Grain microstructural evolution in 2D and 3D polycrystals under triple junction energy and mobility control

Computational Materials Science 118 (2016) 325–337

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Grain microstructural evolution in 2D and 3D polycrystals under triple junction energy and mobility control Dana Zöllner ⇑ Otto-von-Guericke-Universität Magdeburg, Institut für Experimentelle Physik, Universitätsplatz 2, 39106 Magdeburg, Germany

a r t i c l e

i n f o

Article history: Received 18 January 2016 Received in revised form 16 March 2016 Accepted 19 March 2016 Available online 1 April 2016 Keywords: Grain growth Nanocrystalline microstructure Junction mobility Junction energy Monte Carlo simulation

a b s t r a c t Grain growth in two- and three-dimensional nanocrystalline materials is modeled by modifying the Monte Carlo Potts method based on the assumption that grain boundaries as well as triple junctions can be rate limiting during coarsening. To that aim each type of junction-associated lattice points (grain boundary-associated and triple junction-associated) is assigned an own specific mobility and energy. For the first time the influence of triple point (in 2D) and triple line (in 3D) energy and mobility on the grain growth kinetics is shown in detail. The results deviate clearly from normal grain growth kinetics but are in agreement with experimental studies as well as theoretical predictions. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction There are still fundamental as well as unsolved problem associated with the phenomenon of grain growth. This may sound a bit surprising since on one hand the thermally activated migration of grain boundaries was first documented almost a century ago and on the other hand grain growth has been an important research topic for decades investigated by experiments, theories and computer simulations. The importance of understanding grain growth lies in the fact that the grain size has a distinct effect on many materials properties like hardness, yield strength, tensile strength and fatigue strength. During grain growth, where the average grain size may increase by several orders of magnitude, the properties of a material may change dramatically. Hence, much effort has been put into the investigation of grain boundary migration and grain growth for decades, where computer simulations have been found to be of great help spanning a bridge over the gap between theories and experiments. Due to their high value different simulation methods have been established, from which particularly the mesoscopic models can be understood as in-situ (real time) computer experiments permitting the observation of large grain ensembles over long time regions making statistical analyses and comparisons with experiments possible, which explains the high number of high quality

⇑ Tel.: +49 (0)391 67 51894; fax: +49 (0)391 67 18108. E-mail address: [email protected] http://dx.doi.org/10.1016/j.commatsci.2016.03.031 0927-0256/Ó 2016 Elsevier B.V. All rights reserved.

mesoscopic simulations like Monte Carlo Potts model, Vertex method, Surface Evolver and phase-field model. For the classical Monte Carlo (MC) Potts model [1–3] especially in the last decade improvements of the algorithm have been recommended [4–6], which decrease the runtime of the simulations as well as increase the precision of simulation results. This has enabled researchers in the recent past to investigate for example anisotropic grain growth, abnormal sub-grain growth, texturecontrolled grain growth, and even grain boundary migration in 2D materials like polycrystalline graphene in detail. Over the past two decades, however, materials science has shifted its focus to so-called nanocrystalline materials, which are characterized by grain sizes some two to four orders of magnitude smaller than those of conventional specimens. Since they have high values of hardness, fracture strength, yield strength as well as superplastic behavior [7,8] and show steady grain sizes even up to fairly high temperatures coupled with linear or exponential growth kinetics regarding the average grain size [9–12], they also have wide-ranging applications. Here it should be noted that the stabilization of nanocrystalline microstructures is of major interest because nanomaterials are unstable with regard to grain growth—and because the latter phenomenon progressively diminishes any property enhancements associated with the reduced grain size. An overview by Malow and Koch [13] discussed noteworthy investigations of the stabilization of nanocrystalline grain microstructures in different materials and named a number of aspects, which can influence the mobility of grain boundaries in nanocrystalline metals and alloys such as

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grain boundary segregation, pore drag, second phase (Zener) drag, solute drag, as well as chemical ordering. Nonetheless, a general statement cannot be given yet. However, as we will see below particularly the higher order boundary junctions of the polycrystalline microstructures (triple and quadruple junctions) may be a key to unravel the grain growth kinetics in nanocrystalline materials. In general, the specific energies and mobilities of a material could underlie substantial changes caused by fluctuations of the chemical composition of the grain boundary faces, edges and vertices of the boundary network alike! For example, in case of impurity induced reduction of the specific grain boundary energy, which is considered as a main strategy for thermodynamic stabilization of nanocrystallinity [7,14,15], the specific edge and vertex energies can act as the remaining driving force for grain growth already at relatively large average grain sizes. It follows that as a consequence of grain boundary and grain boundary junction engineering, a similar noteworthy effect might well be possible on the specific energy and mobility of all the grain features. In the present paper it is shown that grain growth in two- and three-dimensional nanocrystalline polycrystals can be modeled using a modified Monte Carlo Potts model approach. For the first time a modification of the mobilities and energies of the triple junctions is implemented by attributing each type of junctionadjacent lattice points an own specific mobility and energy resulting in growth kinetics and topologies that deviate clearly from normal grain growth but are in agreement with experimental studies and analytical predictions.

