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Transient Ostwald ripening and the disagreement between steady-state coarsening theory and experiment
Acta mater. 49 (2001) 699–709 www.elsevier.com/locate/actamat
TRANSIENT OSTWALD RIPENING AND THE DISAGREEMENT BETWEEN STEADY-STATE COARSENING THEORY AND EXPERIMENT V. A. SNYDER, J. ALKEMPER and P. W. VOORHEES* Department for Materials Science and Engineering, Northwestern University, Evanston, IL 60208-3108, USA ( Received 2 August 2000; received in revised form 4 October 2000; accepted 4 October 2000 )
Abstract—The coarsening of solid-Sn particles in a Pb–Sn liquid has been studied under microgravity conditions. These experiments permit an unambiguous comparison between theory and experiment to be made. In contrast to steady-state theories, such as those due to Lifshitz and Slyozov and Wagner, the scaled particle size distributions evolve in samples containing 0.1 and 0.2 volume fractions of solid. Steady state was not reached even though the average particle radius increased by a factor of three during the experiment. In addition, the scaled spatial correlation functions were also found to be time dependent in samples containing 0.1, 0.2, and 0.3 volume fractions of solid. The size distributions and correlation functions for all coarsening times at the fractions ⱕ0.3 agree with the predictions of a theory for transient coarsening. We show that the microstructures have not reached the steady-state regime for all volume fractions, are thus not self-similar, and that given our initial experimental conditions the time required to reach steady-state coarsening increases with increasing volume fraction. In these experiments, and we suspect in others as well, the transients are sufficiently long that steady-state theories cannot adequately describe the evolution of the microstructure. 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Ostwald ripening; Phase transformations; Diffusion
1. INTRODUCTION
Ostwald ripening, or particle coarsening, is a competitive growth process in which a system lowers its energy by reducing its interfacial area. During coarsening, large particles grow at the expense of small particles. This leads to an increase in the average particle size and a decrease in the particle number density over time. The process occurs in a number of two-phase mixtures, and is of critical importance for example in precipitate hardened materials, where overaging via coarsening reduces the strength of the alloy. Recognizing the importance of the process, much work has been done on the coarsening problem both theoretically [1–13] and experimentally [14–17]. Unfortunately, a large discrepancy exists between the theoretical predictions and the experimental results. This article addresses this issue and presents one reason for this discrepancy. The classical description of Ostwald Ripening was developed by Lifshitz and Slyozov [1] and Wagner
* To whom all correspondence should be addressed. Fax: ⫹1-847-491-7820. E-mail address: [email protected] (P.W. Voorhees)
[2] (LSW). They examined the coarsening process in a system of infinitely separated spherical particles. In the limit t→⬁ a similarity solution to the system of governing equations was found, and the regime in which this solution is valid has become known as steady state. In this regime, the microstructure is time invariant, or self-similar, when scaled by the timedependent average particle radius. The average particle radius, R¯(t), was also found to increase with time, t, as R¯3(t)⫺R¯3(0) ⫽ KLSWt
(1)
where KLSW is the LSW coarsening rate constant. For an isothermal solid–liquid system the LSW rate constant, KLSW is given by [14]: K ⫽ KLSW ⫽
8 T0⌫D . 9ML(CS⫺CL)
(2)
T0 is the coarsening temperature, ⌫ is the capillary length of the matrix or liquid phase, D is the diffusion coefficient of the solute in the liquid, ML is the slope of the liquidus curve at T0, and CS and CL are the
1359-6454/01/$20.00 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 3 4 2 - 6
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equilibrium compositions of the solid and liquid at a planar solid–liquid interface at T0. KLSW, thus, is a function only of various materials parameters. In 1989 Brown showed that particle size distributions (PSDs) other than those predicted by LSW are possible [18]. This result has recently been confirmed under the constraint that there is no noise present in the system sufficient to perturb the maximum particle scaled size above that predicted by LSW [19–21]. As noise is present in systems undergoing coarsening, only the stable steady-state distributions are expected to be observed experimentally [22, 23]. The LSW theory is only valid in the limit of a zerovolume fraction of coarsening phase. Noting this critical shortcoming, much theoretical work has been done to account for the effects of a finite volume fraction of coarsening phase in the steady-state regime [3–11, 13]. The major difficulty in describing the behavior of a finite volume fraction system lies in the overlap of the diffusion fields surrounding the particles. The theories determine a statistically averaged growth rate for a particle of a given size, but differ in the methodology of this calculation. With the exception of effective medium theories [7, 8], most theories are limited to low volume fractions of particles (ⱕ0.3) where the assumption of spherical nontouching particles is reasonable. All of the theories predict that a finite volume fraction of particles does not alter the temporal exponent of the average particle radius from the LSW prediction of 1/3. A nonzero volume fraction does, however, alter the coarsening rate K: K(f) ⫽ KLSWf(f)
(3)
where f(f) is a system-independent function of the volume fraction, f. K(f) now appears instead of KLSW in equation (1). The theories recover the LSW results at zero volume fraction [f(0) ⫽ 1]. These stable scaled particle size distributions are also predicted to be a function of the volume fraction, becoming broader and more symmetric with increasing volume fraction. Despite this agreement among the various theories, the predictions of f(f) and the form of the steady-state PSDs differ from theory to theory. Although most theories make predictions in the steady-state regime, only a few theories have been developed that can be used to describe the transient behavior of a system undergoing coarsening, i.e. the evolution of the system prior to reaching the selfsimilar regime. Venzl [24] developed a theory in which the evolution of the particle size distribution is determined numerically, and the width of this distribution can be used to calculate the coarsening rate of the system. The theory developed by Chen and Voorhees [12] is a continuum model based on the growth rate equation determined by Marqusee and Ross [6]. They found a strong dependence of the time
required to reach steady state on the form of the PSD at the beginning of coarsening. As a result of the mean field nature of the model, however, it is incapable of accounting for the effects of the local spatial arrangement of particles. The theory developed by Akaiwa and Voorhees (AV) [3] uses numerical simulations to study an ensemble of particles undergoing Ostwald ripening. Individual particle growth rates are determined by calculating the diffusion field surrounding each particle. This theory can be used to determine the evolution of both a nonsteady-state PSD and a non-steady-state spatial arrangement of particles [25, 26]. In reviewing the many experiments on Ostwald ripening in the literature, Ardell claimed that, there is a “chronic disagreement” between experimental results and theoretical predictions [27]. While many experiments show that the average particle radius increases as t1/3 [14–17], it has not been possible to determine if the measured rate constant was correct as some of the material parameters appearing in KLSW are unknown. In addition, many experimentally measured particle size distributions are wider than those predicted by theory [17, 28]. Furthermore, there have been no rigorous tests of the low volume fraction theories. The difficulty in testing Ostwald ripening theory lies in the assumptions employed by the theories, namely: a low volume fraction of particles, mass transport due solely to diffusion within the matrix, spherical particles (an isotropic interfacial energy), and a stress-free matrix. The last requirement is particularly difficult to attain in coherent two-phase solid mixtures. In addition, a rapid coarsening rate is desired such that significant changes in the average particle size are possible in a reasonable amount of time. A number of experiments have been performed using solid–liquid mixtures in the Cu–Co system as it is stress-free, and nearly isopycnic. This density match allows a skeletal structure of particles to form in the system at volume fractions as low as 0.25, so that large-scale sedimentation can be avoided. Although this is too high to test most theories, it permits a relatively wide range of volume fractions to be employed in the experiments. Kang and Yoon studied the coarsening of solid Co-rich particles in a liquid Cu rich matrix at annealing temperatures ranging from 1150 to 1300°C and volume fractions of solid of 0.34, 0.42, and 0.55 [16]. The scaled PSD appeared to be time-independent over a factor of 10 change in R¯, and the rate constant was found to increase with volume fraction. Bender and Ratke [17] recently also studied the coarsening of Co-rich particles using volume fractions of solid ranging from 0.25 to 0.7. In contrast to Kang and Yoon, they found that the volume fraction of coarsening phase did not have a measurable effect on the rate of coarsening and that the particle size distributions are broader than those predicted by theory. They proposed that the solid skeleton of particles was responsible for these
