E Quer Ab

V Baldy

A Desrochers

Ecological drivers of root grafting in balsam fir natural stands

Keywords: Root grafting, Anastomosis, Abies balsamea, Boreal forest

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Although competition for resources is commonly considered as the main tree interaction in forest ecosystems, it is increasingly understood that trees may interact in more complex ways via "root connections" [START_REF] Klein | Belowground carbon trade among tall trees in a temperate forest[END_REF][START_REF] Lortie | Rethinking plant community theory[END_REF][START_REF] Mcintire | Facilitation within Species: A Possible Origin of Group-Selected Superorganisms[END_REF].

Mycorrhizal and suckering networks interconnecting indirectly root systems through fungal hyphae or directly through parental roots have been highlighted to alter forest stand dynamics by allowing exchanges of resources between trees (Adonsou et al., 2016a;[START_REF] Baret | Root connections can trigger physiological responses to defoliation in nondefoliated aspen suckers[END_REF][START_REF] Leake | Networks of power and influence: the role of mycorrhizal mycelium in controlling plant communities and agroecosystem functioning[END_REF][START_REF] Simard | Mycorrhizal networks: a review of their extent, function, and importance[END_REF]. However, root grafting, i.e. the fusion of the vascular system of two or more roots generally from the same species but also rarely between different species, constitutes another underestimated pathway for resource exchanges [START_REF] Graham | Natural root grafts[END_REF][START_REF] Lev-Yadun | Why should trees have natural root grafts?[END_REF][START_REF] Mudge | A history of grafting[END_REF]. This phenomenon has been reported in nearly 200 perennial woody species in diverse habitats worldwide [START_REF] Graham | Natural root grafts[END_REF][START_REF] Rue | Root-Grafting in Tropical Trees[END_REF]. Grafted trees, through an extended common root system, have reinforced mechanical support and are able to exchange water, nutrients, photosynthates but also pathogens [START_REF] Bader | Hydraulic coupling of a leafless kauri tree remnant to conspecific hosts[END_REF][START_REF] Fraser | Carbohydrate transfer through root grafts to support shaded trees[END_REF][START_REF] O'neal | Intraspecific root grafts and clonal growth within Ailanthus altissima stands influence Verticillium nonalfalfae transmission[END_REF]. Grafted trees are thus not only in competition for resources but also in cooperation or facilitation [START_REF] Eis | Root Grafts and their Silvicultural Implications[END_REF][START_REF] Keeley | Population variation in root grafting and a hypothesis[END_REF][START_REF] Loehle | Adaptive significance of root grafting in trees[END_REF][START_REF] Mcintire | Facilitation within Species: A Possible Origin of Group-Selected Superorganisms[END_REF]. It has been shown that the redistribution of resources between trees by a common root system can lead to increased tree growth and to delayed tree mortality (Adonsou et al., 2016b;[START_REF] Tarroux | Effect of natural root grafting on growth response of jack pine (Pinus banksiana; Pinaceae)[END_REF]. Root grafting could thus be considered as an adaptive strategy to cope with environmental heterogeneity and disturbances, or as an adaptation to intense tree competition leading to a complex community structure and competitive ability (Adonsou et al., 2016b;[START_REF] Baret | Root connections can trigger physiological responses to defoliation in nondefoliated aspen suckers[END_REF][START_REF] Jelinkova | Herbivore-simulated induction of defenses in clonal networks of trembling aspen (Populus tremuloides)[END_REF][START_REF] Salomón | Natural root grafting in Picea mariana to cope with spruce budworm outbreaks[END_REF]. In order to better understand the mechanisms determining root grafting and its influence on forest stands dynamics, several studies have tried to describe tree and stand characteristics leading to root grafting [START_REF] Fraser | Age, stand density, and tree size as factors in root and basal grafting of lodgepole pine[END_REF][START_REF] Külla | Influence of cultivation method on root grafting in Norway spruce (Picea abies (L.) Karst.)[END_REF]Tarroux and DesRochers, 2010). Proximity and high density of trees and roots have been shown to increase the probability of root contacts leading to root graft formation [START_REF] Fraser | Age, stand density, and tree size as factors in root and basal grafting of lodgepole pine[END_REF][START_REF] Gaspard | Natural root grafting in hybrid poplar clones[END_REF]Tarroux and DesRochers, 2010). Moreover, soil properties that determine root system architecture can also increase root density in particular soil areas leading to root grafting. For instance in shallow or stoney soils, roots are constrained to a thin layer of soil promoting root encounter [START_REF] Bormann | The occurrence of natural root grafting in eastern white pine, Pinus strobus L., and its ecological implications[END_REF][START_REF] Eis | Root Grafts and their Silvicultural Implications[END_REF]Tarroux and DesRochers, 2010). Finally, mechanical abrasion of root bark by trees swaying in the wind, greater in sandy soils, promotes the fusion of roots' vascular tissues and then graft formation [START_REF] Basnet | Ecological Consequences of Root Grafting in Tabonuco (Dacryodes excelsa) Trees in the Luquillo Experimental Forest, Puerto Rico[END_REF][START_REF] Eis | Root Grafts and their Silvicultural Implications[END_REF]Tarroux and DesRochers, 2010).

Balsam fir (Abies balsamea (L.) Mill) is a common tree species of the boreal forest in Quebec that can grow for prolonged periods of time in dense understory environments [START_REF] Burns | Silvics of North America[END_REF][START_REF] Ruel | Changes in root morphology after precommercial thinning in balsam fir stands[END_REF]. It has shallow rooting and maintains an abundant seedling bank on the ground leading to the formation of very dense stands after disturbances remove canopy trees [START_REF] Morin | Seedling bank dynamics in boreal balsam fir forests[END_REF]. These conditions seem conducive of root grafting and could explain how balsam fir can survive for such prolonged periods in the understory.

However, there is no recent work on root grafting in balsam fir and only anecdotal observations of possible root grafts have been made when studying butt-rotting fungi [START_REF] Redmond | Infection Courts of Butt-rotting Fungi in Balsam Fir[END_REF]. In order to better understand the ecological significance of root grafting in balsam fir it is necessary to investigate the occurrence of root grafts, their frequency and the timing of their initiation.

We make the hypothesis that root grafts formation is common in balsam fir and that their frequency depends on age and size of trees and roots and on the distance between trees. Aims of this study were: (i) to estimate the frequency of root grafting in balsam fir stands of the boreal forest in Quebec, (ii) to describe age and size distributions of roots and trees at graft initiation, (iii) to investigate the effect of the tree proximity, the size of stems, the number and the size of roots on the number of grafts per tree.

Material and method

Study sites and stands characterization

Study sites were located in Abitibi-Témiscamingue in the balsam fir-white birch (Betula papyrifera Marsh.) bioclimatic domain of Quebec boreal forest [START_REF] Grondin | Rapport de classification écologique du sous-domaine bioclimatique de la sapinière à bouleau blanc de l'est[END_REF]. Annual total precipitation for the region averaged 929 mm (253 mm of snow and 675 mm of rain) with a daily temperature of 1.5 ° C and an average of 2366 degree-days above 0 °C (Climate normal 1981-2010, Environment Canada 2018). The three sites were located on the Ontario-Quebec clay belt characterized by post-Wisconsinian lacustrine deposits forming heavy clay soils (at least 60% clay), offering a large physiographic unit with very similar soil conditions over the landscape (Canada Soil Survey Committee, 1987, Vincent and[START_REF] Vincent | L'évolution et l'extension des lacs glaciaires Barlow et Ojibway en territoire québécois[END_REF]. The first site (S1) was located near the Lake Duparquet Research and Teaching Forest (48°52'21.6"N, 79°46'53.0"W), the second site (S2) was located near Kinojévis teaching forest (48°18'26.1"N, 78°84'63.7"W) and the third site (S3) was located near the Harricana teaching forest training center (48°76'27.8"N, 77°77'93.3"W). We selected monospecific, healthy, naturally regenerated and mature stands (> 30 years old) based on their proximity to a water source and a gentle slope in order to be able to carry out the hydraulic excavation of the root systems.

Excavated areas were approximately 50 m 2 and included a minimum of 15 living trees per area, corresponding to stands with a minimum density of 3000 stems ha -1 (Table 1). Before excavation, distance between trees and DBH of each tree were measured. Trees were then cut down with a chainsaw and cross-sectional disks were collected at ground level (0 m). Disks of stems were air dried and sanded (80-400 grit) to reveal growth rings. As balsam fir is known to often present missing growth rings at the trunk base, the age of trees was estimated by counting growth rings and visually cross-dating chronologies of all trees using pointer years such as narrow or wide rings (Parent et al., 2002, Schweingruber, 1988). A total of 125 stem disks were analyzed. Excavated area stand density ranged 6000 stems ha -1 to 14200 stems ha -1 . Age and diameter at breast height (DBH, 1.30 m) of trees differed among sites (ANOVA, Fage = 52.28, FDBH = 18.16, P< 0.001) and ranged 39-74 years old and 7.9-16.5 cm, respectively. Mean age was lowest in S1 and highest in S3 while DBH decreased with increasing density (S1 < S3 < S2) (Table 1).

