Partial pair correlation functions and viscosity of liquid Al–Si hypoeutectic alloys via...

$: Partial pair correlation functions and viscosity of liquid Al–Si hypoeutectic alloys via high-energy X-ray diffraction experiments$
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Partial pair correlation functions andviscosity of liquid Al–Si hypoeutecticalloys via high-energy X-ray diffractionexperimentsPrakash Srirangam a , Manickaraj Jeyakumar a , Mathew J. Kramerb & Sumanth Shankar aa Light Metal Casting Research Centre (LMCRC), Department ofMechanical Engineering , McMaster University , Hamilton, ON,Canada, L8S 4L7b Ames Laboratory, Iowa State University , Ames, IA 50011, USAPublished online: 08 Aug 2011.

To cite this article: Prakash Srirangam , Manickaraj Jeyakumar , Mathew J. Kramer & SumanthShankar (2011) Partial pair correlation functions and viscosity of liquid Al–Si hypoeutectic alloysvia high-energy X-ray diffraction experiments, Philosophical Magazine, 91:30, 3867-3904, DOI:10.1080/14786435.2011.597360

To link to this article: http://dx.doi.org/10.1080/14786435.2011.597360

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Philosophical MagazineVol. 91, No. 30, 21 October 2011, 3867–3904

Partial pair correlation functions and viscosity of liquid Al–Si

hypoeutectic alloys via high-energy X-ray diffraction experiments

Prakash Srirangama, Manickaraj Jeyakumara, Mathew J. Kramerb andSumanth Shankara*

aLight Metal Casting Research Centre (LMCRC), Department of MechanicalEngineering, McMaster University, Hamilton, ON, Canada, L8S 4L7;

bAmes Laboratory, Iowa State University, Ames, IA 50011, USA

(Received 15 August 2010; final version received 10 June 2011)

The liquid structure of Al–Si hypoeutectic binary alloys was characterizedby diffraction experiments using a high-energy X-ray (synchrotron) beamsource. The diffraction experiments were carried out for liquid pure Al, Al–3wt% Si, Al–7wt% Si, Al–10wt% Si and Al–12.5 wt% Si alloys at severaltemperatures. The salient structure information such as structure factor(SF), pair distribution function (PDF), radial distribution function (RDF),coordination number (CN) and atomic packing densities (PD) werequantified as a function of Si concentration and melt temperatures.Reverse Monte Carlo (RMC) analysis was carried out using the diffractionexperimental data to quantify the partial pair correlation functions, such aspartial structure factor, partial pair distribution function (PPDF) andpartial radial distribution function. Furthermore, the partial pair distribu-tion function and the liquid atomic structure information were used ina semi-empirical model to evaluate the viscosity of these liquid alloys atvarious melt temperatures. The results show that the viscosity determinedby semi-empirical methods using the atomic structure information is ingood agreement with the experimentally determined viscosity values.

Keywords: liquid structure; structure factor; pair distribution function;number density; coordination number; viscosity; diffraction

1. Introduction

Aluminum–silicon hypoeutectic alloys are extensively used in the casting ofdomestic, military, automotive and aerospace applications [1]. The high strength-to-weight ratio coupled with good castability has brought the Al–Si alloys to theforefront of commercial shaped casting applications [2]. In recent years, muchresearch has been devoted to understanding and controlling the microstructure ofthe solidified cast part to enhance mechanical and performance properties [3]. Therheological properties (flow behavior) of these alloys in the liquid state should permitprediction of the product quality and performance of the cast components, sincefluid flow during the final stages of solidification dictates the efficiency of liquidmetal feeding the inter-dendritic regions [4]. Understanding how the liquid metal

*Corresponding author. Email: [email protected]

ISSN 1478–6435 print/ISSN 1478–6443 online

� 2011 Taylor & Francis

DOI: 10.1080/14786435.2011.597360

http://www.informaworld.com

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feeds the inter-dendritic regions during solidification is critical to predict and controlthe microstructure of the cast component. Further, during the final stages of thesolidification, the liquid immediately ahead of the dendrites would have a neareutectic composition, which would be the final alloy to solidify in the inter-dendriticregion, thus affecting the evolution of the eutectic phases [5].

Recent experimental results [5–7] have shown that, contrary to popularassumption [8], the Al–Si hypoeutectic alloy liquid exhibits a non-Newtonian fluidflow behavior with a non-thixotropic characteristic [6]. Understanding the liquidstructure of Al–Si eutectic alloy is the key to understand the underlying mechanismof changes in the rheology of Al–Si alloy melt and should enable a better grasp of thesolidification characteristics influencing the morphology of the eutectic phases in thesolidified component.

There have been numerous studies to understand the nucleation and growthmechanisms of the eutectic phases in the solidified Al–Si alloys using transmissionelectron microscopy (TEM), scanning electron microscopy (SEM) and opticalmicroscopy techniques [4,5,9–12]. However, there has been a lack of research toquantify the liquid structure of Al–Si eutectic alloy. Bian et al. [13] performed X-raydiffraction (Mo source) experiments to investigate the structure of liquid Al–Sialloys using reflection geometry. The main drawback of the Mo source X-raydiffraction experiments is that the data is collected from the free surface of the meltand hence is unreliable owing to interference from the surface artifacts such as anoxide layer. Bian et al. [13] studied the liquid structure of Al–12.5wt% Si alloy attemperatures ranging from 898K to 1298K, but could not provide any informationon the pair correlation functions of individual pairs of Al–Al, Al–Si and Si–Siatoms as a function of melt temperature. Wang et al. [14] numerically investigatedthe liquid structure of Al88Si12 alloy using ab initio molecular dynamicsimulations in which they used low and erroneous number density values reportedby Bian et al. [15] (0.04807 atoms/A3 at 898K) for deriving the partial paircorrelation functions.

In the case of binary alloys, the need for three partial structure factors has beenemphasized by several authors in past experimental studies undertaken to under-stand the liquid structure [16–20]. Different methods have been used to separatethese individual partial structure factors, such as the use of three different types ofincident radiation (X-ray, neutrons and electrons), isotope enrichment, anomalousscattering techniques [21,22] and use of the concentration independence rule [17].Enderby et al. [19] derived three partial structure factors for Cu–Sn alloys by neutrondiffraction using isotope–substitution technique. Although this method was consid-ered to be the best for deriving partial pair correlations of individual atom pairs,it had two limitations: (1) the high cost of silicon isotopes; and (2) a lack ofavailability of stable isotopes for aluminum. Halder et al. [23] used the concentrationindependence rule for the first time on Ag–Sn alloys, wherein it was validated thatthe partial structure factors were independent of alloy composition. The threeindependent partial structure factors were derived based on this assumption and bychanging the concentration in a weighting factor equation. Following the work ofHalder et al., this method has been applied by several authors on a number of liquidbinary alloys [17]. In principle, partial interference functions vary with the change inconcentration, and the assumption of concentration independence was not satisfied

3868 P. Srirangam et al.

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for binary alloys such as liquid Cu–Te [17] and liquid Ce–Ni alloys [24]. Followingthis, Bellissent-Funel et al. [25] studied the structure of liquid eutectic Ag–Ge alloyby neutron diffraction to assess the validity of the concentration independence of thestructure factor (SF) and pair structure factor (PSF). They carried out structuralstudies using isotopic substitution as well as the concentration method and foundthat the partial structure factors derived from the concentration method were not ingood agreement with those derived from the more accurate isotopic substitutionmethod.

Considering the difficulties associated with deriving partial pair correlationsarising from the non-availability of isotopes or three sources of radiation, researchersdeveloped an alternative approach to derive partial correlation functions using X-rayand neutron diffraction experimental data along with reverse Monte Carlo (RMC)analysis. RMC analysis is a method of structural modeling based on experimentaldata [26–30]. This method was adopted by Keen and McGreevy [31] for vitreoussilica to generate structural models from X-ray diffraction experiment data using thereflection method. Petkov et al. [32] studied the atomic structures of liquid Sn, Geand Si by generating three-dimensional structure models using RMC analysis fittedwith XRD experimental data. They observed that the RMC results were in excellentagreement with the results of ab initio molecular dynamics simulations. The results ofPetkov et al. [32] demonstrated that the RMC method, which does not require theuse of the interatomic pair potential functions, produced results comparable to thoseobtained from ab initio molecular dynamic simulations, which require the use of theinteratomic potential functions. Recently, Gruner et al. [33] studied atomic clustersin Cu–Sn alloy using X-ray and neutron diffraction data coupled with RMC analysisand observed that the derived partial structure factors were comparable to thoseobtained by Enderby et al. from neutron diffraction experiments with isotopesubstitution [19]. Following this, Takeda et al. [34] studied the atomic structureof liquid Au–Si alloys by transmission method using a high-energy X-ray source,wherein they derived the partial pair correlation functions using RMC analysis andfitted the experimental data. This approach using X-ray and/or neutron data withRMC modeling eliminates the difficulties associated with the non-availability ofisotopes and three radiation sources to derive partial pair correlation functions.

