Steady-State Diffusion

However, there is electrostatic energy between the charges on the nuclei that must be overcome, as shown in Figure 10.6, given by. Finally, several different materials have been used over the years for the laser fusion sphere (Figure 10.7) containing the deuterium and tritium solids (if sufficiently cooled) or gases, including silicate glasses, polymers, and beryllium (Be).

Model in Spherical Coordinates
Polymer Sphere: Approximate
Polymer Sphere: Less Approximate
Silica Glass Sphere

Thus, for the curves in Fig. 10.10, the exact solution flow would be only 67% of the linear or thin-wall flow, although there does not appear to be that much difference between the exact solution and the linear approximation in the figure. Another example of how, in the absence of adequate data, normal conditions, some approximations have to be made.

PASSIVE OXIDATION OF SILICON .1 J uStiFiCation

On the other hand, if 18O is found near the Si-SiO2 interface—as shown in Figure 10.11—then oxygen diffusing through the oxide layer is rate-controlling. Experimentally, a concentration profile similar to that shown in Figure 10.12 is found, and it shows that oxygen diffusion is rate controlling.

Secondary ion mass spectrometry - by sputtering the SiO2 and forming a hole in the oxide layer - usually up to a few microns in depth. If all the 18O is found at the SiO2-O2 interface, silicon diffuses through the oxide layer to form additional oxide.

P arameterS oF the P roblem

For a red blood cell, the total phospholipid bilayer membrane is about 6 nm thick, while the hydrophobic portion is about 3 nm thick (Lodish et al. 2000). Most of this excess CO2 is exhaled and its concentration in exhaled air is about 4% (carbon dioxide).

The partition coefficient is used in a variety of applications from medicinal chemistry or drug design to determine how quickly drugs will penetrate cell walls in the human body to environmental science and modeling the behavior of organic pollutants. Approximate the cell shape with a cylindrical cell with a diameter of d = 7 μm and an average height of h = 1 μm (Figure 10.15).

CVD OF Si FROM SiHCL 3 : DIFFUSION CONTROL

Note that this is identical to the expression obtained for CVD of silicon by a surface reaction in Chapter 5. The surface reaction is assumed to be infinitely fast, so that the surface concentrations are in equilibrium, and there are concentration gradients in the gas which drive diffusion in the gas and control the deposition rate.

For a rate-controlled surface reaction, diffusion was assumed to be infinitely fast, and consequently there were no concentration gradients in the gas phase of either products or reactants: the opposite is true for diffusion. Putting these terms and the initial pressures into the equilibrium constant Ke and letting p(HCl) = x for simplicity, the following equation must be solved to calculate the equilibrium partial pressures for x:.

K inetiC C alCulation

In this case the surface pressures, ps, are not the equilibrium pressures, pe, and, in fact, it is the difference between the surface and equilibrium pressures that drives the surface reaction. This is about a factor of 5 faster than the surface reaction rate-controlled deposition calculated in Chapter 5 under almost identical conditions.

CVD OF SILICON CONSIDERING BOTH DIFFUSION AND SURFACE REACTION

Therefore, because the reaction-controlled deposition is slower, but not by much, diffusion is the rate-controlling step under these conditions. This is usually the case because the surface reaction is exponentially temperature dependent, while the diffusion coefficient depends only on T3 2/.

If the relative humidity were 60%, then it would take about twice as long to evaporate, or about 30 minutes.

C alCulation oF m olar C onCentration

Another approach would assume an ideal solution, so the volume of 1000 g would be. Vi are the molar volumes as a function of concentration of NaCl and H2O, respectively, which may not exist in the literature.

It is difficult to predict that a NaCl–H2O solution behaves close to an ideal solution. Note the discontinuity of NaCl concentration at the NaCl-solution interface: the thermodynamic activity of sodium chloride may be the same in both phases at the interface, but the concentrations are not.

DISSOLUTION OF SPHEROIDIZED CEMENTITE IN AUSTENITE

The next time you put salt in your beer to increase head, check how long the salt particles are effective. However, do not shake the glass as this will generate convection currents and decrease the distance the salt must diffuse and increase the rate of dissolution.

