COMPUTATIONAL MATERIALS ENGINEERING Episode 5 pot

3.5 Applications of the Potts Model So far we have modeled the general features of grain growth common to all microstructures.However, if we are interested in the microstructural evoluti

(two dimensions) or planes (three dimensions) of the lattice sites that immediately adjoin itssubdomain and which are actually owned by neighboring processors as shown in Figure 3-29(c).The important point is that the processors are all synchronized to update the same flavor oflattice site.

The parallel Potts algorithm is shown in Figure 3-30 This algorithm is highly parallel, withthe only communication cost being the local exchanges of boundary spins between neighboring

Allocate initial geometry to subdomains and flavors

For each flavor of site

For each site of that flavor in subdomain

Pick a spin at randon from set 0, Q

Time to output a snapshot?

Output snapshot and other data

Time to end simulation?

End simulation

Y

N

Y N

FIGURE 3-30 The parallel algorithm for the Potts model.

processors These exchanges also serve as synchronization points in the loop over flavors toensure that all processors work on the same flavor at the same time In practice, as long asthe processor subdomains are of reasonable size (50 × 50or larger in two dimensions), thecommunication costs are only a few percent of the total run time and thus the algorithm cansimulate large lattices with parallel efficiencies of over 90% [WPS+97].

PROBLEM 3-27: Parallel Algorithm

Code a 3D square Potts model using a parallel algorithm Perform normal grain growth lations for lattice size 200 × 200 × 200 and107MCS, using 1, 2, and 4 processors Check that identical parabolic kinetics are observed in all cases Calculate the speed-up per processor of the algorithm.

simu-3.4.4 Summary

This concludes the investigation of the Potts model speed-up algorithms The aim of thissection has been to give the reader a wide selection of Potts algorithms with which to simulateindustrial applications By the end of this section the reader should have experience with codingboundary-site models, n-fold way models, and parallel models We have stressed throughoutthat it is important to be organized, systematic, and above all to always verify a new modelbefore applying it to a new phenomenon

3.5 Applications of the Potts Model

So far we have modeled the general features of grain growth common to all microstructures.However, if we are interested in the microstructural evolution of a particular material then it isimportant to accurately simulate the microstructural features of the material such as the grainstructure, texture, and misorientation distribution function We will also want to model the kinet-ics and to compare them with experiment

statisti-In Section 3.3.1 we noted that the domain structure in the Potts model coarsens in a similar manner so that the average domain size increases in time Experimentally, it is observedthat the grain size distribution when normalized by the average grain size remains constantduring grain growth This means that even though some grains grow, while others shrink, thegrain ensemble remains self-similar This type of phenomenon is called normal grain growth.The grain size distribution and the topological distribution derived from 3D Potts model simu-lations of isotropic grain growth are also observed to be time invariant and in agreement withexperimental data, as in Figure 3-31

self-PROBLEM 3-28: Normal Grain Growth

Simulate grain growth in three dimensions using the Potts model for Q= 5, 10, 50, and 100 Plot the grain size distribution as a function of Q and comment on the result.

Measuring grain size in simulations is notoriously laborious for big systems The simplestway of dealing with this is to issue each site with a unique grain identifier as well as a spin when

FIGURE 3-31 Grain size distribution during grain growth, a comparison between an Fe sample and

the 3D Potts model [AGS85].

setting up the initial simulation microstructure This unique identifier is swapped along withthe spin during any successful spin flip, but is not used except when analyzing the snapshots ofthe simulations It provides an easy way to identify all the sites of a particular grain and thuscalculate the grain area/volume

PROBLEM 3-29: Grain Size Pertubations

Simulate 2D grain growth using the Potts model for Q=100 After 100 MCS insert a circular grain into simulation in a random location with a size five times that of the average grain size of the system Plot the size of this grain normalized by the average grain size of the system against time Why does this grain not grow abnormally?

The rate at which the average size increases is another parameter by which experimentalistsmeasure normal grain growth The kinetics of grain growth is characterized by the parabolicequation

R n

whereR0is the initial grain size andA ggis a constant The grain growth exponent,n, has beenthe focus of much of the debate in the grain growth community Hillert’s theoretical derivation[Hil65] givesn= 2but most experiments show grain growth exponents much greater than this;typically the values lie betweenn = 2.5andn = 4 It has been argued that impurity effectsmay be responsible for the deviation from the ideal value However, even data from a widerange of ultrapure metals show considerable deviation fromn= 2 3D Potts model simulations

of isotropic grain growth show grain growth exponents in the range2 < n < 2.5 Why the

range you might ask? The measured exponent depends on many variables of the system, butimportantly on the size of the system,kT s,Q, and on initial distribution of grain size Issuesabout why the grain growth exponent is so sensitive to these variables have yet to be definitelyresolved

PROBLEM 3-30: Effect of Temperature on Grain Growth Exponent

Simulate grain growth in two dimensions using the Potts model for kT s = 0, kT s = 0.5 , and

kT s = 1.0 Plot average grain area as a function of time for each temperature and calculate the grain growth exponent Note the early nonlinear transient at the beginning and end of the simulations Why do these occur?

