_{1}

^{*}

After a silence of three decades, bulk metallic glasses and their composites have re-emerged as a competent engineering material owing to their excellent mechanical properties not observed in any other engineering material known till date. However, they exhibit poor ductility and little or no toughness which make them brittle and they fail catastrophically under tensile loading. Exact explanation of this behaviour is difficult, and a lot of expensive experimentation is needed before conclusive results could be drawn. In present study, a theoretical approach has been presented aimed at solving this problem. A detailed mathematical model has been developed to describe solidification phenomena in zirconium based bulk metallic glass matrix composites during additive manufacturing. It precisely models and predicts solidification parameters related to microscale solute diffusion (mass transfer) and capillary action in these rapidly solidifying sluggish slurries. Programming and simulation of model
is
performed in MATLAB
^{<sup>®</sup>}
. Results show that the use of temperature dependent thermophysical properties yields a synergic effect for multitude improvement and refinement
simulation results.
Simulated values proved out to be in good agreement with prior simulated and experimental results.

Bulk Metallic Glasses [

In order to develop a model consider a dendrite tip evolving out of liquid as it starts cooling in melt pool of additive manufacturing. Its shape could be considered to be resembling a parabola. Using KGT model [

1) The solute field around the dendrite tip is given by Ivantsov solution.

2) The dendrite tip grows at marginal stability limit.

3) The diffusion coefficient d, is (tip) temperature dependent.

4) The segregation/partition coefficient, k, takes into account solute trapping; i-e, k is (dendrite tip) velocity dependent.

5) Initial partition coefficient (k_{o}) is temperature dependent and binary alloy (Zr-Cu) is assumed to behave as multicomponent alloy.

6) The undercooling of tip (ΔT) is the sum of solute undercooling and the curvature undercooling.

7) The effect of convection is ignored.

In present study, however, a further practical approach is adopted which takes into account the calculation of supersaturations of individual constituents/components in an alloy which eliminates the fact that their diffusion fields superimpose and supports the actual conditions in which binary alloy system does not behave as ternary or multicomponent system (BMGMC). This is a contradiction to assumption 5 above in basic KGT model and provides the basis of development of present model to explain microstructural evolution in detail.

There are three main velocities which are of interest here (Figures 1(a)-(c)) when a beam of high energy (electron or laser) travels on surface of specimen in additive manufacturing setup (

1) Moving heat source velocity (V_{b}), 2) solidification front velocity (V_{s}) and 3) dendrite tip velocity (V_{hkl})

Model comprises of three main parts and separate mathematical expression is developed for each segment.

Nucleation

This is based on Oldfield theory of heterogeneous nucleation which describes a relationship between undercooling (ΔT) and grain density at each segment of interest (bulk liquid, mold wall and potent nuclei) in terms of Gaussian distribution to explain solidification. Two most important parameters sought after to be determined are, maximum nucleation density (n_{max}) and grain density (n(ΔT)). Maximum nucleation density may be obtained by integral of nucleation distribution from zero undercooling to infinite undercooling.

n max i = ∫ 0 ∝ d n d Δ T ′ Δ T ′ (1)

Similarly, grain density is given by following equation

n ( Δ T ) = ∫ 0 Δ T n max Δ T σ 2π exp [ − 1 2 ( Δ T ′ − Δ T N Δ T σ ) ] d Δ T ′ (2)

where ΔT_{n} and ΔT_{σ} are mean undercooling and standard deviation of grain density distribution respectively.

With this, probability of happening of one event (nucleation) is given by nucleation probability (p_{v}) as described by Prof. Rappaz in his famous article [

p v ≥ r (3)

i-e if at any instant of time t, p_{v} exceeds r, nucleation will occur. p v = δ n v ⋅ V C A where δn_{v} = grain density increase and V_{CA} = one cell volume (measure by noting all dimensions of a cell assuming it to have square shape) (

Dendrite growth orientation

Second part of the model deals with determination of dendrite growth orientation i-e the direction in which some dendrites preferably grow faster and longer as compared to others due to balance between geometrical and kinetic variables. This also highlights and points towards grain competition and selection mechanisms. Two important parameters are held responsible for assigning and determining grain orientation. a) Growth of first grain as a result of heterogeneous nucleation at mold wall or potent nuclei and b) Location of further subsequent new grain(s) and their crystallographic orientation. For example, for cubic metals, the preferential growth directions of dendrites are given by direction of easy heat flow which is along <100> crystallographic direction/orientation. During early stage of solidification, a nucleus grows at the surface of mold or potent nuclei in the form of hemispherical surface. This surface becomes unstable and then dendritic after a certain incubation time and growth occurs with

main trunk and arms coinciding with <100> crystallographic direction. The location of new grains is assumed to be governed by random process. This specific orientation is described to be controlled by three Euler angles θ, φ and ψ irrespective of grain nucleated at the surface of mold, potent nuclei or bulk liquid. This is described in