controlled by triple points, whereas grain boundary control yields in both cases the well-known parabolic growth law (see also [17,18]). Alternatively, Streitenberger and Zöllner [18] treated grain coarsening as a dissipative process, which was assumed to be driven by the reduction of the Gibbs free interface, edge (triple line) and vertex (quadruple point) energy. For that treatment it has been expected that any polycrystalline grain microstructure can be represented by a polyhedral network, and any and each grain within such network can be approximated by an average N-hedra in 3D and N-gon in 2D [19] (compare Fig. 1). Introducing the specific free energies for each of the three types of grain junctions—the grain boundary energy cgb, the triple line energy or line tension ctj and the quadruple point energy cqj— yields for the total free energy of a network consisting of Ng grains

G¼

Ng  X cgb

Nearly two decades ago, Gottstein and Shvindlerman [16] first analyzed the effect of triple junction mobility on the evolution of polycrystalline grain microstructures with the outcome that a limited junction mobility may exert a drag effect on the grain boundaries attached to the considered triple junctions, which yields deviations of the grain vertex angles at the junction from their equilibrium value of 120°, which is the expected value for normal grain growth. The analytical expression deduced by Gottstein and Shvindlerman [17] defines the rate of grain boundary migration in terms of the inherent mobilities of grain boundary faces and their junctions such that

v ¼ meff cgb j:

ð1aÞ

Here cgb is the grain boundary energy, and j is the curvature of the considered grain boundary segment. The effective mobility meff in Eq. (1a) includes separate mobilities for grain boundaries mgb, triple junctions (triple lines, edges) mtj, and quadruple junctions (quadruple points, vertices) mqj and can be written as

meff ¼

mgb m

m

1 þ amgb þ a2 mgb tj

;

ð1bÞ

qj

where the grain boundary junction spacing a has been introduced as it is shown in Fig. 1. It follows that the effective mobility meff for two-dimensional grain growth is given by the reduced form

meff ¼

mgb m

1 þ amgb

:

ð1cÞ

tj

As a result, it can be shown that three-dimensional grain growth controlled solely by quadruple points and their mobility (of course together with the grain boundary energy in Eq. (1a)) results in an exponential growth law, 3D grain growth controlled by triple lines yields a linear growth law, just as 2D grain growth

 pffiffiffiffiffi cqj ð2p Ni Ri Þ þ ð2Ni  4Þ ; 3 4

ctj

ð2aÞ

where each grain i is categorized by the metrical property grain size Ri (grain radius resp. radius of a grain volume equivalent sphere) as well as by the topological property number of faces or neighboring grains Ni. Additionally, the finite mobilities mgb, mtj and mqj of faces, edges and vertices, respectively, can been used to calculate the dissipation potential in a similar form

Q¼

Ng  X 2p i

2. Theoretical background

2

i

ð4pR2i Þ þ

mgb

R2i þ

 2p pffiffiffiffiffi 1 ðNi  2Þ R_ 2i : N i Ri þ 2mqj 3mtj

ð2bÞ

Deriving an evolution equation analytically is only possible if e _  GðxÞ, the evolution equation is separable in terms of R_ ¼ hRi e _ _ where hRi ¼ hRiðtÞ is the average time-dependent and GðxÞ the size-dependent part of the growth law, for which x ¼ R=hRi is the scaled grain size. Therewith power law behavior of the average growth law, that is

hRi / t1=n ;

ð3aÞ

as well as self-similar scaling of the grain size distribution function,

FðR; tÞ ¼ gðtÞ  f ðxÞ;

ð3bÞ

are observed. This means only if Eqs. (3a) and (3b) hold, then the microstructure is in a quasi-stationary, statistically self-similar growth regime. In case of the above considered kinetics in terms of Eq. (2) this is only true if the time evolution is controlled solely by one particular specific energy together with one particular mobility resulting in different analytical limiting cases. Under the condition of energy conservation this results for the present case of grain growth under grain boundary and triple line mobility and/or energy control in (see also [18])

8 2cgb 9 2cgb l hRix l hRix > > > > pffiffi > > 1 > > N > > 3mtj hRix < mgb = p ffiffi ffi p ffiffi ffi R_ ¼ e e c N c N > > tj > l tj 2 2 > > l3hRi > 2 x2 > 3hRi x > > > pffiffi : 1 ; N mgb

ð4Þ

3mtj hRix

e ¼ Nð1 þ N 0 x=2NÞ2 and N 0 ¼ dN=dx. The using the abbreviation N Euler–Lagrange parameter l follows from the requirement of total area resp. volume conservation. The upper two expressions in Eq. (4) represent two kinds of grain growth kinetics that are driven by the grain boundary energy cgb in combination with one of the finite mobilities mgb (left) and mtj (right). The lower expressions represent two additional kinds of growth laws driven by the specific energy of the triple lines ctj also in combination with one of the finite mobilities mgb or mtj.

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Fig. 1. Schemes of three-dimensional and two-dimensional polyhedral grains pointing out the boundary junctions.

