Prediction of the solid-liquid interface energy of a multicomponent metallic alloy via a solid-liquid interface sublattice model

Journal Pre-proof Prediction of the solid-liquid interface energy of a multicomponent metallic alloy via a solid-liquid interface sublattice model Kewu Bai, Kun Wang, Michael Sullivan, Yong-Wei Zhang PII:

S0925-8388(19)34238-0

DOI:

https://doi.org/10.1016/j.jallcom.2019.152992

Reference:

JALCOM 152992

To appear in:

Journal of Alloys and Compounds

Received Date: 10 July 2019 Revised Date:

29 October 2019

Accepted Date: 11 November 2019

Please cite this article as: K. Bai, K. Wang, M. Sullivan, Y.-W. Zhang, Prediction of the solid-liquid interface energy of a multicomponent metallic alloy via a solid-liquid interface sublattice model, Journal of Alloys and Compounds (2019), doi: https://doi.org/10.1016/j.jallcom.2019.152992. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

Prediction of the solid-liquid interface energy of a multicomponent metallic alloy via a solid-liquid interface sublattice model Kewu Bai*, Kun Wang, Michael Sullivan, and Yong-Wei Zhang Institute of High Performance Computing Singapore 1 Fusionopolis Way, #16-16 Connexis North, Singapore 138632 Abstract Despite central roles in solidification and crystal growth, the solid-liquid interface energy σsl of the multicomponent metallic alloys is scarce and scattered because of the experimental complexity in measurements. Moreover, the theoretical prediction of σsl of the metallic alloy has been long suffered from the incomplete understanding of the solid-liquid interface structures. In this paper, the solid-liquid interface of the metallic alloy is reduced to a two-dimensional layer composed of the solid sublattice and liquid sublattice that is further sandwiched by the bulk solid and liquid phases. This enables a well-defined solid-liquid interface enthalpy and entropy hence the Gibbs energy model of the solid-liquid interface layer. Via a constraint minimization of the solid-liquid interface layer Gibbs energy, the composition- and temperature-dependent σsl of the Al-based and Ag-based alloys are calculated by coupling with the available thermodynamic model parameters of the alloys. The calculation results are validated by the experimental data. It demonstrates that the built interface sublattice model offers a self-consistent framework for the automatic prediction of σsl of the metallic alloy ranging from the binary to a multicomponent system. It thus provides a more compelling solution to the long-standing issue associated with the modeling of the solid-liquid interface thermodynamics of the metallic alloys under both undercooling and equilibrium states.

Keywords: undercooling.

Solid-liquid interface energy, sublattice, Gibbs energy, metallic alloys,

1

Electronic mail: [email protected] Corresponding address: 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632 Tel: (65) 6419-1565, Fax: (65) 6463-0200

1. Introduction The solid-liquid interface energy

is defined as the excess Gibbs energy of a solid-liquid

interface per unit area. It is a fundamental property that controls the nucleation and solidification processes of the metallic alloys. In the classical nucleation theory, it represents the nucleation barrier in the solidification process of an undercooled liquid that drives the transitions between planar, cellular and dendritic growth regimes, and consequently regulates the microstructure of alloys.[1, 2]. Despite central roles in materials science,

of the metallic alloys is scarce and

scattered for a number of reasons. First, the measurement of

of the metallic alloys is

extremely difficult due to the experimental complexities in measurement. Second, the atomic simulation of the solid-liquid interface using molecular dynamics (MD) is computationally demanding and concentrated on the simple metallic system only due to a lack of reliable force fields.[1]Third, the empirical efforts in the prediction of

have been long suffered from the

incomplete understanding of the solid-liquid interface structures in particular for the multicomponent alloys. It is shown that

of metallic alloys normally can be indirectly measured by two

experimental techniques. First is the maximum supercooling technique, in which the liquid-solid interface energy is extracted by measurements of the crystal nucleation rates in undercooled 2

states within an approximation of the classical nucleation theory.[3-5] Due to the occurrence of heterogeneous nucleation induced by the possible impurities with very low concentration, the measured values tend to underestimate the solid-liquid interface energy.[6] Second is the socalled grain boundary groove method[7, 8] in which the Gibbs-Thomson constant obtained from the grain boundary groove shape is used for the estimation of

. As the grain boundary groove

shape may change during sample preparations, the experimental error of this method may range from 15% to 30%.[9] Motivated by the experiment difficulties in measuring the solid-liquid interface energy,[10] there are a lot of theoretical efforts to predict

of metallic systems, which has been reviewed

by Jones, [11] Eustathopoulos [12] and Asta et al.[1], respectively. Most of the efforts concentrated on the solid-liquid interface thermodynamics of pure elements only, in which the of pure metal has been correlated with its fusion entropy by different empiricial relationships. To interpret the empirical relationships, several solid-liquid interface models have been proposed.[5, 13-18] The earliest one accounting for the solid-liquid interface structure was the negentropic model by Spaepen and Thompson.[14, 16] In the model, it was assumed that ( i ) the structure of the solid-liquid structure is atomically smooth. ( ii ) the solid-liquid interface energy is primarily entropic in origin, which can be written as =

(

where point

,

∙

) /



is the molar volume. .

(1) ,



denotes the entropy of fusion of pure metal at the melting

is the Avogadro's number.

is the well-known Turnbull coefficient that depends

on the solid-liquid interface structure. Using this model, the value of

for face-centered cubic

(FCC) and hexagonal close-packed (HCP) structured metal is estimated as 0.86, the value of 3

for body centered cubic (BCC) structured metal is estimated as 0.71.[19] As the solid–liquid interface structure in a metallic system is usually rough in nature,[20] Jian et al. [6, 21] later extended the negentropic model and used the Ising model to illustrate the rough feature of the solid-liquid interface in pure metallic elements. It is interesting to note that the predicted

for

some metallic elements at its melting points ranged from 0.66 to 0.73, which almost equals to the experimental data measured by the grain boundary method (0.66 to 0.75).[22] The predicted value of

(at the maximum undercooling conditions) is 0.52~0.56, which is close to the

experimental data (0.49~0.57) [6] obtained from the indirect measurement of crystal nucleation rates from the undercooled states. As the

of a metallic alloy at any temperature is related to

of the pure elements and the

thermodynamic properties of the solution phases, [12] it was only recently that the CALPHAD technique was frequently used to unravel the inherent solid-liquid interface thermodynamics of the metallic alloys. For examples. Zhang and Du [23] assumed that the solid-liquid interface of the metallic alloy system had a spherical symmetry, and estimated

of the metallic systems

using the concept of effective interface composition and CALPHAD technique. Kaptay [24] treated the solid-liquid interface as a separate thermodynamic phase and analyzed the solid-liquid interface states in metallic systems using the Bulter equation.

of the FCC phases in the Al-

based alloy systems were estimated by using two adjustable model parameters. In contrast, Lippman et al. [25] extended the empirical formula with respect to binary and high order system, The

of pure elements to the

of the FCC phase in the Al-based binary metallic systems

were predicted using the formula by Gránásy et al. [13] and the optimized thermodynamic model parameters of the system. It is evident that above theorteical efforts presented interesting results and new insights into the solid-liquid interface thermodynamics in the metallic alloys system. 4

But in our view, insufficient attention has been given to to the solid-liquid interface structure details. This results in ambiguities in defining the solid-liquid interfacial free energy of the metallic alloys at equilibrium and undercooling states. For this reason, automatic estimation of

a framework for

of the metallic alloys is still missing.