The probability d p ( θ , φ , ψ ) that a newly nucleated grain has its main trunk orientation in the range [ θ , θ + d θ ] and [ φ , φ + d φ ] and one of its set of secondary branches within the orientation [ ψ , ψ + d ψ ] is given by [

d p ( θ , φ , ψ ) = A ⋅ sin θ ⋅ d θ ⋅ d φ ⋅ d ψ (4)

where A = constant which takes into account the fourfold symmetry of the dendrite along its trunk axis and the possible permutations of the <100> directions.

In general, dendrite growth direction of grain nucleated at the mold surface determines the time during which grain can survive competitive growth of its neighbors.

Grain growth/Growth kinetics/Dendrite stability theory

This section concerns kinetics associated with growth of already formed grains in bulk liquid, mold surface and potent nuclei. A unique feature adopted here concerns the determination of supersaturation of individual elements in multicomponent alloy (BMGMC) systems. This approach arises from the notion that in contrast to conventional castings in which undercooling related with thermal diffusion, attachment kinetics and curvature is small, in multicomponent systems (undergoing additive manufacturing treatment) basic KGT model must be used with certain modifications which not only accounts for superimposition of solute fields around each dendrite tip but also incorporate determination of supersaturation for each individual component (Zr, Cu, Ni, Al and Co) of alloy system. This supersaturation Ω_{i} is a function of Peclet number, Pe_{i}

Ω = I v ( P e ) (5)

Ω i = I v ( P e i ) (6)

P e i = R ⋅ V 2 ⋅ D i (7)

Putting in [

Ω i = I v ( R ⋅ V 2 ⋅ D i ) (8)

Ω i = P e i e P e i E 1 ( P e i ) = P e i e P e i ∫ P e i ∞ e − u u d u (9)

Ω i = R ⋅ V 2 ⋅ D i e R ⋅ V 2 ⋅ D i E 1 ( P e i ) = R ⋅ V 2 ⋅ D i e R ⋅ V 2 ⋅ D i ∫ R ⋅ V 2 ⋅ D i ∞ e − u u d u (10)

but

Ω i = c i * − c o , i c i * ( 1 − k i ) (11)

where, c i * = concentration of constituent i in liquid at dendrite tip (to be found), c o , i = initial concentration of constituent i, k i = partition coefficient for this constituent i (velocity dependent).

Comparing (10) and (11)

c i * − c o , i c i * ( 1 − k i ) = R ⋅ V 2 ⋅ D i e R ⋅ V 2 ⋅ D i ∫ R ⋅ V 2 ⋅ D i ∞ e − u u d u (12)

However,

c i * = c o , i 1 − ( 1 − k i ) ( I v ( P e i ) ) (13)

k i = k o + ( a o V D i ) 1 + ( a o V D i ) (14)

where, a_{o} = length scale related to interatomic distance and is estimated to be between 0.5 - 5 nm and

D i = D o e ( − Q R g ⋅ T * ) (15)

where, D o = Proportionality constant, Q = Activation energy, R_{g} = Gas constant, T^{*} = Tip temperature calculated by iterative method (described below) [

In a linearized phase diagram

T * = T L + ∑ i = 1 2 m i ( c i * − c o , i ) − 2 Γ R (16)

where, m_{i} = slope of liquidus surface with respect to constituent i

m i = ∂ T L ∂ c i (17)

T_{L} = Liquidus temperature for initial alloy composition, Γ = Gibbs-Thomson coefficient, 2 Γ R ≈ 0 (negligible) (under normal solidification conditions), 2 Γ R = 1 (under rapid solidification conditions (additive manufacturing)).

Another term can be generated from linearized phase diagram known as fictitious melting point of pure constituent [

T ′ m = T L − ∑ i = 1 2 m i c o , i (18)

Using Equations (13) and (18), Equation (16) becomes

T * = T ′ m + ∑ i = 1 2 m i c o , i 1 − ( 1 − k i ) ( I v ( P e i ) ) − 2 Γ R (19)

where, R = Dendrite tip radius, V = Dendrite tip velocity.