In case of two-dimensional grain growth (under grain boundary and triple point mobility and/or energy control) similar evolution equations follow (see [20,21])

R_ ¼

8 c gb lhRix > > > > m1 < gb

c

gb lhRix N 3pmtj hRix

ctj N 0

0

9 > > > > =

c N > l l tj 2 > > > 3phRi2 x 3phRi x > > > > : m1 ; N

:

ð5Þ

3pmtj hRix

gb

It can be noted that compared to Eq. (2) in Eqs. (4) and (5) only the relevant parameters remain for each limiting case. All these growth laws are separable and represent, therefore, different quasi-stationary, statistically self-similar growth regimes. Now, applying the requirement of total volume respectively area conservation one can derive four different types of average growth laws associated with Eqs. (4) and (5), from which after integration the following power law behaviors for the average grain size follow,

(

hRi2

hRi

hRi3

hRi2

) / t:

ð6Þ

Interestingly Eq. (6) holds for two-dimensional as well as for three-dimensional grain growth, although the evolution equations, Eqs. (4) and (5), are different. Altogether, the growth exponents in the power laws of Eq. (6) vary in the range of 1–3, which has indeed been observed experimentally (e.g., [9–11,13,22]). In the following it will be shown that an uncomplicated modification of the classical Potts model allows the simulation of those four limiting cases in two and three dimensions, and enables not only statements that go beyond the average growth law describing the expected quasi-stationary, statistically self-similar growth regimes, but also yields information on long-time annealing behavior that cannot be predicted from those limiting cases. 3. Modifying the Potts model The traditional Monte Carlo Potts model [1–3] is a sharp interface model, where a continuous polycrystalline grain microstructure characterized by a certain initial grain size, size distribution and topology is mapped onto a triangular or quadratic respectively cubic lattice in two or three dimensions. Each grain takes up a certain volume of the lattice (a certain number of lattice points), and is associated with a certain orientation. Hence, two neighboring lattice points of unlike orientation mark a grain boundary (sharp interface). Details regarding the implementation of the Potts model are provided in Ref. [6]. However, it should be noted that the algorithm contains two essential equations. On one hand, the state of the lattice is characterized by the Hamiltonian

H¼

M X nn 1 X  c  ð1  dðQ i ; Q j ÞÞ: 2 i¼1 j¼1

ð7Þ

The parameter c characterizes the specific grain boundary energy per unit length measuring the interaction of a considered i-th lattice point with all neighboring ones nn. It is assumed to be a function of the misorientation between two adjacent grains [23]. The Kronecker delta function in Eq. (7) is always equal to one if the neighboring orientations are the same and zero for unlike neighboring orientations, and M is the total number of lattice points. On the other hand, a second equation describes the probability p for orientation change of a considered lattice point,

p¼

8 m < mmax

c cmax ;

: m c mmax cmax

 exp



DE cmax kB T c



DE 6 0 ; DE > 0

;

ð8Þ

where DE is the difference in energy, m describes the grain boundary mobility [24], T is the temperature, and kB Boltzmann’s constant. In Eq. (8) the constants mmax and cmax describe the maximum values of mobility and energy, respectively. Setting m = mmax and c = cmax yields a simulation of normal grain growth (compare [25]). The product kB T is generally named simulation temperature, which has originally been introduced in order to prevent unphysical lattice effects such as lattice pinning. kB T should be large enough to avoid lattice pinning, but small enough to avoid lattice break-up (see [26]). Generally, in such a standard MC Potts model the assumption is made that triple lines and quadruple points have no direct influence on the migration kinetics—only grain boundaries control the microstructural evolution via their energy and mobility. However, as mentioned above it has already been demonstrated that both, triple lines as well as quadruple points, may very well have mobilities and energies different from the adjoining grain boundaries, which can be rate limiting during coarsening, changing the growth kinetics. It has been particularly shown in previous works [27–29] that it is possible to modify the standard Potts model such that finite junction mobilities can be taken into account. This has been done by assigning each grain feature—grain boundary, triple point (in 2D) resp. triple line (in 3D) and quadruple point (in 3D)—an own specific mobility. Because the Potts model is a sharp interface model, where grain boundary representations are always implicit, all lattice points adjacent to grain boundaries are characterized by the mobility mgb, adjacent to 3D triple lines and 2D triple points by mtj and adjacent to quadruple points (only in 3D) by mqj (see scheme in Fig. 2). This strategy changes Eq. (8) such that the mobility m depends after modification on the grain feature of the considered lattice point. Since the grain network is constantly changing, it has to be determined anew after every orientation change what

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considerations as in Eqs. (1)–(6). However, it is basically understood that they scale in the same way. 4. Simulation results 4.1. Two-dimensional simulations

Fig. 2. Scheme of three neighboring grains sharing a triple junction, which shows that the mobility mgb is assigned to MCUs next to grain boundaries and mtj is assigned to MCUs surrounding a triple junction.

