What do you call the movement of the molecules or substance from higher concentration to a lower concentration that requires protein?

6.6.4 Diffusion in Biological Solutes in Liquids

The diffusion of small molecules and macromolecules (e.g., proteins) in aqueous solutions plays an important role in microorganisms, plants, and animals. Diffusion is also a major part in food processing and in the drying of liquid mixtures and solutions, such as diffusing aroma constituents in fruit juice, coffee, and tea from solutions during evaporations. In fermentation, nutrients, oxygen, and sugar diffuse to the microorganisms, and products, waste, and sometimes enzymes diffuse away. The kidneys remove waste products like urea, creatinine, and excess fluid from the blood. Kidney dialysis removes waste products from the blood of patients with improperly working kidneys. During the hemodialysis process, the patient's blood is pumped through a dialyzer, and waste diffuses through a semipermeable membrane to the aqueous solution cleaning fluid.

Macromolecules have large molecular weights and various random shapes that may be coil-like, rod-like, or globular (spheres or ellipsoids). They form true solutions. Their sizes and shapes affect their diffusion in solutions. Besides that, interactions of large molecules with the small solvent and/or solute molecules affect the diffusion of macromolecules and smaller molecules. Sometimes, reaction−diffusion systems may lead to facilitated and active transport of solutes and ions in biological systems. These types of transport will be discussed in Chapter 9.

Macromolecules often have a number of sites for interactions and binding of the solute or ligand molecules. For example, hemoglobin in the blood binds oxygen at certain sites. Surface charges on the molecules also affect the diffusion. Therefore, the presence of macromolecules and small solute molecules in solutions may affect Fickian-type diffusion. Most of the experimental data on protein diffusivities have been extrapolated to very dilute or zero concentration since the diffusivity is often a function of concentration. Table 6.4 shows diffusivities of some proteins and small solutes in aqueous solutions. The diffusion coefficients for the macromolecules of proteins are on the order of magnitude of 5 × 10−11 m2/s. For small solute molecules, the diffusivities are around 1 × 10−9 m2/s. Thus, macromolecules diffuse about 20 times slower then small molecules.

Table 6.4. Diffusivities of dilute biological solutes in aqueous solutions

Source: C.J. Geankoplis, Transport Processes and Separation Process Principles, 4th ed., Prentice Hall, Upper Saddle River (2003).

Copyright © 2003

Small solutes such as urea and sodium caprylate often coexist with protein macromolecules in solutions. When these small molecules diffuse the protein solution, the diffusivity of the molecules decreases with increasing protein concentrations. This reduction is partly because of the binding of small molecules to proteins and is partly due to blockage by the large molecules.

Ternary systems

Ternary diffusion is more complicated than binary diffusion, and there is lack of experimental data on ternary diffusion. To obtain estimates for ternary mixtures the interpolation relations, given in Eqs. (46) and (47), are extended as

(48)D′kl∞=1η(D′klk→1ηk)xk (D′kll→1ηl)xl (D′kli→1ηi)x i

(49)D′kl∞=1η(D′klk→1ηkxk+D′kll→1ηlxl+D′kli→1ηixi)

For the complete estimate of the ternary diffusion for nonideal mixtures, six diffusivities at infinite dilution and three diffusivities of the type D′klk→ 1 are needed. The nonelectrolyte multicomponent diffusion coefficients are reduced to binary diffusivities, while the multicomponent electrolyte systems cannot be reduced to a set of binary systems; but to a set of ternary systems. Negative diffusion coefficients can exist in ternary systems and are consistent with the nonequilibrium thermodynamics approach. More experimental and theoretical studies are necessary in the field of multicomponent diffusion.

Some of the molecular theories of multicomponent diffusion in mixtures led to expressions for the mass fluxes of the Maxwell-Stefan form, and predicted the mass fluxes dependent on the velocity gradients in the system. Such dependencies are not allowed in linear nonequilibrium thermodynamics. Mass-flux contains concentration rather than activity as driving forces. In order to overcome this inconsistency, the starting point is Jaumann's entropy balance equation

where ρ is the density of the fluid mixture, S is the entropy per unit mass, s is the entropy-flux vector, and Φ is the rate of entropy production per unit volume. The operator

is the substantial derivative. From the balance equations of mass, momentum, energy, and the Gibbs relation, one obtains explicit expressions for s and Φ, which were derived in chapter 4.

