Diffusion in free space

Consider a molecule diffusing in free space or a solute molecule diffusing in solution. Upon colliding with a surface, assume that the molecule is sufficiently entrained by surface forces that there results a reduction in dimensionality of its diffusion space from d = 3 to d — 2, and that in its subsequent motion the molecule is sterically constrained to follow the pathways defined by the lattice structure of the surface (or, perhaps, the boundary lines separating adjacent domains). If at some point in its trajectory the molecule becomes permanently immobilized, either because of physical binding at a site or because an irreversible reaction has occurred at that site, then, qualitatively, this sequence of events is descriptive of many diffusion-reaction processes in biology, chemistry and physics. 

Diffusion in large pores can be desribed by the laws of diffusion in free space, provided two effects are taken into account ... 

We have up to this point assumed tirat permeation in the pores follows the mechanism of gaseous diffusion in free space (i.e., that the gas molecules xmdergo repeated collisions with each other as they progress through the medium. Figure 3.4c). However, when pore diameter drops below the value of the mean free path X, the diffusional mechanism xmdergoes a change to... 

The diffusion coefficient D0 in free space is obtained from Eqs. (C1)-(C3), so our task turns to finding the perturbed Green function. 

The anomalous diffusivity described by Eq. [13] is due entirely to the fractal nature of the diffusing particle s trajectory in free space. In fractal and multifractal porous media, the diffusing particle s trajectory is further constrained by the geometry of the pore space (Cushman, 1991 Giona et al., 1996 Lovejoy et al., 1998). As a result, when fractional Brownian motion occurs in a two-dimensional fractal porous medium, De becomes scale-dependent, as described by the following equation (Orbach, 1986 Crawford et al., 1993),... 

We know that an ideal polymer chain is a random walk. Consider a distribution function q r, r, t), which tells us the probability that a chain that is t steps long has started at a position r and finished at a position r. We know that a random walker in free space has a distribution function that obeys a diffusion equation, so for an ideal, isolated chain (with no excluded volume interaction) we can write... 

Now suppose the random walk is not in free space, but is affected by a spatially varying potential U(r). This now makes the statistical weights of various possible steps in the random walk unequal via a Boltzmann factor and the effect of this is to modify the diffusion equation thus ... 

The term p/(p-p ) derives from assumption (4.313) representing the effects of Stefan flow. If e = 1, Eq. (4.318) gives the diffusion flow in a free space. 

The orientation of linear rotators in space is defined by a single vector directed along a molecular axis. The orientation of this vector and the angular momentum may be specified within the limits set by the uncertainty relation. In a rarefied gas angular momentum is well conserved at least during the free path. In a dense liquid it is a molecule s orientation that is kept fixed to a first approximation. Since collisions in dense gas and liquid change the direction and rate of rotation too often, the rotation turns into a process of small random walks of the molecular axis. Consequently, reorientation of molecules in a liquid may be considered as diffusion of the symmetry axis in angular space, as was first done by Debye [1],... 

The artificial lipid bilayer is often prepared via the vesicle-fusion method [8]. In the vesicle fusion process, immersing a solid substrate in a vesicle dispersion solution induces adsorption and rupture of the vesicles on the substrate, which yields a planar and continuous lipid bilayer structure (Figure 13.1) [9]. The Langmuir-Blodgett transfer process is also a useful method [10]. These artificial lipid bilayers can support various biomolecules [11-16]. However, we have to take care because some transmembrane proteins incorporated in these artificial lipid bilayers interact directly with the substrate surface due to a lack of sufficient space between the bilayer and the substrate. This alters the native properties of the proteins and prohibits free diffusion in the lipid bilayer [17[. To avoid this undesirable situation, polymer-supported bilayers [7, 18, 19] or tethered bilayers [20, 21] are used. 

The Fokker-Planck equation is essentially a diffusion equation in phase space. Sano and Mozumder (SM) s model is phenomenological in the sense that they identify the energy-loss mechanism of the subvibrational electron with that of the quasi-free electron slightly heated by the external field, without delineating the physical cause of either. Here, we will briefly describe the physical aspects of this model. The reader is referred to the original article for mathematical and other details. SM start with the Fokker-Planck equation for the probability density W of the electron in the phase space written as follows ... 

House, J. E. (1980). "A Proposed Mechanism for the Thermal Reactions in Solid Complexes." Thermochim. Acta 38, 59-66. A discussion of reactions in solids and the role of free space and diffusion. 

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