Root systems excavation and root measurements

The root system of each tree was hydraulically excavated using a firefighter's lance coupled to a high-pressure water pump (Mark III, Wajax, Lachine, Quebec), following Tarroux and DesRochers (2010) (Supplementary material, Figure S1). To uncover all the roots, the excavation depth was between 30 to 60 cm. Complete excavation of root systems took about three weeks per site. Excavation revealed buried rotted stumps that were not included in the analyses. Standing dead or moribund trees and their associated root systems and grafts were analyzed if their state of decomposition allowed a clear visualization of growth rings. For each site, we counted number of grafts and roots (diameter > 1.5 cm) per tree. Grafts and crosssectional disks of each root (> 1.5 cm) were collected for dendrochronological analyses in the laboratory. The age of roots and root grafts were determined by counting and cross-dating the root disks with those of their corresponding stem after they were air-dried and sanded (80-400 grit) (Tarroux and DesRochers, 2010). Root disks were collected nearest to the stump in order to facilitate growth rings analysis and to avoid missing rings that are more common as we move away from the tree stump [START_REF] Krause | Tree-ring patterns in stems and root systems of black spruce (Picea mariana) caused by spruce budworms[END_REF]. The age of roots determined near the stump corresponds to the time when the root was produced by the tree. The grafts then form later in the life of the root, that is why the age of root grafts needs to be determined directly on the graft itself to identify the year when the two roots start producing a single and common wood layer. Root grafts are thus always "younger" than the root itself.

To determine if grafts formed a complete morphological union, they were cut into crosssectional disks throughout their length to confirm if grafted roots shared common growth rings (Supplementary material, Figure S2) [START_REF] Graham | Natural root grafts[END_REF]. Few very intertwined roots were rejected in the lab as they were not grafted and did not share a common wood layer after being cut. Age of grafts was determined by counting the number of common growth rings between the two grafted roots. Age of trees at graft initiation was determined by calculating the difference between the age of tree and the age of the associated graft. Due to the eccentric root growth, root diameter was calculated by averaging perpendicular measurements of the longest and the shortest diameters of the root (Adonsou et al., 2016a). Root diameter at graft initiation was calculated by using the same method but measurements were realized on graft sections, before the first common growth ring between two roots (Supplementary material, Figure S2).

A total of 698 root disks and 58 grafts were analyzed.

Statistical analyses

Statistical analyses were done using R software version 3.5.1 (R core team, 2019).

Differences in mean DBH, age of trees and root diameter between the three sites (S1, S2 and S3) were tested by one-factor ANOVAs, followed by post-hoc Tukey tests. Because of violation of ANOVA assumptions, differences in mean distances, number of roots and graft age were analyzed using Kruskal Wallis test followed by post hoc Wilcoxon tests. The effect of sites on percentage of grafted trees was tested by using Pearson's Chi-squared test. Number of grafted roots and trees according to their diameter, age at graft initiation and distance between trees were also tested using Pearson's Chi-squared test. We developed a generalized linear model with a negative binomial function (GLM.NB.) to identify which predictors between tree age (years), DBH (cm), number of roots, mean root diameter (cm) and distance (m) were the best predictors to explain the number of grafts per tree.

We included mean distance between trees and distance to the nearest tree as predictors; however, after preliminary analysis we only used distance to the nearest tree as it performed better. Full data set were used for this model independently of site because the response variable variance (number of grafts per tree) was similar across sites. Only distance to the nearest tree was considered to calculate the model. The full model (including all the predictors) was simplified in order to determine the most parsimonious model using the function StepAIC (package "MASS", [START_REF] Venables | Modern Applied Statistics with S. Fourth Edition[END_REF]. StepAIC performs stepwise variable selection that adds or removes predictors to produce a model that minimizes Akaike's Information Criterion (AIC). Predictions of the most parsimonious model were represented graphically using "visreg" function (package "visreg", [START_REF] Breheny | Visualization of regression models using visreg[END_REF]. We used conditional inference tree (function "Ctree" in package "partykit") to help interpret the results. It estimates a regression relationship by binary recursive partitioning in a conditional inference framework that is based on maximally selected rank statistics [START_REF] Hothorn | Unbiased recursive partitioning: a conditional inference framework[END_REF]. Conditional inference tree was realized by using the variables of the best model fitted using GLM.NB.

Results

Root grafting was observed in the three excavated stands for a total of 58 grafts per 45 grafted trees corresponding to 1.3 grafts per grafted tree on average (Table 2). In total, 698 roots were excavated and 87 of them were grafted to roots of another tree. Percentage of grafted trees was similar between sites, corresponding to 36% on average (Pearson's χ 2 test, χ 2 = 0.94, P > 0.05) even if tree and root diameters, mean distance between trees, tree and graft ages were different (ANOVAs, FRoot_diameter = 7.09, Kruskal Wallis test, χ 2 min_distance= 13.73, χ 2 Graft_age = 8.31, P < 0.05; Table 1 and2). Mean graft age was 13.57 years (± 0.98 SE), with the youngest graft being 2 years old and the oldest graft 39 years old. Root grafting occurred mainly between healthy trees of the same diameter class (Supplementary material, Table S4). The difference in diameter between two grafted trees generally did not exceed 10 cm. Only 7 grafts were between two dead trees or between a healthy and a dead/moribund tree. Most of grafts were formed near the stumps (< 30 cm). During root graft formation, a callus formed around the fusion point of the vascular tissues deforming the roots making it difficult to estimate one dominant root over the other (Supplementary material, Figure S2). However, most of grafts were constituted by similarly-sized roots (Supplementary material, Table S4). Grafts between two roots occurred from every angle but most of the grafts were formed when two main lateral roots encountered (Supplementary material, Figure S3a). Other dominant types of grafts were formed when a root touched the stump of a neighboring tree, generally involving a dominant and a smaller tree (Supplementary material, Figure S3b). Age of grafted trees at graft initiation ranged 12-106 years old across the sites and averaged 38 years old (± 3.0 SE). Relative distribution of grafted trees did not vary significantly among age classes at graft initiation but 71% of trees were grafted before 45 years old (Figure 1). Mean DBH of grafted trees was 14.1 cm (± 1.8 SE) across sites and it ranged 1.30-77 cm. Proportion of grafted trees was similar across tree DBH classes (Pearson's χ 2 test, χ 2 = 2.36, P > 0.05, results not shown).

Root grafting occurred in roots as young as 5 years old and as old as 64 years old. Sixty-five percent of roots were grafted before 30 years old and mean root age at graft initiation was 26.48 years (± 1.35 SE) (Figure 2A). Mean root diameter at graft initiation was 3.94 cm (± 0.33 SE) and 63% of roots were grafted at a diameter less than 4 cm (Figure 2B). We found no graft from roots smaller than 0.40 cm in diameter or larger than 14.94 cm.

Mean distance between grafted trees was 47.91 cm (± 5.10 SE) and we did not observe any graft beyond a 181 cm distance. Percentages of root grafts did not significantly differ between our 25 cm distance classes but tended to decrease with increasing distance (Figure 3). We showed that 84% of all root grafts occurred between trees < 75 cm apart and that the majority (62%) of grafts formed between trees distant 4-50 cm (Figure 3).

Age of trees, DBH and mean root diameter were not significant predictors of the number of root grafts per tree (GLM.NB, P > 0.05) (Table 3). However, the number of grafts per tree increased with number of roots per tree (GLM.NB, P < 0.05) and decreased with distance between trees (GLM.NB, P < 0.001) (Table 3, Figure 4). The following equation ( 1) predicted the number of root grafts per tree according to the number of roots and the distance between trees (Figure 4): (1) Log (Number of grafts per tree) = 0.61+ (0.11 × number of roots) -2.33 minimal distance between trees.

The conditional inference tree model showed the same significant predictors as the MPM selected and both models had similar R 2 (33% and 44% respectively). When distance between trees was less than 0.13 m, predicted number of root grafts was the highest, around 5 grafts.

When distance between trees was above 0.77 m, trees were generally not grafted. If distance was between 0.13 and 0.77 m and if trees had 10 or less than 10 roots, number of root grafts predicted was around 1. However, for similar distances, trees that had more than 10 roots had predicted numbers of root grafts around 4 grafts (Figure 5).

Discussion

We confirmed here for the first time the occurrence of root grafting in balsam fir. Root grafting has been reported in several fir species such as silver fir (Abies alba Mill.; Elwes and Henry,1909), white fir (Abies concolor (Gordon and Glend.) Lindl. ex Hildebr.; [START_REF] Jepson | The trees of California. Cunningham, Curtis, and Welch[END_REF] and grand fir (Abies grandis (Douglas ex D. Don) Lindl.; [START_REF] Graham | Natural root grafts[END_REF],

suggesting that most species of this genus could form root grafts. Excavated stands were highly interconnected with on average 36% of grafted trees indicating that root grafting is a common occurrence in balsam fir (Table 2). This percentage is lower than in pine stands, usually recognized to have strong root grafting propensity (up to 90% in Pinus resinosa Sol. ex Aiton or to 50% in Pinus strobus L.) [START_REF] Bormann | The structure, function, and ecological significance of root grafts in Pinus strobus L[END_REF]. Root grafting occurrence in balsam fir was however similar to other natural stands of boreal forest species such as Norway spruce (33%) [START_REF] Külla | Influence of cultivation method on root grafting in Norway spruce (Picea abies (L.) Karst.)[END_REF] and balsam poplar (48%) (Adonsou et al., 2016a).

