3 Diffusion

3.1 Review of the Basic Equations

Figure 3.1: Behavior of the error function (from ref. [1]).

Figure 3.2: Representations of Eq. 3.8 for different values of $w=2\sqrt{Dt}$.

figure
figformat % set some defaults so the figures look pretty
z=linspace(-1,1,200);  % These are the z points
w=[0.2,0.4,0.6]; % these are the three values of the normalized diffusion length that we will include in our calculations
c=@(z,w) erf(z/w);  % define a function of two variables, z and w
col={[1,0,0],[0,0.5,0],[0,0,1]};  % these are the three colors (rgb format)
linetype={'-','--','-.'};  % these are the three line types we well used (plain, dashed and dash-dot)
axes
hold on
for i=1:3
    plot(z,c(z,w(i)),'color',col{i},'linestyle',linetype{i})
    legendtext{i}=['$w/L$=' num2str(w(i))];
end
legend(legendtext,'location','best','interpreter','latex')
ylabel('C_{a}')
xlabel('z/L')
ylim([-1.2 1.2])
set(gca,'ytick',[-1,1])
set(gca,'yticklabel',[])  % turn off the y axis tick labls by making 'yticklable' an empty vector
text(-1.15, -1, 'C_{1}','fontsize',16)
text(-1.15, 1, 'C_{2}','fontsize',16)
print(gcf,'../figures/erfsolution.eps','-depsc2') % save as an eps file

• Boundary conditions: $C_A=0$ for $z=\pm \infty$
• Initial condition: All of the A component is confined to a very layer at $z=0$, with a surface coverage (atoms/area) of $C_s$
• Normalization condition: The total amount of material in the sample must be conserved, so if we integrate the concentration profile we must end up with $C_s$:

Figure 3.3: Representations of Eq. 3.8 for different values of the diffusion length, ${\ell }_A^D$.

(a)

(b)

Figure 3.4: Illustration of the interfacial width for the thin-film solution (a) and error function solution (b) to the diffusion equation. In part a, $w^{\prime }=0.83w$ and in part b $w^{\prime }=0.89w\mathrm{.}$

3.3 Vacancy Diffusion Mechanism

Figure 3.5: Diffusion couple in its initial state (a) and after 100,000 hops of the included vacancy (b). The size of the 2d lattice is 30$\times$30..

tic % start a time so that we can see how long the program takes to run
n=30;  % set the number of boxes across the square grid
vfrac=0.01;  % vacancy fraction
matrix=ones(n); 
map=[1,1,1;1,0,0;0,0,1];  % define 3 colors:  white, red, blue
figure
colormap(map)  % set the mapping of values in 'matrix' to a specific color
caxis([0 2]) % range of values in matrix goes from 0 (vacancy) to 2
% the previous three commands set things up so a 0 will be white, a 1 will
% be red and a 2 sill be blue
matrix(:,n/2+1:n)=2;  % set the right half of the matrix to 'blue'
i=round(n/2);   % put one vacancy in the middle
j=round(n/2);
matrix(i,j)=0; 
imagesc(matrix); % this is the command that takes the matrix and turns it into a plot
t=0;
times=[1e4,2e4,5e4,1e5];
showallimages=1; % set to zero if you want to speed things up by not showing images,  set to 1 if you want to show all the images during the simulation

%% now we start to move things around
vacancydiff.matrices={};  % makea blank cell array
while t<max(times)
    t=t+1;
    dir=randi([1 4], 1, 1);  
    if dir==1
        in=i+1;
        jn=j;
        if in==n+1; in=1; end
    elseif dir==2
        in=i-1;
        jn=j;
        if in==0; in=n; end
    elseif dir==3
        in=i;
        jn=j+1;
        if jn>n; jn=n; end
    elseif dir==4
        in=i;
        jn=j-1;
        if jn==0; jn=1; end
    end
    % now we need to make switch
    neighborix=sub2ind([n n],in,jn);
    vacix=sub2ind([n n],i,j);
    matrix([vacix neighborix])=matrix([neighborix vacix]);
    if showallimages
        imagesc(matrix);
        drawnow
    end
    if ismember(t,times)
        vacancydiff.matrices=[vacancydiff.matrices {matrix}]; % append matrix to output file
        imagesc(matrix);
        set(gcf,'paperposition',[0 0 5 5])
        set(gcf,'papersize',[5 5])
        print(gcf,['vacdiff' num2str(t) '.eps'],'-depsc2')
    end
    i=in;
    j=jn;
end
vacancydiff.times=times;
vacancydiff.n=n;
save('vacancydiff.mat','vacancydiff') % writes the vacancydiff structure to a .mat file that we can read in later
toc




    

3.4 Kirkendall Effect

Figure 3.6: Experimental geometry of the original Kirkendall experiment

Figure 3.7: Representative concentration profiles during diffusion in a binary system (a), along with the fluxes for the case where $D_A>D_B$, and the change in local vacancy concentration due to the diffusive fluxes.