Triggered by the sublattice model in the conventional CALPHAD technique,[26] in this paper, a solid-liquid interface sublattice model is proposed to characterize the rough solid-liquid interface of the multicomponent alloy system. The essence of the model is that the solid-liquid layer consists of the interface solid sublattice and interface liquid sublattice. In addition, each species in the alloy systems are allowed to enter into both sublattices. This enables a welldefined solid-liquid interface enthalpy and entropy hence the Gibbs energy model of the solidliquid interface layer. Via minimization of the solid-liquid interface layer Gibbs energy, the sublattice ratio and the element occupancies at the sublattices are optimized, which permits automatic prediction of

of the metallic alloys ranging from the pure to a multicomponent

system. 2. Computational Method In materials, the solid-liquid structure with a thickness of a few atomic layers typically can be classified into two types. The first is atomically flat. Second is atomically rough. The solid-liquid interface structure of the metal alloys belongs to the second type, in which some liquid atoms will adjust themselves and “attach” to crystal surface producing the solid-like atoms assembly. These atoms thus can be defined as “solid interface sublattice”. While the remaining liquid atoms can be defined as “liquid interface sublattice”. [27] Driven by the interactions of the atoms between ( and within) the two sublattices at the interface, a solid-liquid interface structure will be stabilized via a total energy minimization of the solid-liquid interface layer. Figure 1a shows a 5

sketch describing the solid-liquid interface sublattice model in a multi-component metallic system. In this figure, the solid-liquid interface layer is composed of the solid sublattice and the liquid sublattice. Each sublattice can be occupied by any species in the metallic system. The blue and yellow circles respectively represent the possible species or elements in the solid and liquid sublattice, which is sandwiched by the ordered solid phase and the disordered liquid phase as described by the silver-colored cycles and the blank cycles respectively. It should be noted that in the framework of CALPHAD, the number of the sublattice and the species occupying them is generally obtained from the crystallographic information of a stoichiometric compound. Therefore, the number of sublattices is often fixed. Differing from the conventional CALPHAD picture, [26] the sublattice ratio in the solid-liquid interface sublattice as the roughness indicator of the solid-liquid interface [6] is not fixed. The sublattice ratio is obtained by the minimization of the solid-liquid interface layer energy !" #$ a !"

= ∆&

Here ∆&

!"

!" .

multicomponent system at temperature T can be written as[6, 21] !"

− ·

!"

+ ∆

!*+! !"

(2)

is interface enthalpy, which can be expressed as a summation of pairwise nearest

bond energies in the formation of the solid-liquid interface.

!"

is assumed to be entirely

configurational and associated with the different possible occupations of species in the solidliquid interface sublattices. For a multicomponent system, ∆

!*+! !"

is the excess interface

energy accounting for the species interactions in the solid-liquid layer. Supposing that a metallic solid-liquid interface with a particular arrangement is obtained via the energy minimization of !" ,

an energy balance associated with the solid-liquid interface, based on the hypothesis on

the contributions of

by Ewing [28] and Jian et al. [29], can be written as 6

(Δ&

*

+

·-

.

+

· - /) =

·0

!"

+

∑ 2

3

−∆

5 4 !"

(3)

The energy balance at the solid-liquid interface is illustrated in Figure 1b. Among them, ∆Hmix is the fusion enthalpy or mixing latent heat of the solid. Keeping in mind that the excess enthalpy is released or absorbed upon alloy mixing in a multicomponent system, the mixing enthalpy ) must be difference between the solid and liquid phases (or excess mixing latent heat ∆& !*+! * considered in the estimation of ∆Hmix [18]. It means that ∆Hmix needs to be approximated as the composition weighted fusion enthalpy plus ∆& !*+! * . -

.

(- / ) denote the configuration

entropy loss of liquid phase with the formation of the solid (liquid) interface sublattice in the solid-liquid interface layer. In this case,

∙-

.

+

∙ - / , a term proportional to the liquid phase

entropy, accounts for the reduced liquid entropy in the region of the interface. ∆& T· −∆

/

* +

·-

.

+

represent the total energy required for the formation of the solid-liquid interface.[28]

5 4 !"

represents the energy loss with the solid-liquid interface formation.

denotes the vibration entropy contribution of atoms at the solid-liquid interface. solid-liquid interface energy of a phase in the multicomponent alloy system. 0

∑ 2

3

denotes the !" represent

the

corresponding interface area, which is estimated by [6] 0

!"

= (0

!"

+0

!" )/2

(4)

Among them, 0

!"

7

= (

/

∙

) ∙8

= (

/

∙

) ∙8

(5)

And 0

!"

7

(6)

7

Here b equals 1.09, which is regarded as a constant depending on the structure of the crystal. Vs and Vl are the molar volume of the solid phase and liquid phase, respectively. Among them, Vs and Vl arefrom the experimental data or approximated by the composition weighted molar volume of the consituents. [6] In addition, element i upon melting.

3

= 3 ∙ : ∙ ;<(

> > =? @ > / @



3 is



3 denotes

the molar vibration entropy changes of

given by [6]

)

( 7)

Here : is the Gruneison parameter of element i that can be estimated either from experiments or by theoretical calculations.

and

are the molar volume of pure element i in both stable

structures and liquid phase. It is apparent that

can be obtained providing that the energy

balance equation (3) at the solid-liquid layer can be solved by reasonable modeling of the interface layer thermodynamics. As an example, we use (0, A, ) (0, A, ).B to model the solid-liquid interface sublattices of a ternary A-B-C system. Here, each set of parentheses represents one sublattice. The superscript s (or l) outside the parentheses denotes the interface solid (or liquid) sublattice, and the subscript f (or 1-f) outside the parentheses denotes the ratio of the interface solid (or liquid) sublattice. A comma is used to separate atoms in the same sublattice. Based on the nearest-neighbor bond approximation, [6, 30, 31] ∆&

!" can

be written as

∆& !" = C · · $ ∙ (1 − $) ∙ (E ∙ E ∙ F : + EH ∙ EH ∙ FH:H +E+ ∙ EI ∙ FI:I + E ∙ EH ∙ F :H + EH ∙ E ∙ FH: +E ∙ EI ∙ F :I +∙ EI ∙ E ∙ FI: + EI ∙ EH ∙ FI:H + EH ∙ EI ∙ FH:I (8) Where E

and E represent the site fractions of constituent i in liquid and solid interface

sublattice respectively and obey the following conditions, ∑ E

(J" )

=1

(9) 8

and $ ∙ E + (1 − $) ∙ E = 2

( 10 )

where 2 denotes the mole fraction of element i in the solid-liquid interface layer. The term F : denotes the enthalpy contrinution of the solid-liquid interface sublattices with full occupation of the same element i, in which the colon is used to separate the interface solid subalttice and the interface lquid sublattice. In the literature, F : has been correlated with the latent heat (or the fusion entropy ) of element i and its vibration entropy, [6] F: =( ∙