3.5.2 Incorporating Realistic Textures and Misorientation Distributions

Figure 3-32 shows a 2D map of a typical microstructure obtained using an electronback-scattered diffraction (EBSD) method It illustrates clearly that each grain has a uniquecrystallographic orientation and that each grain boundary will have a unique misorientation androtation axis It is essential to capture this level of complexity in the Potts model if we are tosimulate the behavior of real experimental systems

111 100

initial 800-2 Sample

FIGURE 3-32 (a) EBSD map of annealed microstructure of a transformer silicon steel specimen,

(b) Potts model simulated microstructure after grain growth of a transformer silicon steel specimen using as recieved starting microstructure from ESBD, (c) shows the development of the < 111 >,

of the < 111 >, < 110 >, and < 100 > textures fibers as measured from Potts model tions [HMR07].

simula-For 2D simulations the most straightforward way of doing this is to incorporate themicrostructural information and the crystallographic information directly from the EBSD dataset Since each grain in the experimental data set has a unique crystallographic orientation, it

is important to use a unique spin Potts algorithm (as described in Section 3.4) Typically thismeans that each lattice site in the simulation is allocated a unique spin number and a table

is created which correlates the spin number with the Euler angles corresponding to the tallographic orientation of the grain A normal Potts model simulation can then be performedwith the crystallographic information of each lattice site being used to plot the evolution ofmicrostructure in the development of textures as in Figure 3-32

crys-Although this process seems straightforward enough, there are some important issues that wehave omitted to discuss First, we did not incorporate the experimental microstructure with itsassociated crystallographic details directly from the microscope into the model In such EBSDmaps there is a good deal of noise that corresponds to some pixels being identified as singlesite grains, when in fact they are more likely to be a site whose orientation was incorrectlymeasured This kind of noise may not be just due to incorrect measurement; in the case ofdeformed microstructures, the dislocation and other defects may be associated with low anglesubboundaries which are topologically distinct from grain boundaries Also since the map is a2D section of a 3D microstructure, some topological features may appear to be noise when infact they are the tip of a grain protruding into the 2D section For these and many other reasons,the importing of a microstructure into a Potts model often requires a clean-up filter to be applied

so that these effects can be mitigated and features which are not going to be included in themodel can be removed However, it is obvious that using these filters can also distort the dataset in some cases changing the fundamental of the microstructure to be studied For a modelerthe lesson here is to ask for both the filtered and unfiltered data from the microscope, to ensurethat radical massaging of the data is not occurring which would then render the simulationsmeaningless

These 2D microstructure maps are extracted from a small volume of the material This ume is represented in the model only through the boundary conditions Thus choice of theboundary conditions is important when performing simulations and also when interpreting theresults Choosing periodic boundary conditions is not an option since there will not be continuityacross the simulation boundaries The choice of mirror or free surface boundaries is available,and both have implications Furthermore the fact that a 2D simulation is being performed of

vol-a 3D phenomenon needs vol-also to be tvol-aken into vol-account Upshot of these fvol-actors is thvol-at extremecare should be taken when carrying out and interpreting such simulations The best practice is tocarry out a large number of simulations using a large number of different input microstructuresand to measure the evolution of average characteristics, for example, the average texture, mis-orientation distribution function (MDF), and grain size It is when these averaged quantities arecompared with experimental results that meaningful conclusions and predictions may be drawn,see Figure 3-32

PROBLEM 3-31: Incorporating Realistic Textures and Misorientation Distributions

Write a code to import the output from a experimental EBSD orientation map and import it into the Potts model Take a snapshot of the imported microstructure and compare it with the EBSD map Measure the grain size, MDF, and texture of the imported microstructure and compare your results with those calculated by the EBSD software Use your imported microstructure as the starting configuration for an isotropic grain growth simulation using the Potts model.

