COMPUTATIONAL MATERIALS ENGINEERING Episode 7 doc

ALGORITHM 4-5: Two-Dimensional Cellular Automata for Grain Growth 1: Input or generate an initial microstructure 2: forncellular automaton steps do 3: Determine and store which cells are

after the other Such sequencing has been implemented with success by many other authors,for example, Landau and Binder[LB01] or Rollett and Raabe [RR01] to name just two, morerecent implementations The second option is to modify equation (4.22) to include an additionaldriving force [Jan03]:

of the second term depends on the direction of the additional driving forcep aand is positive ifacting in the same direction as the curvature pressure does Equation (4.24) then becomes:

f

1 + f



(4.27)For a given time step∆tthus follows that

r η= 8mγ (1 + f)∆t 3r

which can be rewritten as:

r η=3(2rc 16mγ∆t − p a∆t)andf=2rc p − p a ∆t a∆t ,with∆t < 2 r c

A locally varying driving force thus leads to a locally varying value for f, which provides

an entry point in the update rule to locally correct for the presence of any additional drivingforce Note that in case the value ofp a gets very small, the equations reflect pure curvature-driven grain growth and the maximum value for∆tis limited by the computationally feasiblemaximum value of the neighborhood radiusr η

4.7 Hybrid Cellular Automata Modeling

With hybrid cellular automata we mean combinations of different types of CA and othercomputational methods When a microstructure transforms many different processes can occurconcurrently The process of dynamic recrystallization is an example in which plastic defor-mation, recovery, recrystallization, and precipitation occur simultaneously In this section acombination ofCA with volume finite difference modeling is described, which models boththe grain growth process and the diffusion of solutes

4.7.1 Principle

Before modeling concurrent microstructural processes you need an approach to split up the ferent processes in your computational model Focusing on our case study: it is clear we can usecellular automata to model grain growth and another method to model the diffusion of solute

dif-elements in the microstructure, but how can we model two processes and their interaction

simul-taneously? Well, the answer is rather simple: we do not What we do is compute both processes

sequentially for a small but finite time-step, which in the limiting case of an infinitesimally small

time step is the same as computing them concurrently Both models operate on the same datastructure representing the microstructure, and the interaction between the processes should fol-

low through that data structure An obvious remark to make is that both methods must converge;

in other words a cellular automaton step may not interfere with the numerical convergence ofthe other method and vice versa

Modeling Solute Diffusion

Modeling the diffusion of solute elements can be done using different approaches, for ple, using phase field modeling (see refs [CKYY02], [ZCCT01]), or using Potts-type MonteCarlo simulation [STH97] In this section we propose a method, first published in Janssens

exam-et al [JHF04], which is computationally highly compatible with an irregular, shapeless cellular

automaton, but also works fine with a regular grid one, and is fairly easy to program

Independent of its type, a cellular automaton consists of a+2-dimensional grid of shapelesscells distributed in space Each cellirepresents a volumeV iof material with locally attributedproperties, which also can include a concentration for one or more solute elements The diffusionmodel is discretized as illustrated in Figure 4-25 On the left, two cells are depicted, each given

a certain activitya i = µ i c i = n i /V iof a solute element (n iis the number of solute atoms,µ i

the activity coefficient andc ithe concentration of solute elementi) Solute segregation to grainboundaries can be modeled straightforwardly by introducing different activity coefficients in thebulk and at the grain boundary The cells are at a known distancedapart.Ais the surface areathrough which diffusion can occur The flux of solute atoms through an areaAfrom cell 1 to 2

is defined by Fick’s first law:

with constant total number of atomsn = n1+ n2 and variable timet Special cases can be

formulated, such as a constant concentration in one of the cells These source or sink boundary

conditions lead to an alternative equation:

for c2 constant Next an algorithm is designed to calculate the diffusion process in +2dimensions: For each cell do the following:

1 Determine all neighboring cells within a predetermined radius r η as illustrated inFigure 4-25

2 Sequentially for each cell in this neighborhood, compute the diffusion flux usingequation (4.31)

Note that this algorithm evaluates each pair of cells twice To link a simulation step to a time the simulation needs to be calibrated by performing a computational experiment of ananalytically solvable problem This calibration links the value of the surface areaAto a valuefor the neighborhood radiusr η.r ηcan be chosen arbitrarily within such limits that on average

space-a fespace-asible number of cells is found

In Figure 4-26 the results of a three-dimensional simulation are shown, illustrating theapproach is quantitatively correct The experiment consists of a plane sheet of thickness2lwithinitial solute concentrationC0, surrounded by a gas with a constant solute concentration C1.The non steady state solution is given by J Crank [Cra03]:

n + 1)πx

Modeling Growth

If you have been a hard working student you probably have already programmed Algorithm 4-4

in detail—if not you can download it from this book’s Web site, that is of course if I have beenworking hard as well

Combining Both Computational Models into One, Hybrid CA

Before combining both computational models, you need to make sure the microstructure isrepresented using a data structure on which both models can operate:

• Partition the 2D microstructure into square cells

• The state of a cell can be any grain identification number

• Attribute to a cell is the concentration of a solute atom, which can be any rational numberbetween0and0.1, the latter being the saturated concentration on a grain boundary Notethat this limits the range of applicabillity of the model to low solute concentrations—butthat is the sensible range anyhow

d, A

a1V1n1 a2V2n2

2 1

concen-C _ concen-C0

_ C0+ C1

Dt

1 2 1

we still have to implement is to make the probability that a cell switches grain dependent on itsmomentary solute atom concentration Experiment with making grain boundary mobility and/orenergy a function of the solute atom concentration and see what happens

ALGORITHM 4-5: Two-Dimensional Cellular Automata for Grain Growth

1: Input or generate an initial microstructure

2: forncellular automaton steps do

3: Determine and store which cells are on a grain boundary

4: Compute the diffusion step

5: Make a copy of the current cell states and name it previous microstructure{All computations in this step are made based on this copy}

6: for Each Cell do

7: if Cell is at a grain boundary then

8: Compute local boundary curvature

9: if Curvatureκ >0then

10: Compute new state of cell

11: {i.e., the probability that it switches grain}

simula-4.8 Lattice Gas Cellular Automata

Lattice gas cellular automata, LGCA in short, are a form ofCA in which position and tum of interacting particles are modeled on a discrete grid of points This book is not aboutfluid dynamics, but I feel it is relevant to at least treat the basics of this branch of cellularautomata as LGCA can be straightforwardly combined with conventionalCA This may have

momen-advantages regarding algorithm complexity and/or computational resources over the tation of hybrid modeling, simply because only one type of numerical approach is used instead

implemen-of two or more On the other hand, other particle-type methods (e.g., ref [LL03]) are usedmore frequently and therefore have been developed to a more advanced level Nevertheless,LGCA can be used to model the fluid flow of liquids or gasses, and as such it is of interestfor modeling microstructure processes in which such flow plays a role, as can be the case in,for example, deposition of thin layers from an electrolyte, solidification processes, sintering,and thixo-forming processes The following sections are limited to an explanation of the basicprinciples The reader is referred to other literature (e.g., ref [Raa04b]) for more advanceddevelopments, like the Boltzmann-type LGCA

4.8.1 Principle—Boolean LGCA

In LGCA the material is modeled as a collection of fictive particles, each of which represent asmall volume of liquid or gas The particle state includes the specification of their momentumvector In Boolean LGCA, these fictive particles can only move on a grid of directions, whiletheir mass and the magnitude of the velocity vector are both constant for all particles As a firstexample, consider a 2D square grid as shown in Figure 4-27 The cells are the nodes on thegrid The motion of the particles is limited to the geometry of the grid In a Boolean LGCA allparticles have the same (unit) mass and velocity On a square grid each particle can only move

in one of four directions A conventional LGCA also imposes that at any time, no node on thegrid may contain more than one particle moving in the same direction, that is, not more than