kind of boundary junction every lattice point is associated with. We will see in the present paper that the boundary and junction energies can be treated analogously in Eqs. (7) and (8). Before the simulations can be started the simulation conditions must be set. For all two-dimensional investigations a continuous 2D microstructure with approximately 10,000 grains is mapped onto a quadratic lattice of size 10002 with eight nearest neighbors (first and second nearest) and periodic boundary condition. The microstructure is obtained by a construction algorithm on the basis of a dense circle packing and follows a size distribution, which is similar to that of the quasi-stationary, statistically self-similar state of normal grain growth. For all simulations the same initial microstructure is used, and the simulation temperature is chosen as kBT = 0.6 for kinetics under grain boundary energy control and kBT = 0.12 for kinetics under triple junction energy control. In contrast, for the three-dimensional simulation studies a microstructure with approximately 27,000 grains is simulated on a cubic lattice of size (250 MCUs)3 at a temperature of kBT = 1.95 for kinetics under grain boundary energy control and kBT = 0.3 for kinetics under triple junction energy control. It is important to reduce the simulation temperature for all simulations under triple junction energy control, since those include very low values of energy, which would lead due to Eq. (8) with higher kBT to very rough (unphysical) boundaries. Further details on how the simulation temperature is chosen are based on a one-to-one correspondence between the simulation parameters representing the specific grain boundary energy and mobility, respectively, in Eqs. (7) and (8), and the phenomenological parameters in the equations of motion, Eqs. (4) and (5), and can be found in Ref. [26]. Here two facts should be pointed out. Firstly, for the present work we focus on the four limiting cases that can be predicted analytically as done in [17,18,20]:  For mtj = mgb = mmax and cgb = ctj = cmax the kinetics is controlled solely by the grain boundaries. The triple junctions (points in 2D or lines in 3D) are just geometrical grain features. This case is named here grain boundary–grain boundary-controlled kinetics, and we’d expect normal grain growth to occur.  For mtj  mgb = mmax and cgb = ctj = cmax the kinetics is controlled by the energy of the grain boundaries together with the mobility of the triple junctions (subsequently named grain boundary–triple junction-control).  For mtj = mgb = mmax and cgb  ctj = cmax the kinetics is controlled by the energy of the triple junctions together with the mobility of the grain boundaries (subsequently named triple junction–grain boundary-control).  For mtj  mgb = mmax and cgb  ctj = cmax the kinetics is controlled solely by the triple junctions (subsequently named triple junction–triple junction-control). Secondly, those simulation parameters from Eqs. (7) and (8) are not identical to the mobilities and energies in the theoretical

It follows from the above mentioned theoretical considerations by Gottstein and Shvindlerman [17] as well as Streitenberger and Zöllner [18] that at small average grain sizes the parabolic growth law—as it is usually assumed and observed for normal grain growth—changes to a linear growth law if the kinetics is controlled by the mobility of the triple junctions in conjunction with the energy of the grain boundaries (upper right case in Eqs. (5) and (6). This is indeed shown in Fig. 3a for a parameter variation. While the mobility of the grain boundaries mgb is always kept equal to the maximum value mmax, the value for the triple points decreases systematically from 1  mgb to 1/64  mgb. For mtj = mgb = mmax indeed the expected parabolic growth kinetics can be seen, whereas decreasing the value of mtj reduces the speed of growth and for low values of mtj—apart from an initial period of time that can be ignored because it depends solely on the initial microstructure—a linear kinetics becomes visible, e.g., curve for mtj = 1/64, where it should be noted that the notation mtj = 1/64 means mtj = 1/64  mgb. This kind of behavior has been presented and discussed in detail in, e.g., [27–29]. In contrast, under the condition that the growth kinetics is controlled by the energy of the triple junctions together with the mobility of the grain boundaries (cgb  ctj = cmax and mtj = mgb = mmax) the average growth law changes from hRi / t 1=2 to hRi / t 1=3 as predicted by Eq. (6). This change can indeed be seen for a parameter variation in Fig. 3b, where for cgb = ctj = cmax a fit of the parabolic growth law is obviously a perfect description of the numerical data, whereas for cgb  ctj = cmax the average grain size is proportional to t1/3. The latter is shown for two cases: cgb = 1/4  ctj and cgb = 1/8  ctj, where it becomes apparent that a reduction of cgb leads to not only to a slower increase of the average size, but the initial period of time also becomes much larger. In addition, for the case cgb = 1/4  ctj it can additionally be seen that at large enough average grain sizes the kinetics starts to change back to parabolic behavior—a fact that will be discussed later in Section 4.3. In the following only the theoretical limiting cases will be analyzed. According to Eq. (6) there are four different limiting cases:  For grain boundary–grain boundary-controlled growth the average growth law should show a behavior, where hRi / t 1=2 , which is equivalent to hRi2 / t (upper left case in Eq. (6)). This is shown in Fig. 4a for mtj = mgb = mmax and cgb = ctj = cmax for long time-annealing.  For grain boundary–triple junction-controlled growth the upper right case in Eq. (6) predicts a relation of type hRi / t, which we find indeed to be true in Fig. 4b for mtj = 1/128  mgb and cgb = ctj (Section 2). One may notice the initial period of time (Section 1), which seemingly does not exist in Fig. 4a. This is based on the fact that this early growth regime depends strongly on the differences between the initial microstructure and the structure observed in the quasi-stationary state. Hence, the initial microstructure was apparently close to the self-similar structure of normal grain growth, but quite different from the case of grain boundary–triple junction-controlled growth.  For triple junction–grain boundary-controlled growth, which is the lower left case in Eq. (6), hRi3 / t should be observed. This growth law is shown for mtj = mgb and cgb = 1/8  ctj in Fig. 4c, where it can be seen that after an initial period of time, which is longer than in Fig. 4b, hRi3 / t holds indeed.

D. Zöllner / Computational Materials Science 118 (2016) 325–337

329

Fig. 3. Simulated average growth laws for parameter variations regarding: (a) mtj with cgb = ctj = cmax; (b) cgb with mtj = mgb = mmax for cgb = ctj, cgb = 1/4  ctj and cgb = 1/8  ctj from top to bottom.

Fig. 4. Average growth laws of two-dimensional growth for: (a) grain boundary–grain boundary-control; (b) grain boundary–triple junction-control; (c) triple junction–grain boundary-control; (d) triple junction–triple junction-control.