Tree proximity was the most important driver for root grafting as the number of grafts per tree decreased with increasing distance between trees (Table 3, Figure 4). These results confirm the postulate that distance between trees is the main predictor of root grafting beyond stand density, age or size of trees and roots [START_REF] Gaspard | Natural root grafting in hybrid poplar clones[END_REF][START_REF] Külla | Influence of cultivation method on root grafting in Norway spruce (Picea abies (L.) Karst.)[END_REF]Tarroux and DesRochers, 2010). This could be explained by (i) a shorter time for roots of close trees to meet and a greater probability to cross, (ii) the fact that tree proximity promotes mechanical abrasion and root friction caused by wind swaying and thus root grafting [START_REF] Cook | Backflash damage to residual stands incident to chemi-peeling[END_REF][START_REF] Külla | Influence of cultivation method on root grafting in Norway spruce (Picea abies (L.) Karst.)[END_REF]. Proximity is a better predictor of root grafting than stand density because the latter does not reflect spatial arrangement of trees (Tarroux and DesRochers, 2010). Our results confirmed this assumption, as tree density was three times higher in S1 than in S3 but percentages of grafted trees were similar (Tables 1 and2). Distance between grafted trees averaged 50 cm, as was found for lodgepole pine and Norway spruce, suggesting that close proximity between trees is necessary for root grafting (Figure 3) [START_REF] Fraser | Age, stand density, and tree size as factors in root and basal grafting of lodgepole pine[END_REF][START_REF] Külla | Influence of cultivation method on root grafting in Norway spruce (Picea abies (L.) Karst.)[END_REF]. In our three stands, 80% of grafted trees were linked to their closest neighbor and thus grafted trees were mostly observed between trees in clusters, which is not well reflected by stand density. Dense tree clusters are very common in balsam fir stands that can maintain an abundant seedling bank on the ground conducive to closely growing tree formations [START_REF] Morin | Seedling bank dynamics in boreal balsam fir forests[END_REF][START_REF] Ruel | Changes in root morphology after precommercial thinning in balsam fir stands[END_REF].

The number of grafts per tree increased with the number of roots per tree (Table 3, Figure 4).

As it was reported in Douglas fir (Pseudotsuga menziesii (Mirb.) Franco) and hybrid poplar (Populus maximowiczii × Populus balsamifera), positive correlation between root grafting and number of roots per tree is probably due to the higher probability of intertree root contacts [START_REF] Gaspard | Natural root grafting in hybrid poplar clones[END_REF][START_REF] Reynolds | Estimating probability of intertree root contact in second-growth of Douglas-fir[END_REF]. This effect was particularly important when distance between trees was between 0.13 and 0.77 m; At these distances, the predicted number of grafts for trees with less than 10 roots was around 1 and around 4 if trees had more than 10 roots (Figure 5). Since root density decreases with increasing distance from the stem, trees with a greater number of roots increased root grafting probability [START_REF] Külla | Influence of cultivation method on root grafting in Norway spruce (Picea abies (L.) Karst.)[END_REF]. We found two trees that presented an above-average number of grafts (> 1.3 grafts per tree) even if they were distant of more than 50 cm to the closest tree (Figure 4). These two trees were larger than the average of grafted trees (DBH > 14.1cm) and had lots of roots (> 9 roots per tree). The resulting greater probability of forming root grafts could thus be due to the larger area of soil they occupy. However, the percentage of observed trees with more than 10 roots was low (approx. 13%) so this modeling result has to be considered with care (Tables 1 and2).

When trees increase in size and age, their roots become larger and have more ability to anchor firmly in the soil. During root grafting formation radial growth of the two roots is maintained at the point of contact, creating a pressure point, stronger for larger roots, can possibly facilitating the fusion of vascular tissues [START_REF] Reynolds | Estimating probability of intertree root contact in second-growth of Douglas-fir[END_REF]. In this way, positive correlations are often found between number of grafts per tree, DBH, mean tree age, and mean root diameter [START_REF] Bormann | The structure, function, and ecological significance of root grafts in Pinus strobus L[END_REF][START_REF] Bormann | The occurrence of natural root grafting in eastern white pine, Pinus strobus L., and its ecological implications[END_REF][START_REF] Fraser | Age, stand density, and tree size as factors in root and basal grafting of lodgepole pine[END_REF]Tarroux and DesRochers, 2010). However, these factors did not explain root grafting in balsam fir (Tables 2 and3). Balsam fir growing in close-stems formations responds vigorously to release following disturbances and goes through a self-thinning stage caused by strong competition for light [START_REF] Bégin | Balsam fir self-thinning relationship and its constancy among different ecological regions[END_REF]. During self-thinning, stands undergo a shift in DBH frequency of live trees and diameter distribution becomes increasingly skewed [START_REF] Mccarthy | Self-thinning dynamics in a balsam fir (Abies balsamea (L.) Mill.) insect-mediated boreal forest chronosequence[END_REF]. These changes in tree size hierarchy could explain why DBH was not a good predictor of root grafting for this species. Self-thinning in balsam fir forests of western Newfoundland begun once stands reached stem densities of approximately 31000 stems ha -1 at around 60-65 years old [START_REF] Mccarthy | Self-thinning dynamics in a balsam fir (Abies balsamea (L.) Mill.) insect-mediated boreal forest chronosequence[END_REF]. Mean tree age was 74 years old in Site 3 with a density of 6000 stems ha - 1 ; Self-thinning stage could thus have been in progress at the time of excavation. Trees in sites 1 and 2 seemed too young for being in this self-thinning stage. However, the observations of dead trees and stumps during the excavation and the large distribution of tree DBH suggest that stands could undergo the preliminary stage of self-thinning at excavation time.

Moreover, results on balsam fir age determination have to be considered with care because the physiological age of a balsam fir can differ from its chronological age [START_REF] Bégin | Balsam fir self-thinning relationship and its constancy among different ecological regions[END_REF][START_REF] Morris | How old is balsam tree ? For[END_REF]. Indeed, trees can undergo intermittent radial and height growth interruptions during their understory development leading to absent growth rings at the stem base and consecutively in the root system [START_REF] Desrochers | Is ring count at ground level a good estimation of black spruce age? Can[END_REF][START_REF] Morin | Seedling bank dynamics in boreal balsam fir forests[END_REF][START_REF] Parent | Effects of adventitious roots on age determination in Balsam fir (Abies balsamea) regeneration[END_REF]Parent et al., , 2002)). Missing growth rings could be expected to be particularly numerous in the in closed-stems formations leading to underestimation of the true age of trees and their roots [START_REF] Krause | Changes in radial increment in stems and roots of balsam fir [Abies balsamea (L.) Mill.] after defoliation spruce budworm[END_REF].

In our study, natural root grafts were found in a variety of tree ages and sizes, respectively from 12 to 106 years old and from 1.3 to 77 cm in DBH. Similarly, root grafting occurred from 1 to 90 years in jack pine stands (Tarroux and DesRochers, 2010) and from 2 to 46 years in lodgepole pine stands [START_REF] Fraser | Age, stand density, and tree size as factors in root and basal grafting of lodgepole pine[END_REF]. The oldest root graft that we found was 39 years old and belonged to a 77 and a 74 years old trees (in S3). The youngest tree that was grafted was 36 years old at time of excavation and its oldest graft was 24 years old (in S1). This study is limited to the excavation time and we cannot predict the evolution of the grafts or whether grafts will continue to form over time. Though these large variations in the timing of graft formation means that grafting was not restricted to a particular period but could continue throughout the life of stands and that grafts could persist throughout stand development.

According to our results, root grafts also appear to form in roots of any age: root age at graft initiation averaged 26.5 years but varied from 5 to 64 years. Root age at graft initiation averaged 20 years but ranged between 2-41 years in trembling aspen (Populus tremuloides Michx; [START_REF] Jelinkova | Molecular and dendrochronological analysis of natural root grafting in Populus tremuloides (Salicaceae)[END_REF] and begun between 10-20 years (46% of cases) but ranged between 7-35 years in Norway spruce [START_REF] Külla | Influence of cultivation method on root grafting in Norway spruce (Picea abies (L.) Karst.)[END_REF]. However, root grafting initiated when roots were relatively small: 63% of roots were grafted at a diameter under 4 cm (Figure 2). This is probably because small roots have thinner bark, facilitating the fusion of vascular tissues during graft initiation [START_REF] Eis | Root Grafts and their Silvicultural Implications[END_REF].

Balsam fir is shade tolerant and competition for light is a key factor of tree interactions and community structure [START_REF] Kunstler | Competitive interactions between forest trees are driven by species' trait hierarchy, not phylogenetic or functional similarity: implications for forest community assembly[END_REF][START_REF] Messier | Functional ecology of advance regeneration in relation to light in boreal forests[END_REF]. As resources are redistributed among grafted trees, a shaded tree could receive photosynthates from a well-lit grafted neighbor [START_REF] Fraser | Carbohydrate transfer through root grafts to support shaded trees[END_REF]. Root grafting could thus explain this great shade tolerance in balsam fir, allowing it to survive in the understorey for several decades before release. This notion supports the hypothesis that natural root grafting allows intraspecific cooperative relationships between trees and limits tree competition effects by maintaining tree growth through resources translocation instead of competing for them [START_REF] Loehle | Adaptive significance of root grafting in trees[END_REF][START_REF] Lev-Yadun | Why should trees have natural root grafts?[END_REF]. Our study thus suggests that communal root systems established between grafted trees could sustain tree growth within tree clusters and reduce intraspecific competition for soil resources and light.

It could also explain why competition indices are often inconclusive for this species. In their study, [START_REF] Duchesneau | Effects of light and intraspecific competition on growth and crown morphology of two size classes of understory balsam fir saplings[END_REF] found that growth of balsam fir saplings (height < 200 cm) receiving less than 25% of full sunlight was not affected by intraspecific competition. It suggests that root grafts could allow suppressed grafted trees to persist under conditions where un-grafted trees would be removed by competition [START_REF] Fraser | Carbohydrate transfer through root grafts to support shaded trees[END_REF]. Root grafting could thus delay self-thinning through greater survival rates of suppressed trees, resulting from carbohydrate and water transfers from vigorous and dominant neighboring trees (Adonsou et al., 2016b;[START_REF] Fraser | Carbohydrate transfer through root grafts to support shaded trees[END_REF]. During self-thinning stage, tree density decreases leading to an increase of distance between trees and likely to a decrease in the formation of new grafts.

