- Email: [email protected]
A stochastic grain growth model based on a variational principle for dissipative systems
Physica A 282 (2000) 339–354
www.elsevier.com/locate/physa
A stochastic grain growth model based on a variational principle for dissipative systems Fabrizio Cleria; b; ∗
a Ente
Nazionale Energia e Ambiente, Divisione Materiali Avanzati, Centro Ricerche Casaccia, C.P. 2400, 00100 Roma, Italy b Istituto Nazionale Fisica della Materia, Unit a di Ricerca Roma 1, P.le A. Moro 2, 00185 Roma, Italy Received 16 August 1999; received in revised form 28 December 1999
Abstract A stochastic model for the evolution of a cellular network driven by dissipative forces is presented. The model is based on a variational formulation for the dissipated power, from which we obtain an expression for the transition-rate generating function to be used in kinetic Monte Carlo simulations. The model canonical variables are the positions and velocities of the network vertices where cell walls meet. We apply such a model to the study of grain growth in two dimensions, in which the network represents a cross-section of a polycrystalline microstructure and the cell walls represent grain boundaries. The results of the stochastic grain-growth model for relevant statistical quantities are compared to deterministic model results and analytic theories. c 2000 Elsevier Science B.V. All rights reserved.
PACS: 02.70.Lq; 05.70.Ln; 81.10.Aj Keywords: Grain growth; Variational principles; Dissipative systems; Kinetic Monte Carlo
1. Introduction The structure of many natural systems consists of closed cells with well-de ned boundaries. Examples of such systems include soap froths, ferro uid froths, lipid monolayers, biological tissues, magnetic domains, ame fronts and, possibly, even the largescale structure of the universe. Such cellular structures can display topological disorder in the form of a random connectivity pattern and, in many cases, the structure evolution appears self-similar over some appropriate time scale. In the domain of materials ∗
Correspondence address. Ente Nazionale Energia e Ambiente, Divisione Materiali Avanzati, Centro Ricerche Casaccia, C.P. 2400, 00100 Roma, Italy. Tel.: +39-06-30484825; fax: +39-06-30484729. E-mail address: [email protected] (F. Cleri)
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 0 8 7 - X
340
F. Cleri / Physica A 282 (2000) 339–354
science, the arrangement of grains in a polycrystalline solid has often been modelled as a cellular system [1]. In this latter case the cell walls represent grain boundaries, associated with an excess energy with respect to the bulk monocrystal: grain-boundary migration is driven by the reduction of such an excess energy and, consequently, the grain structure evolves with time in the direction of increasing grain size. From the theoretical point of view the evolution of cellular structures, and in particular grain growth in materials, has been studied by several methods (see the review papers [2– 4] and [5 –11]) often based on computer simulation and in most cases restricted to two dimensions (2D). In the 2D cellular model of a microstructure only the grain boundaries are represented, disregarding the grain interior; grain boundaries are described as generally curved segments connecting the triple junctions, i.e., the points where three grain boundaries meet. In summary, irrespective of the details and approximations for the underlying kinetics, most 2D cellular models appear to give a although the true ext 1=2 scaling behavior at long times for the mean grain radius, R, istence of such a scaling state has long been a subject of debate [2,3]. Such a feature, which is postulated in mean- eld theories [12] and predicted on the basis of simple dimensional arguments [2,3], emerges as a universal dynamic feature of the evolution of any cellular structure connected only through triple junctions. On the other hand, no general agreement has yet been reached on such a basic quantity as the grain-size distribution function, a quantity for which experimental measurements are, furthermore, quite dicult to realize. A conceptually innovative theoretical approach to microstructural evolution was presented some time ago by Needleman and Rice [13], based on a variational principle for dissipative systems. In the case of grain growth Cocks and Gill [14] derived a similar variational functional describing the rate of energy dissipation in a microstructure due to the competition between the reduction of the excess energy and a driving force proportional to the boundary velocity. The variational parameter in such functionals is the continuous grain-boundary velocity eld: by applying D’Alembert’s di erential form of the variational principle (which is, indeed, valid for both conservative and non-conservative dynamical systems [15]), explicit equations for the minimizing velocity eld are obtained, and from these the microstructure evolution. Such an approach stands out as very powerful and exible [16 –18], since any term contributing to energy dissipation in the microstructure (e.g., grain sliding and rotation, matter di usion along grain boundaries, di usion and plastic work in the grain bulk) can be included in the functional, provided a variational principle for each new term can be established. However, the approach is also computationally inecient since it is de ned in terms of a global minimization procedure, which amounts to inverting a large matrix at each time step. In this paper we propose a stochastic model of microstructural evolution based on a di erent interpretation of the Needleman–Rice variational functional, using the well-studied problem of 2D grain growth as a benchmark application. We start from the consideration that, at the microscopic level, in nitesimal grain-boundary displacements are the result of random thermally activated events: the probability of such
F. Cleri / Physica A 282 (2000) 339–354
341
events is proportional to exp(−H=kB T ) with H , the con gurational energy change (i.e., dissipation), playing the role of a transition-rate generating function. Microstructure evolution thus results from a random sequence of local, uncorrelated events, as opposed to the global, non-local solution. We interpret the variational functional itself as the transition-rate generating function for the stochastic model, to be solved numerically by means of a special implementation of the kinetic Monte Carlo (MC) method. A stochastic description of grain growth was rst proposed by Louat [19] and later elaborated by Pande [20], Thorvaldsen [21] and others [2]. By noting that the fundamental results of the classical description of grain growth [12] have only a statistical validity, Louat suggested that the grain growth rate should vary in a random manner, including reversals in the sign. Indeed, in non-ideal systems such as wet soap froth ◦ and lipids it has been observed [22] that the 120 equilibrium relation and the Von Neumann’s law ([23], see below) are veri ed only on average. Indeed, Louat’s work was later criticized [24] and the idea of a stochastic component superimposed to the driving force for grain growth [20,21] was strongly opposed by Mullins [25] on the basis of mean- eld theory arguments. As we will detail in the following, under the hypothesis of locality the random component of the driving force can be interpreted as a structural noise term accounting for correlation e ects beyond the mean- eld solution. In the remainder of this paper we rstly summarize the deterministic variational model; then, we describe the stochastic model of microstructural evolution and its practical implementation as a kinetic MC algorithm. This latter is based on the stochastic migration of the boundaries of a cellular network, which has great advantages in terms of both physical clarity and computational eciency. The merits of the stochastic grain-growth model are discussed by comparing the results of MC simulations with those of the deterministic models [7,11,14]. It is concluded that such a comparison is extremely successful and lends support to further developments.
