Mutual diffusion coefficient concentration dependence

Figure 1 A schematic diagram illustrating a typical concentration dependence of mutual diffusion coefficient in a polymer-penetrant system. (From Ref. 8.)...

When applied to a volume-fixed frame of reference (i.e., laboratory coordinates) with ordinary concentration units (e.g., g/cm3), these equations are applicable only to nonswelling systems. The diffusion coefficient obtained for the swelling system is the polymer-solvent mutual diffusion coefficient in a volume-fixed reference frame, Dv. Also, the single diffusion coefficient extracted from this analysis will be some average of concentration-dependent values if the diffusion coefficient is not constant. 

Here fcd is the infinite-time specific rate, and D is the mutual diffusion coefficient of the reactants. Using the time-dependent specific rates, Schwarz reports an increase of molecular yields that is 2% at low concentration, and 5% at high concentration of solutes. 

Tyn, M.T. and Calus, W.F. Temperature and concentration dependence on mutual diffusion coefficients of some binary liquid... 

Self-difFusion coefficients were measured with the NMR spin-echo method and mutual diffusion coefficients by digital image holography. As can be seen from Figure 4.4-3, the diffusion coefficients show the whole bandwidth of diffusion coefficient values, from 10 m s on the methanol-rich side, down to 10 on the [BMIM][PFg]-rich side. The concentration dependence of the diffusion coefficients on the methanol-rich side is extreme, and shows that special care and attention should be paid in the dimensioning of chemical processes with ionic Hquids. 

There are a number of quantitative features of Eq. (14) which are important in relation to rapid diffusional transport in binary systems. The mutual diffusion coefficient is primarily dependent on four parameters, namely the frictional coefficient 21 the virial coefficients, molecular weight of component 2 and its concentration. Therefore, for polymers for which water is a good solvent (strongly positive values of the virial coefficients), the magnitude of (D22)v and its concentration dependence will be a compromise between the increasing magnitude of with concentration and the increasing value of the virial expansion with concentration. 

The flux of particles is in the opposite sense to the direction of the concentration gradient increase. Equation (6) is Fick s first law, which has been experimentally confirmed by many workers. D is the mutual diffusion coefficient (units of m2 s 1), equal to the sum of diffusion coefficients for both reactants, and for mobile solvents D 10 9 m2 s D = DA + jDb. The diffusion coefficient is approximately inversely dependent upon viscosity and is discussed in Sect. 6.9. As spherical symmetry is appropriate for the diffusion of B towards a spherically symmetric A reactant, the flux of B crossing a spherical surface of radius r is given by eqn. (6) where r is the radial coordinate. The total number of reactant B molecules crossing this surface, of area 4jrr2, per second is the particle current I... 

With typical values for R and D as above, the Smoluchowski rate coefficient (19) is shown in Fig. 3 for a range of times. The time dependence of the rate coefficient is due to the transient concentration of B in excess of the steady-state concentration. As the density distribution of eqn. (16) relaxes to the steady-state distribution (17), so the rate coefficient decreases, because at longer times, B has to diffuse further to A on average. The magnitude of the rate coefficient ( 1010 dm3 mol-1 s-1) is large. In some reactions, the mutual diffusion coefficient of reactants may be nearer 5 x 1CT9 m2 s 1, and the rate coefficient is 3 x 1010 dm3 mol-1 s-1. Under such circumstances, diffusion-limited reactions proceed very rapidly. It is likely that the rates of most chemical reactions are slower than the diffusion-limited rate. Only the most rapid molecular chemical reactions are faster than diffusion-limited rates. Some typical reactions are discussed in Sect. 2 and will be reconsidered in Sect. 5 and later in the volume. 

The concentration dependence of the mutual diffusion coefficient, D. in binary solution can be expressed as... 

In nonideal mixtures, the thermodynamic nonideality of the mixture has to be considered. We still need to predict the concentration dependence of the mutual diffusion coefficient Dt] of a binary pair of nonelectrolytes. The concentration dependency of l)u in liquid mixtures may be calculated by using the Vignes equation or the Leffler and Cullinan equation. Besides these, we may also use a correlation suggested by Dullien and Asfour (1985), given by... 

