COMPUTATIONAL MATERIALS ENGINEERING Episode 4 potx

FIGURE 3-11 Flat boundary in an Ising model with unconserved spins, a square lattice, periodicboundary conditions, Glauber spin dynamics, and a kTs= 0, bkTs = 0.2, and c kTs = 0.8.. By t

FIGURE 3-11 Flat boundary in an Ising model with unconserved spins, a square lattice, periodic

boundary conditions, Glauber spin dynamics, and (a) kTs= 0, (b)kTs = 0.2, and (c) kTs = 0.8.

FIGURE 3-12 General case of a isolated boundary segment, in an Ising model with unconserved

spins, a square lattice.

3.2.8 Temperature

The faceting effects of the previous section are temperature dependant, and the discussionreferred to a low temperature regime In order to understand the effect of temperature it is use-ful to again consider the shrinking circle experiment Figure 3-8 shows the plot ofV 2/3 /V 2/3

o

versust/V o 2/3 for the sphere at various values ofkT s The linearity of the plots confirm thatequation (3.9) is obeyed, but for high values ofkT sthere is nonuniformity at the beginning of

the plots; this is due to the size of the domain initially increasing, corresponding to an increase

in boundary enthalpy This can only occur because there is an increase in the boundary entropyassociated with boundary roughness which more than compensates for the increase in enthalpy.Thus despite the increase in boundary area, the Gibbs free energy of the system is reduced

It should be noted that in the Ising model simulations carried out at finite temperature, there

is a finite probability that any site in the whole system can swap its spin There is rougheningtemperature,T r, where the system becomes disordered (often called the Curie temperature bythose using the Ising model to study magnetism) Figure 3-13(a) shows a shrinking sphere

at kT s = 0; the shape is compact and defined by discrete ledges and facets Figure 3-13(b)shows a shrinking sphere at kT s = kT r, the roughening temperature The system becomesincreasingly disordered The exact roughening temperature associated with the phase transition

to the disordered state depends on the lattice type and neighbors It is possible to prevent suchdisordering while still obtaining high temperatures by confining swap attempts to boundarysites When this is done the effect of roughening the boundary can be investigated independentlyfrom the system roughening, as in Figure 3-13(c)

Swaps that occur on the boundary do increase the number of active sites, but not the netcurvature of the boundary, which is determined by topological considerations of the boundary.The result is that temperature affects the kinetics of the boundary but not the energetics ofcurvature driven growth In effect temperature injects roughness into the boundary, by supplying

kinks, thus reducing the anisotropy of boundary mobility and energy in the system This has the

effect of making the simulations slower as we shall see in the next sections Figure 3-11 shows alinear boundary at different temperatures, which shows the effects of systematically increasingthe temperature

PROBLEM 3-10: Effect of Entropy

Modify your 2D shrinking circle code to disallow spin swaps away from the boundary for zero values of kT s Implement periodic boundary conditions to overcome the issues associated with the lattice boundaries Compare the kinetics of the shrinking circular grains in this model with those measured in Problem 3-5.

non-3.2.9 Boundary Anisotropy

Ising models are performed on lattices, and it seems obvious that the boundary energies andboundary mobilities will have inherent anisotropies that depend on the type of the lattice Forinstance there is an energy anisotropy of boundary plane which can be expressed most conve-niently through a Wulff plot The 2D triangular lattice has a lower anisotropy than the 2D squarelattice, withγ [10]/γ[11] = 1.07atkT s = 0.2 [MSGS02] The presence of such anisotropybrings into question whether equation (3.6) can be used as an accurate description of the sys-tem, and perhaps it should be replaced by the more accurate Herring relation [Her49]:

whereγ is the second derivative of the interface free energy with respect to interface

inclina-tion, and the termγ + γ is referred to as the interface stiffness Furthermore mobility is also ananisotropic function of boundary plane For instance in the 2D square lattice, kink motion occurseasily along the [10] directions but not along [11] directions, for example,M[10]/M[11]≈ 25at

kT s = 0.2[MSGS02]

Given this inherent anisotropy of the system it seems extremely odd that shrinking circles

or spheres show no obvious faceting, nor is it shown in the migration of boundaries in general

(b)

(c)

FIGURE 3-13 Snapshot of sphere shrinking under curvature on a 3D simple cubic lattice

(a) kTs= 0, (b)kTs = kT r , (c) kTs = kT r disallowing grain nucleation.