If the assumption is made that C∞ = 0, then it will take half as long, which is even faster. It is clear that the assumption about the average particle size plays the biggest role in determining the time for decomposition, because the initial particle size is squared, so doubling the initial particle size will increase the time by a factor of 4.

PRECIPITATION AND TOUGHENING IN ZIRCONIA .1 i ntroduCtion

t ranSFormational t oughening
t he m odel
C alCulated h eat -t reat t ime
g roWth r ate

An ideal way to control the tetragonal particle size is by precipitating the tetragonal phase from the cubic phase, which is actually how it is done in industrial practice. The oxygen diffusion coefficient in this system is about 106 times that of the cations (Rhodes and Carter 1966).

OSTWALD RIPENING BY DIFFUSION

The exact solution takes into account the particular particle size distribution resulting from particle burial and that the growth rate of a particle of a given radius depends on the difference between its solubility and the mean particle size distribution. Determining the particle size distribution resulting from burial makes the model much more complex.

CHAPTER SUMMARY

The results for a two-size distribution, Equation 10.56, and Wagner's solution (Wagner 1961), Equation 10.57, are again strikingly similar in the sense that the cubed particle size depends linearly on time and all the parameters are exactly the same. The main point of the simple model is that all the other parameters and predicted time dependence are the same: the particle size grows or shrinks roughly as the square root of time; and the rate is directly proportional to the thermodynamic solubility (over a flat surface), [B∞], the interfacial energy, γ, and the diffusion coefficient in the external phase, D.

APPENDIX

S teAdy -S tAte S olution for C ylindriCAl C oordinAteS

The Model
When Can the “Thin-Wall” Approximation Be Made?

This shows that the steady-state solution in cylindrical coordinates reduces to the linear case if the wall thickness is small compared to the pipe diameter. Again, if CL = 0, gives the ratio of the flux in cylindrical coordinates (at r2) to the ratio in the linear coordinates.

The Model

So if you can live with 10% accuracy, a 5:1 ratio of outer tube radius to wall thickness will be OK. Substituting these values for A and B into Equation A.13 gives the concentration as a function of r from the center of the sphere, .

When Can the “Thin-Wall” Approximation Be Made?

In addition, there may be additional movement in the solution if the molar volume of the solute in the solution, Vsoln, is different from the solute, Vs. There is a change in volume of the solution, Vsoln, given by Vsoln=nsoluteVsolnor4 3/ πr3=(Cs4 3/ πa V3) soln.

EXERCISES

Plot the steady-state concentration of KCl, mol/cm3, in the solid and liquid as a function of distance, r/a, from the center of the now growing sphere to r/a = 10. Plot the concentration of Al2O3 (mol/cm3) from the center of the Al2O3 particles to three times the particle radius into the melt.

Solution of Fick’s Second Law by Variable Substitution 384

Application to Semi-Infinite Boundary Conditions 391

Diffusion Out of a Semi-Infinite Slab 392

INTRODUCTION

Concentration Confined to Positive Values of x 396
Mean Square Distance in One Dimension 401 11.7.4 Mean Square Distance in Three Dimensions 402

It is not intended to imitate the rigor and breadth of these works, but rather to offer some simple and plausible approaches to the solutions of a limited number of problems that a materials scientist or engineer might encounter. The goal of this chapter is to provide this familiarity by modeling solutions to Fick's second law for infinite and semi-infinite boundary conditions.

SOLUTION WITH A DIMENSIONLESS VARIABLE .1 I ntroductIon

d ImensIonless V arIable
s olutIon of f Ick ’ s s econd l aw by V arIable s ubstItutIon

Now the number of moles of boron per unit length along the surface of the crystal only the area, A, of the triangle in Figure 11.2, A = 1/2 C0λ. Therefore, the partial differential equation can be solved as an ordinary differential equation, equation 11.6, with the dimensionless variable y x= / 4Dt, and it gives a specific solution for two integration constants A and B.

SEMI-INFINITE BOUNDARY CONDITIONS .1 m odel : e rror f unctIons

The latter can be used to measure the oxygen diffusion coefficient in aluminum oxide by analyzing the 18O content as a function of depth below the surface and comparing it to Equation 11.10. When discussing diffusion in solids, introductory materials science and engineering books often simply present Equation 11.10 without any background—a black box approach—and use it to calculate the carbon concentration in a piece of steel as a function of time and distance.

