Phenomenological Brownian motion

In this chapter, the motion of solute and solvent molecules is considered in rather more detail. Previously, it has been emphasised that this motion approximates to diffusion only over times which are long compared with the velocity relaxation time (see Chap. 8, Sect. 2.1). At times comparable with or a little longer than the velocity relaxation time, the diffusion equation does not provide a satisfactory description of molecular motion. An alternative approach must be sought. This introduces considerable complications to a theoretical analysis of very fast reactions in solution. To develop an understanding of chemical reactions occurring over very short time intervals, several points need to be discussed. Which reactions might be of interest and over what time scale What is known of the molecular motion of solute and solvent molecules Why does the Markovian (hydrodynamic) continuum analysis fail and what needs to be done to develop a better theory These points will be considered in further detail in this chapter. 

It was pointed out in Chap. 8, Sect. 2.1 that there are primarily two reasons for the failure of the diffusion equation to describe molecular motion on short times. They are connected with each other. A molecule moving in a solvent does not forget entirely the direction it was travelling prior to a collision [271, 502]. The velocity after the collision is, to some degree, correlated with its velocity before the collision. In essence, the Boltzmann assumption of molecular chaos is unsatisfactory in liquids [453, 490, 511—513]. The second consideration relates to the structure of the solvent (discussed in Chap. 8, Sects. 2.5 and 2.6). Because the solvent molecules interact with each other, despite the motion of solvent molecules, some structure develops and persists over several molecular diameters [451,452a]. Furthermore, as two reactants approach each other, the solvent molecules between them have to be squeezed-out of the way before the reactants can collide [70, 456]. These effects have been considered in a rather heuristic fashion earlier. While the potential of mean force has little overall effect on the rate of reaction, its effect on the probability of recombination or escape is rather more significant (Chap. 8, Sect. 2.6). Hydrodynamic repulsion can lead to a reduction in the rate of reaction by as much as 30-40% under the most favourable circumstances (see Chap. 8, Sect. 2.5 and Chap. 9, Sect. 3) [70, 71]. 

Since the complications due to solvent structure have already been discussed, the remainder of this chapter is mainly devoted to a discussion of the complications introduced into the theory of reaction rates when the collision of solvent molecules does not lead to a complete loss of memory of the molecules about their former velocity. Nevertheless, while such effects are undoubtedly important over some time scale, the differences noted by Kapral and co-workers [37, 285, 286] between the rate kernel for reaction estimated from the diffusion and reaction Green s function and their extended analysis were rather small over times of 10 ps or more (see Chap. 8, Sect. 3.3 and Fig. 40). At this stage, it is a moot point whether the correlation of solvent velocity before collision with that after collision has a significant and experimentally measurable effect on the rate of reaction. The time scale of the loss of velocity correlation is typically less than 1 ps, while even rapid recombination of radicals formed in close proximity to each other occurs over times of 10 ps or more (see Chap. 6, Sect. 3.3). 

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The discussion of Kapral s kinetic theory analysis of chemical reaction has been considered in some detail because it provides an alternative and intrinsically more satisfactory route by which to describe molecular scale reactions in solution than using phenomenological Brownian motion equations. Detailed though this analysis is, there are still many other factors which should be incorporated. Some of the more notable are to consider the case of a reversible reaction, geminate pair recombination [286], inter-reactant pair potential [454], soft forces between solvent molecules and with the reactants, and the effect of hydrodynamic repulsion [456b, 544]. Kapral and co-workers have considered some of the points and these are discussed very briefly below [37, 285, 286, 454, 538]. 

Theoretical models of the film viscosity lead to values about 10 times smaller than those often observed [113, 114]. It may be that the experimental phenomenology is not that supposed in derivations such as those of Eqs. rV-20 and IV-22. Alternatively, it may be that virtually all of the measured surface viscosity is developed in the substrate through its interactions with the film (note Fig. IV-3). Recent hydrodynamic calculations of shape transitions in lipid domains by Stone and McConnell indicate that the transition rate depends only on the subphase viscosity [115]. Brownian motion of lipid monolayer domains also follow a fluid mechanical model wherein the mobility is independent of film viscosity but depends on the viscosity of the subphase [116]. This contrasts with the supposition that there is little coupling between the monolayer and the subphase [117] complete explanation of the film viscosity remains unresolved. 

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

In this chapter, we shall first make a brief review of the phenomenological aspect of Brownian motion and we shall then show how the general transport equation derived in Section II allows an exact microscopic theory to be developed. 