Once an appropriate frame of reference is chosen, a two components (A, B) system may be described in terms of the mutual diffusion coefficient (diffusivity of A in B and vice versa). Unfortunately, however, unless A and B molecules are identical in mass and size, mobility of A molecules is different with respect to that of B molecules. Accordingly, the hydrostatic pressure generated by this fact will be compensated by a bulk flow (convective contribution to species transport) of A and B together, i.e., of the whole solution. Consequently, the mutual diffusion coefficient is the combined result of the bulk flow and the molecules random motion. For this reason, an intrinsic diffusion coefficient (Da and Db), accounting only for molecules random motion has been defined. Finally, by using radioactively labeled molecules it is possible to observe the rate of diffusion of one component (let s say A) in a two component system, of uniform chemical composition, comprised of labeled and not labeled A molecules. In this manner, the self-diffusion coefficient (Da) can be defined [54]. Interestingly, it can be demonstrated that both Da and Da are concentration dependent. Indeed, the force/acting on A molecule at point X is [1]... 

Dullien, F. A. L. and Asfour, A-F. A., Concentration Dependence of Mutual Diffusion Coefficients in Regular Binary Solutions A New Predictive Equation, Ind. Eng. Chem. Fundam., 24, 1-7 (1985). 

Tyn, M. T. and Calus, W. F., Temperature and Concentration Dependence of Mutual Diffusion Coefficients of Some Binary Liquid Systems, J. Chem. Eng. Data, 20, 310-316 (1975b). 

The diffusion coefficient of the particles in suspension depends on concentration of particles due to the interparticle interactions. Furthermore, we should distinguish the self-diffusion (or tracer diffusion) coefficient, D, from the collective diffusion (or mutual diffusion) coefficient, The self-diffusion coefficient accounts for the motion of a given particle and can be formally defined as an autocorrelation function of the particle velocity ... 

To examine the concentration dependence, a limiting value of T) q2 at q = 0, (T)lq2)q=0, which was obtained as the intercept of the lines in Figure 11a, is plotted as a function of polymer concentration in Figure lib. There is an increase in the ((T)/q2)g=0(= Df) value with concentration for the fast mode, but no discernible concentration dependence is noted for the slow mode (Ds). The concentration dependence of the fast mode may be compared with the concentration dependence of Kc/R(j in Figure 6. The (mutual) diffusion coefficient of nonaggregating polymer solutions is given by the generalized Stokes-Einstein relation [66,89,90],... 

Vrentas and Duda s theory formulates a method of predicting the mutual diffusion coefficient D of a penetrant/polymer system. The revised version ( 8) of this theory describes the temperature and concentration dependence of D but requires values for a number of parameters for a binary system. The data needed for evaluation of these parameters include the Tg of both the polymer and the penetrant, the density and viscosity as a function of temperature for the pure polymer and penetrant, at least three values of the diffusivity for the penetrant/polymer system at two or more temperatures, and the solubility of the penetrant in the polymer or other thermodynamic data from which the Flory interaction parameter % (assumed to be independent of concentration and temperature) can be determined. An extension of this model has been made to describe the effect of the glass transition on the free volume and on the diffusion process (23.) ... 

The diffusion coefficients used to describe multi-component diffusion are mutual diffusion coefficients. In the multi-component system, mutual diffusion coefficients are defined by Equation 4-13 the matrix of diffusion coefficients depends on the concentration of individual components. The diffusion coefficients used in the earlier sections of the chapter, however, describe solute molecules diffusing in a medium at infinite dilution. The isolated molecule is called a tracer these tracer diffusion coefficients are defined by the physics of random walk processes, as described in Chapter 3. The self-diffusion coefficient, used in Equation 4-11, is a tracer diffusion coefficient in the situation where all of the molecules in the system are identical. The self-diffusion coefficient, T>aa is defined by (recall Equation 3-12) [62] ... 

Figure 4.8 The tracer diffusion coefficient as a function of protein concentration. Tracer and mutual diffusion coefficients measured by a variety of techniques illustrate the dependence of protein diffusion on concentration, (a) Diffusion coefficients for hemoglobin [64] (b) diffusion coefficients for albumin [64] (c) diffusion coefficients for albumin [64, 65].

Figure 4.20. Experimental configurations for measuring self-difFusion and mutual diffusion coefficients. In (a) we make an interface between polymer A containing a trace amount of labelled, but chemically identical, A and pure polymer A. Broadening of this interface is controlled by the self-diffusion coefficient. The self-diffusion coefficient is not dependent on the concentration, so, if the label is truly non-perturbing, situation (b), in which we have an interface between pure labelled A and pure A, also measures the self-diffusion coefficient. However, if the two polymers making the interface are chemically different (c), then what controls broadening of the interface is a mutual difhision coefficient.

Our aim, then, is to take equation (4.4.7), in which the mutual diffusion coefficient is expressed as a product of thermodynamic and kinetic factors, but to replace the self-diffusion coefficient by some suitable weighted average of the concentration-dependent tracer diffusion coefficients of the two species. In fact it is by no means clear on the most fundamental grounds that we can expect to be able to do this at all in order to produce such an expression some additional physical assumptions have to be made, either explicitly or implicitly, and the choice of these assumptions has in the past generated some controversy. 

