle\int d\Omega\; p_o\; cos\theta\; e^{\frac{p_oE\; cos\theta}{k_B T}}}{\displaystyle\int d\Omega\; e^{\frac{p_oE\;cos\theta}{k_B T}}}, |
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\mathbf{E} = -\nabla \Phi |
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= \left(\begin{array}{ccc} |
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\left( 6a_{22}^o -a_{20} \right)\; x &+\; 6a_{22}^e\; y &+\; 3a_{21}^e\; z \\ |
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6a_{22}^e\; x & -\; (a_{20} + 6a_{22}^o)\; y & +\; 3a_{21}^o\; z \\ |
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3a_{21}^e\; x & +\; 3a_{21}^o\; y & +\; 2a_{20}\; z |
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\end{array} \right), |
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\label{eq:MFE} |
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|
\end{equation} |
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where $\bf{E}$ is selected along z-axis. If we consider that the |
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applied field is small, \textit{i.e.} $\frac{p_oE\; cos\theta}{k_B T} << 1$, |
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while the gradient of the electric field in this form is: |
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\begin{equation} \label{eq:grad_e2} |
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\mathsf{G} = |
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\begin{pmatrix} |
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6 a_{22}^o - a_{20} & 6a_{22}^e & 3a_{21}^e\\ |
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6a_{22}^e & -(a_{20}+6a_{22}^o) & 3a_{21}^o \\ |
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3a_{21}^e & 3a_{21}^o & 2a_{20} \\ |
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\end{pmatrix} \\ |
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\end{equation} |
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which is also uniform over the entire space. This form for the |
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gradient can be factored as |
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\begin{gather} |
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\begin{aligned} |
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\mathsf{G} = a_{20} |
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\begin{pmatrix} |
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-1 & 0 & 0\\ |
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0 & -1 & 0\\ |
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0 & 0 & 2\\ |
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\end{pmatrix} |
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+3a_{21}^e |
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\begin{pmatrix} |
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0 & 0 & 1\\ |
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0 & 0 & 0\\ |
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1 & 0 & 0\\ |
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\end{pmatrix} |
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+3a_{21}^o |
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\begin{pmatrix} |
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0 & 0 & 0\\ |
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0 & 0 & 1\\ |
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0 & 1 & 0\\ |
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\end{pmatrix} |
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+6a_{22}^e |
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\begin{pmatrix} |
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0 & 1 & 0\\ |
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1 & 0 & 0\\ |
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0 & 0 & 0\\ |
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\end{pmatrix} |
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+6a_{22}^o |
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\begin{pmatrix} |
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1 & 0 & 0\\ |
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0 & -1 & 0\\ |
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0 & 0 & 0\\ |
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\end{pmatrix} |
