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Materials Research

Print version ISSN 1516-1439On-line version ISSN 1980-5373

Mat. Res. vol.5 no.3 São Carlos July/Sept. 2002 

Stochastic Effects in Microstructure

M.E. Glicksman, K.G. Wang* and P. Crawford

Materials Science and Engineering Department,
Rensselaer Polytechnic Institute, Troy, NY 12180-3590


Received: September 27, 2001; Revised: July 10, 2002


We are currently studying microstructural responses to diffusion-limited coarsening in two-phase materials. A mathematical solution to late-stage multiparticle diffusion in finite systems is formulated with account taken of particle-particle interactions and their microstructural correlations, or "locales". The transition from finite system behavior to that for an infinite microstructure is established analytically. Large-scale simulations of late-stage phase coarsening dynamics show increased fluctuations with increasing volume fraction, Vv, of the mean flux entering or leaving particles of a given size class. Fluctuations about the mean flux were found to depend on the scaled particle size, R/<R>, where R is the radius of a particle and <R> is the radius of the dispersoid averaged over the population within the microstructure. Specifically, small (shrinking) particles tend to display weak fluctuations about their mean flux, whereas particles of average, or above average size, exhibit strong fluctuations. Remarkably, even in cases of microstructures with a relatively small volume fraction (Vv » 10-4), the particle size distribution is broader than that for the well-known Lifshitz-Slyozov limit predicted at zero volume fraction. The simulation results reported here provide some additional surprising insights into the effect of diffusion interactions and stochastic effects during evolution of a microstructure, as it approaches its thermodynamic end-state.

Keywords: microstructural evolution, coarsening processes, simulation, stochastic effect

1. Introduction

Phase coarsening is a spontaneous kinetic process in late-stage microstructural evolution that leads to the decrease in the interfacial energy of two-phase systems. The biphase system consists typically of individual dispersed domains, crystallites, or particles, distributed more-or-less at random through another contiguous matrix phase. During phase coarsening, the transport of atoms via lattice diffusion proceeds through the intervening matrix phase from solute sources at particle-matrix interfaces with high mean curvature towards solute sinks at particle-matrix interfaces with low mean curvature. Typically, in such systems, larger particles tend to grow at the expense of small particles. Over time, as the smallest particles dissolve, the "competitive" multiparticle diffusion process among particles results in an increase in the average size of their population, and in a concomitant decrease in their number. When matrix diffusion is the rate-limiting atomic transport process, the average particle size increases with sub-linear kinetics (cuberoot of time) and the particle number density decreases reciprocally with time. These kinetic "signatures", observed in many aging microstructures, are so robust and fundamental as to be virtually independent of the physical model.

The particle size and microstructure representative of many cast and liquid-phase sintered alloys are known to depend on similar phase coarsening processes. Moreover, the physical and chemical properties of these materials, such as their strength, toughness, ductility, electrical conductivity, and corrosion resistance all depend on the material's average particle size and particle size distribution (PSD). Understanding and controlling microstructures in two-phase systems are materials science topics with broad technological implications. Indeed, predicting microstructure evolution in general remains a cornerstone of modern materials science.

Predicting microstructure evolution quantitatively from first principles has an interesting 40-year history. Todes1 published the seminal paper on formulating the phase coarsening dynamics. However, the initial theory of capillary-mediated phase coarsening was completed by Lifshitz and Slyozov2, and, then independently, by Wagner3. These theories collectively are often referred to as "LSW theory". A major assumption in LSW theory is that particles with equal size experience the same time-rate of change of their volume or mass. LSW theory predicts that (1) the average volume (or cube of the radius) of the particle population increases linearly with time-a fact borne out by numerous careful experiments in many different systems; and (2) that the PSD predicted from LSW theory is self-affine, which implies that the particle patterns at later times appear statistically similar to those observed at earlier times, apart from a uniform global increase of size scale. However, the rate constant controlling the kinetics of the evolving length scales, and the PSD predicted from LSW theory, both disagree with the preponderant body of quantitative experiments on coarsening microstructures4,5.

