The evolution of intergranular networks during grain growth and its effect on percolation behavior

Triple junctions (TJs) are line defects in three-dimensional (3D) polycrystalline materials where three grains meet. Transport along the TJs depends on the connectivity between them. Here, we investigate the connectivity of more than 6000 TJs in a 3D microstructure of pure iron (Fe) through the lens of bond percolation. Our efforts are made possible by synchrotron-based x-ray diffraction tomography, which allows us to resolve the TJ network both temporally and spatially, during grain growth. In the framework of standard percolation theory, we determine a percolation threshold of the TJs, above which there exists a continuous pathway of TJs that travels infinitely far. Our experimental results indicate a surprisingly different percolation threshold (by 16%) compared to models and theories. The reason for the discrepancy is that the real TJ network has a topological disorder that is not present in the idealized microstructures considered in prevailing models and theories. Leveraging the wealth of time-resolved data we trace the origin of this disorder by following the topological transitions that accompany grain growth. Overall, the insights obtained in this study can help guide the design of polycrystalline materials via defect engineering.


Introduction
The microstructure of polycrystals is determined by the arrangement of defects with different dimensionalities: 3D grains, 2D grain boundaries (GBs, or faces), 1D TJ lines (or edges), and finally, 0D nodes (or vertices).These defects are present in certain proportions according to the conservation law, where  , ,  , and  indicate the total numbers of faces, edges, vertices, and grains, respectively.Eq. ( 1) can be further reduced, under the assumption that the microstructure is governed by capillary forces.
Surface tension causes all interfaces to seek the lowest interfacial area, which implies that () all edges are shared by three faces and three grains, and () all vertices are formed by the junction of four edges and four grains, assuming uniform GB energies. 1When these two constraints are imposed on the conservation law, Smith [4] demonstrated that ∑  (6 − )  − 6 ( + 1) = 0 (2) * Corresponding authors.
where   is a face with  edges.Microstructures that satisfy this equality have minimal surface area and are thus topologically stable, i.e., their topological properties are invariant under small deformation [5,6].
There exist many polyhedra that locally conform to Eq. ( 2).Yet if we require that only one type of polyhedron be used to fill all of 3D space, then we are left with a very few options, among them Kelvin cells [7] and -tetrakaidecahedra [8].Both have 14 faces per grain.In contrast, the Platonic solids (i.e., convex, regular polyhedra) may not be space-filling (the cube is the exception) nor do they form minimal surfaces.Nevertheless, researchers [9,10] have so-far relied on tetrahedra to model the network of TJ lines.For example, the diamond lattice is decorated with a motif of two tetrahedrally bonded atoms per unit cell.The atoms are topologically equivalent to nodes, since both have coordination (or degree) of  = 4, see Fig. 1 for a side-by-side comparison.
Such idealized networks are rarely encountered in nature.For instance, real grains do not possess a fixed number of faces; rather, they exhibit a distribution in the number of grain-neighbors [11][12][13][14].Likewise, recent work by two of the authors [15]   Topological similarity between (a) the TJ network in a polycrystal and (b) the bond network in a diamond lattice.In both cases, there are four lines (TJs or bonds) connecting each site (quadruple nodes or atoms).An example of one site with four lines is shown in each case, see legend.The 3D grain map from (a) comes from the experimental data (time-step no.15).There are TJs between grains 1, 2, and 3 (see dotted line); between grains 1, 2, and 4 (not shown, out of the page); between grains 1, 3, and 4; and finally, between grains 2, 3, and 4. All grains are colored randomly.Schematic in (b) adapted with permission from Artem Ogurtsov/Shutterstock.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) in polycrystalline aluminum (Al) do not have a fixed coordination of TJ lines; instead, they display a distribution in .Unfortunately, due to the limited spatial resolution of the measurements in Ref. [15], the mean value of this distribution exceeds the expected value of z = 4 (that is, there exist several small grains below the voxel size, in turn leading to the appearance of hyper-coordinated nodes).It is unclear whether this distribution persists under higher magnification, and if so, how it relates to the topological evolution of grains during coarsening.Due to the dearth of time-resolved information, these questions remain unanswered.
The aim of the present study is to characterize the network topology (such as any scatter in the node coordination); understand its history dependence; and finally, investigate its impact on intergranular transport, particularly on the TJ lines.Owing to the larger free volume in the TJ cores, atomic mobility is generally much higher within TJs than at GBs [16,17].As such, TJ diffusion can have a significant impact on materials properties, e.g., chemical and thermal stability [18] and superplasticity [19], especially in nano-grained materials where the density of TJs is high [20].Likewise, the condition for complete wetting is weaker for TJs compared to GBs [21] and consequently the temperature of the TJ wetting transition is lower than that of the GB wetting transition.
We investigate TJ diffusion using the framework of percolation theory.The word percolation is derived from the Latin word perc ōlāre, meaning filter, and in fact one of the first applications of the percolation concept was to the permeation of a fluid in a porous medium [22,23]: a certain fraction  of the pores are ''open'' to the arrival of the fluid while the remaining 1− pores are ''closed'' and otherwise inaccessible.Above the percolation threshold  =   , there exists an infinitely large cluster of open pores (hereafter called bonds) that travels infinitely far, and it is this continuous pathway that enables the fluid to filter through the material.That is, we are interested in the connectedness of microscale defects that may result in a macroscale weakness [24].For instance, we may assume that the connectivity of a certain fraction of TJs facilitates diffusion, while others do not.Indeed, evidence of this point comes from experimental studies of TJ segregation, which demonstrate a strong variability from junction to junction [25].
Thus far, efforts have focused on calculating the transition threshold for different configurations of bonds in the host medium.For example,   for various types of regular lattices of fixed coordination (including diamond [26][27][28]) have been determined, to high accuracy.Intuitively, bond percolation is made easier in these lattices as the coordination number (and hence the density of available paths) increases, i.e.,   ∝ 1∕ [29].Less explored [30,31] is bond percolation in irregular networks, in which  varies from site to site.Such is presumed to be the case for the TJ network, as noted above.Thus, the question remains open as to the applicability of a standard lattice model to real microstructure.It is further assumed in most studies of percolation that the network is static, i.e., its configuration of bonds does not change in time [32].In reality, however, the TJ network evolves during grain growth [33].We find it of utmost importance to analyze real experimental microstructures, and pinpoint if and how the evolution in that case may deviate from textbook/idealized conditions.
To resolve these issues and to probe intergranular percolation via experiments, we require () 3D and () large-scale data.The first precondition is based on the fact that   is highly sensitive to the dimensionality of the system [23,34].The second arises because smaller datasets lead to artificially low transition thresholds compared to infinitely large ones [35].To this end, early investigations on the topological characteristics of microstructure relied on serial sectioning [36,37] or stereo-radiography [38], by which it can take months to years to obtain a single dataset of only a few hundred grains.For this reason, the exploration of topology and its time-evolution had largely remained outside the realm of experimental endeavors, until recently.
Advances in synchrotron 3D X-ray diffraction microscopy, namely, diffraction contrast tomography (DCT) [39,40], enable us to meet the above two criteria.In this study, we take advantage of a wealth of information generated by a recent synchrotron DCT campaign on grain evolution in pure Fe during isothermal annealing [41].To our advantage, the grains in this sample are relatively equiaxed (as per the definition of Ref. [42]; see the grain aspect ratios presented in Fig. S.1); it follows that there should be no significant anisotropy in the transition threshold   .From this 3D data, we quantify the structure of the TJ network, its evolution over time, and its percolation behavior.These results enable an accurate prediction and control of percolative transport in polycrystalline microstructures.

