Triple Junction Mobility Research Articles

Initial grain orientation controls static recrystallization outcomes in cold-worked iron: Insight from coupled crystal plasticity/vertex dynamics modeling

Recrystallization of metallic materials is an important metallurgical treatment to bring cold-worked alloys to a usable state. Static recrystallization (SRX) occurs after deformation, once the material has built up sufficient plastic strain energy to trigger grain boundary motion. However, we have very limited knowledge about the combined effects on recrystallization kinetics of temperature, amount of cold working, triple junction mobility, and level of recovery in the deformed microstructure. In particular, little is still known about the microstructural factors that control the onset of SRX. Here we develop a comprehensive physical model based on coupling crystal plasticity and grain boundary kinetics utilizing curvature and differential plastic strain energy as driving forces for SRX in polycrystalline iron. The model links finite-element polycrystal plasticity simulations with a two-dimensional vertex dynamics approach, and is used to probe the large parameter space influencing static recrystallization. Our main finding is that the initial texture (defined by the polycrystal’s grain orientation and size distribution) is the dominant feature triggering and controlling SRX kinetics. We find a direct correlation between the degree of dislocation density buildup and the relative grain orientation across a grain boundary at the beginning of the deformation phase. Indeed, analysis of our results points to the maximum Schmid factor of a grain as the best predictor for activation of its grain boundaries during the recrystallization phase. We also find that the amount of post-deformation recovery only has a limited effect on the evolution of the recrystallized microstructure. Finally, using a definition based on the rate of transformation of the material, we find a recrystallization temperature of 600 ∘C, consistent with experimental observations in high-purity iron specimens.

Grain growth kinetics in 2D polycrystals: impact of triple junctions

Abstract Grain growth in random and textured 2D polycrystals is modeled under the supposition that all the triple junctions possess limited mobility. In the random polycrystal, the growth kinetics is at first linear and becomes parabolic at later stages if both the triple-junction mobility and the initial grain size are low. The experimental data on triple-junction mobility are used to show that such an effect could be observed in nanocrystalline materials. In the textured polycrystal, triple junctions with limited mobility are found to shift the evolution of the growth process toward abnormal grain growth.

Hysteresis of Triple Junction Mobility

Hysteresis of Triple Junction Mobility

Influence of displacement constraints to the surface reconstruction of stressed bicrystal thin films

In this work, surface morphology evolution of bicrystal thin films under the combined action of grain boundary and surface diffusion is investigated by considering different mechanical constraints. 2D surface topographies of thin films, that are (a) freestanding, (b) strongly bonded to its substrate and (c) strongly bonded to its substrate and one of sidewalls, are simulated using a numerical implementation of an irreversible thermo-kinetics model. Relationships which give the groove depth as a function of time are obtained. Results show that mechanical loading conditions are effective in determining the morphology and kinetics of grooving. For the three scenarios that had been investigated, it was found that the groove depth evolves linearly with different tip velocities under the same level of uniaxial tension. In freestanding films groove tip evolves faster; i.e. as the film gets constrained from its substrate and/or one of its sidewalls, the tip velocity slows down. It was also observed that high triple junction mobilities at low levels of applied stress hinder the effects of displacement constraints to groove shape, even in the case of asymmetric stress distributions inside the film. On the other hand, low triple junction mobilities at moderate applied stresses allow formation of asymmetric grain boundary grooves due to the induced asymmetry in the driving force for surface diffusion with respect to the grain boundary.

Influence of Finite Mobilities of Triple Junctions on the Grain Morphology and Kinetics of Grain Growth