Root grafting should be considered in forest management since it can influence forest dynamics, by modulating competition relationships between trees or constitute a way for diseases to spread. Considering that balsam fir frequently develops dense tree clusters, pre-commercial or commercial thinning is often prescribed to release growth and reduce competition for resources [START_REF] Ruel | Changes in root morphology after precommercial thinning in balsam fir stands[END_REF]. However, residual grafted trees may not benefit from reduced competition by thinning if roots and stumps of the removed trees keep on living and drain carbohydrates from residual trees, limiting growth of treated stands [START_REF] Fraser | The persistence and function of living roots on lodgepole pine snags and stumps grafted to living trees[END_REF]Tarroux et al., 2010).

Vigorous residual grafted trees could benefit from the acquisition of new functional roots from a removed connected neighbor but if the imbalance between residual root biomass and leaf area is too large, growth of residual trees may suffer. In two naturally regenerated stands of Pinus banksiana, commercially thinned 5 years prior excavation (46% and 56% of trees removed), dendrochronological analyses showed that radial growth of residual trees grafted to removed trees was lower than that of non-grafted tree after 4 years (Tarroux et al., 2010). The amount of resources translocated between two grafted trees is proportional to the number of grafts per tree [START_REF] O'neal | Intraspecific root grafts and clonal growth within Ailanthus altissima stands influence Verticillium nonalfalfae transmission[END_REF]. Since the number of grafts per tree averaged 1.3 in Abies balsamea versus 0.79 in Pinus banksiana, a larger negative effect of thinning on fir growth can be expected. This effect should be particularly noticeable during the mortality phase or the selfthinning phase of natural balsam fir stands development, since this is the period when the number of grafted stumps and suppressed trees is likely to be the highest. Lighter thinning and from below is thus advised in such stands in order to reduce the imbalance between live root biomass and residual leaf area after thinning [START_REF] Eis | Root Grafts and their Silvicultural Implications[END_REF]Tarroux et al., 2010). Thinning from below only removes the smallest trees, hence creating a smaller imbalance between residual leaf area and the "communal" root biomass of a stand. This smaller imbalance allows stands to increase photosynthetic capacity and maintain growth rather than having to invest photosynthates into the support of too much root biomass when larger trees are removed (i.e.

thinning from above) [START_REF] Desrochers | Root biomass of regenerating aspen (Populus tremuloides) stands of different densities in Alberta[END_REF].

In conclusion, although our stand characteristics differed, natural root grafting was found in the three excavated balsam fir stands. Root grafts were formed early or later in tree development (from 12 to 106 years old). Proximity of trees and the number of roots per tree increased the number of root grafts per tree, most likely because these factors are conducive to increased root contacts. These results suggest that studying the spatial arrangement (e.g. local tree and root density near grafted trees) might help to better predict probability of root grafting.

Since balsam fir stands were highly connected through root grafts, trees can redistribute resources within the stand, which could explain how balsam fir can survive for such prolonged times in the understory. Root grafting could reduce intraspecific competition for soil resources and light and allow grafted trees to persist under conditions where un-grafted trees would be removed. Integrating root grafting knowledge into silvicultural practices could improve management of balsam fir stands. For instance, limiting the "waste" of photosynthates in the support roots and stumps of thinned grafted trees by using thinning from below could trigger an increase in photosynthetic capacity and growth resulting from the increase in available resources.

Figures:

Figure 1: Percentage of grafted Abies balsamea trees observed in the 3 sites (± SE) in relation to tree age at graft initiation. Differences between classes were tested by using Pearson's χ 2 test (χ 2 value is represented, ns: non-significant, with P = 0.07 and N = 38).

Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 2345 Figure 2: Percentage of grafted Abies balsamea roots observed in the 3 sites (± SE) in relation to root age (A) and root diameter (B) at graft initiation. Differences between classes were tested by using Pearson's χ 2 test (a > b > c > d, different letters indicate significant differences, χ 2 value are represented with *: P < 0.05 and N = 86, ***: P < 0.001 and N = 81).

Table 1 :

 1 Main characteristics of the three Abies balsamea excavated stands. Mean distance between trees (m) ± SE 0.46 ± 0.06 a 0.81 ± 0.12 b 0.77 ± 0.06 b 0.65 ± 0.05 Note: Totals and means were calculated including all sites ± Standard Error. DBH: Diameter at Breast High. When P < 0.05, different letters indicate significant differences between sites, a < b < c.

	Site	S1	S2	S3
					Total
	Number of excavated trees	71	24	30	125
	Number of excavated roots	352	161	185	698
					Mean
	Stand age (years)	36	51	74	54
	Density (stems.ha -1 )	14 200	4800	6000	8 333
	Mean tree age (years) ± SE	39 ± 15 a	49 ± 10 b	74 ± 14 c	49 ± 1.84
	Mean DBH (cm) ± SE	7.9 ± 0.7 a	16.5 ± 1.6 b	11.9 ± 0.8 b	10.5 ± 0.6

Table 2 :

 2 Root systems and grafts characteristics of the three Abies balsamea excavated stands. .06 a 11.54 ± 1.32 a 19.43 ± 2.51 b 13.57 ± 0.98 Mean diameter of roots (cm) ± SE 6.93 ± 0.27 a 8.27 ± 0.42 a 10.28 ± 0.46 b 8.12 ± 0.21 Mean number of roots per tree ± SE 5.97 ± 0.26 a 6.71 ± 0.30 a 6.61 ± 0.21 a 6.29 ± 0.16 Note: Totals and means were calculated including all sites ± Standard Error. When P < 0.05, different letters indicate significant differences between sites, a < b < c.

	Site	S1	S2	S3
					Total
	Number of grafts	30	12	16	58
	Number of grafted trees	23	10	12	45
	Number of grafted roots	47	18	22	87
					Mean
	Percentage of grafted trees (%) ± SE	32.4 a	41.7 a	40.0 a	36.0 ± 2.86
	Mean number of grafts per grafted tree 1.3	1.2	1.3	1.3 ± 0.03
	Mean graft age (years) ± SE	11.66 ± 1

Table 3 :

 3 Output of negative binomial generalized linear model (GLM.NB) testing the effects of tree age, DBH, number of roots per tree, mean root diameter and distance between trees on the number of root grafts per tree. Note: Values of R 2 and AIC of the general model including all factors (All) and of the most parsimonious model (MPM) are shown. Variables retained in the most parsimonious models are reported in bold (d.f.= degree of freedom, P values = ns: non-significant with P > 0.05, *: P < 0.05, ***: P < 0.001, N = 80).

			Number of grafts per tree
		d.f. Z value	P value
	(Intercept)	1	0.413	ns
	Age (year)	1	0.492	ns
	DBH (cm)	1	0.207	ns
	Number of roots	1	2.582	*
	Mean root diameter (cm)	1	0.597	ns
	Distance (m)	1	-5.411	***
	All R 2 (AIC)		0.16 (232.95)
	MPM R 2 (AIC)		0.33 (230.4)

Acknowledgements

The research was funded by a Natural Sciences and Engineering Research Council discovery grant to AD. We are grateful to Elizabeth Turcotte and Etienne Poisson for their assistance with data collection and fieldwork. We also thank Jordane Gavinet, Hermine Nguena, Eric Meneiri and Mathieu Santonja for their technical support in statistical treatment of the data.

Supplementary material


Hannes Zschiesche
email: hannes.zschiesche@im2np.fr

Ahmed Charai

Claude Alfonso

Dominique Mangelinck

Methods for Gibbs triple junction excess determination -Ti segregation in CoSi 2 thin film

Keywords: Gibbs excess, Triple junction, Atom probe tomography, Ti segregation, CoSi 2 thin film

Methods are presented determining the Gibbs triple junction excess (Γ TJ ) of solute segregation in poly crystalline materials from single atom counting in 3D volumes. One method bases on cumulative profile analysis while two further methods use radial integration of solute atoms. The methods are demonstrated and compared on simulated model volumes which include three grain boundaries joining together at a triple junction with set values for Gibbs grain boundary and triple junction excess. An experimental technique that provides 3D volumes with single atom detection and spatial resolution close to atomic scale is atom probe tomography. An atom probe tomography volume of a CoSi 2 thin film that contains three grain boundaries and a triple junction has been acquired. Ti segregation is found qualitatively at the grain boundaries and triple junction. The quantification of the Ti excess at the investigated CoSi 2 triple junction reveals for the three introduced methods positive Gibbs triple junction excess values. It demonstrates that there is an excess of Ti at CoSi 2 triple junctions and provides opportunities for its quantification.

Introduction

Triple junctions (TJs) are present in poly crystalline material where three grain boundaries (GBs) join together. They influence directly the thermodynamics and kinetics of materials and thereby their properties. Thus, their description by geometric and thermodynamic parameters is of great interest [1][2][3][4]. Experimental measurements of TJ energies were demonstrated using Scanning Tunneling Microscopy [5] or Atom Force Microscopy [6]. Simulations have brought insides to the view on TJ energies from the theoretical approach [7,8].

Segregation of solutes to TJs can also change the energy of the TJ [9]. Direct measurements of TJ segregation were obtained by energy dispersive x-ray spectroscopy (EDS) [10,11]. Thickness measurements by energy filtered transmission electron microscopy (EFTEM) have allowed to conclude the presence of cavities in the TJ region [12] that was interpreted as a segregation of vacancies.

