Boltzmann equation, modified

The generalization to systems of eater complexity is given later. The equilibrium and time Tt) dependent non-equilibrium momentum distributions for species R and A are denoted fg.,(Pp) and g. (P.,t). The time derivative of the hot atom momentm densi IS given by the Boltzmann equation modified for the inclusion of internal states (73). 

One will immediately recognize this as a form of the BOLTZMANN equation, or the GAUSSIAN LAW. We can modify this equation and put it into a form more suitable for our use by making the following definitions. 

The presence of the diffuse layer determines the shape of the capacitance-potential curves. For a majority of systems, models describing the double-layer structure are oversimplified because of taking into account only the charge of ions and neglecting their specific nature. Recently, these problems have been analyzed using new theories such as the modified Poisson-Boltzmann equation, later developed by Lamper-ski. The double-layer capacitanties calculated from these equations are... 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

Because flO+flfIt = the operator flftt is the projection onto the scattering states. Substituting the solution (3.12) into the first equation of the hierarchy, we obtain the modified Boltzmann equation... 

R. F. Snider, /. Chem. Phys., 32, 1051 (1960). Quantum Mechanical Modified Boltzmann Equation for Degenerate Internal States. 

Modified Gouy-Chapman theory has been applied to soil particles for many years (Sposito, 1984, Chapter 5). It postulates only one adsorption mechanism -the diffuse-ion swarm - and effectively prescribes surface species activity coefficients through the surface charge-inner potential relationship contained implicitly in the Poisson-Boltzmann equation (Carnie and Torrie, 1984). Closed-form... 

Borukhov, I., Andelman, D., and Orland, H. (1997). Steric effects in electrolytes A modified Poisson-Boltzmann equation. Phys. Rev. Lett. 79, 435-438. 

Ruckenstein and Schiby derived4 an expression for the electrochemical potential, which accounted for the hydration of ions and their finite volume. The modified Poisson-Boltzmann equation thus obtained was used to calculate the force between charged surfaces immersed in an electrolyte. It was shown that at low separation distances and high surface charges, the modified equation predicts an additional repulsion in excess to the traditional double layer theory of Deijaguin—Landau—Verwey—Overbeek. 

A modified Poisson—Boltzmann equation is obtained by replacing in the Poisson equation for parallel plates... 

Figure 5. Ratio between the double layer force with site exclusion (modified Poisson-Boltzmann) and the double layer force provided by the Poisson-Boltzmann equation (a)constant (small) surface potential (Vs = 0.02 V) (b) constant (small) surface charge density (<7S = 0.032 C/m2), n/V = 1.0 M and T = 300 K.

The corresponding modified Poisson-Boltzmann equations for the planar interface located at x=0 are ... 

In this case, two parameters (Wx, dx) are used to describe the interfacial interactions of the anions. The modified Poisson-Boltzmann equation is (for dx < df). ... 

The modified Poisson-Boltzmann equation for a uni-univalent electrolyte confined between parallel plates is ... 

When the dispersion interaction coefficients are different for anions and cations, the different distributions of the two kinds of ions generate a potential even when the plates are neutral. The potential is obtained from the solution of the correspondingly modified Poisson-Boltzmann equation (Eqs. (1) and (2)) and the interaction free energy can be calculated via the numerical integration of Eq. (7). 

Another mechanism for the hydration repulsion between lipid bilayers was more recently proposed by Marcelja.22 It is based on the fact that in water the ions are hydrated and hence occupy a larger volume. The volume exclusion effects ofthe ions are important corrections to the Poisson— Boltzmann equation and modify substantially the doublelayer interaction at low separation distances. The same conclusion was reached earlier by Ruckenstein and Schiby,28 and there is little doubt that the hydration of individual ions modifies the double-layer interaction, providing an excess repulsion force.28 However, while the hydration of ions affects the double-layer interactions, the hydration repulsion is strong even in the absence of an electrolyte, or double-layer repulsion. 

Eqs. (1), (3) and (4) represent a Modified Poisson-Boltzmann approach, which, when the ions interacts only via the mean field potential ip(z) (e.g. AWa=AWc=0), reduces to the well-known Poisson-Boltzmann equation. The only change in the polarization model is the replacement of the constitutive equation Eq. (1) by Eq. (2), which accounts for the correlation in the orientation of neighboring dipoles. However, since independent functions, four boundary conditions are needed to solve the system composed of Eqs. (2) (3) and (4). For only one surface immersed in an electrolyte, t]/(z-co)-0, m(z-co)=0 whereas for two identical surfaces, the symmetry of ip and m requires that = 0 and m(z=0)=0 where z- 0 represents... 

Eq. (10) represents the self-consistent field equation for the local segment density of the polymer chains subject to an external electrical potential ip, a van der Waals interaction with the plates —UkT and an excluded volume interaction. Eq. (11) is a modified Poisson-Boltzmann equation in which the first term accounts for the charges of the small ions of the salt, the second term for the charges of the polyelectrolyte chains and the third one for the charges of the ions dissociated from the polyelectrolyte molecules. 

If the collisions of molecules produce a chemical reaction, the Boltzmann equation is modified in obtaining the equations of change these problems are addressed and analyzed in the context of quantum theory, reaction paths, saddle points, and chemical kinetics. Mass, momentum, and energy are conserved even in collisions, which produce a chemical reaction. 

The modified Poisson-Boltzmann equation derived by Camie and McLaughhn [5] for the above system then reads... 

MODIFIED POISSON-BOLTZMANN EQUATION which is subject to the following boundary conditions ... 

On the basis of the theory of Camie and McLaughhn [5], the Poisson-Boltzmann equation for j/(x) can be modified in the present case as follows. Consider the end of a rod that carries the +e charge. As in the problem of rod-like divalent cations,... 

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