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effects. They noted that the width of the PSD decreases with coarsening time, however, they considered this effect sufficiently small to claim that the PSDs appeared stationary in time. Mahalingam et al. [29] studied the coarsening of d⬘ precipitates in Al–Li alloys with volume fractions ranging from 0.12 to 0.55. Since this is a solid–solid system, particle sedimentation did not occur. Moreover, unlike many other two-phase solid systems, the misfit is nearly zero and thus elastic stress in the system is very small. By using particles located far from dislocations or grain boundaries the evolution of the average particle size and particle size distributions was determined. Although R¯ was found to increase as t1/3, they concluded that none of the existing theories (comparisons were made to [5, 7, 30, 31]) could predict the coarsening rate or the particle size distributions observed in the experiments. While the size distributions appeared not to evolve during this experiment, the change in average particle size was only a factor of 1.6, and the limited number of particles used to construct the particle size distributions would have made it difficult to detect any evolution. The implications of this small change in R¯ are discussed below. A mixture consisting of solid Sn particles in a Pb– Sn liquid has been identified as an ideal system to study coarsening since it satisfies all of the aforementioned assumptions of theory [14]. The difficulty with the Pb–Sn system at low volume fractions of solid, however, is that the solid and liquid have different densities causing sedimentation in a matter of seconds. Low volume fraction experiments are therefore not possible in the presence of gravity. Hardy and Voorhees [14] performed experiments using a high volume fraction of solid to prevent particle motion and sedimentation. They found that the intercept length distributions were time independent when scaled by the average particle size and in reasonable agreement with the predictions of steadystate theory of Glicksman and Voorhees [5]. They found t1/3 coarsening kinetics, but the measured coarsening rate constants exceeded the theoretical values by factors ranging from 2 to 5. However, it was difficult to determine if steady-state coarsening had been achieved since intercept length distributions are relatively insensitive to the shape of the three-dimensional particle size distribution. It is also possible for the slow sedimentation that was present in the system, and thus the change in the volume fraction, to mask the sometimes subtle changes in the PSD that can accompany a transient ripening process. We examined the coarsening process in Pb–Sn using a range of volume fractions from 0.1 to 0.7. This range includes volume fractions where the theories are believed to be most valid and those where a skeletal structure of particles is present. This gives us a complete, unbiased view of the coarsening process over a large range of volume fractions. To avoid
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the sedimentation of the Sn particles, experiments were performed in microgravity. We first present the results from this coarsening in solid–liquid mixtures experiment (CSLM). PSDs and interparticle spatial correlations are shown for various volume fractions over a range of coarsening times, and are compared to the predictions of transient as well as steady-state theory. Values of KLSW are deduced from the coarsening kinetics and are compared to KLSW computed from the known thermophysical parameters of Pb–Sn solid–liquid mixtures. Finally, we discuss the results of other experiments in light of our results. 2. EXPERIMENT PROCEDURE
A two-phase mixture consisting of solid-Sn particles in a Pb–Sn liquid is an ideal system with which to study Ostwald ripening. The high coarsening rate of the system allows for a large change in R¯ over a reasonable time interval. The interfacial energy is nearly isotropic, the thermophysical parameters of the system have been measured [32], and thus a comparison between experiment and theory is possible with no adjustable parameters. Particle sedimentation was avoided by coarsening samples in the Microgravity Science Laboratory aboard Space Shuttle Columbia, STS83 and STS94. Samples of Pb–Sn were cast and cold-worked using the method employed by Hardy and Voorhees [14]. They were then heated to the solid–liquid regime and held at the coarsening temperature and quenched. This processing procedure took place in a compact isothermal facility in microgravity. Samples were retrieved from the shuttle, sectioned, etched, and photographed. Several images were taken from each section and montaged to form images of entire sample cross-sections, thus avoiding all edge effects in the image analysis. A number of microstructural characteristics were measured using these sample cross-sections. The circular equivalent average particle radius, particle size distributions, and spatial correlation functions were all measured on these plane sections (PS). Results are shown for volume fractions of 0.1, 0.2, 0.3, 0.5. and 0.7 solid. In order to accurately measure particle sizes in the high volume fraction samples, touching particles were cut along the grain boundary. Between 2000 and 10,000 particles were used for each measurement at a given volume fraction and coarsening time. The shortest coarsening time analyzed is 550 s, since the furnace had not reached a thermal equilibrium before this time. Experiment times ranged up to 36,600 s, however, temperature gradients in the furnace effected the coarsening in all samples at long times with the exception of f ⫽ 0.1 and 0.2. At the higher volume fractions, the longest usable coarsening time was 9510 s due to temperature gradients. The effect of temperature gradients on coarsening has since been studied and is published elsewhere [33].
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Microstructures obtained after coarsening for 5810 s in systems with f ⫽ 0.1, 0.3 and 0.7 are shown in Fig. 1. The Sn particles are white and the Pb–Sn matrix is black. The f ⫽ 0.1 microstructure shown in Fig. 1(a) consists of isolated spherical particles and thus is ideal to test the low volume fraction theories of coarsening. The f ⫽ 0.3 microstructure shown in Fig. 1(b) contains spherical particles that are not isolated. Instead, these particles are much closer together. The particle packing that occurs at f ⫽ 0.7, see Fig. 1(c), is clear and induces a clearly nonspherical particle morphology. In our previous report [25] we showed that the spatial correlations were a far more sensitive measure of the presence of steady-state coarsening than PSDs. The spatial correlations are described by the radial distribution function (RDF). The RDF is defined as the ratio of the number of particles per area whose centers lie within a circular shell of radius x and
thickness dx surrounding a particle, to the number of particles per area of the entire system. RDFPS ⫽
NA in a circular shell of radius x to x ⫹ dx N¯A (4)
where NA is the number of particles per area and N¯A is the overall particle density. For nearly all spatial distributions of particles, RDFPS→1 as x→⬁. In order to assess the presence of steady-state coarsening we scaled x by the average particle size. If the system is in steady state, the RDFs measured at different times should superimpose. PSDs measured on plane sections are denoted as PSDPS. However, theoretical predictions for the size distribution are usually made in three dimensions. In order to compare the theory to the experimental
Fig. 1. The microstructure present in the experiment after 5810 s of coarsening. (a) ø ⫽ 0.1, (b) ø ⫽ 0.3, and (c) ø ⫽ 0.7.
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results, the theoretical three-dimensional size distributions were converted to plane section distributions by a method developed by Wicksell [34, 35]. A similar stereological problem was found with the spatial correlation functions. In this case, a computer program was developed to section the simulated microstructures and produce the corresponding RDFPS. A comparison between PSDs and RDFs measured in the experiment was made with those determined from the AV simulations of transient coarsening. In order to account for the transient coarsening seen in the experiment, these simulations used as initial conditions the PSD and RDF similar to those measured experimentally after a short coarsening time. PSDs and RDFs as measured from the experiment at later coarsening times were then directly compared to those calculated from the simulations, see [25]. 3. EXPERIMENTAL RESULTS
PSDs and RDFs were measured for each volume fraction and coarsening time. The coarsening times shown in Figs 2–6 are the initial, middle, and final
Fig. 3. The temporal evolution of (a) the scaled PSD and (b) the scaled RDF for f ⫽ 0.2.
Fig. 2. The temporal evolution of (a) the scaled PSD and (b) the scaled RDF for f ⫽ 0.1.
coarsening times for each volume fraction. The PSDs measured from the experiment at f ⫽ 0.1 have already been published earlier in [26] where a more detailed discussion can be found. In Fig. 2(a), the PSDs measured from the experiment are shown and compared to the AV steady-state prediction for f ⫽ 0.1. The evolution of the scaled PSD over time is “clear” in the sense that the “tail” of the PSDs at large R/R¯PS moves inwards. However, the evolution of the location and value of the maximum of the PSD, which may simply look like noise, is actually due to the initial PSD and RDF found in the experiment. The location of the maximum first moves down as the PSD first broadens and then to higher values of R/R¯PS between 550 and 5810 s. The right side of the PSD then becomes steeper between 5810 and 36,600 s. The PSD evolves towards the steady-state prediction but at the final coarsening time of 36,600 s the scaled PSD has not reached the steady-state prediction, even though the average radius has changed by a factor of 2.7. The scaled RDFs measured at f ⫽ 0.1 over a range of coarsening times is shown in Fig.
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Fig. 4. The temporal evolution of (a) the scaled PSD and (b) the scaled RDF for f ⫽ 0.3.
Fig. 5. The temporal evolution of (a) the scaled PSD and (b) the scaled RDF for f ⫽ 0.5.
2(b). Initially, the RDF has a large peak indicating that the microstructure consists of clusters of particles. Over time the microstructure becomes more dispersed as shown by a decrease in the height of the peak and an increase in the size of the exclusion zone around each of the particles (the RDF shifting to the right). A detailed discussion of the evolution of the correlation function for f ⫽ 0.1 and 0.2 can be found in [25]. The evolution of the PSD and RDF measured from the experiment were in excellent agreement with predictions made using transient ripening simulations [25, 26]. It is clear from Figs 2(a) and (b) that the f ⫽ 0.1 experiments have not reached the steadystate regime. The scaled PSDs and RDFs measured from the experiment at f ⫽ 0.2 are shown in Figs 3(a) and (b). The overall change in the PSDs is very small and can only be seen in the peak becoming more pronounced. However, this behavior is in agreement with simulations of transient coarsening [25]. The steady-state distribution at f ⫽ 0.2, which is also shown in this figure, is clearly different than the PSDs measured in
the experiment. The tail of the scaled theoretical PSD is much shorter than those of the experimental PSDs. The average radius changes during these experiments by a factor of 3.1 after 36,600 s of coarsening, yet the steady-state regime is still not reached. The change in the scaled RDFs is more apparent than in the scaled PSDs. The peak in the initial RDF indicates that the initial microstructure consists of clusters of particles. The microstructure evolves to one which is more dispersed, as can be seen in the decrease with time in the peak height and the growth of the exclusion zone. As with the experiments using f ⫽ 0.1, the evolution of both the RDF and PSD agrees very well with the results of simulations of transient coarsening [25]. Interestingly, the evolution of the PSD and RDF for f ⫽ 0.2 is small in comparison with the evolution of the PSD and RDF at f ⫽ 0.1, even though the average particle size undergoes a greater change. This trend is also seen in the simulations of transient ripening. Figure 4(a) shows the scaled PSDs measured from the experiment at f ⫽ 0.3 and the corresponding AV
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Fig. 6. The temporal evolution of (a) the scaled PSD and (b) the scaled RDF for f ⫽ 0.7.