However, these kind of TEM based measurements are limited through 2D projections. In contrast, atom probe tomography (APT) allows 3D measurements at the atomic scale [13,14]. For example, measurements on a thermally treated Ni-Cu bilayer system have demonstrated that there is a significantly higher Ni diffusion in Cu TJs than in Cu GB [15]. Further studies on the same material system have revealed that additionally there is a lower activation energy for TJ diffusion than for GB diffusion [16,17]. Similar investigations on thermally treated Fe-Cr multilayers have determined a TJ segregation enthalpy of Cr in Fe TJ [18,19]. These experimental results have shown qualitatively some excess at TJ. Moreover, quantification of the excess at a TJ was obtained recently from a theoretical approach that studied GB and TJ segregation in the Fe-Cr system [20]. However, for a quantitative comparison between experimental and theoretical data, methods are required for determination of the Gibbs TJ excess (Γ T J ) [21] by APT similarly to the established method for Gibbs GB excess (Γ GB ) [22].

TJs are of particular importance for poly crystalline thin films, because their proportion is larger due to small grain size. Especially, this is true in silicide thin films that are used in microelectronics. CoSi 2 films on Si substrates are widely used in microelectronics as contact material beside TiSi 2 and NiSi films, depending on the scale. Low electrical resistivity, high thermal stability and the good match of the crystal structure with Si are the main advantages of CoSi 2 [23]. The reaction of thin cobalt films with silicon substrates leads to sequential formation of Co 2 Si, CoSi and CoSi 2 [24,25]. Ti was introduced as an interlayer between the Co layer and the Si substrate to promote a strong texture (epitaxy) in CoSi 2 grown on Si(100) or Si(111) by sputtering deposition and reactive diffusion [26]. The role of the Ti interlayer during the phase formation has been investigated and explanations were proposed, such as the reduction of native oxide at the Si surface [27][28][29], the formation of a diffusion barrier for Co [27,29,30] and the variation of CoSi 2 nucleation due to Ti segregation at the Si interface [31]. The latter one was proven by the influence of a Ti capping layer instead of a Ti interlayer [32,33]. The Ti capping layer induced an increase of the nucleation temperature of CoSi 2 as well as a promoted texture in comparison to the growth without capping Ti. These effects were explained by the diffusion of Ti from the capping layer to the interface. Recently, the presence of Ti segregation at CoSi 2 GBs has been shown [34].

In this work, Ti segregation at CoSi 2 TJs is studied. For a quantitative determination of Γ T J , three methods are developed. They are compared on a model volume with known excess values, before application on an experimentally acquired volume.

Material and Methods

CoSi 2 thin films were grown on Si(111) wafers which were immersed in a 5 % dilute HF for 1 min to remove native oxide before load into the sputtering system. Successive deposition of 5 nm Ti and 100 nm Co from two individual targets of Ti and Co (both 99.99 % purity) was done by magnetron sputtering.

The deposition was performed with a base pressure of about 10 -8 Torr using 99.9999 % pure Ar and a gas flow of 6.7 sccm at room temperature. Thermal annealing were performed in vacuum up to 850 • C to fully transform the asdeposited sample into CoSi 2 .

APT tips were prepared using a focused ion beam (FIB) FEI Helios 600 with a Ga + source [35,36]. APT measurements were performed on a LEAP 3000X HR. Voltage mode at 40 K, 200 kHz, 20 % pulse fraction and 0.2 % detection rate was applied in order to get optimal information as a compromise between resolution and volume size. IVAS software was used for the reconstruction following the common procedure [37,38]. The reconstruction parameters were adjusted with crystallographic information: they were changed in order to match the plane spacing obtained from the observed planes belonging to a pole whose nature was determined by its symmetry [39].

Results

Firstly, the creation of a model volume is described. This model volume contains 3 grains, 3 GBs and the TJ with different solute contents. Then, the methods developed to quantify Gibbs TJ excess are demonstrated on such a model system, before an experimentally acquired volume is presented and Ti excess at the present CoSi 2 TJ is quantified.

Simulation of model volume with known excess values

To test the quantification methods of this Gibbs TJ excess, model volumes were simulated using Matlab. The parameters where chosen close to the ones observed in an experimental volume in order to give a realistic representation. Fig. 1a shows a schematic projection of the described geometry including the geometrical parameters in the projected planes. A cube with CoSi 2 structure (fcc, CaF 2 type) with a single orientation, i.e. a single crystal, was built in order to simulate a 3D volume with atomic resolution. In this volume, a given number of Ti atoms per unit volume (atomic density) were introduced such as:

ρ G Ti = N G Ti V G = x G Ti ω CoSi2 , (1)
where N G Ti is the number of Ti atoms in the considered grain volume V G , x G Ti is the atomic fraction and ω CoSi2 the atomic volume of CoSi 2 . In this definition, the atomic density of Ti is directly linked to the atomic fraction of Ti since the addition of Ti is assumed to not modify the atomic volume of CoSi 2 (i.e. Ti substitutes either Si or Co and does not modify the lattice parameter of CoSi 2 ).

Based on this approximation, ρ G Ti was generated by random substitution of a fraction x G Ti of Co and Si atoms by Ti atoms. It was further assumed that the Ti density was the same in the different grains since it should be the case at equilibrium.

The three GBs inside this cube are defined as rectangular slice volumes that go from the border of the cubic volume to its center (Fig. 1a). They have all the same width d GB and they share a same direction corresponding to the TJ axis. In the limit that the GB width goes to zero (d GB → 0), the GB describes a plane that separates two grains. Note that the grains in our simulated volume are just defined as regions delimited by GBs and do not have different orientations between them in contrary to the real case where their orientation is different. The Gibbs GB excess (Γ GB ) [21] of segregated Ti is given in atoms per area of GB. By assuming that the Ti distribution is uniform inside the GB slice, the Gibbs GB excess can be transformed to an Ti atomic density inside the rectangular slice volume by dividing through d GB . Adding the Ti atomic density of the neighboring grains results in the total Ti atomic density inside the defined GB volume.

ρ GB Ti = ρ G Ti + Γ GB Ti d GB = ρ G Ti + N GB,Exc Ti V GB = N GB Ti V GB = x GB Ti ω CoSi2 (2)
There is thus an additional (excess) number of Ti atoms (N GB,Exc Ti ) inside the GB volume (V GB ) in comparison to the grains. As a uniform atomic density is assumed in the GB, the Gibbs GB excess is realized by random substitution of the fraction x GB Ti of Co or Si by Ti in the GB volume.

The TJ is defined as a cylindrical volume with a radius r TJ along the center line of the cube (Fig. 1a). r TJ was geometrically fixed as the minimal cylinder radius which includes all GB overlap at the TJ (hatched regions in Fig. 1b).

It was thus determined from the largest distance of a GB overlap to the TJ center (Eq. A.2 in Appendix A) which depends on the dihedral angle between adjacent GBs (α, β, γ) and the GB width (d GB ). In the limit that the GB width goes to zero (d GB → 0), the TJ radius goes to zero (r TJ → 0) and the TJ represents the junction line of the three GBs. Similarly to the Gibbs GB excess, the Gibbs TJ excess of Ti segregated at the TJ (Γ TJ ) is given in atoms per length on such a line. A transformation of Γ TJ to a number per 3D unit volume (i.e. a Ti atomic density) can be achieved by dividing through the base area of the TJ cylinder volume. The total Ti atomic density inside this TJ volume depends on the atomic densities of the bordering grains, the atomic density of the GB excess respecting their volume fraction inside the TJ volume (f TJ GB,i , Eq.

A.4 in Appendix A) and the atomic density related to the TJ excess. Note that to calcualte the GB fraction in the TJ cylinder volume (f TJ GB,i ), one has to consider the overlap (hatched regions in Fig. 1a) in order to avoid an artificial TJ excess (Eq. A.3 in Appendix A).

ρ TJ Ti = ρ G Ti + 3 i=1 f TJ GB,i • Γ GB,i Ti d GB,i + Γ TJ Ti π • r 2 TJ = ρ G Ti + 3 i=1 f TJ GB,i • Γ GB,i Ti d GB,i + N TJ,Exc Ti V TJ = N TJ Ti V TJ = x TJ Ti ω CoSi2 (3)
A uniform distribution is also assumed in the TJ and the number of Ti atoms (N TJ Ti ) inside the TJ volume (V TJ ) was made by random substitution of a fraction

x T J Ti of Co or Si by Ti in the TJ volume.

A simulated model volume with set GB and TJ excess is shown in Fig. 1b.

The Ti atoms are visible as dark points in the 3D cubic volume (side length 23 nm). A 2D Ti atomic density map which is projected along the viewing direction (parallel to the TJ) is also shown in Fig. 1b. The color scale of the 2D atomic density map ranges from low Ti atomic density (blue) in grain regions to high Ti atomic density (red) in the TJ region. In this model system, the atomic density for a given atom map is equivalent to an atomic fraction since the atomic density for all atoms (i.e. the inverse of the atomic volume) inside the cube is set to constant in the approximation that there is no change in the atomic volume of CoSi 2 (i.e. 0.0128 nm 3 /at) when Ti is introduced substitutionally. The 2D atomic density map in Fig. 1b is similar to the schematic in Fig. 1a. Knowing the set values inside the model volume will now permit to test Gibbs TJ excess quantification methods.