,

−

∙



3)

∙ KL/(C )

( 11 )

Here, R is the gas constant. Zi is a coefficient correlating with the interface atom coordination, which has been estimated to be 2 in the two-dimensional solid-liquid interface model.[6]

,

denotes the fusion entropy of element under undercooling state, which depends on the fusion entropy of element i, ,

=

,

+N

,

IO,@ BIO,=

, as well as the specific heats of solid and liquid, P

M,

and

M,

, [6] ( 12 )

To work with sublattice models it is also necessary to estimate the interface enthalpy contributions from the so-called fictive end-members, F ,Q , with the full occupations of element i at the interface solid sublattice and element j at the interface liquid sublattice. Here the wellknown enthalpy cycle of a dilute solution of element i in liquid j (shown in Figure 2) is used. It is typically broken into two steps. The first step is the enthalpy required for element i to change its state from a solid to a liquid, which can be quantified by its fusion entropy and vibration entropy using equation (11). The second step is the enthalpy changes associated with the dilution process of the molten i in a liquid of pure j liquid. Based on the enthalpy cycle, F :Q is estimated as

9

F :Q = R ∙ where ∆&^

,

−

>,Y

∆ST>@UVW X>@UVW IZ

,Q _ !` _ !

−

∙



\ 3[ ∙ ]

( 13 )

is the dilute solution enthalpy of liquid atom i in liquid j, which can be

obtained from the thermodynamic model parameters of the binary liquid phase using the CALPHAD technique. [32]

is regarded as a constant accounting for the coordination number

of element i in liquid j, which is approximated as 12. It has been well known that FCC and the hexagonal closest packed (HCP) has a coordination number of 12; The atom coordination number in BCC structure is 8; The atom coordination number of a simple cubic (SC) structure is 6. Consistent with the Thompson-Spaepen model,[19] we assume that the constituents of a solid phase have the same crystal structure in the derivation of the composition dependence of the mixing latent heat ∆Hmix. In other words, the crystal structure of pure elements does affect its latent heat. For example, for ξ phase (an Ag-based solution in the Al-Ag phase diagram with HCP structure), the mixing latent heat of ξ phase is approximated as the enthalpy difference between FCC phase and the liquid phase with the same composition under assumptions ,

,aII

≅

,

,SIc

and

,

g,aII

≅

,

g,SIc

,

. The terms

,aII

and

the fusion entropy of Al in FCC and HCP structure. While

,

,

,SIc

g,aII

respectively respresent

and

,

g,SIc

respectively

respresent the fusion entropy of Ag in FCC and HCP structure. For the Al-Si system, the equilibrium structures of the element Al and Si are FCC and Diamond, respectively. The mixing latent heat of Al-Si alloys is approximated as the enthalpy difference between the FCC phase (rather than the Diamond phase) and the liquid phase with the same composition under the assumption of

,

,`

J ^

≅

,aII

. [19] Here

,

,`

J ^

and

,

,aII

represent the fusion

10

entropy of Si in Diamond and FCC structure, respectively. Such simplifications are also applied for the other phases in the Al-based system. In the same formalism of the sublattice model in the CALPHAD technique,[26] the configuration entropy at the solid-liquid interface, · !" = −C ∙ E+h ;
!" ,

is given by

∙ ($ ∙ ;<($) + (1 − $) ∙ ;<(1 − $) + (1 − $) ∙ (E h ;
At the same time, the so-called excess interface energy, ∆

!*+! !"

, is described by sub-regular

solution model and the Redlich-Kister-Mugganiu polynomial [33] in the CALPHAD technique. It can be written as ∆

!*+! !"

/

= $ ∙ (1 − $) ∙ (E ∙ EH ∙ i; j,H + kE − EH l ∙ ; .,H + kE − EH l ∙ ; /,H + ⋯ n + E ∙ E+ /

∙ i; j,I + kE − EI l ∙ ; .,I + kE − E I l ∙ ; /,I + ⋯ n + EH E+ /

j . / ∙ i;H,I + kEH − EI l ∙ ;H,I + kEH − EI l ∙ ;H,I + ⋯ n + E ∙ EH

∙ k; j,H + (E − EH ) ∙ ; .,H + (E − EH )/ ∙ ; /,H + ⋯ l + EH ∙ E+ j . / ∙ k;H,I + (EH − EI ) ∙ ;H,I + (EH − EI )/ ∙ ;H,I + ⋯l j . / + E ∙ E+ k; ,I + (E − EI ) ∙ ; ,I + (E − EI ) · ; /,I + ⋯ l)

(15)

,o th where ; ,o ,Q (and ; ,Q ) denotes the k order the sub-regular solution model parameters of the bulk

liquid (and bulk solid) phases, which is usually temperature-dependent and optimized by the CALPHAD technique via coupling with phase diagram data. Here the atomic interaction parameters in the solid sublattice are approximated as those of the FCC phase in the phase diagram. This appears to be justified by the argument that the solid-liquid interface structure in the metallic alloy system tends to have a configuration with a high density of topological packing. [34, 35] While the atomic interaction parameters in the liquid sublattice are approximated as those of bulk liquid phase in the phase diagram. it is interesting to note that ( i ) a

“melt-back” process initially occurs at the solid-liquid interface. In other words, , the 11

formation of the solid-liquid interface layer is preceded by the dissolution of the solid in the liquid and followed by the the attachement of the liquid atoms ( with the dissolved solid atom) to the solid phase surface. .[36, 37] ( ii ) The liquid atom attachments or adjustements proceeds with constraints imposed by the solid crystal plane, [15] we, therefore, fix the total composition of the interface layer to be the same as the solid phase composition. This is consistent with the recent report by Lippman et al., [38] in which a reasonable

(0.46) for Al-Cu solution phase

can be obtained by assuming that the composition of solid and liquid at the solid-liquid interface are identical (equal to the overall concentration of the solid phase). In this case, a stable interface structure can be obtained by a constrained minimization of interface Gibbs energy ( ∆

5 4 !" )

subject to the following equality constraints: E + EH + EI = 1 , E + EH + EI = 1, 2 + 2H +

2+ = 1, $ ∙ E + (1 − $) ∙ E = 2 , $ ∙ EH + (1 − $) ∙ EH = 2H , and $ ∙ E+ + (1 − $) ∙ E+ = 2I ,

and the inequality constraints: 0 < E < 1, 0 < E < 1 , 0 < E < 1 , 0 < E < 1, 0 < E < 1, 0 < E < 1. It can be seen that if model parameters 2 , F : , F :Q , 0

!" and

: are known or

estimated from the thermodynamic and physical properties of the metallic system, the values of the unkown solid-liquid interface variabes f, E and E will be derived via minimization of interface Gibbs energy ( ∆

values are f0, E -

.

j

and E j , -

= $ j ∙ (E j ;
j

5 4 !" ).