Grain boundary networks composed of equal microstructural elements were investigated in a recent paper. In this work a more complicated artificial grain topology consisting of one four-sided, two six-sided and one eight-sided grain is designed to further investigate the influence of grain boundary and triple junction mobilities on the kinetics of the system in more detail. Depending on the value of the equal mobility of all triple junctions, the initially square-shaped four-sided grain changes its shape to become more or less rectangular. This indicates that the grain morphology is influenced by the value of the mobility of the triple junctions. It is also demonstrated that a grain arrangement with low mobility triple junctions controlling the kinetics of grain growth enhances growth of the large eight-sided grains. In addition, grain growth is investigated for different values of mobilities of triple junctions and grain boundaries. A strong elongation of several grains is predicted by the modeling results for reduced mobilities of the microstructural grain boundary elements. The two-dimensional modeling results are compared to micrographs of a heat-treated titanium niobium microalloyed steel. This feature, namely the evolution of elongated grains, is observed in the micrograph due to the pinning effect of (Ti, Nb)C precipitates at elevated soaking temperatures of around 1100 °C. Furthermore, the experiments show that a broader distribution of the grain sizes occur at 1100 °C compared to soaking temperatures, where pinning due to precipitates plays a less prominent role. A widening of the distribution of the grain sizes for small triple junction mobilities is also predicted by the unit cell model.

Influence of bulk energy and triple junction mobility on interface kinetics - A tool for interpretation of experiments

A material system consisting of a lamellar grain structure adjacent to a large single grain is investigated. The system evolution is driven by changing of interface energy of the lamellar structure as well as by difference in bulk energies stored in the single grain and the lamellar grains. The triple junctions and the grain boundaries are assumed to have finite mobilities representing kinetic material parameters of system. A complete analysis of the kinetics of this system is provided, which involves several possible scenarios depending on the values of the geometrical and material parameters of the system. The scenarios are fully classified. Moreover, the analysis offers a way, how the values of the material parameters (interface energy densities, difference in bulk energies and mobilities) can be extracted from the measured system kinetics and geometry.

Kinetics of Grain Boundary Networks Controlled by Triple Junction and Grain Boundary Mobility

The kinetics of a triple junction of grain boundaries with distinct specific energies and mobilities and a finite mobility of the triple junction is investigated. The microstructure is approximated by different 2D settings consisting of typical structural elements. First, the migration of the triple point together with the adjacent grain boundaries, is simulated, assuming that the grains are infinitely large. Secondly, growth or shrinkage of finite n-sided grains is simulated by altering the boundary conditions and the results are compared to the already published analytical solution. The numerical results coincide with the corrected analytical solution. This solution can be derived either by applying the principle of maximum dissipation, or by applying the force balance at the triple junction within the framework of linear irreversible thermodynamics. The change of the area of infinite and finite grains is investigated analytically and numerically. By comparing the results of both approaches, the influence of the initial topology of the structural elements on the kinetics of grain growth can be estimated. Furthermore, the kinetics of grain growth of different idealized grain boundary networks is investigated. It is shown that square shaped grains surrounded by hexagons and dodecagons result in a more realistic grain growth scenarios than squares surrounded by octagons. A deviation from idealized grain boundary arrangements is e.g., observed, due to different triple junction mobilities, and the initially n-sided regular grain deforms in a complex manner.

Исследование влияния избыточного свободного объема на подвижность тройных стыков границ зерен

Исследование влияния избыточного свободного объема на подвижность тройных стыков границ зерен

Triple junction drag effects during topological changes in the evolution of polycrystalline microstructures

Experiments, theory and atomistic simulations show that finite triple junction mobility results in non-equilibrium triple junction angles in evolving polycrystalline systems. These angles have been predicted and verified for cases where grain boundary migration is steady-state. Yet, steady-state never occurs during the evolution of polycrystalline microstructures as a result of changing grain size and topological events (e.g., grain face/edge switching - “T1” process, or grain disappearance “T2” or “T3” processes). We examine the non-steady evolution of the triple junction angle in the vicinity of topological events and show that large deviations from equilibrium and/or steady-state angles occur. We analyze $∖tau$ the characteristic relaxation time of triple junction angles τ by consideration of a pair of topological events, beginning from steady-state migration. Using numerical results and theoretical analysis we predict how the triple junction angle varies with time and how τ varies with triple junction mobility. We argue that it is precisely those cases where grain boundaries are moving quickly (e.g., topological process in nanocrystalline materials), that the classical steady-state prediction of the triple junction angle about finite triple junction mobility is inapplicable and may only be applied qualitatively.