Experimental details
The experiment is already specified in Refs.[41,43].As such, we provide only a concise overview here.The polycrystalline Fe (99.9% purity) sample used in this study was cold rolled to 50% reduction and then annealed at 700 • C for 30 min.During annealing, the sample underwent recrystallization and moderate grain growth, with the average grain size (equivalent radius) increasing from about 12 μm right after recrystallization to about 20 μm.This larger grain size was selected to enhance the relative spatial resolution of imaging the 3D grain structure.Cylindrical samples were cut to a diameter of 500 μm for the subsequent imaging experiment.
DCT was conducted at beamline ID11 at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France.The metallic sample was affixed on an alumina holder and mounted on the tomographic stage of the diffractometer.A monochromatic beam with an energy of 40 keV illuminated the sample.The field-of-view (FOV) was defined by a slit of size 400 μm (vertical) by 600 μm (horizontal), placed in proximity to the sample.The transmitted and diffracted beams were recorded simultaneously on a near-field x-ray imaging detector with 2048 × 2048 pixels and an effective pixel size of 1.54 μm.During acquisition, the sample was rotated continuously over 360 • while Xray projection images were collected for every 0.1 • at a frequency of 10 Hz.
The microstructural evolution was captured in an interrupted in situ manner [44], wherein the sample was alternatively imaged at room temperature and annealed at the beamline using a tube furnace operating at 800 • C. Forming gas (Ar+2%H 2 ) was used to prevent oxidation.After each annealling step, the sample was brought to ambient temperature by a jet of forming gas and was allowed to stabilize for 5 min.prior to starting a DCT scan.In total, 15 DCT scans were collected, separated by 5 or 10 min.anneals [41], see Table 1.These data were reconstructed and processed following Refs.[45,46].As shown in Ref. [41], the orientations of the grains and the positions of the GBs were mapped with a resolution of 0.1 • and 1.5-3 μm, respectively.