In the previous section, the phenomenological description of Brownian motion was presented. The Langevin analysis leads to a velocity autocorrelation function which decays exponentially with time. This is characteristic of a Markovian process, as Doobs has shown (see ref. 490). Since it is known heyond question that the velocity autocorrelation function is far from such an exponential function, the effect that the solvent structure has on the progress of a chemical reaction cannot be assessed very reliably by means of phenomenological Langevin description. Since the velocity of a solute is correlated with its velocity a while before, a description which fails to consider solute and solvent velocities can hardly be satisfactory. Necessarily, the analysis requires a modification of the Langevin or Fokker—Plank description. In this section, some comments are made on this new and exciting area of research. 

The theory of relaxation processes for a macromolecular coil is based, mainly, on the phenomenological approach to the Brownian motion of particles. Each bead of the chain is likened to a spherical Brownian particle, so that a set of the equation for motion of the macromolecule can be written as a set of coupled stochastic equations for coupled Brownian particles... 

To say nothing about the different equivalent forms of the theory of the Brownian motion that has been discussed by many authors (Chandrasekhar 1943 Gardiner 1983), there exist different approaches (Rouse 1953 Zimm 1956 Cerf 1958 Peterlin 1967) to the dynamics of a bead-spring chain in the flow of viscous liquid.1 In this chapter, we shall try to formulate the theory in a unified way, embracing all the above-mentioned approaches simultaneously. Some parameters are used to characterise the motion of the particles and interaction inside the coil. This phenomenological (or, better to say, mesoscopic) approach permits the formulation of overall results regardless to the extent to which the mechanism of a particular effect is understood. 

Upon inserting these last equations into Eq. 20, the continuity equation becomes formally identical in form with Eq. 19 obtained by Chandrasekhar. However, it differs in that an expression for C has been obtained which is explicit in the intermolecular forces in contrast to the older Brownian motion theory in which C is a phenomenological constant only. 

The general theory of Brownian motion set forth thus far is a self-contained, purely phenomenological theory which does not depend upon any special hydrodynamic assumptions. If, however, following Einstein, it is further assumed that, on the average, the diffusing particle experiences a hydrodynamic force governed by Stokes equations, the diffusion matrix is then found to be (B26a)... 

There are two forms of phenomenological equations for describing Brownian motion the Smoluchowski equation and the Langevin equation. These two equations, essentially the same, look very different in form. The Smoluchowski equation is derived from the generalization of the diffusion equation and has a clear relation to the thermodynamics of irreversible processes. In Chapters 6 and 7, its application to the elastic dumbbell model and the Rouse model to obtain the rheological constitutive equations will be discussed. In contrast, the Langevin equation, while having no direct relation to thermodynamics, can be applied to wider classes of stochastic processes. In this chapter, it will be used to obtain the time-correlation function of the end-to-end vector of a Rouse chain. 

The phenomenological, stochastic theory of Brownian motion exemplified by (2.8) has been superseded by microscopic [2.12] and "mesoscopic" [2.9,13] approaches. 

Phenomenologically, the viscous stress is the stress which vanishes instantaneously when the flow is stopped. On the other hand the elastic stress does not vanish until the system is in equilibrium. The elastic stress is dominant in concentrated polymer solutions, while viscous stress often dominates in the suspensions of larger particles for which the Brownian motion is not effective. Whichever stress dominates, the rheological properties can be quite complex since both and are functions of the configuration of the beads and therefore depend on the previous values of the velocity gradient. Note that the viscous stress only appears in the system with rigid constraints.t... 

The molecular theory of fluctuations and Brownian motion offers a generalization of the Langevin equations. These equations provide a set of equations of motion that are stochastic in nature and that can be modeled on the basis of phenomenology. For example the velocity of the yth Brownian particle in solution is described by the equation of motion... 

QSM theory provides a firm statistical mechanical basis for the phenomenological theory of irreversible processes as formulated by Onsager and the version of classical Brownian motion discussed in Section 5.2. Moreover, it gives a number of formulas that can be employed in the investigation of the role of microscopic interactions in a diversity of nonequilibrium phenomena. 

As in classical Brownian motion theory, the phenomenological time derivative is a coarse-grained time derivative obtained by time averaging the instantaneous time derivative (d/dt ) Sdj(t )) over the time scale At of macroscopic measurements (time resolution of the macroscopic observer). 

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