steady-state prediction. The distributions from the experiments seem to scale with time. However, the time range employed in the experiments at f ⫽ 0.3 is much shorter than for f ⫽ 0.1 and 0.2, and thus the average radius increases by only a factor of 2.1. The predicted steady-state PSD has a much steeper slope on the right side of the distribution, and the tail is much shorter than for those measured in the experiments. The steady-state distribution also has a much higher peak than the scaled PSDs measured in the experiment. Thus, despite the apparent time independence in the scaled PSDs, we do not obtain agreement between the experiment and the steady-state theory. Figure 4(b) shows the RDFs over the range of coarsening times for f ⫽ 0.3. The height of the peak present in the scaled RDF decreases and the size of the exclusion zone around the particles increases with time from 550 to 2250 s. There appears to be no change in the scaled RDF between 2280 and 9510 s. The initial change in the scaled RDF is an indication that the clustering of particles is reduced. More importantly, the evolution of the scaled correlation
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function is a further indication that the experiments have not reached steady state. This again follows the trend the we saw previously in the evolution of the PSDs and RDFs with volume fraction. With increasing volume fraction there tends to be less evolution in the scaled PSDs and RDFs; at f ⫽ 0.3 there are only small changes in the scaled RDFs, and the scaled PSDs appear to be time independent. The PSDs that appear to scale agree with the simulations of transient coarsening performed at this volume fraction wherein the PSD measured experimentally at 550 s was used as an initial condition. Thus, the time independence of the scaled PSDs is a result of the short time interval employed in the experiments. Indeed, at sufficiently long coarsening times the simulations evolve towards the AV steady-state predictions. Size distributions were also measured at f ⫽ 0.5 and 0.7 and are shown in Figs 5(a) and 6(a), respectively. Here we compare the experimental results with the steady-state Marsh–Glicksman (MG) theory, since most other theories are not valid at these volume fractions. This is not as precise a comparison as is possible at lower volume fractions, however, as the particles seen in Fig. 1(c) are not spherical, and the transformation of the MG distribution to two dimensions assumes that the particles are spherical. We have measured the aspect ratios, the ratio of the two axes of an ellipse fit to the particle cross-section, of the particles on the plane sections and find that at late times the average aspect ratio is ⬇1.15. While this would broaden the steady-state PSDPS prediction, the effect would not account for the observed large discrepancy between the theory and the experiment. As for f ⫽ 0.3, the longest experiment time for f ⫽ 0.5 and 0.7 is 9510 s instead of the 36,600 s used for f ⫽ 0.1 and 0.2. For both f ⫽ 0.5 and 0.7, R¯PS changes by a factor of 2.0. While the scaled PSDs seem to be time independent, the disagreement between the experiment and theory is obvious. The experimental PSDs are broader than the theoretical predictions and have much longer tails. The scaled RDFs for volume fractions of f ⫽ 0.5 and 0.7 are shown in Figs 5(b) and 6(b) . The correlations at such high volume fractions are set by a skeletal structure of particles and the lack of free volume in these samples. Thus, it is not surprising that these scaled RDFs are time independent. The differences seen in the RDF at f ⫽ 0.7 at the earliest coarsening time are believed to be due to the sample manufacturing procedure, since there is a small spatial variation in f in this sample. The trend in the rate of evolution of the RDFs and the PSDs towards steady state with volume fraction is also evident at f ⫽ 0.5 and 0.7: it decreases with increasing f. The kinetics of the coarsening process are examined by plotting the plane section average particle radius, R¯PS, as a function of the cube root of time, t1/3, in Fig. 7. The individual symbols show results from the experiments at each volume fraction. The
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Fig. 7. The temporal evolution of the average radius. The symbols show experiment results. The lines show fits of the equation R¯3PS(t)⫺R¯3PS(0) ⫽ KPSt to the experiment.
lines are least square fits of the results for each volume fraction with the equation, R¯3PS(t)⫺R¯3PS(0) ⫽ KPSt, using KPS and R¯3PS(0) as fitting parameters. It should be noted that this equation is valid only for steady-state coarsening, which was not present in the experiments. Surprisingly, this equation fits the data reasonably well and allows us to draw a few conclusions. It can be seen that the coarsening rate increases with increasing volume fraction of coarsening phase since the slope of the curves at long coarsening times increases. Even at the higher volume fractions, in contrast to the experiments of Bender and Ratke using Cu–Co [17], we find that the coarsening rate is a function of the volume fraction. The average particle size seems to follow the t1/3 behavior even during non-steady-state coarsening. This t1/3 behavior has also been seen in numerous other coarsening experiments. 4. DISCUSSION
When the CSLM experiments were performed, it was believed that the time required for an experiment to reach steady-state coarsening would be small. Indeed, one of the primary tests of any coarsening experiment is to determine if the microstructure is self-similar. In contrast, self-similarity is not present in our experiments: both the scaled PSDs and RDFs clearly evolve at f ⫽ 0.1 and 0.2, and at f ⫽ 0.3 the scaled RDFs evolve. This indicates that the experiments never reach the steady-state coarsening regime, despite a tripling in the average particle size during coarsening at f ⫽ 0.1 and 0.2 and a doubling in average particle size at f ⫽ 0.3. Another indication that these low volume fraction experiments are not at steady-state is the agreement we have found between the experiments and the tran-
sient simulations of Ostwald ripening. In our first report [26], we employed a measured non-steadystate PSD as the initial PSD in simulations based on both the AV and Chen–Voorhees [12] transient coarsening theories. We were able to follow the evolution of the scaled PSDs quite well using these theories. In our second report [25], we included the effects of an initial non-steady-state spatial arrangement of particles using the AV-transient theory. With this approach we were able to follow both the evolution of the scaled RDF and the scaled PSD at f ⫽ 0.1 and 0.2. Transient simulations based on the AV theory were performed at f ⫽ 0.3 using the PSD measured at the earliest time and a random spatial arrangement of particles as initial conditions. We were unable to produce an initial RDF similar to that at the earliest time, most likely due to the asphericity of the particles and the lack of free space at this volume fraction. However, simply using the PSD from an early coarsening time as the initial condition in the simulations we find that the PSDs in the simulation, despite being significantly different from the steady-state predictions, evolve so slowly that in the range of coarsening times employed in the experiment the scaled PSD appears to be time independent. The transient coarsening theory is thus in agreement with these experimental findings as well. The size distribution in the experiment thus may evolve with time, but longer times, such as those that are employed in the simulations, are required to observe this evolution. We believe that the same behavior is occurring at f ⫽ 0.5 and 0.7, although we are not able to run transient simulations at such high volume fractions. While the scaled PSDs and RDFs from the experiment appear to be time independent, the PSDs clearly disagree with the theoretical predictions. To examine this behavior more clearly, we consider the trend that is seen in the evolution of the scaled PSDs and RDFs with volume fraction. At f ⫽ 0.1 the system was clearly not in steady state, as can be seen by the large changes in the scaled PSDs and RDFs, Fig. 2. At f ⫽ 0.2, the scaled PSDs and RDFs evolve, but the evolution is occurring much more slowly than at f ⫽ 0.1. At f ⫽ 0.3 the RDFs evolve only slightly and the PSDs appear to scale. We believe that at f ⫽ 0.5 and 0.7 the scaled PSDs and RDFs are simply evolving too slowly for the changes to be observed in the time-span employed in the experiments. It was always assumed that the time independence of the scaled PSDs that is typically reported is simply due to a very rapid approach to the steady-state regime [14–17]. In our case this would imply that for f ⫽ 0.5 and 0.7, by the time we make our first measurement, the experiments are already in steady state. However, it is highly unlikely that the rate of evolution towards steady state, which slows with volume fraction at fⱕ0.3, suddenly increases to the
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extent that the experiments are at steady state by the time we make our first measurements for f>0.3. It is even more unlikely that the initial PSD and RDF happens to be exactly the steady-state PSD and RDF. We therefore conclude that the rate of evolution towards steady state must be a decreasing function of the volume fraction, and thus the evolution of the scaled PSDs and RDFs at f ⫽ 0.5 and 0.7 is too slow to detect during the experiments. We find that even after a factor of ⬇3 change in R¯ for f ⫽ 0.1 and 0.2, and a factor of two change in R¯ for f ⫽ 0.3–0.7, the steady-state regime is not reached. We predict on the basis of our transient simulations that at f ⫽ 0.1 the system would be close to steady state after a factor of ⬇5 change in R¯PS, given the initial PSD and RDF of the experiment. It is, however, difficult to predict precisely the change in R¯ required to reach steady state due to the strong dependence of the transient evolution process on the shape of the difficult-to-measure tail of the PSD [20, 25]. This required large change in R¯ for the system to reach steady state leads us to question the relevance of steady-state theory. The applicability of steadystate theories has always hinged on the assumption that although the theories are based upon a solution to the equations in the limit t→⬁, the evolution to this long time attractor state in experiments was rapid. This is clearly not the case. For example, in Fig. 3(a) we show that there exists a large difference between the experiment and the theoretical steady-state prediction despite a factor of three change in R¯. In contrast to the differences between the measured PSDs and predicted steady-state PSDs, the differences between the various theoretical steady-state predictions at a given volume fraction are small. In Fig. 8(a), we show several of the theoretical steadystate PSDs at f ⫽ 0.2 in comparison to the PSD measured after a coarsening time of 14,400 s. The theoretical PSDs are very similar. However, the experimental PSD at 14,400 s is clearly different from all of the theories and this difference is much larger than the difference between the predictions of various theories. In contrast, we show in Fig. 8(b) that the AV-transient prediction fits the experimental result quite well. The AV distribution was computed for a coarsening time of 14,400 s using the PSD at 550 s as an initial condition. The comparison is made at 14,400 s instead of the later time of 36,600 s, as the simulation at this latest time becomes highly sensitive to the shape of the initial PSD at high R/R¯PS; see [25]. Transient Ostwald ripening also manifests itself in our experiments in the form of a higher rate constant measured as compared to the steady-state predictions. For different initial PSDs and RDFs it could, however, result in a lower coarsening rate [12]. Traditionally values of the rate constant would be determined using a plot such as that shown in Fig. 7 and compared to those of the various coarsening theories. Unfortunately, this is not possible since the experiments have not reached steady state.
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Fig. 8. Theoretical predictions of the steady-state PSD.