LSW theory is a "limit law", insofar as it applies strictly to microstructures withzero volume fraction of dispersoid phase. The predicted PSD and coarsening rate based on LSW theory therefore ignores all interactions occurring among particles. For any volume fraction of the dispersoid phase, other than zero, the particles would have neighbors positioned at finite distances. Interactions among them clearly become possible. Recently, Glicksman, Wang and Marsh6 reviewed the interaction effects among dispersed particles, and described how such interactions influence the PSD and the kinetics of coarsening. Another source of error in the predicted PSD and coarsening rates at finite volume fractions is that caused by influences of the unique microstructural environment encountered by each particle -i.e., its "locale". Locale differences are ignored when adhering to the tenets of mean-field theory, whereas, in fact, it is known that particles of the same size located at different positions in a microstructure may exhibit different growth rates.

The subject of stochastics in late-stage phase coarsening, although still in an early stage of development, has been discussed recently by Glicksman et al.7,8. The interactionsamong particles and the fluctuations in their individual growth rates induced by their "locale" are complicated influences, so it is not surprising that it is difficult to treat stochastics either analytically or experimentally. It is for this reason that to gain even limited heuristic insight into the stochastics of microstructures, large-scale simulations must play a centralrole. The earliest study of multiparticle phenomena in micorstructures was published in 1973 by Weins and Cahn9, who used a few particles to simulate some basic coarsening interactions during sintering. Their study was followed by Voorhees and Glicksman10 who systematically studied the behavior of several hundred particles randomly placed ina periodic, three-dimensional, unit cell to simulate coarsening. Later, Beenaker11 further improved multiparticle simulation procedures and was able to increase the total number of particles during simulation. More recently, other investigators8,12-15, continued to improve the accuracy of large-scale simulation of microstructural coarsening processes. Studying the interactions among many particles and including the statistics of their fluctuations during growth and dissolution, even in the case of very small volume fractions, can contribute important facts required in the understanding of real microstructures. In this paper, we shall focus on early progress being made on these issues at our laboratory.

This article is organized as follows: The theories for a finite system with nonzero volumefraction, along with that for an infinite system with zero volume fraction (LSW) are presented in section2. In section 3, details of the modeling and numerical simulations are given. The results for the observed growth law for particles, their steady-state PSD's, and the characteristics of the statistical fluctuations in individual particle growth rate are presented in section 4. Finally, a few conclusions are presented in the last section.

2. Theoretical Analysis

2.1 Theory for finite microstructures with nonzero volume fraction

One can simulate only finite microstructural systems, due obviously to the finite nature of computing power and memory. A theoretical approach to treat a finite system comprisesthe first essential step to improve the extent and accuracy of numerical simulations. We proceed with an analysis of a system consisting of a large but finite number of polydisperse spherical precipitate particles suspended throughout the volume of a three-dimensional,isotropic matrix phase. Although we begin this analysis with a set of microstructural definitions more aptly applied to an infinite system, eventually we restrict and apply the mathematical results to finite microstructures.The sizes of the domains comprising the population of dispersed particles are described with a distribution function, F(R, t), defined here conventionally as the number of particles per unit volume at time t, with radii in the range R to R + dR. The normalization for F (R, t) is based on the total number of particles per unit volume, Nv, that is,

where Nv is the number density of spherical particles. With this normalization the corresponding volume fraction, Vv, of the dispersed phase (total volume of dispersoid per unit volume of the microstructure) is defined consistently as

Equation 2 may be rewritten as

where <R3> represents the average of the cube of the particle radii, R.

The biphase microstructure under consideration is aged sufficiently so that the supersaturation of the matrix solution may be assumed to small. Thus, it follows, that nucleation of new particles within the aged microstructure is precluded. This condition constitutes the chief thermodynamic requirement for "late-stage" phase coarsening.

The continuity equation for smoothly growing and dissolving dispersed particles in size space is

Here v(R) is the time-rate of change of the radius, R, of a particle. Equation 4 precludes "non-smooth" changes inthe microstructure such as nucleation, particle splitting, and agglomeration, To obtain v(R), some additional simplifying assumptions are needed: (1) the kinetics of coarsening is determined by volume diffusion through the matrix; and (2) the diffusion transport to or from each particle occurs slowly enough to be considered quasi-static. These additional assumptions justify approximating the diffusion equation with Laplace's equation to describe the concentration fields, C(r), in the matrix. Thus, the diffusion field surrounding a particle may be described by

where C(r) = (c(r) c0)/c0 defines a dimensionless diffusion potential, c(r) is the concentration at any point defined by the position vector, r, and c0 denotes the equilibrium solubility at a flat interface between the matrix and particle phases.