Topological analysis
To resolve the various microstructural features (namely, the GBs, TJs, and nodes), we use the PolyProc function package [47].We begin by constructing a region adjacency graph of the 3D granular data.Throughout, we maintain a unique integer ID (GID) for each grain in the microstructure.For a set of GIDs {  , … ,   }, the graph  is a square  ×  matrix (where again  is the total number of grains) such that an element   = 1 when there is a GB between   and   and   = 0 otherwise.To identify the TJs that border the GB face between grains  and , we must find within this matrix the grains   that satisfy   = 1 and   = 1.The triplet of GIDs {  ,   ,   } then serves as a ''serial number'' for the TJs.
To identify the nodes and their connectivity, we systematically inspect the 3D volume and record the GID of each voxel and its 26 nearest neighbors, considering each voxel as a potential node.We then categorize nodes whose GIDs are wholly shared with another node of larger size to be a subcomponent of the larger node (thus ensuring that our measurements are not redundant).This is done by examining each combination of the GIDs in a node and comparing it with each other node of smaller size.If a node is found to be a subcomponent, it is filtered from the node list.We then take the finalized node list and tag each TJ associated with the node by comparing with the list of TJ serial numbers (see above).The node connectivity  is then determined by counting the number of TJs that satisfy the following conditions: () the TJ is associated with the given node and () it is also associated with at least one other node besides the given node.
There are a few common scenarios that would ''disqualify'' a TJ from our topological analysis.For instance, a TJ may connect to a node that is outside the imaged FOV (Fig. 2(a)), and hence does not comply with condition ().Likewise, a TJ may be only internal to a node (i.e., it starts and ends on the same node) and thus does not meet ().The latter case may be understood as follows: consider a small grain that lies on the path of a long TJ, which can be visualized as a bead on a string of finite length.This would create three TJs that connect to the two nodes on opposite ends of the small grain (Fig. 2(b)).Since nodes are also identified by their GID makeup, the two nodes become  topologically indistinguishable and therefore the associated TJs connect to the single node (Fig. 2(c)).Consequently, the three TJs in question would violate condition () and are excluded from our study.At the same time, the TJ corresponding to the long string (in gray) connects to three unique nodes: the two nodes at its two ends (not pictured), and the single unique node associated with the small grain.

Percolation analysis
We measure   from the experimental 3D data in the following manner: 1. Starting at  = 0, we progressively turn 'on' TJs one-at-a-time, at random.Thus, for each step in our algorithm, the fraction of open TJs increases by an amount  = 1∕ where again  is the total number of TJ edges in the probed microstructure.At the 15th time-step,  = 6323, and thus the discretization of our analysis is  ≈ 0.0002.Once a bond is turned on, it stays on for the remainder of the calculation.2. For each iteration, we determine the sizes of the connected clusters of TJs (in terms of the number of edges per cluster, ) and also assess whether percolation has occurred across the FOV.Clusters are detected using the local table method [48].See Fig. 3(a-c) for an example.Ultimately, percolation is identified if a cluster spans from the top to the bottom of the imaged FOV, see Fig. 3(d) for the network spanning cluster (we examine TJ connectivity in this top-bottom direction because the sample is taller than it is wide).The first instance of percolation gives us the percolation threshold   .3. Because there is no direct or indirect technique to determine the energy of a given TJ [49], we are unable to impose crystallographic constraints here (as was done by Schuh and coworkers for GBs [50,51]).As such, our analysis is centered on studying the impact of topological disorder alone on the percolation properties of the TJ network.4. We use the bond percolation threshold for the diamond lattice,  . , for a standard of comparison.In the thermodynamic limit of an infinite lattice,  . falls within the range of 0.388 to 0.390 [26][27][28].In principle,   for finite systems will be lower than that of the theoretical infinite lattice.Our calculations on a diamond lattice of 5488 bonds gives  . = 0.3874 ± 0.0207; for a larger lattice of 8192 bonds, we find  . = 0.3877 ± 0.0179.
The standard deviations reflect the variation between 5000 independent trials.In comparison, the TJ network has 6312 bonds (Table 1).We measure percolation along one direction, just as we do for the TJs.Our computed threshold on the finite lattice is not significantly different from those values reported in the literature, which helps to validate our methodology.

Basic statistics
Table 1 gives the populations of grains, GBs, and TJs as a function of time during grain growth.We list only those TJs which are at least one voxel in length.The number of TJs falls by 46% as the number of grains decreases by 42%.This can only occur through topological transitions (Section 3.1.3).

Node coordination and its evolution
Based on the procedure in Section 2.2, we calculate and display in Fig. 4 the number of TJs per node after 75 min.of annealing (mean: 3.97, standard deviation: 1.31, sample size: 6323 TJs).Observe that around one-third of the nodes do not have the expected coordination of four.While earlier work [15] showed a similar variance in , the mean in that study was much higher ( z = 6.184) due to the poorer spatial resolution (and consequently, the measured transition threshold   was lower than that of diamond).In our case, spatial resolution is not a limiting factor, contributing under a 5% error in Fig. 4, and therefore z ≈ 4 (see Appendix A for a detailed discussion).This means that we can now make a direct comparison between the percolation characteristics of our microstructure and the diamond lattice (Section 3.2.1).
Fig. 5(a) shows the evolution of the node coordination  during grain growth.Clearly, the TJ network is not static but evolves with time.Even so, the mean coordination remains at z ≈ 4 (as indicated by the blue lines).As grain growth proceeds, the standard deviation of  decreases with time (Fig. 5(b)), from a maximum of 1.37 at the onset of the anneal to a minimum of 0.97 at 70 min.
Since the node coordination is never uniquely 4, it follows that Eq. ( 2) (pertaining to the numbers of faces   with  edges) cannot be satisfied.If we evaluate   outright (see Eq. ( 2) and Fig. S.2 for data at the 15th time-step), we find that the right-hand-side is indeed nonzero.

