Constant friction, diffusion coefficient

The movement of the analyte is an essential feature of separation techniques and it is possible to define in general terms the forces that cause such movement (Figure 3.1). If a force is applied to a molecule, its movement will be impeded by a retarding force of some sort. This may be as simple as the frictional effect of moving past the solvent molecules or it may be the effect of adsorption to a solid phase. In many methods the strength of the force used is not important but the variations in the resulting net force for different molecules provide the basis for the separation. In some cases, however, the intensity of the force applied is important and in ultracentrifugal techniques not only can separation be achieved but various physical constants for the molecule can also be determined, e.g. relative molecular mass or diffusion coefficient. 

Mark-Houwink-Sakurada constant Mass transfer coefficient around gel Fractional reduction in diffusivity within gel pores resulting from frictional effects Solute distribution coefficient Solvent viscosity nth central moment Peak skewness nth leading moment Viscosity average molecular weight Number of theoretical plates Dimensionless number... 

D=mass diffumsion coefficient Z)T=fiiermal diffusion coefficient /=friction coefficient G=(oh (centrifugal acceleration) / =Boltzniann constant meff=particle effective mass r=radius of centrifuge basket s=sedimentation coefficient T = absolute temperature =geometric volume of die channel w=channel thickness y=diermal expansion coefficient p=electrophoretic... 

Nevertheless, the characteristic time constant is roughly proportional to the square of the typical size and to the inverse of collective diffusion coefficient D which is given by the modulus divided by the friction. The porous structures presented here are one of the solutions to achieve a high response material. 

Another surprising result obtained both in the computer simulation studies and in the theoretical analysis is the smaller value of the diffusion coefficient than that predicted by the SE relation for TZ > 15. The SE relation predicts that the friction is proportional to 1Z X whereas this present microscopic calculation shows a near constant value of the friction for 7Z > 15. This leads to a smaller value of the calculated diffusion coefficient than that predicted by the SE relation. 

The coagulation coefficient is a function of the radius of the particle Rp, its mass m , the effective particle Knudsen number k, the temperature of the medium T, and the depth of the interaction potential well between two particles. Using the expression for the overall interaction potential given in Appendix I, the depth of the interaction potential well can be calculated from the knowledge of the Hamaker constant for the particle. The friction coefficient f is related to the diffusion coefficient of the particle, D, through the Einstein equation... 

Here, kn is the Boltzmann constant, T the absolute temperature, and Dto the translational diffusion coefficient extrapolated to zero concentration. The friction coefficient for a sphere of radius r is given by the Stokes law ... 

The temperature dependence of the rate constant is illustrated in Fig. 19. Thble 2 summarizes the apparent activation energies of the ion transfer F fr = -Z 8In(l/T), and of diffusion F a = -Rd nD/Q /T). Ion diffusion coefficients at various temperatures were evaluated from voltammetric data [115]. Provided that the temperature dependencies of the friction coefficient in the bulk of the solution, and at the location a, have equal slopes, a relationship can be derived from Eq. (48) [115] ... 

Here M is the molecular weight and V the partial specific volume of the solute, N the Avogadro number, k the Boltzmann constant, and T the absolute temperature s and D are the sedimentation and translational diffusion coefficients (after extrapolation to infinite dilution). The translational frictional coefficients from both measurements are regarded as identical, i.e., f, = fd. The rotary frictional coefficient, designated as f, can be determined from either flow birefringence or non-Newtonian viscosity measurements. 

The primitive chain reptates along itself with a diffusion constant that can be identified as the diffusion coefficient of the Rouse model. Under the action of a force /, the velocity of the polymer in the tube is v =f /, where is the overall friction coefficient of the chain. It is expected that C is related to the friction coefficient of the individual segments, Q, by the expression... 

The discussion presented above applies only for impulsive collisions for which Pc 1. that is, m. This is not a realistic formulation, except near the threshold for dissociation. In most cases of interest the vibrational frequencies are too high relative to the collision bandwidth cuq defined in Section III C. For a single degree of freedom (n = 1) it is essential to include the frequency dejjendence of the collisional friction discussed in that section. In this case the energy diffusion coefficient D(E) is modeled by Eq. (4.9) which, in the limit of constant friction, is in accord with Eq. (5.1), deduced from the density of states. At low bath pressures, where the collisions are resolved, D(E) is the second moment of P(E, ), ... 