Calibration of a multi-phase field model with quantitative angle measurement

Over the last years, the phase-field method has been established to model capillarity-induced microstructural evolution in various material systems. Several phase-field models were introduced and different studies proved that the microstructure evolution is crucially affected by the triple junction (TJ’s) mobilities as well as the evolution of the dihedral angles. In order to understand basic mechanisms in multi-phase systems, we are interested in the time evolution of TJ’s, especially in the contact angles in these regions. Since the considered multi-phase systems consist of a high number of grains, it is not feasible to measure the angles at all TJ’s by hand. In this work, we present a method enabling the localization of TJ’s and the measurement of dihedral contact angles in the diffuse interface inherent in the phase-field model. Based on this contact angle measurement method, we show how to calibrate the phase-field model in order to satisfy Young’s law for different contact angles.

An advanced level set approach to grain growth – Accounting for grain boundary anisotropy and finite triple junction mobility

The novelty of the proposed level-set approach to grain growth resides in the explicit consideration of structural interfacial elements of the microstructure. The extensions allow to consider anisotropic grain boundary energies and triple junction drag in polycrystalline materials. The simulated predictions were compared to analytical expressions for the growth rate of grains under the influence of a finite triple junction mobility with excellent agreement.

Thermodynamics and Kinetics of 1D Structural Elements and Stability of Nanocrystalline Materials

Grain boundary triple junctions are the structural elements of a polycrystal. Recently it was recognized that they can strongly impact the microstructural evolution, and therefore there engender new opportunities to control and to design the grain microstructure of fine-grained and nanocrystalline materials due to their effect on recovery, recrystallization and grain growth. The measurement of triple junction energy and mobility is thus of great importance. The line energy of a triple junction constructs an additional driving force of grain growth. Taking the triple line energy into account, a modified form of the Zener force and the Gibbs-Thomson relation can be derived to reveal the influence of the triple line energy on second phase particles and the change of the equilibrium concentration of vacancies in the vicinity of voids at a grain boundary. The impact of triple junctions on the sintering of nanopowders is discussed. The role of “grain boundary - free surface” triple lines in the adhesive contact formation between spherical nanoparticles is considered. It is shown that there is a critical value of the triple line energy above which the nanoparticles do not stick together. Based on this result, a new nanoparticle agglomeration mechanism is proposed, which accounts for the formation of large agglomerates of crystallographically aligned nanoparticles during the nanopowder processing.

Grain boundary, triple junction and quadruple point mobility controlled normal grain growth

Reduction in stored free energy provides the thermodynamic driving force for grain and bubble growth in polycrystals and foams. Evolution of polycrystalline networks exhibit the additional complication that grain growth may be controlled by several kinetic mechanisms through which the decrease in network energy occurs. Polyhedral boundaries, triple junctions (TJs), and quadruple points (QPs) are the geometrically distinct elements of three dimensional networks that follow Plateau’s rules, provided that grain growth is limited by diffusion through, and motion of, cell boundaries. Shvindlerman and co-workers have long recognized the kinetic influences on polycrystalline grain growth of network TJs and QPs. Moreover, the emergence of interesting polycrystalline nanomaterials underscored that TJs can indeed influence grain growth kinetics. Currently there exist few detailed studies concerned either with network distributions of grain size, number of faces per grain, or with ‘grain trajectories’, when grain growth is limited by the motion of its TJs or QPs. By contrast there exist abundant studies of classical grain growth limited by boundary mobility. This study is focused on a topological/geometrical representation of polycrystals to obtain statistical predictions of the grain size and face number distributions, as well as growth ‘trajectories’ during steady-state grain growth. Three limits to grain growth are considered, with grain growth kinetics controlled by boundary, TJ, and QP mobilities.

Triple junction controlled grain growth in two-dimensional polycrystals and thin films: Self-similar growth laws and grain size distributions

Triple junction controlled grain growth in two-dimensional nanocrystalline polycrystals is modeled by attributing to each structural feature of a polygonal grain a finite mobility. By considering grain growth as a dissipative process that is driven by the reduction of the Gibbs free interface energy, a general grain evolution equation is derived that separates into two types of possible self-similar growth kinetics. For the case of pure triple junction drag the influence of finite triple junction mobilities on metrical and topological properties is studied. Analytical expressions of the self-similar grain size distribution are derived, which compare very well with results from the modified Monte Carlo Potts model and front-tracking vertex dynamic simulations, taking into account size effects in triple junction limited grain growth. In addition, the analytical grain size distributions are used for a theoretical description of experimental data obtained in nanocrystalline thin films upon annealing.