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\end{aligned} |
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\label{eq:intro_tensors} |
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\end{gather} |
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The five matrices in the expression above represent five different |
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symmetric traceless tensors of rank 2. |
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|
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It is useful to find the Cartesian vectors $\mathbf a$ and $\mathbf b$ |
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that generate the five types of tensors shown in |
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Eq. (\ref{eq:intro_tensors}). If the two vectors are co-linear, e.g., |
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$\psi=0$, $\mathbf{a}=(0,0,1)$ and $\mathbf{b}=(0,0,1)$, then |
| 235 |
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\begin{equation*} |
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\mathsf{G} = \frac{g_0}{3} |
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\begin{pmatrix} |
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-1 & 0 & 0 \\ |
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0 & -1 & 0 \\ |
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0 & 0 & 2 \\ |
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\end{pmatrix} , |
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\end{equation*} |
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which is the $a_{20}$ symmetry. |
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To generate the $a_{22}^o$ symmetry, we take: |
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$\mathbf{a}= (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}},0)$ and |
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$\mathbf{b}=(\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}},0)$ |
| 247 |
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and find: |
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\begin{equation*} |
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\mathsf{G}=\frac{g_0}{2} |
| 250 |
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\begin{pmatrix} |
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1 & 0 & 0 \\ |
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0 & -1 & 0 \\ |
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0 & 0 & 0 \\ |
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\end{pmatrix} . |
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> |
\end{equation*} |
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> |
To generate the $a_{22}^e$ symmetry, we take: |
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$\mathbf{a}= (1, 0, 0)$ and $\mathbf{b} = (0,1,0)$ and find: |
| 258 |
> |
\begin{equation*} |
| 259 |
> |
\mathsf{G}=\frac{g_0}{2} |
| 260 |
> |
\begin{pmatrix} |
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> |
0 & 1 & 0 \\ |
| 262 |
> |
1 & 0 & 0 \\ |
| 263 |
> |
0 & 0 & 0 \\ |
| 264 |
> |
\end{pmatrix} . |
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> |
\end{equation*} |
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> |
The pattern is straightforward to continue for the other symmetries. |
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> |
|
| 268 |
> |
We find the notation of Ref. \onlinecite{Torres-del-Castillo:2006uo} |
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> |
helpful when creating specific types of constant gradient electric |
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> |
fields in simulations. For this reason, |
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Eqs. (\ref{eq:appliedPotential}), (\ref{eq:GC}), and (\ref{eq:CE}) are |
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> |
implemented in our code. In the simulations using constant applied |
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> |
gradients that are mentioned in the main text, we utilized a field |
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> |
with the $a_{22}^e$ symmetry using vectors, $\mathbf{a}= (1, 0, 0)$ |
| 275 |
> |
and $\mathbf{b} = (0,1,0)$. |
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> |
|
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> |