2. Variational functional for microstructural evolution We consider a 2D system consisting of a fully dense pattern of N grains. Grains are represented as irregular polygons whose initial con guration could be obtained, e.g., by a Voronoi construction. Such a con guration should be physically close to the cross-section of a textured microstructure. The ensemble of the border segments between each pair of grains makes up a set of Ngb grain boundaries with length Li , i = 1; : : : ; Ngb , and excess energy i . Grain boundaries can meet only at triple junctions, i.e., fourfold (or higher) junctions are unstable. Given this topological constraint, the number of triple junctions Ntj is given by Ntj = 2=3Ngb and the number of grain boundaries is, in turn, related to the number of grains as Ngb = 3N . In the lack of a more fundamental theory, a macroscopic model of 2D grain growth [2– 4] describes phenomenologically the competition between two terms:
342
F. Cleri / Physica A 282 (2000) 339–354
PNgb (a) the tendency of the system to reduce its excess interfacial energy, i=1 ( i Li ), by reducing the total grain boundary perimeter; the rate of work done under a virtual variation of the boundary length per unit time, ˙∗s = (L∗i =Li )=t can be written as [14] WT∗
=
Ngb Z X i=1
Li
i ˙∗s (s) ds :
(1)
(b) the lattice resistance to grain boundary migration, described as a viscous drag force with an interface mobility coecient, i ; if a linear kinetic relation is assumed between the driving force and the normal component of the boundary velocity, vn =i f, the rate of energy dissipation by the viscous force under a virtual variation of the velocity, v∗ , is given by [14] WM∗ =
Ngb Z X i=1
Li
fvn∗ (s) ds :
(2)
Following Needleman and Rice [13] and Cocks and Gill [14], the above de nitions for WT∗ and WM∗ are combined into a “virtual power principle” (analogous to the virtual work principle) which allows to identify a variational functional representing the power dissipated during grain-boundary migration. The functional can be written in terms of the velocity eld only, v(s), de ned over the collection Ngb of grain boundaries, with components vn and vs parallel and perpendicular, respectively, to the local grain-boundary normal [v(x)] =
Ngb Z X i=1
Li
Z
i i vn (x) ds +
Li
i
@vs (s) ds + @s
Z Li
vn2 (s) ds ; 2i
(3)
where the ith grain boundary has length 06s6Li , excess energy i , curvature i and mobility i , and the vector eld x collectively denotes the grain-boundary coordinates. The three terms in the variational functional can be described as follows [14]: (a) the rst is the contribution from the grain-boundary curvature: it states that the curved sections of the grain boundaries will move at a rate proportional to both the local curvature and excess energy; (b) the second is the contribution coming from the triple junctions: it states that the triple junctions move according to the net force resulting from the sum of the scalar products of the velocity times the tangent vector at the end-points of each grain boundary; (c) the third is the dissipation by the viscous force opposing the migration perpendicular to the boundary, as discussed above. In a deterministic description the functional [v(x)] is made stationary by a varia˜ tional eld v(x) (which can be proven to be unique for the exact solution of the variational problem [14]), thus resulting in the energy-dissipation minimum for a particular con guration of the network. The instantaneous solution eld is found by imposing that the functional be a global minimum for arbitrary variations of v(x) [v(x)] = 0
for v(x) = v˜(x) :
(4)
F. Cleri / Physica A 282 (2000) 339–354
343
Then, the kinetic evolution from the boundary con guration to is obtained from equations of the type ˜ x( ) = x( ) + v(x) t :
(5)
Such a step in the microstructure evolution is associated with a minimal energy change (i.e., minimal dissipation): ˜ t : H = [ v(x)]
(6)
Starting from the new con guration the functional minimization procedure can be carried out over and over, thus describing the deterministic approach of the system to equilibrium. Note that, in practice, an approximate solution to the global variational problem, Eq. (3), is sought, either: (a) by using a discretized grain-boundary network, the continuous velocity eld being replaced by a set of discrete nodal points [25]; or (b) by introducing shape functions describing the local curvature along grain boundaries in order to perform the integrals in Eq. (3) explicitly, thus leaving only the triple-junction velocities as variational parameters [14]. In both cases, the functional is thus turned into a function of a discrete set of velocities, {vi }i=1; :::; M , to be solved, e.g., by nite-elements. In any case, however, such a fully coupled solution of the equations of motion for a system of many grain boundaries for long times can be a daunting task even for moderate system sizes. 3. The velocity Monte Carlo simulation method Although, from a macroscopic point of view, grain growth can be suciently well described as a deterministic, continuous process, it is nevertheless the average result of random, thermally activated events at the microscopic scale. Grain-boundary migration, i.e., the displacement of the interface between two adjoining crystals with di erent orientations, can be microscopically described as a short-range di usive process whose elementary step is the rearrangement of atoms from one crystalline orientation to the other across the interface [26]. In mean- eld theories of grain growth [12,25] a deterministic equation for the rate of change of the grain size is obtained by assuming time scaling and volume conservation; such theories describe the growth of a single grain in an average medium and neglect structural correlations. Several dynamical simulations [3,5,6,11] are based on this same assumption, and often fail to exhibit scaling. On the other hand, a global solution of the nodal equations of motion [14,16 –18] allows to fully include structural correlations. It has been suggested [20] that structural correlations could be introduced in a mean- eld model by adding a stochastic uctuation term to the drift term of the driving force; this leads to a Fokker–Planck equation for the grain size distribution, in which the stochastic term represents a di usive spread about the mean- eld result. In this section we describe a simulation method which will allow a stochastic description of grain growth based on the above variational functional . Grain-boundary
344
F. Cleri / Physica A 282 (2000) 339–354
migration is turned into a random sequence of microscopic con gurations whose macroscopic result is the dissipation of the excess energy and the approach to equilibrium. The probability per unit time of occurrence of a con guration is written [25] in terms of a transition rate between two microscopic con gurations and , W ( → ), as a function of the energy dissipation rate 1 e−(k t=kB T ) if (vk ) ¿ 0 ; (7) W ( → ) = 1 if (vk )0 with a normalizing factor with dimensions of time, obtained as the sum of the waiting times for all the allowed transitions, and T a ctitious temperature related to the amplitude of the random contribution. It is worth noting that the quantity kB T has the role of a scaling parameter which governs the absolute value of the transition probability [25], and does not necessarily coincide with a physical system temperature. The notation vk in Eq. (7) means that the velocity elds of the two con gurations and di er only for the velocity of the kth triple junction and of the three boundaries which share this junction. Hence the global evolution problem, Eq. (3), is decomposed into a stochastic sequence of elementary changes, corresponding to individual, uncorrelated grain-boundary migrations. Consequently, the unknown velocity eld vk in Eq. (7) is no longer the minimizing eld but a set of random variables, with equilibrium distribution given by W , sampled during the simulation. Accordingly, the transition between successive con gurations does not necessarily follow the minimum-dissipation path, as in the deterministic solution, but a stochastic behavior [19 –21]. It can be shown [25] that, in the thermodynamic limit, time-dependent average quantities computed along such a stochastic sequence converge to their deterministic counterparts, i.e., the evolution of the system from a given initial con guration far from equilibrium to the nal equilibrium state can be described in terms of ensemble averages. The kinetic Monte Carlo (MC) simulation method [25] represents the most natural choice to generate a stochastic sequence of microscopic con gurations with the properties described above. Compared to deterministic methods, a MC algorithm is intrinsically local since the trial variations are performed on one random variable at the time. As such, its computational complexity scales linearly with the system size instead of the quadratic scaling (or worse) exhibited by global methods based on matrix inversion or diagonalization. Moreover, it does not require to compute nodal forces but only energy di erences. The special MC algorithm to be described in the following is more ecient than Potts-model MC simulations [27,28] since it describes only grain boundaries instead of the whole grains. Moreover, and similar to the deterministic models, it deals with microscopic properties and transport coecients, such as and , thus allowing a more transparent interpretation of the results. To obtain explicit model equations to be used in a MC simulation we start again from Eq. (3). Since the grain-boundary curvature in real microstructures is generally found to be very smooth, the approximation of straight boundaries has been often adopted in grain-growth models [11]. For the sake of simplicity, this is our choice also in the
F. Cleri / Physica A 282 (2000) 339–354
345
present work and, consequently, the curvature ( rst) term in Eq. (3) is set to zero. By contrast, assuming line-tension equilibrium at the triple junctions (i.e., all angles equal ◦ to 120 ) as in Ref. [14], the driving force is purely curvature and the second term in Eq. (3) is zero. (However, we note that our model presents no diculties in being extended to treat curved [14] or discretized boundaries [29].) Moreover, a linear variation of the velocity along the grain boundaries is assumed. With these simpli cations, it is easy to show that the energy dissipation for an elementary con guration change, Eq. (7), amounts to calculating the value of the function 3 X Li 2 2 [(vn; i ) + (vn; k ) + vn; i vn; k ]
i (vs; i − vs; k ) + (8) (vk ) = 6i i=1
with the sum limited to the three grain boundaries i sharing the triple junction k. The MC implementation of the stochastic grain-growth model has a peculiar feature: sampling of random velocities, instead of random displacements as usual in the kinetic MC method, is required. For this reason we call this method a “velocity” Monte Carlo (VMC). Then, the VMC algorithm works as follows: at each MC step a random velocity variation v0k =vk + vk , with a random number uniformly distributed between −1 and 1, is sampled for each triple junction k. Subsequently, the value of the function in Eq. (8) is calculated for the two cases: (a) with vk equal to the existing junction velocity from the previous time-step, or (b) with vk equal to the randomly variated velocity v0k . The variated velocity v0k is accepted whenever the value of the dissipated power decreases k = (v0k ) − (vk ) ¡ 0
(9)
or, in case k ¿ 0, if a random number uniformly distributed between 0 and 1 satis es the so-called Boltzmann–Metropolis test ¡ e−(k t=kB T ) :
(10)
Otherwise, the old value vk is kept. Thus, at the subsequent time-step the position of the triple junction k (and of the three boundaries sharing it) will be updated according to Eq. (5), by using either vk or v0k . According to Eq. (10), the VMC method statistically allows local uctuations in which positive (“uphill”) energy changes may occur, reminiscent of Louat’s idea of local reversals in the growth rate [19]. We claim that the presence of random uctuations controlled by the parameter kB T should introduce, albeit approximately, the e ect of the structural correlations beyond the mean- eld treatment [20]. The validity of this statement is to be veri ed a posteriori from the simulation results. Due to the particular “velocity” implementation described above, pathological stochastic sequences could be expected, i.e., sequences in which positive energy changes are inde nitely repeated if, for some local con guration, it were dicult to nd a v0k for which k ¡ 0. Such a situation can, indeed, occur for particular combinations of the boundary energy and mobility values, i.e., if the product is of the order of A0 =t0 or
346
F. Cleri / Physica A 282 (2000) 339–354
Fig. 1. De nition of the topological transformations for a 2D network: T1 is the neighbor switching, T2 is the disappearance of a three-sided grain, anti-T2 (only in 3D) is the appearance of a three-sided grain.
larger, with A0 the unit of surface and t0 the unit of time. However, in a deterministic method such values would correspond to impractically small time steps [14], i.e., to unstable solutions as well. During the process of grain-boundary migration, some grains grow while others shrink and disappear. To deal with such an evolving topological structure we must make some assumptions to explicitly provide a solution to topological transformations in the network. As usual in cellular models [3], the transformations that may occur are classi ed as “Tn” (Fig. 1): T1 is a grain-switching process; T2 is a grain-disappearance process; and anti-T2 (possible only in 3D) is a grain-appearance process. All such transformations imply a change in the number of sides of neighboring grains, so the network topology needs to be remapped after either of these occurs. For both T1 and T2 processes in 2D, the most obvious choice is to set a cut-o for the minimum distance between two triple junctions and select the appropriate termination option according to some local rule.