 The last case—lower right term in Eq. (6)—predicts again a parabolic growth law. However, it can be seen quite clearly in Fig. 4d for mtj = 1/64  mgb and cgb = 1/8  ctj that the kinetics of triple junction–triple junction-controlled grain growth is different compared to normal grain growth including a very long early growth regime and very slow growth. As mentioned above a polycrystalline grain microstructure is in a quasi-stationary, statistically self-similar growth regime only if Eqs. (3a) and (3b) are fulfilled. Hence, in addition to the simulated average growth laws following Eq. (6) the grain size distributions have to show scaling, which means that for different annealing times, where Eq. (6) holds, the scaled size distributions f(x) have to be time-independent and therewith unique.

This is shown to be true for all four limiting cases (each for three annealing times) in Fig. 5. It is particularly noteworthy that for the two cases, where the triple junction mobility controls the kinetics in conjunction with the grain boundary or triple junction energy, the size distributions are visibly shifted to smaller relative grain sizes and exhibit a longer tail. The comparison with analytical size distributions as it is shown in Fig. 5 can only be done after further theoretical considerations. For the derivation of Eqs. (2) and (4)–(6) it was assumed in [18] that there exists not only a relation between scaled grain size x and number of neighboring grains N, but this relation is unique. This conclusion can be drawn from the fact that a grain microstructure in a quasi-stationary growth regime implies statistical self-similarity not only regarding the size distribution but also

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D. Zöllner / Computational Materials Science 118 (2016) 325–337

Fig. 5. Scaled grain size distributions for: (a) grain boundary–grain boundary-control; (b) grain boundary–triple junction-control; (c) triple junction–grain boundary-control; (d) triple junction–triple junction-control.

regarding among others the topology in the sense of a unique and time-independent relation between N and x. In order to derive such a theoretical 2D relation between number of neighboring grains N of a grain, which is equivalent to the number of grain edges or triple points, and its relative grain size x ¼ R=hRi an argument based on considerations of Abbruzzese and Lücke [30,31] can be used: As it is presented in Fig. 6a, a central grain (lilac circle) of size (radius) R can be surrounded by N circles (filled in dark lilac) of average size hRi, where it is assumed without loss of generality that N is an integer number. As a result two different relations NðxÞ can be calculated: 1. The perimeter U of a circle through the centers of the surrounding grains (gray circle) can be derived either by U ¼ 2pðR þ hRiÞ or alternatively by U ¼ N  j1  2hRi, where we can set the form factor j1 ¼ 1. It follows

N ¼ p  x þ p:

ð9aÞ

2. The area A of the circle through the centers of the surrounding circles (shaded in light lilac) can be derived either by A ¼ pðR þ hRiÞ2 or by A ¼ pR2 þ N  0:5j2  phRi2 , where holds. Here it follows

N ¼ 4  x þ 2:

j2 ¼ 1 ð9bÞ

We can see that both arguments approximate the number of sides as a linear function of the scaled grain size. It has been shown, e.g., in Ref. [6] that unfortunately those simple arguments do usually not hold for normal grain growth simulations—a finding that is also shown in the present work in

Fig. 7a, where a self-similar, but quadratic relation between N and x is presented for the numerical data divided into and averaged over a significantly large enough number of size classes. Both, Eqs. (9a) and (9b) reflect those numerical data very poorly. Interestingly, a similar quadratic approximation is also a very good description for the self-similar state of grain growth under triple junction–grain boundary-control as in Fig. 7c, which is only slightly more curved. In contrast, although the two cases of grain growth controlled by the mobility of the triple junctions (Fig. 7b and c) also show statistically self-similar behavior, both can clearly be approximated by a linear function N(x). Moreover, in both cases Eqs. (9a) and (9b) seem to be adequate descriptions, and the linear relation is independent of the graphical representation as N(x) or x(N). Comparing the results of grain growth driven by the boundary mobility (Fig. 7a and c) and the two growth kinetics driven by the triple junction mobility (Fig. 7b and d) one can see clear deviations for small grains (x < 0.5) as well as for large grains (x > 1.75), from which the question arises if small and large grains are characterized by different morphologies for the different kinetics. In order to answer this question, in the following we will have a closer look purely at the topology of the statistical self-similar microstructures. This is done in Fig. 8 in form of the Aboav– Weaire-law [32–34]

N  mN ¼ ðhNi  aÞ  N þ ðhNi  a þ l2 Þ

ð10Þ

that relates the number neighboring grains or triple points N of a grain to the average number of neighbors of all neighboring grains mN (see also Fig. 6b). In Eq. (10) the constant a is usually assumed to be close to unity, hNi denotes the mean number of neighbors of the

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D. Zöllner / Computational Materials Science 118 (2016) 325–337

Fig. 6. (a) Scheme of a central grain (lilac circle) surrounded by N smaller circular grains (filled in lilac); (b) part of a polycrystalline microstructure indicating the number of nearest neighboring grains.

Fig. 7. Topology of the grain microstructures for: (a) grain boundary–grain boundary-control; (b) grain boundary–triple junction-control; (c) triple junction–grain boundarycontrol; (d) triple junction–triple junction-control.

grain ensemble, and l2 is the second moment of the neighbor distribution. The latter values (hNi and l2 ) are given in Fig. 8 for each case separately, where it can be seen that the average number of triple points is always nearly identical to six. In addition, at a first look all four cases seem to be very similar. Only on closer inspection small differences can be seen with the result that the two cases, where the triple junction mobility controls the growth together with the boundary or junction energy (Fig. 8b and d), are characterized by a larger slope of the linear least-squares fit to the numerical

data than the other two cases (Fig. 8a and c). Also microstructures obtained by grain boundary energy controlled growth have larger slopes than the corresponding cases obtained by growth under triple junction energy control. Those slopes of the least-squares fits enable the calculation of the constant a from Eq. (10) to



a¼

1:0716 1:0191 1:1246 1:0353



:

ð11aÞ

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D. Zöllner / Computational Materials Science 118 (2016) 325–337

Fig. 8. Aboav–Weaire-law for: (a) grain boundary–grain boundary-control; (b) grain boundary–triple junction-control; (c) triple junction–grain boundary-control; (d) triple junction–triple junction-control.