\section{Point-multipolar interactions with a spatially-varying electric field} |
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> |
|
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> |
This section develops formulas for the force and torque exerted by an |
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> |
external electric field, $\mathbf{E}(\mathbf{r})$, on object |
| 281 |
> |
$a$. Object $a$ has an embedded collection of charges and in |
| 282 |
> |
simulations will represent a molecule, ion, or a coarse-grained |
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> |
substructure. We describe the charge distributions using primitive |
| 284 |
> |
multipoles defined in Ref. \onlinecite{PaperI} by |
| 285 |
> |
\begin{align} |
| 286 |
> |
C_a =&\sum_{k \, \text{in }a} q_k , \label{eq:charge} \\ |
| 287 |
> |
D_{a\alpha} =&\sum_{k \, \text{in }a} q_k r_{k\alpha}, \label{eq:dipole}\\ |
| 288 |
> |
Q_{a\alpha\beta} =& \frac{1}{2} \sum_{k \, \text{in } a} q_k |
| 289 |
> |
r_{k\alpha} r_{k\beta}, |
| 290 |
> |
\label{eq:quadrupole} |
| 291 |
> |
\end{align} |
| 292 |
> |
where $\mathbf{r}_k$ is the local coordinate system for the object |
| 293 |
> |
(usually the center of mass of object $a$). Components of vectors and |
| 294 |
> |
tensors are given using the Einstein repeated summation notation. Note |
| 295 |
> |
that the definition of the primitive quadrupole here differs from the |
| 296 |
> |
standard traceless form, and contains an additional Taylor-series |
| 297 |
> |
based factor of $1/2$. In Ref. \onlinecite{PaperI}, we derived the |
| 298 |
> |
forces and torques each object exerts on the other objects in the |
| 299 |
> |
system. |
| 300 |
> |
|
| 301 |
> |
Here we must also consider an external electric field that varies in |
| 302 |
> |
space: $\mathbf E(\mathbf r)$. Each of the local charges $q_k$ in |
| 303 |
> |
object $a$ will then experience a slightly different field. This |
| 304 |
> |
electric field can be expanded in a Taylor series around the local |
| 305 |
> |
origin of each object. For a particular charge $q_k$, the electric |
| 306 |
> |
field at that site's position is given by: |
| 307 |
|
\begin{equation} |
| 308 |
< |
\braket{p_{mol}} \approx \frac{1}{3}\frac{{p_o}^2}{k_B T}E, |
| 308 |
> |
\mathbf{E}(\mathbf{r}_k) = E_\gamma|_{\mathbf{r}_k = 0} + \nabla_\delta E_\gamma |_{\mathbf{r}_k = 0} r_{k \delta} |
| 309 |
> |
+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma|_{\mathbf{r}_k = 0} r_{k \delta} |
| 310 |
> |
r_{k \varepsilon} + ... |
| 311 |
|
\end{equation} |
| 312 |
< |
where $ \alpha_p = \frac{1}{3}\frac{{p_o}^2}{k_B T}$ is a molecular |
| 312 |
> |
Note that if one shrinks object $a$ to a single point, the |
| 313 |
> |
${E}_\gamma$ terms are all evaluated at the center of the object (now |
| 314 |
> |
a point). Thus later the ${E}_\gamma$ terms can be written using the |
| 315 |
> |
same (molecular) origin for all point charges in the object. The force |
| 316 |
> |
exerted on object $a$ by the electric field is given by,\cite{Raab:2004ve} |
| 317 |
> |
\begin{align} |
| 318 |
> |
F^a_\gamma = \sum_{k \textrm{~in~} a} E_\gamma(\mathbf{r}_k) &= \sum_{k \textrm{~in~} a} q_k \lbrace E_\gamma + \nabla_\delta E_\gamma r_{k \delta} |
| 319 |
> |
+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma r_{k \delta} |
| 320 |
> |
r_{k \varepsilon} + ... \rbrace \\ |
| 321 |
> |
&= C_a E_\gamma + D_{a \delta} \nabla_\delta E_\gamma |
| 322 |
> |
+ Q_{a \delta \varepsilon} \nabla_\delta \nabla_\varepsilon E_\gamma + |
| 323 |
> |
... |
| 324 |
> |
\end{align} |
| 325 |
> |
Thus in terms of the global origin $\mathbf{r}$, ${F}_\gamma(\mathbf{r}) = C {E}_\gamma(\mathbf{r})$ etc. |
| 326 |
> |
|
| 327 |
> |
Similarly, the torque exerted by the field on $a$ can be expressed as |
| 328 |
> |
\begin{align} |
| 329 |
> |
\tau^a_\alpha &= \sum_{k \textrm{~in~} a} (\mathbf r_k \times q_k \mathbf E)_\alpha \\ |
| 330 |
> |
& = \sum_{k \textrm{~in~} a} \epsilon_{\alpha \beta \gamma} q_k |
| 331 |
> |
r_{k\beta} E_\gamma(\mathbf r_k) \\ |
| 332 |
> |
& = \epsilon_{\alpha \beta \gamma} D_\beta E_\gamma |
| 333 |
> |
+ 2 \epsilon_{\alpha \beta \gamma} Q_{\beta \delta} \nabla_\delta |
| 334 |
> |
E_\gamma + ... |
| 335 |
> |
\end{align} |
| 336 |
> |