4. Results for grain growth in two dimensions In order to check whether our stochastic, variational-based model of grain growth gives meaningful results, as a rst application we performed VMC simulations to be compared to the results of various deterministic simulations [7,11,14]. We take r0 , the distance between nearest-neighbor vertices in the hexagonal lattice, as the unit of length. The unit of time t0 represents the (arbitrary) duration of one MC step. Moreover, the system is uniform, i.e., only one value of and exist, common to all grain boundaries. We used values of the product 6 around 6 = 10−2 (r02 =t0 ) which was found to ensure a smooth evolution. The maximum velocity variation allowed in the random sampling is v = 10−3 (r0 =t0 ). The T1-cuto for performing a T1 topological switch is equal √ to r0 =100. The T2-cuto for small-grain forced disappearance is A0 =100, with A0 = ( 27=2)r02 the area of an hexagonal grain in the regular network. A number of di erent simulations of grain growth were performed for this comparison. Typical runs were for systems with initially Ntj = 2160 triple junctions, corresponding to N = 1080 grains, or smaller. Initial con gurations were obtained from a Voronoi construction with 2D-periodic boundary conditions to impose a constant area throughout the simulation. Simulations started by nding a guess set of triple-junction velocities, with the requirement of locally reducing the energy compared to vk = 0. The e ective temperature was xed to a very low value, such as to keep acceptance ratio
F. Cleri / Physica A 282 (2000) 339–354
347
Fig. 2. Sequence of grain growth for a VMC simulation. (a): time = 10 MC step, N = 1080; (b): t = 190, N = 762; (c): t = 420, N = 294; (d): t = 1050, N = 53.
of the Boltzmann–Metropolis test, Eq. (10), at about 1%. Such a low acceptance is necessary in order to preserve a good connectivity of the network (i.e., no “boundary crossing” of one junction across neighboring boundaries) and to avoid too large uctuations in the energy. In Figs. 2a–d we show the network map for a typical VMC run. Practically every theory of grain growth predicts that, once transient e ects are ruled should grow linearly with time and out, the mean area of the grains in the network, A, 1=2 should thus grow with t [2– 4]. Such a result is supported by the mean radius, R, several experiments on soap froths, i.e., systems with perfectly uniform grain boundary energies and mobilities (although its veri cation in material microstructures is hampered by impurity and second-phase e ects). Since the mean grain area is inversely proportional to the number of grains in the network, the scaling state can be de ned also by a law of the type N ∼ t −1 . In fact, it has been shown by several authors that the best representation of the time evolution of the grain number can be obtained from the following equation [30,14]: −2 t (11) N (t) = N (0) 1 + t0
348
F. Cleri / Physica A 282 (2000) 339–354
Fig. 3. Time evolution of the grain number n. The open circles de ne the limits of the linear (time-scaling) region. The slope of the dashed line de nes the scaling exponent, = 0:48 ± 0:01.
to be represented in a log–log plot in which the grain-growth exponent will show up as the slope of the linear region. The parameter t0 is chosen to provide the best linear t. Fig. 3 represents such a log–log plot for a VMC simulation with N (0) = 1080 (here t0 = 2:4). The attainment of a scaling state is clearly evident in a rather large interval, corresponding to times during which the system size goes from N =850 to 350 grains (indicated by the two open circles). The slope is −2 = −0:96 ± 0:03, thus the grain-growth exponent is =0:48±0:01. Notably, most computer simulations [5 –12,14] give = 0:48 to 0:52. It is worth noting that in the absence of random uctuations in the driving force (i.e., if we set kB T = 0) the growth exponent is about = 0:8. This is a rst proof that stochastic uctuations, including reversals in the growth direction, can introduce non-local structural correlations in the local evolution model. A supplementary indication of the scaling state is given by the second-moment, 2 , of the frequency distribution fn of n-sided grains in the microstructure [3,31]: 2 =
X
fn (n − 6)2 :
(12)
n=3
In Fig. 4 we plot the behavior of 2 for a N (0) = 512 simulation as a function of time, between the initial state and a time at which the number of grains is reduced by one order of magnitude. The 2 uctuates around a constant value with average 2 = 1:39 ± 0:14. Such a value compares well to the typical values between 1.4 and 1.5 reported in the literature [3,9,14], whereas lower and higher values are usually considered signatures of a bad scaling behavior even when coupled to values of ∼ 0:5.
F. Cleri / Physica A 282 (2000) 339–354
349
Fig. 4. Time evolution of the second moment of the frequency distribution of n-sided grains. The horizontal line de nes the average value of 2 = 1:39 ± 0:14; the shaded region de nes the interval comprised around ±1 one standard deviation from the average.
In addition to fn there are a number of di erent topological properties which characterize the cellular network. Such properties should be self-similar in the scaling time range. We found a good scaling behavior and excellent agreement between the results of VMC simulations and deterministic models [11,14] for fn itself [31], for the distribution of the average number of sides of the neighbors of a n-sided grain [32,33], for the average normalized area of a n-sided grain [34], and for the grain-size distribution (see below). Based on purely geometrical considerations, Von Neumann [23] established that for any 2D uniform network the growth rate of the area of a n-sided grain, An , is given by dAn = 0 (n − 6) ; (13) dt where 0 is a coecient depending on the grain-boundary curvature. Although Von Neumann’s law is dicult to verify experimentally [4], it nevertheless represents a severe test for any theory or simulation of grain growth, and for our model in particular since it has been claimed [3] that vertex models with straight boundaries should violate it. For a N (0)=512 VMC simulation, we report in Fig. 5a the time evolution of the area of a n-sided grain, An (t) − An (0), for a sample of several n-sided grains. Furthermore, in Fig. 5b the same data are summarized as growth rates, by plotting the average slope of the curves in Fig. 5a as a function of the number of sides n. It can be seen that for 4 ¡ n ¡ 10 Von Neumann’s law is strictly obeyed and, in particular, for n = 6 the growth rate is zero. For values n ¿ 10 the grain population is very small and the growth rate becomes erratic, while for n=3 the growth rate is in uenced by the special treatment of T2 processes. Coupled to the values of and 2 above, the veri cation of Von Neumann’s law represents a strong con rmation that the stochastic grain-growth model with straight boundaries actually displays a scaling state. Notably, the growth rates in Fig. 5 have a measurable dispersion around the average at long times, as a result of the stochastic sampling in the VMC simulation.
350
F. Cleri / Physica A 282 (2000) 339–354
Fig. 5. (a) Time evolution of the grain area An (t) for a sample of several n-sided grains; grain areas are normalized by subtracting the value An (0) at the time of appearance of each grain; grains are followed until they change number of sides; (b) growth rates of n-sided grains, obtained from the average slopes of the curves in (a).
We now turn to the topological property which has been the subject of the largest debate, namely the grain size distribution P(x), expressed, as usual, as a function of Experimentally, the form of this distribution has been the scaled grain radius x = R= R, suggested to be close to log-normal [3,12,19]. Moreover, there has been a long debate about P(x) truly being a self-similar distribution [3,5 –7]. In Fig. 6 the VMC distribution, obtained as an average over 200 con gurations within the scaling time range, is compared with two theoretical distributions: the Rayleigh, a special case the log-normal rst proposed by Louat [19], and Hillert’s distribution, obtained by assuming scaling in a mean- eld theory [12]. It can be seen that the x ¡ 1:1 portion follows rather well the Hillert distribution, while the x ¿ 1:25 portion appears to decrease less steeply than Hillert’s, and rather follows the Rayleigh distribution. Such a behavior gives support to the analysis by Pande and Dantsker [35,36], who showed that these distributions are
F. Cleri / Physica A 282 (2000) 339–354
351
Fig. 6. Grain size distribution as a function of the scaled grain radius x = R= R for the VMC simulation (histogram), compared with the Rayleigh (dashed curve) and Hillert (full curve) theoretical distributions.