Topological transitions
The existence of nodes with  < 4 is not totally unexpected, given how we define the node coordination (see Section 2.2).As noted previously, the external surfaces of the sample may intercept some of the TJ lines, resulting in a subcritical node coordination.To this point, for the present dataset, 60% of such nodes are located within 10 voxels of the sample surfaces, as shown visually in Fig. S.3.The remainder of the nodes with  < 4 have re-entrant TJs, i.e., the TJ line starts and ends on the same node, and thus does not contribute to the connectivity of the TJ network.We do not include re-entrant TJs in our calculation of .This particular topology is associated with very small grains, such as the so-called dyohedral grain form, for which  = 2,  = 2, and  = 1 [36].Finally, we note that nodes with  = 3 (so-called ''triple point junctions'') have been observed in phase-field simulations of 3D grain growth with a strong nonuniformity of GB energy [3].
On the other hand, the presence of nodes with  > 4 defies the arguments outlined in Section 1 and therefore warrants careful consideration.While our conventional understanding is that four TJs converge at a node, there are transients in grain growth during which this ''rule'' is temporarily broken (but the initial and final configurations are topologically correct).As such, we propose that such supercritical nodes are associated with topological transformations, such as the disappearance of grains with more than four faces in 3D (vide infra).Of course, it is mathematically impossibe that a decrease in the total number of grains per unit volume (here, from 1327 to 776 over 75 min.)can take place without grain disappearance.
Before considering grain disappearance in 3D, we briefly review the state of our understanding in 2D systems.According to Smith [4], if a shrinking grain in 2D has 3 <  < 6 sides, its shortest GB disappears first.This means that the evanescence of a five-sided grain proceeds through a cascade of topological transitions where  evolves as 5 → 4 → 3, after which the three-sided grain disappears.Yet the traditional picture differs from the real behavior captured via in situ experiments in quasi-2D films: grains with  = 4 or 5 sides can shrink to vanishing without losing sides in the process [52,53].One can imagine that the vertices of the shrinking grains move along the directions of the outgoing GBs, eventually collapsing on a single point or node.The quadrivalent or pentavalent node is, however, short-lived: eventually GBs will appear between the grains that were originally second neighbors, thereby restoring the network to the ''correct'' topology.
Far fewer studies [54] have examined the topological evolution in bulk systems, due to the difficulty of obtaining this kind of information.Even so, one may anticipate an analogous behavior in 3D.For example, an  -faced polyhedral grain that maintains its shape and topology will asymptotically produce a -valent node upon elimination, where  = 2 − 4 (see Fig. B.2 for a schematic illustration).In the usual case, a tetrahedral grain has  = 4 and hence  = 4; if instead  > 4, then  > 4. It is also possible to momentarily produce a supercritical node during ''neighbor-switching'' events (items 2-3 below, wherein the number of grains is conserved).
To lend credence to this idea, we quantify the three types of topological transformations, which gives a mechanism for the emergence and annihilation of the supercritical nodes.Following Refs.[55][56][57], these are 1.Grain disappearance, as noted previously.We count the number of grains that vanish within a given time-interval , at the resolution of synchrotron DCT.That is, the ultimate topological behavior of a vanishing grain may preclude observation, and it is possible that grains with  > 4 lose GBs immediately before disappearing.Since such events may occur when the grain volumes are already much smaller than the average volume in the system, we suggest that there is no practical difference between these two situations.2. Face creation.As a result of capillary-driven GB motion, two nodes may approach each other.This will temporarily produce a supercritical node, before giving way to a new face between two grains that had not previously been in contact.We monitor this topological event by counting all GBs that are present at time  +  but absent at time . 3. Grain separation.This is topologically the reverse of item 2 above: a face will vanish, giving rise to a supercritical node; this configuration is relatively unstable and will subsequently dissociate into two nodes with  = 4 separated by a TJ.To keep track of grain separation events, we count the number of faces lost, being careful to exclude in our summation those faces of the disappearing grains (item 1).
Fig. 1 of Ref. [57] provides a schematic illustration of each event type, in 3D.The results from our calculations are given in Fig. 6(a) as a function of time.To evaluate the topological evolution independent of length-scale, we divide the number of events by the number of grains at each time-step.Theory and simulation show that the number of events per grain (for each of the three categories) approaches an (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)Fig. 6.Topological transitions give rise to nodes with less-stable configurations, i.e.,  ≠ 4. (a) We divide the transitions or so-called events into three types, termed grain disappearance, face creation, and grain separation, see Section 3.1.3for details.The number of events is normalized by the number of grains at a given time-step.The total number of events per grain does not reach an asymptote, but decreases almost monotonically with time.(b) Correlation of the total number of topological events, from (a), against the total number of less-stable nodes, from Fig. 5.Each data-point is associated with a given time-step (see color-bar).Coefficient of determination: 0.894.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)asymptotic limit during steady-state grain growth [57].Yet that is not the case here: we see a decay in the numbers of events per grain, which may be expected if initially the grain structure is not typical of normal grain growth but instead solidification or recrystallization (see, e.g., Ref. [58]); the topology of recrystallized networks is a topic worthy of future study.That is, while the Fe sample underwent complete recrystallization and some grain growth before imaging (as described in Section 2), its microstructure is influenced by the thermomechanical history, including solidification and recrystallization.As time progresses during grain growth, the topological operations from above restore the TJ network to its lowest-energy topology.It follows that the microstructure must be in a transition stage.Earlier work arrives at the same conclusion, based upon the lack of self-similarity in the distributions of grain sizes and faces [41].
With this data in hand, we plot in Fig. 6(b) the number of topological events (unscaled) against the number of unstable nodes (those with  ≠ 4).Each data-point represents a time-step where the two quantities are calculated.These measurements reveal a strong positive correlation ( 2 = 0.90), which suggests a direct relationship between the unstable nodes and the topological transitions that are thought to produce them.