We note that the Levi-Civita symbol can be eliminated by utilizing the matrix cross product as defined in Ref. \onlinecite{Smith98}: |
| 337 |
> |
\begin{equation} |
| 338 |
> |
\left[\mathsf{A} \times \mathsf{B}\right]_\alpha = \sum_\beta |
| 339 |
> |
\left[\mathsf{A}_{\alpha+1,\beta} \mathsf{B}_{\alpha+2,\beta} |
| 340 |
> |
-\mathsf{A}_{\alpha+2,\beta} \mathsf{B}_{\alpha+1,\beta} |
| 341 |
> |
\right] |
| 342 |
> |
\label{eq:matrixCross} |
| 343 |
> |
\end{equation} |
| 344 |
> |
where $\alpha+1$ and $\alpha+2$ are regarded as cyclic permuations of |
| 345 |
> |
the matrix indices. Finally, the interaction energy $U^a$ of object $a$ with the external field is given by, |
| 346 |
> |
\begin{equation} |
| 347 |
> |
U^a = \sum_{k~in~a} q_k \phi_k (\mathrm{r}_k) |
| 348 |
> |
\end{equation} |
| 349 |
> |
Performing another Taylor series expansion about the local body origin, |
| 350 |
> |
\begin{equation} |
| 351 |
> |
\phi({\mathbf{r}_k}) = \phi|_{\mathbf{r}_k = 0 } + r_{k \alpha} \nabla_\alpha \phi_\alpha|_{\mathbf{r}_k = 0 } + \frac{1}{2} r_{k\alpha}r_{k\beta}\nabla_\alpha \nabla_\beta \phi|_{\mathbf{r}_k = 0} + ... |
| 352 |
> |
\end{equation} |
| 353 |
> |
Writing this in terms of the global origin $\mathrm{r}$, we find |
| 354 |
> |
\begin{equation} |
| 355 |
> |
U(\mathbf{r}) = \mathrm{C} \phi(\mathbf{r}) - \mathrm{D}_\alpha \mathrm{E}_\alpha - \mathrm{Q}_{\alpha\beta}\nabla_\alpha \mathrm{E}_\beta + ... |
| 356 |
> |
\end{equation} |
| 357 |
> |
These results have been summarized in Table \ref{tab:UFT}. |
| 358 |
> |
|
| 359 |
> |
\begin{table} |
| 360 |
> |
\caption{Potential energy $(U)$, force $(\mathbf{F})$, and torque |
| 361 |
> |
$(\mathbf{\tau})$ expressions for a multipolar site at $\mathbf{r}$ in an |
| 362 |
> |
electric field, $\mathbf{E}(\mathbf{r})$ using the definitions of the multipoles in Eqs. (\ref{eq:charge}), (\ref{eq:dipole}) and (\ref{eq:quadrupole}). |
| 363 |
> |
\label{tab:UFT}} |
| 364 |
> |
\begin{tabular}{r|C{3cm}C{3cm}C{3cm}} |
| 365 |
> |
& Charge & Dipole & Quadrupole \\ \hline |
| 366 |
> |
$U(\mathbf{r})$ & $C \phi(\mathbf{r})$ & $-\mathbf{D} \cdot \mathbf{E}(\mathbf{r})$ & $- \mathsf{Q}:\nabla \mathbf{E}(\mathbf{r})$ \\ |
| 367 |
> |
$\mathbf{F}(\mathbf{r})$ & $C \mathbf{E}(\mathbf{r})$ & $\mathbf{D} \cdot \nabla \mathbf{E}(\mathbf{r})$ & $\mathsf{Q} : \nabla\nabla\mathbf{E}(\mathbf{r})$ \\ |
| 368 |
> |
$\mathbf{\tau}(\mathbf{r})$ & & $\mathbf{D} \times \mathbf{E}(\mathbf{r})$ & $2 \mathsf{Q} \times \nabla \mathbf{E}(\mathbf{r})$ |
| 369 |
> |
\end{tabular} |
| 370 |
> |
\end{table} |
| 371 |
> |
|
| 372 |
> |
\section{Boltzmann averages for orientational polarization} |
| 373 |
> |
If we consider a collection of molecules in the presence of external |
| 374 |
> |
field, the perturbation experienced by any one molecule will include |
| 375 |
> |
contributions to the field or field gradient produced by the all other |
| 376 |
> |
molecules in the system. In subsections |
| 377 |
> |
\ref{subsec:boltzAverage-Dipole} and \ref{subsec:boltzAverage-Quad}, |
| 378 |
> |
we discuss the molecular polarization due solely to external field |
| 379 |
> |
perturbations. This illustrates the origins of the polarizability |
| 380 |
> |
equations (Eqs. 6, 20, and 21) in the main text. |
| 381 |
> |
|
| 382 |
> |
\subsection{Dipoles} |
| 383 |
> |
\label{subsec:boltzAverage-Dipole} |
| 384 |
> |
Consider a system of molecules, each with permanent dipole moment |
| 385 |
> |
$p_o$. In the absense of an external field, thermal agitation orients |
| 386 |
> |
the dipoles randomly, and the system moment, $\mathbf{P}$, is zero. |
| 387 |
> |
External fields will line up the dipoles in the direction of applied |
| 388 |
> |
field. Here we consider the net field from all other molecules to be |
| 389 |
> |
zero. Therefore the total Hamiltonian acting on each molecule |
| 390 |
> |
is,\cite{Jackson98} |
| 391 |
> |
\begin{equation} |
| 392 |
> |
H = H_o - \mathbf{p}_o \cdot \mathbf{E}, |
| 393 |
> |
\end{equation} |
| 394 |
> |
where $H_o$ is a function of the internal coordinates of the molecule. |
| 395 |
> |
The Boltzmann average of the dipole moment in the direction of the |
| 396 |
> |
field is given by, |
| 397 |
> |
\begin{equation} |
| 398 |
> |
\langle p_{mol} \rangle = \frac{\displaystyle\int p_o \cos\theta |
| 399 |
> |