A phenomenological approach to investigate nanocrystalline grain growth

A phenomenological approach to model the size effect observed during grain growth in nanocrystalline materials is presented. To that aim, the standard Monte Carlo Potts model is modified such that the mobility of grain boundaries depends on their triple junction distance. For initially very small grains growth kinetics are observed that are in agreement with experimental results, theoretical predictions and another simulation approach using a direct modification of the triple junction mobilities.

Investigating the von Neumann–Mullins relation under triple junction dragging

The influence of triple junction mobility on the rate of size change is investigated by square-lattice Monte Carlo Potts model simulations. For normal, grain-boundary-controlled grain growth the classical von Neumann–Mullins relation (Ȧ∝n-nc) is fulfilled for individual polyhedral grains as well as for polycrystalline grain microstructures. For triple-junction-controlled grain growth the number of neighboring grains is related self-similarly to the radius change rate (Ṙ∝(n-nc)/n), which is also shown to be fulfilled for individual grains and polycrystalline microstructures.

A phase-field model for grain growth with trijunction drag

A phase-field model has been developed to study the effect of triple junction (TJ) mobility on 2-D grain growth kinetics. The method captures the results of past work such as a linear increase in the average grain size with time, but can also follow the transition from TJ-limited to grain boundary energy-limited growth. The distribution of grain boundary curvature is examined. In the low TJ mobility simulations the distribution has a peak at zero curvature and approaches the grain boundary mobility-limited steady-state distribution at larger sizes. Even for extremely low TJ mobility, a small fraction of the grain boundary length has non-zero curvature and thus a lack of self-similarity is observed for all TJ-limited simulations, even when the average size is increasing linearly in time. We find that the topology of the grain structure is independent of the degree of TJ drag, within the range of parameters employed in the simulation. The effects of TJ mobility increase as the grain size decreases, suggesting that TJ mobility can play a significant role in nanocrystalline grain growth kinetics.

Control of microstructure in the manufacturing of aluminum foil

Experimental results from studying of grain growth in two-dimensional aluminum foils are correlated with results from the generalized theory of two-dimensional grain growth, which considers the limited mobility of triple junctions.

A predictive model for microstructure evolution in metallic multilayers with immiscible constituents

Focussing on the thermal stability of layered structures, we developed a predictive model to study the microstructure evolution of metallic multilayers with different morphologies including aligned and classical staggered grain geometries. We found that the zig-zag microstructure experimentally observed in multilayers forms when grains in each upper layer have a relative shift less than half the in-plane grain size to the lower layer. During this formation process, the non-equilibrium triple junctions move, driven by the imbalance of tensions of interphase and grain boundaries, corresponding to an extension to classical grooving theory. Numerical simulations show that a finite mobility of the triple junction can effectively impede the development of grooves, suggesting that the classical t1/4 dependence of groove depth with the assumption of an infinite triple junction mobility might be questionable at low temperatures to predict the time to pinch-off. Further, a map for the stability of layered structure in Cu/Nb system is developed in terms of the aspect ratio of grain dimensions and the ratio of the distance between two nearest triple junctions to the in-plane grain size. A criterion for this stability is also proposed for multilayers with similar grain boundary energies based on simplified geometrical consideration. Both the map and the simple criterion are in good agreement with the experiments for Cu/Nb multilayers.

Modelling the kinetics of a triple junction

The grain structure in a one phase system involves grain boundaries (surfaces), three-grain junctions (lines) and four-grain junctions (points). A certain Gibbs energy and mobility can be assigned to each object. System evolution, driven by a decrease in the total Gibbs energy, occurs by migration of objects constrained by rather complex contact conditions. A system with cylindrical symmetry is assumed, where three grain boundaries with different mobilities and different specific Gibbs energies are in contact at a triple junction line of given mobility. The equations of evolution of the system are derived by means of the thermodynamic extremal principle. New general contact conditions at the triple junction are derived, including the mobilities of all objects and the energies, contact angles and curvatures of the grain boundaries. Special contact conditions are also provided for the cases where the triple junction mobility is infinite and/or in the case of one of the grain boundaries having zero mobility. The model is demonstrated by several examples.