Diffusion from Brownian motion

The theories in this paper are first-principles statistical mechanics theories used to calculate static thermodynamic and molecular ordering properties (including solubilities of LCPs in various kinds of solvents) and dynamic properties (diffusion from Brownian motion). The diffusion of the LCP molecules constitutes a lower limit for the speed of processing of the LCPs. The static theory is used to calculate the packing of the bulky relatively rigid side chains of SS LCPs these calculations indicate that head-to-tail polymerization of the monomers of these SS LCPs will be very strongly favored. The intermolecular energies and forces calculated from the static theory are used in the dynamic theory. 

Plotting U as a function of L (or equivalently, to the end-to-end distance r of the modeled coil) permits us to predict the coil stretching behavior at different values of the parameter et, where t is the relaxation time of the dumbbell (Fig. 10). When et < 0.15, the only minimum in the potential curve is at r = 0 and all the dumbbell configurations are in the coil state. As et increases (to 0.20 in the Fig. 10), a second minimum appears which corresponds to a stretched state. Since the potential barrier (AU) between the two minima can be large compared to kBT, coiled molecules require a very long time, to the order of t exp (AU/kBT), to diffuse by Brownian motion over the barrier to the stretched state at any stage, there will be a distribution of long-lived metastable states with different chain conformations. With further increases in et, the second minimum deepens. The barrier decreases then disappears at et = 0.5. At this critical strain rate denoted by ecs, the transition from the coiled to the stretched state should occur instantaneously. 

Diffusion Random migration of particles in a favored direction resulting from Brownian motion or turbulent eddy motion of the suspending gas. 

The motion along the one-dimensional resonance line called Arnold diffusion is prominent at lower-order resonances when nonlinearity is weak. In fact, the motion with the residence time distribution of the power 3/2 is observed for low-order resonance with weak nonlinearity. On the other hand, overlapped resonances allow the motion across resonances which leads to Brownian motion at a two-dimensional region. Indeed, the distribution with the power 2 is observed at higher-order resonances, and it is more frequently observed with stronger nonlinearity. Hence, one can distinguish clearly the Arnold diffusion from the motion induced by resonance overlaps by the power of the residence time distribution at each resonance condition. 

The self-diffusion coefficients of toluene in polystyrene gels are approximately the same as in solutions of the same volume fraction lymer, according to pulsed field gradient NMR experiments (2fl). Toluene in a 10% cross-linked polystyrene swollen to 0.55 volume fraction polymer has a self-diffusion coefficient about 0.08 times that of bulk liquid toluene. Rates of rotational diffusion (molecular Brownian motion) determined from NMR spin-lattice relaxation times of toluene in 2% cross-linked ((polystytyl)methyl)tri-/t-butylphosphonium ion phase transfer catalysts arc reduced by factors of 3 to 20 compai with bulk liquid toluene (21). Rates of rotational diffusion of a soluble nitroxide in polystyrene gels, determined from ESR linewidths, decrease as the degree of swelling of the polymer decreases (321. 

Since similar approach vas used in [37] for Brownian diffusions, it should be noted the principal difference of turbulent diffusion from Brownian one. In the process of Brownian diffusion, the particles perform random thermal motion due to collisions with molecules of ambient liquid. In [37] the appropriate force acting on the considered particle, is taken into account as quasi-elastic force proportional to the particle s displacement Fcontr = —c(x. As a result, the form of the equation (11.60) changes, there appears a term proportional to x, and from the condition of thermodynamic equUibrium of the system it follows that... 

If two reactants A and B with radii and r, respectively, diffuse together and react at an interaction distance (r + r ), then theories developed from Brownian motion predict that the diffusion-controlled second-order rate constant is given by... 

Equation 4.52 can be used to calculate the collision radii of particles when perikinetic and orthokinetic rates of flocculation are equal, and flocculation is in transition from Brownian-motion (diffusion) dominated kinetics to fluid-motion kinetics. For example ... 

Diffusion results from Brownian motion, the random battering of a molecule by the solvent. Let s apply the one-dimensional random walk model of Chapter 4 (called random flight, in three dimensions) to see how far a peirticle is moved by Brownian motion in a time t. A molecule starts at position x = 0 at time t = 0. At each time step, assume that the particle randomly steps either one unit in the +x direction or one unit in the -x direction. Equation (4.34) gives the distribution of probabilities (which we interchangeably express as a concentration) c(x, N) that the particle will be at position x after N steps,... 

One can hope that these will not greatly affect comparisons of macroscopic relaxation functions rather than microscopic functions and that better treatments as from Fulton s methods for example will clarify these questions. Even so, It seems fair to claim that a better basis now exists for extracting useful information from Kerr effect measurements and to explore questions of whether rotational reorientations in time are diffusion or Brownian motion like at one extreme infrequently by large jumps at the other or something in between. With developments in instrumentation of the sort suggested above there appear to be real possibilities for studies of dynamics of simpler molecules to complement those by other methods. 

Gravitational sedimentation techniques have limited worth for particles smaller than about a micron due to the long settling times required. In addition most sedimentation devices suffer from the effect of convection, diffusion and Brownian motion. These difficulties may be reduced by speeding up the settling process by centrifuging the suspension. 

If this sedimentation velocity is small compared to the average velocity from Brownian motion of the particles, then the particles will remain in suspension. As we saw in Chapter 1, the diffusion coefficient of a particle in solution can be given by the Stokes-Einstein equation ... 

When submicron particles are used to trace slow flows, one must consider errors due to particle diffusion resulting from Brownian motion. A first-order estimate of this error relative to the displacement in the jc-direction is given by Santiago et al. (1998) ... 

Another example is that of the self-diffusion and Brownian motion of chains in solution which results in a fluctuation of their concentration on a time scale of about 10 s. As mentioned in Chapter 6, quasi-elastic light scattering is the ideal technique to determine the diffusion coefficient D from the self-correlation function of the scattered intensity. 

Figure 5 relates N j to collection efficiency particle diffusivity from Stokes-Einstein equation assumes Brownian motion same order of magnitude or greater than mean free path of gas molecules (0.1 pm at... 

Stokes diameter is defined as the diameter of a sphere having the same density and the same velocity as the particle in a fluid of the same density and viscosity settling under laminar flow conditions. Correction for deviation from Stokes law may be necessary at the large end of the size range. Sedimentation methods are limited to sizes above a [Lm due to the onset of thermal diffusion (Brownian motion) at smaller sizes. 

Diffusion is a stochastic process associated with the Brownian motion of atoms. For simplicity we assume a one-dimensional Brownian motion where a particle moves a lattice unit <2 in a short time period Td in a direction either forward or backward. After N timesteps the displacement of the particle from the starting point is... 

From the image sequences, information on the velocities of nano-particles can be extracted. The statistical effect of Brownian motion on the flowing speed of the mixed liquid is found small enough to be ignored as shown in Fig. 37 where most of the particles trajectories in the liquid are straight lines and parallel with the wall basically. Therefore, Brownian diffusive motion is ignorable. 

The hydrodynamic drag experienced by the diffusing molecule is caused by interactions with the surrounding fluid and the surfaces of the gel fibers. This effect is expected to be significant for large and medium-size molecules. Einstein [108] used arguments from the random Brownian motion of particles to find that the diffusion coefficient for a single molecule in a fluid is proportional to the temperature and inversely proportional to the frictional coefficient by... 

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