For multicomponent diffusion, and the mass-flux expressions, we mainly use the Fick and the Maxwell-Stefan forms. Using the symmetric-diffusivity, in length2/time, we have definition

where ρi is the density of component i, the Lij are the phenomenological coefficients, and c is the total molar concentration

where Mi shows the molecular weight of component i. The diffusivity coefficients have the following properties

(55)∑i=1nwiD′ij=0 j=1,2,…n

In the diffusivity matrix, and there are ½n(n–1) independent diffusivities D′ij, which are also the coefficients in a positive definite quadratic form, since according to the second law of thermodynamics, the internal entropy for an uncoupled process never decreases. In terms of these symmetric diffusivities, the mass-flux expression becomes

(56)ji=−DiT∇lnT−ρi∑j=1 nD′ijXji=1,2,…,n

where DiT is the generalized thermal diffusion coefficient in mass/(length)(time). The generalized driving force Xi is given by

(57)cRTXi=∇Pi−wi∇P−ρiFi+wi∑j=1nρjFj

where Fi is the force per unit mass acting on the i th species. Using Eqs (56) and (57) we can express the mass-flux in terms of general driving force that is the Fick form

(58)ji=−DiT∇lnT −ρicRT∑j=1nD′ij(∇Pj−wj∇P−ρjFi+wj ∑k=1nρkFk)

Instead of expressing the mass-flow vector ji in terms of the driving forces Xi, it is sometimes convenient to express the Xi as a linear function of the ji that is in the Maxwell-Stefan form

(59)cRT∑k≠inC′ik(jkρk−jiρi)=∇Pi−wi∇P−ρiFi+wi∑j=1nρjFj −cRT∑k≠inC′ik(DkTρk−DiTρi)(∇lnT)

where C′ij is the inverse diffusivity, and sometimes is expressed as C′ij=xixk/D′ik, and D′ik are the Maxwell-Stefan diffusivities. The expressions in Eqs (58) and (59) contain the same information and are interrelated through the connection between the multicomponent diffusivities D′ij and the multicomponent inverse diffusivities C′ij. For low density gases ∇Pi = ∇(ciRT), and standard results are obtained. For polymeric liquids, a similar form to Eq. (59) can be found from a molecular theory [3] by replacing the pressure P and the partial pressure Pi with the total stress tensor and the partial stress tensor. The mass flux is related to the velocity gradient via the stress tensor, temperature, and concentration gradients. The linear nonequilibrium thermodynamics is able to generalize these expressions to include thermal, pressure, and forced diffusion.

Introduction

Most kinetic phenomena in solids involve diffusion, or a unit step very similar to that operating in diffusion. It is appropriate therefore to start our study of kinetic phenomena with a study of the fundamentals involved in diffusion in various materials. We cannot provide a complete description of diffusion because, as with nearly every other chapter in this book, such a complete description requires a book in itself. This chapter begins with a phenomenological treatment of linear processes that is based on the thermodynamic theory of irreversible processes. This very general approach is made specific by application to diffusion in a binary alloy. Diffusion in ionic crystals and semiconductors follows on the basis of the relations developed for metals and the thermodynamics of defects in these materials. Then diffusion along high diffusivity regions, such as grain boundaries, is discussed. Finally, the current computer assisted strategies for extending the basic concepts described in this chapter to problems involving multicomponent diffusion in technological applications are discussed. Also, diffusion of polymer strands is treated briefly. Diffusion in liquids is not considered in this chapter.

6.4.1 Gas Diffusion in Meso- and Macroporous Media

Modeling of diffusion of gases in porous media involves averaging mass and momentum balances and considering the three-dimensional nature of the medium. In a practical engineering approach, we consider the counterdiffusion of gases through a porous medium, and assume that we can describe the geometry by means of a single effective pore radius, the porosity ε, and a tortuosity factor. The flux of a species with respect to a unit area of the medium is (Kerkhof and Geboers, 2005)

(6.68)Ni,av′=ετ2Ni,av

where Ni,av′=vi,avci is the cross-section averaged molar fluxes based on cross-section averaged velocities, which depend on driving forces defined by

Bi=dPidx

The averaged molar flux of a gas is

Ni,av′=ετ2(rp28ηPav+DK)ΔPRTL

where L is the length of the tube, η is the dynamic viscosity, rp is the channel radius, and DK is the Knudsen coefficient, and is approximated by

DK≈0.89[ 23rp(8RTπM)1/2 ]

where M is the molar mass.

