Modelling and Simulation of Solidification Phenomena during Additive Manufacturing of Bulk Metallic Glass Matrix Composites ( BMGMC ) — A Brief Review and Introduction of Technique

Despite a wealth of experimental studies focused on determining and improving mechanical properties and development of fundamental understanding of underlying mechanisms behind nucleation and growth of ductile phase precipitates from melt in glassy matrix, still, there is dearth of knowledge about how these ductile phases nucleate during solidification. Various efforts have been made to address this problem such as experiments in microgravity, high resolution electron microscopy and observation in synchrotron light after levitation but none have proved out to be satisfactory. In this study, an effort has been made to address this problem by modelling and simulation. Current state of the art of development, manufacturing, characterisation and modelling and simulation of bulk metallic glass matrix composites is described in detail. Evolution of microstructure in bulk metallic glass matrix composites during solidification in additive manufacturing has been presented with the aim to address fundamental problem of evolution of solidification microstructure as a result of solute partitioning, diffusion and capillary action. An overview is also presented to explain the relation of microstructure evolution to hardness and fracture toughness. This is aimed at overcoming fundamental problem of lack of ductility and toughness in this diverse class of materials. Quantitative prediction of solidification microstructure is done with the help of advanced part scale modelling and simulation techniques. It has been systematically proposed that 2-dimensional cellular automaton (CA) method combined with finite element (for thermal modelling) tools (CA-FE) programmed on FORTRAN and parallel simulated on ABAQUS would best How to cite this paper: Rafique, M.M.A. (2018) Modelling and Simulation of Solidification Phenomena during Additive Manufacturing of Bulk Metallic Glass Matrix Composites (BMGMC)—A Brief Review and Introduction of Technique. Journal of Encapsulation and Adsorption Sciences, 8, 67-116. https://doi.org/10.4236/jeas.2018.82005 Received: February 6, 2018 Accepted: June 22, 2018 Published: June 25, 2018 Copyright © 2018 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access DOI: 10.4236/jeas.2018.82005 Jun. 25, 2018 67 Journal of Encapsulation and Adsorption Sciences