In this paper, the liquid structure of Al–Si hypoeutectic alloys determined using ahigh-energy synchrotron radiation beam source at different melt temperatures abovethe liquidus temperature of the respective alloy is presented. The diffractionexperiments were performed by transmitting high-energy X-rays from a synchrotronbeam source through the bulk of liquid alloy sample. The results from scatteringexperiments were used in RMC modeling to derive the partial pair functions of Al–Al, Al–Si and Si–Si as a function of melt temperature. Furthermore, the viscosity ofthese alloys was determined by semi-empirical methods using liquid atomic structureinformation.

2. Brief theory on diffraction of liquid binary alloys

Knowledge of liquid atomic structure is essential for the advancement of condensedmatter in developing predictive models, in establishing structure-property

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relationships in metallurgy and materials sciences, and to understand phase

transitions and the fundamental properties of an alloy melt [17,20,35–37]. Liquid

structure is commonly determined by X-ray, neutron diffraction and X-ray

absorption fine spectroscopy (XAFS) methods [17,20,38,39]. Gingrich has presented

a literature review on the contributions by various investigators on the diffraction of

X-rays by liquid elements [40]. Later, several studies have been carried out on the

atomic structure of various liquid metals, non-metals and binary alloy systems by

X-ray and neutron diffraction studies as a function of melt temperature and alloy

composition [41–47].The important parameter used in the study of liquids and amorphous materials

is the pair distribution function (PDF) represented by g(r). The PDF corresponds

to the probability of finding another atom at a distance r from an atom at the origin

position (at the point r¼ 0) [17,20]. In a binary alloy, the origin atom and the atom at

a distance, r could be either of the two atom types in the melt. The PDF is a time-

averaged structure of overall atoms in the melt [48,49] and provides the magnitude

and distribution probability of any pair-wise correlation as a function of distance but

not their orientation dependence [20].The raw data from diffraction experiments collects information on the total

intensity, I, as a function of the scattering angle, �. After applying corrections [17]

for multiple scattering, Compton scattering and geometric corrections, the X-ray

intensity scattered coherently, Icoh (in electron units, eu) could be derived as a

function of the wave vector, Q, as:

Icoh ¼ h f2i þ h f i 2

Z 10

4�r 2 �ðrÞ � �o� � sinQr

Qrdr, ð1Þ

where

Q ¼4�sin�

�, ð2Þ

f 2� �¼ cif

2i þ cjf

2j

� �, h f i 2 ¼ cifi þ cjfj

� 2, ð3Þ

and

�ðrÞ ¼Xi

Xj

ci fi fj�ijðrÞ=h f i2: ð4Þ

In Equations (2)–(4), � is the wavelength of the incident beam, ci is the atomic

fraction of i-type atoms, fi is the atomic scattering factor of species i, �(r) is the

average number density function, �ij(r) is the number of i-type atoms found at a

radial distance r from j-type atom at the origin.From Equation (1), the SF, S(Q), could be evaluated [17] as represented in

Equation (5). S(Q) defines the structure of the liquid and is directly measured from

the diffraction experiments:

SðQÞ ¼ IcohðQÞ � h f2i þ h f i 2

� �=h f i2

� : ð5Þ


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The Fourier transformation of S(Q) provides the atomic PDF, g(r), as:

gðrÞ ¼ 1þ1

2� 2r�o

Z 10

Q½SðQÞ � 1�sinðQrÞdQ: ð6Þ

The PDF represents the time-averaged probability of atom distribution in thestructure of the liquid alloy and when the PDF is multiplied by the linear atomdensity in two-dimensional space, the radial distribution function (RDF) can beobtained as:

RDF ¼ 4�r2�0gðrÞ: ð7Þ

Figure 1 shows a schematic of the SF, PDF and RDF of a typical metallic liquidsystem along with the notations used in this publication: Q1, Q2, S(Q1) and S(Q2) inthe SF; r1, r2, g(r1) and g(r2) in the PDF; and r0, rmax and rmin in the RDFrepresenting the cut-off position, maximum probability position and outer positionof finding an atom in the first coordination shell from an atom placed at the origin,respectively.

The total number of atoms in the first coordination shell, CN was determined byfive different methods [17,50,51] as described below:

Method (1): If the quantity rg(r) is symmetrical about the first peak, the CN couldbe evaluated by

CN ¼ 2

Z r0max

r00

4�r�0½rgðrÞ�dr: ð8Þ

In Equation (8), r00 and r0max are the left hand edge and the maximum position of therg(r) curve, as shown by the shaded area in Figure 2a.

Method (2): The CN could be estimated as twice the area under the (4�r2�0g(r))(RDF) curve up to its first peak maximum position rmax, as in:

CN ¼ 2

Z rmax

r0

4�r2�0gðrÞdr, ð9Þ

where r0 and rmax are the left hand edge and the maximum peak positions of the firstpeak of the RDF curve, as shown by the shaded area in Figure 2b.

Method (3): The area under the first peak of the RDF curve subtracted by the areaunder the extrapolated edge of the second peak [50], as shown by the shaded area inFigure 2c.

Method (4): A method was based on a direct relationship between the firstcoordination number (CN) and the packing density (PD) of liquid metals asgraphically proposed by Cahoon [51] could be used to determine CN. The packingdensity (PD) of atoms in the first coordination shell of the liquid structure is the ratioof the total volume of the liquid to the volume occupied by the atoms in that liquidassuming that the atoms are hard spheres [17]. The expression to evaluate PD isgiven by:

PD ¼�

6�0�

3: ð10Þ


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In Equation (10), �0 is the number density and � is the hard sphere diameter [17].

For most metal systems, the hard sphere diameter could be evaluated from r1 in the

PDF as given by:

� ¼r1

1:145: ð11Þ

Q1 Q2 Q

S(Q2)

S(Q1)

S(Q

)

(a)

(b)

ro rmax rmin r

RD

F

(c)

rmin

g(r

)

g(r1)

r2

r1

r

g(r2)

Figure 1. Schematic representation of the (a) SF, S(Q), (b) PDF, g(r), and (c) RDF for atypical liquid metal.


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Method (5): The CN could be evaluated by the area under the first peak of the RDF

curve between r0, and rmin as presented by the expression in Equation (12) and the

shaded area in Figure 2d:

CN ¼

Z rmin

r0

4�r2�0gðrÞdr: ð12Þ

In a binary alloy with two components i and j, there are three atom pair

interactions, namely, i–i, i–j and j–j. The PDF obtained from the diffraction

experiments provided the total structure information of liquid metals and composed

of three partial pair distribution functions (PPDF); one for each type of atom pair

interaction. The PPDF presents a more detailed understanding of the liquid structure

of binary alloys [20,38,52,53]. In the case of a binary alloy with components i and j,

three PPDF i.e. gii(r), gij(r) and gjj(r) are required for a complete description of

liquid structure. The three PPDF could be evaluated by solving three versions of

Equation (13) simultaneously:

4�r2�0½ gijðrÞ � 1� ¼ rGijðrÞ ¼2

�

Z 10

Q½SijðQÞ � 1�sinðQrÞdQ, ð13Þ

Figure 2. Schematic representation of RDF for evaluating the coordination number byvarious methods.


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where Sij(Q), Gij(r) and gij(r) represent the partial structure factor (PSF), reducedpartial pair distribution function and PPDF, respectively.

The total structure factor S(Q) obtained from diffraction experiment can beexpressed in terms of three PSFs as represented by three versions of:

SðQÞ ¼WiiSiiðQÞ þWjjSjjðQÞ þ 2WijSijðQÞ, ð14Þ

where

Wij ¼ cicj fi fj= ci fi þ cj fj� �2

: ð15Þ

The separation of these individual structures is one of the most important subjects inthe structural study of liquids. As shown in Equation (14), the structure of a binaryalloys with components i and j is expressed by two like atom pairs (i–i and j–j) andone unlike atom pair (i–j). Equation (14) for determination of the PSFs can berewritten in a matrix form, as shown in Equations (16)–(18):

Aj j ¼ Wj j�1 STj j, ð16Þ

where

Aj j ¼

SiiðQÞSjjðQÞSijðQÞ

0@

1A, Wj j ¼

w11w12w13

w21w22w23

w31w32w33

0@

1A and STj j ¼

S1ðQÞS2ðQÞS3ðQÞ

0@

1A: ð17Þ

The CN for each of the three atom pairs could be evaluated by Equation (18) (bymethod (5) mentioned earlier in this section), wherein the definitions of r0, rmax andrmin are shown schematically in Figure 1:

ðCNÞij ¼ 4�

Z rmin

ro

�ijr2gijðrÞdr, ð18Þ

where �ij is the partial number density of the atom pair ij and evaluated by:

�ij ¼ �offiffiffiffiffiffifficicjp

ð19Þ

where �o is the total number density of the alloy at any given temperature.

3. RMC modeling

This section elaborates on the procedure used for the reverse Monte Carlo (RMC)analysis. Figure 3 represents the flow chart of the RMC analysis proceduredeveloped in this study. The RMC procedure has been divided into five steps and atotal of 10,000 Al–Si alloy atoms have been used in the analysis. The Al–12.5wt% Sialloy would be used as an example alloy to describe the various steps in the RMCanalysis procedure.

1200 Si atoms for the Al–12.5wt% Si alloy were randomly placed in a box andthe dimensions of the box were determined by the total number density of the Al–Sialloy. The Si configuration file was created using the RANDOM1 program.