We proceed therefore by fitting the average particle size as a function of time predicted by the transientAV simulations to the experimental data using KLSW as a fitting parameter. By comparing KLSW determined using the simulations with that calculated from material parameters we can test the accuracy of the theory even though the experiments are not at steady state. This is only possible at f ⫽ 0.1, 0.2, 0.3, since the AV theory (to our knowledge the only transient theory that includes the effects of spatial correlations) is only valid at these low volume fractions. The values of KLSW obtained by fitting the data can then be compared to the KLSW ⫽ 1.1±0.05 µm3/s* obtained by Hardy et al. from grain boundary groove experiments [32]. The fit between the experiment and the predicted
* The original value reported in the work by Hardy et al. was KLSW ⫽ 8/9A ⫽ 1.01±0.05 µm3/s. However, the fit in Fig. 13 of their paper was not done with equations (21) and (22), but instead allowed for a constant offset. By fixing this error, KLSW becomes 1.1 µm3/s.
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Rˆ using the transient simulations is shown in Fig. 9. The agreement between the experiment and the transient theory is excellent. In the transient regime the microstructure is not self-similar and thus one way of accounting for this transient behavior is to consider K to he a function of time. Since we find that the steady-state equation R¯3PS(t)⫺R¯3PS(0) ⫽ KPSt, see Fig. 7, is able to fit the data, K must only slowly change with time in the transient regime. The predictions of the AV-transient theory for the PSDs, RDFs and R¯(t) agree quite well with those measured in the experiment. Thus, the AV-transient theory is an excellent model for Ostwald ripening in this ideal system. In principle, we should obtain data at longer coarsening times to determine if the systems achieve steady state. As reported previously [3], if we assume that the f ⫽ 0.1 experiments had reached steady state, we calculate a value of KLSW ⫽ 2.3 µm3/s based on the AV theory [3], KLSW ⫽ 2.1 µm3/s based on the Marsh and Glicksman theory [8], and KLSW ⫽ 2.5 µm3/s based on the Marqusee and Ross theory [6]. The effect of transient coarsening was then included in the AV simulations by using the experimentally observed PSD at 550 s and a random spatial distribution of particles as initial conditions for f ⫽ 0.1, 0.2 and 0.3. In each case this yields a lower value of KLSW ⫽ 1.9 µm3/s. Including the effects of spatial correlations in the transient calculations we find KLSW ⫽ 1.6 µm3/s for f ⫽ 0.1 and KLSW ⫽ 1.7 µm3/s for f ⫽ 0.2. Both of these values are much closer to the value from the grain boundary groove experiment than the values obtained from any of the steadystate theories. The absence of steady-state coarsening observed in our experiments can be seen in other experiments as well. Mahalingam et al. [29] found that none of the theories could predict the coarsening rate or the PSDs found in their experiments. This may not be surpris-
Fig. 9. AV simulation results were fit to the CSLM experiment, results for f ⫽ 0.1, 0.2, and 0.3 to determine KLSW.
ing, however, as R¯ changed by only a factor of 1.6 during these experiments, and thus the distributions may have been evolving towards a steady state. Hardy and Voorhees [14] measured coarsening rates that were much higher than those predicted by steady-state theories. Both of these behaviors are clear signs that these experiments, like CSLM, were not in the steady-state regime. Chen and Voorhees [12] also found agreement between the predictions of transient coarsening theory and experiments on the coarsening of residual pores in a sintered Cu matrix [28]. Bender and Ratke noted that the width of the PSD measured in their experiments decreased with coarsening time, however, they considered this effect sufficiently small to claim that the PSDs appeared stationary in time [17]. When viewed in the context of this work it is possible to assert that the evolution of the PSD was sufficiently slow that even though the average particle changed by a factor of 10 in their experiments, steady state was never reached. The large change in average particle size present in their experiments shows clearly that the evolution towards steady state can be extremely slow. There have been many experiments performed in solid-state systems, such as g–g⬘ alloys, where large changes in R¯ have also been observed [36, 37]. Unfortunately, it is not possible to address these experiments due to the presence of elastic stress that accompanies the coarsening process in these systems. All of the experiments in stress-free systems lead us to believe that the transient regime is much longer than previously expected. We thus suspect that transients are sufficiently long in many experiments that steady-state theory cannot describe the evolution of the microstructure. 5. CONCLUSIONS
Contrary to the commonly held belief that microstructural scaling always occurs during coarsening, we found that PSDs and spatial correlation functions evolve at f ⫽ 0.1 and 0.2. Even though the average particle radius changed by a factor of three in these experiments, the scaled PSDs were not in agreement with any prediction of steady-state theory. However, all the measured PSDs and spatial correlation functions agreed with the predictions of the AV-transient coarsening theory for f ⫽ 0.1 and 0.2, and for f ⫽ 0.3 the PSDs agreed with the predictions of the AVtransient theory. We find that the time independence of the scaled PSDs at f ⫽ 0.5 and 0.7 follows from the very slow evolution of the scaled PSDs and RDFs at these high volume fractions. None of the experiments had reached the regime in which the microstructure becomes self-similar. We also find that the rate constant changes on a sufficiently long time scale that to within the scatter of our data and the limited time span of our experiments, R¯ increases as t1/3. The coarsening process is dominated by long duration transients; much longer than previously assumed. Many other experiments also disagree with predic-
SNYDER et al.: TRANSIENT OSTWALD RIPENING
tions of the steady-state theories, and show signs of transient coarsening. Thus we conclude that in these, as well as many other experiments, steady-state theories cannot adequately describe the evolution of the microstructure during coarsening.
Acknowledgements—The financial support of the Microgravity Sciences Research Division of NASA and a National Science Foundation Graduate Fellowship (V.S.) are gratefully acknowledged. We would like to thank Norio Akaiwa for making his simulation program available for this study. We would also like to thank several undergraduate students for their help in the image analysis of the samples: Stacy Sakai, Sey-Hee Kim, Matt Kreiner, Melissa Anyetei, Apieska Shah, Stan Chou, Ratchatee Techapiesancharoenkij, Diane Kim, and Linda Chi.
REFERENCES 1. Lifshitz, I. and Slyozov, V., J. Phys. Chem. Solids, 1961, 19, 35. 2. Wagner, C., Z. Elektrochem., 1961, 65, 581. 3. Akaiwa, N. and Voorhees, P., Phys. Rev. E, 1994, 49, 3860. 4. Voorhees, P. W. and Glicksman, M. E., Acta metall., 1984, 32, 2001. 5. Voorhees, P. W. and Glicksman, M. E., Acta metall., 1984, 32, 2013. 6. Marqusee, J. A. and Ross, J., J. Chem. Phys., 1984, 80, 536. 7. Brailsford, A. and Wynblatt, P., Acta metall., 1979, 27, 489. 8. Marsh, S. P. and Glicksman, M. E., in, ed. S. P. Marsh. TMS, Warrendale, 1992. 9. Beenakker, C. W. J., Phys. Rev. A, 1986, 33, 4482. 10. Tokuyama, M. and Kawasaki, K., Physica A, 1984, 123, 386. 11. Marder, M., Phys. Rev. Lett., 1985, 55, 2953.
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12. Chen, M. K. and Voorhees, P. W., Modell. Sim. Mater. Sci. Engng, 1993, 1, 591. 13. Mandyam, H., Glicksman, M. E., Helsing, J. and Marsh, S. P., Phys. Rev. E, 1998, 58, 2119. 14. Hardy, S. C. and Voorhees, P. W., Metall. Mater. Trans., 1988, 19 A, 2713. 15. Seyhan, I., Ratke, L., Bender, W. and Voorhees, P., Metall. Mater. Trans., 1996, 27, 2470. 16. Kang, C. and Yoon, D., Metall. Mater. Trans., 1981, 12, 65. 17. Bender, W. and Ratke, L., Acta metall., 1998, 46, 1125. 18. Brown, L. C., Acta metall., 1989, 37, 71. 19. Giron, B., Meerson, B. and Sasorov, P. V., Phys. Rev. E, 1998, 58, 4213. 20. Niethhammer, B. and Pego, R., J. Stat. Phys., 1999, 95, 867. 21. Velazquez, J. J. L., J. Stat. Phys., 1995, 92, 195. 22. Meerson, B., Phys. Rev. E, 1999, 60, 3072. 23. Rubinstein, I. and Zaltzman, B., Phys. Rev. E, 2000, 61, 709. 24. Venzl, G., Ber. Buns. Phys. Chem., 1983, 87, 318. 25. Snyder, V., Alkemper, J. and Voorhees, P., Acta mater., in press. 26. Alkemper, J., Snyder, V., Akaiwa, N. and Voorhees, P., Phys. Rev. Lett., 1999, 82, 2725. 27. Ardell, A. J., in, ed. G. W. Lorimer. The Institute of Metals, Cambridge, 1988, pp. 485–494. 28. Watanabe, R., Tada, K. and Masuda, Y., Z. Metall., 1976, 67, 619. 29. Mahalingam, K., Gu, B., Liedl, G., T.S. Jr., Acta metall., 1987, 35, 483. 30. Ardell, A. J., Acta metall., 1972, 20, 31. 31. Davies, C., Nash, P. and Stevens, R., Acta metall., 1980, 28, 179. 32. Hardy, S. C. et al., J. Cryst. Growth, 1991, 114, 467. 33. Snyder, V., Akaiwa, N., Alkemper, J. and Voorhees, P., Metall. Mater. Trans., 1999, 30, 2341. 34. Wicksell, S., Biometrica, 1925, 17, S4. 35. Weibel, E., Vol. 2, Academic Press, London, 1980. 36. Ardell, A. and Nicholson, R., J. Phys. Chem. Solids, 1966, 27, 1793. 37. Ardell, A. and Nicholson, R., Acta metall., 1966, 14, 1295.
