Isotropic translational

Figure 4 Variation of the dimensionless effective rate constant k ) with dimensionless rotational diffusion resistance ( ) for a fixed value of the dimensionless intrinsic surface reactivity a) obtained theoretically for isotropic translational diffusion (solid line). The dashed line gives the results of the asymptotic analysis, equation (23) (Agarwal and Khakhar, [46]).

We consider application of the above to specific cases in the following sections. 6.2 Isotropic translational diffusion... 

Pairwise Brownian dynamics has been primarily used for the analysis of diffusion controlled reactions involving the reaction between isotropic molecules with complex reactive sites. Since its introduction by Northrup et al. [58], the pairwise Brownian dynamics method has been considerably refined and modified. Some of the developments include the use of variable time steps to reduce computational times [61], efficient calculation methods for charge effects [63], and incorporation of finite rates of reaction [58,61,62]. We review in the following sections, application of the method to two example problems involving isotropic translational diffusion reaction of isotropic molecules with a spherical reaction surface containing reactive patches and the reaction between rodlike molecules in dilute solution. 

Figure 15 Variation of with b q for reaction between rodlike molecules for the case of isotropic translational difiision. (Gupta and Khakhar [59]).

Thus, rather than the average reaction probability, Pq), the weighted average (jD(jPqj is required in this case. As in the case of isotropic translational diffusion, the magnitude of the reaction probability is small for small 9, and thus the expression for dimensionless rate constant may be simplified as... 

J. Srinivasalu Gupta and D. V. Khakhar, Brownian dynamics simulation of diffusion-limited polymerization of rodlike molecules Isotropic translational diffusion, J. Chem. Phys., 107 (1997) 3289-3294. 

For particles involved in free isotropic translational diffusion, G. is described by Pick s equation for translational diffusion ... 

Isotropic translational diffusion has been simulated by a simple random walk process in which each spin — representing one or more nematic molecules — jumps to one of its nearest neighbor sites with equal probability [11]. After the diffusion jump has been performed, the spin acquires the orientation of the local director at the new coordinates. Calculating G t) we have, like in the diffusion-less case, updated from the MC data the spin configuration inside the droplet 8 times per NMR cycle. Now additional diffusion steps have been added in between these structural updates, with their number A ranging from 1 to 32. In this last case the spectra are completely motionally averaged due to dififiision effects since for A = 32 each of the spins exhibits a total of 256 jumps within the duration of one NMR cycle. This already corresponds to the fast diffusion limit with C to-... 

The scattering law for isotropic translational diffusion, such as that found in a normal liquid, is a single Lorentzian function L whose width increases with Q squared [49]... 

All the theory developed up to this point has been limited in the sense that translational motion (the continuum degree of freedom) has been restricted to one dimension. In this section we discuss the generalization of this to three dimensions for collision processes where space is isotropic (i.e., collisions in homogeneous phases, such as in a... 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

When an isotropic material is subjected to planar shock compression, it experiences a relatively large compressive strain in the direction of the shock propagation, but zero strain in the two lateral directions. Any real planar shock has a limited lateral extent, of course. Nevertheless, the finite lateral dimensions can affect the uniaxial strain nature of a planar shock only after the edge effects have had time to propagate from a lateral boundary to the point in question. Edge effects travel at the speed of sound in the compressed material. Measurements taken before the arrival of edge effects are the same as if the lateral dimensions were infinite, and such early measurements are crucial to shock-compression science. It is the independence of lateral dimensions which so greatly simplifies the translation of planar shock-wave experimental data into fundamental material property information. 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

EXAMPLES (1) Isotropic Peripheral PGA - consider the isotropic version of the peripheral PCA defined by equations 7.61 and 7.63 i.e. take a-2 — os = 02.3- In this case, the detailed balance condition is satisfied when the 3-spin coupling constant hi23 = 0. From equation 7.96, we see that this condition translates to... 

Chapter 3 is devoted to pressure transformation of the unresolved isotropic Raman scattering spectrum which consists of a single Q-branch much narrower than other branches (shaded in Fig. 0.2(a)). Therefore rotational collapse of the Q-branch is accomplished much earlier than that of the IR spectrum as a whole (e.g. in the gas phase). Attention is concentrated on the isotropic Q-branch of N2, which is significantly narrowed before the broadening produced by weak vibrational dephasing becomes dominant. It is remarkable that isotropic Q-branch collapse is indifferent to orientational relaxation. It is affected solely by rotational energy relaxation. This is an exceptional case of pure frequency modulation similar to the Dicke effect in atomic spectroscopy [13]. The only difference is that the frequency in the Q-branch is quadratic in J whereas in the Doppler contour it is linear in translational velocity v. Consequently the rotational frequency modulation is not Gaussian but is still Markovian and therefore subject to the impact theory. The Keilson-... 

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

NMR Self-Diffusion of Desmopressin. The NMR-diffusion technique (3,10) offers a convenient way to measure the translational self-diffusion coefficient of molecules in solution and in isotropic liquid crystalline phases. The technique is nonperturbing, in that it does not require the addition of foreign probe molecules or the creation of a concentration-gradient in the sample it is direct in that it does not involve any model dependent assumptions. Obstruction by objects much smaller than the molecular root-mean-square displacement during A (approx 1 pm), lead to a reduced apparent diffusion coefficient in equation (1) (10). Thus, the NMR-diffusion technique offers a fruitful way to study molecular interactions in liquids (11) and the phase structure of liquid crystalline phases (11,12). 

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