Evolution of pattern complexity in the Cahn–Hilliard theory of ... Acta Met...The ﬁrst model for...

Acta Materialia 53 (2005) 693–704

www.actamat-journals.com

Evolution of pattern complexity in the Cahn–Hilliard theory ofphase separation

Marcio Gameiro a, Konstantin Mischaikow b, Thomas Wanner c,*

a School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USAb Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, Atlanta, GA 30332, USA

c Department of Mathematical Sciences, George Mason University, 4400 University Drive, MS 3F2, Fairfax, VA 22030, USA

Received 21 September 2004; accepted 14 October 2004

Available online 11 November 2004

Abstract

Phase separation processes in compound materials can produce intriguing and complicated patterns. Yet, characterizing the

geometry of these patterns quantitatively can be quite challenging. In this paper we propose the use of computational algebraic

topology to obtain such a characterization. Our method is illustrated for the complex microstructures observed during spinodal

decomposition and early coarsening in both the deterministic Cahn–Hilliard theory, as well as in the stochastic Cahn–Hilliard–Cook

model. While both models produce microstructures that are qualitatively similar to the ones observed experimentally, our topolog-

ical characterization points to significant differences. One particular aspect of our method is its ability to quantify boundary effects in

finite size systems.

� 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Dynamic phenomena; Microstructure; Phase field models; Spinodal decomposition; Coarsening

1. Introduction

The kinetics of phase separation in alloys has drawn

considerable interest in recent years. Most alloys of

commercial interest owe their properties to specific

microstructures which are generated through special

processing techniques, such as phase separation mecha-

nisms. In quenched binary alloys, for example, one typ-

ically observes phase separation due to a nucleation andgrowth process, or alternatively, due to spinodal decom-

position [4,7,8]. While the former process involves a

thermally activated nucleation step, spinodal decompo-

sition can be observed if the alloy is quenched into the

unstable region of the phase diagram. The resulting

1359-6454/$30.00 � 2004 Acta Materialia Inc. Published by Elsevier Ltd. Adoi:10.1016/j.actamat.2004.10.022

* Corresponding author. Tel.: +1 703 993 1472; fax: +1 703 993

1491.

E-mail address: [email protected] (T. Wanner).

inherent instability leads to composition fluctuations,and thus to instantaneous phase separation. Common

to both mechanisms is the fact that the generated micro-

structures usually are thermodynamically unstable and

will change in the course of time – thereby affecting

the material properties. In order to describe these and

other processes, one generally relies on models given

by nonlinear evolution equations. Many of these models

are phenomenological in nature, and it is therefore offundamental interest to study how well the equations

agree with experimental observations.

In this paper, we propose a comparison method

based on the geometric properties of the microstruc-

tures. The method will be illustrated for the microstruc-

tures generated during spinodal decomposition. These

structures are fine-grained and snake-like, as shown

for example in Fig. 1. The microstructures are computedusing two different evolution equations which have been

proposed as models for spinodal decomposition: The

ll rights reserved.

mailto:[email protected]

Fig. 1. Microstructures obtained from the Cahn–Hilliard–Cook model (4). The top row shows solution snapshots for the deterministic case r = 0,the bottom row is for noise intensity r = 0.01. In both cases we used c1/2 = 0.005 and m = 0. The set X+(t) defined in (5) is shown in dark blue. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

694 M. Gameiro et al. / Acta Materialia 53 (2005) 693–704

seminal Cahn–Hilliard model, as well as its stochastic

extension due to Cook.

The first model for spinodal decomposition in binary

alloys is due to Cahn and Hilliard [4,7]; see also the sur-veys [6,16]. Their mean field approach leads to a nonlin-

ear evolution equation for the relative concentration

difference u = .A � .B, where .A and .B denote the rel-ative concentrations of the two components, i.e.,

.A + .B = 1. The Ginzburg–Landau free energy is givenby

EcðuÞ ¼ZX

WðuÞ þ c2jruj2

� �dx; ð1Þ

where X is a bounded domain, and the positive param-eter c is related to the root mean square effective interac-tion distance. The bulk free energy, W, is a double wellpotential, typically

WðuÞ ¼ 14u2 � 1� �2

: ð2Þ

Taking the variational derivative dEc/du of the Ginz-burg–Landau free energy (1) with respect to the concen-tration variable u, we obtain the chemical potential

l ¼ �cDuþ oWou

ðuÞ;

and thus the Cahn–Hilliard equation ou/ot = Dl, i.e.,

ouot

¼ �D cDu� oWou

ðuÞ� �

; ð3Þ

subject to no-flux boundary conditions for both l and u.Due to these boundary conditions, any mass flux

through the boundary is prohibited, and therefore mass

is conserved. We generally consider initial conditions for

(3) which are small-amplitude random perturbations of

a spatially homogeneous state, i.e., u(0,x) = m + e(x),where e denotes a small-amplitude perturbation with to-tal mass �Xe = 0. In order to observe spinodal decompo-sition, the initial mass m has to satisfy (o2W/ou2)(m) < 0.