INFINITE BOUNDARY CONDITIONS .1 m athematIcal m odel

The reason for giving all three forms of the same solution is that each one exists in the literature. Note that the distance for significant concentration change (e.g. up to 10% silver or gold) is much larger for this system than was seen in Section 11.2.2 for boron to silicon diffusion due to the larger diffusion coefficient here.

APPLICATION TO SEMI-INFINITE BOUNDARY CONDITIONS .1 c onstant s urface c oncentratIon

A very practical application of external diffusion via Equation 11.15 is the intrinsic gain in silicon used for integrated circuits. At elevated temperatures, where many integrated circuit processing steps are performed, such as dopant diffusion to form p-n junctions (Section 11.2.2), the remaining oxygen is precipitated as small SiO2 particles.

FINITE SOURCE SOLUTIONS .1 I ndIVIdual s ources

e xtended s ources
m ethod of I maGes

Introduction
Concentration Confined to Positive Values of x

s urface s ource and m easurement of d
I on I mplantatIon

For example, in Figure 11.11, the influence of the source at x = 1 extends beyond the origin at negative values of x. For a single source in Figure 11.14 located at the origin, half of the material is reflected back at positive values of x and the solution is done.

FINITE SOURCE AND RANDOM WALK .1 I ntroductIon

m ean d Isplacement Equation 11.17 for a single source at x 0 is,
m ean s quare d Istance In o ne d ImensIon
m ean s quare d Istance In t hree d ImensIons
r andom w alk and d IffusIon

Example with 15 Steps
Example with Three Steps
General Result

The first term in equation 11.36 is obvious, while the second term is harder to see. To obtain the mean position of all the diffusing atoms, the average of equation 11.38 must be taken thus, .

CHAPTER SUMMARY

Another change of variables in Equation A.3 gives a powerful result that naturally leads to important solutions for infinite and semi-infinite boundary condition diffusional mass transfer problems. The limits of integration for Equation A.4 actually extend from −∞ < x0 < ∞ because this equation was obtained assuming that C = 0 for x0 < 0.

In the limit as Δx0 → 0, this is the definition of the derivative of the integral, so in the limit of a point source, Q(x0), Equation A.6 becomes. Equation A.10 is a very good approximation and is plotted in Table A.1 together with the error between the approximation and the value of erf(x) and erfc(x) calculated with a spreadsheet.

Introduction

The error function is shown graphically by the shaded area under the curve in Figure 11.3. Other error function approximation tables are available in the literature (Abramowitz and Stegun 1965).

Recursion Formula

The complementary error function, erfc x( ) = −1 erf x( ), is depicted to the right of the shaded area in the figure. The error function is found in many mathematics textbooks, in many software packages, and on graphing calculators.

Some Values for the Gamma Function Because,

Infinite Boundary Condition Solution 417

Example: Homogenization Anneal of Cu–Sn Alloy 419 12.4 Diffusion Out of a Sheet of Finite Thickness 420
Example: Removal of Carbon from a Steel Sheet 428

INTRODUCTION
CORING IN A CAST ALLOY .1 I ntroductIon
Other Important Finite Boundary Conditions 428 .1 Transient in a Membrane and Interdiffusion over Finite

Diffusion through a Wall or Membrane 428 12.5.3 Interdiffusion of A and B with Finite Thicknesses 430

Finite Difference Numerical Solutions of Fick’s Second Law 432 .1 Fick’s Second Law in Finite Difference Form 432

W hat I s c orIng ?

CORING BY SEPARATION OF VARIABLES .1 s eparatIon of V arIaBles

a pplIed to c orIng

DIFFUSION OUT OF A SHEET OF FINITE THICKNESS .1 I ntroductIon

c omparIng s olutIons
d IffusIon through a W all or m emBrane

FINITE DIFFERENCE NUMERICAL SOLUTIONS OF FICK’S SECOND LAW

s olVIng the f InIte d Ifference e quatIon

CHAPTER SUMMARY

However, in doing so, Equation 12.6 requires that the left and right sides of the equation remain the same. Note also that the above solution is the sum of both the equilibrium term, C(x, ∞) = 0, and the transition term of Equation 12.12.