.

and -

Suppose that the derived solid-liquid interface varaible /

in equation (3) will be written as

+ EHj ;
(16)

and -

/

= (1 − $ j ) ∙ (E j ;
2+ ;<2+ )

(17)

12

where 2 is the mole fraction of element i in the bulk liquid. The

of the metallic system can

be obatined consequently. We concentrate on the Al-based and Ag-based alloy systems to explore the solid-liquid interface thermodynamics and predict the corresponding

of the phases in the systems. A

major reason is that the solid-liquid interface thermodynamics of the alloy systems has been widely investigated in the literature. The input model parameters include the optimized thermodynamic model parameters of the bulk phases such such as the FCC phase and liquid phase, molar volume, heat capacity, and the Gruneison parameter of the different constituents in the alloy system. These model parameters are listed in Table 1 and Table 2 respectively. It needed to be pointed out that all model parameters without reference mark in Table 1 are from references [6, 21, 31]. The model parameters without reference mark in Table 2 are from reference [39].

3. Results and Discussions 3.1 rst of unary systems We first evaluate

of elements Al and Si at its melting points and undercooling states

considering that Al and Si are the most common constituents in many alloys. For pure Al, the measured experimental value of solid-liquid

by Turnbull et al. using the maximum

supercooling experiment (at ∆T = 130 K, ∆T =Tm-T) is 93 mJ/m2. [3] The derived

from the

maximum cooling technique by Kelton [4] (at ∆T = 175 K ) is 102 mJ/m2. In contrast,

of

pure Al at its melting point is 170.5 mJ/m2, which is extrapolated from the measured binary AlSi and Al-Cu binary solid-liquid energy using the grain boundary groove method.[7, 8] Moreover, the

of pure Al at the melting point is predicted as 172.6 mJ/m2 [40] by MD

13

simulations using the modified embedded-atom method, which is believed to offer a more reasonable estimation of

of pure Al. Figure 3a shows the calculated temperature-dependent

of pure Al in comparison with experimental data. Consistent with Jian et al., [6] the calculated

of Al at its melting point is in good agreement with the extrapolated value from the

experimental data [7, 8] and MD simulation result. [40] It is, however, larger than the values obtained from the supercooling experiment.[3, 4] This apparent discrepancy may reflect the high activity of Al with oxygen in the maximum cooling experiment, which tends to underestimate due to the possible heterogeneous nucleation. [6] The possible rational is that, due to the high affinity of Al with oxygen, some residue amounts of oxygen in the enviroment with react with Al melt, and resulting in aluinum oxide at the surface of the Al melt during measurements. The aluminum oxide will act as heterogenous nucelation sites and limit the accessible undercooling range. [41] Because of its high melting point,

of pure Si is often derived from the maximum cooling

experiment. The first measurement was carried out by Stiffler et al. [42] using a pulsed-laser melting method, which yields a deep cooling (∆T = 505 K) and

of 340±20 mJ/m2. Using the

maximum container-less undercooling technique (∆T = 420 K ),

of pure Si at the

undercooling state was estimated as 438 mJ/m2 by Li et al.[43] Considering that the containerless method is more effective in achieving near homogenous nucleation, the measured

of

pure Si by Li et al. [43] is therefore believed to be more accurate than that measured by Stiffler et al. [42]. In contrast, by examining the critical undercooling temperature for Si from lateral growth to intermediary growth and from intermediary growth to continuous growth, Jian et al. [21] estimated

of pure Si at 1178 K as 446.5 mJ/m2 and 448.9 mJ/m2 respectively.

Furthermore,

of pure Si at 1583 K and 1473 K was estimated as 683.7mJ/m2 and 686.9

14

mJ/m2, respectively. Using the grain boundary method, Gündüz et al. [8] measured

of Si in

contact with eutectic Al-Si liquid at 850.1 K, which is 352.41±38.41 J/m2. Based on the measured value by grain boundary method,

of pure Si at the eutectic temperature of 850.1 K

was extrapolated by Jian et al. [21] to be 316.8 mJ/m2. This value is close to 302 mJ/m2 estimated from the critical undercooling for silicon from lateral growth to intermediary growth and 304.1 mJ/m2 derived from intermediary growth to continuous growth.[21] In contrast with metallic elements, Si has two unusual characteristics. one is that the molar volume of the diamond structured Si ( melting point (

) is much larger than that of the liquid phase around the

), which had been assessed by Liu et al recently.[44] Hence, the vibration

entropy of Si at the solid-liquid interface can be rewritten as:[45]

3

=3∙:

Here :

∙ ;<(

u> u> u ? @ u> / u

)

( 18 )

denotes the Gruneisen parameter of the diamond structured Si. A recent inelastic

neutron scattering of silicon powder yielded a mean isobaric Gruneisen parameter of 6.95±0.67 in contrast with the isothermal Gruneisen parameter of 0.98. Furthermore, the mode Gruneisen parameter of Si would have a value of around 4.4 when considering the phonon anharmonicity. [46] A well-fitting the

of pure Si at different temperatures yield an optimized value of :

(4.85) that is close to the measured mode Gruneisen parameter of Si (4.4). A comparison of the calculation results with the experimental values is shown in Figure 3b. It is noticeable that our calculation results, on the whole, are in reasonable agreement with the estimated

of pure Si

by Jian et al. [21] and that derived using the data by Gündüz et al.,[8] but somewhat larger than those measured by the maximum cooling method, [42, 43] which are usually regarded as the lower bound of the solid-liquid energy of materials. 15

3.2 rst of binary and ternary systems Using the same set of model parameters for the solid-liquid interface energy of pure Al and Si, we predict

between silicon and binary silicon-aluminum melt. Without using adjustable

model parameters, we first calculate

between solid Si and silicon-aluminum melt at liquidus

of Al-Si phase diagram. The calculation results are shown in Figure 3c. In the figure, the black line denotes our calculation results. The blue line represents those derived by Jian et al. [29] based on the critical undercooling temperatures for Si growth from lateral mode to intermediary mode. The pink line represents those derived by the same author [29] based on the critical undercooling temperature for Si growth from intermediary mode to continuous mode. It can be seen that our calculated solid-liquid interface energies of Si at liquidus of Al-Si phase diagram are in a reasonable agreement with that predicted by Jian et al. [29] when the mole fraction of silicon in Si-Al melt is in the range from 0.12 to 0.7, but show some deviations when the composition of Si of Al-Si melts is above 0.7. In addition, we calculate

between Si and the

undercooled Si-Al melt of different compositions at the eutectic temperature of the Al -Si binary system (850.1 K). The results are shown in Figure 3d. In the figure, the black line represents our calculation results. The open square with error bars denotes that obtained from the grain boundary groove method. [8] While the black squares denote that derived from the critical undercooling technique by Jian et al. [21] Based on the critical cooling temperatures for Si with different growth modes in Si-20 at.%, The blue line with error bars denotes the calculation results by Jian et al. [29] It is found that our calculated

between Si and silicon-aluminum

melt at 850.1 K is about 6% lower than those predicted by Jian et al., [29] but is in a reasonable agreement with those measured by the grain boundary groove method.[8] It should be mentioned

