Diffusion coefficients dependence on concentration

To calculate the theoretical value of the diffusion coefficient of sodium chloride in 0.01 molar aqueous solution and to compare this with the experimental value. 

Measurements of the diffusion coefficient of aqueous sodium chloride at several concentrations have been made at 25 C by Hamed and Hildreth (J. Amer. Chem. Soc. 1951, 73, 650). At 0.01 molarity they found for the Pick s law coefficient 

The value of D at infinite dilution, calculated from conductance and transport numbers is (see problem 97) 

The molar conductance A of NaCl and the transport numbers, all at infinite dilution, are (Macinnes, Principles of electrochemistry , Reinhold, 1939) 

The viscosity of water at 25 °C is (Dorsey, Properties of ordinary water-substance , Reinhold, 1940, p. 183) 

---

If the diffusion coefficient depends on time, the diffusion equation can be transformed to the above type of constant D by defining a new time variable a = jDdt (Equation 3-53b). If the diffusion coefficient depends on concentration or X, the diffusion equation in general cannot be transformed to the simple type of constant D and cannot be solved analytically. For the case of concentration-dependent diffusivity, the Boltzmann transformation may be applied to numerically extract diffusivity as a function of concentration. 

The experimental evidence presented here and in the literature (15) show that the real diffusion coefficient depends on concentration. These results are incompatible with the notion of concentration-independent diffusion coefficients for the dissolved and Langmuir sorbed molecules [D and Djj in equation (15)] as proposed by the dual-mode sorption and transport model ( 13). 

Numerical simulations of vapor bubble growth in a superheated solution of polymer were performed,using iterative algorithm to account for the diffusion coefficient dependence on concentration in the interval (k, ko). The results are reproduced in Figures 7.2.10-7.2.12,... 

Various forms of diffusion coefficients are used to establish the proportionality between the gradients and the mass flux. Details on determination of the diffusion coefficients and thermal diffusion coefficients is found in Chapter 12. Here, however, it is appropriate to summarize a few salient aspects. In the case of ordinary diffusion (proportional to concentration gradients), the ordinary multicomponent diffusion coefficients Dkj must be determined from the binary diffusion coefficients T>,kj. The binary diffusion coefficients for each species pair, which may be determined from kinetic theory or by measurement, are essentially independent of the species composition field. Calculation of the ordinary multicomponent diffusion coefficients requires the computation of the inverse or a matrix that depends on the binary diffusion coefficients and the species mole fractions (Chapter 12). Thus, while the binary diffusion coefficients are independent of the species field, it is important to note that ordinary multicomponent diffusion coefficients depend on the concentration field. Computing a flow field therefore requires that the Dkj be evaluated locally and temporally as the solution evolves. 

The diffusion coefficient depends on a number of factors, including the molecular properties of solutes the structures of tissues, and temperature. The temperature-dependence is less critical for drug delivery, since the temperature in tumors is stable and close to the body temperature. The dependence of D on tissue structures is significant (Netti et al., 2000 Pluen et al., 2001). It is mediated through the size and the volume fraction of pores, the tortuosity of diffusion pathways, and the connectedness of pores (Yuan et al., 2001). Diffusion of macromolecules is faster in tissues with a lower collagen type I content (Pluen et al., 2001) or tissues treated with collagenase (Netti et al., 2000). However, there is no correlation between D and the concentration of total or sulfated glycosaminoglycans (Netti et al., 2000). 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

Because of volume changes due to the reaction, pressure gradients may occur inside the catalyst pellet. This can give rise to two effects. First, it influences the effective diffusion coefficients, since the gas-phase diffusion coefficients depend on pressure. Second, the pressure gradients affect the concentrations (or more accurately, chemical activities), which determine the reaction rate. Hence pressure gradients must directly influence the effectiveness factor. 

In this equation ep is the porosity of the catalyst pellet and yp the tortuosity of the catalyst pores as discussed in Chapter 3 (the rest of the symbols are as defined before). From this formula it follows that the effective diffusion coefficient depends on both the gas composition and the pressure. Since we know the pressure as a function of the concentration, Equation 7.74 provides the effective diffusion coefficient as a function of the concentration. If we define... 

If the effective diffusion coefficient depends on the concentration, the derivation for Ant given in Appendix B does not hold. An expression for Anv for concentration-de-pendent effective diffusion coefficients is therefore needed. The discussion is for arbitrary kinetics in an infinite slab derivations for other catalyst geometries occur in a similar way. 

The diffusion coefficient D has appeared in both the macroscopic (Section 4.2.2) and the atomistic (Section 4.2.6) views of diffusion. How does the diffusion coefficient depend on the structure of the medium and the interatomic forces that operate To answer this question, one should have a deeper understanding of this coefficient than that provided hy the empirical first law of Tick, in which D appeared simply as the proportionality constant relating the flux / and the concentration gradient dc/dx. Even the random-walk intapretation of the diffusion coefficient as embodied in the Einstein-Smoluchowski equation (4.27) is not fundamental enough because it is based on the mean square distance traversed by the ion after N stqis taken in a time t and does not probe into the laws governing each stq) taken by the random-walking ion. 

A good agreement is generally obtained between the models based on transport equations and the SDE for mass and heat molecular transport. However, as explained above, the SDE can only be applied when convective flow does not take place. This restrictive condition limits the application of SDE to the transport in a porous solid medium where there is no convective flow by a concentration gradient. The starting point for the transformation of a molecular transport equation into a SDE system is Eq. (4.108). Indeed, we can consider the absence of convective flow in a non-steady state one-directional transport, together with a diffusion coefficient depending on the concentration of the transported property... 

