ME 599 Course Project

Simulation of the Facet Formation by the Finite Element Method

Background and Theory

On the basis of classical theory and recent work by Sun et al (1997), I focused onthe facet formation. When considering the systems in which an interface isseparating a vapor phase, solid phase, and grains, it is often assumed that vaporatoms diffuse so quickly compared to the rate of interface reaction such asevaporation or condensation, which causes the vapor phase to have a spatiallyuniform chemical potential at all times. This condition can be described by weakstatement as follow.

where p represents the reduction in total free energy per unit interface area

moving per unit distance, is the magnitude of interface displacement, m is themobility of interface, and G is the total free energy of the system. Adopting thekinetic law stating that the normal velocity of interface migration is proportional to

the driving pressure p ( = mp), a weak statement can be given as

The finite element method determines an approximate normal velocity ofinterface that satisfies a weak statement. In our work, an interface is modeled byan assembly of straight line elements and we followed the procedure given inSun et al to characterize linear geometries and seed the interface with nodalpoints. From the straight line element shown below,

The variation of total free energy associated with the virtual motion of singleelement is given as

The left-hand side of weak statement can be expressed in terms of variations ofnodal positions, velocities of nodal points, and angle theta.

where viscosity matrix H is given as

.

For n nodal points, the weak statement becomes

and because the above weak statement is valid for arbitrary virtual change , itfollows

Since the viscosity matrix and the force column are dependent on the nodalpositions, the above equation is a set of non-linear ordinary differential equationsthat should be solved numerically.

Simulation of Facet Formation

In present study, I consider one facet formation case using 41 line elements: with

anisotropic surface tension. The free energy densities of the two bulk phases

and are assumed to be the same, which makes the third term of equation (1)negligible. In my case, I only simulated the one fourth of cycle because of thesymmetry.

The following is the results and program.

Results

0 0.5 1 1.5 2 2.5 3 3.5 4x 10-3

0

0.5

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1.5

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2.5

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3.5

4 x 10-3 2D facet formation after

Initial position

0 0.5 1 1.5 2 2.5 3 3.5 4x 10-3

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1.5

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3.5

4 x 10-3 2D facet formation after

After 2*e-8 time

0 0.5 1 1.5 2 2.5 3 3.5 4x 10-3

0

0.5

1

1.5

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2.5

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3.5

4 x 10-3 2D facet formation after

After 3*e-8 time

0 0.5 1 1.5 2 2.5 3 3.5 4x 10-3

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0.5

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1.5

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2.5

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3.5

4 x 10-3 2D facet formation after

After 4*e-8 time

0 0.5 1 1.5 2 2.5 3 3.5 4x 10-3

0

0.5

1

1.5

2

2.5

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3.5

4 x 10-3 2D facet formation after

After 5*e-8 time

0 0.5 1 1.5 2 2.5 3 3.5 4x 10-3

0

0.5

1

1.5

2

2.5

3

3.5

4 x 10-3 2D facet formation after

After 6*e-8 time

0 0.5 1 1.5 2 2.5 3 3.5 4x 10-3

0

0.5

1

1.5

2

2.5

3

3.5

4 x 10-3 2D facet formation after

After 7*e-8 time

0 0.5 1 1.5 2 2.5 3 3.5 4x 10-3

0

0.5

1

1.5

2

2.5

3

3.5

4 x 10-3 2D facet formation after

After 8*e-8 time

0 0.5 1 1.5 2 2.5 3 3.5 4x 10-3

0

0.5

1

1.5

2

2.5

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3.5

4 x 10-3 2D facet formation after

After 9*e-8 time

0 0.5 1 1.5 2 2.5 3 3.5 4x 10-3

0

0.5

1

1.5

2

2.5

3

3.5

4 x 10-3 2D facet formation after

After 10*e-8 time

Program

% simulation of facet formation% radius is 0.004clear all;Delta_x=10^-4; % Equal spacingr_s0=1.0; % Surface Tensionne=40; % No. of Total Elementsnn=ne+1; % No. of Total Nodesms=10^2; % Mobilityma=10^5; % Additional mobility value to prevent singulareps=1/ma; % Diagonal additiont=10^-12; % Initial timedt=5*10^-9; % Time stepIter=30; % Number of time stepg=1.2*10^-7; % Free energytime=5; % Time step to reach similarity solution regionK=0.3; % r_s=r_s0(1+Ksin(theta)): K: anisotropic coefficientv=zeros(nn*2-1,1); % Velocity of positionsH=zeros(ne*2-1,ne*2-1); % Viscosity Matrixx=zeros(nn,Iter); % X positiony=zeros(nn,Iter); % Y position

Frs=zeros(4,ne); % Surface Energy Vector of Each ElementDelta_g=zeros(4,ne); % Gibbs Free Energy Vector of Each Element