This model is iterative model which is based on assigning final values to original value thus generating a loop whose explanation will be given in next section. 100 iterative cycles are used to generate homogeneous and normalized data based upon best software engineering practice. In general while writing the program, reading it and executing it, Ω depends on I v ( P e i ) and c i * , c i * depends on k_{i} and I v ( P e i ) , k_{i} depends on D i , D i depends on Tip temperature and finally, Tip temperature depends on c i * . Thus, a loop is generated which accounts for “to and fro” motion of information and iterative handling of data. This is the essence of generation of refined outputs and results. A schematic flow chart describing the working of model and interdependence of parameters is presented in

Finally, total undercooling (ΔT) is related to supersaturation (Ω) by [

Δ T i = m i ⋅ c o , i ⋅ [ 1 − 1 1 − Ω i ⋅ ( 1 − k i ) ] (20)

The criteria used to specify radius of dendrite tip (R) is assumed to be given by marginal stability wavelength of planar wave front (as given in Mullins and Sekerka [

Accordingly, one has for a ternary system

R = 2 π ( Γ ∑ i = 1 2 m i G c , i ε c ( P e i ) − G ) 1 / 2 (21)

where, G_{c,i} = solute gradient of constituent i in the liquid near tip which can be written as

G c , i = − υ D i C i * ( 1 − k i ) (22)

G = Average thermal gradient near tip and ε c ( P e i ) = f ( P e i ) = 1 (low speed/ low P_{ei}).

Putting values of P e i = R ⋅ V 2 ⋅ D i and Equation (22) in Equation (21) [

4 π 2 Γ R 2 + 2 R ∑ i = 1 2 P e i m i C o , i ( 1 − k i ) ξ c ( P e i ) [ 1 − ( 1 − k i ) I v ( P e i ) ] + G = 0 (23)

where, Γ = Gibbs-Thomson coefficient, m_{i} = the slope of liquidus, ξ c = a function of the Peclet number and segregation coefficient, G = Thermal gradient.

A computer program was written in MATLAB^{®}. Instead of fixing the Peclet number as was done in previous approaches [

Sr. No. | Parameter | Description | Value |
---|---|---|---|

1 | T_{L} | Liquidus temperature | 2128 K |

2 | C_{o}_{Zr} | Initial concentration of Zr in alloy | 0.65 wt % |

3 | R_{i} | Initial value of tip radius | 0.001 mm |

4 | V_{i} | Initial value of dendrite tip velocity | 2 mm/sec |

5 | D_{o} | Proportionality constant | 0.492 mm^{2}/sec |

6 | a_{o} | Length scale related to interatomic distance | 0.000005 mm |

7 | Q | Activation energy for diffusion | 67,700 J∙mol^{−1} |

8 | Γ | Gibbs Thomson Coefficient | 1.90 × 10^{−4} (K mm) |

9 | R_{g} | Gas Constant | 8.314 (J∙mol^{−1}∙K^{−1}) |

10 | G | Thermal Gradient | 100 K/mm |

11 | H_{f} | Heat of fusion | 21,000 (J∙mol^{−1}) |

12 | C | Constant related with unit thermal undercooling θ_{t} | 1 |

front. Performing simulations employing data from other elements in a BMGMC system e.g. Cu, Al, Co, Ni is left for reader as an exercise and home work task.

BMGMC samples were produced in two ways. Firstly, they are made in form of wedge using vacuum suction casting system in lab scale Vacuum Arc Melting (VAM) button furnace at CSIRO-Manufacturing. The process consists of carefully calculating raw material based on weight percentage of each element in the alloy system. These powders/granules/chucks are subsequently mixed using hand spatulas to a homogeneity observable by naked eye. For their positioning, handling and control inside enclosed chamber of VAM furnace, they are wrapped in an aluminum foil which not only protects the powders but also serve as alloying element in sufficient quantity in original mix. This Aluminum foil wrapped toffee is placed in horizontal slot in the water cooled copper hearth of furnace at appropriate time after which, it is melted to get solid chuck/button for subsequent research. During second approach, casted wedge samples were subjected to laser solid forming (LSF) [

Model works by explaining dendritic growth in cast alloys during solidification by manipulating physical process parameters with the change of heat and mass transfer coefficients. Its unique feature is it explains the behavior of multicomponent alloys in terms of transient state variables. An effort is made to keep constant values to a minimum to get real picture of actual physical processes. Boundary conditions of solidification phenomena are kept open which makes model more rigorous and robust and it is possible to apply this to a variety of alloys systems under various conditions. Following results and graphs have been generated after writing script of solidification code and running it in MATLAB^{®}.