The boundary conditions at the spherical interface of the ith particle are specified through the Gibbs-Thomson (or Thomson-Freundlich) local equilibrium solubility relation, namely,

where lcis a capillary length, usually of near atomic dimensions, defined by

InEq. 7, g denotes the specific interfacial free energy between the particle and matrix, Wa is the particle's atomic volume, kB is Boltzmann's constant, and T is the absolutetemperature.

The solution to Laplace's equation for n particles (some acting as sources and someas sinks) distributedas a three-dimensional polydispersion throughout the matrix may be represented as the superposition of n dimensionless concentration fields summed over themicrostructure.

The vector r appearing in Eq. 8 is the field point, always located in the matrix, whereas ri is a vector that locates the center of the ith particle in the microstructure.

The ith particle's volume flux equals 4pBi, and the far-field potential C¥, comprise together,for all n particles, a total of n + 1 unknowns that must be determined in solvingEq. 8 for the microstructure's diffusion field. The far-field potential C¥ is found by using global mass conservation. The following conservation law may be written for a discrete systemconsisting of n spherical particles each competing for, but also conserving, thesolutetransported via diffusion through the matrix phase:

Substituting the expression for the diffusion potential, Eq. 8, along with the mass conservation condition, Eq. 9, one obtains after a few steps of algebra

where rjk is the distance between the centers of any pair of particles j and k. The relationship, Eq. 10, used to define C¥, is employed to interpret microstructure responses for the firsttime. Eq. 10 clearly demonstrates that the far-field potential, C¥, depends explicitly on the size of the microstructural system, i.e. , on the number of particles,n, andon local information concerning particle positions and the distances between pairs of them. Infact, Eq. 10 establishes an important bridge between detailed computer simulations carried out for a finite, discrete system of n particles, and the averaged behavior predictedfor an infinite, continuous systems. Equation 10, in fact, includes enough detailed environmental information to describe the "locale" of every particle, and, most importantly, its subtle influence on the particle's diffusion-limited growth or shrinkage. The environmental information built into Eq. 10 is normally excluded from mean-field descriptions of microstructure evolution, but if included, adds important microstructural physics to the diffusion solution.

The assumption of describing C(r) using a quasistatic (conservative) diffusion field requiresthat the mass transfer to or from a particle is equal to its change of mass. Thus, using Eq. 8, the time-rate of change of a particle's radius, dR/dt, is directly connected to the volume flux, Bi, as

where D0 , appearing on the right-hand side of Eq.(11), is the interdiffusion coefficient for solute inthe matrix.

Inprinciple, the volume fluxes, Bi, can be determined by substituting both the Gibbs-Thomson local equilibrium relation, Eq. 6, and the far-field potential, Eq. 10, into Eq. 8. However, one cannot obtain a useful analytical result for a finite system, excepting the limiting case of an infinite system with zero volume fraction. We can determine numerically the volume fluxes from Eqs. 6, 8, and 10, and then substitute the values of the Bi' s into the growth rate relationship, Eq. 11. In this manner, one may solve Eq. 11 numerically, and find the so-called "kinetic" equation for particle growth and shrinkage, enabling prediction of the particle size distribution. The details for accomplishing this are provided inthe next section.

2.2 Theory for infinite systems with zero volume fraction

If the number of particles comprising a biphase microstructure rises without limit, so n ® ¥, but the matrix volume is allowed to increase sufficiently to become diluted toan infinitesimal particle density, so that Vv ® 0, (i.e., distances, rjk, between particle pairs increase indefinitely) then Eq. 10 reduces to the limiting form

Eq. 12 was derived originally by Todes1 under his assumption of an "infinite system" with zero volume fraction of particles. For the case of such an infinite, yet infinitely diluted system, one may substitute Eq. 12 and Eq. 6 into Eq. 8 and show that the volume fluxes per steradian, Bi' s, reduce to the linear form

If it is assumed that <R> = R*, where R* is the "critical radius" defined later in Lifshitzand Slyozov's paper2, then Eq. 13 reduces to the usual LSW formulation. Substituting Eq. 13 into Eq. 11, the LSW growth rate for a particle becomes