Figure 4.24. Diffusion coefficients as functions of the composition in the miscible blend polystyrene-poly(xylenyl ether) (PS-PXE) at a temperature 66 °C above the (concentration-dependent) glass transition temperature of the blend, measured by forward recoil spectrometry. Squares represent tracer diffusion coefficients of PXE (VpxE = 292), circles the tracer diffusion coefficients of PS and diamonds the mutual diffusion coefficient. The upper solid line is the prediction of equation (4.4.11) using the smoothed curves through the experimental points for the tracer diffusion coefficients and an experimentally measured value of the Flory-Huggins interaction parameter. The dashed line is the prediction of equation (4.4.11), neglecting the effect of non-ideality of mixing, illustrating the substantial thermodynamic enhancement of the mutual diffusion coefficient in this miscible system. After Composto et al. (1988).

Figure 4.25. (a) The mutual diffusion coefficient in the miscible polymer blend poly(vinyl chloride)-polycaprolactone (PVC-PCL) at 91 °C, as measured by x-ray microanalysis in the scanning electron microscope (Jones et al. 1986). The solid line is a fit assuming that the mutual diffusion coefficient is given by equation (4.4.11), with the composition dependence of the tracer diffusion coefficient of the PCL given by a combination of equations (4.4.9) and (4.4.10). The tracer diffusion coefficient of the PVC is assumed to be small in comparison, (b) The calculated profile of diffusion between pure PVC and pure PCL, on the basis of the concentration dependence of the mutual diffusion coefficient shown in (a). The reduced length u — where the... 

Tables 7.2 and 7.3 display the heats of transports and thermal diffusion ratio (Kj) of chloroform in binary mixtures with selected alkanes and of toluene (1), chlorobenzene (2), and bromobenzene at 30 °C and 1 atm. Concentration-dependent thermal conductivity, mutual diffusion coefficients, and heats of transport of alkanes in chloroform and in carbon tetrachloride are given by Rowley et al. (1988). The polynomial fits to these coefficients for the alkanes in chloroform and in carbon tetrachloride are used to estimate the degree of coupling and the thermal diffusion ratio Kn from Eqns (7.46) and (7.47), and shown in Figures 7.1 and 7.2 (Demirel and Sandler, 2002). The thermal conductivity and the thermodynamic factors for the hexane-carbon tetrachloride mixture have been predicted by the local composition model of NRTL.

The diffusivity of organic vappurs and of organic liquids in polymers is mosdy far from ideal and frequently concentration dependent. Show how the thermodynamic diffusion coeffient, or rather the ratio of the thermodjrnamic diffusion coefficient versus mutual diffusion coefficient changes as the volume fraction of the penetrant changes from = 0.02 to = 0.5 while the binary interaction parameter of polymer/penetrant is % = 0.6. 

The concentration fluctuation-relaxation fxmction depends only on the mutual-diffusion coefficient and the square of the scattering vector. Because larger values of q can be obtained by light scattering, the rate of decay of concentration fluctuations can be increased to the kilohertz range. Even when the typical averaging times of 1 min are included, the measurement of the mutual-diffusion coefficient becomes a routine rapid procedure To ensure the observation of mutual diffusion, the decay constant is measured as a function of q, and the q dependence is verified. The mutual-diffusion... 

The concentration dependence of D (c) in the concentrated regime depends on the solvent quality and the local viscosity. It is often observed to go through a maximum with concentration due to a large increase of local viscosity with concentration. However, if the polymer is well above its glass-transition temperature, the mutual-diffusion coefficient can even increase throughout the concentrated regime. Solutions of poly(dimethyl siloxane) in dioxane exhibit a continuously increasing mutual-diffusion coefficient. 

The 1/(1 - CO a) term is commonly referred to as the frame of reference term. For many cases of importance in polymeric systems such as in gas permeation, coa is relatively small, and the 1/(1 - >a) factor can safely be neglected so that the flux relative to fixed coordinates is equal to the flux relative to moving coordinates. Even for intermediate concentrations (0.1 < coa < 0.5), this factor may often be of second-order importance compared to difficulties in accurately determining the mutual diffusion coefficient due to strong concentration dependencies. However, not accounting for the factor 1/(1 — coa) can lead to very significant errors in flux calculations in highly swollen systems (eg, 90-95% solvent), even if the mutual diffusion coefficient is accurately determined (6). 

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