Effect of heat dissipation on dendrite tip velocity

heat loss due to solidification. Thermal conductivity of Zirconium is also not very high which suggest that very little heat loss will occur as a result of presence and movement of Zirconium through fluid. Overall large atomic weight remains as the most favorable factor for its increased dendrite tip velocity through thick slurry. A small effect of gravity also contributes towards this overall effect (not considered in detail here). Second element of interest in this multicomponent system is Ni, which is not been able to attain enough velocity in growing dendrite tip due to its lower atomic weight. Thus it remains ineffective in creating shear zone around it which can allow it to penetrate and gain speed in thick slurry. It also does not have very high thermal conductivity thus not a lot of heat gets dissipated to surrounding because of it and again it reflects its inability to create low viscosity zone around it which may facilitate it to gain high speeds. Drastically high speed and very high Peclet number over a range is depicted by copper. This was not astonishing, as it is the element of highest thermal conductivity with an intermediate position with respect to its atomic weight in periodic table of elements. These both features; specially high thermal conductivity gives it an edge to create a region of very high heat dissipation around it facilitating its motion at a high speed over a range of Peclet number in a thick slurry of multicomponent alloy systems. The atomic size and configuration of copper is also favorable for attaining this high speed. Thus for this thermal gradient, it remains as the most important element contributing towards overall speed/dendrite tip growth velocity of system. The last element is Aluminum, which despite of its high thermal conductivity does not attain very speed. This happens because; the atomic size and weight of aluminum is not very high which does not help it to gain high velocity in the thick slurry of multicomponent systems at tip of dendrite or spheroid. Also, as the heat keeps on getting dissipated from its surrounding and it has relatively low atomic weight as compared to others, it does not attain high speed and gets stuck in its own region. This reasoning is based on atomic size, weight, electronic configuration and thermal conductivity of individual elements. However, no data exists about actual dendrite growth velocity of this complex BMGMC system and further experimental research is needed to measure this.

Effect of tip temperature on dendrite tip velocity

Below graph (_{tip} at slow V_{f} because dendrites are equiaxed in nature and due to their rapid mechanical interaction with each other a lot of heat is accumulated in small area. This is more evident in multicomponent alloys. Here, since only Zirconium is under consideration, so the large atomic size and weight of Zr also contribute towards increase of this value. This is early/initial stage of solidification. Another reason for this is there is planar wave front during initial stages which does not allow the development of high surface area i-e surface area/volume ratio remain low and not a lot of heat gets dissipated. This is also shown in previous works by Rappaz et al. [_{tip}

with increase in velocity is observed. This happens as Zr dendrite gain speed; it starts creating regions of very high surface/volume ratio around it in the body of melt and at the tip of advancing dendrite. In other words fast moving dendrites become source of high heat dissipation. Owing to this, temperature drops. Discrepancy between experimental values and simulation results is due to lesser number of iterative cycles which create low convergence. Hence, a temperature field with a deviation from actual values is observed. This anomaly could be removed by increasing iterations which enhances simulation efficiency and helps in model refinement. A decreasing trend in dendrite tip temperature is also expected at higher thermal gradients (G) as this will provide a large sink for the heat to get dissipated away from liquid melt pool which eventually will lower the tip temperature even at low growth velocities.

Effect of tip temperature on dendrite tip radius

The effect of dendrite tip temperature on the evolved radius is shown in

thermal gradient which is final description of microstructural phenomena in solidifying alloys [

Effect of ξ (a function of P_{ei}) on P_{ei}

Below graph (_{e}) shows almost decreasing linear relationship with the increase of rate of heat transfer from system to surrounding for all major alloying elements of hypoeutectic system at a fixed thermal gradient. It shows the effectivity of heat transfer process for BMGMC and describes that dissipation was proper and homogenous. The curves are generated by plotting solution of present model by the use of indigenous MATLAB code incorporating different transient thermo-physical data of each individual alloy system from literature [

Evolution of segregation/partition coefficient with temperature

This is very interesting graph (

when studied over a temperature range. It evolves with the change/evolution of temperature. Although assumed to be, and observed to be almost linear, its evolution is highly dependent on the gap chosen to calculate the values. The smaller the temperature gap, better will be the representation of actual behavior or evolution of partition coefficient over that period. For simplicity reasons, this effect is not studied in detail and general assumption that it shows linearity of evolution over temperature and time in the range of interest is made and adopted. Again, thermal gradient (G) is kept constant at 100 K/mm.