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The remaining 8800 Al atoms were randomly placed in the box containing the initialSi configuration file using the ADDRAND1 program. This would give the Al–Siconfiguration file containing all the 10,000 Al and Si atoms together in a square boxof 57.02 A length per side for the Al–12.5wt% Si alloy temperature of 867K asdictated by the number density of the alloy (0.05393 atoms/A3).

Moved OutAl-Si Configuration

START

Type of atom: 1No. of Si atoms,

Total DensityRANDOM.exe

ADDRANDOM.exe

Si Configuration

Al-Si Configuration

RMCA.exe

Type of atom: 2No. of Al Atoms

100% move out for given cut offs?

No

RMCA.exe

χ2 =minimum

(Asymtotic)

Program FileF(Q) data file

Max. moves: 0.05Std. Dev.= 0.05

Program File

RMCA.exe

χ2 = minimum(Asymtotic)


A

A

RMCA.exe

χ2 =minimumAsymtotic


RMCA.exe

χ2 = minimum(Asymtotic)


RMCA.exe

χ2 =minimum

(Asymtotic)


RMCA.exe

χ2 =minimum

(Asymtotic)


STOP

yes

yes

yes

yes (24 hours)

yes

yes

yes

No

No

No

No

No

ST

EP

1S

TE

P 2

ST

EP

3

ST

EP

4

ST

EP

4

ST

EP

5

No

Figure 3. Flow chart representing the RMC analysis procedure in this study.


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A program file was created as shown in Figure 4 to carry out an RMC program2

on the Al–Si atom configuration obtained from Step 2. The aim of this procedurewas to move out all the atoms in the initial box such that any two atoms were at thefarthest possible distance without changing the number density of the alloy and thesize of the box containing the atoms. The coordination constraints presented inFigure 4 dictates that any Al–Al atom pair would be at a minimum distance of 2.35 Afrom each other, any Al–Si atom pair would be at a minimum distance of 2.4 A fromeach other and any Si–Si atom pair would be at a minimum distance of 2.4 A fromeach other. This value of minimum distances was typically about 7 to 9% of the cutoff distance in the PDF (r0 in Figure 1) as obtained from the diffraction experimentdata. This ensured that the initial condition of the atom arrangement in the box usedto fit the experiment data in the next step was without any atom clustering ortrapping. It should be noted that if this step was not carried out, many atoms wouldhave been found trapped and unable to move when the RMC program wasperformed on the initial box of atoms coupled with the experiment data. Such atomtrappings would be reflected by abnormal peaks before the first peak in therespective partial pair distribution functions (PPDF). Further, moving the Al and Siatoms from an initially larger value of r, rather than the cut-off value of r0 to fit anexperimental data with RMC proved to be more effective (with respect to simulationtime and definitions of the three PPDFs).

The output from Step 3 yielded a box of atoms with the favorable initialconfiguration, which was coupled with the experiment data for the reduced structurefactor, (S(Q) – 1) and the RMC modeling was carried out with an input program file,as shown in Figure 5. It must be noted that the PDF could also be used for input asexperiment data; however, the reduced structure factor has been used to avoid errorsthat could arise from the Fourier transformation used to obtain the PDF.Furthermore, our experience showed that the use of reduced structure factor yieldedmore repeatable and accurate results for the PPDF. Additionally, the weighting

! number density0.053928835! cut offs2.4 2.4 2.35! maximum moves0.5 0.5 0.5! nswap, swapfrac00! r spacing0.01! moveout option.true.! number of configurations to collect0! step for printing9000 1! Timelimit,stepforsaving300 3! No.of g(r), S(Q), F(Q), EXAFS expts0 0 0 0! no. of coordination constraints4

2 2 0.0 2.35 0 1.0 0.000011 1 0.0 2.4 0 1.0 0.000011 2 0.0 2.4 0 1.0 0.000012 1 0.0 2.4 0 1.0 0.00001

! no. of average coordination constraints0! no. of bvs constraints0! no. of triplet constraints0! whether to use a potential.false.

Figure 4. Initial Al–Si configuration for move out (Al–12.5wt% Si).


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function Wij (Equation (14)) varied with Q, whereas this function would have to beassumed constant if PDF had been used as the input data file in the RMC programinstead of the reduced structure factor. Traditionally, PDF has been used in theRMC procedure because most of the work in metallic alloys had been carried out inthe reflection mode with the Mo X-ray source and this would not yield a highlyrefined S(Q) data for use in the RMC program. In the RMC program shown inFigure 5, two parameters needed to be critically controlled for a successfulsimulation: the moveout and the standard deviation. The moveout values were inAngstrom units and represented the maximum mobility of the atoms in thesimulation box. The standard deviation represents the level of accuracy required forfitting the experiment data. Initially, the moveout and standard deviation valueswere set at 0.05 and 0.05, respectively. The RMC program was allowed to run atthese values until the chi-squared (�2) stabilized to a nearly constant value. Themoveout value was maintained at 0.05 and the standard deviation was lowered to0.03, 0.01 and 0.001 in subsequent iterations of the program. Each iteration wasstopped when the �2 reached a nearly constant value. At the standard deviation valueof 0.001, the moveout was reduced to 0.01 and the RMC program was again carriedout until the �2 reached a nearly constant value at which point the simulation wouldbe deemed complete. Typically it took about a day or two to complete one successfulrun of the RMC analysis procedure, as shown in Figure 3.

! number density0.053928835! cut offs222! maximum moves0.01 0.01 0.01! nswap, swapfrac00! r spacing1.0! moveout option.false.! number of configurations0

to collect! step for printing1000 1! Time limit, step for saving6000 10! No. of g(r), S(Q), F(Q),0 0 1 0

EXAFS exptsx594fq.fq

! Range of points used52311! Standard deviation100.03! whether to convolute.false.! whether to renormalise.true.! beta1! whether to offset.false.! nback1! bcoeff.0!.false.! no. of coordination0

constraints! no. of average coordination0

constraints! no. o f bvs constraints0! no. of triplet constraints0! whether to use a potential.false.

Figure 5. RMC program file to model the alloy with the experimental data (Al–12. 5wt% Si).


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The RMC simulations quantified the partial structure factor values, Sij(Q) for the

three types of atom pairs in the binary alloy. Each of these partial structure factor

values has been used as input into a program to evaluate the respective PPDF

by Fourier transformation (Equation (13)). The values of PPDF along with the

respective values of �ij have been used to evaluate the partial RDF for the three

Al–Al, Al–Si and Si–Si atom pairs and subsequently, the partial CN and partial PD

were also quantified.

4. Viscosity

The viscosity of liquid metals and alloys has been studied extensively by various

experiments [54–56] and theoretical models [17,20,57–62]. Quantified viscosity is

extensively used in studies of solidification, solidification simulations, metallic glass

formation and various other metallurgical applications [17,63]. The physical

properties of liquid metals could be evaluated from the atomic structure of liquids

by coupling experiment diffraction data with appropriate theories [17]. Rice and

Allnat (RA) [58] used the concept of pair potential and pair distribution functions

to evaluate the viscosity of liquid metals and binary alloys. Kitajima et al. [59]

extended the concept of pair potential and pair distribution function method to

evaluate the viscosity of Na–K binary alloy. Later, Bhuyan et al. [60] had calculated

the viscosity of Ag–In alloys using the RA theory. The main difficulty associated

with this approach was to ascertain the partial pair potential functions by molecular

dynamics simulations, which was cost prohibitive, and the number of atoms used

in such simulations were about 100 at most [64,65]. Further, obtaining valid

and reliable pair potential functions for binary metallic alloy at various melt

temperatures is still an ongoing research study. The following sections elaborate the

popular theories of evaluating viscosity from fundamental atom arrangement

information.

4.1. Born–Green theory

Born and Green [57] derived an expression for evaluating the viscosity

of liquid metals based on the PDF and pair interatomic potential function as

given by:

� ¼2�

15

m

kT

� �1=2�20

Z 10

gðrÞ@ðrÞ

@rr4dr: ð20Þ

Using Equation (20), the viscosity for various liquid metals have been evaluated and

it was found that the viscosity values match with the experimental data for most

metals at their melting point [20]. However, the temperature dependence of viscosity

could not be satisfactorily validated using the Born–Green theory owing to lack of

confidence in determining the pair potential functions by the simulation

techniques [17].


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4.2. Hard-sphere theory

The packing density (PD) of a liquid metal could be calculated from the hard-sphere

diameter and the viscosity of dense fluids could be expressed in terms of the packing

density as shown in [66]:

� ðPa sÞ ¼ 3:8� 10�8ðMTÞ

12

V23

ðPDÞ43 1� PD

2

� ð1� PDÞ3

, ð21Þ

where M is the molecular weight and V is the molar volume.This method has been shown to under-predict the shear viscosity of the alloy melt

by a factor of 0.3. This method was used to evaluate the shear viscosity of the alloys

in this study by assuming the alloy as one integral fluid with a specific PD.