Introduction
Discovered in 1960 by Duwartz et al. [1] at Caltech, Metallic Glasses [2] may be defined as disordered atomic-scale structural arrangement of atoms formed as a result of rapid cooling of complex alloy systems directly from their melt state to below room temperature with a large undercooling and a suppressed kinetics in such a way that the supercooled state is retained/frozen [3] [4] [5] [6]. This results in the formation of "glassy structure". The process is very much similar to inorganic/oxide glass formation in which large oxide molecules (silicates/borides/aluminates/and sulphides) form a regular network retained in its frozen/supercooled liquid state [7]. The only difference is that metallic glasses are comprised of metallic atoms rather than inorganic metallic compounds.
Their atomic arrangement is based on mismatch of atomic size and quantity (minimally three) [8], is based on short [9] [10] [11] to medium range order [12] [13] [14] or long range disorder [15] (unlike metals-well defined long range order) and can be explained by other advanced theories or mechanisms such as liquid-liquid transition [16] [17] [18], phase separation [19], confusion [20], order in disorder [13] [21] [22] [23] [24], frustration [21], and vitrification [25] [ 26]. Important features characterizing them are their amorphous microstructure and unique mechanical properties. Owing to absence of mechanisms promoting development and movement of dislocations, no plasticity is exhibited by bulk metallic glass matrix composites. This results in very high yield strength and elastic strain limit as there is no easy plane of flow. From fundamental definition standpoint, metallic glasses are typically different from bulk metallic glasses, in that the former is glassy structure less than 1 mm thick, while later is greater than 1 mm [27] [28]. Till date, the largest bulk metallic glass produced in "as cast" condition is 80 mm diameter and 85 mm in length [29]. There are reports of making large thin castings as casing of smart phones but they are typically less than 1 mm [30]. Furthermore, they are characterised by special properties such as glass forming ability (GFA), and metastability. ticity in glassy structure while retaining its high strength at the same time [63] [64] [65] [66]. This can be done by various mechanisms including exploitation of intrinsic ability of glass to exhibit plasticity at very small (nano) length scale [67] [68], introduction of external impulse (obstacles) to shear band formation and propagation (ex-situ composites) [69] [70], self or externally assisted multiplication of shear bands [71] [72], formation of ductile phases in brittle glassy matrix during solidification (in-situ composites) [73] [74] [75] [76] and transformation inside a ductile crystalline phase e.g. B2 -B19' transformation in Zr based systems (stress/transformation induced plasticity (TRIP) [77] [78] [79] [80]. It was known thermodynamically and observed after simulation [81] and experimentation [82] [83] [84] [85] since early days that structurally constrained glass either relaxes (losses free volume) or underwent crystal structure change (devitrification) during heating. The drive for devitrification [86] [87] did not came out of ingenious effort but was an outcome of natural impulse of stressed structure to attain a state of lower stress by undergoing structural change and evolution of new structures (phases) when subjected to temperature effect similar to heat treatment for crystalline metallic alloys. This result in new class of bulk metallic glass called ductile bulk metallic glass [88]- [96]. Other approach which encompasses nucleation and growth of primary ductile phase in brittle glassy matrix during solidification gave rise to second type of ductile material known as ductile bulk metallic glass matrix composites. These were reported first time in 2001 by Prof. William L. Johnson's group at Caltech [63] when they successfully incorporated primary ductile phase reinforcements in glassy matrix in the form of precipitates in-situ nucleating during solidification thus giving birth to "so called" family of in-situ dendrite/metallic glass matrix composites [97]- [112] Later reinforcements were observed to exhibit different morphologies other than dendrites (e.g. spheroid, whiskers and plates) but fundamental mechanism of reinforcements remained same (i.e. metal matrix composites). These materials are formed as a result of conventional solute partitioning mechanisms as observed in other metallurgical alloys resulting in copious formation of ductile phase (Ti-Zr-Nb β in case of Ti based composites [63], B2 CuZr in case of Zr based composites [113]- [118] or transformed B2 (B19' martensite) in case of shape memory alloys (a special class of bulk metallic glass matrix composites) [73] [116] [119] [120] [121] [122] [123]). This ductile phase can appear in the form of three dimensional dendrites or spheroids emerging directly from liquid during solidification. Structural changes in these alloys can be explained by "phase separation" [124]- [131] before liquid-solid or solid-solid transformation or formation and retention of "quenched in" nuclei during solidification. This is another very important route for the manufacture of these alloys. Very recently, another important phenomenon known as liquid-liquid transition (LLT) [17] has been observed in these alloys. This has become observable due to ability to characterize these alloys under Synchrotron radiation [132] [133] [134] [135] and in microgravity conditions [17] [136] employing container less levitation techniques [136].

Additive Manufacturing of Bulk Metallic Glasses and their Composites
Although, considerable progress has been made in advancing "as cast" sizes in bulk metallic glass and their composites, still, maximum possible diameter and length which has been produced by conventional means till date [29], is far from limit of satisfaction to be used in any structural engineering application. This primarily is associated with mechanical cooling rate achievable as a result of quenching effect from water cooled walls of Cu container which in itself in not enough to overcome critical cooling rate (R c ) of alloy (~0.067 K/s [29]) to produce a uniform large bulk glassy structure. In addition to this, occurrence of this bulk glassy structure is limited to compositions with excellent inherent glass forming ability (GFA) [ [190]- [195]), and CAFE [196]- [201]) at part scale are very helpful in explaining the evolution of microstructure and grain size development in metals and alloys. They have been extensively used in predicting solidification behaviour of various types of melts during conventional production methods [201] [ 3) Additive manufacturing: Use of very high cooling rate inherently available in the process to (a) not only form glassy matrix but use liquid melt pool formed at very high temperature to trigger nucleation (liquid-solid transformation) of ductile phase in the form of dendrites from within the pool "in-situ" (This is done by controlling machine parameters in such a way that optimised cooling rate satisfying narrow window of "quenching" bulk metallic glasses is achieved)