Effect of dendrite tip growth velocity on supersaturation

This is final and second most important graph (

Primarily, present model is formulated as a result of combination of following [

1) Gandin, C.-A., M. Rappaz, and R. Tintillier, Three-dimensional probabilistic simulation of solidification grain structures: Application to superalloy precision castings. Metallurgical Transactions A, 1993. 24(2): p. 467-479.

2) Rappaz, M., et al., Analysis of solidification microstructures in Fe-Ni-Cr single-crystal welds. Metallurgical Transactions A, 1990. 21(6): p. 1767-1782.

3) Rappaz, M., et al., Development of microstructures in Fe-15Ni-15Cr single crystal electron beam welds. Metallurgical Transactions A, 1989. 20(6): p. 1125-1138.

4) Kurz, W., B. Giovanola, and R. Trivedi, Theory of microstructural development during rapid solidification. Acta Metallurgica, 1986. 34(5): p. 823-830.

5) Zhang, J., et al. Probabilistic simulation of solidification microstructure evolution during laser-based metal deposition. In Proceedings of 2013 Annual International Solid Freeform Fabrication Symposium―An Additive Manufacturing Conference. 2013.

Apart from these, its salient features are;

1) Supersaturation of individual elements was measured to account for overall behavior of multicomponent system―an approach missing in previous studies

2) Due to scarcity, dispersion and unavailability of data, a correlation from nearest possible element in same group in periodic table was used.

3) An effort was made to remove/reduce error by use of iteration based approach to refine model.

4) Programming of model was done in MATLAB^{®}―not done elsewhere previously.

5) Temperature dependent properties (transient heat transfer conditions) were used.

6) A unique approach based on segregation coefficient (k) as a function of temperature was adopted (Previously [

7) Slope of liquids (m) is taken to be concentration (C^{*}) dependent.

8) Peclet number (P_{e}) & ξ are not taken as constant like previous studies [Bobadilla, M., J. Lacaze, and G. Lesoult, Journal of Crystal Growth, 1988. 89(4): p. 531-544] in which it is assumed

a) ξ = 1 (low growth rate) (low P_{e})

b) ξ = 0 (very fast cooling rate―typical Additive Manufacturing conditions)

9) 2Γ/R = 1 (high velocity AM conditions).

10) New relation for dendrite tip temperature was developed.

NOTE: No physical microstructure was either reported previously or tried in present approach. Physical simulation is “not possible” at deterministic stage as it is numerical/analytical model which defines parameters for next stage probabilistic studies only.

Following conclusions may be drawn out of this study.

a) There is significant effect of initial metal temperature, composition, type of alloying elements, temperature gradient and thermo-physical properties on final microstructure developed as a result of heat and mass transfer phenomena.

b) Determination of supersaturation of individual elements yield best possible strategy for its correlation with superimposition of solute field around each dendrite tip.

c) Determination of dimensionless solutal Peclet number is the main factor responsible for accurate quantitative prediction of microstructure in solidifying alloys.

d) Dependence and evolution of ξ on, and with respect to solutal Peclet number is decisive in explaining transient nature transport phenomena in additive manufacturing processes.

e) Employment of iterative process helps in refining the model and generates accurate results.

f) Final microstructure evolution is expressed in the form of dendrite tip temperature and dendrite tip radius as a function of growth rate/dendrite tip velocity and must be carefully measured.

In essence, model comprises of extension of KGT theory for multicomponent systems beyond BLL model [

This work is partially supported by scholarship provided by RMIT University for tuition and living of author. Further, author would like to acknowledge moral support and encouragement provided by his primary supervisor (Dr. Dong Qiu) and Prof. Milan Brandt throughout the work and helpful discussion and constructive criticism by Prof. Mark Easton. The authorization of gracious use of experimental facilities during short term stay at CSIRO and above average moral support by Dr. Daniel East is also gratefully acknowledged.

The author(s) declare no competing financial interests.

Rafique, M.M.A. (2018) Simulation of Solidification Parameters during Zr Based Bulk Metallic Glass Matrix Composite’s (BMGMCs) Additive Manufacturing. Engineering, 10, 85-108. https://doi.org/10.4236/eng.2018.103007