Chapter 15 Interdiffusion in Compounds 519
Chapter 16 Spinodal Decomposition Revisited 563
447
13
Fluxes, Forces, and Diffusion
13.1 Introduction 448
13.2 Flux Density of Moving Particles 448
13.3 Mobility and Forces 449
13.3.1 Absolute Mobility 449
13.3.2 Relaxation Time—Time to Reach the Drift Velocity 450
13.4 Stokes Law and Particle Size Measurement 451
13.4.1 Stokes Law 451
13.4.2 Particle Settling 452
13.5 Electron Mobility 452
13.6 Absolute Mobility and Diffusion 453
13.7 Diffusion in Liquids 454
13.7.1 Introduction 454
13.7.2 Self-Diffusion in Water 455
13.7.3 Other Diffusion Examples 456
13.7.3.1 Self-Diffusion in Water at 300 K 456 13.7.3.2 Diffusion in Liquid Copper at Its Melting Point 456 13.7.3.3 Diffusion in Liquid Lead at Its Melting Point 457
13.7.3.4 Oxygen Diffusion in Water 457
13.7.3.5 Summary 457
13.1 INTRODUCTION
Previously, diffusion and the steady-state and transient diffusion models have all assumed that transport occurs in thermodynamically ideal systems. In fact, Fick’s laws are based on ideal systems.
Unfortunately, the world is far from ideal and in many cases transport needs to be considered under nonideal conditions. This does not imply that all of the previous models are invalid. Certainly, under normal conditions, most gases can be considered ideal as can many liquids. And there are many systems that form solid solutions close enough to ideal that can be used to understand the kinetics of important materials science and engineering processes. In reality, examining nonideal systems helps to reinforce understanding of kinetic processes in general, and provides additional tools to investigate more complex systems and processes. In this chapter, some generalizations of transport are considered that have implications beyond kinetic processes and lead naturally to an understand- ing of charge transport—electrical conductivity—in materials as well.
13.2 FLUX DENSITY OF MOVING PARTICLES
It is useful to obtain a fundamental relationship between the number of particles per unit volume η (particles/m3), moving with a uniform or drift velocity, vd (m/s), and the particle flux density, J′ (particles/
m2 s). The resulting general equations can be applied to the modeling of electrical conductivity—both
13.8 Ionic Conductivity and Diffusion 458
13.8.1 Nernst–Einstein Equation 458
13.8.2 Ionic Conductivity of a 10 % Sodium Chloride Solution 458 13.8.3 CaO-Doped Zirconium Oxide as a Solid Electrolyte 459
13.9 Coupled Diffusion in Ionic Systems 461
13.9.1 Electrochemical Potential 461
13.9.2 Diffusion in an Electrolyte 462
13.9.3 Diffusion in an Ionic Solid 462
13.10 Diffusion of Polymers in Liquids 463
13.10.1 Introduction 463
13.10.2 Molecular Weight Distribution 463
13.10.3 Solvent Effects 464
13.10.4 End-to-End Distance of a Freely Jointed Chain 465 13.10.5 Radius of Gyration and Hydrodynamic Radius 466
13.10.5.1 Introduction 466
13.10.5.2 Mass Distribution and Radius of Gyration 467
13.10.5.3 Polymer Chain with n = 15 468
13.10.5.4 Example: Diffusion of Poly(Ethylene Glycol) in Water 469
13.10.6 Diffusion in Polymer Melts 470
13.11 Chapter Summary 470
Appendix 471
Exercises 474
References 476
13.3 Mobility and Forces 449 electronic and ionic—the thermal conductivity of solids and liquids, as well as
diffusion.
Consider a box of particles shown in Figure 13.1. If these were atoms, molecules, ions, or electrons moving in gases, liquids, or solids, they would be moving with high thermal energies, colliding, and leading to random overall motion. However, if there is some kind of force applied in a given direction, say +x, the particles will all move with some constant net velocity, vd, in the +x direction. At time t = L/vd, all of the particles in the box, NT = ηL3, will have passed through the end of the box with the cross section A = L2. So the particle number flux density J′ (units: par- ticles/m2 s) is given by
′ = =
( )
=J N At
L
L L v v
T
d d
η 3 η
2 . (13.1)
By dividing Equation 13.1 by Avogadro’s number, NA, the equivalent expression in terms of the molar flux and the concentration (mol/m3) is obtained:
J J
N N v Cv
A A
d d
= ′
= η = . (13.2)
Thus, not surprisingly, in addition to a flux due to diffusion, there is also one due to flow because of the applied force, and these fluxes just add to get the total molar flux:
J DdC dx Cvd
= − + . (13.3)
13.3 MOBILITY AND FORCES 13.3.1 A
bsoluteM
obilityConsider a particle (solid, liquid, or gas) settling (or rising) in a liquid under the influence of gravity as shown in Figure 13.2. The particle* will reach a constant drift or terminal velocity, vd, in the (posi- tive or negative) vertical direction when the forces acting on it are balanced. These forces are gravity, Fg, the buoyancy force, Fb, due to the weight of the fluid displaced, and a drag force caused by the viscous forces between the moving particle and the fluid, Fd. As a result, the equation of motion of this particle, F = ma (Newton’s first law, one of the few things that one needs to remember in this world), becomes
ma F F= = g+ Fb+Fd (13.4) where the resulting acceleration will be in the –z direction if z is the vertical direction and the gravitational force exceeds the drag and buoyancy force.
Viscous drag forces occur in many different areas of physics and engineer- ing and they typically are assumed to be proportional to the velocity of the moving particle: for example, Fd = –γ v. For a particle moving in a fluid in which there is simple laminar flow around the particle, this is indeed the case, and is a good starting point in developing a model for diffusion in liquids. Consider the general result in Equation 13.4 with all the forces
* A “particle” in this case usually does not refer to the type of particles considered by nuclear physicists. However, it does include things as small as electrons up to actual macroscopic
“chunks” of material that are nanometers, or micrometers, or larger in size.
vd
y x z
L
L L
A F
FIGURE 13.1 A particle in a box with sides of length L moving with a constant drift velocity, vd, in the positive x-direction under an influence of some force, F.
FbuoyancyFdrag
Fgravity rm
FIGURE 13.2 A particle of radius r and mass m moving vertically in a liquid under the influence of buoyancy, drag, and gravity forces.
that might be acting on this particle lumped into a single force, F, with the exception of the viscous force. The result is the following simple differential equation for the velocity of the particle as a function of time:
ma mdv
dt F v
= = − γ . (13.5)
This too is a very general equation in that F can be any type of net applied force: mechanical, electri- cal, chemical, and so on. For now, it will be left as a general force. Rearranging Equation 13.5 gives
dv
F v mdt
− =
γ 1
which is easily integrated by substituting u F= − γv so that dv−
( )
1γ du, resulting in−1γln F
(
−γv)
=m1 t A+and rearranging
ln F v
mt A
(
−γ)
= − γ + ′leading to
F−γv A= ′′e−mγt
where A A A, ,′ ′′ are all integration constants. Solving for the velocity with v = 0 at t = 0, then
v=F − mt
−
γ
γ
1 e (13.6)
Equation 13.6 is plotted in Figure 13.3 and at t = ∞, dv/dt = 0 and v = vd = F/γ.* In any event, the drift velocity can be more conveniently written as
v F
BF
d= =
γ (13.7)
where B is called the absolute mobility with units of m/s N and B=1γ.