the two limiting cases for P(x), depending on the amplitude of the stochastic noise (Hillert’s corresponding to zero noise). Notably, the VMC distribution overlaps quite well the deterministic distribution obtained from the global variational solution [14]. It has been observed [20,25] that Hillert’s distribution does not represent well the experimental results but, on the other hand, Rayleigh’s distribution (which is frequently obtained both in theory and simulations [19,29,37] and ts some experiments) is unbounded and does not satisfy volume conservation. From Fig. 7 it can be seen that the VMC distribution is clearly cut o around xC ∼ 2:2. In mean- eld theories [12,22,25] the average number of sides of a grain, n(x), is assumed to be a unique function of the scaled radius x. In this way a well-de ned relation between An and n is imposed and Eq. (8) becomes strictly deterministic. For example, Hillert’s model assumes a linear n(x), while Fradkov et al. [37] have shown that a parabolic n(x) leads to the Rayleigh distribution. As shown in Fig. 7, the n(x) averaged over 200 VMC con gurations is linear, n(x) = 3:85 + 2:05x, in the range x ¡ 1:1; instead, for 1:1 ¡ x ¡ 2:3 a parabolic law, n(x) = 4:63 + 1:22x2 , very close to that corresponding to the Rayleigh distribution [25], is obtained. Trying to settle this issue within more rigorous bounds, Mullins [25] developed an elegant analysis in terms of the reduced growth rate, G(x) Z 2 xC P(x0 ) d x0 (14) G(x) = x − P x and proved that any theory or simulation based on curvature-driven growth and timescaling must satisfy the constraint n(x) = 6 + 6 xG(x), with a structural constant [25]. Firstly, we note that our model gives an average value of ∼ 0:136, well within the range predicted by most analytical models. Secondly, in Fig. 7 the best t for n(x)
352
F. Cleri / Physica A 282 (2000) 339–354
Fig. 7. Plot of the average number of sides of a grain, n(x) (best t to 200 VMC con gurations, full curve) as a function of the scaled grain radius x. The full circles represent the Mullins constraint, 6 + 6 xG(x), with the integral in G(x) calculated from the histogram in Fig. 6.
is compared to the r.h.s. of the constraint equation (obtained by integrating the VMC histogram for P(x) in Fig. 6): it can be seen that the VMC simulation satis es Mullins’ constraint in a wide range, 0:5 ¡ x ¡ 2:3, with a marked failure only in the small grain-size region. Such a result is even more meaningful, when noting that (a) Mullins’ analysis was originally intended to show that no stochastic term should be necessary beyond mean- eld arguments, and (b) our stochastic, curvature-less, straight-boundary model can reproduce the results of deterministic, curvature-driven models [14]. This latter observation suggests that the unbalanced line-tension at the triple junctions may represent an e ective curvature, which exactly replaces the contribution of the curvature term in Eq. (3).
5. Conclusions We proposed a stochastic model for the time evolution of a cellular network driven by dissipative forces, such as a polycrystalline microstructure undergoing grain growth. The model is based on a variational formulation of the dissipated power, rstly introduced by Needleman and Rice [13] in a study of cavitation at grain boundaries and subsequently extended to many other problems of microstructural evolution in two dimensions, including grain growth [14]. We gave a new interpretation to the variational functional established in those works, by using it as the transition-rate generating function in a stochastic model of microstructure evolution. The stochastic point of view here developed (see also [19 –21,35,36]) provides a model that can be used in kinetic Monte Carlo simulations (in the present case,
F. Cleri / Physica A 282 (2000) 339–354
353
“velocity” Monte Carlo, VMC). The model variables are the positions and velocities of the network vertices where cell walls meet. We applied such a model to the well-studied case of grain growth in two dimensions, in which the cellular network represents a cross-section of a polycrystalline microstructure and the cell walls represent grain boundaries. The VMC simulation results were found to compare well to deterministic results [7,11,14] for the most relevant statistical quantities. Our simulations give support to the existence of a time-scaling regime, as determined from the time evolution of the grain number and of the second moment of the frequency distribution of n-sided grains. We obtained a value of the dynamical exponent = 0:48 ± 0:01 and 2 = 1:39 ± 0:14, in good agreement with the best values obtained in deterministic simulations [3,14]. Notably, the VMC grain-size distribution displays a cross-over from Hillert to a Rayleigh-like distribution, as does the function n(x) describing the average number of sides of a grain, as a function of the scaled grain radius; such a behavior lends support to the suggestion [20,35,36] that a stochastic noise superimposed to the deterministic drift can interpolate between the two extremes of strictly deterministic growth (Hillert) and pure random walk (Rayleigh). Moreover, we veri ed that both the Von Neumann law [23] and the Mullins constraint for n(x) [25] are nicely obeyed by our straight-boundary model, whose convergence to the global equilibrium properties and scaling behavior have been amply proved. Such results give the a posteriori proof that stochastic uctuations can, indeed, account for non-local structural correlations. An important result is that VMC simulations can reproduce all the main features of curvature-driven models [3,7,14]: although our model can be easily extended to include curved boundaries, these results prove that the unbalance of the line-tension at triple junctions is, indeed, equivalent to the driving force from boundary curvature. From the point of view of physical realism, this model inherits all the exibility of the original variational formulation [13,14], which allows to extend this same description to include other dissipative mechanisms such as grain sliding and rotation and di usive processes [16 –18]. Such a capability of dealing with di erent dissipative mechanisms is extremely promising in view the study of real-material microstructures in which, moreover, it is essential to include energy and mobility anisotropy. We have already obtained interesting results on abnormal and oriented grain growth by including the grain-misorientation dependence of the energy and mobility, which will be described in a forthcoming paper.
Acknowledgements I am grateful to D. Wolf (ANL, Argonne) for prompting my interest in this area and for his kind hospitality at the ANL Materials Science Division. Useful discussions with G. D’Agostino (ENEA, Roma) are also acknowledged. This work was performed under the INFM-Forum “MUSIC” Project.
354
F. Cleri / Physica A 282 (2000) 339–354
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
C.S. Smith, in: Metal Interfaces, ASM, Cleveland, 1965, p. 65. H.V. Atkinson, Acta Metall. 36 (1988) 469. J.A. Glazier, D. Weaire, J. Phys. Condens. Matter 4 (1992) 1867. J. Stavans, Rep. Prog. Phys. 56 (1993) 733. D. Weaire, J.P. Kermode, Philos. Mag. B 48 (1983) 245. D. Weaire, J.P. Kermode, Philos. Mag. B 50 (1984) 379. F.J. Humpreys, Mater. Sci. Technol. 8 (1992) 135. S.M. Allen, J.W. Cahn, Acta Metall. 27 (1979) 1085. H.J. Frost, C.V. Thompson, Acta Metall. 35 (1987) 529. H.J. Frost, C.V. Thompson, C.L. Howe, J. Wang, Scr. Metall. 22 (1988) 65. K. Kawasaki, T. Nagai, K. Nakashima, Philos. Mag. B 60 (1989) 399. M. Hillert, Acta Metall. 13 (1965) 227. A. Needleman, J.R. Rice, Acta Metall. 28 (1980) 1315. A.C.F. Cocks, S.P. Gill, Acta Mater. 44 (1996) 4765, 4777. B.D. Vujanovic, S.E. Jones, Variational Methods in Nonconservative Phenomena, Academic Press, Boston, 1989. J. Pan, A.C.F. Cocks, Compos. Mater. Sci. 1 (1993) 95. J. Pan, A.C.F. Cocks, Acta Mater. 43 (1995) 1395. A.C.F. Cocks, J. Mech. Phys. Solids 44 (1996) 1429. N.P. Louat, Acta Metall. 22 (1974) 721. C.S. Pande, Acta Metall. 35 (1987) 2671. A. Thorvaldsen, Acta. Metall. 41 (1993) 1347. J. Stavans, J.A. Glazier, Phys. Rev. Lett. 62 (1989) 1318. J. Von Neumann, in: Metal Interfaces, ASM Cleveland, 1965, p. 106. N. Ryum, O. Hunderi, Acta Metall. 37 (1989) 1375. W.W. Mullins, Acta Mater. 46 (1998) 6129; K. Binder, M.H. Kalos, in: K. Binder (Ed.), Monte Carlo Methods in Statistical Physics, Topics in Current Physics, Vol. 7, Springer, Berlin, 1986 (Chapters 1, 6). A.P. Sutton, R.W. Ballu, Interfaces in Crystalline Materials, Oxford Science Publications, Oxford, 1995 (Chapter 9). M.P. Anderson, D.J. Srolovitz, G.S. Grest, P.S. Sahni, Acta Metall. 32 (1984) 783, 793. M.P. Anderson, D.J. Srolovitz, G.S. Grest, P.S. Sahni, Philos. Mag. B 59 (1989) 293. J. Pan, A.C.F. Cocks, S. Kucherenko, Proc. Roy. Soc. London A 453 (1997) 2161. T. Herdtle, H. Aref, J. Fluid Mech. 241 (1992) 233. E. Carnal, A. Mocellin, Acta Metall. 29 (1981) 135. D.A. Aboav, Metallography 3 (1970) 383. D. Weaire, Metallography 7 (1974) 157. D. Lewis, Anat. Rec. 38 (1928) 351. C.S. Pande, E. Dantsker, Acta Metall. 38 (1990) 945. C.S. Pande, E. Dantsker, Acta Metall. 39 (1991) 1359. V. Fradkov, L. Schvindlerman, D. Udler, Scr. Metall. 19 (1985) 1285.
www.elsevier.com/locate/physa
A stochastic grain growth model based on a variational principle for dissipative systems Fabrizio Cleria; b; ∗
a Ente
Nazionale Energia e Ambiente, Divisione Materiali Avanzati, Centro Ricerche Casaccia, C.P. 2400, 00100 Roma, Italy b Istituto Nazionale Fisica della Materia, Unit a di Ricerca Roma 1, P.le A. Moro 2, 00185 Roma, Italy Received 16 August 1999; received in revised form 28 December 1999
Abstract A stochastic model for the evolution of a cellular network driven by dissipative forces is presented. The model is based on a variational formulation for the dissipated power, from which we obtain an expression for the transition-rate generating function to be used in kinetic Monte Carlo simulations. The model canonical variables are the positions and velocities of the network vertices where cell walls meet. We apply such a model to the study of grain growth in two dimensions, in which the network represents a cross-section of a polycrystalline microstructure and the cell walls represent grain boundaries. The results of the stochastic grain-growth model for relevant statistical quantities are compared to deterministic model results and analytic theories. c 2000 Elsevier Science B.V. All rights reserved.