One drawback of the deterministic partial differential

equation (3) is that it completely ignores thermal fluctu-

ations. To remedy this, Cook [10] extended the model by

adding a random fluctuation term n, i.e., he consideredthe stochastic Cahn–Hilliard–Cook model

ouot

¼ �D cDu� oWou

ðuÞ� �

þ r � n; ð4Þ

where

hnðt; xÞi ¼ 0; hnðt1; x1Þnðt2; x2Þi ¼ dðt1 � t2Þqðx1 � x2Þ:

Here, r > 0 is a measure for the intensity of the fluctua-tion and q describes the spatial correlation of the noise.

In other words, the noise is uncorrelated in time. For the

special case q = d we obtain space-time white noise, butin general the noise will exhibit spatial correlations. See

also [23].

Both the deterministic and the stochastic model pro-

duce patterns which are qualitatively similar to themicrostructures observed during spinodal decomposi-

tion [5]. Recent mathematical results for (3) and (4) have

identified the observed microstructures as certain ran-

dom superpositions of eigenfunctions of the Laplacian,

and were able to explain the dynamics of the decompo-

sition process in more detail [3,27,28,33,34,38].

Even though both models are based on deep physi-

cal insight, they are phenomenological models. Many

M. Gameiro et al. / Acta Materialia 53 (2005) 693–704 695

researchers have therefore studied how well the models

agree with experimental observations. See for example

[2,13,17,30], as well as the references therein. Most of

these studies have been restricted to testing scaling

laws. Exceptions include for example Ujihara and

Osamura [37], who perform a quantitative evaluationof the kinetics by analyzing the temporal evolution of

the scattering intensity of small angle neuron scattering

in Fe–Cr alloys. Another approch was pursued by

Hyde et al. [18], who consider spinodal decomposition

in Fe–Cr alloys. Using experimental data obtained by

an atom probe field ion microscopy PoSAP analysis,

they study whether the observed microstructures are

topologically equivalent to the structures generated byseveral model equations.

Two structures are topologically equivalent, if one

can be deformed into the other without cutting or glu-

ing. While it is difficult in general to determine when

two structures are equivalent, one can easily obtain a

measure for the degree of similarity by studying topo-

logical invariants. These are objects (such as numbers,

or other algebraic objects) assigned to each structure,which remain unchanged under deformations. In other

words, if these invariants differ for two structures, the

structures cannot be topologically equivalent. Yet, sim-

ilarities between the invariants indicate similarities be-

tween the considered structures.

The topological invariant used in [18] is the number

of handles in the microstructure occupied by one of

the two material phases, which is introduced as a char-acteristic measure for the topological complexity of per-

colated structures. Since moving a dislocation through a

sponge-like interconnected microstructure generally re-

quires cutting through the handles of the structure, the

handle density can be viewed as an analogue of the par-

ticle density for systems containing isolated particles.

From a theoretical point of view, the number of handles

in a microstructure is a special case of topological invar-iants called Betti numbers. In this paper, we demon-

strate how the information contained in the Betti

numbers can be used to further quantify the geometry

of complex microstructures. We rely on recent progress

in computational homology which makes efficient com-

putation of the Betti numbers possible. Specifically, we

use the recently released public-domain software pack-

age CHomP [20,21].Our results are illustrated for the morphologies gen-

erated by the deterministic Cahn–Hilliard model (3)

and the stochastic model (4). We show that knowledge

of the Betti numbers can detect significant differences

in the dynamic behavior of these two models. By com-

bining the information provided by different Betti num-

bers, we are able to quantify boundary effects in finite

size systems. In addition, our method can be used to de-tect fundamental changes in the microstructure topology

due to parameter variation, such as the transition from

percolated structures to isolated particles. Many struc-

ture–property relationships in materials science are

based on fundamental assumptions concerning the

underlying microstructure, and our topological analysis

can aid in unveiling the underlying topology. To sim-

plify the presentation, we only consider two-dimensionalmicrostructures generated by spinodal decomposition.

Nevertheless, the presented software works in arbitrary

dimensions. Aside from the study of 3D microstruc-

tures, this also allows for the analysis of spatio-temporal

structure changes, by including time as a fourth dimen-

sion [12].

2. Homology and Betti numbers

As is indicated in the introduction we claim that

homology provides a useful technique for identifying

and distinguishing the evolving microstructures of (3)

and (4). While this paper is not the appropriate forum

for an in-depth description of algebraic topology and

homology, four points do need to be discussed: (i) Whatstructures do we want to understand the geometry of,

(ii) what geometric features does homology measure,

(iii) how is the homology being computed, and (iv) what

additional information does homology provide when

compared to existing methods of microstructure

analysis?

