Solution motion

One can paraphrase these definitions by saying that a solution (or motion) is stable if all solutions (or motions) which were initially close to it, continue to remain in its neighborhood a solution (or motion) is asymptotically stable if all neighboring solutions (motions) approach it asymptotically. 

I) The existence of a stable singular point of (6-126) is the criterion for the existence of a stable periodic solution motion) of the original system (6-112). 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

Even if we consider a single solvent, e g., water, at a single temperature, say 298K, depends on the solute and in fact on the coordinate of the solute which is under consideration, and we cannot take xF as a constant. Nevertheless, in the absence of a molecular dynamics simulation for the solute motion of interest, XF for polar solvents like water is often approximated by the Debye model. In this model, the dielectric polarization of the solvent relaxes as a single exponential with a relaxation time equal to the rotational (i.e., reorientational) relaxation time of a single molecule, which is called Tp) or the Debye time [32, 347], The Debye time may be associated with the relaxation of the transverse component of the polarization field. However the solvent fluctuations and frictional relaxation occur on a faster scale given by [348,349]... 

A fully realistic picture of solvation would recognize that there is a distribution of solvent relaxation times (for several reasons, in particular because a second dispersion is often observable in the macroscopic dielectric loss spectra [353-355], because the friction constant for various types or modes of solute motion may be quite different, and because there is a fast electronic component to the solvent response along with the slower components due to vibration and reorientation of solvent molecules) and a distribution of solute electronic relaxation times (in the orbital picture, we recognize different lowest excitation energies for different orbitals). Nevertheless we can elucidate the essential physical issues by considering the three time scales Xp, xs, and Xelec-... 

Solute motions follow a quite different pattern. The solute is perturbed after a given number of motions of the solvent. Hypothetically, the solute could remain motionless, which would correspond to place the system of coordinates in the solute molecule. However, from a statistical point of view it is better to allow some motion given the fact that if a solvent molecule is displaced, only one solute-solvent interaction is perturbed whereas if it is the solute that moves all the solute solvent interactions are modified. 

If we further assume that the solute motion can be neglected and that the solvent velocity autocorrelations are independent of the presence of the solute, we get in the case of (quasi)linear solvent molecules... 

To derive an approximate solution we start by transforming the space variable x to one that moves by the convective transport mechanism— i.e., a coordinate that remains fixed with the solute motion when diffusion can be neglected. This coordinate will be denoted by and is... 

A microscopic calculation of the size-dependent diffusion is presented here. The calculation is based on the well-known mode coupling theoretical approach. The theoretical calculation is shown to give an excellent agreement with the simulation results and provides a physical interpretation of the enhanced diffusion. It is found that the enhanced diffusion of smaller solutes arises from the decoupling of the solute motion from the density mode of the solvent. 

For neat liquids the contribution from the current term is found to be much less at high densities [59]. Thus there is a decoupling of the solute motion from the current mode of the solvent even for dense neat liquids. For smaller solutes the current term contribution further reduces and in addition there takes place this decoupling of the solute motion from the solvent density mode that gives rise to the enhanced diffusion of smaller solutes. 

Figure 9. Time dependencies of the single-particle and the collective intermediate scattering functions compared for two different solute sizes at a particular wavenumber q = 6.001 at reduced temperature T" = 0.75 and in the normal density regime (p = 0.89). The solid line represents the collective intermediate scattering function. The long-dashed line is the single-particle intermediate scattering function for solute-solvent size ratio 1.0 and the short-dashed line is for solute-solvent size ratio 0.5. The plots show that the decoupling of the solute motion from the solvent dynamics increases as the solute size is decreased. The time is scaled by rsct where TJC = [mo2/kBT] 2.

This sharp decoupling also explains the saturation of the diffusion for TZ > 10. For 7Z > 10, due to almost full decoupling of the solute motion from the solvent dynamics there is hardly any contribution from the density mode and the friction is determined by the binary part. Thus in the time scale at which the solute dynamics takes place, the solvent remains nearly static. Now once the solute becomes very small, it does not feel the static structure of the solvent around it and can diffuse through the interparticle distances. Thus further decrease in the size of the solute does not decrease the binary friction leading to the near saturation of the diffusion when plotted against the solute-solvent size ratio. 

In the usual implementation of the continuum theories of SD, one assumes that the surrounding solvent is sufficiently weakly perturbed by the presence of the solute that the system response to the solute electronic transition is well approximated by the dielectric susceptibility of the pure solvent. Further, one usually assumes that the contributions of solute motion to SD can be neglected. As shown in Section 3.4.3, continuum theories can be quite successful in predicting the solvation response in highly polar liquid solvents. It is worth examining the reasons for their success in greater detail and discussing their likely limitations. 