e^{~p_o E \cos\theta /k_B T}\; d\Omega}{\displaystyle\int e^{~p_o E \cos\theta/k_B |
| 400 |
> |
T}\; d\Omega}, |
| 401 |
> |
\end{equation} |
| 402 |
> |
where the $z$-axis is taken in the direction of the applied field, |
| 403 |
> |
$\bf{E}$ and |
| 404 |
> |
$\int d\Omega = \int_0^\pi \sin\theta\; d\theta \int_0^{2\pi} d\phi |
| 405 |
> |
\int_0^{2\pi} d\psi$ |
| 406 |
> |
is an integration over Euler angles describing the orientation of the |
| 407 |
> |
molecule. |
| 408 |
> |
|
| 409 |
> |
If the external fields are small, \textit{i.e.} |
| 410 |
> |
$p_oE \cos\theta / k_B T << 1$, |
| 411 |
> |
\begin{equation} |
| 412 |
> |
\langle p_{mol} \rangle \approx \frac{{p_o}^2}{3 k_B T}E, |
| 413 |
> |
\end{equation} |
| 414 |
> |
where $ \alpha_p = \frac{{p_o}^2}{3 k_B T}$ is the molecular |
| 415 |
|
polarizability. The orientational polarization depends inversely on |
| 416 |
< |
the temperature and applied field must overcome the thermal agitation. |
| 416 |
> |
the temperature as the applied field must overcome thermal agitation |
| 417 |
> |
to orient the dipoles. |
| 418 |
|
|
| 419 |
|
\subsection{Quadrupoles} |
| 420 |
|
\label{subsec:boltzAverage-Quad} |
| 421 |
< |
Consider a system of molecules with permanent quadrupole moment |
| 422 |
< |
$q_{\alpha\beta} $. The average quadrupole moment at temperature T in |
| 423 |
< |
the presence of uniform applied field gradient is given |
| 424 |
< |
by,\cite{AduGyamfi78, AduGyamfi81} |
| 421 |
> |
If instead, our system consists of molecules with permanent |
| 422 |
> |
\textit{quadrupole} tensor $q_{\alpha\beta}$. The average quadrupole |
| 423 |
> |
at temperature $T$ in the presence of uniform applied field gradient |
| 424 |
> |
is given by,\cite{AduGyamfi78, AduGyamfi81} |
| 425 |
|
\begin{equation} |
| 426 |
< |
\braket{q_{\alpha\beta}} \;=\; \frac{\displaystyle\int d\Omega\; e^{-\frac{H}{k_B T}}q_{\alpha\beta}}{\displaystyle\int d\Omega\; e^{-\frac{H}{k_B T}}} \;=\; \frac{\displaystyle\int d\Omega\; e^{\frac{q_{\mu\nu}\;\partial_\nu E_\mu}{k_B T}}q_{\alpha\beta}}{\displaystyle\int d\Omega\; e^{\frac{q_{\mu\nu}\;\partial_\nu E_\mu}{k_B T}}}, |
| 426 |
> |
\langle q_{\alpha\beta} \rangle \;=\; \frac{\displaystyle\int |
| 427 |
> |
q_{\alpha\beta}\; e^{-H/k_B T}\; d\Omega}{\displaystyle\int |
| 428 |
> |
e^{-H/k_B T}\; d\Omega} \;=\; \frac{\displaystyle\int |
| 429 |
> |
q_{\alpha\beta}\; e^{~q_{\mu\nu}\;\partial_\nu E_\mu /k_B T}\; |
| 430 |
> |
d\Omega}{\displaystyle\int e^{~q_{\mu\nu}\;\partial_\nu E_\mu /k_B |
| 431 |
> |
T}\; d\Omega }, |
| 432 |
|
\label{boltzQuad} |
| 433 |
|
\end{equation} |
| 434 |
< |
where $\int d\Omega = \int_0^{2\pi} \int_0^\pi \int_0^{2\pi} |
| 435 |
< |
sin\theta\; d\theta\ d\phi\ d\psi$ is the integration over Euler |
| 436 |
< |
angles, $ H = H_o -q_{\mu\nu}\;\partial_\nu E_\mu $ is the energy of |
| 437 |
< |
a quadrupole in the gradient of the |
| 438 |
< |
applied field and $ H_o$ is a function of internal coordinates of the molecule. The energy and quadrupole moment can be transformed into body frame using following relation, |
| 434 |
> |
where $H = H_o - q_{\mu\nu}\;\partial_\nu E_\mu $ is the energy of a |
| 435 |
> |
quadrupole in the gradient of the applied field and $H_o$ is a |
| 436 |
> |
function of internal coordinates of the molecule. The energy and |
| 437 |
> |
quadrupole moment can be transformed into the body frame using a |
| 438 |
> |
rotation matrix $\mathsf{\eta}^{-1}$, |
| 439 |
> |
\begin{align} |
| 440 |
> |
q_{\alpha\beta} &= \eta_{\alpha\alpha'}\;\eta_{\beta\beta'}\;{q}^* _{\alpha'\beta'} \\ |
| 441 |
> |
H &= H_o - q:{\nabla}\mathbf{E} \\ |
| 442 |
> |
&= H_o - q_{\mu\nu}\;\partial_\nu E_\mu \\ |
| 443 |
> |
&= H_o |
| 444 |
> |
-\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu |
| 445 |
> |
E_\mu. \label{energyQuad} |
| 446 |
> |
\end{align} |
| 447 |
> |
Here the starred tensors are the components in the body fixed |
| 448 |
> |
frame. Substituting equation (\ref{energyQuad}) in the equation |
| 449 |
> |
(\ref{boltzQuad}) and taking linear terms in the expansion we obtain, |
| 450 |
|
\begin{equation} |
| 451 |
+ |
\braket{q_{\alpha\beta}} = \frac{\displaystyle \int q_{\alpha\beta} \left(1 + |
| 452 |
+ |
\frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu |
| 453 |
+ |
E_\mu }{k_B T}\right)\; d\Omega}{\displaystyle \int \left(1 + \frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu }{k_B T}\right)\; d\Omega}. |