Stability of nodes
The above results support the general idea that supercritical nodes come about when ''the Euclidean dimensions of some part of the system chance to pass through zero'' as noted by Ref. [36], nearly 50 years ago.For instance, such nodes may come about via grain disappearance.In this scenario, we can evaluate the Gibbs energy   of a -valent node (taken as the final state) at constant temperature and pressure, relative to that of the shrinking grain with  = ( + 4) ∕2 faces (the initial state).For an isotropic GB energy,   is therefore proportional to the difference between the surface areas of the initial and final configurations, denoted  0 and   , respectively.In Appendix B, we derive an exact expression for   (  0 ,   ) , under the assumption that regular polyhedral grains shrink to a point.Ultimately, given  or , we can compute the corresponding   via Eq.(B.7).Our efforts are informed in part by an earlier treatment of the same problem in Ref. [59].
The above analysis assumes that only the GBs contribute to the energy change of grain elimination.If we instead assume that the TJ energy governs node formation, we arrive at a different expression (compare Eqs. (B.7)-(B.8)).Even so, the behavior of the two equations is surprisingly similar, see Appendix B. It can be seen that the energy change increases monotonically with  and passes through the abscissa at  ∼ 13.4.That is, it is thermodynamically plausible for grains with fewer than 13.4 faces (in 3D) to shrink to a point since   < 0. This establishes an upper-bound on the node coordination, given by  = 2 − 4 = 22.8.Indeed, according to Fig. 5, there are no nodes in the microstructure with coordination greater than this value.
In Fig. 7(a), we plot the theoretical   as a function of  for  > 4 (see purple line).On the same figure, we show also the probability  of the node coordinations  obtained experimentally (see colored data-points).These data come from Fig. 5(a).Observe that the most probable nodes in the microstructure are those with the lowest Gibbs energy   .A similar trend has been reported for 2D defects in metals and ceramics: the GB character distribution is determined by the anisotropy of the GB energy [60,61].In fact, there is an inverse correlation between the two quantities.
We might expect the same behavior for 0D defects here.To test for a relation between node population and energy, we correlate  against   in Fig. 7(b), using all of the experimental datapoints from Fig. 7(a).We fit these data to an empirical, Boltzmann-like distribution of the form where  ≡   −  4 (such that a quadruple junction has  = 0).We pick Eq. ( 3) as our model function since it has been suggested that GB populations approach this distribution during grain growth through a mechanism that preferentially eliminates the high-energy GBs from the microstructure as coarsening proceeds [60].Likewise, we see in Fig. 7(a) that the population of  = 6 nodes steadily decreases in time (the time-evolution of the other nodes is less clear, presumably due to their far lower populations in the microstructure).Ultimately, we find that Eq. ( 3) is a reasonable fit to the data, with  2 = 0.842.We anticipate that the inverse correlation between  and   would become even stronger if we were to use an improved model for   , specifically, one that takes into account the anisotropy of the GB energies.Since we do not have this information a priori, we resort to our simplified expression for   in Eq. (B.7) as a first-order approximation.