Since the constant force acting on the particle results in a constant velocity, there must be an equal and opposite viscous drag force of the liquid acting on the particle with magnitude Cv. The diffusion coefficient D and the friction coefficient C are related through the Einstein relation ... 

The proportionality constant D is called the diffusion coefficient (S.I. unit m2-s ). Einstein also derived that D = kBT//, and taking Stokes s expression for the friction factor / for spheres, the relation becomes... 

The characteristic time for diffusion inside a cloud or rain drop follows from the mathematical solution of Eq. (1-19). Diffusion coefficients in the aqueous phase are of the order of DL= 1.8 x 10-9m2/s. The resulting time constants for the approach to uniform concentrations are shown in Fig. 8-10. For large drops the equilibration takes longer than the fall time over a distance of 100 m. Large drops, however, also develop an internal circulation due to frictional drag that enhances mass transport by mixing. The time constant for diffusion inside a water drop thus represents an upper limit to the true mixing time. 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation Dx lkT)a = C holds, which forms the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the t5q)e of boundary conditions used in deriving Stokes law. It follows that the diffusion coefficient ratio is given by ... 

In chemical kinetics the concept of the order of a reaction forms the basis of a kinematics which constitutes a frame for most of the molecular theories of chemical reactions. The fundamental magnitudes of this kinematics are the concentrations and the specific rate constants. In simple cases only the time enters as an independent variable, whereas in a diffusion process both time and space are involved. Diffusion processes are generally described in terms of diffusion coefficients, volume concentrations and thermodynamic potential or activity factors. Partial volume factors and friction coefficients associated with the components of the diffusing mixture are also essential in the description. A feature of the macro-dynamical theory is that it covers any region of concentration. Especially simple equations are connected with the differential diffusion process (diffusion with small concentration differences), for which the different coefficients or factors mentioned above are practically constant. 

A stochastic dynamics simulation requires a value to be assigned to the collision frequency friction coefficient 7. For simple particles such as spheres this can be related to the diffusion constant in the fluid. For the simulation of a rigid molecule it may be possible to derive 7 via the diffusion coefficient from a standard molecular dynamics situation. In the more general case we require the friction coefficient of each atom. For simple molecules such as butane the friction coefficient can be considered to be the same for all atoms. The optimal value for 7 can be determined by trial and error, performing a stochastic dynamics simulation for different values of 7 and comparing the results with those from experiment (where available) or from standard molecular dynamics simulations. For large molecules the atomic friction coefficient is considered to depend upon the degree to which each atom is in contact with the solvent and is usually taken to be proportional to the accessible surface area of the atom (as defined in Section 1.5). 

Note that, in addition to translational motion, other kinds of motion will be subject to random thermal fluctuations as well. In particular, the tumbling of (macro)molecules in solution can be described in terms of rotational diffusion, for which equivalent rotational diffusion constants can be defined. By analogy with translational diffusion described above, the rotational diffusion coefficient (/>rot) and rotational friction coefficient (/rot) are related by the expression ... 

If the electric field is applied in a solution, there is force acting on a charged object, which will accelerate the charged object. Hydrodynamic friction is counteracting the force of the electric field. The hydrodynamic friction is proportional to the velocity, and the friction coefficient according to Einstein s equation can be determined from the diffusion coefficient [23]. Eventually both forces will be balanced and the object/molecule will move with a constant velocity. This steady state is reached very quickly on the time scale of the PEG NMR experiment, which is on the order of tens of milliseconds. Therefore it is justified to assume this force balance for the entire experiment and to calculate from the force balance the effective number of charges per molecule or complex [22]. 

The diffusion coefficient D is also related to the frictional coefficient /. The frictional force increases with the velocity dr/dt of the particles the proportionality constant is the frictional coefficient fj) ... 

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