Actually, this last statement might seem quite counterintuitive, especially for small solutes. The vibrational energy relaxation of diatomics and triatomics, for example, can be quite slow when judged by the ps and sub-ps time scales of intermolecular motion. Though ps relaxation times have been seen (39,40), there are also some well-known examples of Tis in the ps and even ms ranges (5). Yet, the vibrational friction apparently begins its work very quickly, even in these slowest of cases. As one can see from the behavior of a typical vibrational friction (Fig. 1) (Y. Deng and R. M. Stratt, unpublished), the solvent retains its memory of the solute dynamics for only a short time, so the time lag between solute motion and... 

In the column of the second type the solution under treatment moves from bottom to top and is removed through the drain along the perimeter of the column. When the solution motion is interrupted ion exchanger is moved from the tank by the action of excess pressure. 

The results are presented in Fig. 36. The dependence of h on V + W in columns with continuous motion of ion exchanger is similar to the dependence shown in Fig. 35. The dependence of h upon the linear rate of solution motion for both types of column with phases moving alternately and for columns with a fixed bed are virtually the same when performing the operations of ion-exchanger bed transfer under expansionless... 

The study in 3.0-4.0 mol/dm solutions of KCl and NH4CI, of ion-exchange dynamics in countercurrent columns with solution motion from bottom to top (Fig. 37) has shown that in this system the HTU values are the same as in the hxed bed (from 2 to 3 cm over a flow rate ranging from 8 to 50 cm/min). 

He observed activation energies for molecular motions of 11.0 and 12.5 kcal/mol, respectively. These energies are similar to those for chain melting within the same cholesteric esters(19). Although more experimentation is required before a firm conclusion can be reached, the activation energies for chain melting probably represent upper limits to activation energies for solute motions in a cholesteric phase. 

Ganapathy and Welss(22), demonstrated that the selective photoproduction of one atropisomer from a racemic solution of 1,1 -binaphthyl (BN.) requires the presence of both a chiral and macroscopically ordered cholesteric reaction medium. No measurable change in atropisomeric content was observed upon heating racemic N in a variety of cholesteric mixtures. Thus, photoequilibration is more selective than thermal equilibration BN the isomerlzing excited state, and solvent are able to interact more strongly than ground state N. and the solvent matrix. Since the maximum atropisomeric excess observed upon irradiation of Bfi. was 1.1%, once again the solvent influence on the solute motions is very weak. 

As a result, the traditional picture of liquid phase reaction dynamics —solute motion in the potential of mean force damped by friction— holds only for slow variable regime reactions. For fast variable regime reactions the nonclassical picture of Eq. (3.36) and Figure 3.5 applies. 

Since Stratt s work is well explained in the literature, we only note here that his ideas are closely related to ours. Especially our equivalent chain equations [20] provide an exact one-dimensional harmonic solid representation of the solvent effect on solute motions. We developed this representation in order to bring out the solid-like physics of the solvent effect evident on the shortest timescales. 

This sequence of collision events clearly shows how collisions of the solute molecules with the solvent can lead to a coupling between the motions of the solute molecules. This notion of solvent coupling of solute motion in the liquid is reminiscent of the hydrodynamic interaction effect on the friction coefficient of a pair of molecules, briefly discussed in the preceding section. In fact, the terms explicitly written in (10.11) simply represent, from a microscopic collisional point of view, the effects of such hydrodynamic interactions on the rate kernel. 

If reactant velocity does not influence the rate of reaction when an encounter pair is formed (see Sect. 2.4), the effect of velocity may be removed from an analysis of the solute motion. Davies [447] showed that, when the velocity distribution is of no interest, the position and time distribution of a solute is described by the telegrapher s equation. It is a diffusion-like process, but one where the particle has a limiting velocity so that a wave of solute probability spreads out with a... 

The characteristics of the fluid velocity depend on the design of the hydrodynamic cell and the flow pattern. The latter is said to be laminar when the solution flows smoothly and constantly in parallel layers such that the predominant velocity is that in the direction of the flow. Laminar flow conditions are desirable since accurate descriptions of the solution hydrodynamics are available. On the other hand, under turbulent flow conditions the solution motion is chaotic and the velocities in the directions perpendicular to that of the flow are significant. The transition between the laminar and turbulent regimes is defined in terms of the dimensionless Re5molds number, Re, that is proportional to the relative movement rate between the electrode and solution, and the electrode size, but inversely proportional to the kinematic viscosity of the solution. Thus, for low Re values the flow pattern is laminar and it transits to turbulent as Re increases. For example, in a tubular channel the laminar regime holds for Re < 2300. 

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