A

FIGURE 4-27 Two-dimensional lattice gas cellular automata.

one particle can move along the same vertex in the same direction at the same time This is to beguaranteed by the initial distribution of particles and velocity vectors, and by the state changerules Algorithm 4-6 is a simplified example of an LGCA algorithm Assuming Figure 4-27 isthe initial state, the following events occur in positionAin the grid:

ALGORITHM 4-6: Simple Boolean LGCA

1: Define a cell’s state as the number of particles and their direction of motion

2: Define a cell’s neighborhood as the four neighboring cells

3: Populate the square grid with cells{vertex occupation limits!}

4: fornLGCA steps do

5: for Each cell do

6: Transfer incoming particles from neighboring cells—hop

7: Transform incoming particle directions to outgoing directions using the state change

rules—scatter

8: end for

9: end for

Line 3: Initially, there are no particles in the cells at locationA

Line 6: Both from the cell above and the cell below a particle is incoming.

Line 7: A state change rule is needed As the rule must be conservative, we note that the sum

of direction of motion at positionAis zero, which must remain so when the particles are

transformed (one could also say scattered) to an outgoing set On a square grid that leaves

us with only two possibilities for the current configuration atA, as depicted in the rightpart of Figure 4-27 One can thus define either a deterministic state change rule, pickingonly one of the possibilities, or one can define a probabilistic state change rule and leteither occur by chance with probability1/2.

PROBLEM 4-10: Two-Dimensional LGCA

Program a simple 2D LGCA Add some features like choices of boundary conditions and cles (“solid” particles at a fixed position from which particles bounce back).

obsta-Hop Scatter

FIGURE 4-28 Hexagonal LGCA.

The example of an LGCA on a square grid was first published by Hardy et al [HPdP73] in

1973 The model has three major problems preventing it from correctly modeling the behavior

of fluids: a lack of Galilean invariance, a lack of rotational isotropy, and a crossover dimension

problem These problems are described in detail by Frisch et al [FHP86] In the same article the authors show that switching to a hexagonal LGCA, see Figure 4-28, solves the problem

of rotational isotropy to a satisfactory level, while the violation of Galilean invariance is notrelevant at low Mach numbers, and the crossover dimension problem no longer exists when oneconstructs a 3D model

In conclusion to this very brief introduction to LGCA, here is a partial quote from a verycomplete book by Rothman and Zaleski [RZ97] on the matter (the authors write about a hexag-onal grid LGCA with particular state change rules):

Although the microscopic makeup of fluids ranges from the simplest mon-atomic gas to, say, a plex mixture such as milk, nearly all fluids flow in a way that obeys the same equations of fluid mechanics we introduce a microscopic model of a fluid that is far simpler than any natural fluid —at a macroscopic scale it flows just like them!

com-Indeed, a lattice gas cellular automaton, given a particular combination of grid and a specific set

of state change rules, does behave as a real fluid—the secret of the trade is what rules to apply.The interested reader is referred to, for example, reference [RZ97] for those secrets

4.8.2 Boolean LGCA—Example of Application

Assume we have an electrolyte which flows over a substrate with existing thin layer structures,and that from this electrolyte an additional thin layer is to be deposited Simplifying the prob-lem to the maximum, assume one has direct control over the amount of electrolyte that flows.Also assume that, given the composition of the electrolyte, one knows (e.g., setting the depo-sition current) exactly how much material is deposited The situation described is depicted inFigure 4-29, and can be put into formula as follows:

1 The (incoming) electrolyte contains a deposition-elementαwith concentrationc α < 1per volume unit

2 The flow rate of the incoming electrolyte isfvolume units per time units

3 The deposition rate is linearly dependent on the element concentration, and, as anexample, is specified byd = d∞(1 − exp −sc α), withd ∞andsconstants

4 Theαlayer to be deposited is pureα

Thin Layer Structures Deposited Layer

Electrolyte Flow

FIGURE 4-29 A simplified deposition process.