Spatial correlations
It is important to bear in mind that the TJ network is irregular (in the sense that  is variable) and not random (in its arrangement of the nodes).To demonstrate this point, we define (  ) as the probability of finding a node with coordination   .This information can be retrieved directly from Fig. 1.Likewise, (  ,   ) is the probability of a configuration of two nearest neighbor nodes with coordination numbers   and   .For a random network, the nodes are statistically independent and hence In Fig. 8, we plot the correlation function (  ,   )∕ ( (  )(  ) ) .The data indicate that node neighbors are indeed correlated, e.g., if   is large, then   is more likely to also be large (or vice versa) than independence would imply.This property is sometimes referred to as assortative mixing [62], i.e., the preference of low  nodes to be linked to other low  nodes.The trend shown here is contrary to that of a Voronoi tesselation, in which unlike nodes (  ≠   ) are positively correlated while like nodes (  =   ) are negatively correlated [30].The Voronoi network is hence disassortative.Finally, we note that spatial correlations persist even in the early stages, i.e., preceding grain growth, with a marked preference for assortative mixing (almost indistinguishable from that conveyed in Fig. 8).
To understand why the spatial correlations should exist, we must consider the topological arrangements of grain structures.Patterson and others [63,64] have introduced an affinity term to describe the tendency of grains of different face classes to contact each other, relative to random behavior.Generally, large grains with many faces and small grains with far fewer faces show the highest affinity for Fig. 9. Illustration of spatial correlations in node coordination.Assume a relatively large grain (in gray) is initially surrounded by several small grains.Eventually these small grains will shrink to a point during coarsening, giving rise to supercritical nodes (numbered 1-4, in blue).This situation gives rise to nearest-neighbor nodes with like coordination.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)mutual contact.Thus, one can imagine several small grains decorating the periphery of a large grain.During coarsening, these smaller grains may reduce in size until they become points (nodes), and the points themselves will be connected by edges (TJs) of the large grain.See Fig. 9.That is, these nodes have a similar coordination and are also adjacent to each other.As such, spatial correlations among the nodes are inevitable given the topological characteristics of the grain growth microstructure.This example also illustrates another property of the TJ network: the spatial segregation of like-coordinated nodes (in this example, at the periphery of the large grain).Indeed,

Transition threshold
Having elucidated the structure of the TJ network, we now direct our attention towards analyzing its percolation characteristics.We determine We arrive at this result using the microstructure annealed for 75 min.The error represents the standard deviation of the mean   across 1000 trials, each employing a random order of TJs (see point 1 above).Importantly, we find that     >  .
by a surprising 16% despite having z = 4 in the TJ network and also controlling for the total number of bonds in the diamond lattice (see item #4 in Section 2.3).This result implies that the real TJ network is more resilient to intergranular percolation compared to the idealized diamond lattice (considering only the differences in the topology of the two networks).
The transition threshold varies by ≲ 8% over time during grain growth, although we caution that   is sensitive to the system size (namely, the number of bonds or TJs), as noted in Section 2.3.Thus, it is not advisable to compare the experimental   values to each other since the total number of TJs decreases steadily with time (Table 1).Stated differently, any variation in the transition threshold will be due to a convolution of two factors: differences in the network topology (Fig. 5) as well as differences in the system size, and it is nontrivial to disentangle the contributions of each to   .
According to a universal, empirical law developed by Galam and Mauger [65], systems of equal dimension  and coordination number  should have equal bond percolation thresholds too.However, Refs.[66,67] present several counterexamples of regular lattices with identical  and  but dissimilar   values.In a similar vein, we show here that despite having equal (average) coordination numbers, the diamond lattice and the TJ network exhibit different bond percolation thresholds.This is another piece of evidence to prove that  and  are necessary but insufficient to predict percolation thresholds.

Cluster distributions
The disparity in   must stem from fundamental differences in the structures of the diamond and TJ networks.To better visualize the network structure, we quantify the cluster size distribution (, ), which gives the probability that a bond is ''on'' and a member of a cluster (or connected component) of  bonds.In principle, (, ) can be calculated from the configuration of node neighbors.For example, in the simplest case where  = 1, That is, the configuration of nodes (encoded by  (   ,   ) ) determines the cluster structure and, by extension, the percolation properties.In the limit that  → 0, there exist no open bonds and hence (1, 0) = 0; in the other extreme  → 1, all bonds are open to percolation, i.e., the concept of a discrete cluster becomes obsolete, and so (1, 1) = 0.In between these two limits, (1, ) will pass through a maximum.For the diamond lattice,   =   = 4 and  (   ,   ) = 1, leading to Eq. (C.1).We have derived expressions for not only (1, ) but also (2, ) and (3, ), see Appendix C. Fig. 10 compares the cluster size distributions for the TJ network (denoted exp.) and the diamond lattice (dia.).The former is measured directly from the experimental data in Fig. 3 while the latter is calculated with the aid of Eqs.(C.2)-(C.3).For  < 0.15 the curves for the TJ network and the diamond lattice nearly overlap.Yet for higher  approaching   , they diverge.In this regime, the TJ network has a greater tendency to form small clusters of  = 2 and  = 3 bonds.Since the number of bonds is fixed, it follows that larger clusters (that may span the domain) are less likely, and this would account for the higher percolation threshold     .A similar logic is used in Ref. [31] to explain the higher   of random networks in comparison to regular grids of the same coordination number, for  ≥ 3.
The topology of the TJ network determines its percolative behavior.In networks that are both assortative and spatially segregated, such as ours (refer again to Section 3.1.5),it has been found via simulation [68] that   can be substantially higher than that of networks with random (uncorrelated) connections.The inhomogeneous distribution of the low-and high-coordinated nodes in space (see Fig. S.3(a)) leads to the formation of densely connected clusters separated by depleted areas (as visualized in Fig. 3(d)).These clusters will readily disconnect from each other when bonds are randomly removed, thus accounting for the relatively high   .If, hypothetically, the network contained more long-range connections, it would be more robust to link removal and consequently   would be lower [69].
The above logic provides some general guidance on how to engineer microstructures with desired percolative properties.For example, to increase   (and hence, render the material more ''resistant'' to percolation of liquid metal or another diffusant along the TJs), we should aim to increase the spatial segregation of like nodes in the microstructure.This could be accomplished in practice by annealing under conditions that favor abnormal grain growth, see, e.g., Refs.[70][71][72][73].In that case, the percolation pathway is forced to ''go around'' the exceptionally large grains in the microstructure.