16

again that the predicted

of binary Al-Si by our model is extrapolated from those of pure Al

and Si without any adjustable parameters. In the following, we demonstrate that our model can be easily extended to the ternary system to estimate

of both solutions and intermetallic compounds in the alloy system. To the best of

our knowledge, most of the empirical models on the solid-liquid interface thermodynamics are only valid for simple crystalline phases such as the FCC phase, few theoretical efforts are carried out to study the solid-liquid interface thermodynamics of intermetallic compounds and other phases in a self-consistent way. Table 3 shows the calculation results in comparison with the experimental data from different authors by the grain boundary grove method. In the Table, TAl2Cu and T-Al2Ag represent the eutectic Al2Cu and Al2Ag phases in the ternary Al-Cu-Ag systems. ξ represents an Ag-based solution in HCP structure in Al-Ag binary system. A Pearson correlation coefficient close to 0.9 (Pearson’s r shown in Figure 4) indicates that calculated



of the different phases, either in the form of solution or intermetallics, are in a reasonable agreement with the experimental data. We next show that our calculation results may rationalize the conflicts in experimental values in the alloy systems. First is the measured

of FCC phase (Al-16.42 at% Ag-4.97 at. %

Cu) at 775 K in ternary Al-Cu-Ag system, which varied from 67 ± 15 mJ/m2 [47] to 136 ± 17 mJ/m2.[48]. Engin et al. [49] noted that the Gibbs-Thomson coefficient, ΓGT, of the Al-Cu-Ag FCC solution derived by Bulla et al. [47] is about three times smaller than those obtained for the similar solid phase by themselves and others.[7, 8] As the Gibbs-Thomson coefficient ΓGT is one of the important parameters for the estimation of the solid-liquid interface energy in the grain boundary method (σsl = ΓGT ∙ ∆Sz f , where ∆Sz f is the entropy of fusion per unit volume), they speculated that the measured

by Bulla et al. [47] is doubtful. It is interesting to note that our 17

calculated

(108.2 mJ/m2) of FCC phase is more close to the calculation result by Lippman et

al. [25], which is within the lower bound of the experimental data derived by Keşlioğlu et al [48] and somewhat larger than the measured by Bulla et al. [47] Second is the scattered experimental data of

of T-Al2Ag in the ternary Al-Cu-Ag system from different authors [9,

47, 50] have a difference of about 420 %. Ocak et al.[50] stated that their measured

of T-

Al2Ag is more reliable than those estimated by Bulla et al. [47]. The reason is that the derived ∆ | for T-Al2Ag by Bulla et al. [47] is four times smaller than those by themselves. However, we found that ∆Sf used in the estimation of ∆ | by Ocak et al. [50, 51] is 45.32 J/K, which is much higher than those of similar solid phases in the Al-based and Ag-based systems[8, 47, 49], and beyond a magnitude range of the fusion entropy for most metallic compounds from R to 2R.[52] Therefore, we believe that the measured

values of T-Al2Ag by Bulla et al. [9, 47] are more

reliable instead. In fact, our calculated value of

(13.3 mJ/m2) for T-Al2Ag is more close to the

reported by Bulla et al. ( 28 ± 7mJ/m2) [47] and much smaller than those measured by Ocak et al. (145.9±17mJ/m2).[50] Now we examine the solid-liquid interface thermodynamic in metallic alloys. In the literature, it is generally believed that

in a metallic system originated from the entropy penalty with the

solid-liquid interface layer formation. [28] For close-packed pure metals, its

is linearly

correlated with the melting point because that its fusion entropy is constant (9.7 J/mol K) [5] For metallic alloys, Eustathopoulos [12] suggested that a temperature-dependent the fusion entropy of the solution phase have to be used to replace that of pure elements. A test carried out by Lippman et al. [25] indicated that mere adoption of the existing correlations of for the

of pure

elements cannot lead to the full prediction of

of the FCC phases in the Al-based system.

Here we analyzed different contributions to

of the metallic alloys using the solid-liquid 18

interface sublattice model. The results are shown in Figure 4b. In the upper panel of the figure, the excess mixing latent heat, ∆& !*+! * , of different solid phases are displayed vertically as columns. The composition of each solid phase is shown on the left of the columns. The minimized energies of the solid-liquid interface layer,

!" ,

of each solid phase are shown in

the middle panel of the figure, which is purposely sorted in order from the largest to the smallest. The corresponding

the solid phases are shown in the lower panel of the figure. It is

interesting to note that ( i ), the excess mixing latent heat did not show a clear correlation with its . (ii) In contrast, the orange and blue arrows in the figure indicate a rough correlation between and

!" .

It means that -

!" ,

the energy loss will significantly compensate the total

energy required for the solid-liquid interface formation. (iii) The abnormal increase of Al2Cu (with only 0.04 at. % Ag doped Al2Cu phase, in comparison that of shows the non-negligible effects of the liquid phase composition on

of T-

of binary Al2Cu )

of the multicomponent

alloys. It is tempting to ask whether the built interface sublattice model (shown in Figure 1a) may unravel the physical picture of the solid-liquid structure of the metallic systems. It is interesting to note that the advanced transmission microscopy had enabled direct observations of the interface liquid and interface solid at some types of solid/liquid interface.[53] Based on the experimental observation, a solid-liquid interface atomistic model, with the interface liquid and interface solid coexisting within an interfacial monoatomic layer, was used to describe the representative faceted chemically heterogeneous solid/liquid interface of Al(111)/Pb(liquid).[54] It is apparent the built solid-liquid sublattice model is quite similar to the experimental observation and the solid-liquid interface model in these two reference papers. At the same time, we assumed that the configuration of the interface solid sublattice for a solid-liquid interface of

19

the metallic alloy may be similar to that of the metastable phase with topologically close-packed structure. This has been partly confirmed by the characterized dynamic solid-liquid interface structure of Al in contact with the eutectic Al-Si by Arai et al. [55] using in-situ high-resolution transmission microscopy. It was found that hat the interface between crystal Si and eutectic Al-Si liquid is composed of a transient Si (1x1x1) layer rather than the one with the diamond structure in the Al-Si binary phase diagram. Furthermore, the strong preference of silicon was found experimentally at the interface between crystal Si and eutectic Al-Si liquid, which is consistent with our calculation results. This partly supported our assumption that the interface composition is dominated by the solid-liquid composition. Despite these, the atomistic and first principles simulations are expected to present more insights into the roles of the “chemical interaction” and the interface solid configurations in governing thermodynamics of the solid-liquid interface energy of the metallic alloy system in particular for that with BCC structure. Finally, it also needs to be mentioned that, in the present work,

was assumed to be isotropic

and the solid-liquid interfacial energy anisotropy is not considered in this work. This is because that little experimental data of the orientation-dependent

is

available in the literature.[31,

56] In addition, it should be noted that the fundamental problem with the sub-regular solution model used in this paper is that the model does not work well for the system with pronounced short-range ordering. It means that this model is not applicable to Al3Ni in the Al-Ni system owing to the pronounced short-range ordering at the Al-rich side of Al-Ni melts.[57] Further investigation is clearly required to clarify the roles of short-range ordering in the solid-liquid interface thermodynamics of the alloy system.

4. Conclusion 20

The sublattice model in the conventional thermodynamics picture has been extended to model the solid-liquid interface thermodynamics of a multicomponent alloy system, and to estimate its . It is shown that the experimental results of both solid solution and intermetallic phases in the Al-based and Ag-based system are reproduced without any adjustable parameters. The calculation results indicate that the established solid-liquid interface model will provide a selfconsistent framework for the calculation of

of metallic alloys ranging from pure to the high

order system, thereby unravelling the intricate thermodynamics relevant to the solid-liquid interface of the alloy system.