Apart from the microstructure obtained by grain growth under triple junction energy–grain boundary mobility-control the values of a are indeed very close to unity. From the constant term in Eq. (10) together with Eq. (11a) we can estimate the widths of the neighbor distributions and the second moments l2 , respectively, as



l2 ¼

2:1641 2:9011 1:7377 2:3944

 ;

ð11bÞ

which are in near-perfect agreement with the numerically calculated second moments as they are given in Fig. 8. It follows that triple junction mobility controlled growth kinetics yield distributions of the number of neighboring grains that are broader than under grain boundary mobility control, and the distributions are also broader under grain boundary energy control. All in all, these topological correlations are needed in the evolution equations of Eq. (5) and have, therefore, to be characterized analytically as N(x). In the two cases, where the growth is driven by the grain boundary mobility in conjunction with one of the energies as shown in Fig. 7a and c, we have seen above that the relation N(x) can be described by the quadratic function

NðxÞ ¼ c2 x2 þ c1 x þ c0 :

ð12aÞ

In contrast, we found for the other two cases of triple junction mobility driven grain growth as presented in Fig. 7b and d a linear dependence of type

NðxÞ ¼ c1 x þ c0 ;

ð12bÞ

which can alternatively be written as xðNÞ ¼ a1 N þ a0 as it is done for triple junction energy–triple junction mobility-control in Fig. 7d.

It has been shown in [20] that the knowledge of the topology in terms of Eq. (12a) or (12b) enables the calculation of the corresponding scaled grain size distribution functions (see also [21]) f ðxÞ ¼

e ÞD=2 ðx þ x1 Þ Dðxc C e Þx þ xc C e Þ1þD=2 ðx2 þ ðx1  C " " ! !## e e e x1 þ C 2x þ x1  C x1  C pffiffiffiffi pffiffiffiffi  arctan ;  exp D pffiffiffiffi arctan D D D

ð13Þ

e  ðx1  C e Þ2 P 0. In Eq. (13) the factor D where it holds D ¼ 4xc C denotes the Euclidian dimension of the microstructure. The critical R grain size xc is a result of the scaling condition hxi ¼ xf ðxÞdx ¼ 1. e is defined in the relative growth law [35] The parameter C

R_ e x  xc ¼C _ x þ x1 hRi

ð14Þ

and can be used as a free (fitting) parameter. In particular, the parameter x1 follows from topology such that for the two cases of triple junction mobility driven grain growth it is given by x1 ¼ c0 =c1 ¼ a0 using the linear topological function Eq. (12b) as fitted to the numerical data in Fig. 7b and d, where with x þ x1 ¼ c11 N. On the other hand, for the two cases of grain growth under grain boundary mobility control, where we have described NðxÞ by a quadratic polynomial, Eq. (12a), as presented in Fig. 7a and c, it can be shown that D=2 Dðxc e CÞ x the size distribution function follows to f ðxÞ ¼ 1þD=2 ðx2 xe C þxc e CÞ     e e effiffiffi pffiffiffiC  arctan  pC exp D pCffiffiDffi arctan 2x , which is interestingly D D identical to Eq. (13) if x1 is set equal to zero.

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333

Fig. 9. Average growth laws of three-dimensional growth for: (a) grain boundary–grain boundary-control; (b) grain boundary–triple junction-control; (c) triple junction– grain boundary-control; (d) triple junction–triple junction-control.

The analytical size distributions are shown in Fig. 5 in comparison with the numerical data of the time independent distributions. Although there are small deviations for the case of normal grain growth visible particularly regarding the maximum in the distribution, Eq. (13) describes the simulation results within the temporarily limited, statistically self-similar regime altogether very well.

Since the quasi-stationary, statistically self-similar growth regime is—as discussed above—characterized not only by an average growth law according to Eq. (6), but also by a timeindependent scaled grain size distribution function, Eq. (3b), the associated standard deviation of the scaled grain size sd½xðtÞ should be constant in this growth regime. This is tested in Fig. 10, where four facts become apparent:

4.2. Three-dimensional simulations Analogously just as in the above discussed two-dimensional growth also in three dimensions the average growth law takes one of the dependencies according to Eq. (6) for the four limiting cases:  In case of grain boundary–grain boundary-controlled kinetics the average growth law follows the relation hRi2 / t as presented in Fig. 9a for mtj = mgb = mmax and cgb = ctj = cmax over the total observation time.  In case of grain boundary–triple junction-controlled kinetics the relation of type hRi / t is shown in Fig. 9b for mtj = 1/32  mgb and cgb = ctj. Again an initial period of time with deviating behavior becomes visible.  In case of triple junction–grain boundary-controlled kinetics hRi3 / t holds as shown for mtj = mgb and cgb = 1/16  ctj in Fig. 9c.  Finally, in case of triple junction–triple junction-controlled kinetics Eq. (6) predicts also a parabolic type of growth law, for which the relation hRi2 / t can clearly be found in Fig. 9d for mtj = 1/32  mgb and cgb = 1/16  ctj.