A Brief Introduction to Modelling and Simulation
Although, in use since ancient Roman times [209], modelling and simulation Many difficult, or in some cases, impossible to envisage problems can now be simulated using these computing platforms. These include simulation of water flow and its patterns in rivers and channels, simulation of interior of sun, stars and other heavenly bodies, cosmic events and nuclear engineering problems.
However, despite of these advantages, there are still situations and applications which limits the use of modelling and simulation techniques. These include, unavailability of strong efficient computing algorithms (with lesser approximations) needed for the replication of actual real-world situations, unavailability of real world experimental data (physical constants and thermo-physical properties) needed to simulate a particular problem, unavailability of more accurate deterministic or accurate (non-probabilistic) models using actual situations rather than basing their outcome on statistics. Owing to these reasons, there is still need for further investigation and removal of bottlenecks from modelling and simulation techniques and it is envisaged that their popularity is still at arm's length. However, progress in these is notable and healthy.

Current State of art in Modelling and Simulation of Additive Manufacturing of Bulk Metallic Glass and their Composites
Nucleation and growth phenomena in single component (pure metals), binary and multicomponent alloys is rather well understood. Classical nucleation theory (CNT) [210] provides many answers to the behaviour of these melts. Traditionally, bulk metallic glass and their composites were produced using conven-  [213] and twin roll casting [214]) in which their metastable phase (glass) and any in-situ ductile precipitates (stable phase) are nucleated based on their ability to surpass activation energy barrier. In addition, these processes, impart very high cooling rate to castings which is essential for retention of supercooled liquid (glass) at room temperature. However, they have their limitations which pose limits to their applicability and use for making complex and intricate parts and components at large scale.
Very recently, with the advent and popularity of additive manufacturing (AM), interest has sparked to exploit the inherent and fundamental advantages present in this unique process to produce BMG and BMGMCs. Additive manufacturing techniques are useful in achieving this objective as very high cooling rate in fused liquid melt pool is already present inherently to assist the formation of glassy structure which is suppression of "kinetics" and prolonging of undercooling ("thermodynamics")-two main phenomena responsible for any phase transformation. However, the in-situ nucleation of primary phase equiaxed dendrites or spheroids during solidification and then microstructural evolution (solute diffusion and capillary action assisted) is not satisfactorily explained by classical nucleation theory alone.
Either some modification is needed in classical nucleation theory (CNT) or more reliable probabilistic microstructure evolution models (e.g. J-M-A-K Correction [215] or Rappaz modification) are needed to explain nucleation and growth (and other phenomena e-g liquid-liquid transition (LLT) [17] [18] [19] and phase separations [128]) in BMGMCs. In this work, an effort has been made to meet both requirements. Following are propositions and state of art; 1) At present scenario, there is no single hybrid/combined model which explain phenomena of heat transfer (liquid melt pool formation as a result of laser-matter interaction and its evolution-solidification) and coupled this with nucleation and growth (NG) (solute diffusion [216] and capillary action driven) at microscale to predict microstructure and grain size in BMGMC as melt cools in additive manufacturing melt pool. a) Only one study has been conducted to model the same phenomena (solidification only) during Cu mold suction casting which will serve as base [217] in addition to very recent attempts [215] in which emphasis is laid on development of generalised theory rather than solving a problem.
b) Only one study has been reported on microstructure formation during twin roll casting using cellular automaton-finite element (CAFÉ) [218] but that is not aimed at BMGMCs, is carried out using commercial software package and does not involve any mathematical modelling at the back end. Software embedded (nucleation and growth and heat transfer) models are used only.
2) No effort has been made to correlate the effect of edge-to-edge matching (E2EM) [223] [224] with direction of easy heat flow and crystallographic orientation (an effect that can be inherently used in additive manufacturing).
3) No substantial study has been reported about evolution of microstructure in three dimensions in BMGMCs in additive manufacturing. In light of this research gap, present review is compiled. An effort has been put together to overcome these shortcomings and propose a methodology for the modelling and simulation of solidification phenomena during additive manufacturing of BMGMCs. A model system Zr 47.5 Cu 45.5 Al 5 Co 2 has been proposed owing to evolution of distinct ductile phase (CuZr-B2) in it during solidification and its tendency to show shape memory effect (exhibiting two types of martensitic phase evolution from B2 phase). Further, the method is proposed to be applied to conventional wedge shape casting geometry along with its final application to melt pool in additive manufacturing making use of powers of deterministic, probabilistic and their coupled modelling approaches. This route is proposed to get maximum benefit from application of modelling and simulation to understand nucleation and growth phenomena during solidification both in conventional as well as modern processing technologies (AM). It is envisaged that application of hybrid CAFE model by programming on FORTRAN  /MATLAB  and parallel simulation on ABAQUS  will help understand solidification in BMGMC in much better way not done elsewhere previously.