Acknowledgments The authors acknowledged the financial support from a research project titled “Integrated Digital Twin Platform for Additive Manufacturing” and grant A1898b0043 sponsored by the Agency for Science, Technology and Research, Singapore. Kewu Bai would like to thank useful discussions with Dr. Xie Qingge.

References [1] M. Asta, C. Beckermann, A. Karma, W. Kurz, R. Napolitano, M. Plapp, G. Purdy, M. Rappaz, R. Trivedi, Solidification microstructures and solid-state parallels: Recent developments, future directions, Acta Mater., 57 (2009) 941-971. [2] S.K. Eswaramoorthy, J.M. Howe, G. Muralidharan, In Situ Determination of the Nanoscale Chemistry and Behavior of Solid-Liquid Systems, Science, 318 (2007) 1437-1440. [3] D. Turnbull, R.E. Cech, Microscopic Observation of the Solidification of Small Metal Droplets, J Appl Phys., 21 (1950) 804-810. [4] K.F. Kelton, Crystal Nucleation in Liquids and Glasses, in: H. Ehrenreich, D. Turnbull (Eds.) Solid State Phys., Academic Press, 1991, pp. 75-177. 21

[5] B.B. Laird, The solid–liquid interfacial free energy of close-packed metals: Hard-spheres and the Turnbull coefficient, J. Chem. Phys., 115 (2001) 2887-2888. [6] Z. Jian, K. Kuribayashi, W. Jie, Solid-liquid Interface Energy of Metals at Melting Point and Undercooled State, Mater. Trans., 43 (2002) 721-726. [7] M. Gündüz, J.D. Hunt, Solid-liquid surface energy in the Al-Mg system, Acta Metall., 37 (1989) 1839-1845. [8] M. Gündüz, D.J. Hunt, The measurement of solid-liquid surface energies in the Al-Cu, Al-Si and Pb-Sn systems, Acta Metall., (1985). [9] A. Bulla, Measurement of the solid-liquid interface energy in ternary alloy, Ph.D Thesis, RWTH Aachen University, (2012) 134. [10] H. Jones, An Evaluation of Measurements of Solid/Liquid Interfacial Energies in Metallic Alloy Systems by the Groove Profile Method, Metall. Mater. Trans. A, 38 (2007) 1563-1569. [11] H. Jones, The solid–liquid interfacial energy of metals: calculations versus measurements, Mater Lett., 53 (2002) 364-366. [12] N. Eustathopoulos, Energetics of solid/liquid interfaces of metals and alloys, Int. Met. Rev., 28 (1983) 189-210. [13] L. Gránásy, M. Tegze, A. Ludwig, Solid-liquid interfacial free energy, Mater. Sci. Eng. A, 133 (1991) 577-580. [14] F. Spaepen, A structural model for the solid-liquid interface in monatomic systems, Acta Metall., 23 (1975) 729-743. [15] F. Spaepen, Homogeneous Nucleation and the Temperature Dependence of the Crystal-Melt Interfacial Tension, in: H. Ehrenreich, D. Turnbull (Eds.) Solid State Phys., Academic Press, 1994, pp. 1-32. [16] F. Spaepen, R. B. Meyer, The Surface Tension in a Structural Model for the Solid-Liquid Interface, Scr. Metall., 10 (1976) 257-263. [17] Y. Waseda, W. A. Miller, Calculation of the Crystal-Melt Interfacial Free Energy from Experimental Radial Distribution Function Data, Trans. Jpn. Inst. Met. , 19 (1978) 546-552. [18] D. Nason, W.A. Tiller, Solid/liquid interfacial free energies in binary systems, Surf. Sci., 40 (1973) 109-124. [19] C.V. Thompson, F. Spaepen, Homogeneous crystal nucleation in binary metallic melts, Acta Metall., 31 (1983) 2021-2027. [20] R.E. Smallman, A.H.W. Ngan, Chapter 3 - Solidification, in: R.E. Smallman, A.H.W. Ngan (Eds.) Modern Physical Metallurgy (Eighth Edition), Butterworth-Heinemann, Oxford, 2014, pp. 93-119. [21] Z. Jian, K. Kuribayashi, W. Jie, F. Chang, Solid–liquid interface energy of silicon, Acta Mater., 54 (2006) 3227-3232. [22] W.A. Miller, G.A. Chadwick, On the magnitude of the solid/liquid interfacial energy of pure metals and its relation to grain boundary melting, Acta Metall., 15 (1967) 607-614. [23] Z. Cong, D. Yong, A Novel Thermodynamic Model for Obtaining Solid–Liquid Interfacial Energies, Metall. Mater. Trans. A, 48 (2017) 5766-5770. [24] G. Kaptay, On the solid/liquid interfacial energies of metals and alloys, J. Mater. Sci., 53 (2018) 3767-3784. [25] S. Lippmann, I.-H. Jung, M. Paliwal, M. Rettenmayr, Modelling temperature and concentration dependent solid/liquid interfacial energies, Philos. Mag., 96 (2016) 1-14.

22

[26] J.O. Andersson, A.F. Guillermet, M. Hillert, B. Jansson, B. Sundman, A compound-energy model of ordering in a phase with sites of different coordination numbers, Acta Metall., 34 (1986) 437-445. [27] K.A. Jackson, Liquid Metals and Solidification ASM, Cleveland-Ohio, 1958. [28] R.H. Ewing, The free energy of the crystal-melt interface from the radial distribution function, J. Cryst. Growth, 11 (1971) 221-224. [29] Z. Jian, X. Yang, F. Chang, W. Jie, Solid-Liquid Interface Energy between Silicon Crystal and Silicon-Aluminum Melt, Metall. Mater. Trans. A, 41 (2010) 1826-1835. [30] L. Gránásy, M. Tegze, Crystal-Melt Interfacial Free Energy of Elements and Alloys, Mater. Sci. Forum, 77 (1991) 243-256. [31] Z. Jian, N. Li, M. Zhu, J. Chen, F. Chang, W. Jie, Temperature dependence of the crystal– melt interfacial energy of metals, Acta Mater., 60 (2012) 3590-3603. [32] T. Uesugi, S. Miyamae, K. Higashi, Enthalpies of Solution in Ti-X (X = Mo, Nb, V and W) Alloys from First-Principles Calculations, Mater. Trans., 54 (2013) 484-492. [33] COST 507: Thermochemical database for light metal alloys in: I. Ansara, A.T. Dinsdale, M.H. Rand (Eds.), Luxembourg: Office for Official Publications of the European Communities, 1998. [34] D.M. Herlach, P. Galenko, D. Holland-Moritz, Chapter 4 Solid-liquid interface, in: D.M. Herlach, P. Galenko, D. Holland-Moritz (Eds.) Pergamon Materials Series, Pergamon, 2007, pp. 39-40. [35] D. Holland-Moritz, Short-range order and solid–liquid interfaces in undercooled metallic melts, Mater. Sci. Eng. A, 304-306 (2001) 108-113. [36] A. Mahata, M. Asle Zaeem, M. I. Baskes, Understanding Homogeneous Nucleation in Solidification of Aluminum by Large Scale Molecular Dynamics Simulations, Model. Simul. Mater. Sc., 26 (2018) 025007. [37] N. Eustathopoulos, Wetting by Liquid Metals—Application in Materials Processing: The Contribution of the Grenoble Group, Metals, 5 (2015) 350-370. [38] S. Lippmann, C. Simon, S. Zechel, M. Seyring, U.S. Schubert, G. Wilde, M. Rettenmayr, Determining solid/liquid interfacial energies in Al-Cu by curvature controlled melting point depression, Acta Mater., 147 (2018). [39] M.W. Chase, Jr., NIST-JANAF Themochemical Tables, Fourth Edition, J. Phys. Chem. Ref. Data, Monograph 9, (1998) 1-151. [40] E. Asadi, M. Asle Zaeem, S. Nouranian, M.I. Baskes, Two-phase solid–liquid coexistence of Ni, Cu, and Al by molecular dynamics simulations using the modified embedded-atom method, Acta Mater., 86 (2015) 169-181. [41] D.M. Herlach, Solidification of Containerless Undercooled Melts, in: D.M. Herlach, D.M. Matson (Eds.) Containless Cooling of Dops and Droplets, Wiley, 2012, pp. 2. [42] S.R. Stiffler, M.O. Thompson, P.S. Peercy, Supercooling and Nucleation of Silicon after Laser Melting, Phys. Rev. Lett., 60 (1988) 2519-2522. [43] D. Li, D. Herlach, High undercooling of bulk molten silicon by containerless processing, Europhys. Lett., 34 (2007) 423. [44] X. Liu, K. Oikawa, Assessment of Temperature and Pressure Dependence of Molar Volume and Phase Diagrams of Binary A-Si Systems, Mater. Trans., 55 (2014) 1673-1682. [45] T. Ujihara, G. Sazaki, K. Fujiwara, N. Usami, K. Nakajima, Physical model for the evaluation of solid–liquid interfacial tension in silicon, J Appl Phys., 90 (2001) 750-755.

