Dimensionless interaction energy

The denominator K( /st) is the equivalent to the characteristic dimension in Pick s law, proportional to the diffuse layer thickness k" and depends exponentially on the dimensionless interaction energy vj7(x). The physical idea is that slow coagulation, as well as ion adsorption, are controlled by diffusion through an electrostatic barrier, which slows down with increasing barrier height. 

The dimensionless interaction energies dipmd 7 > quadcha 7 especially quadind 1 7 Tc vciy Sensitive to the mean diameter a, y however, as a rule, the contributions and are very small in comparison to the four... 

The dimensionless interaction energy between two point-dipoles can be expressed as [23, 26]... 

A direct, complete calculation of the interaction of an individual atom with a three-dimensional piece of metal presents great mathematical difficulties. However, this calculation is hardly of much interest since both the dependence of the interaction energy on the distance and its order of magnitude can be found (with accuracy to within dimensionless factors of order unity) from simpler considerations. We may now define the function I(p, p") as... 

For crystals containing unequal numbers of multiple cations and anions, the definition of a Madelung constant becomes more complicated, but the Madelung (or lattice) energy can always be calculated by simple computerized methods [76]. In principle, a dimensionless interaction curve can be derived for any structure type. 

This dimensionless group, introduced for conciseness in rate correlations, has no simple physical interpretation. It is the product of several ratios A/kT represents the ratio of the characteristic London interaction energy to the thermal energy of the particle, R is the aspect ratio, while the Peclet number may be considered as the ratio of a characteristic energy for drag losses to the thermal energy possessed by the particle. This interpretation for the Peclet number becomes evident by using the relation D = tnkT to write... 

As can be seen, X is a dimensionless free-energy parameter which is sometimes called a generalized interaction parameter. 

For electrostatically stabiUzed suspensions, this maximum volume fraction will depend primarily on the dimensionless double layer thickness, Ka, and the electrostatic interaction energy through the dimensionless surface potential, T g(= etpelksT). Because the dimensionless double layer thickness can be large (i.e., 10-100) when the salt concentration is low, the effective volume fraction at which the maximum volume fraction is reached can be very small. 

Dimensionless free energy of interaction of some adsorbate i with the adsorbed phase (a) or any phase j... 

Dimensionless free energy of interaction of a mobile-phase molecule (m) or solute molecule (x) with the nonsorbed phase (n) [Eq. (4)1... 

This dimensionless quantity is the polymer-solvent interaction energy per mole of solvent, divided by RT, which itself has the dimensions of energy. Since = N (Ni +N2/"2), Eq, (12-19) can be recast to give the enthalpy of forming a mixture with volume fraction 2 of polymer in yV moles of solvent as... 

Fig. 4 Energy distribution function, (p(s t) (cmol kJ mol ), against the dimensionless product of the lateral interaction energy (P) and the local isotherm (0)P0, for carbon monoxide adsorption over a bimetallic Pto.25-Rho.75 silica-supported catalyst, at 698 K. (From Ref. [12].)...

Defined in this fashion, x is a dimensionless measure of the differences in the strength of pairwise interaction energies between species in a mixture (compared with the same species in their pure component states). Using this definition, we write the energy of mixing per lattice site as... 

Here, is the dimensionless surface potential and is the value of d>j for h o°. Equation 5.179 expresses the dependence riei(/ ) in a parametric form riei(0), hifd). Fixed surface potential or charge means that or s, does not depend on the film thickness h. The latter is important to be specified when integrating H(h) or f(h) (in accordance with Equations 5.162 to 5.165) to calculate the interaction energy. 

Note that the dimensionless units defined in Table I are used, so that the curvature along the X direction is renormalized to 1. Here Uq is the two-body interaction potential defined in Eq. (2.13). The two terms linear in X are the dipolar interaction energy (with Uj and U2 two unit vectors, respectively, along the z-axis of the fixed frame for the solute and the solvent body, cf. Fig. 1). Finally a quadratic term in X has been added in order to confine the fluctuations of the stochastic field. 

The bare interaction energy is invariant with respect to scaling of the wave vector. This can be illustrated by selecting an arbitrary scale k and introducing the dimensionless group parameter... 

In addition, the model delivers an expression for the dimensionless interaction parameter x, which describes the free-energy cost per monomer of contacts between... 

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