Modelling-Introduction
This analysis is divided into two sections. The first section deals with the evolution of the melt pool as a result of the interaction of highly localised, focused laser light with matter (metal powder). This results in the formation of a melt pool whose shape, size, geometry and transient behaviour is very much a function of the heat transfer coefficients (HTC) evolving at every step of its formation (melting and homogenisation) and dissipation (solidification). Solidification in this section is considered by a modified general (classical) nucleation theory (CNT) [225]. Once formed, this pool travels as the laser traverses its path all along the powder bed dictated by CAD geometry at the back end. The second

Modelling and Simulation of Heat Transfer in Liquid Melt Pool-Solidification
As the microstructure formed during selective laser melting (SLM) is mostly columnar [226], it is a good indicator that heat flux transfer from melt is highly unidirectional thus heat transfer from bottom is a transient 1D process. Although, heat is lost from the material in x-y plane i.e. perpendicular to the z-direction (perpendicular to build direction), its contribution is so low that it can be safely ignored. However, this was an old concept. New experimental observations have proposed a new concept according to which during SLM, a melt pool is formed, where the shape of this pool is a function of; 1) Laser power (laser beam intensity).
In even more advanced and recent models, [227] [228] the transfer of heat after its generation is considered by three main parameters: 1) Heat transfer due to convection.

3) Conduction from the bottom and the side walls.
This is very recent and advanced approach which, however, ignores marangoni convection effects. Overall, the heat transfer phenomena associated with the solidification of metal in a liquid melt pool in AM is associated with three processes: 1) Generation of heat (laser matter interaction).
2) Assimilation of heat (melting and stages of solidification).

Generation of Heat (Laser-Matter Interaction)
This is the first stage of additive manufacturing in which heat is generated. The

Beer Lambert Law for Additive Manufacturing
Consider a thin layer of powder with thickness d 1 , on a flat disk substrate of refractory metal with thickness d 2 and radius r uniformly illuminated by light of intensity I.
For absorptivity of powder (or melt) assuming uniform temperature throughout the disk, the temperature evolution is Heat generated by this process is used for melt pool generation (its morphology, homogenisation, and holding (generation of supercooled liquid (SCL) region and its progression)).

Assimilation of Heat (Melting and Stages of Solidification)
As the heat generated above interacts with metal powder, it causes its melting and generation of liquid melt pool. The behaviour of a certain metal/alloy in the melt pool can be explained by its cooling curve which is briefly described below.