23

[46] D.S. Kim, H.L. Smith, J.L. Niedziela, C.W. Li, D.L. Abernathy, B. Fultz, Phonon anharmonicity in silicon from 100 to 1500 K, Phys. Rev. B, 91 (2015) 014307. [47] A. Bulla, C. Carreno-Bodensiek, B. Pustal, R. Berger, A. Bührig-Polaczek, A. Ludwig, Determination of the Solid-Liquid Interface Energy in the Al-Cu-Ag System, Metall. Mater. Trans. A, 38 (2007) 1956-1964. [48] K. Keşlioğlu, Y. Ocak, S. Aksöz, N. Maraşlı, E. Çadırlı, H. Kaya, Determination of interfacial energies for solid al solution in equilibrium with Al-Cu-Ag liquid, Met Mater Int, 16 (2010) 51-59. [49] S. Engin, U. Böyük, N. Maraşlı, Determination of interfacial energies in the Al–Ag and Sn– Ag alloys by using Bridgman type solidification apparatus, J. Alloys Compd., 488 (2009) 138-143. [50] Y. Ocak, S. Aksöz, K. Keşlioğlu, N. Maraşlı, E. Çadırlı, H. Kaya, Interfacial energies of solid CuAl2 in the CuAl2–Ag2Al pseudo binary alloy, Thin Solid Films, 518 (2010) 43224327. [51] Y. Ocak, S. Akbulut, K. Keslioglu, N. Maraşlı, E. Çadırlı, H. Kaya, Experimental determination of interfacial energies for Ag 2 Al solid solution in the CuAl 2 —Ag 2 Al system, Chin. Phys. B, 18 (2009) 3952. [52] A. Takeuchi, K. Amiya, T. Wada, K. Yubuta, W. Zhang, A. Makino, Entropies in Alloy Design for High-Entropy and Bulk Glassy Alloys, Entropy, 15 (2013) 3810. [53] S.H. Oh, M.F. Chisholm, Y. Kauffmann, W.D. Kaplan, W. Luo, M. Rühle, C. Scheu, Oscillatory Mass Transport in Vapor-Liquid-Solid Growth of Sapphire Nanowires, Science, 330 (2010) 489. [54] H. Liang, B.B. Laird, M. Asta, Y. Yang, In-plane characterization of structural and thermodynamic properties for steps at faceted chemically heterogeneous solid/liquid interfaces, Acta Mater., 143 (2018) 329-337. [55] S. Arai, S. Tsukimoto, S. Muto, H. Saka, Direct Observation of the Atomic Structure in a Solid-Liquid Interface, Microsc. Microanal., 6 (2000) 358-361. [56] J.J. Hoyt, M. Asta, A. Karma, Method for Computing the Anisotropy of the Solid-Liquid Interfacial Free Energy, Phys. Rev. Lett., 86 (2001) 5530-5533. [57] Ivan Egry, Louis Hennet, Mirko Kehr, Gerhard Mathiak, Simone De Panfilis, Irina Pozdnyakova, D. Zanghi, Chemical short-range order in liquid Al–Ni alloys, J. Chem. Phys., 129 (2008) 064508. [58] L.A. Girafalco, Statistical Mechanics of Solids, in, Oxford University Press, 2000, pp. 140. [59] J. Gröbner, H.L. Lukas, F. Aldinger, Thermodynamic calculation of the ternary system AlSi-C, Calphad, 20 (1996) 247-254. [60] H. Ohtani, M. Yamano, M. Hasebe, Thermodynamic analysis of the Co–Al–C and Ni–Al–C systems by incorporating ab initio energetic calculations into the CALPHAD approach, Calphad, 28 (2004) 177-190. [61] S.S. Lim, P.L. Rossiter, J.E. Tibballs, Assessment of the Al-Ag binary phase diagram, Calphad, 19 (1995) 131-141. [62] V. Witusiewicz, U. Hecht, S. G. Fries, S. Rex, The Ag–Al–Cu system, J. Alloys Compd., 385 (2004) 133-143. [63] N. Marasli, D.J. Hunt, Solid-liquid surface energies in the Al-CuAl2,, Al-NiAl3, and Al-Ti sytems, Acta Mater., 44 (1996) 1085-1096.

24

Captions of Figures

Figure 1 (a) Schematic of the solid-liquid interface structure of a metallic system, in which the solid-liquid interface is composed of the solid sublattice and liquid interface sublattice and sandwiched by bulk solid and liquid phases. The blue and yellow circles represent the atoms in the solid sublattice and liquid sublattice of the solid-liquid interface, respectively. The silvercolored cycles and the blank cycles denote the atoms in the bulk solid and liquid phase, respectively. (b) The energy balance existing at the solid-liquid interface of a multicomponent metallic system. .