The driving forces may be defined by

B1=dP1dx=− RT[gD1D12′(x2N1,av′−x1N2,av′)+f1m N1,av′]τ2ε

B2=dP2dx=−RT[gD1D12 ′(x1N2,av′−x2N1, av′)+f2mN2,av′]τ2 ε

where gD is the diffusion averaging factor, and fim is the wall-friction factors. These equations show that the force on a component per unit volume is due to friction with the other components and due to shearing friction with the tube wall. These equations may be solved in an iterative manner. These driving forces can be extended to multicomponent mixtures.

Bi=dPidx=−RT[∑j=1n1Dij′(xjNi ,av′−xiNj,av′)+f1m Ni,av′]τ2ε

Here, fim is obtained from the binary friction model.

For isobaric counter-diffusion, from Eq. (6.68), we have

N1,av′N2,av′=−f1mf2m

For the case of equimolar diffusion through a porous medium, we have a net total pressure gradient defined by

dPdx=−RTN1,av′(f1m−f2m)τ2ε

For counter-diffusion in large pores, the friction term dominates against to the wall-friction term.

Publisher Summary

This chapter discusses physiological properties of diffusion. Concentration diffusion deals with the mixing of gases at constant total pressure. The diffusion coefficient, D12, for species in a mixture also containing molecules of other component is defined by Fick's First Law of Diffusion. The negative sign indicates that the direction of diffusion is opposite to that of the concentration gradient. If the two types of molecules have the same values of the mean free path and the mean speed, then diffusion coefficient is reduced to simple form. This equation is applicable if m1 = m2 and σ1 = σ2, and it is accordingly known as the self-diffusion equation, since it describes the diffusion of molecules through identical molecules. The course of the diffusion may be followed particularly easily if one of the components is radioactive, but otherwise the process may be followed by mass spectrometric analysis. In these pairs of substances, the molecular force fields of the two species differ somewhat and hence, their collision diameters, which are determined by these force fields, are also slightly different.

6.7.1 The Effective Friction Coefficient

In chapter 2 on diffusion of non-interacting Brownian particles, we have seen that the diffusion coefficient D0 is related through the Einstein relation D0 = kBT/γ with the friction coefficient γ of the Brownian particle with the solvent. The mean-squared displacement of a Brownian particle (without an external force) is thus related to the stationary velocity that the particle attains when subjected to an external force. Now suppose that the Brownian particle interacts with neighbouring Brownian particles. The pure solvent is thus replaced by a dispersion, and the friction coefficient is now an “effective friction coefficient” γeff, the numerical value of which is affected by the interactions of the tracer particle with the host particles. It is tempting toassume an Einstein relation between the long-time self diffusion coefficient and the effective friction coefficient, that is,

(6.112)Dsl=kBT/γeff·

That this is indeed a valid relation can be seen from the Langevin equation approach as described in chapter 2. In the Langevin equation (2.2,3 for the position and momentum coordinate of the tracer particle, the friction coefficient is now replaced by the effective friction coefficient, and the fluctuating force is now the “effective force”, which is due to interactions with both the fluid molecules and the Brownian host particles. The analysis given in chapter 2 to derive eq.(2.21) for the mean squared displacement can now be carried over to the effective Langevin equation, provided that the time scale is taken much larger than the time scale of fluctuations of the position coordinates of the host particles. The effective fluctuating force is delta correlated in time (see eq.(2.5)) only on this larger time scale. The analysis of chapter 2 can now be copied to arrive at eq.(2.21), where the friction coefficient is equal to the effective friction coefficient. Comparison with the definition (6.30) of the long-time self diffusion coefficient immediately leads to eq.(6.112). The time scale on which the effective Langevin equation with a delta correlated effective fluctuating force is valid, is the interaction time scale τI that was discussed in section 6.3 (see eq.(6.31)).

The problem is thus to calculate the stationary average velocity < vt > of the tracer particle due to an external force Fext. The brackets < ⋯ > denote ensemble averaging over fluctuations of the actual velocity due to interactions with the host Brownian particles. We have seen in chapter 5 that the velocity of the tracer particle (particle number 1 say) is related linearly to the hydrodynamic forces Fj h on all Brownian particles in the suspension,

vt=−βΣj=1N D1j·Fjh·

On the other hand, the total force on each of the particles is zero on the Brownian time scale. The hydrodynamic force is just one of the various forces that a Brownian particle experiences. In addition to the hydrodynamic force, there is the direct interaction force −∇jΦ, with Φ the total potential energy of the assembly of Brownian particles, and the Brownian force −kBT∇j In {P}, with P the probability density function (pdf) of the position coordinates. The tracer particle is the only Brownian particle that is subject to the external forceFext. Since the total forces are equal to zero, the hydrodynamic forces are equal to minus the sum of the remaining forces. Hence (Δij is the Kronecker delta),