Fig. 10. Standard deviation of the scaled grain size as a function of annealing time for grain boundary–grain boundary-control (blue), grain boundary–triple junctioncontrol (red), triple junction–grain boundary-control (green) and triple junction– triple junction-control (lilac). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 11. Size distribution of the scaled grain size for three different annealing times for: (a) grain boundary–grain boundary-control @ t ¼ f500; 1000; 1500g; (b) grain boundary–triple junction-control @ t ¼ f3000; 5000; 7000g; (c) triple junction–grain boundary-control @ t ¼ f2000; 4000; 6000g; (d) triple junction–triple junction-control @ t ¼ f4000; 6000; 8000g; each in comparison with Eq. (13).

 Each kinetics reaches indeed a regime, where sd½x is constant apart from some fluctuations around an average value.  Each kinetics needs a different amount of time (initial period) to reach this regime.  The length of the initial period is not the same as in Fig. 9.  Particularly, the average values are different for each kinetics. It follows that each of the four kinetics should be characterized by a unique self-similar grain size distribution function of the scaled grain size in the time region, where the standard deviation is (more or less) constant. This is visualized in Fig. 11. It can be seen that the numerical size distributions are indeed time-independent as shown for three different annealing times each. Also as presented for the above 2D cases in Fig. 5 we find for the two cases under triple junction mobility control (grain boundary or triple junction energy controlled growth) that the size distributions are visibly shifted to smaller relative grain sizes (compare Fig. 11b and d). The analytical size distribution function, Eq. (13), is found to be e are a very good description of the numerical data, where x1 and C both used as fitting parameters. This is done because in contrast to the two-dimensional cases above the growth kinetics in 3D are characterized by different topologies as we will see in the following. It is especially noteworthy that Eq. (13) yields a very good description of the simulation data if we take into consideration the fact that the analytical size distribution has been obtained in 2D under the assumption that x1 ¼ 0 characterizes a kinetics,

where the topology can be described by a quadratic relation between number of faces of a grain N and scaled grain size x, whereas x1 > 0 characterizes microstructures with a linear function NðxÞ, where it is possible to obtain x1 from NðxÞ. However, in contrast to microstructures in two dimensions we find in 3D that the number of neighboring grains is always a quadratic function of the scaled grain size from a similar argument as it has been used above to determine Eq. (9). If we assume to that aim a spherical grain of size R surrounded by N grains of average size hRi it follows: 1. The surface S of a sphere through the centers of the surrounding grains/spheres can be derived either by S ¼ 4pðR þ hRiÞ2 or by S ¼ N  phRi2 j1 , where we can set the form factor follows

N ¼ 4x2 þ 8x þ 4:

j1 ¼ 1. It ð15aÞ

2. Alternatively, the volume V of the sphere through the centers of the surrounding grains can be derived by V ¼ 4=3pðR þ hRiÞ3 or by V ¼ 4=3pR3 þ N  0:5  4=3phRi3 j2 , where yielding

N ¼ 6x2 þ 6x þ 2:

j2 ¼ 1 holds, ð15bÞ

Both arguments approximate N as a quadratic function of x, which we find to be true indeed for all four self-similar growth kinetics as shown in Fig. 12 though the least-squares fits yield different constants than in Eq. (15).

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Fig. 12. Number of neighboring grains or faces as a function of the scaled grain size for three different annealing times: (a) for grain boundary–grain boundary-control; (b) grain boundary–triple junction-control; (c) triple junction–grain boundary-control; (d) triple junction–triple junction-control; each together with a quadratic least-squares fit.

Fig. 13. Average grain sizes changing with annealing time for: (a) grain boundary energy–triple junction mobility-control; (b) triple junction energy–grain boundary mobility-control.

Nevertheless, although the situation is therewith different compared to the two-dimensional case, where the parameter x1 in the scaled size distribution function, Eq. (13), was a result of the topology in form of Eq. (12) with x1 ¼ 0 for the case that the microstructures were described by the quadratic relation N(x) and x1 > 0 for the linear dependence of type N(x), in 3D all four kinetics show a quadratic dependence (Fig. 12), while the values of x1 as found from the least-squares fit of Eq. (13) to the numerical data are somewhat similar as in the 2D cases.

4.3. A remark on long-time annealing The above presented results focus on those growth kinetics, which can be described as limiting cases. This means that mtj  mgb and cgb = ctj holds for grain boundary–triple junctioncontrol, mtj = mgb and cgb  ctj for triple junction–grain boundarycontrol and mtj  mgb and cgb  ctj for triple junction–triple junction-control, which can be found particularly for small average grain sizes (see [17,18,20]). However, since during grain growth

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Fig. 14. Grain boundary energy–triple junction mobility-controlled grain growth showing temporal development of: (a) standard deviation of the scaled grain size; (b) skewness of the scaled size distribution.