PACS: 02.70.Lq; 05.70.Ln; 81.10.Aj Keywords: Grain growth; Variational principles; Dissipative systems; Kinetic Monte Carlo
1. Introduction The structure of many natural systems consists of closed cells with well-de ned boundaries. Examples of such systems include soap froths, ferro uid froths, lipid monolayers, biological tissues, magnetic domains, ame fronts and, possibly, even the largescale structure of the universe. Such cellular structures can display topological disorder in the form of a random connectivity pattern and, in many cases, the structure evolution appears self-similar over some appropriate time scale. In the domain of materials ∗
Correspondence address. Ente Nazionale Energia e Ambiente, Divisione Materiali Avanzati, Centro Ricerche Casaccia, C.P. 2400, 00100 Roma, Italy. Tel.: +39-06-30484825; fax: +39-06-30484729. E-mail address: [email protected] (F. Cleri)
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 0 8 7 - X
340
F. Cleri / Physica A 282 (2000) 339–354
science, the arrangement of grains in a polycrystalline solid has often been modelled as a cellular system [1]. In this latter case the cell walls represent grain boundaries, associated with an excess energy with respect to the bulk monocrystal: grain-boundary migration is driven by the reduction of such an excess energy and, consequently, the grain structure evolves with time in the direction of increasing grain size. From the theoretical point of view the evolution of cellular structures, and in particular grain growth in materials, has been studied by several methods (see the review papers [2– 4] and [5 –11]) often based on computer simulation and in most cases restricted to two dimensions (2D). In the 2D cellular model of a microstructure only the grain boundaries are represented, disregarding the grain interior; grain boundaries are described as generally curved segments connecting the triple junctions, i.e., the points where three grain boundaries meet. In summary, irrespective of the details and approximations for the underlying kinetics, most 2D cellular models appear to give a although the true ext 1=2 scaling behavior at long times for the mean grain radius, R, istence of such a scaling state has long been a subject of debate [2,3]. Such a feature, which is postulated in mean- eld theories [12] and predicted on the basis of simple dimensional arguments [2,3], emerges as a universal dynamic feature of the evolution of any cellular structure connected only through triple junctions. On the other hand, no general agreement has yet been reached on such a basic quantity as the grain-size distribution function, a quantity for which experimental measurements are, furthermore, quite dicult to realize. A conceptually innovative theoretical approach to microstructural evolution was presented some time ago by Needleman and Rice [13], based on a variational principle for dissipative systems. In the case of grain growth Cocks and Gill [14] derived a similar variational functional describing the rate of energy dissipation in a microstructure due to the competition between the reduction of the excess energy and a driving force proportional to the boundary velocity. The variational parameter in such functionals is the continuous grain-boundary velocity eld: by applying D’Alembert’s di erential form of the variational principle (which is, indeed, valid for both conservative and non-conservative dynamical systems [15]), explicit equations for the minimizing velocity eld are obtained, and from these the microstructure evolution. Such an approach stands out as very powerful and exible [16 –18], since any term contributing to energy dissipation in the microstructure (e.g., grain sliding and rotation, matter di usion along grain boundaries, di usion and plastic work in the grain bulk) can be included in the functional, provided a variational principle for each new term can be established. However, the approach is also computationally inecient since it is de ned in terms of a global minimization procedure, which amounts to inverting a large matrix at each time step. In this paper we propose a stochastic model of microstructural evolution based on a di erent interpretation of the Needleman–Rice variational functional, using the well-studied problem of 2D grain growth as a benchmark application. We start from the consideration that, at the microscopic level, in nitesimal grain-boundary displacements are the result of random thermally activated events: the probability of such
F. Cleri / Physica A 282 (2000) 339–354
341
events is proportional to exp(−H=kB T ) with H , the con gurational energy change (i.e., dissipation), playing the role of a transition-rate generating function. Microstructure evolution thus results from a random sequence of local, uncorrelated events, as opposed to the global, non-local solution. We interpret the variational functional itself as the transition-rate generating function for the stochastic model, to be solved numerically by means of a special implementation of the kinetic Monte Carlo (MC) method. A stochastic description of grain growth was rst proposed by Louat [19] and later elaborated by Pande [20], Thorvaldsen [21] and others [2]. By noting that the fundamental results of the classical description of grain growth [12] have only a statistical validity, Louat suggested that the grain growth rate should vary in a random manner, including reversals in the sign. Indeed, in non-ideal systems such as wet soap froth ◦ and lipids it has been observed [22] that the 120 equilibrium relation and the Von Neumann’s law ([23], see below) are veri ed only on average. Indeed, Louat’s work was later criticized [24] and the idea of a stochastic component superimposed to the driving force for grain growth [20,21] was strongly opposed by Mullins [25] on the basis of mean- eld theory arguments. As we will detail in the following, under the hypothesis of locality the random component of the driving force can be interpreted as a structural noise term accounting for correlation e ects beyond the mean- eld solution. In the remainder of this paper we rstly summarize the deterministic variational model; then, we describe the stochastic model of microstructural evolution and its practical implementation as a kinetic MC algorithm. This latter is based on the stochastic migration of the boundaries of a cellular network, which has great advantages in terms of both physical clarity and computational eciency. The merits of the stochastic grain-growth model are discussed by comparing the results of MC simulations with those of the deterministic models [7,11,14]. It is concluded that such a comparison is extremely successful and lends support to further developments.