The explanation is that the anisotropy of the interface stiffness and the mobility compensatefor each other, producing almost isotropic reduced mobility,M ∗ = M(γ + γ ) This result,however, only applies to the case where the driving force is due to curvature driven growth.When the driving force includes an external field, such as in equation (3.3), whereH is non-zero, the reduced mobility is no longer isotropic and indeed the shrinking circles are faceted,see Figure 3-14 This result is explained by the fact that interface facets are nearly immobile

at low temperatures, due to the rarity of kink nucleation, while interfaces with a high density

of geometrically necessary kinks are highly mobile As a result the grain shape reflects the

FIGURE 3-14 Snapshot of circle shrinking under curvature on a 2D triangular lattice, (a) H = 0 ,

a curvature driving force only leads to isotropic shape, (b) H = 0.2 , external and curvature driving force leads to highly faceted shape.

underlying anisotropy of the lattice The reason why the anisotropy of the interface mobilityand the interface stiffness cancel out in the case of curvature driven growth is that the entropicpart of the stiffness is large for inclinations where the mobility is low and vica versa [MSGS02].What this shows is that boundary mobility is a function of the type of driving force as alsoobserved in experimental systems [WGS02]

PROBLEM 3-11: The Effect of a Volume Driving Force

Investigate the e ffect of non-zero values of H on the evolution of a shrinking circular domain in the 2D Potts model using a square lattice.

3.2.10 Summary

This concludes the investigation of the Ising model The model is simple and yet encapsulates

a great deal of complex physics Using the model provides an insight into motion by curvaturewithout having to worry about the complex topological issues associated with multidomainsystems By the end of this section the reader should have experience with coding their ownIsing models, using different lattices, using different boundary conditions, and visualizing andanalyzing simple systems; a fundamental understanding of why the Potts model encapsulates thephysics of boundary motion by curvature; an appreciation of lattice effects; and an appreciation

of the role of simulation temperature

3.3 Q-State Potts Model

Soap froths, such as that shown in Figure 3-15(a), are easy to make because again, surface sion does all the work for you Take a soap bubble, add another soap bubble, and they are imme-diately attracted to each other because by being together they minimize their surface energy.Add a few more bubbles and you notice something else; they actually rearrange their interfaces

ten-to create a minimum internal surface area The characteristic Y junctions where three bubblesmeet have perfect120◦angles You never observe four bubbles meeting with90◦angles It doesnot matter how big or small the bubbles are, you always get these Y junctions This behavior is

a direct result of the minimization of isotropic interface energy Thus soap froths are not just ajumble of bubbles; they have form, and the form is dictated by surface energy considerations.The Potts model simulates the effects of interface energy on the topology of the boundary net-works, and so it is a tool to investigate self-ordering behavior Figure 3-16 shows theevolution of such a system simulated using the Potts model Instead of bubbles we have domains

(b) (a)

FIGURE 3-15 (a) A soap froth—the structure is self-ordering, (b) a Potts model 2D simulation of a

soap froth.

and instead of liquid membranes we have domain boundaries In this simulation the domainboundaries are associated with an isotropic excess energy, which has a profound influence onthe network topology because it implies that in 2D the number of boundaries impinging on avertex is always equal to three In other words, only triple points with120◦vertex angles arestable Fourfold and higher vertices, if formed, will always dissociate into the relevant number

3.3.1 Uniform Energies and Mobilities

The Potts model is a generalization of the Ising model The state of the system is described

in terms of the set ofQspins, which are associated with each lattice site,s i ∈ {0, Q}, where

ilabels the lattice site The system defines a boundary between unlike spins and no interfacebetween like spins In the isotropic case the energy associated with this boundary is described

by an energy functionγ:

γ (s i , s j) =

0 fors i = s j J

FIGURE 3-16 Microstructural evolution of an initially random distribution of spins on a 2D square

lattice using the Potts model on a simple 2D square lattice, periodic boundary conditions, Metropolis spin dynamics, and kTs = 0 The initial configuration of spins was set by allocating each lattice a random spin si ∈ {0, Q}