FF=zeros(2*ne-1,1); % Combined Force Vector

% Initialinzing Positionfor i=1:nnx(i,1)=Delta_x*(i-1);y(i,1)=sqrt(1.6*10^-5-x(i,1)^2);end

%Calculate Viscosity Matrix Componentfor ii=1:Iterfor k=1:neleng(k)=sqrt((x(k+1,ii)-x(k,ii))^2+(y(k+1,ii)-y(k,ii))^2);co(k)=(x(k+1,ii)-x(k,ii))/leng(k);si(k)=(y(k+1,ii)-y(k,ii))/leng(k);fac(k)=leng(k)/(6*ms);

ke(1,1,k)=2*si(k)^2*fac(k);ke(2,1,k)=-2*si(k)*co(k)*fac(k);ke(3,1,k)=si(k)^2*fac(k);ke(4,1,k)=si(k)*co(k)*fac(k);ke(1,2,k)=-2*si(k)*co(k)*fac(k);ke(2,2,k)=2*co(k)^2*fac(k);ke(3,2,k)=-si(k)*co(k)*fac(k);ke(4,2,k)=co(k)^2*fac(k);ke(1,3,k)=si(k)^2*fac(k);ke(2,3,k)=-si(k)*co(k)*fac(k);ke(3,3,k)=2*si(k)^2*fac(k);ke(4,3,k)=-2*si(k)*co(k)*fac(k);ke(1,4,k)=-si(k)*co(k)*fac(k);ke(2,4,k)=co(k)^2*fac(k);ke(3,4,k)=-2*si(k)*co(k)*fac(k);ke(4,4,k)=2*co(k)^2*fac(k);r_s(k)=r_s0*(1+K*abs(si(k)));end

% Force Vector due to r_s, gfor e=1:neFrs(1,e)=r_s(e)*co(e);Frs(2,e)=r_s(e)*si(e);Frs(3,e)=-r_s(e)*co(e);Frs(4,e)=-r_s(e)*si(e);

Delta_g(1,e)=-g*si(e);Delta_g(2,e)=g*co(e);Delta_g(3,e)=-g*si(e);Delta_g(4,e)=g*co(e);end

% Combining Force Vector (FF: Frs+g)for k=2:ne-1for i=1:4FF(i+2*(k-1)-1)=FF(i+2*(k-1)-1)+Frs(i,k)+Delta_g(i,k);endendfor i=1:3FF(i)=FF(i)+Frs(i+1,1)+Delta_g(i+1,1);endfor i=1:2FF(2*ne-3+i)=FF(2*ne-3+i)+Frs(i,ne)+Delta_g(i,ne);end

% Combining Viscosity Matrix (H: combine ke)for k=2:ne-1for i=1:4for j=1:4H(i+2*(k-1)-1,j+2*(k-1)-1)=H(i+2*(k-1)-1,j+2*(k-1)-1)+ke(i,j,k);endendend

for i=1:3for j=1:3H(i,j)=H(i,j)+ke(i+1,j+1,1);endend

for i=1:2for j=1:2H(2*ne-3+i,2*ne-3+i)=H(2*ne-3+i,2*ne-3+i)+ke(i,j,ne);endend

% Add small eps to remove singularityfor i=1:ne*2-1H(i,i)=H(i,i)+eps;end

% Solve for Velocityv=H\FF;

% Calculate New Positionfor e=1:nn

if e==1x(e,ii+1)=x(e,ii)+v(i)*dt;y(e,ii+1)=y(e,ii)+v(1)*dt;elseif e==nnx(e,ii+1)=x(e,ii);y(e,ii+1)=y(e,ii);elsex(e,ii+1)=x(e,ii)+v(2*e-2)*dt;y(e,ii+1)=y(e,ii)+v(2*e-1)*dt;end

% Calculate Non-Dimensionalized Positionxx(e,ii+1)=x(e,ii+1)/sqrt(ms*r_s0*t);yy(e,ii+1)=y(e,ii+1)/sqrt(ms*r_s0*t);

end

t=t+dt % Step Forward to Next Timeend

% Just remove first few time steps to reach similarity solutionxxx=xx(:,time:Iter);yyy=yy(:,time:Iter);

% Plot the Position According to timefor i=1:20axis([0 4*10^-3 0 4*10^-3]);figureplot(x(:,i),y(:,i));title(['2D facet formation after'])gridend

% Just remove first few time steps to reach similarity solutionxxx=xx(:,time:Iter);yyy=yy(:,time:Iter);

% Plot the Non-Dimensionalized Positionfigureaxis([0 4*10^-3 0 4*10^-3]);plot(xxx,yyy);title(['2D facet formation after'])gridhold on;