Since we are interested in phase separation and u(t,x)

represents the relative concentration difference betweenthe two materials at time t and location x, the simplest

decomposition of the domain X consists of the two sets

XþðtÞ :¼ x 2 Xjuðt; xÞ > mf g andX�ðtÞ :¼ x 2 Xjuðt; xÞ < mf g; ð5Þ

where m denotes a suitable threshold value, such as thetotal mass. In this situation, the sets X±(t) represent the

regions in the material where one or the other element

dominates. Of course, u is the solution of a (stochastic)

nonlinear partial differential equation and hence we can-

not expect to have an explicit representation of the sets

X±. For the simulations in this paper, we use a finite dif-

ference scheme to numerically solve (3) and (4) and thus

given the domain X = (0,1) · (0,1) with a spatial gridsize N · N we obtain values Uðtk; x1‘ ; x2nÞ where xj =1/(2N) + (j � 1)/N. We give a geometric interpretationto this numerical grid via the squares Q‘, n :¼ [(‘ � 1)/N, ‘/N] · [(n � 1)/N,n/N]. In particular we define a cubi-cal approximation to X±(t) by

UþðtkÞ :¼ Q‘; njUðtk; x1‘ ; x2nÞ > m� �

and

UþðtkÞ :¼ Q‘; njUðtk; x1‘ ; x2nÞ < m� �

:

Fig. 1 indicates sets U±(tk) at various times steps for a

particular solution to (3) and (4). It is easy to observe

that patterns produced by U±(tk) are complicated, time


dependent, and appear at intermediate time steps to

differ qualitatively for the deterministic and stochastic

models. Before moving on to describe how homology

can be used to quantify these observations, notice that

if the computations had been performed in a three-

dimensional domain, then the same approach could beused except that the two-dimensional squares Q‘, nwould be replaced by three-dimensional cubes.

This leads us to the question of what geometric prop-

erties homology can measure. We begin with the fact

that for any topological space X there exist homology

groups Hi(X), i = 0,1,2, . . . (see [20] and the referencestherein). For a general space Hi(X) takes the form of

an arbitrary abelian group. However, in the context ofour investigations there are two simplifying conditions.

The first is that from the point of view of materials sci-

ence we are only interested in structures that can occur

in three-dimensional Euclidean space. The second is that

the spaces we compute the homology of are represented

in terms of a finite number of cubes (squares if we re-

strict our attention to a two-dimensional model as is

being done in this paper). In this setting the homologygroups are much simpler; first, Hi(X) = 0 for i P 3 andsecond, for i = 0,1,2,

HiðX Þ ffi Zbi ;where Z denotes the integers and bi is a non-negativeinteger called the ith Betti number. Thus, for our pur-

poses the homology groups are characterized by theirBetti numbers.

Each of the homology groups measures a different

geometric property. H0(X) counts the number of con-

nected components (pieces) of the space X. More pre-

cisely, if b0 = k, then X has exactly k components. Aspecific example of this is shown in the left diagram in

Fig. 2, where b0 = 26 corresponding to the 26 differentconnected components. Observe that the size or the

Fig. 2. Betti numbers for the darker region of the r = 0 and t = 0.0036 snadifferent colors, the right diagram indicates the location of the b1 = 4 loops.reader is referred to the web version of this article.)

shape of the components does not play a role in the va-

lue of b0. Depending on the application this can be ta-ken as a strength or weakness of this approach. For

the problems being discussed in this paper we see it as

a strength, since we have no a priori knowledge of the

specific geometry of the material microstructures.H1(X), or more precisely b1, provides a measure of

the number of tunnels in the structure, though the cor-

respondence is slightly more complicated. In a two-di-

mensional domain, such as that indicated in Fig. 2,

tunnels are reduced to loops. In particular, as is shown

in the right diagram of Fig. 2, the green structure en-

closes exactly four regions indicated by the black curves

and hence b1 = 4. Notice that each of the remainingwhite regions hits the boundary. As before, size or shape

plays no role in the definition of a loop.

For a three-dimensional structure loops become tun-

nels, where again the length or width of the tunnel is irrel-

evant. For example a washer has one wide but very short

tunnel while a garden hose has a narrow but long tunnel.

In either case b1 = 1. As mentioned earlier we are onlyconsidering a two-dimensional domain in which caseH2(X) = 0. However, for a general three-dimensional

domain b2 equals the number of enclosed volumes orcavities within the material.

From a computational point of view, determining the

Betti numbers of a complex 3D structure is far from triv-

ial. In fact, it was pointed out explicitly in [18] that ‘‘de-

signing a computer program to directly count the handle

density in a complex structure would be extremely diffi-cult [18, p. 3419]’’. The authors therefore compute the

number of handles indirectly, using techniques from dig-

ital topology [22]. These techniques relate the number of

handles to the Euler characteristic of the structure,

which can be computed easily in three space dimensions.

Our study is made possible by recent progress in compu-

tational homology, which does allow for the direct com-

pshot from Fig. 1. The left diagram shows the b0 = 26 components in(For interpretation of the references to color in this figure legend, the


putation of the Betti numbers. As we already indicated

in the introduction, we use the software package

CHomP [21]. An elementary introduction to homology

and the underlying algorithms can be found in [20]. Typ-

ical computation times of the software in our situation

are described in the following section.To close this section, we would like to place our

methodology in the proper context and comment on

its merits. Characterizing material properties is an old

subject, and numerous tools have been devised over

the course of time. Many of these tools use spatial aver-

aging, such as the structure factor or point correlation

functions. While these quantities do provide informa-

tion on predominant wavelengths and length scales,the averaging process used in their definition removes

much of the local connectivity information of the micro-

structure. This fact is illustrated in [36], where several

different correlation functions are used to reconstruct

microstructures.

In order to obtain quantitative connectivity informa-

tion on microstructures, several authors have therefore

employed topological methods. See for example[1,9,15,18,19,29], as well as the references therein.