General form of Cooling Curve
A cooling curve of a metal/alloy is a plot of the variation of temperature with time. It has different regions which embodied various types of information.
Cooling curves can have different shapes depending on the metal or alloy type. A schematic cooling curve is shown in Figure 1 for a single component pure metal (without any inoculants). Its distinct regions are explained as follows; Region above A 1 : This is the region in which metal is in its complete liquid-state and can be described by only melting and liquid-state homogenisation. Heat carried by metal in this region is "super heat" only and lost in the form of specific heat (mc p ΔT). This homogenisation in turn depends on type of melting (gas/solid (coal)/liquid (oil) fired crucible furnace melting, electric (resistance/induction/arc) melting) and subsequent melt treatment. (Note: Homogenisation is required by some external means in case of all modes of melting. Only induction furnace is manifested by self-homogenisation due to phenomena of induction currents).  (melting range). In BMGMCs/multicomponent alloys, it is also called start of the super cooled region (SCL). This region is followed by undercooling (ΔT n ) region which is described below Region A -D: This is the most important region of cooling curve (present case) for pure metals. In this region, metal cools down to a specific temperature characterised by a certain minimum amount of energy (activation energy for nucleation) needed to overcome a barrier of energy (energy barrier to nucleation) to create a liquid-solid (L-S) interface eventually leading to formation of a stable nuclei out of the melt. This region is further divided into two regions A -C and C -D.
Region A -C: This is region in which undercooling occurs, heat is extracted, the temperature drops and shape of cooling curve goes down. This is characterised by two energies described in the above paragraph.
Region C -D: This is the region in which heat energy is absorbed, temperature is gained and shape of curve goes up. This is called recalescence.

Cooling Curve for Well Inoculated Zr-Based in-situ Dendrite
BMGMCs Shape of cooling curve changes its form as melt is changed from single component to binary to multicomponent alloys. This can be explained in the form of various cases. 2) It is followed by region of constant temperature cooling which is called local solidification. This is only visible in case of very fluid alloys in which mushy region is very fluid/less viscous (not BMGMCs). This region is absent in most multicomponent (industrial) alloys as their solidification is dominated by mushy zone. (Note: BMGMCs are special case of alloys in which mushy region is extensively dominated but another phenomenon known as "sluggishness" governs the solidification. In these alloys, three laws [8] which describe BMGMC formation and evolution make sure that not only sluggishness dominates kinetics but it also ensures "glass formation" (i.e. retaining supercooled liquid at room temperature).  2) This is followed by region of constant temperature at which all liquid get transformed into solid. However, in these alloys, this region is very small (because of presence of marked mushy zone).
3) Instantly, after this region, alloy enters in "solidification range". As the alloy is very fast cooled, this region is again not very clearly identified which is typical behaviour in case of fast cooled castings.

Extraction of Heat-Determination of Heat Transfer Coefficients (HTCs)
In the development of model, heat transfer coefficients will be determined at every point of cooling curve following earlier defined one dimensional (1D) schemes [229]. These will ensure, time of solidification calculation during cooling following above cooling curve and helps in determining shape of melt pool and its transient behaviour during cooling.

Final Time of Solidification
Final time of solidification is sum of time in each region/section of cooling curve of a particular alloy/melt. It will be determined using standard transport equations and will be used empirically to assess the conformability of additive manufacturing process. Time of solidification gives other parameters as well such as fraction of mass solidified after a time, t, which is direct measure of microstructure evolved during that time. It can be qualitatively (extrapolation) used to predict further (type (equiaxed, columnar, mix, CET) and amount) evolution of microstructure with time.

Modelling and Simulation of Nucleation (Heterogeneous) in Liquid Melt Pool-Microstructural Development
Modelling and simulation of microstructural development in liquid melt pool can be described by macroscopic and microscopic models of heat and mass transfer depending on type of alloy, its nature, number of elements, cooling curve, undercoolings (constitutional (solute/particulate), thermal, curvature, interfacial), thermal and kinetic limitations, behaviour of mushy zone, presence or absence of inoculants. These can be broadly divided into macroscopic and microscopic models [196] which are explained as follows; M. M. A. Rafique Journal of Encapsulation and Adsorption Sciences

Macroscopic Models
By following the regimes of macroscopic models, finite element method (FEM) and finite difference methods (FDM) can be used to explain microstructural development both during steady and transient state transport processes.