=β Σj=1N·

For identical Brownian particles this expression reduces to,

(6.113)=β·Fext++< vtBr>,

where the direct interaction velocity is the contribution to the velocity due to direct interactions,

(6.114)=−β,

and the Brownian velocity vtBr is the contribution to the velocity due to Brownian motion,

(6.115)=−·

The ensemble averages are with respect to a pdf P, which is affected by the external force that acts on the tracer particle (see fig.6.12). The probability of finding a host particle just in front of the translating tracer particle is expected to be larger than in its wake. The first problem to be solved is the evaluation of this distorted pdf. This is done in the next subsection for hard-sphere suspensions, up to leading order in interactions. In subsection 6.7.3, each of the ensemble averages in eq.(6.113) is evaluated, and with it, the proportionality constant between the velocity < vt > and the external force. The resulting expression for the long-time self diffusion coefficient, up to first order in volume fraction, then follows immediately from eq.(6.112).

7.7.2 Diffusion data searching

Diffusion data up to the end of the 1980s were well compiled in the chapters of a Landort–Börnstein book (Mehrer H (ed), 1990). It contains data for impurity diffusion (LeClaire A D and Neumann G 1990), self-diffusion in binary alloys (Bakker, 1990), chemical diffusion in binary alloys (Murch and Bruff 1990), diffusion in ternary alloys (Dayananda, 1990) and boundary diffusion (Kaur and Gust, 1990). Diffusion data reported thereafter can be found in the review journal Defect and Diffusion Forum published in Zürich-Uetikon by Trans Tech Publications.

11.2.4.1 The mass transport regime

Diffusion to a microelectrode is notably different than that to a conventional electrode, including mm-size electrodes. For a simple fast electron transfer process, the application of a potential step from a value where no current flows to one where the reaction is diffusion controlled produces different diffusion regimes. At short times, the diffusion layer is very thin relative to the electrode. Diffusion to and from the electrode is planar (irrespective of the electrode geometry) and the microelectrode behaves like a conventional electrode. As time increases, the diffusion layer thickness becomes comparable, and then larger than the dimensions of the electrode. The diffusion regime evolves from planar to spherical (quasi-hemispherical for a microdisc) (Figure 11.11) and yields a steady-state rate of mass transport to the electrode. So in contrast to the constant planar diffusion regime observed with large electrodes, a microelectrode shifts from planar to spherical diffusion. This, of course, depends on the characteristic dimension of the electrode. Below 50 µm, the microelectrode can fully develop its diffusion layer without influence from natural convection. Larger electrodes produce thicker diffusion layers which soon become affected by natural convection. In this case, the steady rate of mass transport observed is greater than that for diffusion alone.

12.10.1 Qualitative considerations

Diffusion has great practical significance for composites. We need to know how much of a solvent diffuses into a composite material, how rapidly and to what extent. The amount of liquid absorbed is a useful but not an infallible indication of the magnitude of the change in mechanical properties.

Diffusion of liquids into well-bonded composites occurs chiefly by activated diffusion, with very little contribution from wicking along fibres. The rate of diffusion is indicated by the diffusivity or diffusion coefficient. The presence of fibres means that the composite is anisotropic, and we have to deal with several different diffusion coefficients. The reason why diffusion parallel with the fibre direction in unidirectional laminates is usually faster than in other directions is probably simply a question of the simplicity of the diffusion path, although internal stresses cannot be entirely neglected.

Diffusion

Diffusion is the movement of the electroactive species due to a concentration gradient. Each electroactive species has a characteristic diffusion rate through a diffusion layer on a stationary electrode in an unstirred solution. It is dependent upon the species properties and those of the solvent and is nearly independent of the surrounding electrolytes. The limiting current, ilim, flowing during electrolysis is given by

ilim=nFDCAd

where n is the number of electrons, F the Faraday, D the diffusion coefficient, C the concentration, A the area of the electrode, and d is the diffusion layer thickness at the electrode surface.

When some other means of mass transfer is operative, such as reproducible stirring, the diffusion layer reaches a study-limiting value. Diffusion is relatively slow when compared to mechanical stirring.