Figure 3.8: Schematic representation of the interface between $\alpha$ and $\beta$ phases.

• Like mobilities, tracer diffusion coefficients are purely kinetic parameters.
• Tracer diffusion coefficients will depend on the local composition (the relative amount of A and B atoms in a binary alloy, for example).
• A binary A-B alloy there are two, independent, composition-dependent tracer diffusion coefficients. If the tracer species is chemically identical to atom A, then we refer to the tracer diffusion coefficient as $D_A^{*}$. If the tracer species is chemically identical to atom B, then we refer to the tracer diffusion coefficient as $D_B^{*}$ .

Figure 3.9: Schematic Representation of an tracer atom (center, green) in an alloy of A (red) and B (atoms).

3.8 Summary of Diffusion in a Binary System

1. $\tilde{D}$: The interdiffusion coefficient (often referred to as the mutual diffusion coefficient). If you are interested in the time-dependent evolution of the composition profile, this is the diffusion coefficient that you use when you are solving the diffusion equation:
   
   $$\frac{\partial C_B}{\partial t}=\frac{\partial }{\partial x}\tilde{D}\left(C_B\right)\left(\frac{\partial C_B}{\partial z}\right)$$
   note that for a binary system, we only need to specify one of the compositions, since $C_A=C_0-C_B$. Also note that in general, $\tilde{D}$ depends on the composition, so cannot be treated as a constant.
2. $D_A$ and $D_B$: The intrinsic diffusion coefficients for the individual components. These are important for two reasons. First, they are needed if you want to describe motions of atomic planes relative to the external boundaries of the sample (the Kirkendall effect). This motion was determined from the the atomic fluxes relative to atomic planes, as opposed to fixed points in space. These fluxes are determined by the appropriate intrinsic diffusion coefficient. For example, for the B component, we have:
   
   $$J_B=-D_B\frac{\partial C_B}{\partial z}$$
   Also, predictive models of interdiffusion are generally based on the relationship between these intrinsic diffusion coefficients and the interdiffusion coefficient through the following expression:
   $$\begin{array}{cc}\tilde{D}& =\left[X_B{D}_A+X_A{D}_B\right]\end{array}$$
3. $D_A^{*}$ and $D_B^{*}$: The tracer diffusion coefficients for the individual components. Imagine a single atom in a homogeneous material. The tracer diffusion coefficient describes the probability that the atom has diffused a certain distance in a given period of time. These diffusion coefficients are purely kinetic parameters, and can be expressed in terms of a mobilty:
   
   $$D_B^{*}=RT{M}_B$$
   Unlike the interdiffusion and intrinsic diffusion coefficients, they are not affected by the thermodynamics of the system. In general, values of the tracer diffusion coefficients will depend on the concentration of the material in which the tracer atoms are diffusing. The special cases of $D_A^{*}$ at $X_A=1$ and $D_B^{*}$ at $X_B=1$ are self diffusion coefficients. The intrinsic diffusion coefficients are related to the tracer diffusion coefficients through the following relationship:

3.9 Diffusion in Ternary Systems

Figure 3.10: Chemical composition and carbon chemical potential across a ternary Si-C-Fe diffusion couple, showing the situation before diffusion (a) and after diffusion (b).

Figure 3.11: Examples of 1 and 2-dimensional defects that act as high-diffusivity paths.

Figure 4.1: Slip bands in single crystal aluminum undergoing tensile deformation.

Figure 4.2: Relationship between an applied tensile stress, $\sigma$ and the resolved shear stress, ${\tau }_{rss}$.

Figure 4.3: Sliding of close packed planes on top of one another.

Figure 4.4: Application of a shear strain to a material.

1. The stress is a periodic function, with the stress repeating every time the displacement is increased by an amount equal to $b$, the distance between atoms along the slip direction.
2. The stress is equal to zero at the stable equilibrium positions at $u=0,b,2b$, etc.
3. For $u<b/2$ the stress is positive because we need to apply a stress to move the atoms out of their stable equilibrium positions.
4. At $u=b/2$ the system is at an unstable equilibrium. The stress is also equal to zero at this position, but the equilibrium is unstable because any slight perturbation in the displacement will cause the atomic plane to fall back into an equilibrium position at $u=0$ or $u=b$.
5. The maximum stress is at $u=b/4$ . The stress actually reverses sign for $u>b/2$, since a stress must be applied to avoid having the atoms fall into the equilibrium position at $u=b$.

Figure 4.5: Schematic representation of the stress vs. displacement as the atomic planes in Figure 4.3 slide over one another.

Figure 4.6: Moving a carpet by propagating a defect along its length.

Figure 4.7: Illustration of the boundary between regions that have slipped from the application of a shear stress. The dislocation in this example refers to the boundary between the slipped and unslipped regions.

Figure 4.8: Edge dislocation in a simple cubic lattice.