A Monte Carlo method is used to sample different states: choosing a random change to aspin (to one of theQother states) at a random lattice site and accepting or rejecting the changebased on the change to the total energy of the system, ∆E, computed via the Hamiltonian

in equation (3.15) Glauber or Kawasaki dynamics can be employed using the Metropolis orsymmetric probability functions as described in Section 3.2.2 As in the Ising model, the timerequired to attempt a single spin flip whether successful or unsuccessful is defined arbitrarily

asτand so 1 MCS is defined asNattempted flips The same lattice types (e.g., 2D square, 2Dhexagonal, 3D simple cubic) and boundary conditions (e.g., periodic, symmetric) can be used

In fact in the case where the energy and mobility of the boundaries is isotropic, the only change

in the model is the switch from two states toQstates; thus Figure 3-6 is the basic algorithm for

a vanilla Potts model using Glauber dynamics

Nevertheless, as Figure 3-16 shows, there is an important difference in the observed ior of the model Note how, despite an initial random allocation of Qspins, the system self-organizes in a cellular pattern which coarsens in a self-similar manner This is an importantresult, and reasons for such self-organization will be considered in the next subsection, for

behav-it concerns the competbehav-ition between the minimization of interfacial energy wbehav-ith the need forspace filling But before going on to consider why the model works, it is important to get a feelfor the model, and thus the reader is encouraged to attempt Problems 3-12 and 3-13

PROBLEM 3-12: Cellular Systems

Write a basic Potts model code with a 2D triangular lattice, Glauber dynamics using a lis probability function Set the initial geometry of the simulation by allocating each lattice site

Metropo-a rMetropo-andom spin between 1 Metropo-and Q Use periodic boundary conditions Save snapshots of the spin configurations every 100 MCS Use an imaging tool to visualize these snapshots and to make

a movie of the simulation Show that you get a familiar Ising system when Q= 2, and that as

Q increases the system transitions to a self-ordering cellular system, the structure of which is independent of Q

PROBLEM 3-13: Lattice Effects

Write a Potts model code with a 2D square lattice, Glauber dynamics using a Metropolis ability function, and periodic boundary conditions Set the initial geometry of the simulation by allocating each lattice site a random spin, where Q= 50 Show that when kT s= 0, you get a self-organizing network which has straight boundaries at45◦ angles to the orthogonal lattice Run a series of simulations for kT s = 0.25, 0.5, 0.75, 1.0 Observe how the boundaries become rougher and less straight as temperature increases.

prob-3.3.2 Self-Ordering Behavior

In 2D networks formed through the action of the minimization of isotropic surface energy, theaverage number of boundaries per grain is six Therefore the only stable network is a hexagonalarray of grains, where each grain has six neighbors and the 120◦ vertex angles at the triplepoints can be satisfied by straight boundaries These boundaries having no curvature have nonet force acting on them and so remain static Any networks that deviate from this regular arrayinevitably contain some grains with less than six sides and some with more than six sides

If the triple points maintain their120◦angles then the array must contain curved boundaries.Curvature driven migration given by equation (3.6) then causes the system to evolve, as shown

in Figure 3-16 The boundaries of grains with less than six sides are concave (curved towardthe center of a grain), and so boundary migration makes these grains shrink Grains with morethan six sides have convex boundaries and so these grains grow In other words, the competingrequirements of space filling and surface tension cause large grains to grow and small grains

to shrink This forms the basis of a remarkable law proposed by von Neumann [vN52], whichstates that the growth rate of a 2D cell with area,A, andN ssides is given by

dA

wherecis a constant This result has been shown to be correct for both 2D soap froths and 2Dgrain structures [Mul56] and has more recently been generalized to 3D

PROBLEM 3-14: Self-Ordering Behavior

Perform 2D Potts model simulations with Q = 100, and plot dA/dt versus N s Show that Potts model simulations obey equation (3.16) despite the fact that no topological rules are employed Perform the simulations for lower values of Q and show that at small values of Q equation (3.16) breaks down Why is this? Examine the movies of your simulations to discover