4.3. Semi-empirical model

A semi-empirical method to evaluate viscosity of liquid metals had been proposed

by Iida et al. [20] by expanding the model proposed by Andrade [67] to obtain

an expression for melt viscosity using fundamental properties such as pair

distribution function and average inter-atomic potentials. Viscosity is a structure

sensitive property and microscopically viscosity has been related to the transfer of

momentum between the nearest-neighbor atoms in the liquids (Newtonian fluids)

[20]. The information about the nearest neighboring atoms could be readily obtained

from the pair distribution function, g(r) [20]. Iida et al. [20] proposed an equation to

evaluate the viscosity of Newtonian liquids using the concept of pair distribution

function g(r) as represented by:

� ¼8�

90PðTÞm�

20

Z 10

gðrÞr4dr: ð22Þ

In Equation (22), the upper limit of the integral could be changed to the distance

over which the nearest neighbor interactions occur in the liquid metals by assuming

that the momentum transfer occurs mostly between nearest neighboring atoms,

which is represented by the rmin in the g(r) curve, as shown in Figure 1b. In Equation

(22), P(T) is the probability function that the atom will stay in a state of oscillation

around a fixed coordinate position. Equation (22) could be rewritten as:

� ¼8�

90PðTÞm�

20

Z rmin

0

gðrÞr4dr, ð23Þ

where 0 is a constant and calculated from the Lindemann’s melting point atomic

frequency formula modified for metals by Iida et al. and given by:

0PðTÞ ¼ �L ¼ �cRTm

MV23m

!1=2

, ð24Þ


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where c is a constant (9.0� 108), P(T) is dependent on temperature and decreaseswith increasing temperature and evaluated by:

PðTÞ ¼ 1�

Z 1’

2�ð Þ1=2 exp �’2

2

� �d’, ð25Þ

and � is a correction factor, which is related to the surface tension (�m) and molarvolume (Vm) of the liquid at the melting point and evaluated by:

� ¼ 2:2� 103 V1=3m

� �mRTm

� �1=2

: ð26Þ

In Equation (25),

’ ¼

ffiffiffi3p

2

Tb � T

T

� �, ð27Þ

where Tb is the boiling point temperature of the metal.Based on Equation (22), Djemili et al. [68] proposed an equation for evaluating

the viscosity of binary alloys as shown in Equation (28). Using Equation (28), theviscosity of Cd-In binary alloy was evaluated and validated by viscosity experiments[68], i.e. via

� ¼8�

90�

20 x2i PðiÞm

� Z a

0

giiðrÞr4drþ x2j Pð j Þmj

�

Z b

0

gjjðrÞr4drþ xixj miPðiÞ þmjPð j Þ

� Z c

0

gijðrÞr4dr

�, ð28Þ

where P is the proportion of vibrating atom in the liquid and the upper limits ofthe integrals, as represented by a, b and c, correspond to the rmin values of gii(r), gij(r)and gjj(r), respectively.

In this study, the viscosities of Al–Si alloys were determined by the semi-empiricalmethod using Equation (28).

5. Experiments

Alloy samples were prepared from 99.999% purity Al ingots and electronic grade(99.9999% purity) elemental Si. All alloy compositions in this publication areexpressed as percent weight of each component in the alloy. Each alloy sample wasplaced in a clean alumina crucible, melted in an electric furnace and poured into acylindrical copper mold of 15mm diameter and 70mm height to form a castspecimen. A cylindrical sample of 1.5mm diameter and 10mm length was machinedout of the casting and placed in a transparent quartz tube of 2mm in diameter and10mm in length. The tube was subsequently evacuated and sealed in Ar gas tominimize the oxidation. The alloy chemistry was measured with the inductivelycoupled plasma (ICP) method and confirmed with the glow discharge opticalemission spectroscopy (GDOES). Thermal data during solidification was collectedfrom each alloy composition at a solidification rate of about 48�C/min to evaluatethe respective the liquidus temperatures. The alloy compositions and correspondingliquidus temperatures (TL) are presented in Table 1.


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X-ray diffraction experiments were carried out using high-energy synchrotronX-ray source at Sector 6ID-D, MUCAT (�-CAT) beam line, Advanced PhotonSource (APS), in Argonne National Laboratory (ANL), Argonne, IL, USA. Amonochromatic (98.099 KeV) beam (0.1264 A wavelength) had been chosen toprovide high penetration through the bulk and minimize multiple scattering. Silicondouble-crystal monochromators were employed to attain the required wavelength.The distance from the sample to the detector had been calibrated using NationalInstitute of Standards and Technology Si 640C standard providing an effectiverange of wave vector, Q, range of about 15 A�1. The Debye rings obtained asdiffraction data were integrated into a single one-dimensional plot of intensity andscattering angle using the FIT2D [69] software program, which also provided ageometric correction for the flat plate detector geometry.

The quartz tube with the alloy sample was placed vertically inside a furnacecapable of heating samples to a temperature of 1723K. The sample was heated to1023K to ensure complete melting. The diffraction data was collected at various melttemperatures ranging between 5K and 250K above the respective liquidustemperature for the Al–Si alloys. Table 1 shows the various liquid temperaturesfor the respective alloys for which diffraction experiments were carried out in thisstudy.

Ten scans of 60 s each were obtained at each melt temperature using a MARCCDdetector (41 cm� 41 cm CsI coated solid state detector manufactured by GE3). Thediffraction data was also obtained for the empty container (quartz tube) and theempty furnace for purposes of background subtraction. The sample data along withbackground and empty container data were analyzed using PDFgetX2� software[70] to obtain the SF, S(Q). The analysis included corrections for multiple scattering,Compton and geometric corrections to the raw intensity data to obtain the IcohðQÞ,S(Q) and g(r), respectively.

6. Results and discussion

The experimental results of the total structure factor and total pair correlationfunctions for liquid Al–Si hypoeutectic alloys were presented in another recent

Table 1. Alloy compositions and temperatures at which diffraction experiments were carriedout. The respective liquidus temperatures of the alloys are also shown.

Al(TL¼ 933K)

Al–3wt %Si (TL¼ 909K)



Al–12.5wt %Si (TL¼ 849K)

938K 920K 898K 885K 867K963K 942K 920K 907K 889K988K 964K 942K 929K 911K1013K 1052K 964K 973K 933K1038K 986K 995K 956K1063K 1008K 1017K 978K1088K 1030K 1039K 1000K1113K 1052K 1022K


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publication [71]. A few of these graphical results of the total structure factor andtotal pair correlation functions have been repeated here to ensure a continuity ofthought and discussion. Figure 6 shows the structure factor S(Q) for Al–12.5wt% Sialloy and is typical for all the alloy compositions. In Figure 6a, the plot at 867Ktemperature is the true evaluated data and the other temperature plots areprogressively incremented by a fixed integer for better clarity and visualization.Figure 6b represents the magnified image of Figure 6a showing the effect of melttemperature on the height and width of the first peak of S(Q). The height of the firstpeak decreased with increasing temperature from 867K to 1065K, as shown inFigure 6b. This is indicative of the loss of order in the atomic structure withincreasing temperature of the alloy melt [17].

Figure 7 shows the features of the first peak, S(Q1) of the SF plot, plotted as afunction of melt temperature for the various alloys. Figure 7a shows the height andFigure 7b shows the position of the first peaks. It was observed that the height of

0 2 4 6 8 10 12 140

2

4

6

8

10

Str

uctu

re fa

ctor

, S(Q

)

Q(A–1) Q(A–1)

867K

889K

911K

933K

956K

978K

1000K

1022K

1065K

2.4 2.6 2.8 3.01.6

1.8

2.0

2.2867K

1065K

S(Q

)

(a) (b)

Figure 6. Structure factor S(Q) vs. Q for (a) Al–12.5wt% Si; (b) magnified region of firstpeaks shown in (a) [71].

1.95

2.00

2.05

2.10

2.15

2.20

2.25

2.30

2.35

12.5%Si

7%Si

Firs

t pea

k he

ight

. S(Q

1)

Temperature (K)

0%Si3%Si

10%Si

850 900 950 1000 1050 1100 1150 850 900 950 1000 1050 1100 11502.655

2.660

2.665

2.670

2.675

2.680

2.685

Firs

t pea

k po

sitio

n, Q

1

Temperature (K)

3%Si 0%Si

10%Si

7%Si 12.5%Si

(a) (b)

Figure 7. Structure factor, S(Q), (a) first peak height, S(Q1) [71] and (b) first peak position,Q1, plotted as a function of melt temperature for Al–Si hypoeutectic alloys.


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S(Q1) decreased with increasing melt temperature and increased Si content,

independently. The level of atomic order in the liquid decreased with decreasing

peak heights; alternately, the atomic order in the liquid decreased with increasing

temperature and with increasing Si content in the alloy as well. In Figure 7b, the first

peak position of structure factor, Q1 moved to lower values of Q with increasing melt

temperatures for any specific alloy.In Figure 7b, large variations in Si content, for example from 0 to 12.5wt%,

show an observable increase in the position of the first peak. However, smaller

changes in Si content of about 3wt% did not show any appreciable changes in the

position of the first peak. Reasonably higher difference in Si levels of about 5wt% or

more would be required to observe a change in the position of the first peak of the

SF curve in these alloys. The first peak position Q1 for pure Al near the liquidus

temperature was 2.677 A�1, whereas for Al–12.5wt% Si it was 2.684 A�1. Similar

observations have been made for pure Al by previous researchers as well [17,20].