2. Variational functional for microstructural evolution We consider a 2D system consisting of a fully dense pattern of N grains. Grains are represented as irregular polygons whose initial con guration could be obtained, e.g., by a Voronoi construction. Such a con guration should be physically close to the cross-section of a textured microstructure. The ensemble of the border segments between each pair of grains makes up a set of Ngb grain boundaries with length Li , i = 1; : : : ; Ngb , and excess energy i . Grain boundaries can meet only at triple junctions, i.e., fourfold (or higher) junctions are unstable. Given this topological constraint, the number of triple junctions Ntj is given by Ntj = 2=3Ngb and the number of grain boundaries is, in turn, related to the number of grains as Ngb = 3N . In the lack of a more fundamental theory, a macroscopic model of 2D grain growth [2– 4] describes phenomenologically the competition between two terms:
342
F. Cleri / Physica A 282 (2000) 339–354
PNgb (a) the tendency of the system to reduce its excess interfacial energy, i=1 ( i Li ), by reducing the total grain boundary perimeter; the rate of work done under a virtual variation of the boundary length per unit time, ˙∗s = (L∗i =Li )=t can be written as [14] WT∗
=
Ngb Z X i=1
Li
i ˙∗s (s) ds :
(1)
(b) the lattice resistance to grain boundary migration, described as a viscous drag force with an interface mobility coecient, i ; if a linear kinetic relation is assumed between the driving force and the normal component of the boundary velocity, vn =i f, the rate of energy dissipation by the viscous force under a virtual variation of the velocity, v∗ , is given by [14] WM∗ =
Ngb Z X i=1
Li
fvn∗ (s) ds :
(2)
Following Needleman and Rice [13] and Cocks and Gill [14], the above de nitions for WT∗ and WM∗ are combined into a “virtual power principle” (analogous to the virtual work principle) which allows to identify a variational functional representing the power dissipated during grain-boundary migration. The functional can be written in terms of the velocity eld only, v(s), de ned over the collection Ngb of grain boundaries, with components vn and vs parallel and perpendicular, respectively, to the local grain-boundary normal [v(x)] =
Ngb Z X i=1
Li
Z
i i vn (x) ds +
Li
i
@vs (s) ds + @s
Z Li
vn2 (s) ds ; 2i
(3)
where the ith grain boundary has length 06s6Li , excess energy i , curvature i and mobility i , and the vector eld x collectively denotes the grain-boundary coordinates. The three terms in the variational functional can be described as follows [14]: (a) the rst is the contribution from the grain-boundary curvature: it states that the curved sections of the grain boundaries will move at a rate proportional to both the local curvature and excess energy; (b) the second is the contribution coming from the triple junctions: it states that the triple junctions move according to the net force resulting from the sum of the scalar products of the velocity times the tangent vector at the end-points of each grain boundary; (c) the third is the dissipation by the viscous force opposing the migration perpendicular to the boundary, as discussed above. In a deterministic description the functional [v(x)] is made stationary by a varia˜ tional eld v(x) (which can be proven to be unique for the exact solution of the variational problem [14]), thus resulting in the energy-dissipation minimum for a particular con guration of the network. The instantaneous solution eld is found by imposing that the functional be a global minimum for arbitrary variations of v(x) [v(x)] = 0
for v(x) = v˜(x) :
(4)
F. Cleri / Physica A 282 (2000) 339–354
343
Then, the kinetic evolution from the boundary con guration to is obtained from equations of the type ˜ x( ) = x( ) + v(x) t :
(5)
Such a step in the microstructure evolution is associated with a minimal energy change (i.e., minimal dissipation): ˜ t : H = [ v(x)]
(6)
Starting from the new con guration the functional minimization procedure can be carried out over and over, thus describing the deterministic approach of the system to equilibrium. Note that, in practice, an approximate solution to the global variational problem, Eq. (3), is sought, either: (a) by using a discretized grain-boundary network, the continuous velocity eld being replaced by a set of discrete nodal points [25]; or (b) by introducing shape functions describing the local curvature along grain boundaries in order to perform the integrals in Eq. (3) explicitly, thus leaving only the triple-junction velocities as variational parameters [14]. In both cases, the functional is thus turned into a function of a discrete set of velocities, {vi }i=1; :::; M , to be solved, e.g., by nite-elements. In any case, however, such a fully coupled solution of the equations of motion for a system of many grain boundaries for long times can be a daunting task even for moderate system sizes. 3. The velocity Monte Carlo simulation method Although, from a macroscopic point of view, grain growth can be suciently well described as a deterministic, continuous process, it is nevertheless the average result of random, thermally activated events at the microscopic scale. Grain-boundary migration, i.e., the displacement of the interface between two adjoining crystals with di erent orientations, can be microscopically described as a short-range di usive process whose elementary step is the rearrangement of atoms from one crystalline orientation to the other across the interface [26]. In mean- eld theories of grain growth [12,25] a deterministic equation for the rate of change of the grain size is obtained by assuming time scaling and volume conservation; such theories describe the growth of a single grain in an average medium and neglect structural correlations. Several dynamical simulations [3,5,6,11] are based on this same assumption, and often fail to exhibit scaling. On the other hand, a global solution of the nodal equations of motion [14,16 –18] allows to fully include structural correlations. It has been suggested [20] that structural correlations could be introduced in a mean- eld model by adding a stochastic uctuation term to the drift term of the driving force; this leads to a Fokker–Planck equation for the grain size distribution, in which the stochastic term represents a di usive spread about the mean- eld result. In this section we describe a simulation method which will allow a stochastic description of grain growth based on the above variational functional . Grain-boundary
344
F. Cleri / Physica A 282 (2000) 339–354
migration is turned into a random sequence of microscopic con gurations whose macroscopic result is the dissipation of the excess energy and the approach to equilibrium. The probability per unit time of occurrence of a con guration is written [25] in terms of a transition rate between two microscopic con gurations and , W ( → ), as a function of the energy dissipation rate 1 e−(k t=kB T ) if (vk ) ¿ 0 ; (7) W ( → ) = 1 if (vk )0 with a normalizing factor with dimensions of time, obtained as the sum of the waiting times for all the allowed transitions, and T a ctitious temperature related to the amplitude of the random contribution. It is worth noting that the quantity kB T has the role of a scaling parameter which governs the absolute value of the transition probability [25], and does not necessarily coincide with a physical system temperature. The notation vk in Eq. (7) means that the velocity elds of the two con gurations and di er only for the velocity of the kth triple junction and of the three boundaries which share this junction. Hence the global evolution problem, Eq. (3), is decomposed into a stochastic sequence of elementary changes, corresponding to individual, uncorrelated grain-boundary migrations. Consequently, the unknown velocity eld vk in Eq. (7) is no longer the minimizing eld but a set of random variables, with equilibrium distribution given by W , sampled during the simulation. Accordingly, the transition between successive con gurations does not necessarily follow the minimum-dissipation path, as in the deterministic solution, but a stochastic behavior [19 –21]. It can be shown [25] that, in the thermodynamic limit, time-dependent average quantities computed along such a stochastic sequence converge to their deterministic counterparts, i.e., the evolution of the system from a given initial con guration far from equilibrium to the nal equilibrium state can be described in terms of ensemble averages. The kinetic Monte Carlo (MC) simulation method [25] represents the most natural choice to generate a stochastic sequence of microscopic con gurations with the properties described above. Compared to deterministic methods, a MC algorithm is intrinsically local since the trial variations are performed on one random variable at the time. As such, its computational complexity scales linearly with the system size instead of the quadratic scaling (or worse) exhibited by global methods based on matrix inversion or diagonalization. Moreover, it does not require to compute nodal forces but only energy di erences. The special MC algorithm to be described in the following is more ecient than Potts-model MC simulations [27,28] since it describes only grain boundaries instead of the whole grains. Moreover, and similar to the deterministic models, it deals with microscopic properties and transport coecients, such as and , thus allowing a more transparent interpretation of the results. To obtain explicit model equations to be used in a MC simulation we start again from Eq. (3). Since the grain-boundary curvature in real microstructures is generally found to be very smooth, the approximation of straight boundaries has been often adopted in grain-growth models [11]. For the sake of simplicity, this is our choice also in the
F. Cleri / Physica A 282 (2000) 339–354
345
present work and, consequently, the curvature ( rst) term in Eq. (3) is set to zero. By contrast, assuming line-tension equilibrium at the triple junctions (i.e., all angles equal ◦ to 120 ) as in Ref. [14], the driving force is purely curvature and the second term in Eq. (3) is zero. (However, we note that our model presents no diculties in being extended to treat curved [14] or discretized boundaries [29].) Moreover, a linear variation of the velocity along the grain boundaries is assumed. With these simpli cations, it is easy to show that the energy dissipation for an elementary con guration change, Eq. (7), amounts to calculating the value of the function 3 X Li 2 2 [(vn; i ) + (vn; k ) + vn; i vn; k ]
i (vs; i − vs; k ) + (8) (vk ) = 6i i=1
with the sum limited to the three grain boundaries i sharing the triple junction k. The MC implementation of the stochastic grain-growth model has a peculiar feature: sampling of random velocities, instead of random displacements as usual in the kinetic MC method, is required. For this reason we call this method a “velocity” Monte Carlo (VMC). Then, the VMC algorithm works as follows: at each MC step a random velocity variation v0k =vk + vk , with a random number uniformly distributed between −1 and 1, is sampled for each triple junction k. Subsequently, the value of the function in Eq. (8) is calculated for the two cases: (a) with vk equal to the existing junction velocity from the previous time-step, or (b) with vk equal to the randomly variated velocity v0k . The variated velocity v0k is accepted whenever the value of the dissipated power decreases k = (v0k ) − (vk ) ¡ 0
(9)
or, in case k ¿ 0, if a random number uniformly distributed between 0 and 1 satis es the so-called Boltzmann–Metropolis test ¡ e−(k t=kB T ) :
(10)
Otherwise, the old value vk is kept. Thus, at the subsequent time-step the position of the triple junction k (and of the three boundaries sharing it) will be updated according to Eq. (5), by using either vk or v0k . According to Eq. (10), the VMC method statistically allows local uctuations in which positive (“uphill”) energy changes may occur, reminiscent of Louat’s idea of local reversals in the growth rate [19]. We claim that the presence of random uctuations controlled by the parameter kB T should introduce, albeit approximately, the e ect of the structural correlations beyond the mean- eld treatment [20]. The validity of this statement is to be veri ed a posteriori from the simulation results. Due to the particular “velocity” implementation described above, pathological stochastic sequences could be expected, i.e., sequences in which positive energy changes are inde nitely repeated if, for some local con guration, it were dicult to nd a v0k for which k ¡ 0. Such a situation can, indeed, occur for particular combinations of the boundary energy and mobility values, i.e., if the product is of the order of A0 =t0 or
346
F. Cleri / Physica A 282 (2000) 339–354
Fig. 1. De nition of the topological transformations for a 2D network: T1 is the neighbor switching, T2 is the disappearance of a three-sided grain, anti-T2 (only in 3D) is the appearance of a three-sided grain.
larger, with A0 the unit of surface and t0 the unit of time. However, in a deterministic method such values would correspond to impractically small time steps [14], i.e., to unstable solutions as well. During the process of grain-boundary migration, some grains grow while others shrink and disappear. To deal with such an evolving topological structure we must make some assumptions to explicitly provide a solution to topological transformations in the network. As usual in cellular models [3], the transformations that may occur are classi ed as “Tn” (Fig. 1): T1 is a grain-switching process; T2 is a grain-disappearance process; and anti-T2 (possible only in 3D) is a grain-appearance process. All such transformations imply a change in the number of sides of neighboring grains, so the network topology needs to be remapped after either of these occurs. For both T1 and T2 processes in 2D, the most obvious choice is to set a cut-o for the minimum distance between two triple junctions and select the appropriate termination option according to some local rule.