TRANSIENT OSTWALD RIPENING AND THE DISAGREEMENT BETWEEN STEADY-STATE COARSENING THEORY AND EXPERIMENT V. A. SNYDER, J. ALKEMPER and P. W. VOORHEES* Department for Materials Science and Engineering, Northwestern University, Evanston, IL 60208-3108, USA ( Received 2 August 2000; received in revised form 4 October 2000; accepted 4 October 2000 )
Abstract—The coarsening of solid-Sn particles in a Pb–Sn liquid has been studied under microgravity conditions. These experiments permit an unambiguous comparison between theory and experiment to be made. In contrast to steady-state theories, such as those due to Lifshitz and Slyozov and Wagner, the scaled particle size distributions evolve in samples containing 0.1 and 0.2 volume fractions of solid. Steady state was not reached even though the average particle radius increased by a factor of three during the experiment. In addition, the scaled spatial correlation functions were also found to be time dependent in samples containing 0.1, 0.2, and 0.3 volume fractions of solid. The size distributions and correlation functions for all coarsening times at the fractions ⱕ0.3 agree with the predictions of a theory for transient coarsening. We show that the microstructures have not reached the steady-state regime for all volume fractions, are thus not self-similar, and that given our initial experimental conditions the time required to reach steady-state coarsening increases with increasing volume fraction. In these experiments, and we suspect in others as well, the transients are sufficiently long that steady-state theories cannot adequately describe the evolution of the microstructure. 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Ostwald ripening; Phase transformations; Diffusion
1. INTRODUCTION
Ostwald ripening, or particle coarsening, is a competitive growth process in which a system lowers its energy by reducing its interfacial area. During coarsening, large particles grow at the expense of small particles. This leads to an increase in the average particle size and a decrease in the particle number density over time. The process occurs in a number of two-phase mixtures, and is of critical importance for example in precipitate hardened materials, where overaging via coarsening reduces the strength of the alloy. Recognizing the importance of the process, much work has been done on the coarsening problem both theoretically [1–13] and experimentally [14–17]. Unfortunately, a large discrepancy exists between the theoretical predictions and the experimental results. This article addresses this issue and presents one reason for this discrepancy. The classical description of Ostwald Ripening was developed by Lifshitz and Slyozov [1] and Wagner
* To whom all correspondence should be addressed. Fax: ⫹1-847-491-7820. E-mail address: [email protected] (P.W. Voorhees)
[2] (LSW). They examined the coarsening process in a system of infinitely separated spherical particles. In the limit t→⬁ a similarity solution to the system of governing equations was found, and the regime in which this solution is valid has become known as steady state. In this regime, the microstructure is time invariant, or self-similar, when scaled by the timedependent average particle radius. The average particle radius, R¯(t), was also found to increase with time, t, as R¯3(t)⫺R¯3(0) ⫽ KLSWt
(1)
where KLSW is the LSW coarsening rate constant. For an isothermal solid–liquid system the LSW rate constant, KLSW is given by [14]: K ⫽ KLSW ⫽
8 T0⌫D . 9ML(CS⫺CL)
(2)
T0 is the coarsening temperature, ⌫ is the capillary length of the matrix or liquid phase, D is the diffusion coefficient of the solute in the liquid, ML is the slope of the liquidus curve at T0, and CS and CL are the
1359-6454/01/$20.00 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 3 4 2 - 6
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equilibrium compositions of the solid and liquid at a planar solid–liquid interface at T0. KLSW, thus, is a function only of various materials parameters. In 1989 Brown showed that particle size distributions (PSDs) other than those predicted by LSW are possible [18]. This result has recently been confirmed under the constraint that there is no noise present in the system sufficient to perturb the maximum particle scaled size above that predicted by LSW [19–21]. As noise is present in systems undergoing coarsening, only the stable steady-state distributions are expected to be observed experimentally [22, 23]. The LSW theory is only valid in the limit of a zerovolume fraction of coarsening phase. Noting this critical shortcoming, much theoretical work has been done to account for the effects of a finite volume fraction of coarsening phase in the steady-state regime [3–11, 13]. The major difficulty in describing the behavior of a finite volume fraction system lies in the overlap of the diffusion fields surrounding the particles. The theories determine a statistically averaged growth rate for a particle of a given size, but differ in the methodology of this calculation. With the exception of effective medium theories [7, 8], most theories are limited to low volume fractions of particles (ⱕ0.3) where the assumption of spherical nontouching particles is reasonable. All of the theories predict that a finite volume fraction of particles does not alter the temporal exponent of the average particle radius from the LSW prediction of 1/3. A nonzero volume fraction does, however, alter the coarsening rate K: K(f) ⫽ KLSWf(f)
(3)
where f(f) is a system-independent function of the volume fraction, f. K(f) now appears instead of KLSW in equation (1). The theories recover the LSW results at zero volume fraction [f(0) ⫽ 1]. These stable scaled particle size distributions are also predicted to be a function of the volume fraction, becoming broader and more symmetric with increasing volume fraction. Despite this agreement among the various theories, the predictions of f(f) and the form of the steady-state PSDs differ from theory to theory. Although most theories make predictions in the steady-state regime, only a few theories have been developed that can be used to describe the transient behavior of a system undergoing coarsening, i.e. the evolution of the system prior to reaching the selfsimilar regime. Venzl [24] developed a theory in which the evolution of the particle size distribution is determined numerically, and the width of this distribution can be used to calculate the coarsening rate of the system. The theory developed by Chen and Voorhees [12] is a continuum model based on the growth rate equation determined by Marqusee and Ross [6]. They found a strong dependence of the time
required to reach steady state on the form of the PSD at the beginning of coarsening. As a result of the mean field nature of the model, however, it is incapable of accounting for the effects of the local spatial arrangement of particles. The theory developed by Akaiwa and Voorhees (AV) [3] uses numerical simulations to study an ensemble of particles undergoing Ostwald ripening. Individual particle growth rates are determined by calculating the diffusion field surrounding each particle. This theory can be used to determine the evolution of both a nonsteady-state PSD and a non-steady-state spatial arrangement of particles [25, 26]. In reviewing the many experiments on Ostwald ripening in the literature, Ardell claimed that, there is a “chronic disagreement” between experimental results and theoretical predictions [27]. While many experiments show that the average particle radius increases as t1/3 [14–17], it has not been possible to determine if the measured rate constant was correct as some of the material parameters appearing in KLSW are unknown. In addition, many experimentally measured particle size distributions are wider than those predicted by theory [17, 28]. Furthermore, there have been no rigorous tests of the low volume fraction theories. The difficulty in testing Ostwald ripening theory lies in the assumptions employed by the theories, namely: a low volume fraction of particles, mass transport due solely to diffusion within the matrix, spherical particles (an isotropic interfacial energy), and a stress-free matrix. The last requirement is particularly difficult to attain in coherent two-phase solid mixtures. In addition, a rapid coarsening rate is desired such that significant changes in the average particle size are possible in a reasonable amount of time. A number of experiments have been performed using solid–liquid mixtures in the Cu–Co system as it is stress-free, and nearly isopycnic. This density match allows a skeletal structure of particles to form in the system at volume fractions as low as 0.25, so that large-scale sedimentation can be avoided. Although this is too high to test most theories, it permits a relatively wide range of volume fractions to be employed in the experiments. Kang and Yoon studied the coarsening of solid Co-rich particles in a liquid Cu rich matrix at annealing temperatures ranging from 1150 to 1300°C and volume fractions of solid of 0.34, 0.42, and 0.55 [16]. The scaled PSD appeared to be time-independent over a factor of 10 change in R¯, and the rate constant was found to increase with volume fraction. Bender and Ratke [17] recently also studied the coarsening of Co-rich particles using volume fractions of solid ranging from 0.25 to 0.7. In contrast to Kang and Yoon, they found that the volume fraction of coarsening phase did not have a measurable effect on the rate of coarsening and that the particle size distributions are broader than those predicted by theory. They proposed that the solid skeleton of particles was responsible for these