When analyzing the ratio (Q2/Q1) of the second peak position to the first peak

position of S(Q), it was observed that the ratio remained around that for pure Al

(1.81) and did not vary significantly from this value with increasing silicon content in

the alloy. The Q2/Q1 ratio falls between 1.8–1.9 for most metal systems which satisfy

the hard-sphere model with isotropic metallic bonding in the melt for determining

S(Q). Elements such as Si, Ge and Sb have anisotropic bonding and hence, largely

deviates from hard-sphere model reflected by a small hump on the right-hand side of

first peak position in the plot for the S(Q) and with a Q2/Q1 ratio significantly

different from 1.8. It is evident from the results for S(Q) in Figures 6 and 7 that

although the atomic structure of liquid silicon is different from that of the aluminum

melt, the addition of silicon did not show any anomalous behavior or anisotropic

bonding in the Al–Si hypoeutectic alloy melts. Further, there was no evidence of any

pre-peaks before the first peak of S(Q) in any of the S(Q) plots for the alloys

investigated in this study. The presence of a pre-peak before the first peak of S(Q) in

the low-Q region could represent the compound formation tendency in the liquid

alloy melt [15] and Figure 6a shows that there is no compound formation in the

Al–Si hypoeutectic alloys with 0wt% to 12.5wt% Si composition range.The PDF obtained by Fourier transformation of SF for Al–12.5wt% Si is shown

in Figure 8. Similar plots have been obtained for pair distribution function for all

alloy compositions by Fourier transformation of their respective SF functions using

the respective alloy number density evaluated and presented in Table 2. The number

density (�o) at each alloy temperature was evaluated based on the molar weight and

molar volume data obtained from the FactSage thermodynamic database software

[72,73]. An empirical equation was formulated using linear regression analysis

(R2¼ 0.99) to evaluate the number densities of Al–Si hypoeutectic alloys as a

function of melt superheating and Si concentration as shown in:

�0 ¼ �6� 10�6 T� TLð Þ þ 7:495� 10�3 CSið Þ þ 0:05315, ð29Þ

where �0 is in atoms per A3, CSi is in atom fraction; the terms T and TL represent the

alloy melt temperature and the liquidus temperature of the alloy in K, respectively.Figure 9 shows that the first peak height of g(r) decreased with increasing melt

temperature for all alloy compositions. The peak height was higher in the case of


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pure Al and decreased with increasing silicon content from 0 to 12.5wt% Si, asrepresented in Figure 9a. The first peak position of g(r) is shown in Figure 9b. Thevariation of r1 with melt temperature was not significant and almost constant withincreasing melt temperatures for a given Si level in the alloy. However, the effect ofSi content on r1 shown in Figure 9b is similar to the effect of Si content on Q1

presented in Figure 7b. Large variations in Si content, for example from 0 to

Table 2. Evaluated number density values for Al–Si hypoeutectic alloy compositions atvarious melt temperatures.

Alloy Temperature (K) �0 (atoms/A3) Alloy Temperature (K) �0 (atoms/A3)

938 0.05309 920 0.05332963 0.05293 942 0.05318988 0.05278 Al–3Si 964 0.05304

Al 1013 0.05262 1052 0.05251038 0.05246 898 0.053591063 0.05231 920 0.053451088 0.05215 942 0.053311113 0.052 964 0.05318867 0.05393 Al–7Si 986 0.05304889 0.05379 1008 0.05291911 0.05366 1030 0.05277933 0.05352 1052 0.05263

Al–12.5Si 956 0.05366 885 0.05376978 0.05325 907 0.053621000 0.05311 929 0.053491022 0.05298 Al–10Si 973 0.053221065 0.05272 995 0.05308

1017 0.052951039 0.05281

0

2

4

6

8

10P

air

dist

ribut

ion

func

tion,

g(r

)

r (A°) r (A°)

1065K

1022K

1000K

978K

956K

933K

911K

889K

867K

2 4 6 8 10 12 14 16 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.31.4

1.6

1.8

2.0

2.2

2.4

g(r)

1065K

867K

(a) (b)

Figure 8. Pair distribution function, g(r), for (a) Al–12.5 wt% Si; (b) magnified region of firstpeaks shown in (a) [71].


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12.5wt% show an observable increase in the value of r1. However, smaller changesin Si content of about 3wt% did not show any appreciable changes in the position ofthe first peak. Reasonably higher difference in Si levels of about 5wt% or morewould be required to observe a change in the position of the first peak of the g(r)curve in these alloys.

Figure 10 shows the radial distribution function (RDF) for pure Al and Al–Sihypoeutectic alloys at a temperature of 963K. The addition of silicon contentdecreased the peak height of RDF curve and moved the rmin position to lower rvalues, as shown in Figure 10. Figure 11 shows the variation of coordination number(CN) as a function of melt temperature for various Al–Si hypoeutectic alloys.

0 1 2 3 4 5 6 70

5

10

15

20

25

0% Si 3% Si 7% Si 10% Si 12.5% Si

RD

F

r (Å)

3.4 3.6 3.8 4.06.0

6.2

6.4

6.6

6.8

2.7 2.8 2.9 3.0 3.111.0

11.5

12.0

12.5

13.0

Figure 10. Radial distribution function (RDF) at 963K for various Al–Si hypoeutectic alloys.The change in rmax and rmin positions with increasing silicon content is shown in the insets.

2.10

2.15

2.20

2.25

2.30

2.35

2.40

2.45

12.5%Si

10%Si

7%Si

Firs

t pea

k he

ight

, g(r

) 1

Temperature (K)

0%Si3%Si

850 900 950 1000 1050 1100 1150 850 900 950 1000 1050 1100 11502.760

2.765

2.770

2.775

2.780

2.785

2.790

2.795

2.800

2.805

2.810

Firs

t pea

k po

sitio

n, r

1

Temperature (K)

3%Si

0%Si

10%Si

7%Si

12.5%Si

(a)(b)

Figure 9. The pair distribution function, g(r), (a) first peak height, g(r1), and (b) first peakposition, r1, as a function of melt temperature for Al–Si hypoeutectic alloys [71].


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The values of CN in Figure 11 were determined by integrating the area under the firstpeak of RDF between r0 and rmin (method (5) and Equation (12) in Section 2). It isevident from Figure 11 that the coordination number increased with decreasing melttemperature for the alloys. Increase in silicon content in the Al resulted in a decreaseof the coordination number at any given temperature. Further, Table 3 shows thevalues of CN estimated by various methods (as given in Section 2) for Al and Al–Sihypoeutectic alloy compositions as a function of melt temperature.

Figure 12 shows the CN evaluated by various methods for pure Al and Al–Sihypoeutectic alloy compositions. From Table 3, it is evident that there is no uniquemethod to determine the CN and its value depends significantly on the method usedto evaluate the same. Waseda [17] reported similar evaluations for the CN of liquidAg and liquid Au. An empirical equation based on linear regression analysis hasbeen formulated to enable evaluation of CN by the various methods described inSection 2 and shown in Equations (30)–(34) along with the respective R2 values of theregression analysis:

Method ð1Þ : CN ¼ �0:00185 T� TLð Þ � 1:308 CSið Þ þ 7:778 ðR2 ¼ 0:96Þ; ð30Þ




Method ð5Þ : CN ¼ �0:0017 T� TLð Þ � 2:4352 CSið Þ þ 11:5185 ðR2 ¼ 0:99Þ: ð34Þ

In Equations (30)–(34), CSi is in atom fraction; T and TL are in K. The CN value forpure Al as determined by method (5) at 938K was 11.51 and this value validated thevalue of 11.5 evaluated by Waseda [17].

850 900 950 1000 1050 1100 115010.8

10.9

11.0

11.1

11.2

11.3

11.4

11.5

11.6

12.5%Si

10%SiCoo

rdin

atio

n nu

mbe

r

Temperature (K)

0% Si3% Si

7%Si

Figure 11. Coordination number as a function of melt temperature for various Siconcentrations in the Al–Si hypoeutectic alloys.


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Figure 13 shows the packing density of Al–Si hypoeutectic alloys as a function ofmelt temperature. It is evident from Figure 13 that the packing density of any givenalloy composition decreased with increase in melt temperature. Further, at any givenmelt temperature, the packing density of the alloy melt decreased when comparingalloys with 0wt% Si and 12.5wt% Si. However, the change is not so significant forincrease in Si levels of about 3wt%.

Table 3. Evaluated coordination number (CN) values for Al–Si hypoeutectic alloycompositions at various melt temperatures.