Figure 2. the enthalpy cycle of element i dilute solution in liquid j. It breaks down into the melting process of element i and the subsequent dilution process of the molten i in a solution of pure j liquid. The atoms in light grey color and dark grey color denote the atom i and atom j respectively. Figure 3. Assessment of of pure elements. (a) Calculated of pure Al at different temperatures in comparison with experimental data. (b) Calculated of pure Si at different temperatures in comparison with experimental data. (c) Calculated of pure Si in contact with Al-Si melts at liquidus of Al-Si phase diagram in comparison with calculation results in the literature. The blue and pink lines are the prediction results by Jian et al. [29] (d) Calculated of pure Si in contact with the undercooled Al-Si melts at 850.1 K in comparison with experimental data. The blue line with errors bars denotes the calculation results by Jian et al. [29]. The blank square with error bar and black square represent the experiment data by grain boundary groove [8] and critical cooling method [21], respectively.

of the solid solutions and intermetallic compounds in Al-based and Ag-based Figure 4. metallic systems. (a) Experimental vs predicted solid-liquid interface energy σsl of different phases at various temperatures showing overall agreement. (b) Column chart showing the 25

complex solid-liquid interface thermodynamics of different phases in the alloy system The upper panel shows the calculated ∆& !*+! of different phases. The composition of each phase is shown * on the left of each column. The middle panel shows the calculated ∆ 45 !" of the solid-liquid interface layer for each phase. The lower panel shows the calculated of each phase in the Albased and Ag-based alloy systems. The numerical numbers on the top of different columns are the calculated σsl of different phases. Here all the column data are sorted by the values of ∆ 45 !" in order of the smallest value to the largest. The magenta and blue arrows in the figure indicate a rough correlation between the solid-liquid interface energy and ∆ 45 !" . T-Al2Cu and T-Al2Ag at the bottom of the figure represent the eutectic Al2Cu and Al2Ag phases in the ternary Al-Cu-Ag systems. ξ represents the Ag-based solution in the HCP structure in the Al-Ag binary system

Table 1. Thermo-chemical property parameters of element aluminum, silicon, magnesium, nickel, copper, and silver.

Table 2. Thermodynamic model parameters used in the calculations.

of phases in Al and Ag-based systems in comparison with experimental Table 3. Calculated data. ξ specifically represents an Ag-based solution in HCP structure. T-Al2Cu and T-Al2Ag represent the eutectic Al2Cu and Al2Ag phases in the ternary Al-Cu-Ag system.

26

Table 1. Thermo-chemical property parameters of element aluminum, silicon, magnesium, nickel, copper, and silver. Element Aluminum Silicon Magnesium Nickel Copper Silver

Gruneisen parameters : 4.84 4.4[46] 1.71[58] 2.01 1.96 2.40

6 l ×10 3 -1

(m mol ) 11.41 [44] 15.70 7.93 7.91 11.54

×106 (m3 mol-1) 10.59 [44] 13.98 7.56 7.61 11.16

,

-1

-1)

(Jmol K 11.45 27.63[21] 9.43 10.22 9.59 9.16

Melting point (K) 933.47 1687.00 923.00 1728.00 1357.77 1234.93

Table 2. Thermodynamic model parameters used in the calculations. Element

Aluminum Silicon

The heat capacity of liquid Cl (J mol -1 K-1) 31.80 27.196

Magnesium

34.39

Nickel Copper Silver

39.30 33.30 230.56

The heat capacity of solid Cs (J mol -1 K-1) 20.68+5.18×10-3 T 22.81719-3.89951×10-3 T0.082885×10-6 T2 +0.42×10-9 T3-3.5406×105/T2 26.54083-1.5333×10-3 T8.0624×10-6 T2 +0.57×10-9 T31.74×105/T2 15.08+7.52×10-3 T 22.65+6.28×10-3 T 21.31+8.54×10-3 T+1.51×105T-2

Binary and Ternary

Al-Si[59]

Al-Mg[33]

Al-Cu[33] Al-Ag[61]

Al-Ni[60] Cu-Ag[62]

27

Table 3 Calculated σsl in Al and Ag-based systems in comparison with the experimental data. ξ specifically represents an Ag-based solution in HCP structure. T-Al2Cu and TAl2Ag represent the eutectic Al2Cu and Al2Ag phases in the ternary Al-Cu-Ag systems System

Phase

σsl (mJ m-2)

Temperature (K)

Experiment Al-Ag

Al-Si

Al-Ni Al-Cu

Al-Mg Al-Cu-Ag

ξ (Ag-42 at. % Al) LIQUID (Al- 37.5 at.% Ag) FCC (Al–23.8 at. % Ag) LIQUID (Al- 37.5 at.% Ag) FCC (Al–1.6 at. % Si) LIQUID (Al-12.1 at.% Si) DIMOND (100 at% Si) LIQUID (Al-12.1 at.% Si) FCC (Al–0.023 at. % Ni) LIQUID (Al–3.06 at.% Ni) FCC (Al–2.5 at. % Cu) Ed p

839

64.7±8.4 [49]

Current Calculations 80.0

839

166.3±21.6[49]

140.8

851

168.95 ± 21.96 [8]

201.5

851

352.41±38.41[8, 46]

318.2

913

174.6±21 [63]

213.3

821

200.9

Al2Cu (Al-32 at. % Cu) LIQUID (Al-17.3 at.% Cu)

821

FCC (Al- 18.9 at. % Mg) LIQUID (Al-37.4 at. Mg) FCC (Al-16.42 at% Ag-4.97 at. % Cu) LIQUID(Al-16.57 at% Ag11.87 at.% Cu) T-Al2Cu (Al-32.31 at. %Cu0.04 at. % Ag) LIQUID(Al-16.57 at% Ag11.87 % Cu)

723

160.1±19 [63] 163.40±21.24 [8] 190±48 [9] 87.78±11.41 [8] 88.36±10.60 [63] 80±15 [9] 149.2±19.4 [7]

775

67 ± 15 [47] 136 ± 17 [48] 51± 8 [9]

108.2

775

96±17[47] 106±17 [9]

109.7

T-Al2Cu (Al-32.31 at. %Cu0.04 at. % Ag) LIQUID(Al2Cu-23.6 Ag2Al) T-Al2Ag (Al-38.60 at. %Ag– 2.93 at. % Cu) LIQUID(Al-16.57 at% Ag11.87 % Cu)

799

86.9±10.4 [50]

775

28±7 [47] 59±11[9]

799

145.9±17[50]

T-Al2Ag (Al-38.60 at. %Ag– 2.93 at. % Cu) LIQUID(Al2Cu-23.6 Ag2Al)

78.9

146.4

13.4

28

(a)

(b) Figure 1 29

Figure 2

30

(a)

(b) Figure 3

31

(c)

(d) Figure 3

32

(a)

(b) Figure 4

33

Highlights: The sublattice model in the conventional thermodynamics is extended to model the solid-liquid interface thermodynamics of the metallic alloys. The model enables a well-defined solid-liquid interface enthalpy and entropy hence the Gibbs energy model of the solid-liquid interface layer thereby providing a more compelling solution to the long-standing issue associated with the modeling of the solid-liquid interface thermodynamics in the multicomponent metallic systems. The built interface sublattice model offers a self-consistent framework for the automatic prediction of the solid-liquid interface energy of the metallic alloys ranging from the binary to a multicomponent system.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☒The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Kewu Bai

Institute of High Performance Computing Agency for Science, Technology, and Research (A*STAR) 1 Fusionopolis Way, #16-16 Connexis North Tower, Singapore 138632 Phone: +65 6419 1565 Email: [email protected]