Interaction, characteristic radius

Let us consider a slit-like pore of width D along whose walls the ip(x) potential is localized (Fig. 4). We shall regard the interaction of monomers with the walls as a short-range interaction and the characteristic radius of interaction as being of the order of the segment size a. The exact assignment of the form of the potential is immaterial for our purposes, since it describes the effective interaction of units with the pore walls, renormalized by the solvent molecules. Conditions are to be as follows ... 

Fig. 4a and b. Distribution of the segment density at different values of the energy 0 (a) and schematic picture of lattice-like chain of length N in a slit-like pore of width D (b). 0cis the critical energy characteristic of the case when the entropy losses of the macromolecule in the pore are compensated by the energy of interaction with the wall. attractive potential of a depth 0 and with a characteristic radius of interaction r0 of the order of the segment size a... 

An unexpected feature of Table 5.1 is the remarkable similarity between the energies calculated from the characteristic radius rc and those calculated from the ionization radius r0, for the same interactions, but with bond orders increased by unity. It means that the steric factor which is responsible for the increase in bond order i.e. screening of the internuclear repulsion) is also correctly described by an adjustment to r o to compensate for modified valence density. Calculating backwards from first-order D0 = 210 kjmol-1, an effective zero-order C-C bond length of 1.72 A is obtained. 

Molecular-mechanics force fields distinguish between general and 1,3 non-bonded interactions. The obvious reason for this distinction is that the distance between ligands is affected when linked to the same central atom. Their final non-bonded separation depends, not only on ligand type, but also on the size of the central atom. In such a three-atom system the relevant parameters are the characteristic radius (rc) of the central atom, together with the... 

Values of the Lennard-Jones potential well depth F (J/mol) and characteristic radius R (nm) for calculating the Lennard-Jones potential coefficients Cl i = 2FR and B = FR for the interaction of alkane atomic species with various surface atomic species... 

At ultralow temperatures, when the de Broglie wavelength of atoms greatly exceeds the characteristic radius of interatomic interaction forces, atomic collisions and interactions are generally determined by the 5-wave scattering. Therefore, in two-component Fermi gases one may consider only the interaction between atoms of different components, which can be tuned by using Feshbach resonances. 

For a > 0 one has weakly bound molecular states (it is certainly assumed that the characteristic radius of interaction R a), and for such molecular systems the criterion of the wide resonance is different [48,50], The binding energy of the weakly bound molecule state is determined by the pole of the scattering amplitude (Equation 10.3). One then finds [48,50] that this state exists only for a > 0, and under the condition... 

We thus see that for analyzing macroscopic properties of the molecular Bose gas, one should first solve the problem of elastic interaction (scattering) between two molecules. In this section we present the exact solution of this problem for homo-nuclear molecules formed by fermionic atoms of different components (different internal states) in a two-component Fermi gas. The case of Af m will be discussed in Section 10.3. The solution for M = m was obtained in Refs. [49] and [50] assuming that the atom-atom scattering length a greatly exceeds the characteristic radius of interatomic potential ... 

The third solvent-resin spreadsheet on the computer disk is the DECITREE.WKl file (Figure 19.5), which allows one to compare the solubility parameters of possible substitute solvents with a selected resin and solvents that will be replaced. The values for the selected resin are supplied by the user while the solvent data is available from the lookup tables. The typical R values are calculated and the location of individual solvents and resin are detailed on a polarity versus hydrogen bonding plot (Figure 19.6). The keyboard commands for the plot are the same as in the previous data files. This spreadsheet allows the calculation of the relative energy difference (RED) term as a measure of the relative affinity of a solvent for a substance. The RED number is defined as the ratio R/ R or the solvent-substance radius of interaction value divided by the characteristic radius of the substance. If the RED value is less than 1.0, the solvent is expected to be compatible with the substance while progressively larger RED values indicate poorer solvents. A RED value of zero indicates a perfect match between the solubility values of the solvent and substance under study. 

In addition, the droplets have a hydrodynamic radius, th, which is obtained from the diffusion coefficient extrapolated to infinite dilution. As will be shown below, the droplet interactions can, to a very good approximation, be described in terms of hard spheres. A third characteristic radius, the hard-sphere radius, ths, then enters to describe the interactions. Associated with the three radii, there are three different characteristic droplet volumes, and therefore three different characteristic droplet volume fractions. If 0 = 0 -t- 0o denotes the total volume fraction of surfactant and oil, the hard sphere volume fraction, 0hs, can be written as follows ... 

What seems to be happening is that, when prevented from establishing third-order interaction, a rearrangement of the combined valence density occurs in such a way that a more efficient lower-order interference pattern is promoted. Such a rearrangement exists in a modification of the atomic valence spheres. An outward flow of electron density causes a decrease in characteristic radius, and vice versa. A decrease of ro(0) 1.36 A, balanced by an increase of ro(C)->-1.784 A, is found to promote the formation of 2 -order interaction at = Vl-36x 1.784 = 1.56 A, to match the observed = 1.56 x 0.724 = 1.128 A and dissociation energy = 1,389t - X 1.784 /1.36 = 1,076kJmol as observed. 

Equation (1.38) corresponds to the Mei -Saupe approximation in the theory of low-molecular-weight liquid crystals with respect to the order of magnitude, mq, urjd, where is the characteristic energy of the intermolecular interaction, and is the characteristic radius of the attractive forces (cf. [60]). Equality (1.38) can also be rewritten as... 

Application of Eqs. (21)-(27) to the calculations of the nucleation rates J for various alloy models revealed a number of interesting results, in particular, sharp dependence of J and embryo characteristics on the supersaturation, temperature, interaction radius, etc. These results will be described elsewhere. 

These are general equations that allow calculating solvatochromic shifts if we know electric characteristics of the solute (dipole moments in the ground and excited states, pg and /Onsager radius a, and the function of interactions/(fio, o). 

Any fundamental study of the rheology of concentrated suspensions necessitates the use of simple systems of well-defined geometry and where the surface characteristics of the particles are well established. For that purpose well-characterized polymer particles of narrow size distribution are used in aqueous or non-aqueous systems. For interpretation of the rheological results, the inter-particle pair-potential must be well-defined and theories must be available for its calculation. The simplest system to consider is that where the pair potential may be represented by a hard sphere model. This, for example, is the case for polystyrene latex dispersions in organic solvents such as benzyl alcohol or cresol, whereby electrostatic interactions are well screened (1). Concentrated dispersions in non-polar media in which the particles are stabilized by a "built-in" stabilizer layer, may also be used, since the pair-potential can be represented by a hard-sphere interaction, where the hard sphere radius is given by the particles radius plus the adsorbed layer thickness. Systems of this type have been recently studied by Croucher and coworkers. (10,11) and Strivens (12). 

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