3.3.3 Boundary Energy

So far we have treated boundaries as if they are continuous isotropic interfaces We can relaxthis condition by assuming that the energy of each grain boundary is anisotropic Althoughsuch a system has little relevance to a soap bubble froth, it is extremely important for modelingcrystalline microstructures and biological cellular structures, since in almost all cases thesesystems possess anisotropic interfaces

Let us assume thatγis a function of the crystallographic misorientation across the boundary.This changes the equilibrium condition at the nodes (where three or more boundaries meet) Ifthese are to remain in a state of local equilibrium and maintain the equilibrium angles defined bythe boundary energies, then neglecting torque forces, the angles at the nodes in two dimensionsare given by the relation:

Figure 3-17(b) shows how a node is represented in the Potts model on a square lattice Notethat the triple point angles are discrete quantities, which depend not just on boundary energiesbut also on the type of lattice Although it may seem that there must be a problem in implement-ing anisotropic energies since at any one time the node angles can only be either90◦or180◦inthe square lattice and120◦or30◦in the triangular lattice, in fact this type of lattice effect is nottoo problematic, since the effective angles are integrated over time during simulations and so doallow a wide range of node angles to be simulated More in depth discussion of lattice effectswill be dealt with in Section 3.5

Implementing anisotropic energies into the Potts model is relatively straightforward since

it only requires the modification of Hamiltonian The easiest way to code these systems is tointroduce a new identifier, which along with the spin is associated with each lattice site Thus

we may give each lattice site both a spin identifier,s i, and component identifier,η i, as shown inFigure 3-17(c) The component identifier carries information necessary to calculate the

FIGURE 3-17 The relationship between boundary energy and node angle, (a) a continuum system,

(b) Monte Carlo Potts model: each grain orientation is represented by a different gray scale, the boundaries are sharp being implicitly defined between sites of different orientations, (c) showing the implementation of components and spins into the model.

anisotropic nature of the boundary, while the spin identifier continues to be used to calculatewhether a particular site is on a boundary In such a system the Hamiltonian becomes:

different The simplest system we can consider is a single phase polycrystal with only two types

of component, A and B, which then results in a system with three types of boundary, A–A, B–B,and A–B boundaries, with three boundary energies,JAA,JBB, andJABwhich gives:

This is the simplest anisotropic system, but already it is getting complicated The behavior

of the system can be most easily understood by considering the the dimensionless parameters

RA = JAA/JABandRB = JBB/JAB

Clearly whenRA = RB = 1 the system is isotropic WhenRA = RB > 1the relativeenergy of the A–B boundaries decreases in relation to the A–A and B–B boundaries; thus duringevolution the system will try to minimize the area or length of A–A and B–B boundaries in favor

of A–B boundaries and so minimize the energy of the system Figure 3-18 shows such evolution

of such a 2D system Notice how it self-organizes into a mosaic structure which minimizes thelength of A–A and B–B boundaries The mosaic structure is itself then able to coarsen in aself-similar manner

FUNCTION 3-3: Calculate Anisotropic Energy Dynamics Using Metropolis Scheme

spin old= existing spin of site

spin new= new proposed spin of site

η old= existing component of site

η new= new component of site

Continued

energy old= sum of bond energies over neighbor shell with spin =spin oldand

component=η old

energy new= number of like neighbor sites with spin =spin newand component=η new

∆E = energy old − energy new

IF∆E <= 0then

spin newandη newis accepted

else ifT >0then

probability = exp −∆E/kT s

random= A random number unformly distributed between 0, 1

ifrandom < probabilitythen

spin newandη newis accepted

end if

end if

The basic algorithm to determine whether a spin change is accepted or not using anisotropicenergy in the Metropolis scheme is shown in Function 3-3 Note that this implementation slowsthe algorithm up quite a bit, and that speed-up can be achieved by a more sophisticated algorithmdescribed in Section 3.4

PROBLEM 3-15: Anisotropic Energy

Write a Potts model code to simulate the grain growth of a two component system Write the code in such a way that the system undergoes an initial period of isotropic growth for a time,

t init , to establish an equiaxed grain structure with fractions fAand fBof the two components For

t > t init allocate energies JAA, JBB, JABto the boundaries Experiment with varying the ratios