Substituting Eq. 14 into Eq. 4, finding the self-similar solution to Eq. 4, and then applying the LSW stability argument based on mass conservation for the particle population that 1 there exists some maximum particle size and 2 that the timerate of changeof the maximum sized particle, relative to the growing average, must vanish, yields the (dimensional) growth law of any particle of radius R as

Equation 14, which provides the basis for Equation 15, is a deterministic growth rate law, insofaras the rate of growth or shrinkage of every particle is solely a function ofitsradius. Indeed, in such an infinitely-diluted microstructure, each particle would exist perhaps somewhat illogically-in total isolation from all its "neighbors." LSW theory also predicts that the particle size distribution (PSD) is affine, or self-similar. If the particle sizes are expressed as radii normalized to the growing average, or critical, radius, then the affine form of the PSD does not change with time. Self-similarity of coarsening quasi-spherical microstructures has been demonstrated experimentally in Pb-Sn alloys by Hardy and Voorhees14.

2.3 Microstructural interactions

In order to extend LSW theory to cases of non-zero volume fraction, a number of markedly different approaches were developed. Marqusee and Ross16 were the first to model the effects of non-zero volume fraction on phase coarsening kinetics by using active-medium theory to describe the quasi-static diffusion fields. The emission of solute from dissolving particles, or absorption of solute from growing ones, are modeled by using point sources or sinks of solute distributed within the matrix. Furthermore, those investigators limited the spatial extent of diffusional interactions by allowing "diffusional screening" to occuractive-medium approach led to the use of Poisson's equation to replace the Laplace approximation for quasi-static diffusion,

where the source or sink density, s, appearing in Eq.(16) is given by

Re-organizing these Eqs. 16 and 17 yields the diffusion analog of the Debye-Hückel equation, namely13

Here k º (4pNv<R>)1/2 is introduced as the diffusion analog of the reciprocal of the Debyescreening length, and C¥ = lc/R* » l c/<R> is the far-field diffusion potential. Eq.(18) is well known from theories of ionic solutions and plasmas. Inthree dimensions, the general solution to the Debye-Hückel equation for diffusion around a spherical source or sink,subject to the Gibbs-Thomson boundary condition, may be expressed in the form of the well-known Yukawa potential. The spherically symmetric diffusion solution being sought may be written interms of the volume fluxes as

where now the Bi' sare defined as

The diffusion fields obeying Eqs. 19 and 20 surrounding spherical particles embedded in an"active" matrix phase are shown plotted in Fig.1. For illustrative purposes, the critical particle size, R*, used in Fig. 1 to normalize all length scales, was chosen to be lc , so that the mean concentration, or diffusion potential, C¥ = 1. (In reality, dispersoids in aged microstructures usually have critical radii between circa 103 lc to 105 lc , so that our choice of R*/lc merely introduces a convenient scale factor of unity into the diffusionpotential). Inaddition, the dispersoid volume fraction is arbitrarily set to be 0.05, which is sufficiently small so that R* » <R>, but large enough to have interactions occurringamong the particles. Note that the fields plotted in Fig. 1 show that the gradients are negative around all particles smaller than the average, i.e., those for which r <1, and that these gradients increase sharply in magnitude as the particles become smaller and more curved. Also, the gradients adjacent to larger particles-those for which r >1-remainrelatively small when compared to the gradients around smaller particles. The gradients surrounding "critical" particles are, by definition, zero, because at the instant depicted in Fig. 1, "critical" particles-those for which r= l -are conditionally stable, and wouldneither be growing nor shrinking. The scheme of relative diffusion potentials and their gradients depicted in Fig. 1 makes clear the "competitive" nature of multiparticle diffusion. Specifically, at every instant, small particles (relative to the average-size particle) are forced to shrink rapidly; larger than average-size particles grow at the expense of dissolving smaller particles, and average-size (critical) particles themselves are conditionally stableand neither grow nor shrink. What makes the overall kinetics of competitive multiparticle aging so interesting is that the average-size particle in the population itself increases with time, in accord with Eq. 15. Thus, the polydisperse population of particles is constantly in "competition" with average-size particles that inexorably grow as the cube-root of time. The net result of this coupled competitive diffusion process is a microstructural populationthat maintains an affine distribution for all time. That is, the PSD remains identical except for a scale factor that increases as the cube-root of time.