The diffusion step becomes important for polymer-modified electrodes. Thus, the apparent diffusion coefficient depends on the concentration of redox groups because the acceleration of the electron exchange decreases with the ion distance. These conclusions were drawn from a series of polyvalent ions anchored electrochemically to poly(4-vinyl pyridine) on graphite [47]. 

It follows that the diffusion coefficient depends on the concentration of the diffusing species according to the relationship... 

Protein diffusion coefficients depend on the pH and ionic composition of the medium. For example, in a 1% bovine serum albumin (BSA) solution, the diffusion coefficient of BSA increased by a factor of 4 when KCl concentration in the solution was decreased below 0.01 M[54]. 

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] ... 

Equation (III.3.21) does not apply to higher concentrations. The equivalent conductivity decreases proportionally to the square root of concentration (see Eq. III.3.8), but the diffusion coefficient depends on the ionic strength of the solution, according to Eq. (III.3.18). Using the Debye-Huckel limiting law for a l l-valent electrolyte [3] ... 

Substitution of this equation to replace D b in Eq. Q5-4a) allows calculation of the diffusion of the dilute component A. Note that the effective diffusion coefficient depends on the concentrations of the other species. Although the diffusion of the dilute conponent can be calculated accurately, the diffusion of the concentrated components B and C probably will not be accurate (see Problem 1 S.R hk Even for this special limiting case (ideal ternary with dilute conponent A), the use of the Maxwell-Stefan equations is certainly preferable and not any more difficult than the Fickian method. 

As the diffusion coefficient depends on the hydrodynamic and thermodynamic factors (Equation 2.4-43), its behaviour in a wide concentration range significantly differs for solulion.s in a good or poor solvent. 

When multiple scattering is discarded from the measured signal, DLS can be used to study the dynamics of concentrated suspensions, in which the Brownian motion of individual particles (self-diffusion) differs from the diffusive mass transport (gradient or collective diffusion), which causes local density fluctuations, and where the diffusion on very short time-scales (r < c lD) deviates from those on large time scales (r c D lones and Pusey 1991 Banchio et al. 2000). These different diffusion coefficients depend on the microstructure of the suspension, i.e. on the particle concentration and on the interparticle forces. For an unknown suspension it is not possible to state a priori which of them is probed by a DLS experiment. For this reason, a further concentration limit must be obeyed when DLS is used for basic characterisation tasks such as particle sizing. As a rule of thumb, such concentration effects vanish below concentrations of 0.01-0.1 vol%, but certainty can only be gained by experiment. 

As shown above, we can easily determine a hydro-dynamic radius at high dilution where interactions are negligible. The situation, however, becomes more problematic at higher concentrations, where, for example, diffusion coefficients depend on both size and on interactions. As seen in Figure 17.14, the droplet self-diffusion coefficient decreases with increasing volume fraction. Is this decrease due to interactions only or is it also a consequence of droplet growth ... 

The diffusion coefficient depends on both the temperature of the system and the concentration of volatiles, and can be estimated through the Vrentas-Duda free-volume theory [10]. 

The diffusion coefficient depends on the temperature of the system and on the concentration of volatiles. An increase in the temperature results in an increase in diffusivity and a decrease in viscosity, both beneficial for devolatilization. However, many polymers are thermally sensitive, so there may be a practical upper limit on the temperature to which the polymer may be exposed, as higher temperatures may degrade the polymer. Heat stabilizers have been used to enhance the thermal stabilization [35]. 

De Smedt, etal. used FRAP to examine diffusion of fluorescein-labeled dextrans and polystyrene latex spheres through hyaluronic acid solutions(17). Dextrans had molecular weights 71, 148, and 487 kDa. The hyaluronic acid had and of 390 and 680 kDa. The dextran diffusion coefficients depend on matrix polymer c as stretched exponentials in c, as seen in Figure 9.9b. Hyaluronic acid solutions are somewhat more effective at retarding the larger dextran probes. Viscosities for these solutions were reported by De Smedt, et a/. (18). The concentration dependence of rj is stronger than the concentration dependence of Dp of the polystyrene spheres, which is in turn stronger than the concentration dependence of Dp of the dextrans. Spheres and dextrans both diffuse more rapidly than expected from the solution viscosity and the Stokes-Einstein equation. 

Chapter 8 treats single-chain motion including measurements of polymer self-diffusion and tracer diffusion, and measurements that track the motions of individual polymers. It is almost uniformly found that a stretched exponential in polymer concentration and a joint stretched exponential in c, P, and M describe how the single-chain diffusion coefficient depends on matrix concentration and molecular weight and on probe molecular weight. On log-log plots, these functions appear as smooth curves that almost always agree with measurements of D (c, P, M) at concentrations extending from dilute solution to the melt. 

A step up in complexity from the linear diffusion equation is a nonlinear version of the diffusion equation where the diffusion coefficient depends on the concentration of diffusing species. The equation to be considered is then... 

Solutions for a number of typical cases are reported below. To simplify our task we use the assumption that reactant migration is not observed (a large excess of foreign electrolyte), that the diffusion coefficients Dj do not depend on concentration, and that for the reactant v = 1. (The subscript j is dropped in what follows.)... 

Koizumi and Higuchi [18] evaluated the mass transport of a solute from a water-in-oil emulsion to an aqueous phase through a membrane. Under conditions where the diffusion coefficient is expected to depend on concentration, the cumulative amount transported, Q, is predicted to follow the relationship... 

---