Limitations
Both FEM and FDM based models cannot fully describe mushy region, its behaviour and evolution during solidification as they do not account for microscopic 1) solute diffusion and 2) capillary effects which are primarily responsible for scale at which microstructure forms (which is very small as compared to macroscopic methods based on average continuity equations [230] [231] [232] [233] in which it is assumed that solidification starts at liquidus and finishes at solidus/eutectic temperatures (A case of BMGMCs having good match of GFA and eutectic temperature [234] [235]). In order to overcome these limitations, microscopic models were proposed.

Microscopic Models of Microstructure Evolution/Formation during Solidification
Stage 1 Model: These models take into account the mechanism of 1) grain nucleation and 2) grain growth in alloys which are solidifying with equiaxed dendrite or eutectic microstructures [236]. These do not account for alloys which are solidifying with columnar dendritic and planar interfaces. A modification of these accounts for equiaxed-columnar (at mould wall) and columnar to equiaxed transition (CET) in bulk of liquid. These can be used to "describe microstructures" and "prediction of grain size" in case of eutectic compositions of BMGMC. Majority of these is based on "analytical/deterministic approaches" which can be described as follows; f(ΔT n ) is difficult to be found from theoretical considerations alone. It needs to be found experimentally i.e. form a set of experiments e.g. Method 1 Measurement of cooling curve. This has been explained in detail in Section 7.2.1. and 7.2.2. Method 2 Measurement of grain density (optical micrograph of cross section (using Image J  /ASTM 562 -11 Method)) for specimens solidified at various cooling rates [196].

Growth
As soon as grain has nucleated, and its growth can be explained by special modi-Journal of Encapsulation and Adsorption Sciences fied case of classical nucleation theory (CNT) for BMGMC (A detailed treatment of modified CNT for BMGMC is given in Appendix A) and its distribution can be explained by constitutional supercooling zone/Interdependence theory (propagation of liquid-solid interface/liquid-solid spherical front)-a possibility which is still under investigation by author for suitability for additive manufacturing processes), it grows with an interface velocity which is also a function of undercooling.

Velocity of Growth
Velocity of growth may be written as In this case, there is no need to determine solidification kinetics of dendrite tip/eutectic (spherical front) interface by cooling curve or grain size but it can be determined by theoretical models developed (by using basic laws of physics) [237] [238] as applied to BMGMC only under transient condition.

Impingement and Columnar to Equiaxed Transition (CET)
Impingement of grains as they grow is another important phenomenon which for all practical reasons governs the shape of grain after columnar to equiaxed transition (CET) (CET in additive manufacturing is recently explained by Amrita Basak et al. [239] which is combined with present model and is explained in detail in Appendix B). This phenomenon is not remarkably present in bulk metallic glass and their composites due to their sluggish nature and slightly formed crystal grains as compared to huge glassy matrix. However, despite these drawbacks, this is mainly responsible for equiaxed dendritic grain formation even in glassy alloys, especially in eutectic compositions which is assumed to be the case for present research.
This has been typically treated by

1) Standard J M A K [210] [240] correction or by 2) Geometrical [241] [242] or
3) Random grain arrangement models [196] These "microscopic" solidification models have been coupled with "macroscopic" transient one-dimensional (1D) heat flow calculations to successfully predict "microstructural features" specially "grain size" at the scale of whole process (part scale) [243] [244]. cleation) in case of well inoculated melts (present case)) giving rise to onset of columnar dendritic microstructure (at a very small length scale) since they almost neglect any aspect which is related to crystallographic effects.