Due to the computational issues mentioned above,

the easily computable Euler characteristic takes a pre-

dominant role in these studies. Yet, the Euler charac-

teristic, which is the alternating sum of the Betti

numbers, provides considerably less topological infor-

mation than the complete set of Betti numbers. This

is true even in the two-dimensional case considered inthis paper. In Section 5 it will be shown that the Euler

characteristic is closely related to boundary effects,

whereas the Betti numbers also contain information

on the bulk structure.

3. Effects of noise on the pattern morphology

In this section we use computational homology to

study the effects of the stochastic forcing term in (4)

on the temporal evolution of the pattern complexity.

We consider the case of total mass m = 0 and W as in(2), and to begin with restrict ourselves to c1/2 =0.005. The simulations are performed on the unit

square X = (0,1) · (0,1). The deterministic part of theevolution equation (4) is approximated using a finitedifference scheme with linearly implicit time-stepping,

similar to schemes described in [14,31], for the stochas-

tic noise term we follow [35]. This numerical scheme is

implemented efficiently using the fast Fourier trans-

form. For the simulations in this paper we used an

implementation in C in combination with the fast Fou-

rier transform package FFTW [11]. As for the noise

term, we consider the case of cut-off noise which guar-antees a spatial correlation function q which closely

approximates d.

For varying values of the noise intensity r, we numer-ically integrate the Cahn–Hilliard–Cook equation (4)

starting from a random perturbation of the homogene-

ous state m = 0 with amplitude 0.0001 up to time

tend ¼160c

W00ðmÞ2; where W00ðmÞ ¼ 3m2 � 1; ð6Þ

which in our situation reduces to tend = 0.004. This time

frame covers the complete spinodal decompositionphase, as well as early coarsening. The domain is discre-

tized by a 512 · 512-grid, the time interval [0, tend] is cov-ered by 10,000 integration steps. Every 50 time steps, the

sets X±(t) introduced in the last section are determined,

and finally their Betti numbers are computed using [21].

On a 2 GHz Dual Xeon Linux PC it took about 15 min

to create the 402 sets X±(t) for t/tend = 0, 0.005,

0.010, . . ., 0.995, 1, as well as a total of 46 min to com-pute their Betti Numbers.

Fig. 3 contains the results of these computations for

two different values of the noise intensity. The solid

red curves show the Betti number evolution for the

deterministic case r = 0, the dashed blue curves are fornoise intensity r = 0.01; the solution snapshots in Fig. 1are taken from these two simulations. At first glance,

the graphs in Fig. 3 are not surprising. For the initialtime t = 0 both b0 and b1 are large, since the initial per-turbation was chosen randomly on the computational

grid. In fact, the actual values lie well outside the dis-

played range. The smoothing effect of (4) leads to a ra-

pid decrease of the Betti numbers for times t close to 0,

and coarsening behavior in the Cahn–Hilliard–Cook

model is responsible for the decrease observed towards

the end of the time window. All shown evolutions exhi-bit small-scale fluctuations, which are most likely arti-

facts caused by the cubical approximation of the sets

X±(t). Even though these fluctuations are not necessarily

desirable, they are a more or less automatic consequence

of the numerical approximation of the partial differen-

tial equation (4).

Despite these similarities, there are some obvious dif-

ferences between the deterministic and the stochasticevolution. In the deterministic situation, the initial com-

plexity decay occurs sooner than in the stochastic case.

In addition, it appears that the Betti numbers of X+(t)

for r = 0.01 decay more or less monotonically, providedwe ignore the above-mentioned small-scale fluctuations.

In contrast, in the deterministic case the initial decrease

seems to be followed by a period of stagnation or even

growth of the Betti numbers, see for example b0 forX+(t) or b1 for X

�(t).

The observations made in the last paragraph could

indicate fundamental differences between the determinis-

tic and the stochastic Cahn–Hilliard model. However,

these observations are based on a single randomly cho-

sen initial condition, and it is therefore far from clear

Fig. 3. Evolution of the Betti numbers for the solutions of Fig. 1. In each diagram, the solid red curve corresponds to r = 0, the dashed blue curve isfor r = 0.01. In both cases we used c1/2 = 0.005 and m = 0. The diagrams in the left column show the results for X+(t), the right column is for X�(t).The top row contains the evolutions of b0, the bottom row shows b1. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)


whether they represent behavior typical for either of the

models. For this we have to observe ensembles of solu-

tions for each of the models and study the statistics oftheir complexity evolutions. For the purposes of this pa-

per, we concentrate on the averaged complexity evolu-

tion. Fig. 4 shows the averaged Betti number evolution

Fig. 4. Evolution of Betti number averages for a sample size of 100, with ccorrespond to the values r = 0.1, 0.03, 0.01, 0.003, 0.001, and 0, from left to

for six values of the noise intensity r ranging from 0to 0.1, in each case based on solution ensembles of size

100. The qualitative form of the complexity evolutioncurves differs substantially. For large noise the complex-

ity decays monotonically, while it shows a surprising in-

crease in the deterministic situation. The change

1/2 = 0.005 and m = 0. In the top row at dimension level 60 the curves

right; analogously in the bottom row at level 25.


between the two behaviors occurs gradually, and there

seems to be a specific threshold rcr for the noise intensitybeyond which monotone decay is observed. Notice also

that despite the small ensemble size, the evolution curves

of X+(t) and X�(t) are in good agreement, which for our

choice of m = 0 and W as in (2) has to be expected. Inthis sense, the ensemble behavior is a reflection of typical

solution dynamics, reinforcing our above observations.