| 454 |
+ |
\end{equation} |
| 455 |
+ |
Recall that $\eta_{\alpha\alpha'}$ is the inverse of the rotation |
| 456 |
+ |
matrix that transforms the body fixed co-ordinates to the space |
| 457 |
+ |
co-ordinates. |
| 458 |
+ |
% \[\eta_{\alpha\alpha'} |
| 459 |
+ |
% = \left(\begin{array}{ccc} |
| 460 |
+ |
% cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & -cos\theta\; cos\psi\; sin\phi - cos\phi\; sin\psi & sin\theta\; sin\phi \\ |
| 461 |
+ |
% cos\psi\; sin\phi + cos\theta\; cos\phi \; sin\psi & cos\theta\; cos\phi\; cos\psi - sin\phi\; sin\psi & -cos\phi\; sin\theta \\ |
| 462 |
+ |
% sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta |
| 463 |
+ |
% \end{array} \right).\] |
| 464 |
+ |
|
| 465 |
+ |
Integration of the first and second terms in the denominator gives |
| 466 |
+ |
$8 \pi^2$ and |
| 467 |
+ |
$8 \pi^2 ({\nabla} \cdot \mathbf{E}) \mathrm{Tr}(q^*) / 3 $ |
| 468 |
+ |
respectively. The second term vanishes for charge free space (where |
| 469 |
+ |
${\nabla} \cdot \mathbf{E}=0$). Similarly, integration of the first |
| 470 |
+ |
term in the numerator produces |
| 471 |
+ |
$8 \pi^2 \delta_{\alpha\beta} \mathrm{Tr}(q^*) / 3$ while the second |
| 472 |
+ |
produces |
| 473 |
+ |
$8 \pi^2 (3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'} - |
| 474 |
+ |
{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'})\partial_\alpha E_\beta / |
| 475 |
+ |
15 k_B T $. |
| 476 |
+ |
Therefore the Boltzmann average of a quadrupole moment can be written |
| 477 |
+ |
as, |
| 478 |
+ |
\begin{equation} |
| 479 |
+ |
\langle q_{\alpha\beta} \rangle = \frac{1}{3} \mathrm{Tr}(q^*)\;\delta_{\alpha\beta} + \frac{{\bar{q_o}}^2}{15k_BT}\;\partial_\alpha E_\beta, |
| 480 |
+ |
\end{equation} |
| 481 |
+ |
where $\alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a molecular |
| 482 |
+ |
quadrupole polarizablity and |
| 483 |
+ |
${\bar{q_o}}^2= |
| 484 |
+ |
3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'}-{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'}$ |
| 485 |
+ |
is the square of the net quadrupole moment of a molecule. |
| 486 |
+ |
|
| 487 |
+ |
\section{Gradient of the field due to quadrupolar polarization} |
| 488 |
+ |
\label{singularQuad} |
| 489 |
+ |
In section IV.C of the main text, we stated that for quadrupolar |
| 490 |
+ |
fluids, the self-contribution to the field gradient vanishes at the |
| 491 |
+ |
singularity. In this section, we prove this statement. For this |
| 492 |
+ |
purpose, we consider a distribution of charge $\rho(\mathbf{r})$ which |
| 493 |
+ |
gives rise to an electric field $\mathbf{E}(\mathbf{r})$ and gradient |
| 494 |
+ |
of the field $\nabla\mathbf{E}(\mathbf{r})$ throughout space. The |
| 495 |
+ |
gradient of the electric field over volume due to the charges within |
| 496 |
+ |
the sphere of radius $R$ is given by (cf. Ref. \onlinecite{Jackson98}, |
| 497 |
+ |
equation 4.14): |
| 498 |
+ |
\begin{equation} |
| 499 |
+ |
\int_{r<R} \nabla\mathbf{E} d\mathbf{r} = -\int_{r=R} R^2 \mathbf{E}\;\hat{n}\; d\Omega |
| 500 |
+ |
\label{eq:8} |
| 501 |
+ |
\end{equation} |
| 502 |
+ |
where $d\Omega$ is the solid angle and $\hat{n}$ is the normal vector |
| 503 |
+ |
of the surface of the sphere, |
| 504 |
+ |
\begin{equation} |
| 505 |
+ |
\hat{n} = \sin\theta\cos\phi\; \hat{x} + \sin\theta\sin\phi\; \hat{y} + |
| 506 |
+ |
\cos\theta\; \hat{z} |
| 507 |
+ |
\end{equation} |
| 508 |
+ |
in spherical coordinates. For the charge density $\rho(\mathbf{r}')$, the |
| 509 |
+ |
total gradient of the electric field can be written as,\cite{Jackson98} |
| 510 |
+ |
\begin{equation} |
| 511 |
+ |
\int_{r<R} {\nabla}\mathbf {E}\; d\mathbf{r}=-\int_{r=R} R^2\; {\nabla}\Phi\; \hat{n}\; d\Omega =-\frac{1}{4\pi\;\epsilon_o}\int_{r=R} R^2\; {\nabla}\;\left(\int \frac{\rho(\mathbf r')}{|\mathbf{r}-\mathbf{r}'|}\;d\mathbf{r}'\right) \hat{n}\; d\Omega |
| 512 |
+ |
\label{eq:9} |
| 513 |
+ |
\end{equation} |
| 514 |
+ |
The radial function in the equation (\ref{eq:9}) can be expressed in |
| 515 |
+ |
terms of spherical harmonics as,\cite{Jackson98} |
| 516 |
+ |
\begin{equation} |
| 517 |
+ |
\frac{1}{|\mathbf{r} - \mathbf{r}'|} = 4\pi \sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\frac{{r^l_<}}{{r^{l+1}_>}}\;{Y^*}_{lm}(\theta', \phi')\;Y_{lm}(\theta, \phi) |