PROBLEM 3-32: Comparing the Effect of Boundary Conditions

Use your imported microstructure as the starting configuration for an isotropic grain growth simulation using the Potts model Compare the grain growth kinetics and grain size distributions obtained using mirror boundary conditions with those obtained using free-boundary conditions.

There are no routine methods for extracting the necessary 3D information from experiment

It is possible to combine EBSD with serial sectioning, but this a very labor intensive task andstill leaves the problem of how to extrapolate between the sections 3D X-ray tomographymethods have more recently become possible using high energy focused synchrotron X-raysources, but at the moment the resolution is low and again the method is not widely available.Another approach to this problem is to use computation methods to reconstruct an equivalent3D microstructure with the grain size, grain size distribution, texture, and MDF, since obtainingthese characteristics of the 3D microstructures from experiment is straightforward

The first step is to obtain a 3D microstructure with the right grain size and grain size tribution This is done by using a 3D Potts model and using anisotropic mobility to grow anappropriate microstructure using trial and error, see Figure 3-33(a) This is easy for equiaxedmicrostructures and less easy for more complicated microstructures Next the experimental tex-ture is discretized intoQorientations and allocated randomly to the spins of the grains of the 3Dmicrostructure This produces a 3D microstructure with the correct texture but random MDF.This MDF is calculated and quantized inton bbins, such thatS k is the number of boundaries

dis-with misorientations betweenk ∆θand(k + 1)∆θ, k = 0, 1, , n b A system Hamiltonian isdefined as the sum of the squared differences betweenS m

Desired (i) Model (i) Desired (ii) Model (ii) Model (ii) 4

3.5 3 2.5 2 1.5 1 0.5 0

FIGURE 3-33 (a) Three-dimensional equiaxed microstructure grown using the Potts model,

(b) Showing the desired and the achieved MDFs generated by discretizing a texture, allocating entations to the grains, and then using the swapmethod to achieve the desired MDF [MGHH99].

ori-whereS m

k defines the MDF of the model andSexpk defines the experimental MDF.H mdf is a

state variable providing a measure of the difference between the model MDF and the mental MDF It is equal to zero when the model MDF and the experimental MDF are identical

experi-We use a Monte Carlo algorithm in order to minimize H mdf and in doing so construct the

desired MDF The method is as follows: two grains are chosen at random, and theH mdf due

to swapping their orientations is calculated The probabilityp(H mdf) that the swap is accepted

is a Metropolis function Figure 3-33(b) shows the wide range of MDFs that can be achievedusing this algorithm (Read ref [MGHH99] for more information.)

This swap method is effective and produces a starting 3D microstructure with a texture andMDF that are identical to the experiment It is not elegant More ambitious ways of reconstruct-ing 3D microstructures from 2D metrics, which integrate the microstructure generation, texturegeneration, and MDF optimization steps into one step have been proposed Unfortunately noneyet have been shown to work Progress on 3D X-ray methods may make the swap methodredundant in the future It will be interesting to see

3.5.3 Incorporating Realistic Energies and Mobilities

Read and Shockley [RS50] derived an analytical expression for the energy (per unit area) of alow angle grain boundary The boundary is assumed to comprise of a regular array of disloca-tions The boundary energy can be expressed as a function of the misorientation:

The parametersγ0 andA are related to elastic constants and properties of the dislocationcores:γ0sets the overall energy scale, andAadjusts the angle of the maximum grain boundaryenergy For large angle grain boundaries, this model would not be expected to be valid, asthe dislocation cores would overlap substantially, and their interaction could not be neglected.Nevertheless, this formula has been successfully fit to experimental grain boundary energies forwide misorientation angles Thus a normalized version of equation (3.28) can be used to modelthe functional form of a general grain boundary in the Potts model:

Clearly in most real systems mobility is also a function of the boundary character:

µ (s i , s j , O i , O j) Thus we must modify the probability transition function so that probability

of a spin flip is proportional to the mobility of that boundary The Metropolis probability tion function then becomes:

By including or not including such factors in a model we are making assumptions about whichare the important factors in a system We are also making the Potts model more complex Thus

it is best practice in such situations to carry out simulations on simple geometry to validate themodel before going on to tackle the full 3D polycrystalline system The simplest of such sys-tems, but which nevertheless still contains boundaries and triple points, is discussed in the nextsection