First tackle the problem of the electrolyte flow by solving Problem 4-11

PROBLEM 4-11: Thin Layer Deposition from an Electrolyte

Program a 2D LGCA modeling the flow of electrolyte over thin layer structures as shown in Figure 4-29 Simulate the deposition process by stopping the particles’ motion, for example, when they hit a nucleation site on the substrate and when they hit particles which already stopped moving (i.e., these are the “solidified” particles).

To model the deposition process we add a Boolean attribute to the state of the LGCAwhich can take the values of either solid or liquid Algorithm 4-7 is a possible solution toProblem 4-11 When running this algorithm you may end up with holes in the deposited layer

To solve this problem you will need to add a step which corrects this, for example, by shufflingsolid cells until the hole ends up at the edge of the layer

ALGORITHM 4-7: A Solution to Problem 4-11

1: Define a cell’s state to include the direction of motion of particles and the attribute

solid/liquid

2: Define a cell’s neighborhood as the four neighboring cells

3: Initialize the substrate shape including the thin layer structures by setting the appropriatecells’ attribute accordingly (i.e., substrate and structures solid, the rest liquid4: Populate the square grid with cells{vertex occupation limits!}

5: fornLGCA steps do

6: for Each cell do

7: if cell is liquid then

8: Transfer incoming particles from neighboring cells

9: Compute whether the cell changes state to solid

4.9 Network Cellular Automata—A Development

for the Future?

Network cellular automata (NCA) are a fairly recent development of cellular automata [Wol02].Here is an attempt at a definition:

• Cells are defined as nodes in a network of connections Different from conventional lular automata is that cells do not necessarily constitute a volume partition of space; theyare abstracted to a higher level As a matter of fact, a network can usually be mapped ontoany number of dimensions In Figure 4-30 the cells are drawn on a 2D-periodical grid, but

cel-it should be noted that this grid specification is not part of the defincel-ition of the network, cel-it

is merely a way of drawing the network orderly

• A cell’s neighborhood definition is determined by the local configuration of the network

An example of a definition could be: the cell itself plus all cells directly connected to itand all the connections between these cells, as illustrated for the gray cell in the cutout inFigure 4-30

• The state of a network cell is the configuration of the connections within its neighborhood

• A state change rules operates on the network configuration within the neighborhood nections may be added, deleted, and modified The only restriction is that no connectionsshould be added to cells outside of the neighborhood To do so one would need infor-mation on the location in the network of cells outside the neighborhood, which is thesame as considering a larger neighborhood, for example, by includingnthlevel, indirectlyconnected cells

Con-4.9.1 Combined Network Cellular Automata

Combined network cellular automata (CNCA) are a logical extension of NCA The concept

is straightforward: the state of a cell is a combination of the network configuration within

the neighborhood and a separate state, optionally with attributes, assigned to the cell itself.

As depicted in Figure 4-31, the main difference with standard NCA is that a state changerule now also operates on these separate cell states and their attributes This figure also hints

at the tremendous amount of possibilities that can be used to define the state transformation

State Change

Neighborhood

FIGURE 4-30 Network cellular automata.

Cell State

Network Connection

B

A A

B A

B

B

State Change

FIGURE 4-31 Combined network cellular automata.

function f, as the combination of network configuration and separate cell states has a highnumber of possible combinations It is to be expected that, when using CNCA, one will limitthe possible combinations, but the openness of the definition allows one to select those needed.Taking Figure 4-31 as an example:

• ftakes as input the states of the cells in the neighborhood and/or the local configuration

of the connections between the center cell and its neighbors

• fneeds to be defined for all possible combined state-network configurations, for example,the initial configuration on the left in the cutout in Figure 4-31 leads to the modifiedsituation in the right part of the figure:

—The state of the cell is changed fromAtoB

—Some network connections are deleted while others are created, thereby changing thelocal network layout but also redefining the neighborhood for the subsequent CNCAstep

It is important to note that, depending on the type of network reconfiguration, in some casesthe state transformation cannot be performed on all cells in parallel, as conflicts may appearwhen the local network is changed simultaneously for neighboring cells This topic needs moreresearch on the mathematical method level, but a practical approach is to either construct a statetransformation function which makes sure such conflicts cannot occur, or either not to performstate transformations on all cells in parallel

The crucial question now is, Can such CNCA be useful to model microstructure evolution,

or are they only useful in the search for a grand unified theory [Wol02]?

4.9.2 CNCA for Microstructure Evolution Modeling

Different network-like structures can be observed in microstructures:

• Microstructures are grains connected by grain boundaries.

• A 2D microstructure is a collection of triple junctions connected by grain boundaries

• Possibly, a 3D microstructure is representable as a collection of quadruple and triple

junc-tions connected by grain boundaries

• Some microstructures of composite or multiphase materials clearly show networks in theirstructure.

Taking the first example in the list, here is a plausible way to model grain growth using CNCA:

• Give all the grains in the microstructure a unique identification number

• The state of a cell can be any of these identification numbers, and represents one wholegrain in the microstructure

• Attribute to a cell’s state is its crystallographic orientation Additional attributes can be thevolume of the grain, its average chemical composition, temperature, dislocation density,phase, etc

• The network configuration is such that each grain cell is connected to the grain cells withwhich it is in contact through a grain boundary surface (one could add triple junctions)

An example of such a model is given in Figure 4-32

It is without question that the usefulness of CNCA for the analysis and modeling of ing microstructures still has to be investigated But as a last act in this chapter about cellularautomata, I would like to put forward the vision that:

evolv-1 It is possible to analyze the evolution of complex 3D microstructures using CNCA, and

that much more detail is to be found about the dependence of microstructure evolution onthe local neighborhood of grains, which are, after all, the building blocks of microstruc-tures

2 Furthermore, that using the results of such analysis, at least in some simplified cases, it is

possible to predict the evolution of such microstructures.

?

10 mm

FIGURE 4-32 Combined network cellular automata applied to a simple microstructure of aluminum

oxide The inset shows a more complex microstructure of a rolled duplex steel, where determining the network is also much more complex.

List of Symbols

η neighborhood

f state transformation function

γ grain boundary energy

k Boltzmann constant (= 1.38065812 e−23J/K)

κ curvature

lCA one-dimensional length of a cell

m grain boundary mobility

m0 temperature independent prefactor of grain boundary mobility

R grain boundary radius

r η radius of the neighborhood

r c equivalent radius of a cell

T temperature

t time

Υ global state function of a cellular automaton

v grain boundary velocity

ξ i state of a cell

4.10 Further Reading

Cellular Automata in General

Although the book A New Kind of Science by Stephen Wolfram [Wol02] is quite controversial,

could have been written without the first 200 pages, and is written in a new kind of style, whichone may find irritatingly arrogant, I consider Stephen Wolfram one of the brightest pioneers ofcellular automata and his book teaches a lot about the method If you find yourself looking forreferences to other works in that book, there are none! Stephen Wolfram tells you to go look forthose on your own on page 850 Pity

More Advanced Literature on Cellular Automata and Microstructure

Evolution

After reading this chapter, which really only covers the very basics, you are ready to become anexpert Here is a list of books that may be a help in doing so:

• D Raabe Computational Materials Science Wiley, New York, 1998.

• D Raabe, F Roters, F Barlat, and L.-Q Chen (ed.) Continuum Scale Simulation of

Engi-neering Materials Wiley-VCH, New York, 2004.

• B Chopard and M Droz Cellular Automata Modeling of Physical Systems Cambridge

University Press, Cambridge, 1998 This book also gives an overview of the history ofcellular automata, but quickly focuses its attention on LGCA-type cellular automata