Gibbs TJ excess quantification methods applied on simulated model volumes

Three different methods will be applied to simulated model volumes in order to quantify Γ TJ . In the simulated model volumes, Ti Gibbs TJ and GB excess values have been set to Γ TJ Ti = 4 at/nm and Γ GB Ti = 1 at/nm 2 and the Ti atomic density in the grains to ρ G Ti = 0.18 at/nm 3 . The geometry of the GBs has been varied from narrow GBs (d GB = 0.5 nm) to wide GBs (d GB = 3.0 nm) and equal dihedral angles (α = β = γ = 120 • C). Accordingly to Eq. A.2, r TJ was set to 0.3 nm (when d GB = 0.5 nm) and 1.7 nm (when d GB = 3.0 nm). The values are close to the one observed in an experimental volume (see Sec. 3.3).

Cumulative profile method -linear integration

The method proposed in this paragraph bases on a cumulative profile similarly to the method used for determination of the Gibbs GB excess [22]. Therefore, a cuboid volume is centered along a GB crossing the TJ perpendicular (Fig. 2a). The width of the cuboid can be varied (dotted arrows) to optimize the geometry around the TJ in order to get the highest sensitivity to the Gibbs TJ excess in the 1D atomic fraction and cumulative profiles. Fig. 2b shows the Ti atomic fraction along the linear integration inside the selected cuboid.

A separation of the different features inside the cuboid is possible from varying constant Ti atomic fraction regions. As expected, the highest Ti atomic fraction is observed at the region where the TJ is present. However, the values are not directly related to the atomic fractions inside the labelled features, i.e. GBs and/or TJ, but to the average over the width of the cuboid that can include grains, GBs and TJ at the referring location. Because of such averaging effect, it is difficult to extract the value of the TJ excess from such a 1D Ti atomic fraction profile. The cumulative profile allows a better quantification of the TJ excess: it is achieved by plotting the cumulative number of Ti atoms over the cumulative number of all atoms along the analysis direction (black arrow in Fig. 2a). Fig. 2c shows such a cumulative profile on the simulated model volume with d GB = 0.5 nm. The slope in the cumulative profile refers directly to the Ti atomic fraction in the according region. The region corresponding to the TJ has the highest Ti atomic fraction due to the TJ excess and thus the largest slope: it can be separated from the regions before and after the TJ that are less rich in Ti. The constant atomic fraction regions (slopes) on the left and on the right of the TJ region can be linearly fitted (green dashed lines). In the Gibbs approach, the TJ is considered as a line between three GB planes, i.e. r TJ is null: thereby, the slope in the plot of the TJ region is infinite large and the segregation excess at the TJ can be represented as a step (red vertical line in Fig. 2c) between the fitted constant atomic fractions on the left and the right of the TJ in the cumulative profile (green dashed line in Fig. 2c). The step height refers directly to the amount of Ti atoms in excess at the TJ. The Gibbs TJ excess value, Γ TJ , is calculated by dividing this number of Ti atoms through the length of the TJ included in the cuboid. The Gibbs TJ excess determined by this method is Γ TJ Ti = 3.6 at/nm which is slightly lower than the set value of Γ TJ Ti = 4 at/nm.

The relatively large variation might be due to limitations of the cumulative method. Indeed, as detailed by Jenkins et al. [40], the cumulative method suffers from limitations even in the simple case of GB. Furthermore, there are additional limitations for the TJ. Firstly, there is a high sensitivity of the measured value to the choice of the step location: the step height varies strongly when the position of the step is changed (±0.6 at/nm for a variation of the TJ center by 25%) due to the large difference between Ti atomic fraction in the regions on the left and on the right of the TJ. This difference comes from the presence of two GBs on the right of the TJ and only one on the left. In addition, the random substitution of Co or Si by Ti can introduce fluctuations in the distribution of Ti. These local variations influence the fits and thereby the final result (up to ±1.0 at/nm for variations in selected volume, including possible small variations of TJ center selection). In addition, this method is not applicable when the TJ excess is relatively low or the GB width is large since the change corresponding to the TJ region becomes less pronounced. In order to overcome these limitations, two other methods which are based on radial integration instead of linear integration are now considered.

Radial integration methods

The second and third methods base on a radial integration of the Ti atoms.

In these methods, a cylinder with height z and an increasing radius r is defined with its axis on the TJ line (center line of defined cylindrical TJ region in Fig. 1a) and the number of Ti atoms inside the cylinder is counted (as shown schematically in Fig. 3a). The total number of Ti atoms inside the cylinder depends on r as following:

N Ti (r) = 3 i=1 N G,i Ti (r) + 3 j=1 N GB,j,Exc Ti (r) + N TJ,Exc Ti (r). ( 4
)
The contributions of Ti atoms in the TJ due to the grains, GBs and TJ are related to the geometry and the Ti atomic densities in these different regions.

Dependencies of the number of Ti atoms with the radius r are now considered using the previous assumptions (homogeneous atomic densities in the related volumes, same atomic volume in the whole volume,...). While the number of Ti atoms of the grains changes as a square function in r

(N G Ti (r) = z • ρ G Ti • πr 2 ),
the expressions for the GB excess and TJ excess are more complex and two approaches can be distinguished depending on the value of radius: i.e. if r r TJ

N Ti (r)/z = i ρ G,i Ti • πr 2 + j Γ GB,j Ti • r + Γ TJ Ti . (5)
Fig. 3b shows a radial 1D Ti atomic fraction plot of the simulated model volume used for the cumulative profile method. The Ti atomic fraction is highest in the very tight TJ region (smallest radius r). Close to the TJ region the atomic fraction is much lower, but not immediately constant since the approximation for Eq. 5 does not hold until a certain radius r is reached where the atomic fraction is about constant. This region of about constant atomic fraction is appropriate for the second order polynomial fit and chosen in the radial integral plot. In Fig. 3c, the radial integration of Ti atoms divided through the height of the cylinder (black dots) together with a second order polynomial fit in the region r r TJ (green dashed line) are shown. The TJ excess is determined as the constant from this second order polynomial fit. The result of Γ TJ Ti = 3.9 at/nm is in accordance with the set Gibbs TJ excess in the simulated volume (4 at/nm).

Nevertheless, a difficulty of this method is that it bases on the constant coefficient of the second order polynom: as this term is usually small compared to other contributions, its value depends critically on the fit quality (variations up to ±1.0 at/nm observed for varying fit region). Moreover, if the GB thicknesses is increased, and consequently r TJ , a significant discrepancy between the quantified result and the set value appears. In this case, the approximation of the GB as a rectangular projection does not hold anymore since the assumption, r r TJ , is not fulfilled and Eq. 5 is no more valid. Therefore, another quantification method is now developed which is applied in the range r ≤ r TJ . over r should give a linear plot.

N Ti (r)/(z • r) = π • ρ TJ Ti • r (6)
Comparing Eq. 6 with Eq. 1, 2 and 3 shows that the slope constant a is directly proportional to ρ G Ti , Γ GB Ti , Γ TJ Ti and the geometry of GBs and TJ. The values of ρ G Ti and Γ GB,i Ti can be determined independently, as well as d GB,i and thereby r TJ . Thus, the TJ excess is given as:

Γ TJ Ti = a π -ρ G Ti - 3 i=1 f TJ GB,i Γ GB,i Ti d GB,i • πr 2 TJ . (7)
Fig. 4a shows the Ti atomic fraction of a model volume with d GB = 3 nm in a radial 1D plot binning on equal radius increments. The change of the GB width causes that the TJ region with about constant atomic fraction (r TJ = 1.7 nm) is much broader than the one studied in Fig. 3 (r TJ = 0.3 nm, x-scale changed).

This broader TJ region is appropriate for an analysis of the TJ atomic density by radial integration in r ≤ r TJ .

Fig. 4b shows a plot of the integrated number of Ti atoms divided by z•r over r. A linear fit (green dashed line) is applied at low cylinder radius (r ≤ r TJ ).

Using the set values for the Ti concentration inside the grains, the Gibbs GB excess and the width of the GBs as well as the radius of the TJ volume gives a value for the Gibbs TJ excess of Γ TJ Ti = 4.1 at/nm. Even though this result is very close to the set value of 4 at/nm, it is noticed that other created volumes with the same setting configuration can reveal values with higher discrepancy to the TJ excess set value (up to ±1 at/nm). This can be due to very limited statistics inside the selected cylinder. Thereby, small variations in the Ti atom distribution inside the TJ volume can have significant effects on the linear fit 275 and thus on the determined Gibbs TJ excess value.

After the exploration of the methods on model volumes in this section, an application on an experimental volume is of interest. A qualitative view on the Ti excess at the GBs and at the TJ is given in a colored 2D Ti atomic density map projected along the TJ (Fig. 5b). Grain regions are in blue, because the Ti atomic density is low in comparison to the ones of GBs where excess of Ti is present (green to yellow). The projected GB planes appear straight from the border of the selected volume towards the TJ and parallel to the viewing direction (minimum of d GB ). Notably, the highest Ti atomic density in the colored 2D map is visible at the TJ region (red). The Ti atomic density at the TJ region is two orders of magnitude higher than the Ti atomic density inside the grains as the atomic densities selected for the simulated model volume. However, the atomic density in tomographic reconstructions can vary artificially due to local magnification and detection efficiency effects when crossing interfaces. The atomic density can also depend on the methodology used to establish the 2D map (sampling volume, vortex, size,...). Consequently, the atomic density obtained from the 2D map are only semi quantitative and more precise methods should be used.

For the GB excess, the cumulative profile analysis [22] allows an accurate determination: the Gibbs GB excess were determined to Γ GB1 Ti = 0.9 at/nm 2 , Γ GB2 Ti = 0.8 at/nm 2 and Γ GB3 Ti = 0.8 at/nm 2 by this method.