The simulations discussed so far consider the specific

value c1/2 = 0.005. We performed analogous simulationsfor various values of c, for various grid sizes, and a vari-ety of time steps. In each case, the time window ex-

tended from 0 to tend as defined in (6). These results

show that the behavior shown in Fig. 4 is typical. Inall cases, the time window covers the complete spinodal

decomposition phase, as well as early coarsening; the

resulting evolution curves for the deterministic case

show the characteristic non-monotone behavior; for suf-

ficiently large noise intensity we observe monotone de-

cay. The only parameters changing with c appear to bethe absolute height of the evolution curves and (possi-

bly) the critical noise intensity rcr. While these scalingswill be addressed in more detail in Section 5, we close

this section with a brief discussion of the dependence

of rcr on c.To get a more accurate picture of the monotonicity

properties of the evolution curves we compute the aver-

aged Betti number evolution from larger solution

ensembles and for times up to only tend/4. As before, tendis given by (6), and we consider the case m = 0. The noiseintensity r is chosen equal to c1/2, and the ensemble sizeis taken as 1000. The results of these simulations for

c1/2 = 0.005, 0.006, 0.007, and 0.01 are shown in Fig. 5.Notice that each of these curves exhibits monotone de-

cay of the Betti numbers with a pronounced plateau,

so one would expect that these curves are for values

r � rcr. Yet, it appears that the length of the plateau de-creases with increasing c. This latter observation could

0 0.05 0.1 0.15 0.2 0.250

10

20

30

40

50

60

70

80

t/tend

1

1

2

2

3

3

4

4

b0

Fig. 5. Betti number averages for X+(t) with varying c1/2 = r and m = 0. Fromand 0.01, respectively. The sample size for each of these simulations is 1000

indicate that for the larger c-values the noise intensitiesused in Fig. 5 are too large. In fact, our scaling analysis

in Section 5 will show that this is the case.

The results of this section demonstrate that Betti

numbers can be used to quantitatively distinguish be-

tween microstructure morphologies generated by differ-ent models. Ultimately we hope that this quantitative

information can be used to match models to actual

experimental data. In fact, the limited experimental data

in [18] seems to indicate that the handle density of the

microstructures generated through spinodal decomposi-

tion decreases monotonically. Thus, these experimental

results favor the stochastic Cahn–Hilliard–Cook model

with sufficiently large noise intensity.

4. Morphology changes due to mass variation

So far we restricted our study to the case of equal

mass, i.e., we assumed m = 0. Yet, spinodal decomposi-

tion can be observed as long asW00(m) < 0, which for ourchoice of W is equivalent to |m| < 3�1/2 � 0.577. Totalmass values outside this range lead to nucleation and

growth behavior which produces microstructures con-

sisting of isolated droplets – in contrast to the micro-

structures shown in Fig. 1. In this section we use

computational homology to quantify the pattern mor-

phology changes during spinodal decomposition in the

deterministic Cahn–Hilliard model (3) as the total mass

is increased from m = 0 towards m � 3�1/2.Before presenting our numerical results, we have to

address one technical issue. Even though the homogene-

ous state m is unstable as long asW00(m) < 0, the strengthof the instability changes with m. One can easily show

that the growth rate of the most unstable perturbation of

the homogeneous state is close to W00(m)2/(4c) [27]. Thus,as the total mass approaches the boundary of the spi-

nodal region, the time frame for spinodal decomposition

0 0.05 0.1 0.15 0.2 0.250

5

0

5

0

5

0

5

0

5

t/tend

b1

top to bottom the curves correspond to c1/2 = r = 0.005, 0.006, 0.007,.


grows. In order to compare the pattern morphology for

different values of the total mass we therefore scale the

considered time window. As in the last section, we com-

pute the solutions up to time tend defined in (6), whose

scaling is motivated by largest growth rate mentioned

above.How do changes in the total mass m affect the micro-

structures generated through spinodal decomposition?

Fig. 6 contains typical patterns for m = 0, 0.1, . . ., 0.5.In each case, the pattern was generated by solving (3)

up to time t = 0.4 Æ tend, with tend as in (6). Notice thateven though all of these microstructures are a conse-

quence of phase separation through spinodal decompo-

sition, the last two microstructures resemble the onesgenerated by nucleation and growth. In other words,

Fig. 6 indicates a gradual change from the highly inter-

connected structures observed in the equal mass case to

the disconnected structures observed in nucleation.

In order to further quantify this gradual change, we

use computational homology as in the last section. For

c1/2 = 0.005 and various values of m between 0 and0.55 we computed the averaged Betti number evolutionof the sets X±(t) in (5) from t = 0 up to time t = tend. The

resulting two-dimensional surfaces are shown in Fig. 7.