3.5.4 Validating the Energy and Mobility Implementations

Although the implementation of the Read–Shockley energy function seems a straightforwardextension of the model to change the boundary energy, it has another implicit effect, which is

to change the node angles of the boundaries As discussed in Section 3.3.3, this changes theboundary curvature acting on a boundary and so the driving force on that boundary If we are

to simulate systems with a continuous range of boundary energies and so a continuous range

of node angles, we need to make sure that the discrete nature of the simulation lattice does not

affect these angles

One way to do this is to consider a model geometry such as that shown in Figure 3-34

We consider a system with a constant driving force for motion and in which the triple pointshave invariant geometry A similar approach is taken by experimentalists studying boundaryand triple point mobility The grain structure is columnar, with two grains, B and C, capped by athird grain, A Boundary conditions are periodic in thex-direction and fixed in they-direction.There are two boundary misorientations in the system:θ1is the misorientation angle of the A–Band A–C boundaries, andθ2is the misorientation angle of the B–C boundaries There are twotriple junctions in the system, and the geometry is arranged such that these two are identical andsymmetric From equation (3.17) the equilibrium junction angle whereθ1is the energy of theA–B and A–C boundaries, andθ2is the energy of the B–C boundaries

The driving force acting on the boundary is γ2/D Assuming that the driving force isproportional to the velocity of the boundary, the boundary velocity in they-direction

dy

dt = µ1γ2

whereµ1is the intrinsic mobility of the A–B and A–C boundaries.

To examine the validity of theQ-state Potts method, a nominalγ2is set anddy/dtis sured with time By finding the regime in whichdy/dtis constant, and using equation (3.33),the effectiveγ2can be extracted Figure 3-35 compares the measuredγ2to the nominalγ2 Itcan be seen that for largeγ2(i.e., high misorientations) there is good agreement between the

of the boundary energies of the boundaries that intersect at the triple point, γ(θ1 )and γ(θ2 ).

Simulation Read-Shockley

Misorientation ( θ 2 ) (a)

105

1041000 100 10 1

1 10 100 1000 104 10 5

µ1

Nominal µ1 (b)

0 0 0.2 0.4 0.6 0.8 1 1.2

γ /γ 2max

FIGURE 3-35 (a) Measured γ2versus nominal γ2for Potts model simulations of boundary motion

in the system illustrated in Figure 3-34, kT s = 0.5 (b) Measured µ1 versus nominal µ1 for Potts model simulations of boundary motion in the system illustrated in Figure 3-34 with µ2 = 1.

simulation and the theory But asγ2decreases, a deviation from theory is observed; the effective

γ2becomes constant as the nominalγ2continues to decrease This deviation occurs aroundγ2

= 0.4γ1, corresponding to a misorientation angleθ2= 2◦whenθ1= 15◦

This behavior has its origin in the discrete nature of the lattice As θ2 gets smaller tive toθ1, the equilibrium triple junction angle,φ12, gets larger until it approaches180◦andthe A–B and A–C boundaries become nearly flat Because junction angles must be changed

rela-by the addition or removal of an entire step in a discrete lattice, small differences in the tion angle cannot be resolved That is, at some point, the last step is removed, the boundary

junc-becomes flat, and the triple junction angle cannot change with further decreases inγ2 Becausethe triple junction angle defines boundary curvature, it also defines the driving force Thus ifthis angle becomes invariant at someγ2, so does the driving force acting on the boundary Thiseffect is unavoidable in these discrete lattice simulations and hence there is a limit to the range

of anisotropies that the model can simulate For simulations on the square lattice, the limit isreached aroundγ2= 0.4γ1, whenφ12= 157◦; larger triple junction angles cannot be resolved.

Note that this effect limits only the maximum triple junction angle and thus the range ofboundary energies (anisotropy) that may be resolved It does not limit the absolute value of theboundary energy For example, a system ofθ= 1◦boundaries, each with energyγ = 0.25, has

120◦triple junctions and can be successfully simulated by theQ-state Potts model The triplejunction limitation need be considered only if a higher angle boundary (in this case,θ > 4◦)

must be included in the system

The limitation on energetic anisotropy does not affect the model’s ability to simulate form boundary mobility Since mobility is independent of curvature, it is unaffected by triplejunction angles Figure 3-35 shows the linear relationship between mobility and velocity in the

nonuni-Q-state Potts model over four orders of magnitude (Read ref [HMR03] for further information.)

PROBLEM 3-33: Validating a 3D Potts Model

Validate the energy and mobility implementation of a 3D Potts model using a 3D version of the geometry shown in Figure 3-36.