Conclusions
We examine the connectivity of a real TJ network in 3D and for the first time, thanks to new strides in synchrotron-based DCT and microanalysis.Based on our efforts, we can draw a number of conclusions: 1.There exists a scatter in the node coordination () distribution, which is largely due to topological transients occurring during grain growth.Importantly, these transient configurations are not captured in idealized networks used to model polycrystalline microstructures.In comparison, we plot the same two distributions for the diamond ('dia.')lattice, see Appendix C for details on their calculation.
2. The population of a -valent node scales inversely with its Gibbs free energy.This means that highly coordinated nodes are exponentially less likely to be found in the microstructure (at long times).Similar to the GBs, we find a Boltzmann-like equation best describes the relationship between node population and energy.3. We calculate a percolation threshold (  ) of the TJ network that is significantly different (by 16%) to that of the diamond lattice, despite the topological similarity between the two structures.The disparity in   can only be attributed to the irregular structure of the TJ network, namely the assortativity and spatial segregation of nodes.
This work paves the way for several future investigations.For one, we aim to characterize by experiment the topology of just the recrystallized microstructure (i.e., prior to any grain growth), as we believe it is ''inherited'' by the material at the onset of imaging.The recrystallized structure is strongly affected by the spatial distribution of active nucleation sites, and the competition between nucleation and growth.We would also like to measure the percolative properties of other types of microstructures, including those with abnormally large grains or with a pronounced texture.In addition, we hope to measure   considering also the GB and TJ energies.At present, however, this calculation is nontrivial given that the TJ energy is a function of 25 degrees-of-freedom [49].Finally, we have identified two characteristics of the (normal) grain growth microstructure -assortativity and spatial segregation of nodes -that can be incorporated into future simulations of intergranular transport.Theoretical analysis on the number of hypercoordinated nodes (those with  > 4) that result from missing grains with radius less than the spatial resolution ( min ).The number of missing grains is determined by fitting a lognormal distribution to the experimental data and truncating it at various values of  min .As a conservative estimate, we assume that each missing grain gives rise to one hypercoordinated node.The effect of spatial resolution is less pronounced at the later time-steps where R is higher.grain, for simplicity.We further assume that the GB energy is constant and isotropic.
Let  0 be the initial surface area of the polyhedral grain.As the grain shrinks, the outer GBs (connected to the TJs that bound the grain) increase in area.These outer GBs have a total surface area   , which can be computed from the areas of the 3 − 6 triangles connected to point .The number of TJ edges that are generated in this process is  =  = 2 − 4. See Hence, we the node energy   relative to that of the polyhedral grain.
By definition, the faces of a regular polyhedron are themselves congruent regular polygons.The area of a regular polygon is  2 ∕ (4 tan (∕)) where  is the side length and  is the number of sides.The latter is related to the number of faces  from above as  = 6 − 12∕ .Accordingly, Likewise,   can be found as

Appendix C. Bond cluster distribution for diamond
Consider the bond between two sites, labeled  and , in a lattice with  = 4, such as diamond.Let  be the probability that this particular bond is 'on' in the context of percolation.To form a cluster of size  = 1, it follows that all six neighboring bonds must be 'off'.Three of the neighboring bonds are associated with site  , and the other three with .See