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effects. They noted that the width of the PSD decreases with coarsening time, however, they considered this effect sufficiently small to claim that the PSDs appeared stationary in time. Mahalingam et al. [29] studied the coarsening of d⬘ precipitates in Al–Li alloys with volume fractions ranging from 0.12 to 0.55. Since this is a solid–solid system, particle sedimentation did not occur. Moreover, unlike many other two-phase solid systems, the misfit is nearly zero and thus elastic stress in the system is very small. By using particles located far from dislocations or grain boundaries the evolution of the average particle size and particle size distributions was determined. Although R¯ was found to increase as t1/3, they concluded that none of the existing theories (comparisons were made to [5, 7, 30, 31]) could predict the coarsening rate or the particle size distributions observed in the experiments. While the size distributions appeared not to evolve during this experiment, the change in average particle size was only a factor of 1.6, and the limited number of particles used to construct the particle size distributions would have made it difficult to detect any evolution. The implications of this small change in R¯ are discussed below. A mixture consisting of solid Sn particles in a Pb– Sn liquid has been identified as an ideal system to study coarsening since it satisfies all of the aforementioned assumptions of theory [14]. The difficulty with the Pb–Sn system at low volume fractions of solid, however, is that the solid and liquid have different densities causing sedimentation in a matter of seconds. Low volume fraction experiments are therefore not possible in the presence of gravity. Hardy and Voorhees [14] performed experiments using a high volume fraction of solid to prevent particle motion and sedimentation. They found that the intercept length distributions were time independent when scaled by the average particle size and in reasonable agreement with the predictions of steadystate theory of Glicksman and Voorhees [5]. They found t1/3 coarsening kinetics, but the measured coarsening rate constants exceeded the theoretical values by factors ranging from 2 to 5. However, it was difficult to determine if steady-state coarsening had been achieved since intercept length distributions are relatively insensitive to the shape of the three-dimensional particle size distribution. It is also possible for the slow sedimentation that was present in the system, and thus the change in the volume fraction, to mask the sometimes subtle changes in the PSD that can accompany a transient ripening process. We examined the coarsening process in Pb–Sn using a range of volume fractions from 0.1 to 0.7. This range includes volume fractions where the theories are believed to be most valid and those where a skeletal structure of particles is present. This gives us a complete, unbiased view of the coarsening process over a large range of volume fractions. To avoid
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the sedimentation of the Sn particles, experiments were performed in microgravity. We first present the results from this coarsening in solid–liquid mixtures experiment (CSLM). PSDs and interparticle spatial correlations are shown for various volume fractions over a range of coarsening times, and are compared to the predictions of transient as well as steady-state theory. Values of KLSW are deduced from the coarsening kinetics and are compared to KLSW computed from the known thermophysical parameters of Pb–Sn solid–liquid mixtures. Finally, we discuss the results of other experiments in light of our results. 2. EXPERIMENT PROCEDURE
A two-phase mixture consisting of solid-Sn particles in a Pb–Sn liquid is an ideal system with which to study Ostwald ripening. The high coarsening rate of the system allows for a large change in R¯ over a reasonable time interval. The interfacial energy is nearly isotropic, the thermophysical parameters of the system have been measured [32], and thus a comparison between experiment and theory is possible with no adjustable parameters. Particle sedimentation was avoided by coarsening samples in the Microgravity Science Laboratory aboard Space Shuttle Columbia, STS83 and STS94. Samples of Pb–Sn were cast and cold-worked using the method employed by Hardy and Voorhees [14]. They were then heated to the solid–liquid regime and held at the coarsening temperature and quenched. This processing procedure took place in a compact isothermal facility in microgravity. Samples were retrieved from the shuttle, sectioned, etched, and photographed. Several images were taken from each section and montaged to form images of entire sample cross-sections, thus avoiding all edge effects in the image analysis. A number of microstructural characteristics were measured using these sample cross-sections. The circular equivalent average particle radius, particle size distributions, and spatial correlation functions were all measured on these plane sections (PS). Results are shown for volume fractions of 0.1, 0.2, 0.3, 0.5. and 0.7 solid. In order to accurately measure particle sizes in the high volume fraction samples, touching particles were cut along the grain boundary. Between 2000 and 10,000 particles were used for each measurement at a given volume fraction and coarsening time. The shortest coarsening time analyzed is 550 s, since the furnace had not reached a thermal equilibrium before this time. Experiment times ranged up to 36,600 s, however, temperature gradients in the furnace effected the coarsening in all samples at long times with the exception of f ⫽ 0.1 and 0.2. At the higher volume fractions, the longest usable coarsening time was 9510 s due to temperature gradients. The effect of temperature gradients on coarsening has since been studied and is published elsewhere [33].
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Microstructures obtained after coarsening for 5810 s in systems with f ⫽ 0.1, 0.3 and 0.7 are shown in Fig. 1. The Sn particles are white and the Pb–Sn matrix is black. The f ⫽ 0.1 microstructure shown in Fig. 1(a) consists of isolated spherical particles and thus is ideal to test the low volume fraction theories of coarsening. The f ⫽ 0.3 microstructure shown in Fig. 1(b) contains spherical particles that are not isolated. Instead, these particles are much closer together. The particle packing that occurs at f ⫽ 0.7, see Fig. 1(c), is clear and induces a clearly nonspherical particle morphology. In our previous report [25] we showed that the spatial correlations were a far more sensitive measure of the presence of steady-state coarsening than PSDs. The spatial correlations are described by the radial distribution function (RDF). The RDF is defined as the ratio of the number of particles per area whose centers lie within a circular shell of radius x and
thickness dx surrounding a particle, to the number of particles per area of the entire system. RDFPS ⫽
NA in a circular shell of radius x to x ⫹ dx N¯A (4)
where NA is the number of particles per area and N¯A is the overall particle density. For nearly all spatial distributions of particles, RDFPS→1 as x→⬁. In order to assess the presence of steady-state coarsening we scaled x by the average particle size. If the system is in steady state, the RDFs measured at different times should superimpose. PSDs measured on plane sections are denoted as PSDPS. However, theoretical predictions for the size distribution are usually made in three dimensions. In order to compare the theory to the experimental
Fig. 1. The microstructure present in the experiment after 5810 s of coarsening. (a) ø ⫽ 0.1, (b) ø ⫽ 0.3, and (c) ø ⫽ 0.7.
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results, the theoretical three-dimensional size distributions were converted to plane section distributions by a method developed by Wicksell [34, 35]. A similar stereological problem was found with the spatial correlation functions. In this case, a computer program was developed to section the simulated microstructures and produce the corresponding RDFPS. A comparison between PSDs and RDFs measured in the experiment was made with those determined from the AV simulations of transient coarsening. In order to account for the transient coarsening seen in the experiment, these simulations used as initial conditions the PSD and RDF similar to those measured experimentally after a short coarsening time. PSDs and RDFs as measured from the experiment at later coarsening times were then directly compared to those calculated from the simulations, see [25]. 3. EXPERIMENTAL RESULTS
PSDs and RDFs were measured for each volume fraction and coarsening time. The coarsening times shown in Figs 2–6 are the initial, middle, and final
Fig. 3. The temporal evolution of (a) the scaled PSD and (b) the scaled RDF for f ⫽ 0.2.
Fig. 2. The temporal evolution of (a) the scaled PSD and (b) the scaled RDF for f ⫽ 0.1.
coarsening times for each volume fraction. The PSDs measured from the experiment at f ⫽ 0.1 have already been published earlier in [26] where a more detailed discussion can be found. In Fig. 2(a), the PSDs measured from the experiment are shown and compared to the AV steady-state prediction for f ⫽ 0.1. The evolution of the scaled PSD over time is “clear” in the sense that the “tail” of the PSDs at large R/R¯PS moves inwards. However, the evolution of the location and value of the maximum of the PSD, which may simply look like noise, is actually due to the initial PSD and RDF found in the experiment. The location of the maximum first moves down as the PSD first broadens and then to higher values of R/R¯PS between 550 and 5810 s. The right side of the PSD then becomes steeper between 5810 and 36,600 s. The PSD evolves towards the steady-state prediction but at the final coarsening time of 36,600 s the scaled PSD has not reached the steady-state prediction, even though the average radius has changed by a factor of 2.7. The scaled RDFs measured at f ⫽ 0.1 over a range of coarsening times is shown in Fig.
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Fig. 4. The temporal evolution of (a) the scaled PSD and (b) the scaled RDF for f ⫽ 0.3.
Fig. 5. The temporal evolution of (a) the scaled PSD and (b) the scaled RDF for f ⫽ 0.5.
2(b). Initially, the RDF has a large peak indicating that the microstructure consists of clusters of particles. Over time the microstructure becomes more dispersed as shown by a decrease in the height of the peak and an increase in the size of the exclusion zone around each of the particles (the RDF shifting to the right). A detailed discussion of the evolution of the correlation function for f ⫽ 0.1 and 0.2 can be found in [25]. The evolution of the PSD and RDF measured from the experiment were in excellent agreement with predictions made using transient ripening simulations [25, 26]. It is clear from Figs 2(a) and (b) that the f ⫽ 0.1 experiments have not reached the steadystate regime. The scaled PSDs and RDFs measured from the experiment at f ⫽ 0.2 are shown in Figs 3(a) and (b). The overall change in the PSDs is very small and can only be seen in the peak becoming more pronounced. However, this behavior is in agreement with simulations of transient coarsening [25]. The steady-state distribution at f ⫽ 0.2, which is also shown in this figure, is clearly different than the PSDs measured in
the experiment. The tail of the scaled theoretical PSD is much shorter than those of the experimental PSDs. The average radius changes during these experiments by a factor of 3.1 after 36,600 s of coarsening, yet the steady-state regime is still not reached. The change in the scaled RDFs is more apparent than in the scaled PSDs. The peak in the initial RDF indicates that the initial microstructure consists of clusters of particles. The microstructure evolves to one which is more dispersed, as can be seen in the decrease with time in the peak height and the growth of the exclusion zone. As with the experiments using f ⫽ 0.1, the evolution of both the RDF and PSD agrees very well with the results of simulations of transient coarsening [25]. Interestingly, the evolution of the PSD and RDF for f ⫽ 0.2 is small in comparison with the evolution of the PSD and RDF at f ⫽ 0.1, even though the average particle size undergoes a greater change. This trend is also seen in the simulations of transient ripening. Figure 4(a) shows the scaled PSDs measured from the experiment at f ⫽ 0.3 and the corresponding AV
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Fig. 6. The temporal evolution of (a) the scaled PSD and (b) the scaled RDF for f ⫽ 0.7.