The final solution of equation A.2 with infinite boundary conditions methods is obtained by adding equations A.4 and A.12.

Infinite Series Solution

Therefore, the infinite series solution to the finite sheet boundary problem with initial concentration of C(x, 0) = C0 and C(x, ∞) = 0 is. Calculate the percentage differences in time for parts c and d of this problem and those in Problem 12.4.

Fluxes, Forces, and Diffusion 447 Chapter 14 Interdiffusion and Metals 479

Relaxation Time—Time to Reach the Drift Velocity 450

Self-Diffusion in Water at 300 K 456 13.7.3.2 Diffusion in Liquid Copper at Its Melting Point 456

INTRODUCTION
FLUX DENSITY OF MOVING PARTICLES

Ionic Conductivity of a 10 % Sodium Chloride Solution 458 13.8.3 CaO-Doped Zirconium Oxide as a Solid Electrolyte 459
End-to-End Distance of a Freely Jointed Chain 465 13.10.5 Radius of Gyration and Hydrodynamic Radius 466

MOBILITY AND FORCES .1 A bsolute M obility

R elAxAtion t iMe —t iMe to R eAch

At time t = L/vd, all particles in the box, NT = ηL3, will have passed through the end of the box with cross section A = L2. These forces are gravity, Fg, the buoyancy force, Fb, due to the weight of the displaced fluid, and a drag force caused by the viscous forces between the moving particle and the fluid, Fd.

STOKES LAW AND PARTICLE SIZE MEASUREMENT .1 s tokes l Aw

P ARticle s ettling

For particles (or bubbles) in a liquid that have reached final or drift velocity, equation 13.4 applies and. For example, Figure 13-4 shows how a particle size distribution can be measured by settling in a liquid.

ELECTRON MOBILITY

Moreover, the relaxation time ≅ 10–7 seconds is such a small number that the time before reaching the drift velocity can certainly be neglected.

ABSOLUTE MOBILITY AND DIFFUSION

Therefore, this must mean that there is a relationship between the diffusion coefficient and the absolute mobility; namely, This is known as Einstein's relation, and it is a particularly important result because Equation 13.19 states that the diffusion coefficients for atoms, ions, electrons, molecules, and even dirt or pollen particles are related to their absolute mobilities and vice versa.

DIFFUSION IN LIQUIDS .1 i ntRoDuction

s elf -D iffusion in w AteR
o theR D iffusion e xAMPles

Self-Diffusion in Water at 300 K
Diffusion in Liquid Copper at Its Melting Point
Diffusion in Liquid Lead at Its Melting Point
Oxygen Diffusion in Water
Summary

This is the Stokes-Einstein equation for diffusion in liquids, where r is the hydrodynamic radius of the diffusing species—atoms, ions, molecules, or actual particles. As a result, many additional models exist in the literature to generate more accurate models than the simple Stokes-Einstein relation in Equation 13.21.

IONIC CONDUCTIVITY AND DIFFUSION .1 n eRnst –e instein e quAtion

cao-D oPeD Z iRconiuM o xiDe As A s oliD e lectRolyte

To calculate the ionic conductivity of this solid solution at 1000°C, it is the number of oxygen ions/m3, ηO, that should be used for η in Equation 13.24, in this case, because the oxygen ions conduct. By replacing D O( 2−) in Equation 13.24 with this expression, the Nernst–Einstein equation becomes. 13.26) which implies that the electrical conductivity can be considered as a result of either the oxygen ions moving or the oxygen vacancies moving.

COUPLED DIFFUSION IN IONIC SYSTEMS .1 e lectRocheMicAl P otentiAl

D iffusion in An e lectRolyte
D iffusion in An i onic s oliD

That is, the overall rate is controlled by the slower diffusing ion, but with a diffusion coefficient twice that of the slower ion. In this case, if DNA+ DCl−, then Deff =DCl−; that is, the rate of deformation will be directly proportional to the diffusion coefficient of the slower diffusing ion and it only takes one Cl– ion to move a molecule of NaCl.