4. Results for grain growth in two dimensions In order to check whether our stochastic, variational-based model of grain growth gives meaningful results, as a rst application we performed VMC simulations to be compared to the results of various deterministic simulations [7,11,14]. We take r0 , the distance between nearest-neighbor vertices in the hexagonal lattice, as the unit of length. The unit of time t0 represents the (arbitrary) duration of one MC step. Moreover, the system is uniform, i.e., only one value of and exist, common to all grain boundaries. We used values of the product 6 around 6 = 10−2 (r02 =t0 ) which was found to ensure a smooth evolution. The maximum velocity variation allowed in the random sampling is v = 10−3 (r0 =t0 ). The T1-cuto for performing a T1 topological switch is equal √ to r0 =100. The T2-cuto for small-grain forced disappearance is A0 =100, with A0 = ( 27=2)r02 the area of an hexagonal grain in the regular network. A number of di erent simulations of grain growth were performed for this comparison. Typical runs were for systems with initially Ntj = 2160 triple junctions, corresponding to N = 1080 grains, or smaller. Initial con gurations were obtained from a Voronoi construction with 2D-periodic boundary conditions to impose a constant area throughout the simulation. Simulations started by nding a guess set of triple-junction velocities, with the requirement of locally reducing the energy compared to vk = 0. The e ective temperature was xed to a very low value, such as to keep acceptance ratio
F. Cleri / Physica A 282 (2000) 339–354
347
Fig. 2. Sequence of grain growth for a VMC simulation. (a): time = 10 MC step, N = 1080; (b): t = 190, N = 762; (c): t = 420, N = 294; (d): t = 1050, N = 53.
of the Boltzmann–Metropolis test, Eq. (10), at about 1%. Such a low acceptance is necessary in order to preserve a good connectivity of the network (i.e., no “boundary crossing” of one junction across neighboring boundaries) and to avoid too large uctuations in the energy. In Figs. 2a–d we show the network map for a typical VMC run. Practically every theory of grain growth predicts that, once transient e ects are ruled should grow linearly with time and out, the mean area of the grains in the network, A, 1=2 should thus grow with t [2– 4]. Such a result is supported by the mean radius, R, several experiments on soap froths, i.e., systems with perfectly uniform grain boundary energies and mobilities (although its veri cation in material microstructures is hampered by impurity and second-phase e ects). Since the mean grain area is inversely proportional to the number of grains in the network, the scaling state can be de ned also by a law of the type N ∼ t −1 . In fact, it has been shown by several authors that the best representation of the time evolution of the grain number can be obtained from the following equation [30,14]: −2 t (11) N (t) = N (0) 1 + t0
348
F. Cleri / Physica A 282 (2000) 339–354
Fig. 3. Time evolution of the grain number n. The open circles de ne the limits of the linear (time-scaling) region. The slope of the dashed line de nes the scaling exponent, = 0:48 ± 0:01.
to be represented in a log–log plot in which the grain-growth exponent will show up as the slope of the linear region. The parameter t0 is chosen to provide the best linear t. Fig. 3 represents such a log–log plot for a VMC simulation with N (0) = 1080 (here t0 = 2:4). The attainment of a scaling state is clearly evident in a rather large interval, corresponding to times during which the system size goes from N =850 to 350 grains (indicated by the two open circles). The slope is −2 = −0:96 ± 0:03, thus the grain-growth exponent is =0:48±0:01. Notably, most computer simulations [5 –12,14] give = 0:48 to 0:52. It is worth noting that in the absence of random uctuations in the driving force (i.e., if we set kB T = 0) the growth exponent is about = 0:8. This is a rst proof that stochastic uctuations, including reversals in the growth direction, can introduce non-local structural correlations in the local evolution model. A supplementary indication of the scaling state is given by the second-moment, 2 , of the frequency distribution fn of n-sided grains in the microstructure [3,31]: 2 =
X
fn (n − 6)2 :
(12)
n=3
In Fig. 4 we plot the behavior of 2 for a N (0) = 512 simulation as a function of time, between the initial state and a time at which the number of grains is reduced by one order of magnitude. The 2 uctuates around a constant value with average 2 = 1:39 ± 0:14. Such a value compares well to the typical values between 1.4 and 1.5 reported in the literature [3,9,14], whereas lower and higher values are usually considered signatures of a bad scaling behavior even when coupled to values of ∼ 0:5.
F. Cleri / Physica A 282 (2000) 339–354
349
Fig. 4. Time evolution of the second moment of the frequency distribution of n-sided grains. The horizontal line de nes the average value of 2 = 1:39 ± 0:14; the shaded region de nes the interval comprised around ±1 one standard deviation from the average.
In addition to fn there are a number of di erent topological properties which characterize the cellular network. Such properties should be self-similar in the scaling time range. We found a good scaling behavior and excellent agreement between the results of VMC simulations and deterministic models [11,14] for fn itself [31], for the distribution of the average number of sides of the neighbors of a n-sided grain [32,33], for the average normalized area of a n-sided grain [34], and for the grain-size distribution (see below). Based on purely geometrical considerations, Von Neumann [23] established that for any 2D uniform network the growth rate of the area of a n-sided grain, An , is given by dAn = 0 (n − 6) ; (13) dt where 0 is a coecient depending on the grain-boundary curvature. Although Von Neumann’s law is dicult to verify experimentally [4], it nevertheless represents a severe test for any theory or simulation of grain growth, and for our model in particular since it has been claimed [3] that vertex models with straight boundaries should violate it. For a N (0)=512 VMC simulation, we report in Fig. 5a the time evolution of the area of a n-sided grain, An (t) − An (0), for a sample of several n-sided grains. Furthermore, in Fig. 5b the same data are summarized as growth rates, by plotting the average slope of the curves in Fig. 5a as a function of the number of sides n. It can be seen that for 4 ¡ n ¡ 10 Von Neumann’s law is strictly obeyed and, in particular, for n = 6 the growth rate is zero. For values n ¿ 10 the grain population is very small and the growth rate becomes erratic, while for n=3 the growth rate is in uenced by the special treatment of T2 processes. Coupled to the values of and 2 above, the veri cation of Von Neumann’s law represents a strong con rmation that the stochastic grain-growth model with straight boundaries actually displays a scaling state. Notably, the growth rates in Fig. 5 have a measurable dispersion around the average at long times, as a result of the stochastic sampling in the VMC simulation.
350
F. Cleri / Physica A 282 (2000) 339–354
Fig. 5. (a) Time evolution of the grain area An (t) for a sample of several n-sided grains; grain areas are normalized by subtracting the value An (0) at the time of appearance of each grain; grains are followed until they change number of sides; (b) growth rates of n-sided grains, obtained from the average slopes of the curves in (a).
We now turn to the topological property which has been the subject of the largest debate, namely the grain size distribution P(x), expressed, as usual, as a function of Experimentally, the form of this distribution has been the scaled grain radius x = R= R, suggested to be close to log-normal [3,12,19]. Moreover, there has been a long debate about P(x) truly being a self-similar distribution [3,5 –7]. In Fig. 6 the VMC distribution, obtained as an average over 200 con gurations within the scaling time range, is compared with two theoretical distributions: the Rayleigh, a special case the log-normal rst proposed by Louat [19], and Hillert’s distribution, obtained by assuming scaling in a mean- eld theory [12]. It can be seen that the x ¡ 1:1 portion follows rather well the Hillert distribution, while the x ¿ 1:25 portion appears to decrease less steeply than Hillert’s, and rather follows the Rayleigh distribution. Such a behavior gives support to the analysis by Pande and Dantsker [35,36], who showed that these distributions are
F. Cleri / Physica A 282 (2000) 339–354
351
Fig. 6. Grain size distribution as a function of the scaled grain radius x = R= R for the VMC simulation (histogram), compared with the Rayleigh (dashed curve) and Hillert (full curve) theoretical distributions.