To avoid the large Betti numbers close to the random

initial state, these graphs do not include the times t = 0

and t = 0.005 Æ tend. Of particular interest is the topologyof the sets X+(t) which correspond to the dominant

phase. The graphs in the left column of Fig. 7 show that

in terms of the quantitative topological information gi-ven by the averaged Betti numbers, one observes a grad-

ual and continuous change from the interconnected

Fig. 6. Microstructures obtained from the Cahn–Hilliard model (3) for c1/2 =are shown for total mass m = 0, 0.1, . . ., 0.5 at time t = 0.4 Æ tend, i.e., shortlyshown in dark blue. (For interpretation of the references to color in this fig

microstructures for m = 0 to the nucleation morphology.

In addition, the graph of b0 for X+(t) indicates that for

t/tend > 0.25, i.e., after the completion of spinodal decom-

position, and for m > 0.2 the dominant phase forms a

connected structure; that is, the complementary phase

breaks up completely. Yet, the typical droplet structurereminiscent of nucleation and growth behavior can only

be observed for values of the total mass larger than 0.3.

In the surfaces of Fig. 7 this is reflected by the fact that

for fixed time t, the Betti number b1 of the dominantphase X+(t) continues to increase with m until m � 0.3.As the total mass approaches 3�1/2, the first Betti num-

ber then decreases again, corresponding to an increasing

droplet size.In addition to the results shown in Fig. 7, we per-

formed analogous simulations for c1/2 = 0.0025, 0.0075,and 0.0100. In each case, the qualitative shape of the

computed surfaces matched the one for c1/2 = 0.005,only the absolute height of the surfaces changed. In fact,

our computations indicate that the units on the vertical

axes of Fig. 7 scale proportionally to c�1 for these valuesof c.

The computations of this section indicate how

homology can be used to quantify global morphological

changes, such as connectivity, due to mass variation. On

a more general level, given a model of a particular mate-

rial it would be interesting to seek correlations between

macroscopic properties and the type of geometric infor-

mation expressed in Fig. 7. Such studies would be in the

spirit of structure–property relationships which havebeen obtained for certain types of microstructures, such

as for systems of isolated particles using particle distri-

0.005 for varying m. From top left to bottom right solution snapshots

after completion of the spinodal decomposition phase. The set X+(t) is

ure legend, the reader is referred to the web version of this article.)

Fig. 7. Evolution of Betti number averages for a sample size of 100, with c1/2 = 0.005 and varying mass m. The left column corresponds to X+(t), theright one to X�(t); the top row contains the graphs for the 0th Betti number, the bottom row for the first.


butions, or for interconnected structures [24,25] using

the notion of matricity. Comparisons of this nature

would probably be of even greater importance in the

case of three-dimensional simulations, since visualiza-

tion of these phenomena would be extremely

complicated.

5. Boundary effects and scalings

While our study so far concentrated on the Betti

numbers b0 and b1, significant additional informationcan be obtained by combining the topological informa-

tion for X+(t) with the information for the complemen-

tary sets X�(t). To illustrate this, consider again the

microstructure in the left diagram of Fig. 2 consisting

of 26 components. Most of these components are intro-

duced through boundary effects – only 6 components areinternal. Even though the Betti number b0 of X

+(t) does

not distinguish between these two types of components,

only internal components of X+(t) introduce loops in

X�(t). In fact, if we denote the number of internal com-

ponents of X+(t) by bint, 0(X+(t)) and the number of com-

ponents touching the boundary of X by bbdy, 0(X+(t)),

then we have

bint; 0ðXþðtÞÞ ¼ b1ðX�ðtÞÞ andbbdy; 0ðXþðtÞÞ ¼ b0ðXþðtÞÞ � b1ðX�ðtÞÞ:

For c1/2 = 0.005 and m = r = 0 the averaged temporalevolution of these quantities is shown in the left diagram

of Fig. 8, for a sample size of 100. While the number of

internal components shows the typical non-monotone

behavior described in the last sections, the number of

boundary components remains basically constant, wellinto the coarsening regime. Notice also that due to

m = 0 and our choice of W, the ensemble averages ofb1(X

+(t)) and b1(X�(t)) have to be equal. Thus, the num-

ber bbdy, 0(X+(t)) of boundary components equals in fact

the Euler characteristic of the set X+(t). Similarly, the

right diagram of Fig. 8 shows the evolutions of bint, 0and bbdy, 0 for the corresponding stochastic case withr = 0.01. While the number of internal componentsexhibits the monotone decay described in Section 3,

the number of boundary components reaches the same

level as in the deterministic situation, but starts its decay

earlier.

In view of Fig. 8, one can rightfully question the con-

clusions obtained so far. During the spinodal decompo-

sition regime, bint, 0 and bbdy, 0 are comparable, furtherinto the coarsening regime the number of boundarycomponents clearly dominates. In order to validate

our results of the last two sections, it is therefore neces-

sary to consider various system sizes. If instead of the

unit square we consider the domain X = (0, ‘) · (0, ‘),then a combination of the spatial rescaling x ´ x/‘and the temporal rescaling t´ t‘2 transforms both (3)

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

t/tend

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

t/tend

Fig. 8. Averaged temporal evolution of the number bint, 0 of internal components of the set X+(t), and of the number bbdy, 0 of components of X