steady-state prediction. The distributions from the experiments seem to scale with time. However, the time range employed in the experiments at f ⫽ 0.3 is much shorter than for f ⫽ 0.1 and 0.2, and thus the average radius increases by only a factor of 2.1. The predicted steady-state PSD has a much steeper slope on the right side of the distribution, and the tail is much shorter than for those measured in the experiments. The steady-state distribution also has a much higher peak than the scaled PSDs measured in the experiment. Thus, despite the apparent time independence in the scaled PSDs, we do not obtain agreement between the experiment and the steady-state theory. Figure 4(b) shows the RDFs over the range of coarsening times for f ⫽ 0.3. The height of the peak present in the scaled RDF decreases and the size of the exclusion zone around the particles increases with time from 550 to 2250 s. There appears to be no change in the scaled RDF between 2280 and 9510 s. The initial change in the scaled RDF is an indication that the clustering of particles is reduced. More importantly, the evolution of the scaled correlation
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function is a further indication that the experiments have not reached steady state. This again follows the trend the we saw previously in the evolution of the PSDs and RDFs with volume fraction. With increasing volume fraction there tends to be less evolution in the scaled PSDs and RDFs; at f ⫽ 0.3 there are only small changes in the scaled RDFs, and the scaled PSDs appear to be time independent. The PSDs that appear to scale agree with the simulations of transient coarsening performed at this volume fraction wherein the PSD measured experimentally at 550 s was used as an initial condition. Thus, the time independence of the scaled PSDs is a result of the short time interval employed in the experiments. Indeed, at sufficiently long coarsening times the simulations evolve towards the AV steady-state predictions. Size distributions were also measured at f ⫽ 0.5 and 0.7 and are shown in Figs 5(a) and 6(a), respectively. Here we compare the experimental results with the steady-state Marsh–Glicksman (MG) theory, since most other theories are not valid at these volume fractions. This is not as precise a comparison as is possible at lower volume fractions, however, as the particles seen in Fig. 1(c) are not spherical, and the transformation of the MG distribution to two dimensions assumes that the particles are spherical. We have measured the aspect ratios, the ratio of the two axes of an ellipse fit to the particle cross-section, of the particles on the plane sections and find that at late times the average aspect ratio is ⬇1.15. While this would broaden the steady-state PSDPS prediction, the effect would not account for the observed large discrepancy between the theory and the experiment. As for f ⫽ 0.3, the longest experiment time for f ⫽ 0.5 and 0.7 is 9510 s instead of the 36,600 s used for f ⫽ 0.1 and 0.2. For both f ⫽ 0.5 and 0.7, R¯PS changes by a factor of 2.0. While the scaled PSDs seem to be time independent, the disagreement between the experiment and theory is obvious. The experimental PSDs are broader than the theoretical predictions and have much longer tails. The scaled RDFs for volume fractions of f ⫽ 0.5 and 0.7 are shown in Figs 5(b) and 6(b) . The correlations at such high volume fractions are set by a skeletal structure of particles and the lack of free volume in these samples. Thus, it is not surprising that these scaled RDFs are time independent. The differences seen in the RDF at f ⫽ 0.7 at the earliest coarsening time are believed to be due to the sample manufacturing procedure, since there is a small spatial variation in f in this sample. The trend in the rate of evolution of the RDFs and the PSDs towards steady state with volume fraction is also evident at f ⫽ 0.5 and 0.7: it decreases with increasing f. The kinetics of the coarsening process are examined by plotting the plane section average particle radius, R¯PS, as a function of the cube root of time, t1/3, in Fig. 7. The individual symbols show results from the experiments at each volume fraction. The
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Fig. 7. The temporal evolution of the average radius. The symbols show experiment results. The lines show fits of the equation R¯3PS(t)⫺R¯3PS(0) ⫽ KPSt to the experiment.
lines are least square fits of the results for each volume fraction with the equation, R¯3PS(t)⫺R¯3PS(0) ⫽ KPSt, using KPS and R¯3PS(0) as fitting parameters. It should be noted that this equation is valid only for steady-state coarsening, which was not present in the experiments. Surprisingly, this equation fits the data reasonably well and allows us to draw a few conclusions. It can be seen that the coarsening rate increases with increasing volume fraction of coarsening phase since the slope of the curves at long coarsening times increases. Even at the higher volume fractions, in contrast to the experiments of Bender and Ratke using Cu–Co [17], we find that the coarsening rate is a function of the volume fraction. The average particle size seems to follow the t1/3 behavior even during non-steady-state coarsening. This t1/3 behavior has also been seen in numerous other coarsening experiments. 4. DISCUSSION
When the CSLM experiments were performed, it was believed that the time required for an experiment to reach steady-state coarsening would be small. Indeed, one of the primary tests of any coarsening experiment is to determine if the microstructure is self-similar. In contrast, self-similarity is not present in our experiments: both the scaled PSDs and RDFs clearly evolve at f ⫽ 0.1 and 0.2, and at f ⫽ 0.3 the scaled RDFs evolve. This indicates that the experiments never reach the steady-state coarsening regime, despite a tripling in the average particle size during coarsening at f ⫽ 0.1 and 0.2 and a doubling in average particle size at f ⫽ 0.3. Another indication that these low volume fraction experiments are not at steady-state is the agreement we have found between the experiments and the tran-
sient simulations of Ostwald ripening. In our first report [26], we employed a measured non-steadystate PSD as the initial PSD in simulations based on both the AV and Chen–Voorhees [12] transient coarsening theories. We were able to follow the evolution of the scaled PSDs quite well using these theories. In our second report [25], we included the effects of an initial non-steady-state spatial arrangement of particles using the AV-transient theory. With this approach we were able to follow both the evolution of the scaled RDF and the scaled PSD at f ⫽ 0.1 and 0.2. Transient simulations based on the AV theory were performed at f ⫽ 0.3 using the PSD measured at the earliest time and a random spatial arrangement of particles as initial conditions. We were unable to produce an initial RDF similar to that at the earliest time, most likely due to the asphericity of the particles and the lack of free space at this volume fraction. However, simply using the PSD from an early coarsening time as the initial condition in the simulations we find that the PSDs in the simulation, despite being significantly different from the steady-state predictions, evolve so slowly that in the range of coarsening times employed in the experiment the scaled PSD appears to be time independent. The transient coarsening theory is thus in agreement with these experimental findings as well. The size distribution in the experiment thus may evolve with time, but longer times, such as those that are employed in the simulations, are required to observe this evolution. We believe that the same behavior is occurring at f ⫽ 0.5 and 0.7, although we are not able to run transient simulations at such high volume fractions. While the scaled PSDs and RDFs from the experiment appear to be time independent, the PSDs clearly disagree with the theoretical predictions. To examine this behavior more clearly, we consider the trend that is seen in the evolution of the scaled PSDs and RDFs with volume fraction. At f ⫽ 0.1 the system was clearly not in steady state, as can be seen by the large changes in the scaled PSDs and RDFs, Fig. 2. At f ⫽ 0.2, the scaled PSDs and RDFs evolve, but the evolution is occurring much more slowly than at f ⫽ 0.1. At f ⫽ 0.3 the RDFs evolve only slightly and the PSDs appear to scale. We believe that at f ⫽ 0.5 and 0.7 the scaled PSDs and RDFs are simply evolving too slowly for the changes to be observed in the time-span employed in the experiments. It was always assumed that the time independence of the scaled PSDs that is typically reported is simply due to a very rapid approach to the steady-state regime [14–17]. In our case this would imply that for f ⫽ 0.5 and 0.7, by the time we make our first measurement, the experiments are already in steady state. However, it is highly unlikely that the rate of evolution towards steady state, which slows with volume fraction at fⱕ0.3, suddenly increases to the
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extent that the experiments are at steady state by the time we make our first measurements for f>0.3. It is even more unlikely that the initial PSD and RDF happens to be exactly the steady-state PSD and RDF. We therefore conclude that the rate of evolution towards steady state must be a decreasing function of the volume fraction, and thus the evolution of the scaled PSDs and RDFs at f ⫽ 0.5 and 0.7 is too slow to detect during the experiments. We find that even after a factor of ⬇3 change in R¯ for f ⫽ 0.1 and 0.2, and a factor of two change in R¯ for f ⫽ 0.3–0.7, the steady-state regime is not reached. We predict on the basis of our transient simulations that at f ⫽ 0.1 the system would be close to steady state after a factor of ⬇5 change in R¯PS, given the initial PSD and RDF of the experiment. It is, however, difficult to predict precisely the change in R¯ required to reach steady state due to the strong dependence of the transient evolution process on the shape of the difficult-to-measure tail of the PSD [20, 25]. This required large change in R¯ for the system to reach steady state leads us to question the relevance of steady-state theory. The applicability of steadystate theories has always hinged on the assumption that although the theories are based upon a solution to the equations in the limit t→⬁, the evolution to this long time attractor state in experiments was rapid. This is clearly not the case. For example, in Fig. 3(a) we show that there exists a large difference between the experiment and the theoretical steady-state prediction despite a factor of three change in R¯. In contrast to the differences between the measured PSDs and predicted steady-state PSDs, the differences between the various theoretical steady-state predictions at a given volume fraction are small. In Fig. 8(a), we show several of the theoretical steadystate PSDs at f ⫽ 0.2 in comparison to the PSD measured after a coarsening time of 14,400 s. The theoretical PSDs are very similar. However, the experimental PSD at 14,400 s is clearly different from all of the theories and this difference is much larger than the difference between the predictions of various theories. In contrast, we show in Fig. 8(b) that the AV-transient prediction fits the experimental result quite well. The AV distribution was computed for a coarsening time of 14,400 s using the PSD at 550 s as an initial condition. The comparison is made at 14,400 s instead of the later time of 36,600 s, as the simulation at this latest time becomes highly sensitive to the shape of the initial PSD at high R/R¯PS; see [25]. Transient Ostwald ripening also manifests itself in our experiments in the form of a higher rate constant measured as compared to the steady-state predictions. For different initial PSDs and RDFs it could, however, result in a lower coarsening rate [12]. Traditionally values of the rate constant would be determined using a plot such as that shown in Fig. 7 and compared to those of the various coarsening theories. Unfortunately, this is not possible since the experiments have not reached steady state.
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Fig. 8. Theoretical predictions of the steady-state PSD.
