Variable dielectric coefficient

Pack, G., G. Garrett, L. Wong and G. Lamm. (1993). The Effect of a Variable Dielectric Coefficient and Finite Ion Size on Poisson-boltzmann Calculations of Dna-electrolyte Systems. Biophysical Journal. 65 1363-1370. 

This result has been extended to a variable dielectric coefficient by Bucher " and Ehrenson. In the presence of a point-ion electrolyte, the Debye-... 

Analytical approximations can be developed Physical insight can guide and/or result from the solution Computationally fast for large systems Variable dielectric coefficient can be included... 

Let denote the potential obtained from a standard PB calculation assuming a constant bulk dielectric coefficient and < )var( ) denote that obtained by assuming a variable dielectric coefficient (i.e., Eqs. [385]-[388]). Now let Vb z) be the MC potential (a sum of Lennard-Jones and electrostatic terms) that a cation in the system would experience in a bulk dielectric continuum. The approximate effect of including a variable dielectric coefficient in the Monte Carlo simulation can be found by replacing the bulk MC potential Vb z) with the PB-corrected potential VB(r)- -< )var(r) — < )B(r). Of course the resulting ion distribution will not be self-consistent with the assumed (PB-derived) dielectric coefficient map, but it will be a noticeable improvement on the original MC distribution using a bulk dielectric. 

The ambiguity of their interpretation has already been pointed out in Sect. 6.3 They could be seen to represent the dependence either of a dielectric coefficient on a mechanical variable (and in this way related to the piezooptic effect) or of a piezoelectric coefficient on an electric variable (and associated with electrostriction). 

The effect of a variable local dielectric coefficient based on Eqs. [385]-... 

Garrett and Poladian demonstrate the uniqueness of the nonlinear PB equation for the case of a constant dielectric coefficient. The extension for a variable (and positive) dielectric coefficient is readily shown by a suitable modification of Green s theorem based on Eq. [3]. However, care must be taken in the numerical solution of some modified PB theories as nonuniqueness has been observed. ... 

Equations (6) and (7) express these relationships. are the elastic compliance constants OC are the linear thermal expansion coefficients 4 and d jj,are the direct and converse piezoelectric strain coefficients, respectively Pk are the pyroelectric coefficients and X are the dielectric susceptibility constants. The superscript a on Pk, Pk, and %ki indicates that these quantities are defined under the conditions of constant stress. If is taken to be the independent variable, then O and are the dependent quantities ... 

Note that the reference state for the activity coefficient depends on the variable under consideration. The reference state with respect to the dielectric constant is a hypothetical solvent with infinite dielectric constant... 

The quantities Pjk are also dielectric material coefficients. Their dimension is obviously reciprocal to the dimension of the permittivity. They are also the coordinates of a symmetric second rank tensor for which several names have been in use. Following Thurston (1974) we call it impermittivity. (Alternatively, the names dielectric impermeability (Cady 1964, p. 163), dielectric permeability (Grindlay 1970, p. 56), vetivity , derived from the latin word vetare as the opposite to the latin permittere (Voigt 1910, p. 441) have been proposed.) The effect described by Eq. (4.31) is the inverse effect to Eq. (4.30). It is obvious that E and D have exchanged their roles as independent and dependent variables and, therefore, Sij and Pij are material coefficients derived from different thermodynamic potentials. 

Vj is the intrinsic (van der Waals) molecular volume, n is the solute ability to stabilize a neighboring charge or dipole by nonspecific dielectric interactions, and and a , are the solute ability to accept or donate a hydrogen in a hydrogen bond. The coefficients m, s, b, and a are constants for a particular set of conditions, determined by multiple linear regression of the LSER variable values for a series of chemicals with the measured value for a particular chemical property. This equation was used to estimate cation solubility and predict cation toxicity to Vihrio fischeri, Daphnia magm, and Leuciscus idus melanotus. 

By insertion of Eq. [52b] for j(r, ) into this equation, it is easy to see that a Poisson-type equation is obtained (actually a Laplace equation, with the rhs equal to zero). If the diffusion coefficient is a constant over the domain, then that variable drops out. Notice that, in Eq. [52b], an effective dielectric constant [exp(—pct)(r))] appears, and this expression can vary over orders of magnitude due to large variations in the potential. Finally, if we seek a solution to the time-dependent problem, we must solve the full Smoluchowski equation iteratively. The particle density then depends explicitly on time. 

There are other physical measurements which reflect molecular mobility and can be related to relaxation times and friction coefficients similar to those which characterize the rates of viscoelastic relaxations. Although such phenomena are outside the scope of this book, they are mentioned here because in some cases their dependence on temperature and other variables can be described by reduced variables and, by means of equation 49 or modifications of it, free volume parameters can be deduced which are closely related to those obtained from viscoelastic data. These include measurements of dispersion of the dielectric constant, nuclear magnetic resonance relaxation, diffusion of small molecules through polymers, and diffusion-controlled aspects of crystallization and polymerization. 

These are called the first piezoelectric equations, Sij are elastic compliance coefficients, are dielectric constants and dmi are piezoelectric coefficients (or piezoelectric moduli). Ifwe taking E and [S] as variables, the second piezoelectric equations will be as follows ... 

The boundary conditions for these piezoelectric equations are important (a) The condition mechanically free stipulates specifically that boundaries of a piezoelectric sample (e.g., a piezoelectric vibrator) can move, i.e., the vibrator vibrates with a variable strain and zero (or constant) stress. Under this condition, the coefficients in these equations carry a superscript T e.g., is the dielectric constant at constant stress, (b) The condition mechanically clamped stipulates specifically that the boundaries of a vibrator cannot move. This condition means that, when the frequency of the applied voltage is much higher than the resonance frequency of the vibrator, the strain is constant (or zero), while the stress varies. In this case, the coefficients in these equations carry a superscript S e.g., is the dielectric constant at constant strain, (c) The condition of electrical short circuit implies specifically that the electric field = 0 (or a constant), while the electric displacement D 0 inside the vibrator. This is the case when the two electrodes on the surface of the crystal sample are electrically connected (or the electric potential on the entire surface of the sample is constant). Under this condition, the coefficients in these equations carry a superscript E e.g., sfj (or c ) is the elastic compliance (or stiffness) coefficient at constant electric field, (d) The condition of electrical open circuit corresponds to the case when aU the free charges are kept on the electrodes of the sample (electrically insulated) and the internal electric field / 0, while = 0 in the sample, hi this case, the coefficients in these equations carry a superscript D e.g., sjj (or c ) is the elastic compliance (or stiffness) coefficient at constant polarization. 

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