| 518 |
+ |
\label{eq:10} |
| 519 |
+ |
\end{equation} |
| 520 |
+ |
If the sphere completely encloses the charge density then $ r_< = r'$ and $r_> = R$. Substituting equation (\ref{eq:10}) into (\ref{eq:9}) we get, |
| 521 |
+ |
\begin{equation} |
| 522 |
|
\begin{split} |
| 523 |
< |
&q_{\alpha\beta} = \eta_{\alpha\alpha'}\;\eta_{\beta\beta'}\;{q}^* _{\alpha'\beta'} \\ |
| 524 |
< |
&H = H_o - q:\vec{\nabla}\vec{E} = H_o - q_{\mu\nu}\;\partial_\nu E_\mu = H_o -\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu. |
| 523 |
> |
\int_{r<R} {\nabla}\mathbf{E}\;d\mathbf{r} &=-\frac{R^2}{\epsilon_o}\int_{r=R} \; {\nabla}\;\left(\int \rho(\mathbf r')\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\frac{{r'^l}}{{R^{l+1}}}\;{Y^*}_{lm}(\theta', \phi')\;Y_{lm}(\theta, \phi)\;d\mathbf{r}'\right) \hat{n}\; d\Omega \\ |
| 524 |
> |
&= -\frac{R^2}{\epsilon_o}\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\int \rho(\mathbf r')\;{r'^l}\;{Y^*}_{lm}(\theta', \phi')\left(\int_{r=R}\vec{\nabla}\left({R^{-(l+1)}}\;Y_{lm}(\theta, \phi)\right)\hat{n}\; d\Omega \right)d\mathbf{r} |
| 525 |
> |
' |
| 526 |
|
\end{split} |
| 527 |
< |
\label{energyQuad} |
| 527 |
> |
\label{eq:11} |
| 528 |
> |
\end{equation} |
| 529 |
> |
The gradient of the product of radial function and spherical harmonics |
| 530 |
> |
is given by:\cite{Arfkan} |
| 531 |
> |
\begin{equation} |
| 532 |
> |
\begin{split} |
| 533 |
> |
{\nabla}\left[ f(r)\;Y_{lm}(\theta, \phi)\right] = &-\left(\frac{l+1}{2l+1}\right)^{1/2}\; \left[\frac{\partial}{\partial r}-\frac{l}{r} \right]f(r)\; Y_{l, l+1, m}(\theta, \phi)\\ &+ \left(\frac{l}{2l+1}\right)^{1/2}\left[\frac |
| 534 |
> |
{\partial}{\partial r}+\frac{l}{r} \right]f(r)\; Y_{l, l-1, m}(\theta, \phi). |
| 535 |
> |
\end{split} |
| 536 |
> |
\label{eq:12} |
| 537 |
|
\end{equation} |
| 538 |
< |
Here the starred tensors are the components in the body fixed |
| 539 |
< |
frame. Substituting equation (\ref{energyQuad}) in the equation (\ref{boltzQuad}) |
| 136 |
< |
and taking linear terms in the expansion we get, |
| 538 |
> |
where $Y_{l,l+1,m}(\theta, \phi)$ is a vector spherical |
| 539 |
> |
harmonic.\cite{Arfkan} Using equation (\ref{eq:12}) we get, |
| 540 |
|
\begin{equation} |
| 541 |
< |
\braket{q_{\alpha\beta}} = \frac{ \int d\Omega \left(1 + \frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu }{k_B T}\right)q_{\alpha\beta}}{ \int d\Omega \left(1 + \frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu }{k_B T}\right)}, |
| 541 |
> |
{\nabla}\left({R^{-(l+1)}}\;Y_{lm}(\theta, \phi)\right) = [(l+1)(2l+1)]^{1/2}\; Y_{l,l+1,m}(\theta, \phi) \; \frac{1}{R^{l+2}}, |
| 542 |
> |
\label{eq:13} |
| 543 |
|
\end{equation} |
| 544 |
< |
where $\eta_{\alpha\alpha'}$ is the inverse of the rotation matrix that transforms |
| 545 |
< |
the body fixed co-ordinates to the space co-ordinates, |
| 142 |
< |
\[\eta_{\alpha\alpha'} |
| 143 |
< |
= \left(\begin{array}{ccc} |
| 144 |
< |
cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & -cos\theta\; cos\psi\; sin\phi - cos\phi\; sin\psi & sin\theta\; sin\phi \\ |
| 145 |
< |
cos\psi\; sin\phi + cos\theta\; cos\phi \; sin\psi & cos\theta\; cos\phi\; cos\psi - sin\phi\; sin\psi & -cos\phi\; sin\theta \\ |
| 146 |
< |
sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta |
| 147 |
< |
\end{array} \right).\] |
| 148 |
< |
Integration of 1st and 2nd terms in the denominator gives $8 \pi^2$ |
| 149 |
< |
and $8 \pi^2 /3\;\vec{\nabla}.\vec{E}\; Tr(q^*) $ respectively. The |
| 150 |
< |
second term vanishes for charge free space |
| 151 |
< |
(i.e. $\vec{\nabla}.\vec{E} \; = \; 0)$. Similarly integration of the |
| 152 |
< |
1st term in the numerator produces |
| 153 |
< |
$8 \pi^2 /3\; Tr(q^*)\delta_{\alpha\beta}$ and the 2nd term produces |
| 154 |
< |
$8 \pi^2 /15k_B T (3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'} - |
| 155 |
< |
{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'})\partial_\alpha E_\beta$, |
| 156 |
< |
if $\vec{\nabla}.\vec{E} \; = \; 0$, |
| 157 |
< |
$ \partial_\alpha E_\beta = \partial_\beta E_\alpha$ and |
| 158 |
< |
${q}^*_{\alpha'\beta'}= {q}^*_{\beta'\alpha'}$. Therefore the |
| 159 |
< |
Boltzmann average of a quadrupole moment can be written as, |
| 160 |
< |
|
| 544 |
> |