Another visualization of segregation is a 2D atomic fraction map (Fig. 5c).

It shows the number of Ti atoms relative to all atoms in the local environment.

Again, the Ti atomic fraction is lowest in grain regions, increased in the GBs and highest in the TJ region. Thus, an additional Ti segregation to the TJ is observed in the 2D maps that requires more precise quantification.

Ti Gibbs TJ excess quantification at experimentally measured CoSi 2 TJ

For the quantification of the Ti Gibbs TJ excess at the experimentally acquired CoSi 2 TJ, the methods introduced in section 3.2 were applied and the corresponding plots are shown in Fig. 6.

Fig. 6a shows a cumulative profile (black line) along GB3 with the complementary linear 1D Ti atomic fraction plot in Fig. 6d. The TJ region can be recognized as the region with the highest average slope (in between two vertical dotted lines) that refers to the highest atomic fraction in the plot of Fig. 6d.

It is much less pronounced than in the model system (Fig. 2). The visibility of these variations in the cumulative profile (Fig. 6a) is optimal for a cuboid width between 8 nm and 11 nm. From the step (red vertical line) at the center of the TJ region between the linear fits (green dashed lines) of the regions at the left and the right of the TJ region, a Ti Gibbs TJ excess is determined to

Γ TJ Ti = 2.2 at/nm.
Plots of the radial integration methods are shown in Fig. 6b and6c together with the radial Ti atomic fraction plot (Fig. 6d). The radial integral plot of Ti atoms divided by the height of the cylinder (black dots) in Fig. 6b is fitted by a second order polynomial in the region r r TJ (green dashed line). The constant of this fit reveals a Ti Gibbs TJ excess of Γ TJ Ti = 1.5 at/nm. The radial integration of Ti atoms divided by the product of the height (z) and the radius of the cylinder (r) inside the TJ region is linear (black dots in Fig. 6c). The slope of the linear fit (green dashed line) for r < r TJ is a = 3.1 at/nm 3 . With the knowledge of the previously given data (Ti grain atomic density, Gibbs GB excess, GB width), a Ti Gibbs TJ excess at the CoSi 2 TJ is Γ TJ Ti = 1.7 at/nm.

A positive Ti Gibbs TJ excess value at the CoSi 2 TJ is revealed from all three methods confirming the qualitative observation. However, there is a difference in the excess values from the three methods. The value determined by the cumulative profile is the highest (2.2 at/nm) while the radial integration method in the range r r TJ delivers the lowest value (1.5 at/nm) and the radial integration method in the range r ≤ r TJ a value in between (1.7 at/nm). Approximations and limitations of the three methods, which are partly explained in the section above, need further discussion for better understanding of the applicability of the methods as well as the meaning of the resulting value itself.

Discussion

Higher concentrations of solutes at TJs have already been reported in literature for solute diffusion [15][16][17][18][19] or segregation [10,11], but without quantification of the Gibbs TJ excess. Our work provides three methods for a quantifica-350 tion of Gibbs TJ excess: one based on a cumulative profile and two on a radial integration profile. The advantages and limitations of these methods will now be discussed.

A cumulative profile method in which the number of a specific atom is plotted as a function of the number of all atoms is sensitive to the atomic fraction since it is the slope of such a plot. The radial integration methods that use the number of solutes over a distance are sensitive to the atomic density since they count atoms per volume instead per total number of atoms (slope is proportional to the atomic density). State-of-the-art APT reconstruction procedures can deliver local spatial resolution in the range of sub nanometers resolving crystal lattice planes. However, their precision can suffer from detection efficiency variations or local magnification effects, especially at interfaces. When the detection efficiency varies for different elements, the reconstructed atomic fractions can be affected and thus cumulative profile analysis. On the other hand, local magnification effects can change the reconstructed atomic densities affecting the analysis using the presented radial integration methods. In principle, cumulative profile analysis could also be applied in the radial geometry in order to determine the effects of possible detection efficiency variation and local magnifications. However, there is no direct link to the Gibbs TJ excess in the range r r TJ when a radial cumulative profile is used instead of a radial integration profile. In the range r ≤ r TJ , the atomic fraction of Ti in the TJ region can be determined. It is related to the excess number of atoms at the TJ (Eq. 3).

Nevertheless, knowledge about the TJ area is needed among other geometries in order to calculate the Gibbs TJ excess from the atomic fraction. This area is related to r 2 TJ and thus again related to an absolute distance (atomic density) in the reconstruction, even to the power of two, making a direct comparison of the effects of artificial variations in the reconstruction of atomic fraction or atomic density very difficult.

There are other advantages and short comings for each method than being sensitive to atomic fraction or atomic density in the reconstruction. For the cumulative profile made from a linear integration, there is more freedom in the choice of the investigated region. The geometry of the cuboid which selects the region of investigation can be varied as well as its orientation relative to the GBs allowing to avoid obvious inhomogeneities. High statistics can be reached when a large cuboid analysis volume is chosen. Furthermore, there is no definition of the TJ geometry intrinsically necessary. One of the limitations of this method is the high sensitivity of the TJ excess on the choice of a center location between the neighboring regions of constant atomic fractions. Moreover, this method is not appropriate when the close environment of the TJ does not provide regions of constant solute atomic fraction in the chosen cumulative profile analysis. The radial integration method in the range r r TJ has similar advantages such as good statistics from a large investigated volume and no assumptions on the exact TJ structure are necessary. However, the determination of the TJ excess from the constant of a second order polynomial fit depends highly on the quality of the fit and can vary for changes in the fit region. In addition, this method is not applicable for inhomogeneous radial solute distributions around the TJ region. The radial integration method in the range r ≤ r TJ is complementary to the previous methods and so are its advantages. A linear fit instead of a second order polynomial is a priori more reliable. In addition, inhomogeneities around the TJ region do not directly influence the result revealed by this method.

However, the assumed geometry of the TJ is important for the result and needs to be representative for the investigated TJ as the assumed homogeneous radial solute distribution inside the TJ region.

While the investigated volume for the cumulative profile analysis can be optimized for best sensitivity, the volumes of the radial integration are fixed by definition. The radial integration methods can balance this drawback due to their complementarity requirements on the TJ geometry of the two radial integration analyses (r r TJ : tight GBs and thereby TJ, r ≤ r TJ : wide GBs and thereby large TJ region) allowing to choose the optimal method depending on the present TJ geometry in extreme cases or applying both in medium cases as in our present experimental volume. For all three methods, a correct alignment of the investigating volume with the TJ axis is of high importance. Misalignments can lead to variations in the fitted atomic fractions or atomic densities and thus to deviations in the determined Gibbs TJ excess value.

Furthermore, the dihedral angles have been set in the model volume to about 120 • with the argument of the representation of an equilibrium state. For significant variations of the dihedral angle values from 120 • , it is possible that the TJ region is not as homogeneously filled with GB regions in difference to the schematic visualization in Fig. 1a. A highly asymmetric GB overlap can be formed which would question the validity of the TJ volume approximation as a cylinder including the three GB overlap. The validation of this model is of specific importance for the radial integration method in the range r ≤ r TJ (third method), because the atomic fraction can become inhomogeneously (or just not representative) when the defined TJ volume is not homogeneously filled with GB contributions. Furthermore, such an asymmetry can affect strongly the cumulative method (first method) since a choice for the position of the step between the linear fits from bordering regions with constant atomic fraction is necessary.

Correlative structural and chemical information at sub Ångström length scale [20] and our experimental work can be the different material system or different crystallography (misorientation of adjacent GBs, orientation of GB planes and TJ line) of the considered TJs. In future experimental studies, it may be possible to determine the TJ crystallography in addition to the chemistry either by using complementary techniques (such as transmission kikuchi diffraction in scanning electron microscope [41,42] or diffraction analysis in transmission electron microscope [43,44]). APT could also provide enough crystallographic information [45,46] if special acquisition conditions are fulfilled such as: TJ line centered in APT tip along analysis direction, acquisition parameter setting optimized for crystallographic information. This will allow a better quantitative comparison with theoretical calculations on the same material system and the same crystallography of the TJ.

Such future studies could bring more light into the question of the nature of the TJ excess which can have different origins such as segregation to structural defects, formation of another phase at the junction or simply geometrical descriptions. A geometric estimation of the Gibbs TJ excess was given in the discussion of Eich et al. [20] (Eq. 18):

Γ TJ = √ 3/4 • d GB • Γ GB by considering
the overlap of the GB regions (Fig. 1 in [20]). It is an approximation which bases on the geometry of the adjacent GBs width assuming dihedral angles of 120 • . From the match between estimation and calculated values, it was concluded that Gibbs TJ excess can be described as a geometric effect of the finite width of the GB zones and a result of the GB overlapping zones [20].

In our work, such an GB overlap depended TJ excess can be estimated by the multiplication of the projected overlap areas (Eq. A.3 in Appendix A) with the average GB Ti atomic densities (Eq. 2). For our experimental volume it gives a TJ excess of about Γ TJ,overlap Ti = 0.9 at/nm. This value is lower than the determined excess at the TJ suggesting that a geometrical explanation from GB overlap cannot account for the full value of the TJ excess. This opens the question for more precise explanations of TJ excess origins than confinement of finite GB widths.