Alloy(wt%) Temp (K)

CN(Method I)

CN(Method II)

CN(Method III)

CN(Method IV)

CN(Method V)

Al 938 7.76 8.74 10.37 7.08 11.51963 7.73 8.68 10.35 7.05 11.47988 7.69 8.64 10.30 7.03 11.421013 7.66 8.60 10.22 7.00 11.381038 7.63 8.56 10.16 6.96 11.341063 7.59 8.50 10.10 6.92 11.301088 7.56 8.46 10.08 6.88 11.251113 7.53 8.42 9.95 6.85 11.20

Al–3%Si 920 7.71 8.66 10.32 7.11 11.41942 7.68 8.62 10.26 7.06 11.37964 7.65 8.58 10.22 7.03 11.331052 7.52 8.42 9.85 7.00 11.21

Al–7%Si 898 7.67 8.61 10.29 7.03 11.31920 7.63 8.56 10.23 7.01 11.30942 7.61 8.52 10.18 6.98 11.27964 7.57 8.48 10.13 6.93 11.22986 7.55 8.44 10.09 6.89 11.181008 7.37 8.42 10.02 6.85 11.141030 7.35 8.40 9.98 6.80 11.101052 7.33 8.36 9.93 6.77 11.08

Al–10%Si 885 7.63 8.54 10.25 7.06 11.23907 7.60 8.50 10.21 7.02 11.19929 7.57 8.46 10.16 7.00 11.15973 7.52 8.42 10.11 6.97 11.14995 7.49 8.36 10.07 6.94 11.101017 7.46 8.32 10.03 6.92 11.071039 7.46 8.30 9.99 6.89 11.03

Al–12.5%Si 867 7.58 8.48 10.17 6.98 11.19889 7.54 8.44 10.12 6.95 11.14911 7.51 8.40 10.06 6.91 11.10933 7.48 8.36 10.02 6.88 11.06956 7.44 8.31 9.97 6.86 11.02978 7.41 8.27 9.89 6.82 10.981000 7.38 8.24 9.78 6.78 10.941022 7.34 8.19 9.74 6.76 10.901065 7.29 8.12 9.65 6.72 10.85


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It can be observed from Figures 5–13 that the disorder in the alloy meltsincreased with increase in melt temperatures for Al and Al–Si alloy melts. This wasrepresented by decrease in peak height of S(Q) and g(r), decrease in CN and PD.In the case of the Al–7wt% Si alloy, a sudden change in peak heights, peak positionsand PD was observed at 986K and this could result from the temperature-induced

6

7

8

9

10

11

12

Method I

Method II

Method III

Method IV

Coo

rdin

atio

n nu

mbe

r

Temperature (K)

Method V

6

7

8

9

10

11

12

Method I

Method II

Method III

Method IV

Coo

rdin

atio

n nu

mbe

r

Temperature (K)

Method V

(a) (b)

6

7

8

9

10

11

12

Method I

Method II

Method III

Method IV

Coo

rdin

atio

n nu

mbe

r

Temperature (K)

Method V

900 950 1000 1050 1100 1150 900 950 1000 1050 1100

850 900 950 1000 1050 1100

850 900 950 1000 1050 1100

850 900 950 1000 1050 1100

6

7

8

9

10

11

12

Method I

Method II

Method III

Method IV

Coo

rdin

atio

n nu

mbe

r

Temperature (K)

Method V

(c) (d)

6

7

8

9

10

11

12

Method I

Method II

Method III

Method IV

Coo

rdin

atio

n nu

mbe

r

Temperature (K)

Method V

(e)

Figure 12. Variation of CN as a function of melt temperature evaluated by various methodsfor (a) Al, (b) Al–3wt% Si, (c) Al–7wt% Si, (d) Al–10wt% Si and (e) Al–12.5wt% Si.


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structural changes, where some clusters break to form free atoms or result in newclusters in the melt. However, these changes were not so significant as to beattributed to anomalous behavior in the melt as the trend in the plots for structureparameters show a linear behavior with temperature. Addition of silicon content(0wt% to 12.5wt%) to the Al melt also caused disorder in the melt. Thecoordination number of pure Al near the melting point has been observed to be 11.5and it was 6.5 for pure silicon, which shows that the Al melt is more densely packedas compared to the silicon melt. The addition of silicon resulted in decreasing theCN and PD, representing a decrease albeit nominal in the atomic close packing inthe melt.

The structure information obtained from the SF, PDF and RDF from high-energy X-ray diffraction experiments as presented in Figures 5–13 provide only thetotal structural changes as a function of melt temperature and silicon content. Thelimitation of studying the total structure information alone is good for pure metalsand it cannot explain the atomic structure information of binary alloys. In order tounderstand the effect of silicon and temperature on the liquid structure of Al–Sihypoeutectic alloys, we have carried out RMC (reverse Monte Carlo) analysis toobtain the partial pair correlations using the experimental data obtained fromdiffraction experiments.

The RMC analysis has been carried out for all the diffraction data obtained inthis study as presented in Table 1. The results of the RMC analysis showed that themodeling of the atoms has been found to be valid for all the diffraction data. Theeutectic alloy (Al–12.5wt% Si) at 867K has been used to demonstrate the validity ofthe RMC analysis in this section. The results for all the other alloys at varioustemperatures are presented in Tables 4–7.

Figure 14 shows the comparison of experimental S(Q) (open circles) with theS(Q) obtained from RMC analysis (dark black line) at 867K. The total structure

850 900 950 1000 1050 1100 11500.385

0.390

0.395

0.400

0.405

0.410

0.415

Al Al-3% Si Al-7% Si Al-10% Si Al-12.5% Si

Pac

king

den

sity

Temperature (K)

Figure 13. Variation of packing density as a function of melt temperature for various Siconcentrations in the Al–Si hypoeutectic alloys.


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Table 5. The liquid structure information for partial pair correlations of Al–Al, Al–Si and Si–Si of Al–7wt% Si obtained from high-energy diffraction experiments and RMC analysis.

(a)

g(r)1 g(r)2 r1 (A) r2 (A)

Temp (K) Si–Si Al–Si Al–Al Si–Si Al–Si Al–Al Si–Si Al–Si Al–Al Si–Si Al–Si Al–Al

898 2.16 2.12 2.17 1.16 1.17 1.18 2.88 2.78 2.80 4.93 5.10 5.13920 2.20 2.10 2.16 1.22 1.20 1.17 2.86 2.77 2.80 5.11 5.14 5.13942 2.15 2.14 2.14 1.23 1.19 1.17 2.74 2.77 2.80 5.05 5.10 5.12964 2.27 2.07 2.14 1.20 1.14 1.17 2.72 2.77 2.80 5.33 5.10 5.12986 2.04 2.06 2.12 1.36 1.18 1.16 2.83 2.76 2.80 5.07 5.08 5.121008 2.23 2.06 2.09 1.29 1.18 1.16 2.78 2.76 2.78 5.20 5.07 5.121030 2.18 2.03 2.06 1.16 1.18 1.16 2.72 2.75 2.78 5.19 5.10 5.121052 2.08 2.00 2.06 1.16 1.14 1.16 2.69 2.74 2.77 5.13 5.11 5.12

(b)

Coordination number Packing density

Temp (K) Si–Si Al–Si Al–Al Si–Si Al–Si Al–Al

898 0.63 2.66 9.60 0.030 0.101 0.378920 0.61 2.65 9.55 0.029 0.099 0.377942 0.73 2.60 9.52 0.026 0.099 0.375964 0.86 2.58 9.49 0.025 0.098 0.373986 0.82 2.58 9.45 0.028 0.097 0.3721008 0.71 2.52 9.40 0.027 0.097 0.3681030 0.86 2.53 9.35 0.025 0.096 0.3651052 0.60 2.51 9.31 0.024 0.095 0.364

Table 4. The liquid structure information for partial pair correlations of Al–Al, Al–Si and Si–Si of Al–3wt % Si obtained from high-energy diffraction experiments and RMC analysis.

(a)

g(r)1 g(r)2 r1 (A) r2 (A)


920 2.81 2.16 2.23 1.19 1.18 1.185 2.73 2.83 2.83 5.37 5.15 5.08942 2.12 2.15 2.22 1.51 1.18 1.183 3.07 2.82 2.82 4.51 5.05 5.09964 2.65 2.17 2.2 1.43 1.2 1.18 2.85 2.85 2.82 5.24 5.17 5.091052 2.43 2.07 2.11 1.26 1.19 1.171 2.73 2.81 2.82 5.4 5.06 5.09

(b)



920 0.38 1.80 10.23 0.010 0.071 0.409942 0.34 1.80 10.19 0.015 0.070 0.406964 0.44 1.82 10.14 0.012 0.071 0.4051052 0.36 1.77 10.06 0.010 0.068 0.403


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factor from X-ray diffraction experiments has been used as input data in the RMCanalysis. The S(Q) from RMC analysis, SRMC(Q) is the summation of weightedpartial structure factors (Equation (14)) and shown in Figure 14. The SRMC(Q) is inexcellent agreement with the experimental S(Q) as shown in Figure 14. The weightingfactors have been calculated using Equation (15). It could be worth mentioning herethat no smoothening of the data was necessary on the results obtained from theRMC analysis.

The three PPDF, gSi-Si(r), gAl-Si(r) and gAl-Al(r) have been obtained from theFourier transformation of the three respective PSFs: SSi-Si(Q), SSi-Al(Q) andSAl–Al(Q). Figure 15 shows the variation of total and partial pair distributionfunctions with r for Al—12.5wt% Si alloy at 867K. All three graphs in Figure 15had the same variables for the respective abscissa and ordinates. Figures 15b and 15care magnified sections of Figure 15a to show the first two peaks and the gSi-Si(r),respectively. The total PDF from RMC analysis have been in good agreement withthe experimental g(r), as shown in Figure 15 without any data smoothing applied tothe curves. The gSi-Si(r) was much smoother showing first and second peaksvery clearly than those reported by Wang et al. [14] and de Jong et al. [29].Figures 16a–c show the typical PPDF of Al–Al, Al–Si and Si–Si pairs of atoms,respectively, obtained from RMC analysis for the Al–12.5wt% Si alloy at variousmelt temperatures. The first peak position, r1 of gAl-Al(r) varies between 2.80 to 2.83

Table 6. The liquid structure information for partial pair correlations of Al–Al, Al–Si and Si–Si of Al–10wt% Si obtained from high energy diffraction experiments and RMC analysis.