+(t)

touching the boundary of the base domain X. In each diagram, the solid red curve corresponds to bint, 0, the dashed blue curve is for bbdy, 0. The leftdiagram is for the deterministic case r = 0, the right one for the stochastic case with noise intensity r = 0.01. In both cases we used c1/2 = 0.005 andm = 0. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


and (4) into equations on the unit square, but with new

interaction parameter c/‘2. In the stochastic case (4), therescaling leaves the noise intensity r unchanged. In otherwords, we can consider larger system sizes by consider-

ing smaller values of c.The results of such a system size scaling analysis are

shown in Fig. 9, where we consider both the determin-

istic case r = 0 and the stochastic case with r = 0.01 for

0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

t/tend

0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

t/tend

σ = 0

σ = 0.01

b int, 0

Fig. 9. Evolution of internal and boundary component averages of X+(t) for

case r = 0, the bottom row is for r = 0.01. The left column is for the numberbint, 0 Æ c/c*, with c

1=2� ¼ 0:0015; from top to bottom in each of the diagrams

column is for the number of components touching the boundary and shows th

of c1/2.

m = 0 and a variety of c-values. The resulting evolutioncurves for bint, 0 have been multiplied by c/c*, withc1=2� ¼ 0:0015, which is the smallest considered valueof the interaction parameter. Thus, the left column of

Fig. 9 indicates a scaling of bint, 0 proportional toc�1, i.e., proportional to the area of the underlying do-main. For the number of boundary components shown

in the right column of Fig. 9, we used the scaling factor

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

160

t/tend

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

160

t/tend

bbdy, 0

varying c and a sample size of 100, with m = 0. The top row shows theof internal components and shows the evolutions of the scaled quantity

the curves are for c1/2 = 0.0015, 0.0025, 0.005, 0.0075, 0.01. The righte evolutions of the scaled quantity bbdy, 0 Æ (c/c*)

1/2, for the same values


(c/c*)1/2. The diagram therefore indicates that bbdy, 0

scales proportional to c�1/2, i.e., proportional to thelength of the boundary of the domain. Due to the spe-

cific form of the involved scaling factors, the absolute

numbers given on the vertical axes in Fig. 9 are the

correct component numbers for c1/2 = 0.0015. In thissituation, the number of internal components clearly

dominates the number of boundary components

throughout spinodal decomposition and early coarsen-

ing. Notice also that the results for the stochastic case

indicate that the critical noise intensity rcr discussed atthe end of Section 3 appears to be independent of the

system size. This explains the findings of Fig. 5.

While the scaling in the left column of Fig. 9 showsmore deviation than the one in the right column, these

deviations are a consequence of the small system sizes

for large values of c. Yet, we find it remarkable thatthe non-monotone evolution of bint, 0 can be observedeven for c1/2 = 0.01, despite the fact that in this casewe have bint, 0 � 0.5bbdy, 0 during spinodal decomposi-tion and early coarsening, and that the amplification

factor used to scale the interior component curve is0.012/c* = 44.4. We believe that the accuracy of theabove scalings over the considered c-range indicatestheir validity also for smaller values of c.

The results of this section demonstrate that the infor-

mation provided by the collection of Betti numbers for

X±(t) can be used to derive detailed quantitative state-

ments about the relative sizes of boundary effects in fi-

nite size systems, when compared to internal (or bulk)effects. This is in sharp contrast to what can be extracted

from other (easier computable) topological invariants,

such as the Euler characteristic. In the above situation,

the Euler characteristic only describes the number of

boundary components, but can provide no insight into

the number of internal components.

6. Conclusion

In this paper we proposed the use of computational

homology as an effective tool for quantifying and distin-

guishing complicated microstructures. Rather than dis-

cussing experimental data, we considered numerical

simulations of the deterministic Cahn–Hilliard model,

as well as its stochastic extension due to Cook. The ob-tained topological characterizations were used to (a) un-

cover significant differences in the temporal evolution of

the pattern complexity during spinodal decomposition

between the deterministic and the stochastic model, as

well as to (b) establish the existence of a gradual transi-

tion of the spinodal decomposition microstructure mor-

phology as the total mass approaches the boundary of

the spinodal region. Furthermore, by combining differ-ent topological information, we managed to (c) obtain

detailed information on boundary effects for various

system sizes. In order to simplify our presentation, the

results were presented in a two-dimensional setting.

Nevertheless, the computational tools are available for

and effective in arbitrary dimensions.

We believe that the presented methods can serve as

practical tools for assessing the quality of continuummodels for phase separation processes in materials. This

is achieved by providing quantitative topological infor-

mation which can readily uncover differences in models

as in (a). In addition, this quantitative information can

be used to compare the computed microstructure topol-

ogy to experimental observations. For example, the

study in [18] determined the handle density of spinodally

decomposing iron–chromium alloys. This quantity cor-responds to the first Betti number, and the results of

[18] indicate a monotone decay – which in combination

with (a) favors the stochastic Cahn–Hilliard–Cook

model over the deterministic one. Finally, combining

topological information as in (c) makes it possible to

quantify boundary effects, which are present in any finite

size system.

A second possible application is the identification ofmodel parameters based on the microstructure topol-

ogy. For example, our considerations in (b) resulted in

characteristic evolution curves as a function of the

parameter m, as well as in scaling information with re-

spect to c. Combined, these graphs could be used todetermine specific values of these parameters for certain

experimental situations.