Another approach which is commonly used for the determination of excess is the application of proximity histograms (proxigrams [47]). Thereby, interfaces are created by the calculation of local concentrations (atomic fractions) and connecting a net (surface) for an isoconcentration value (similar for atomic density with isodensity value). The advantage of such an approach is that the shape of the interfaces can be complex [48]. For segregation at GBs or interphase boundaries, it is appropriate to develop procedures which map interfacial excess and surface concentrations [49]. Reliable reconstruction of atomic density or atomic fraction is still crucial. Improvements on this approach were recently reached by subdividing the interface for more accurate excess analysis in terms of spatial distribution of segregates and automatization of analysis [50]. Even though a TJ does not represent an interface in a classical sense, the proxigram approach may provide details in TJs about local distributions of segregation in correlation with structures or structural defects. However, the definition of useful isoconcentration interfaces can be very delicate. Furthermore, such analyses are limited in their representativity since they characterize subregions of the TJ while the Gibbs TJ excess gives an average value for the characterized TJ region. The methods presented in our work are convenient approaches to quantify a Gibbs TJ excess. This value is of high interest for various applications [51]. It is also of great interest from a fundamental point of view in relationship with crystallographic parameters like misorientations of the adjacent grains and relative orientation of the TJ axis as well as for different materials or different material processing [52].

Ongoing work aims for improvements of the presented methods. In the linear integration method higher reproducibility and reliability could be achieved by finding automatically optimal fit regions as well as the center of the TJ region similarly to optimization on interface segregation quantifications [50]. For the radial integration methods, improvements on even more reliable reconstruction of atomic densities and thereby distances in the tomographic reconstruction, especially at interfaces, are essential. The half of each of the GB overlap rectangles to the adjacent GBs needs to be In general, d GB is not necessarily the same for all three GBs and r TJ depends on both the GB widths and the dihedral angles of the adjacent GBs. The overlapping areas are than more complex and three cases can be distinguished.

Conclusion

In the case that the widths of the GBs are different and d GB,i > d GB,j and 655

Figure 1 :

 1 Figure 1: a) Geometries used in the simulated model volume for a projected view along TJ and GBs. b) Simulated model volume (cube) with Gibbs GB and TJ excess together with a projected 2D Ti atomic density map that visualizes qualitatively the excess at GBs and TJ (set values: ρ G Ti = 0.18 at/nm 3 , ρ GB Ti = 0.51 at/nm 3 , ρ TJ Ti = 0.94 at/nm 3 ; d GB = 3 nm; Γ GB,1 Ti = Γ GB,2 Ti = Γ GB,3 Ti = 1 at/nm 2 ; Γ TJ Ti = 4 at/nm).

Figure 2 :

 2 Figure2: First method to quantify Γ TJ . a) A cuboid is defined with two sides in parallel to one GB and the TJ. b) 1D Ti atomic fraction profile shows the different regions of constant atomic fraction: GB1 + grain on the left, GB2 + GB3 + grain on the right of the TJ and pure grain on the outer right. The tight TJ region has the highest Ti atomic fraction. Note that the atomic fraction values do not refer directly to the included features, but to an average inside the selected cuboid at the related integration location. c) A cumulative profile perpendicular to the TJ allows to determine the absolute number of atoms in excess at the TJ from the step height in the plot (inset). Γ TJ is revealed by dividing this step height through the height of the cuboid (length of considered TJ).

  (a) or if r ≤ r TJ (b). 11 a) When r r TJ , the TJ volume is completely included in the cylinder: the number of Ti atoms related to the TJ excess (N TJ,Exc Ti ) remains thus constant under variation of r. In this case, the GB volume inside the cylinder can be approximated as a rectangular slice and the number of Ti atoms related to the GB excess varies linearly in r (N GB, Exc Ti = z • Γ GB Ti • r). The variation of the number of Ti atoms with r can thus be expressed as

Figure 3 :

 3 Figure3: Second method for measureing Γ TJ : Radial integration. a) Defining a cylinder along the TJ allows a radial integration of the Ti atoms. b) A radial 1D Ti atomic fraction plot shows the high Ti atomic fraction inside the TJ region (left), a transition region with much lower and almost constant, but slightly decreasing Ti atomic fraction and finally a region appropriate for the second order polynomial fit. c) In the approach r r TJ , the contribution of the Ti atoms in excess at the TJ is constant (Eq. 5) and determined by a second order polynomial fit.

250b)

  When r ≤ r TJ , the Ti atom number contributions of grain, GBs and TJ are proportional to r 2 (N Ti (r) = z •πr 2 •ρ TJ Ti ) since the Ti atomic density was set as constant in the TJ of the model volume. It means that plotting N Ti (r)/(z • r)

Figure 4 :

 4 Figure 4: Third method for TJ excess determination: a) Ti atomic fraction in a radial 1D plot equivalent to Fig.3b, but on the volume with larger GB width (d GB = 3 nm instead of d GB = 0.5 nm). Note that fluctuations in the atomic fraction from a constant value are present due to low statistics in the radial 1D profile close to the center. b) In the approach r ≤ r TJ , the atomic densities of grain, excess at GBs and TJ contribute to a linear slope when the number of Ti atoms per z divided by r is plotted over r. Γ TJ is revealed from this slope and independent determinations of Ti atomic density in grains, GBs and the geometries of GBs and TJ.

3. 3 .Figure 5 :

 35 Fig.5ashows a reconstructed APT volume of the CoSi 2 thin film con-

Figure 6 :

 6 Figure 6: Quantification of Ti excess at CoSi 2 TJ in experimentally acquired APT volume from Fig. 5a. a) Cumulative profile method. b) Radial integration method in the approach r r TJ . c) Radial integration method in the approach r ≤ r TJ . d) Linear 1D Ti atomic fraction plot related to a). e) Radial 1D Ti atomic fraction plot related to b) and c).

  using, for example, high angle annular dark field scanning transmission electron microscopy measurements could give insights to the usability of the proposed model and methods. However, they are limited regarding the possibility to project the structure along the TJ axis. On the other hand, such asymmet-ric GB arrangements do not represent equilibrium states. In our experimental volume, the dihedral angles are around 120 • which represent a state close to equilibrium and the usage of the three proposed methods with the geometric assumptions should be justified.A similar geometry was used in a theoretical study on GB and TJ segregation in a binary Fe-Cr alloy by Eich et al.[20]. Segregation to GBs and TJ was investigated for a bulk concentration ranging from high Fe concentrations to high Cr concentrations. Different temperature and different chemical potentials have been set. Gibbs GB excess values up to the order of 100 at/nm 2 were found. For the Gibbs TJ excess, values up to the order of 10 at/nm were obtained. The excess values vary depending on the bulk concentration. In our work, the determined experimental Ti Gibbs TJ excess at a CoSi 2 TJ is one order of magnitude lower than the maximum Gibbs TJ excess calculated in the Fe-Cr alloy. These lower excess values in our study can explain the high relative variations mentioned for the three presented methods. Reasons for the difference in the order of magnitude between the theoretical study ofEich et al.

  Three methods have been presented to determine the Gibbs TJ excess in 3D volumes obtained by APT that has an atomic resolution and the capability to count single atoms. Their functionality has been demonstrated on simulated model volumes with well-defined geometries and concentrations of GBs and TJ. These methods based on cumulative profile and radial integration of solute atoms have been compared. An experimental APT volume of CoSi 2 containing 3 GBs with Ti segregation joining together in a TJ has been analyzed. Ti excess at the TJ can be visualized with 2D Ti atomic density or atomic fraction maps. Applying the three methods, the Ti Gibbs TJ excess at the CoSi 2 TJ is quantified revealing positive excess values. To conclude, there is Gibbs TJ excess of Ti at the investigated CoSi 2 TJ which influences its structure and properties and further the properties of the entire thin film. The demonstrated quantification of Gibbs TJ excess opens the door for a more complete characterization of poly crystalline materials. It provides a more accurate correlation between chemical characterization at the atomic scale and materials properties. For such a correlation, a systematic study of selected TJ would be needed since poly crystalline materials, depending on their texture, consists of various TJs that can have different influence on the properties.

Figure A. 7 : 4 )

 74 Figure A.7: Schema of GB overlap in TJ region for visualization of TJ radius and its calculation as well as calculating the overlap area.

Figure A. 8 :

 8 Figure A.8: Visualization of single GB fraction in TJ cylinder (no GB overlap). The area of a single GB in the TJ projected along the TJ axis is the sum of the horizontally stripped rectangular area Arect and the vertically stripped arc area Aarc.

=

  A rect,i + A arc,i -Σ j =i S ij /2. (A.5)Knowing this projected GB area in the TJ, the volume fraction is calculated by 650 the ratio to the base area of the TJ cylinder

Acknowledgment This work was supported by the French government through the program "Investissements d'Avenir A*MIDEX" (Project APODISE, no. ANR-11-IDEX-0001-02) managed by the National Agency for Research (ANR). The authors would like to thank Maxime Bertoglio for assistance with sample preparation and Marion Descoins for assistance with APT tip preparation and APT measurements.

Conflict of interest

The authors declare that they have no conflict of interest. The projected GB1 appears in a green frame while GB2 is framed in orange.

The center of their junction represents the TJ. The frame of the TJ cylinder projection is drawn as a blue circle. The radius of the TJ cylinder (r TJ ) was defined as the radius of the circle that includes all occurring GB overlap at the TJ. The overlapping area is a tetragon and marked in yellow. The illustration shows that r TJ is thereby defined as the diagonal of the overlapping area of the adjacent GBs in the considered case. The angle of the diagonal of the GB overlap area can be calculated to

and with the knowledge of the GB width, the radius of the TJ is

Furthermore, the overlapping area of the adjacent GBs is

GB fraction in TJ

The volume fraction of a single GB in the TJ is related to the area of the GB slice in the TJ cylinder projected along the TJ axis (Fig.