Glicksman, Wang and Marsh6 recently proved that the diffusional Debye screening length LD (k-1), which effectively limits the distance over which the diffusion potential of one particle affects another, is related to the ratio of moments of the PSD and to the reciprocal square-root of the volume fraction of the system,

Equation 21, in fact, can be employed to gauge the interaction length scale in a three-dimensional microstructure undergoing diffusion-limited coarsening, and compare the results with computer simulation or experiment. Moreover, as indicated in Eq. 21, thevolume fraction of the microstructure, Vv , is the main factor that acts to reduce the Debye screening length. The population of dissolving and growing particles collectively act to "cuts off" the diffusion field emanating from each particle beyond the Debye screening distance, LD. This interesting screening effect is shown in Fig. 2, where the diffusion potentials for a small particle (r < 1), a critical particle (r= l) , and a large particle (r >1) are plotted for several values of the volume fraction. A volume fraction of zerodenotesthe behavior expected in the LSW model, where the screening distance becomes infinite, and the Yukawa potential reduces to the Laplace potential. At non-zero volume fractions, the gradients steepen, especially around larger particles, and the diffusion fields drop off to the mean potential (C¥= 1) much faster than for the limiting case where Vv = 0. Thus, Debye screening tends to speed up diffusive transport in a microstructure, and increasesthe kinetics of phase coarsening.



3. Modeling and Simulation of Microstructures

3.1 Modeling

The microstructure is modeled by placing n particles of the dispersoid phase in a cubic box.The contiguous spaces between the particles represents the matrix phase in which the dispersoid population is embedded. Particles are located by specifying the positions of their centers with three random coordinates representing the Cartesian vector, ri, and by their radii, Ri, chosen initially from a relatively narrow Gaussian distribution. These radii are non-dimensionalized by the capillary length,lc, a material property based on thermodynamic data, and the microstructure evolution time is non-dimensionalized by a characteristic diffusion time, another property based on the materials transport coefficients, definedas t d º - lc2/(WaD0 c0). The dimensionless form of the growth rate, Eq. 11, can be written as

where Ri is known at the time t. Assuming that all the Bi's are known, numerical integration of the growth rate, Eq. 22, can be used in a forward marching scheme to evolve the radii. The Runge-Kutta technique, was used to evaluate Ri (t + Dt). Here, Dt represents each subsequent time step in the marching integral. One, of course, must re-determine the volume fluxes, Bi at each subsequent time step.

Introducing Eq. 8 into Eq. 6, along with the non-dimensionalized system of linear equations, one may cast these equations into matrix form as follows:

where A is the (n + 1) by (n + 1) matrix

and where rij represents the separation between a pair of particles i and j. B and U are (n + 1) by 1 column matrices, namely,


Equation 23 represents (n + 1) linear equations. Substituting Eq. 10 into Eq. 23 and eliminating the (n + l)th row and column in matrix A, allows reduction of Eq. 23 to n linear equations, which may be written as the matrix equation

where A' is the n by n matrix


The Gauss-Seidel method was employed to solve this system of linear equations, Eq. 27, yielding at each time step values for the Bi's. Substitution of the up-dated Bi's back into Eq. 22 dynamically advances the microstructure by up-dating the radii of all the particles and their coordinates at any time step.

3.2 Simulation of microstructures

As described briefly above, the simulation of the evolving microstructure is initiated by defining an assembly of n particles obeying some initially chosen size distribution. These initial particles are placed randomly within a cubic box to comprise the initial microstructure. The box size may be determined from the specified volume fraction, Vv, the initial number of particles, n(0), and their starting distribution.

The choice of time-step in performing the integration of Eq. 22, is crucial. Large time-steps that allow the system to evolve quickly do not permit sufficiently accurate solutions to the microstructure evolution equation, Eq. 22. Too large a time step may result in an unsatisfactory PSD. At each time step, the computational program checks the radius of every particle. If any became smaller than 0.1l<R>, they are removed from the simulation box and considered "dissolved". Equivalently, the column and row corresponding to a dissolvedparticle is struck from the matrix, Eq. 28. The initial number of particles forming the microstructure influences the total runtime. Specifically, data are collected from each simulation throughout the time period for which mass is conserved and the volume fraction remains constant to within a prescribed tolerance. It is observed, not surprisingly, that the greater the number of particles forming the initial microstructure, the better is the accuracy of the simulation, the longer the run may be evolved, and the faster the statistics for the data settle to steady trends. However, numerical difficulties are eventually encountered as the number of particles,n, and thus the number of linear equations, n2, increase.Thelarger the number of particles, the greater is the required computer memory, and the longer is the CPU time required to evolve the system. Ultimately, the increasing CPU time limits the practicality of these simulations for large microstructural assemblages.