DIFFUSION OF POLYMERS IN LIQUIDS .1 i ntRoDuction

M oleculAR w eight D istRibution
s olVent e ffects
e nD - to -e nD D istAnce of A f Reely J ointeD c hAin
R ADius of g yRAtion AnD h yDRoDynAMic R ADius

Introduction
Mass Distribution and Radius of Gyration
Polymer Chain with n = 15
Example: Diffusion of Poly(Ethylene Glycol) in Water

D iffusion in P olyMeR M elts

Therefore, the root mean square of the edge-to-edge distance for a loosely bound polymer is simply The significance of equation 13.45 is that it is a relation between the length and molecular weight of the polymer, and its radius of gyration can be used as the hydrodynamic radius in Stokes–.

CHAPTER SUMMARY

Again, because n ∞ M, D∝M−2, or the diffusion coefficient is inversely proportional to the square of the molecular weight, which is what is observed (Green 2005). For polymers, the best measure for r is the root mean square of the radius of gyration of a polymer molecule, R2 1 2g.

General Case
Dissolution of A α B β

Calculation of the Parabolic Rate Constant 483
Interdiffusion in Metals and the Kirkendall Effect 491
INTRODUCTION
OXIDATION OF METALS .1 I ntroductIon
OSMOSIS .1 I ntroductIon
INTERDIFFUSION IN METALS AND THE KIRKENDALL EFFECT
CASE STUDY: RHODIUM–COPPER INTERDIFFUSION * .1 I ntroductIon
CHAPTER SUMMARY

Consequently, copper is a more rapidly diffused species in the rhodium-copper alloy on the copper side of the interface. This is consistent with the rough diameter of the molten spot on rhodium shown in Figure 14.20.

Introduction

The interdiffusion in a binary metallic system is modeled in some detail, including Darken's analysis of the Kirkendall effect: the movement of inert markers at non-equal diffusion rates of the interdiffusion atomic species. Many approximations are made by analyzing the failure mode of the pipe to illustrate how many technological problems need to be solved, even though some of the important data are not available.

Coupled Fluxes: Charge Neutrality

The model used is NiO, and to reduce the algebra, it is assumed that the diffusion of oxygen is slow compared to that of the nickel ion, so that only Ni+2 and electron transport need be considered. This does not detract from the main results of the final result, but saves a lot of writing with a small loss overall.

Thermodynamics

Substituting equation A.5 for eE into equation A.4 for the nickel ion flux density gives So it would be nice if the oxidation was done in pure oxygen at pO2=1bar, its standard state, then it just becomes the gradient in the nickel chemical potential.

Conversion to Wagner’s Equation

Interdiffusion with Mobile Oxygen Ions: Kirkendall Effect 526 .1 Expression for Marker Velocity 526
Interdiffusion with Mobile Electrons: High Electronic Conductivity 529 .1 MgO–NiO Interdiffusion with Fixed Oxygen Ions 529

Formation of ABO 2 from AO and BO 533

PHENOMENA OF INTEREST

Reaction of MgO and Al 2 O 3 to Form MgAl 2 O 4 Spinel 536
Reactions with Intermediate Products 541
The Linear Model Again for MgAl 2 O 3 Formation 543

INTERDIFFUSION IN NiO–MgO .1 S everal P oSSible M odelS

e xPected l ack of o xygen d iffuSion
M odel with i MMobile o xygen
v alueS of D
i nterdiffuSion with M obile o xygen i onS : k irkendall e ffect

INTERDIFFUSION WITH MOBILE ELECTRONS: HIGH ELECTRONIC CONDUCTIVITY

Mgo–nio i nterdiffuSion with f ixed o xygen i onS
k eePing the e lectric f ield b ut with M obile e lectronS
M obile o xygen i onS and M obile e lectronS

INTERDIFFUSION AND SOLID-STATE REACTIONS .1 i MPortance

This is essentially the same situation as in Figure 14.12 for the interdiffusion of metals, but now there is an interdiffusion of the oxides. The same applies to the interdiffusion coefficient, Equation 15.18: if the oxygen diffusion coefficient is two orders of magnitude lower than that of the cations, the second term in D can be neglected and it returns to the immobile anion solution, Equation 15.7.