Using Clebsch-Gordan coefficients $C(l+1,1,l|m_1,m_2,m)$, the vector |
| 545 |
> |
spherical harmonics can be written in terms of spherical harmonics, |
| 546 |
|
\begin{equation} |
| 547 |
< |
\braket{q_{\alpha\beta}}\; = \; \frac{1}{3} Tr(q^*)\;\delta_{\alpha\beta} + \frac{{\bar{q_o}}^2}{15k_BT}\;\partial_\alpha E_\beta, |
| 547 |
> |
Y_{l,l+1,m}(\theta, \phi) = \sum_{m_1, m_2} C(l+1,1,l|m_1,m_2,m)\; Y_{l+1}^{m_1}(\theta,\phi)\; \hat{e}_{m_2}. |
| 548 |
> |
\label{eq:14} |
| 549 |
|
\end{equation} |
| 550 |
< |
where $ \alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a molecular quadrupolarizablity and ${\bar{q_o}}^2= |
| 551 |
< |
3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'}-{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'}$ is a square of the net quadrupole moment of a molecule. |
| 550 |
> |
Here $\hat{e}_{m_2}$ is a spherical tensor of rank 1 which can be expressed |
| 551 |
> |
in terms of Cartesian coordinates, |
| 552 |
> |
\begin{equation} |
| 553 |
> |
{\hat{e}}_{+1} = - \frac{\hat{x}+i\hat{y}}{\sqrt{2}},\quad {\hat{e}}_{0} = \hat{z},\quad and \quad {\hat{e}}_{-1} = \frac{\hat{x}-i\hat{y}}{\sqrt{2}}. |
| 554 |
> |
\label{eq:15} |
| 555 |
> |
\end{equation} |
| 556 |
> |
The normal vector $\hat{n} $ is then expressed in terms of spherical tensor of rank 1 as shown in below, |
| 557 |
> |
\begin{equation} |
| 558 |
> |
\hat{n} = \sqrt{\frac{4\pi}{3}}\left(-Y_1^{-1}{\hat{e}}_1 - Y_1^{1}{\hat{e}}_{-1} + Y_1^{0}{\hat{e}}_0 \right). |
| 559 |
> |
\label{eq:16} |
| 560 |
> |
\end{equation} |
| 561 |
> |
The surface integral of the product of $\hat{n}$ and |
| 562 |
> |
$Y_{l+1}^{m_1}(\theta, \phi)$ gives, |
| 563 |
> |
\begin{equation} |
| 564 |
> |
\begin{split} |
| 565 |
> |
\int \hat{n}\;Y_{l+1}^{m_1}\;d\Omega &= \int \sqrt{\frac{4\pi}{3}}\left(-Y_1^{-1}{\hat{e}}_1 -Y_1^{1}{\hat{e}}_{-1} + Y_1^{0}{\hat{e}}_0 \right)\;Y_{l+1}^{m_1}\; d\Omega \\ |
| 566 |
> |
&= \int \sqrt{\frac{4\pi}{3}}\left({Y_1^{1}}^* {\hat{e}}_1 +{Y_1^{-1}}^* {\hat{e}}_{-1} + {Y_1^{0}}^* {\hat{e}}_0 \right)\;Y_{l+1}^{m_1}\; d\Omega \\ |
| 567 |
> |
&= \sqrt{\frac{4\pi}{3}}\left({\delta}_{l+1, 1}\;{\delta}_{1, m_1}\;{\hat{e}}_1 + {\delta}_{l+1, 1}\;{\delta}_{-1, m_1}\;{\hat{e}}_{-1}+ {\delta}_{l+1, 1}\;{\delta}_{0, m_1} \;{\hat{e}}_0\right), |
| 568 |
> |
\end{split} |
| 569 |
> |
\label{eq:17} |
| 570 |
> |
\end{equation} |
| 571 |
> |
where $Y_{l}^{-m} = (-1)^m\;{Y_{l}^{m}}^* $ and |
| 572 |
> |
$ \int {Y_{l}^{m}}^* Y_{l'}^{m'}\;d\Omega = |
| 573 |
> |
\delta_{ll'}\delta_{mm'} $. |
| 574 |
> |
Non-vanishing values of equation \ref{eq:17} require $l = 0$, |
| 575 |
> |
therefore the value of $ m = 0 $. Since the values of $ m_1$ are -1, |
| 576 |
> |
1, and 0 then $m_2$ takes the values 1, -1, and 0, respectively |
| 577 |
> |
provided that $m = m_1 + m_2$. Equation \ref{eq:11} can therefore be |
| 578 |
> |
modified, |
| 579 |
> |
\begin{equation} |
| 580 |
> |
\begin{split} |
| 581 |
> |
\int_{r<R} {\nabla}\mathbf{E}\;d\mathbf{r} = &- \sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\int \rho(r')\;{Y^*}_{00}(\theta', \phi')[ C(1, 1, 0|-1,1,0)\;{\hat{e}_{-1}}{\hat{e}_{1}}\\ &+ C(1, 1, 0|-1,1,0)\;{\hat{e}_{1}}{\hat{e}_{-1}}+C( |
| 582 |
> |
1, 1, 0|0,0,0)\;{\hat{e}_{0}}{\hat{e}_{0}} ]\; d\mathbf{r}' \\ |
| 583 |
> |
&= -\sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\int \rho(r')\;d\mathbf{r}'\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right)\\ |
| 584 |
> |
&= - \sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\;C_\mathrm{total}\;\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right). |
| 585 |
> |
\end{split} |
| 586 |
> |
\label{eq:19} |
| 587 |
> |
\end{equation} |
| 588 |
> |
In the last step, the charge density was integrated over the sphere, |
| 589 |
> |
yielding a total charge $C_\mathrm{total}$.Equation (\ref{eq:19}) |
| 590 |
> |
gives the total gradient of the field over a sphere due to the |
| 591 |
> |
distribution of the charges. For quadrupolar fluids the total charge |
| 592 |
> |
within a sphere is zero, therefore |
| 593 |
> |
$ \int_{r<R} {\nabla}\mathbf{E}\;d\mathbf{r} = 0 $. Hence the quadrupolar |
| 594 |
> |
polarization produces zero net gradient of the field inside the |
| 595 |
> |
sphere. |
| 596 |
|
|
| 167 |
– |
|
| 597 |
|
\bibliography{dielectric_new} |
| 598 |
|
\end{document} |
| 599 |
|
% |