2) "Equiaxed-Columnar" Transition
They cannot predict the co-called "equiaxed-columnar" transition which occurs very near to mould wall [245] or variation of transverse size of columnar grains [246] (also known as columnar dendritic arm branching). This is explained in detail in individual cases for each type of metal (crystal structure).

Case 1: Cubic Metals
It is well established facts that for cubic metals, this "grain selection" is based These methods are also ineffective in predicting "equiaxed-columnar" and then "branching of dendrite arms" in bcc metals (i.e. grain selection) as best alignment between heat flow and crystallographic direction is not well known.
Only assumptions are possible (i.e. in case of bcc best heat flow direction could be assigned to close packed direction).

Evolution of Probabilistic Models
The solution to above four problems is presented first by Brown and Spittle [248] [249]. They developed probabilistic models. They used monte carlo (MC) procedure for explaining solidification phenomena developed in earlier research [250]. MC method is based upon minimising of interfacial energy (which is practically calculated by using physical properties of material (Zr-and Fe-based BMGMC)) from literature and earlier published data or inference from extrapolation or interpolation of data as needed). Procedurally, this minimisation is achieved by 1) Considering the energy of "unlike sites" (e.g. (a) "liquid/solid sites" or (b) "sites belonging to different grains" and 2) By allowing transition between these states to occur according to randomly generated numbers By using this method, Brown and Spittle merely able to produce computed 2 D microstructures which resembled very closely to those observed in real micrographic cross section. In particular a) The selection of grains in the columnar zone and b) Columnar to equiaxed transition (CET) were nicely reproduced using this technique also a) the effect of solute concentration or b) melt superheat upon the resultant microstructure was determined "qualitatively" in a nice way. Their quantitative representation was not achieved.

Limitations
These methods suffer consistently from lack of physical basis and thus cannot be used to analyse quantitatively the effect of various physical phenomena (happening within the phase transformations). For example, to illustrate this, con-M. M. A. Rafique Journal of Encapsulation and Adsorption Sciences sider the following example. 1) During one monte carlo (MC) time step, Consider N sites where N is number of sites whose evolution is calculated and is chosen from another N (total number) sites. Therefore, not all sites of interest (i.e. those located near to solid-liquid interface) are investigated. This in turn, leads to algorithm predicted grain competition in columnar region, which does not at all reflect the physical mechanisms observed in organic alloys.
2) Furthermore, the results are sensitive to type of Monte Carlo (MC) network itself which is used for computations. Thus, a single powerful model is presented in present work which combines "advantages of probabilistic methods with those of deterministic approaches" to predict more accurately the grain structure in a casting.

Two-Dimensional Cellular Automaton (CA) Method
For this purpose, for now, a 2D Cellular Automaton model is developed which is based upon physical mechanisms of nucleation and growth of dendritic grains.
Its salient features are as follows; 1) Heterogeneous nucleation; which was modelled by means of a nucleation site distribution in deterministic solidification models, is treated in a similar way in present probabilistic approach.
2) If total density of grains which nucleate at a given undercooling is obtained from an average distribution (d c = average (distribution)), the location of these sites is chosen randomly 3) Crystallographic orientation of a newly nucleated grain is also taken into account at random.
4) The growth kinetics of (a) dendrite tip and (b) of side branches are also incorporated into the model in such a way that final simulated microstructure is independent of the "cellular automaton network" which is used for computations.
Although, it produces micrographic cross sections very much similar to those already obtained by Brown and Spittle, present model has a "sound physical basis" and can thus reflect effects of (a) cooling rate" or (b) "solute concentration" quantitatively.

Detailed Description (Phase 1-Application of CAFE to
Conventional Casting) Physical background: Consider a BMGMC wedge shape casting as shown in   solidified in water cooled wedge shape Cu-mould [78] [251]. Their dendritic grains which have various crystallographic orientations appear as zones of different colours (Figure 3(b)). Most common regions encountered in any casting appear here [245] [246] and are marked all along cross-section. On the top end of wedge shape ingot coarse grains are present as this region was exposed to air. Its more detailed explanation will follow after characterizing region chronologically from bottom to top.