(a)

g(r)1 g(r)2 r1 (A) r2 (A)


885 2.21 2.11 2.19 1.20 1.17 1.16 2.83 2.80 2.81 5.17 5.05 5.07907 1.90 2.12 2.17 1.19 1.16 1.17 2.83 2.80 2.81 5.22 5.03 5.06929 1.87 2.12 2.15 1.16 1.16 1.16 2.79 2.81 2.81 5.08 5.09 5.08973 1.84 2.01 2.10 1.20 1.15 1.16 2.74 2.80 2.81 4.94 5.08 5.10995 1.84 2.01 2.10 1.17 1.15 1.15 2.70 2.80 2.81 4.99 5.09 5.071017 1.79 1.99 2.09 1.16 1.17 1.15 2.75 2.80 2.81 4.98 5.08 5.081039 1.86 2.02 2.07 1.16 1.15 1.157 2.76 2.79 2.817 5.05 5.07 5.08

(b)



885 0.91 2.87 8.66 0.041 0.123 0.379907 0.86 2.88 8.63 0.041 0.122 0.378929 0.91 2.84 8.61 0.039 0.123 0.377973 0.87 2.79 8.52 0.037 0.121 0.376995 0.82 2.77 8.48 0.035 0.120 0.3751017 0.89 2.77 8.43 0.037 0.120 0.3741039 0.88 2.76 8.41 0.037 0.118 0.372


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for all the alloy compositions and this value matches with pure Al (2.80 A); however,r1 of gSi-Si(r) varies between 2.72 A to 2.83 A, which is significantly higher than r1 ofpure silicon (2.59 A). Further, the nearest-neighbor distances of the Al–Si atom pairswere shorter than Al–Al and Si–Si atom pairs. The statistics of Si–Si withtemperature have not been good because of the low volume fraction in the alloyresulting in a low number of atoms in the RMC analysis.

Figure 17 shows the first shell coordination number of Al–Al, Al–Si and Si–Sipairs of atoms for Al–Si hypoeutectic alloys. The coordination number of Al–Al andAl–Si atoms decreased uniformly with increasing melt temperature, whereas in thecase of Si–Si atoms, the change was not uniform with temperature due to the smallvolume fraction of Si atoms (1200 atoms) in the total of 10,000 atoms used in theRMC analysis. Further, the coordination number of Al–Al atoms decreased withincreasing silicon content from 0wt% to 12.5wt%, whereas, in the case of Al–Si andSi–Si, the coordination number increased with increasing silicon content in the alloyat any given melt temperature, as one would expect. The partial coordinationnumbers results in this study were compared with the results of ab initio moleculardynamics by Wang et al., as shown in Figure 18. The data from Wang et al. [14] were

Table 7. The liquid structure information for partial pair correlations of Al–Al, Al–Si and Si–Si of Al–12.5 wt% Si obtained from high-energy diffraction experiments and RMC analysis.

(a)

g(r)1 g(r)2 r1 (A) r2 (A)


867 1.97 2.12 2.14 1.15 1.15 1.17 2.82 2.79 2.81 5.07 5.06 5.03889 1.97 2.02 2.13 1.11 1.15 1.16 2.76 2.78 2.79 5.06 5.05 5.02911 2.27 2.00 2.11 1.17 1.15 1.16 2.74 2.78 2.79 5.05 5.04 5.02933 1.94 2.03 2.09 1.16 1.15 1.16 2.79 2.78 2.79 4.92 5.04 5.02956 1.92 2.00 2.08 1.14 1.14 1.15 2.75 2.78 2.79 5.14 5.03 5.02978 1.87 2.01 2.05 1.14 1.14 1.15 2.78 2.78 2.79 5.11 5.02 5.011000 1.85 1.98 2.04 1.16 1.14 1.14 2.77 2.78 2.79 5.08 5.02 5.011022 1.83 1.99 2.02 1.15 1.14 1.14 2.74 2.78 2.78 5.10 5.02 5.011065 1.80 1.93 1.98 1.13 1.11 1.14 2.76 2.78 2.78 5.00 5.01 4.99

(b)



867 1.15 3.16 8.53 0.051 0.134 0.367889 1.15 3.15 8.49 0.047 0.132 0.366911 1.21 3.13 8.46 0.050 0.132 0.365933 1.15 3.11 8.43 0.048 0.131 0.364956 1.13 3.12 8.39 0.047 0.130 0.363978 1.16 3.10 8.35 0.047 0.130 0.3611000 1.13 3.09 8.32 0.047 0.130 0.3601022 1.13 3.08 8.29 0.047 0.130 0.359


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0.0

0.5

1.0

1.5

2.0

2.5 g(r) - Experiment

g(r) - RMC Simulation

gSi - Si

(r)*WSi-Si

gSi - Al

(r)*2WSi-Al

Pai

r di

strib

utio

n fu

nctio

n, g

(r)

at 8

67K

r (A°)

(a)

0.00

0.01

0.02

0.03

0.04(c)

gSi - Si

(r)*WSi-Si

2 4 6 8 10 12 140 2 4 6 8 10 12 14 16 18 20

1 2 3 4 5 60

1

2

gAl - Al

(r)*WAl-Al

(b)

Figure 15. The variation of total and partial pair distribution functions with r, for Al–12.5wt% Si alloy at 867K. All three graphs have the same variables in the respective axes.(b) and (c) are magnified sections of (a) to show the first two peaks and the gSi-Si(r),respectively.

0 2 4 6 8 10 12 14

0

1

2

(c)

(b) S(Q) - RMC Simulation

S(Q) - Experiment

SSi - Al

(Q)*2WSi-Al

SAl - Al

(Q)*WAl-Al

Str

uctu

re fa

ctor

, S(Q

) at

867

K

Q, (A–1)

(a)

0 2 4 6 8 10 12 140.00

0.01

0.02

0.03

0.04

SSi - Si

(Q)*WSi-Si

2 4 60

1

2

SSi - Si

(Q)*WSi-Si

Figure 14. The variation of total and partial structure factors for Al–12.5wt% Si alloy at867K. All three graphs have the same variables in the respective axes. (b) and (c) are magnifiedsections of (a) to show the first two peaks and the SSi-Si(Q), respectively.


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digitized from the original publication and plotted along with the results from thisstudy. Wang et al. [14] reported that the CNAl-Al and CNAl-Si decreased withincreasing melt temperature as observed in the present study. In the case of CNSi-Si,they stated that the CNSi-Si increased with increasing melt temperature though theplot shown in the Figure 18c presents no such trend. Even though, the trends lookedsimilar in both the studies, the actual CN values show a difference and this could bedue to the method of evaluation in determining the rmin positions in RDF curve aswell as due to the significant differences in the number density values used in both thestudies.

Figure 19 shows the packing density of Al–Al, Al–Si and Si–Si pairs of atoms inunmodified Al–Si hypoeutectic alloys at various melt temperatures. The packingdensity of Al–Al and Al–Si decreased uniformly with increase in melt temperaturefor any given alloy composition. However, in the case of Si–Si the variation inpacking density with temperature is not smooth due to the small volume fraction ofSi atoms (1200 atoms) in the total of 10,000 atoms used in the RMC analysis.Further, the packing density of Al–Al decreased with increasing silicon content atany given melt temperature, whereas, in the case of Al–Si and Si–Si packing densityincreased with increasing Si content in the alloy, as one would expect. The numerical

0 2 4 6 8 10 12 14

0

2

4

6

8

10

g Al-A

l(r)

r (A°)

1065K

1022K

1000K

978K

956K

933K

911K

883K

867K

0 2 4 6 8 10 12 14

0

2

4

6

8

10

g Al-S

i(r)

r (A°)

1065K

1022K

1000K

978K

956K

933K

911K

883K

867K

(a) (b)

0 2 4 6 8 10 12 14

0

2

4

6

8

10

g Si-

Si(r

)

r (A°)

1065K

1022K

1000K

978K

956K

933K

911K

883K

867K

(c)

Figure 16. Partial pair distribution functions obtained from RMC for Al–12.5wt% Si atvarious melt temperatures: (a) gAl-Al(r), (b) gSi-Al(r), (c) gSi-Si(r).


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850 900 950 1000 1050 1100

850 900 950 1000 1050 1100

850 900 950 1000 1050 1100

8.0

8.5

9.0

9.5

10.0

10.5

11.0

Coo

rdin

atio

n nu

mbe

r (A

l-Al)

Temperature (K)

Al-3% Si Al-7% Si Al-10% Si Al-12.5% Si

(a)

1.50

1.75

2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.75 Al-3% Si Al-7% Si Al-10% Si Al-12.5% Si

Coo

rdin

atio

n nu

mbe

r (A

l-Si)

Temperature (K)

(b)

0.2

0.4

0.6

0.8

1.0

1.2

1.4


Coo

rdin

atio

n nu

mbe

r (S

i-Si)

Temperature (K)

(c)

Figure 17. Coordination number as a function of melt temperature for Al–Si hypoeutecticalloys: (a) Al–Al, (b) Al–Si, (c) Si–Si.