Figure 4.9: Schematic representation edge dislocation motion in the slip plane, illustrating the Burgers vector, $\overrightarrow{b}$ for an edge dislocation.

Figure 4.10: Deformation from Fig. 4.9, but resulting from an edge dislocation of opposite sign moving in the opposite direction.

Figure 4.11: Annihilation of two dislocations of opposite sign that are moving in the same glide plane.

4.2 Screw Dislocations

Figure 4.12: Schematic representation of a screw dislocation.

1. When a dislocation line travels through a material, the motion of the line traces out a plane.
2. The relative displacement between the material on either side of this plane is given by the Burgers vector $\overrightarrow{b}$.

Figure 4.13: Motion of a screw dislocation The dislocation moves from the front of the crystal to the back. The net result is the production of a step edge with magnitude and direction equal tot he Burgers vector, $\overrightarrow{b}$.

1. Move four steps down (a to b)
2. Move three steps to the right (b to c)
3. Move four steps up (c to d)
4. Move four steps to the left (d back to a)

Figure 4.14: Determination of $\overrightarrow{b}$ for an edge dislocation. In this case $\hat{s}$ is defined so that it is pointing into the plane of the figure. The vector ${\overrightarrow{n}}_d$ is defined in Eq. 4.8.

Figure 4.15: Determination of $\overrightarrow{b}$ for an edge dislocation with an opposite sign to the dislocation from Figure 4.14.

1. Draw a circuit in the clockwise direction (viewed from the top, so we are looking in the direction of $\hat{s}$) around the dislocation line. The circuit begins and ends at point a.
2. Repeat the circuit in a perfect part of the crystal. The circuit begins at point $s$ and ends at point $f$.
3. The Burgers vector is obtained as the vector that starts at $s$ and ends at $f\mathrm{.}$

Figure 4.16: Burgers circuit for a right-handed screw dislocation, with $\overrightarrow{s}$ defined so that the the positive direction of the dislocation line is toward the bottom of the crystal (along the negative z direction).

Figure 4.17: Left-handed version of the dislocation from Figure 4.16.

4.4 The $\overrightarrow{b}\times \hat{s}$ cross product

Figure 4.18: Definition of ${\overrightarrow{n}}_d$.

• Right-handed screw dislocation: $\hat{s}$ and $\overrightarrow{b}$ point in opposite directions.
• Left-handed screw dislocation: $\hat{s}$ and $\overrightarrow{b}$ point in the same direction.
• Edge dislocation: $\hat{s}$ perpendicular to $\overrightarrow{b}$, $\overrightarrow{b}\times \hat{s}$ points to the extra half plane.

4.5 Connection to the Crystal Structure

Figure 4.19: Burgers vectors for the BCC and FCC crystal structures.

Figure 4.20: Dislocation loop with screw, edge and mixed character at different points along the loop. The direction of the dislocation line, $\overrightarrow{s}$, is indicated by the arrows surrounding the dislocation loop.

Exercise: What happens to the shape of the crystal in Figure 4.20 if the loop expands and exits the crystal on all sides?

Solution: When a dislocation line moves, it introduces a net displacement of $\overrightarrow{b}$ between parts of the crystal on either side of the plane defined by the motion of the dislocation line. So if the dislocation exits the crystal, it will result net translation of $\overrightarrow{b}$ of the two parts of the crystal. The trick here is to figure out if the top half moves by an amount $\overrightarrow{b}$ or if the bottom half moves by this amount. In this case the top half must be shifted by $\overrightarrow{b}$ because the extra half plane at the top of the crystal ends up at the right edge of the crystal, and the extra half plane in the bottom part of the crystal ends up at the left edge. The final situation is as follows:

4.7 Dislocation Density

Figure 4.21: Glide of an edge dislocation.

Figure 4.22: Cross slip of a screw dislocation.

Figure 4.23: Schematic representation of dislocation climb.

4.9 Dislocation Energy

4.9.1 Screw Dislocations

Figure 4.24: Strain field around a screw dislocation that is centered along the z axis.

• In the slip plane itself the material is in a state of pure shear.
• Above the slip plane there is compressive component to the strain field.
• Below the slip plane there is a tensile component to the strain field.

Figure 4.25: Schematic representation of an edge dislocation, showing the regions of tensile and compressive strain.

Figure 4.26: 2-dimensional stress tensor

Figure 4.27: Stress distribution around an edge dislocation.

4.10 Dislocation Line Tension

Figure 4.28: Relationship between the curvature and the the one-dimensional pressure acting on a dislocation line.

Figure 4.29: Force force acting on a dislocation.

Figure 4.30: Force force acting on a dislocation.

4.12 Dislocation Multiplication

Figure 4.31: Schematic representation of a Frank-Read dislocation source. Note that all sections of all of the dislocations have the same Burgers vector.

5 Thermodynamics of Interfaces

5.1 A Brief Review of the Thermodynamic Potentials