Characterization
Bottom region glass: The tip of casting is 100% glass (monolithic BMG). This region is classified as glass and no crystal structure is observed here because cooling rate is maximum here which results in extraction of heat at a very high rate resulting in retaining supercooled liquid state at room temperature.

Conclusions
Despite their recent popularity and emblem to be exploited as potential struc- N can be neglected as during initial stages there is no nucleation event.
According to CNT, a minimum energy value is needed to create a solid-liquid interface eventually leading to stable nuclei out of melt. This is known as "activation energy". This activation energy is the energy to overcome ΔG * -the energy barrier to nucleation. Now, as solid-liquid interface grows to form stable nuclei, atoms must be transported through liquid thus another temperature dependent activation energy must be overcome known as ΔG d (activation energy for diffusion).
The net effect is that CNT predicts a nucleation rate (I) given by Total No. of atoms in contact with substrate initial nucleation rate total time = Definition used in Equation (7).
2) The difference between frequency and rate is that frequency is "occurrence of an event per unit time" while rate is total number of that event (in terms of numerical value) per unit time.
Thus, from Equation (6) N o = Total number of heterogeneous substrate particles originally available per unit volume; N = Number that have already nucleated; I o = constant.
Value of I o can be calculated from Equation (5) using another term known as "liquid diffusion coefficient".
where a = atomic diameter = 0.4 nm; υ = frequency which gives where u is a constant The value of u can be measured from Method 1: T (heterogeneous nucleation temperature). This is defined as temperature, where there is an initial nucleation rate of one nucleus/cm 3 /sec.  Time is user defined input and temperature comes from user defined value initially as well. Then its every new value is assigned back to Equation 5. With temperature and time, k changes and assigned back to Equation (15). Also, with time, υ (vibrational frequency) changes and assigned back to original equation (5). Similarly, the value of u also changes with time and temperature. Below table (Table 1) summarises the values which are user defined and which change as a function of transience as programs runs. Note: 1) In BMGMCs, in some cases due to slow motion of large atoms, only nucleation happens, and growth never happens. In these cases, a new phenomenon known as soft impingement effects of crystals must be considered. These could be solutal/thermal. However, this is quite rare.
2) In general, in case of BMGMC, classical nucleation theory (CNT) cannot be applied alone to describe nucleation and growth.

Appendix B
1) Special case of growth of "Columnar microstructures" The growth of columnar dendrites, which is initiated by nuclei that form at the mould interface (Cu mould casting/Twin roll casting of BMGMC) (only if constitutional supercooling zone (CSZ) is suppressed-not present case) is usually simulated in a much simpler way. Again, in this case, there is no need to use cooling curve measurements or grain size measurement but same growth kinetics models [7] [8] can be used to determine.
Undercooling of eutectic front (ΔT n eutectic) or 3) Columnar to equiaxed transition (CET) [239] Growth rate of solid-liquid interface cos V S θ = (16) where S = Scan speed Temperature gradient parallel to dendrite growth direction can be calculated using cos hkl G G ψ = (17) where ψ = Angle between "normal vector" and "possible dendrite growth orientation" at the solid-liquid interface. This is evaluated by CFX-Post in Ansys  .
A modification known as Rappaz modification is applied to predict CET. This is as follows ( ) ( ) [259], it is possible to get full map of microstructure evolution and grain size.

Comparison
Below a comparison of "strengths and capabilities" and "evolution of different theories over time" which have enabled a better understanding of nucleation and growth phenomena in bulk metallic glass matrix composites, is tabulated. The aim is to present reader with a concise smart workable data for first hand use and reference for solving nucleation and growth problems in bulk metallic glass matrix composites by modelling and simulation. This will help professional programmer, working engineer and a researcher to effectively find previously done research till now with its strengths and capabilities at one platform.