3.5.5 Anisotropic Grain Growth

Having validated the model we are now free to simulate anisotropic grain growth using realistictextures, misorientation distributions using Read–Shockley energies, and anisotropic mobili-ties Figure 3-37 shows the evolution of such a system in which the initial microstructure has astrong texture< 100 >cube texture The system undergoes normal grain growth, which causes

a tightening of the texture The boundaries are colored to show their misorientation, black beinghigh misorientation and white being low misorientation Note how all the high misorientationboundaries (dark colored) are removed from the system during grain growth with all the bound-aries becoming white This causes a reduction in the average misorientation and a narrowingmisorientation distribution This effect is observed experimentally and is due to the high energyboundaries being replaced by low misorientation boundaries

FIGURE 3-36 The 3D hexagonal geometry used to validate the Potts model for anisotropic energies

and mobilities [HMR03].

FIGURE 3-37 The evolution of microstructure during a Potts model simulation of anisotropic grain

growth of a single texture component, using Read–Shockley energies and uniform mobilities The simulation was performed using a square (1,2) lattice, Glauber dynamics, Metropolis transition prob- ability function, and kT s = 0.5

Being able to understand what factors influence such changes in texture and MDF is one ofthe main advantages of simulation Turning off anisotropic mobility is an impossible experiment

to perform, but it is trivial to perform the same set of simulations with Read–Shockley energiesand uniform mobilities, or anisotropic energies and uniform energies Comparing the results inthis case reveals that mobility has little effect on the texture changes, and the energy function isthe dominant factor (for more information read ref [HHM01])

What if we were to explore the effect of different energy functions, in particular the effect

of energy cusps, such as those due to coincidence site lattice (CSL) boundaries? This requireschanging the energy function and thus the Hamiltonian of the system

PROBLEM 3-34: Anisotropic Grain Growth

Write an anisotropic Potts model simulation code which incorporates Read–Shockley energies, isotropic mobilities, and Euler angle description of orientations Investigate the e ffect of grain growth on the MDF of an initial random texture.

The CSL description of grain boundaries is a geometric model based on the fact that forcertain misorientation rotations, a fraction1/Σof atomic lattice sites will be coincident Theresulting boundary is termed a CSL boundary and is characterized byΣ Exact CSL boundariesare seldom observed in general materials because the CSL relationship requires three indepen-dent boundary degrees of freedom to assume particular values Brandon [19] introduced theconcept of an acceptance criterion, which admits a wider range of CSL boundaries Misorienta-tions with angular deviations of less than∆θΣ = 15 ◦ /Σ1/2from the true CSL misorientationare assumed to be within theΣCSL

The Read–Shockley derivation requires that the array of boundary dislocations be spaced

uniformly by some multiple of the Burgers vector b A CSL boundary can be viewed as a secondary array of dislocations with spacing b/Σimposed on this primary array As such, thecontribution to the grain boundary energy from the CSL can be modeled:

bound-in equation (3.29) The total energy of a boundary bound-in our system is thus given by the sum

of equations (3.29) and (3.34) Note that for non-CSL boundaries, the contribution fromequation (3.34) is zero Figure 3-38(a) shows the form of such an energy function

This type of simulation shows some interesting differences between modeling the full 3Dcrystallographic orientations of a crystal and the 2D crystallographic orientation In the lat-ter case each grain requires only a scalar index to denote its orientation, and the misorienta-tionθis then easily calculated as a sum In such a system, energy cusps of CSL as shown inFigure 3-38(a) have a profound effect on grain growth with the MDF produced mirroring theenergy function and a large fraction of the boundaries to forming multijunctions as shown inFigure 3-38(b) However, if the same simulations are carried out in which each grain requiresthree Euler angles to denote its orientation, then the evolution is very different The extra degrees

of freedom that exist in Euler space mean that the chances of forming a boundary within the CSLlimit become much smaller The vast majority of boundaries that are classified as CSLs do nothave significantly reduced energy; in fact, 95% of nominalΣ5boundaries have energy within5% of the random, high angle boundary energy Even if the misorientation angle of the genericCSL is close to that of the exact CSL, the axis need not be close to the true axis Therefore,most nominal CSL boundaries have energy near that of non-CSL boundaries and should not bemorphologically enhanced during grain growth (for more information read ref [HHM03])

PROBLEM 3-35: The Effect of CSL Boundaries on Grain Growth

Write an anisotropic Potts model simulation code which incorporates Read–Shockley energies, CSL boundaries, and isotropic mobilities Show that when the orientations of the grain are denoted by a scalar quantity, grain growth increases the number of CSLs in the system Also show that this does not happen when the grain orientations are represented by Euler angles.