It would be interesting to apply the methods pre-sented in this paper to more elaborate models which in-

clude additional effects, such as for example anisotropic

elastic forces [26] or polycrystalline structures [32].

Moreover, we believe that combining our methods with

the matricity concept introduced in [24,25] can further

increase the classification potential of the method.

Acknowledgements

The work of M.G. was partially supported by

CAPES, Brazil; the work of K.M. and T.W. was par-

tially supported by NSF grants DMS-0107396 and

DMS-0406231, respectively. We thank E. Sander for

helpful discussions concerning the presentation of this

material. We also thank the anonymous referee whosesuggestions led to significant improvements.

References

[1] Arns CH, Knackstedt MA, Pinczewski WV, Mecke KR. Phys

Rev E 2001;63(3):031112.

[2] Binder K. In: Arnold L, editor. Stochastic nonlinear sys-

tems. Berlin: Springer; 1981. p. 62–71.

[3] Blömker D, Maier-Paape S, Wanner T. Commun Math Phys

2001;223(3):553–82.


[4] Cahn JW. J Chem Phys 1959;30:1121–4.

[5] Cahn JW. J Chem Phys 1965;42:93–9.

[6] Cahn JW. Trans Metall Soc AIME 1968;242:166–80.

[7] Cahn JW, Hilliard JE. J Chem Phys 1958;28:258–67.

[8] Cahn JW, Hilliard JE. J Chem Phys 1959;31:688–99.

[9] Cerezo A, Hetherington MG, Hyde JM, Miller MK. Scripta

Metall Mater 1991;25:1435–40.

[10] Cook H. Acta Metall 1970;18:297–306.

[11] Frigo M, Johnson SG. Proc ICASSP 1998 1998;3:1381–4.

[12] Gameiro M, Kalies W, Mischaikow K. Phys Rev E 2004;

70(3):035203 .

[13] Garcke H, Niethammer B, Rumpf M, Weikard U. Acta Mater

2003;51(10):2823–30.

[14] Hairer E, Wanner G. Solving ordinary differential equations

II. Berlin: Springer; 1991.

[15] Hilfer R. Transport Porous Med 2002;46(2-3):373–90.

[16] Hilliard JE. In: Aaronson HI, editor. Phase transforma-

tions. Metals Park (OH): American Society for Metals; 1970.

p. 497–560.

[17] Hyde JM, Miller MK, Hetherington MG, Cerezo A, Smith

GDW, Elliott CM. Acta Metall Mater 1995;43:3403–13.

[18] Hyde JM, Miller MK, Hetherington MG, Cerezo A, Smith


[19] Ioannidis MA, Chatzis I. J Colloid Interface Sci

2000;229(2):323–34.

[20] Kaczynski T, Mischaikow K, Mrozek M. Computational homol-

ogy. Applied mathematical sciences, vol. 157. New York: Sprin-

ger; 2004.

[21] Kalies W, Pilarczyk P. Computational homology program.

Available from: http://www.math.gatech.edu/~chom/, 2003.

[22] Kong TY, Rosenfeld A. Comput Vision Graph Image Process

1989;48(3):357–93.

[23] Langer JS. Ann Phys 1971;65:53–86.

[24] Leßle P, Dong M, Schmauder S. Comput Mater Sci

1999;15(4):455–65.

[25] Leßle P, Dong M, Soppa E, Schmauder S. Scripta Mater

1998;38(9):1327–32.

[26] Löchte L, Gitt A, Gottstein G, Hurtado I. Acta Mater

2000;48(11):2969–84.

[27] Maier-Paape S, Wanner T. Commun Math Phys

1998;195(2):435–64.

[28] Maier-Paape S, Wanner T. Arch Ration Mech Anal

2000;151(3):187–219.

[29] Mendoza R, Alkemper J, Voorhees PW. In: Rappaz M, Beck-

ermann C, Trivedi R, editors. Solidification processes and

microstructures: a symposium in honor of Wilfried Kurz. TMS;

2004. p. 123–29.

[30] Miller MK, Hyde JM, Hetherington MG, Cerezo A, Smith


[31] Quarteroni A, Valli A. Numerical approximation of partial

differential equations. Berlin: Springer; 1994.

[32] Ramanarayan H, Abinandanan TA. Acta Mater

2003;51(16):4761–72.

[33] Sander E, Wanner T. J Stat Phys 1999;95(5-6):925–48.

[34] Sander E, Wanner T. SIAM J Appl Math 2000;60(6):

2182–2202.

[35] Shardlow T. Numer Func Anal Opt 1999;20(1–2):121–45.

[36] Torquato S. Annu Rev Mater Res 2002;32:77–111.

[37] Ujihara T, Osamura K. Acta Mater 2000;48(7):1629–37.

[38] Wanner T. Trans Am Math Soc 2004;356(6):2251–79.

http://www.math.gatech.edu/~chom/

Evolution of pattern complexity in the Cahn -- Hilliard theory of phase separationIntroductionHomology and Betti numbersEffects of noise on the pattern morphologyMorphology changes due to mass variationBoundary effects and scalingsConclusionAcknowledgementsReferences

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