We proceed therefore by fitting the average particle size as a function of time predicted by the transientAV simulations to the experimental data using KLSW as a fitting parameter. By comparing KLSW determined using the simulations with that calculated from material parameters we can test the accuracy of the theory even though the experiments are not at steady state. This is only possible at f ⫽ 0.1, 0.2, 0.3, since the AV theory (to our knowledge the only transient theory that includes the effects of spatial correlations) is only valid at these low volume fractions. The values of KLSW obtained by fitting the data can then be compared to the KLSW ⫽ 1.1±0.05 µm3/s* obtained by Hardy et al. from grain boundary groove experiments [32]. The fit between the experiment and the predicted
* The original value reported in the work by Hardy et al. was KLSW ⫽ 8/9A ⫽ 1.01±0.05 µm3/s. However, the fit in Fig. 13 of their paper was not done with equations (21) and (22), but instead allowed for a constant offset. By fixing this error, KLSW becomes 1.1 µm3/s.
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Rˆ using the transient simulations is shown in Fig. 9. The agreement between the experiment and the transient theory is excellent. In the transient regime the microstructure is not self-similar and thus one way of accounting for this transient behavior is to consider K to he a function of time. Since we find that the steady-state equation R¯3PS(t)⫺R¯3PS(0) ⫽ KPSt, see Fig. 7, is able to fit the data, K must only slowly change with time in the transient regime. The predictions of the AV-transient theory for the PSDs, RDFs and R¯(t) agree quite well with those measured in the experiment. Thus, the AV-transient theory is an excellent model for Ostwald ripening in this ideal system. In principle, we should obtain data at longer coarsening times to determine if the systems achieve steady state. As reported previously [3], if we assume that the f ⫽ 0.1 experiments had reached steady state, we calculate a value of KLSW ⫽ 2.3 µm3/s based on the AV theory [3], KLSW ⫽ 2.1 µm3/s based on the Marsh and Glicksman theory [8], and KLSW ⫽ 2.5 µm3/s based on the Marqusee and Ross theory [6]. The effect of transient coarsening was then included in the AV simulations by using the experimentally observed PSD at 550 s and a random spatial distribution of particles as initial conditions for f ⫽ 0.1, 0.2 and 0.3. In each case this yields a lower value of KLSW ⫽ 1.9 µm3/s. Including the effects of spatial correlations in the transient calculations we find KLSW ⫽ 1.6 µm3/s for f ⫽ 0.1 and KLSW ⫽ 1.7 µm3/s for f ⫽ 0.2. Both of these values are much closer to the value from the grain boundary groove experiment than the values obtained from any of the steadystate theories. The absence of steady-state coarsening observed in our experiments can be seen in other experiments as well. Mahalingam et al. [29] found that none of the theories could predict the coarsening rate or the PSDs found in their experiments. This may not be surpris-
Fig. 9. AV simulation results were fit to the CSLM experiment, results for f ⫽ 0.1, 0.2, and 0.3 to determine KLSW.
ing, however, as R¯ changed by only a factor of 1.6 during these experiments, and thus the distributions may have been evolving towards a steady state. Hardy and Voorhees [14] measured coarsening rates that were much higher than those predicted by steady-state theories. Both of these behaviors are clear signs that these experiments, like CSLM, were not in the steady-state regime. Chen and Voorhees [12] also found agreement between the predictions of transient coarsening theory and experiments on the coarsening of residual pores in a sintered Cu matrix [28]. Bender and Ratke noted that the width of the PSD measured in their experiments decreased with coarsening time, however, they considered this effect sufficiently small to claim that the PSDs appeared stationary in time [17]. When viewed in the context of this work it is possible to assert that the evolution of the PSD was sufficiently slow that even though the average particle changed by a factor of 10 in their experiments, steady state was never reached. The large change in average particle size present in their experiments shows clearly that the evolution towards steady state can be extremely slow. There have been many experiments performed in solid-state systems, such as g–g⬘ alloys, where large changes in R¯ have also been observed [36, 37]. Unfortunately, it is not possible to address these experiments due to the presence of elastic stress that accompanies the coarsening process in these systems. All of the experiments in stress-free systems lead us to believe that the transient regime is much longer than previously expected. We thus suspect that transients are sufficiently long in many experiments that steady-state theory cannot describe the evolution of the microstructure. 5. CONCLUSIONS
Contrary to the commonly held belief that microstructural scaling always occurs during coarsening, we found that PSDs and spatial correlation functions evolve at f ⫽ 0.1 and 0.2. Even though the average particle radius changed by a factor of three in these experiments, the scaled PSDs were not in agreement with any prediction of steady-state theory. However, all the measured PSDs and spatial correlation functions agreed with the predictions of the AV-transient coarsening theory for f ⫽ 0.1 and 0.2, and for f ⫽ 0.3 the PSDs agreed with the predictions of the AVtransient theory. We find that the time independence of the scaled PSDs at f ⫽ 0.5 and 0.7 follows from the very slow evolution of the scaled PSDs and RDFs at these high volume fractions. None of the experiments had reached the regime in which the microstructure becomes self-similar. We also find that the rate constant changes on a sufficiently long time scale that to within the scatter of our data and the limited time span of our experiments, R¯ increases as t1/3. The coarsening process is dominated by long duration transients; much longer than previously assumed. Many other experiments also disagree with predic-
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tions of the steady-state theories, and show signs of transient coarsening. Thus we conclude that in these, as well as many other experiments, steady-state theories cannot adequately describe the evolution of the microstructure during coarsening.
Acknowledgements—The financial support of the Microgravity Sciences Research Division of NASA and a National Science Foundation Graduate Fellowship (V.S.) are gratefully acknowledged. We would like to thank Norio Akaiwa for making his simulation program available for this study. We would also like to thank several undergraduate students for their help in the image analysis of the samples: Stacy Sakai, Sey-Hee Kim, Matt Kreiner, Melissa Anyetei, Apieska Shah, Stan Chou, Ratchatee Techapiesancharoenkij, Diane Kim, and Linda Chi.
REFERENCES 1. Lifshitz, I. and Slyozov, V., J. Phys. Chem. Solids, 1961, 19, 35. 2. Wagner, C., Z. Elektrochem., 1961, 65, 581. 3. Akaiwa, N. and Voorhees, P., Phys. Rev. E, 1994, 49, 3860. 4. Voorhees, P. W. and Glicksman, M. E., Acta metall., 1984, 32, 2001. 5. Voorhees, P. W. and Glicksman, M. E., Acta metall., 1984, 32, 2013. 6. Marqusee, J. A. and Ross, J., J. Chem. Phys., 1984, 80, 536. 7. Brailsford, A. and Wynblatt, P., Acta metall., 1979, 27, 489. 8. Marsh, S. P. and Glicksman, M. E., in, ed. S. P. Marsh. TMS, Warrendale, 1992. 9. Beenakker, C. W. J., Phys. Rev. A, 1986, 33, 4482. 10. Tokuyama, M. and Kawasaki, K., Physica A, 1984, 123, 386. 11. Marder, M., Phys. Rev. Lett., 1985, 55, 2953.
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12. Chen, M. K. and Voorhees, P. W., Modell. Sim. Mater. Sci. Engng, 1993, 1, 591. 13. Mandyam, H., Glicksman, M. E., Helsing, J. and Marsh, S. P., Phys. Rev. E, 1998, 58, 2119. 14. Hardy, S. C. and Voorhees, P. W., Metall. Mater. Trans., 1988, 19 A, 2713. 15. Seyhan, I., Ratke, L., Bender, W. and Voorhees, P., Metall. Mater. Trans., 1996, 27, 2470. 16. Kang, C. and Yoon, D., Metall. Mater. Trans., 1981, 12, 65. 17. Bender, W. and Ratke, L., Acta metall., 1998, 46, 1125. 18. Brown, L. C., Acta metall., 1989, 37, 71. 19. Giron, B., Meerson, B. and Sasorov, P. V., Phys. Rev. E, 1998, 58, 4213. 20. Niethhammer, B. and Pego, R., J. Stat. Phys., 1999, 95, 867. 21. Velazquez, J. J. L., J. Stat. Phys., 1995, 92, 195. 22. Meerson, B., Phys. Rev. E, 1999, 60, 3072. 23. Rubinstein, I. and Zaltzman, B., Phys. Rev. E, 2000, 61, 709. 24. Venzl, G., Ber. Buns. Phys. Chem., 1983, 87, 318. 25. Snyder, V., Alkemper, J. and Voorhees, P., Acta mater., in press. 26. Alkemper, J., Snyder, V., Akaiwa, N. and Voorhees, P., Phys. Rev. Lett., 1999, 82, 2725. 27. Ardell, A. J., in, ed. G. W. Lorimer. The Institute of Metals, Cambridge, 1988, pp. 485–494. 28. Watanabe, R., Tada, K. and Masuda, Y., Z. Metall., 1976, 67, 619. 29. Mahalingam, K., Gu, B., Liedl, G., T.S. Jr., Acta metall., 1987, 35, 483. 30. Ardell, A. J., Acta metall., 1972, 20, 31. 31. Davies, C., Nash, P. and Stevens, R., Acta metall., 1980, 28, 179. 32. Hardy, S. C. et al., J. Cryst. Growth, 1991, 114, 467. 33. Snyder, V., Akaiwa, N., Alkemper, J. and Voorhees, P., Metall. Mater. Trans., 1999, 30, 2341. 34. Wicksell, S., Biometrica, 1925, 17, S4. 35. Weibel, E., Vol. 2, Academic Press, London, 1980. 36. Ardell, A. and Nicholson, R., J. Phys. Chem. Solids, 1966, 27, 1793. 37. Ardell, A. and Nicholson, R., Acta metall., 1966, 14, 1295.