Fig. 1 .
Fig. 1.Topological similarity between (a) the TJ network in a polycrystal and (b) the bond network in a diamond lattice.In both cases, there are four lines (TJs or bonds) connecting each site (quadruple nodes or atoms).An example of one site with four lines is shown in each case, see legend.The 3D grain map from (a) comes from the experimental data (time-step no.15).There are TJs between grains 1, 2, and 3 (see dotted line); between grains 1, 2, and 4 (not shown, out of the page); between grains 1, 3, and 4; and finally, between grains 2, 3, and 4. All grains are colored randomly.Schematic in (b) adapted with permission from Artem Ogurtsov/Shutterstock.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2 .
Fig. 2. Illustration of a few scenarios that would disqualify TJs from our topological analysis: (a) TJ (red) crosses the imaged FOV; (b-c) three TJs (red) start and end on the same node (blue).In (b), the two nodes are topologically equivalent if the gray TJs (at top-right and the bottom-left) have the same serial number.In this case, the two nodes are consolidated into a single node (c) during analysis.See Section 2.2 for details.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3 .
Fig. 3. 3D visualization of TJ connectivity at time-step no. 15.We turn 'on' a number fraction  of the TJs at random, such that they are now available for percolation (conversely, the TJs that remain 'off' are inconsequential and are thus rendered transparent): (a)  = 0.158, (b)  = 0.475, and (c)  = 0.633 (≫   ).In each snapshot, the connected components or clusters of TJs are colored according to their size, i.e., the number of TJs within the cluster.Sample boundaries are translucent gray.For  → 0, the clusters are small and comprised of only single TJs, while for  ≥   the clusters are relatively large and span the domain, see (d) for an example of a single spanning cluster.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 4 .
Fig. 4. Distribution of the number of TJs per node, , obtained from the experimental microstructure.Probability indicates the number fraction of nodes with a given coordination of TJ lines.Mean: 3.97, standard deviation: 1.31.

Fig. 5 .
Fig. 5. Evolution of node coordination  during isothermal coarsening.(a) Violin plot of  at each recorded time-step.The width of the violins reflects the frequency of nodes with a given coordination.The blue horizontal line gives the mean coordination, which is approximately four for all time-steps.(b) Corresponding standard deviations of  vs. time.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7 .
Fig. 7.The energetics of highly-coordinated ( > 4) nodes.(a) Probability of such nodes for all recorded time-steps (see color-bar).As before, probability indicates the number fraction of nodes with a given coordination.For all highly-coordinated nodes, we plot also the Gibbs energy of forming a node with  TJs,   ≈   ∕ 0 − 1, see Appendix B. The higher the coordination, the higher is the energetic cost.(b) Same data as in (a), but now plotted against  ≡   −  4 .With this definition, a node with the requisite four TJs would have  = 0. We fit this data to a nonlinear equation of the form probability ∝ exp (−), see the dashed line.Coefficient of determination: 0.842.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. S.3(a) depicts the segregation of nodes by coordination  in the microstructure.

Fig. 10 .
Fig. 10.Bond cluster distributions (, ) as a function of , for clusters with  = 2 and  = 3 TJs.We compute (, ) using the experimental ('exp.')data gleaned from Fig. 3.In comparison, we plot the same two distributions for the diamond ('dia.')lattice, see Appendix C for details on their calculation.

Fig. A. 1 .
Fig. A.1.Theoretical analysis on the number of hypercoordinated nodes (those with  > 4) that result from missing grains with radius less than the spatial resolution ( min ).The number of missing grains is determined by fitting a lognormal distribution to the experimental data and truncating it at various values of  min .As a conservative estimate, we assume that each missing grain gives rise to one hypercoordinated node.The effect of spatial resolution is less pronounced at the later time-steps where R is higher.

Fig. B. 2 .
Fig. B.2. Elimination of a cuboidal grain.Initially the grain has  = 6 faces (left) where each face has  = 4 sides.As the grain shrinks to a point at location  (right), 12 triangular faces (GBs) are created, as are  = 8 lines (TJs, in orange).(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig. B.2 for an illustration.Thus, our task is to calculate the quantity   ≈   ∕ 0 − 1 (B.1)

Fig. B. 3 .
Fig. B.3.Gibbs energy of grain elimination for grains with  faces, in 3D.The orange curve corresponds to Eq. (B.7) and the blue curve to Eq. (B.8).(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

6 )
By virtue of Eq. (B.6), the ratio of the lengths ∕ℎ can be calculated for any  -sided regular polyhedron.Combining Eqs.(B.1)-(B.6),we obtain  is the right-hand-side of Eq. (B.6).The above analysis does not consider the contribution of the TJs to the Gibbs energy of node formation.We can repeat the calculation with attention to the change in line length instead of face area during grain elimination (refer again to Fig. B.2).We label this energy change  ′  to avoid any confusion with   .Much like how we approached the GBs, we assume that the TJ energy is constant and isotropic.In this scenario, B.7)-(B.8) are plotted as a function of  in Fig. B.3.

Fig. C. 4 .
If this is not true, then the cluster would have size

Fig. C. 4 .
Fig. C.4.Connectivity of bonds on a 2D square lattice ( = 4).Two sites are labeled  and .The open bond between them is highlighted in green, while the closed bonds are in gray.This configuration gives rise to a cluster of size  = 1.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) revealed that nodes https://doi.org/10.1016/j.actamat.2024.119987Received 8 March 2024; Received in revised form 1 May 2024; Accepted 6 May 2024

Table 1
Overview of grain statistics, with a focus on the numbers of grains, GBs, and TJs as a function of time.See Section 2.2 for details on their calculation.