the two limiting cases for P(x), depending on the amplitude of the stochastic noise (Hillert’s corresponding to zero noise). Notably, the VMC distribution overlaps quite well the deterministic distribution obtained from the global variational solution [14]. It has been observed [20,25] that Hillert’s distribution does not represent well the experimental results but, on the other hand, Rayleigh’s distribution (which is frequently obtained both in theory and simulations [19,29,37] and ts some experiments) is unbounded and does not satisfy volume conservation. From Fig. 7 it can be seen that the VMC distribution is clearly cut o around xC ∼ 2:2. In mean- eld theories [12,22,25] the average number of sides of a grain, n(x), is assumed to be a unique function of the scaled radius x. In this way a well-de ned relation between An and n is imposed and Eq. (8) becomes strictly deterministic. For example, Hillert’s model assumes a linear n(x), while Fradkov et al. [37] have shown that a parabolic n(x) leads to the Rayleigh distribution. As shown in Fig. 7, the n(x) averaged over 200 VMC con gurations is linear, n(x) = 3:85 + 2:05x, in the range x ¡ 1:1; instead, for 1:1 ¡ x ¡ 2:3 a parabolic law, n(x) = 4:63 + 1:22x2 , very close to that corresponding to the Rayleigh distribution [25], is obtained. Trying to settle this issue within more rigorous bounds, Mullins [25] developed an elegant analysis in terms of the reduced growth rate, G(x) Z 2 xC P(x0 ) d x0 (14) G(x) = x − P x and proved that any theory or simulation based on curvature-driven growth and timescaling must satisfy the constraint n(x) = 6 + 6 xG(x), with a structural constant [25]. Firstly, we note that our model gives an average value of ∼ 0:136, well within the range predicted by most analytical models. Secondly, in Fig. 7 the best t for n(x)
352
F. Cleri / Physica A 282 (2000) 339–354
Fig. 7. Plot of the average number of sides of a grain, n(x) (best t to 200 VMC con gurations, full curve) as a function of the scaled grain radius x. The full circles represent the Mullins constraint, 6 + 6 xG(x), with the integral in G(x) calculated from the histogram in Fig. 6.
is compared to the r.h.s. of the constraint equation (obtained by integrating the VMC histogram for P(x) in Fig. 6): it can be seen that the VMC simulation satis es Mullins’ constraint in a wide range, 0:5 ¡ x ¡ 2:3, with a marked failure only in the small grain-size region. Such a result is even more meaningful, when noting that (a) Mullins’ analysis was originally intended to show that no stochastic term should be necessary beyond mean- eld arguments, and (b) our stochastic, curvature-less, straight-boundary model can reproduce the results of deterministic, curvature-driven models [14]. This latter observation suggests that the unbalanced line-tension at the triple junctions may represent an e ective curvature, which exactly replaces the contribution of the curvature term in Eq. (3).
5. Conclusions We proposed a stochastic model for the time evolution of a cellular network driven by dissipative forces, such as a polycrystalline microstructure undergoing grain growth. The model is based on a variational formulation of the dissipated power, rstly introduced by Needleman and Rice [13] in a study of cavitation at grain boundaries and subsequently extended to many other problems of microstructural evolution in two dimensions, including grain growth [14]. We gave a new interpretation to the variational functional established in those works, by using it as the transition-rate generating function in a stochastic model of microstructure evolution. The stochastic point of view here developed (see also [19 –21,35,36]) provides a model that can be used in kinetic Monte Carlo simulations (in the present case,
F. Cleri / Physica A 282 (2000) 339–354
353
“velocity” Monte Carlo, VMC). The model variables are the positions and velocities of the network vertices where cell walls meet. We applied such a model to the well-studied case of grain growth in two dimensions, in which the cellular network represents a cross-section of a polycrystalline microstructure and the cell walls represent grain boundaries. The VMC simulation results were found to compare well to deterministic results [7,11,14] for the most relevant statistical quantities. Our simulations give support to the existence of a time-scaling regime, as determined from the time evolution of the grain number and of the second moment of the frequency distribution of n-sided grains. We obtained a value of the dynamical exponent = 0:48 ± 0:01 and 2 = 1:39 ± 0:14, in good agreement with the best values obtained in deterministic simulations [3,14]. Notably, the VMC grain-size distribution displays a cross-over from Hillert to a Rayleigh-like distribution, as does the function n(x) describing the average number of sides of a grain, as a function of the scaled grain radius; such a behavior lends support to the suggestion [20,35,36] that a stochastic noise superimposed to the deterministic drift can interpolate between the two extremes of strictly deterministic growth (Hillert) and pure random walk (Rayleigh). Moreover, we veri ed that both the Von Neumann law [23] and the Mullins constraint for n(x) [25] are nicely obeyed by our straight-boundary model, whose convergence to the global equilibrium properties and scaling behavior have been amply proved. Such results give the a posteriori proof that stochastic uctuations can, indeed, account for non-local structural correlations. An important result is that VMC simulations can reproduce all the main features of curvature-driven models [3,7,14]: although our model can be easily extended to include curved boundaries, these results prove that the unbalance of the line-tension at triple junctions is, indeed, equivalent to the driving force from boundary curvature. From the point of view of physical realism, this model inherits all the exibility of the original variational formulation [13,14], which allows to extend this same description to include other dissipative mechanisms such as grain sliding and rotation and di usive processes [16 –18]. Such a capability of dealing with di erent dissipative mechanisms is extremely promising in view the study of real-material microstructures in which, moreover, it is essential to include energy and mobility anisotropy. We have already obtained interesting results on abnormal and oriented grain growth by including the grain-misorientation dependence of the energy and mobility, which will be described in a forthcoming paper.
Acknowledgements I am grateful to D. Wolf (ANL, Argonne) for prompting my interest in this area and for his kind hospitality at the ANL Materials Science Division. Useful discussions with G. D’Agostino (ENEA, Roma) are also acknowledged. This work was performed under the INFM-Forum “MUSIC” Project.
354
F. Cleri / Physica A 282 (2000) 339–354
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
C.S. Smith, in: Metal Interfaces, ASM, Cleveland, 1965, p. 65. H.V. Atkinson, Acta Metall. 36 (1988) 469. J.A. Glazier, D. Weaire, J. Phys. Condens. Matter 4 (1992) 1867. J. Stavans, Rep. Prog. Phys. 56 (1993) 733. D. Weaire, J.P. Kermode, Philos. Mag. B 48 (1983) 245. D. Weaire, J.P. Kermode, Philos. Mag. B 50 (1984) 379. F.J. Humpreys, Mater. Sci. Technol. 8 (1992) 135. S.M. Allen, J.W. Cahn, Acta Metall. 27 (1979) 1085. H.J. Frost, C.V. Thompson, Acta Metall. 35 (1987) 529. H.J. Frost, C.V. Thompson, C.L. Howe, J. Wang, Scr. Metall. 22 (1988) 65. K. Kawasaki, T. Nagai, K. Nakashima, Philos. Mag. B 60 (1989) 399. M. Hillert, Acta Metall. 13 (1965) 227. A. Needleman, J.R. Rice, Acta Metall. 28 (1980) 1315. A.C.F. Cocks, S.P. Gill, Acta Mater. 44 (1996) 4765, 4777. B.D. Vujanovic, S.E. Jones, Variational Methods in Nonconservative Phenomena, Academic Press, Boston, 1989. J. Pan, A.C.F. Cocks, Compos. Mater. Sci. 1 (1993) 95. J. Pan, A.C.F. Cocks, Acta Mater. 43 (1995) 1395. A.C.F. Cocks, J. Mech. Phys. Solids 44 (1996) 1429. N.P. Louat, Acta Metall. 22 (1974) 721. C.S. Pande, Acta Metall. 35 (1987) 2671. A. Thorvaldsen, Acta. Metall. 41 (1993) 1347. J. Stavans, J.A. Glazier, Phys. Rev. Lett. 62 (1989) 1318. J. Von Neumann, in: Metal Interfaces, ASM Cleveland, 1965, p. 106. N. Ryum, O. Hunderi, Acta Metall. 37 (1989) 1375. W.W. Mullins, Acta Mater. 46 (1998) 6129; K. Binder, M.H. Kalos, in: K. Binder (Ed.), Monte Carlo Methods in Statistical Physics, Topics in Current Physics, Vol. 7, Springer, Berlin, 1986 (Chapters 1, 6). A.P. Sutton, R.W. Ballu, Interfaces in Crystalline Materials, Oxford Science Publications, Oxford, 1995 (Chapter 9). M.P. Anderson, D.J. Srolovitz, G.S. Grest, P.S. Sahni, Acta Metall. 32 (1984) 783, 793. M.P. Anderson, D.J. Srolovitz, G.S. Grest, P.S. Sahni, Philos. Mag. B 59 (1989) 293. J. Pan, A.C.F. Cocks, S. Kucherenko, Proc. Roy. Soc. London A 453 (1997) 2161. T. Herdtle, H. Aref, J. Fluid Mech. 241 (1992) 233. E. Carnal, A. Mocellin, Acta Metall. 29 (1981) 135. D.A. Aboav, Metallography 3 (1970) 383. D. Weaire, Metallography 7 (1974) 157. D. Lewis, Anat. Rec. 38 (1928) 351. C.S. Pande, E. Dantsker, Acta Metall. 38 (1990) 945. C.S. Pande, E. Dantsker, Acta Metall. 39 (1991) 1359. V. Fradkov, L. Schvindlerman, D. Udler, Scr. Metall. 19 (1985) 1285.