RAand RBand the fraction fA Note how the most interesting e ffects occur when fA= fB= 0.5

Figure 3-19 shows the equilibrium structures formed under a variety of other conditions If

RA= RB>1the A–A and B–B boundaries are favored over A–B boundaries, and the systemself-orders the phases to segregate the A and B components and thus minimize boundary energy.Figure 3-19(b) shows such a structure which orders the A component and the B componentinto separate enclaves and can be contrasted with Figure 3-19(a), which shows the randomdistribution of A and B components which comes in the isotropic case whenRA = RB = 1.Figure 3-19(c) shows what happens whenRA> RB; the system gets rid of the high energy A–Aboundaries altogether Figure 3-19(d) shows another example of the type of mosaic structuresthat are formed whenRA = RB< 1.0 Figure 3-19(e) shows another example ofRA > RBbutthis time whereγBB = γAB, here the A component grains are not removed because it is onlythe A–A boundaries which are high energy; however, they do become an isolated component.Figure 3-19(f) shows the effect of usingkT s= 0, with anisotropic energies Because of the highlattice pinning present, the structure shows a very high degree of planarity in the low-boundaryplanes which are at45◦to the simulation lattice.

Note that in many of these 2D anisotropic energy systems, four grain junctions tions), are possible The angles at which boundaries meet in a quadrijunction are not uniquelydetermined by an energy balance Instead, the angle of a grain corner in a stable quadrijunctionmust be greater than or equal to the angle of the same corner in the trijunction formed when thequadrijuction fluctuates into two trijunctions This angular flexibility has an important effect onthe kinetics Systems in which quadrijunctions are unstable undergo normal grain growth Whenquadrijunctions are stable (due to the ratios ofRAandRB) grain growth can stop due to the

(quadrijunc-FIGURE 3-18 The evolution of microstructure during a Potts model simulation of a two component

system in which the initial distribution of components is equal and RA= RB= 0.5 The A and B components are differentiated by the gray scale The simulation was performed using a square (1,2) lattice, Glauber dynamics, Metropolis transition probability function, and kTs= 0.

flexibility of these junctions to change their angles and thus eliminate boundary curvature.

A systematic analysis of such congruent point grain growth has been carried out by Holm

et al [HSC93].

Both greater number of texture components and more sophisticated (and realistic) energyfunctions such as the Read–Shockley function can be incorporated into the model as we shallsee in the applications in Section 3.5

PROBLEM 3-16: Anisotropic Energy

Model the grain growth of a two component system in which RA= RB= 0.5 using kT s= 0, a triangular lattice, Glauber dynamics, and a Metropolis transition probability function Compare your results with the square lattice simulations shown in Figure 3-18.

PROBLEM 3-17: Anisotropic Energy

Modify your code for Problem 3-16 so that the system energies become isotropic when the average grain size reaches a 10th of the system dimensions You should see the grain structure self-segregate and then de-segregate.

3.3.4 Boundary Mobility

To simulate the case where the mobility is also a function of the boundary character,

µ (s i , s j , η i , η j), then we must modify the probability transition function so that probability of aspin flip is proportional to the mobility of that boundary The Metropolis probability transitionfunction then becomes:

µ m andµ mis the maximum mobility in the system

For the simplest model system with two phases or components A and B,p ois reduced to asimple binary function:

µ (s i , s j , η i , η j) =







0 in the grain interiors (s i = s j,η i = η j)

MAAfor A–A boundaries (s i = s j,η i = η j = A)

so are not favored to grow, the mobility advantage acts to shrink these grains In systems whereone phase is initially in the minority, this leads to a phenomenon called abnormal grain growthwhere the minority component grows to become the majority component, as in Figure 3-20.More sophisticated mobility functions can be incorporated into the model to simulate the

effect of temperature gradients, since mobility of interfaces are often a function of temperature.This is easily implemented into the model by making the mobility a function of the latticedimensions, for example,µ (s i , s j , x, y)

The odd thing one notices about Potts model simulations as soon as one becomes familiarwith them is that increasing temperature does not increase the mobility of the boundaries This