4. Results

Simulation of microstructural coarsening was carried out for volume fractions covering the range 10-4 < Vv < 10-1.Figure 3 shows that for values of the time, t, larger than about 0.3, the cube of the average radius of the particles increases linearly with time. The "cube-root of time" kinetics, is a well-known prediction from LSW theory for diffusion-limited coarsening, and is confirmed by numerous experiments. Figure 4 displaysthePSD developed in the simulated microstructure. The radii plotted here are scaled by the average radius of the particle population for a volume fraction of Vv = 10-4.The PSD predicted from LSW theory (Vv = 0) is shown in Fig. 4 for comparison. Neither prior theoretical, experimental, nor simulation results are available for Vv = 10-4.Figure 4 clearly indicates that microstructures with even a small volume fraction of particles exhibit detectable broadening of the PSD and a marked decrease its peak height, as compared withthat predicted from LSW theory.





Figure 5 displays the form of the computed volume fluxes, B(r)for zero volume fraction and for Vv = 10-4. LSW theory predicts that the volume flux has a linear variation with the scaled particle radius, that is, B(r) = 1 - r. However,as is also indicated in Fig. 5, the simulation for Vv = 104 shows clearly that particles of identical size actually exhibit a small range of flux values scattered around the LSW prediction. The growth rate experienced by a particle of radius r, even at a relatively small volume fraction, is not a purely deterministic quantity as predicted by mean-field theory.



Figures 6 through 8, show the volume fluxes B(r) simulated for Vv = 10-3, Vv = 10-2, and 10-1, respectively. These figures demonstrate that the B(r)'s steadily deviate from the linearLSWprediction. It is also evident from these simulations that the fluctuation, or "noise" band for the volume fluxes increases with increasing volume fraction.







5. Conclusions

Microstructure evolution in a two-phase system was simulated by solving a large linear matrix equation that describes multiparticle diffusion. Particles interacted in the simulated microstructure under conditions close to steady-state, or affine, coarsening. The far-field diffusion potential used in these simulations is derived here for a finite system [c.f. Eq. 10] in this paper. The effects of system size and microstructure locale information for each particle were included in the far-field diffusion potential. As the volume fraction becomes small, our results approach those of Todes1 and LSW2,3, exclusive of the weak interactions that persist among particles.

Simulations of diffusion-limited coarsening carried out on sparse to moderately dense microstructures (10-4 < Vv <10-1) display classical coarsening kinetics that is the cube of the average radius increases linearly with time. However, the PSD computed for a biphase microstructure as sparse as Vv = 10-4 remains somewhat broader and more symmetric than that predicted by LSW. This is the first PSD derived using simulation at such a smallvolume fraction.

Mean-field theory treats the growth rates of particles as a deterministic kinetic process thatdepends only on the size of a particle relative to the average size of the entire microstructural population. In addition, mean-field theories ignore environmental information for individual particles. From the simulations reported here, however, we find that the fluctuation band fora particle's volume flux widens with the increase of volume fraction. Larger particles experience stronger fluctuations, or locale noise, in their growth rates than do smaller particles. The non-linear character of the particle volume flux versus particle size increases steadily with increasing volume fraction. Finally, this work suggests that stochastic effects must always be present in real biphase microstructures and may play an important role in microstructure evolution. Once the statistical nature of microstructural fluctuations are understood, then theories of microstructure evolution can be modified to include their effect. The authors hope that this work will stimulate quantitative experiments that focus on systems at small volume fraction, to help achieve firmer understanding of multiparticle interactions within evolving biphase microstructures.


The authors are pleased to acknowledge partial financial support received from the National Aeronautics and Space Administration, Marshall Space Flight Center, Huntsville, AL, under Grant NAG-8-1468.


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