the average grain size will always increase, in real systems (and computer simulations) the junctions lose their influence with increasing grain size. While Fig. 4 shows indeed the limiting cases for the 2D simulations just as predicted analytically in Eq. (6), the deviating behavior for long-time annealing became apparent already in Fig. 3 and can be seen again in Fig. 13 for grain boundary energy–triple junction mobility-control (Fig. 13a) and triple junction energy–grain boundary mobility-control (Fig. 13b). According to Eq. (6) we expect hRi / t respectively hRi / t1=3 , which agrees with the simulation results for quite lengthy annealing times (fitted curves in red1 for thousands of simulation time steps after initial periods of time). Nevertheless, for larger average grain sizes and therewith long-time annealing we can see that the kinetics tends back to normal grain growth with hRi / t 1=2 (fitted curves in blue). Specifically, for the case, where the triple junction energy controls the growth kinetics (Fig. 13a), it can be seen that there is a clear transition period between linear growth (exponent 1/1) and parabolic growth (exponent 1/2), which can approximately be described by an exponent of 1/1.5 (lilac curve). For this case Fig. 14 shows in addition the standard deviation and skewness of the scaled grain size distribution for even longer annealing times, where it can be seen again that after an initial period of time the values fluctuate around an average value, while hRi / t. However, while the average growth law changes very quickly to parabolic growth reaching it at approximately t ¼ 12; 000 (see Fig. 13a), standard deviation and skewness decrease steadily but slowly over very long annealing times such that they have not reached the self-similar regime of normal grain growth even after 40,000 time steps. In particular, the standard deviation is (more or less) constant for far longer annealing times (up to approximately 20,000 time steps) than the average grain size follows a linear growth law, which is only up to approximately 7000 time steps. This kind of transient growth behavior is surely related to the behavior of grain growth in the early regime as it has been analyzed, e.g., for normal grain growth [36] and will be a point of future interest. 5. Summary and conclusions In the present paper for the first time in-depth investigations have been carried out analyzing the influence of triple point (in 2D) and triple line (in 3D) energy and mobility on the grain growth 1 For interpretation of color in Fig. 13, the reader is referred to the web version of this article.

kinetics since they may be rate limiting during coarsening. To that aim, grain growth in two- and three-dimensional nanocrystalline polycrystals has been modeled by a modification of the Monte Carlo Potts method, where each type of boundary junction (grain boundary faces and triple junctions) has been assigned an own specific mobility and energy. As a result, we find in two as well as three dimensions four different kinetics, where the average grain size (Figs. 4 and 9) follows the limiting cases that have been predicted analytically (e.g., [17,18,20]), where  the kinetics controlled by grain boundaries yields normal grain growth including a parabolic growth law, hRi / t1=2 ,  the kinetics controlled by the energy of the grain boundaries together with the mobility of the triple junctions yields hRi / t,  the kinetics controlled by the energy of the triple junctions together with the mobility of the grain boundaries yields hRi / t1=3 , and  the kinetics controlled solely by the triple junctions yields again hRi / t1=2 . The latter could be problematic, if someone were to analyze only the average grain size. For example, as shown in Fig. 9a and d the average size increases according to hRi / t1=2 for long annealing times for both cases. However, the associated standard deviation of the scaled grain size takes with 0.4192 and 0.4945 visibly different values, from which it follows that the two kinetics are characterized by different grain size distributions; a fact that can be seen in Figs. 5 (for 2D) and 11 (for 3D). This shows the importance of analyzing not only the average value. In addition, also the topology of those two cases is visibly different. Hence, it can be concluded that, e.g., for grain boundary and grain boundary junction engineering it is important to investigate the polycrystalline microstructures and their temporal development also beyond the average grain size in order to produce specimens with well-defined grain microstructures, which is of great importance since, e.g., also the distribution of the size of grains within a specimen can very well influence materials processing. Generally speaking, all four kinetics are characterized by their individual, self-similar grain size distributions, which agree very well with an analytical distribution that has been calculated among others from topological considerations (see [20]). The distributions for growth under triple junction mobility control (together with grain boundary or triple junction energy) show a longer tail and are shifted to a higher number of small grains regarding their average size. The statistical self-similarity is shown

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in Figs. 5, 7, 11 and 12 by plotting the numerical data for different annealing times. Nevertheless, it should be noted that apart from normal grain growth the other three kinetics hold only for small average grain size and represent limiting cases. Whenever the average grain size increases to larger values (as it will always happen during coarsening) the triple junctions lose their influence and the grain boundaries will always control the kinetics for long-time annealing as shown in Fig. 13. In particular, the average growth laws are in agreement with experimental measurements. For nanocrystalline iron Krill et al. [10] found linear relations between grain size and annealing time just as in Fig. 9b and the upper right case in Eq. (6) tending for larger grain sizes back to parabolic growth as seen above in Fig. 13a. A similar behavior has also been observed for nanocrystalline nickel powder [11]. Malow and Koch [13] calculated also for nanocrystalline iron additionally the growth exponents n from hRi1=n / t observing exponents in the range between approximately 0.05 and 0.35 depending on the annealing temperature. A growth exponent of n ¼ 1=3 is, e.g., in agreement with the lower left case in Eq. (6). All in all, it has been shown that the four theoretically predicted growth kinetics can be modeled using the Monte Carlo Potts model yielding valuable information not only on the average grain size and size distribution, but also on the topology of the structure, e.g., in terms of the Aboav–Weaire-law as well as on long-time annealing. Acknowledgement The author would like to thank Peter Streitenberger from Magdeburg University, Germany for helpful discussions. References [1] M.P. Anderson, D.J. Srolovitz, G.S. Grest, P.S. Sahni, Acta Metall. 32 (1984) 783–791.

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