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8.0

8.2

8.4

8.6

8.8

9.0

9.2

9.4

9.6

9.8

Coo

rdin

atio

n nu

mbe

r (A

l-Al)

Temperature (K)

Present work Wang et al. [14]

(a)

3.0

3.1

3.2

3.3

3.4

3.5

Coo

rdin

atio

n nu

mbe

r (A

l-Si)

Temperature (K)


(b)

800 900 1000 1100 1200 1300 1400

800 900 1000 1100 1200 1300 1400

800 900 1000 1100 1200 1300 1400

0.4

0.5

0.6

0.7

1.1

1.2

1.3

Coo

rdin

atio

n nu

mbe

r (S

i-Si)

Temperature (K)


(c)

Figure 18. Comparison of partial coordination numbers, (a) Al–Al, (b) Al–Si and (c) Si–Si,for Al–12.5wt% Si alloy with ab initio results obtained by Wang et al. [14].


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850 900 950 1000 1050 1100

850 900 950 1000 1050 1100

850 900 950 1000 1050 1100

0.35

0.36

0.37

0.38

0.39

0.40

0.41

0.42


Pac

king

den

sity

(A

l-Al)

Temperature (K)

(a)

0.070

0.080

0.090

0.100

0.110

0.120

0.130

0.140

0.150


Pac

king

den

sity

(A

l-Si)

Temperature (K)

(b)

0.010

0.020

0.030

0.040

0.050


Pac

king

den

sity

(S

i-Si)

Temperature (K)

(c)

Figure 19. Packing density as a function of melt temperature for Al–Si hypoeutectic alloys:(a) Al–Al, (b) Al–Si, (c) Si–Si.


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values for partial pair correlation functions of all alloy compositions at various melttemperatures obtained from RMC analysis are presented in Tables 4–7.

From the results of partial pair correlations, it has been observed that the unlikeatom pairs were more favorable to coordinate in the melt as represented by theirshort nearest neighbor distances, increase in CN and PD values with increase insilicon content. Si–Si atom pairs also exist in the melt, though less likely asrepresented by low CN and PD values. Further, the first peak height of PPDF of Al-Si pair of atoms falls between those of Al–Al and Si–Si pair of atoms for all the alloycompositions at most of the melt temperatures. This feature along with non-appearance of compound forming tendencies in the melt imply that liquid structureof the Al–Si alloy melt behaves like a random mixing of Al and Si atoms as observedin the Na–K alloy [17].

Figure 20 shows the variation of viscosity of Al and Al–Si hypoeutectic alloys asa function of melt temperature evaluated by using the hard sphere theory. It isevident from Figure 20 that the viscosity calculated by the hard sphere method didnot follow any trend because of competing effects of PD and molar volume factors asevaluated from Equation (21). Further, it can be seen that the viscosity increasedwith increasing melt temperature, which is against the laws of physics, which dictatesan exponential decrease in melt viscosity with increasing temperature. Due to thelack of accuracy in the viscosity evaluated by the hard sphere theory, the semi-empirical model was used to evaluate melt viscosity and is believed to be a morerepresentative of the melt behavior.

Figure 21 represents the viscosity of Al and Al–Si alloys evaluated by the semi-empirical approach using Equation (28). The alloy melt viscosity decreased withincreasing temperature for all the hypoeutectic alloy compositions. Further, withincreasing silicon content in Al melt, the viscosity decreased at any giventemperature. Similar observations have been found experimentally by Song et al.[56] and Sklyarchuk et al. [74], as shown in Figure 22. It is evident from Figure 22

850 900 950 1000 1050 1100 11500.50

0.51

0.52

0.53

0.54

0.55

0.56

0.57

Vis

cosi

ty (

mP

a.s)

Temperature (K)

3Si

Al

10Si

7Si12.5Si

Figure 20. Viscosity of Al and Al–Si hypoeutectic alloys at various melt temperaturesevaluated by hard sphere theory.


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that the viscosity values from this study were higher than the experimentallydetermined viscosity values. This shows that determining accurate viscosity values isreally a challenging task, and from the past literature, it has been observed that thereis a 400% spread in the reported viscosity values determined by different techniquesfor Al and Al–Si alloys [20]. The values used for � (correction factor), Vm (molarbolume) and 0 (constant) in Equation (28) are presented in Table 8. The � valuesfor pure Al and Si were obtained from Iida et al. [20] and the � values for alloycompositions were evaluated as a weighted average of � for the respective puremetals.

In Equation (28), P(i) was assumed to be P(T) for Al and P(j) the P(T) for Si.Since P(T) is dependent on temperature, P(i) and P(j) for a particular Si compositionof Al–Si at a given melt superheat above the alloy liquidus temperature wereevaluated by assuming a similar melt superheat temperature above the melting pointsof Al and Si, respectively. Iida et al. used the boiling point of Al, Tb(Al)¼ 2333K toevaluate P(T) for pure Al using Equation (28). Using the same boiling point, theviscosity evaluated by the semi-empirical approach was 2.21 mPa s at 938K and thisvalue verifies the viscosity of 2.2 mPa s at 943K evaluated by the Born–Green theoryusing the potential functions and pair distribution functions [17], thus, showing thatthe semi-empirical approach is fairly reliable. However, recent findings show that theboiling point of pure Al, Tb(Al)¼ 2467.15K [75]. This value has been used for Tb(Al)

in this study to obtain the P(T) values at various melt temperatures. Table 9 presentsthe P(i) and P(j) values for various Al–Si alloy compositions as a function of melttemperature used in this study.

Viscosity is a structure-sensitive property and the results obtained from the liquidatomic structure measurements aid in evaluating the viscosity as well as under-standing the variation of viscosity of Al–Si alloys as a function of melt temperatureand silicon content. Similar behavior would be observed for packing density andcoordination number of these hypoeutectic alloys, as well. The free volume in the

850 900 950 1000 1050 1100 11501.4

1.6

1.8

2.0

2.2

2.4

2.6

Vis

cosi

ty (

mP

a.s)

Temperature (K)

Al

3Si

7Si

10Si

12.5Si

Figure 21. Viscosity of Al and Al–Si hypoeutectic alloys at various melt temperatures by semi-empirical model.


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liquid alloy increased with increasing silicon content as shown by the decrease in therespective packing density and coordination number values, and thus, the freemobility of atoms in the liquid increased resulting in a decrease in the frictionalresistance between atom pairs, which decreased melt viscosity. In binary alloy

850 900 950 1000 1050 1100 1150 1200 12500.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

7% Si [74]

12.5% Si [56]

0% Si [74]12.5% Si

Vis

cosi

ty (

mP

a.s)

Temperatrue (K)

0% Si

0% Si [20]

7% Si

Figure 22. Comparison of the experimental viscosity values for Al and Al–Si hypoeutecticalloys with the literature values.

Table 9. P(i) and P(j) values for Al–Si hypoeutectic alloy compositions at various melttemperatures.

Aluminum Al–3wt% Si Al–7wt% Si Al–10wt% Si Al–12.5wt% Si

Temp(K) P(i)

Temp(K) P(i) P( j)

Temp(K) P(i) P( j)

Temp(K) P(i) P( j)

Temp(K) P(i) P(j)

938 0.951 920 0.948 0.681 898 0.948 0.681 885 0.948 0.681 867 0.948 0.681963 0.945 942 0.942 0.675 920 0.942 0.675 907 0.942 0.675 889 0.942 0.675988 0.937 964 0.935 0.669 942 0.935 0.669 929 0.935 0.669 911 0.935 0.6691013 0.930 1052 0.907 0.532 964 0.928 0.663 973 0.921 0.657 933 0.928 0.6631038 0.922 986 0.921 0.657 995 0.914 0.648 956 0.921 0.6571063 0.914 1008 0.914 0.648 1017 0.907 0.532 978 0.914 0.6481088 0.907 1030 0.907 0.532 1039 0.907 0.532 1000 0.907 0.5321113 0.897 1052 0.899 0.473 1022 0.899 0.473

Table 8. �, Vm and 0 values for Al and Al–Si hypoeutectic alloy compositions.

Alloy Al Al–3Si Al–7Si Al–10Si Al–12.5Si Si

� value 0.52 0.5159 0.5105 0.5065 0.5031 0.38Vm 9.9E-06 1.28E-05 1.21E-05 1.18 E-05 1.11 E-05 —0 4.09E12 3.87E12 3.81E12 3.79E12 3.75E12 —


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systems, with compound forming tendencies, viscosity increased with increasingsolute content [55,76]. Alloys with compound forming tendencies exhibit a pre-peakbefore the first large peak in the low Q region of the structure factor curve, S(Q)versus Q. As seen in Figure 6, Al–Si alloys do not exhibit such compound-formingtendencies.

The following have been deemed as the limitations in the semi-empirical methodto evaluate the viscosity of Al–Si hypoeutectic alloys.

. The liquid alloy in this theory has been treated as a Newtonian liquid. Thisassumption may not be completely true as suggested by recent findings[6,61,62,76,77], which have shown that liquid metals, specifically the Al–Sialloys, could behave as non-Newtonian fluids [6,76].

. The constant factor � for all alloy compositions has been calculated basedon their weight fractions in the alloy. However, there is no theoretical reasonto verify the validity of this assumption, yet.

. o was a constant and assumed that there has been no net diffusion of atomsin the structure at any temperature.

. It has been assumed that P(i) and P(j) as the values of P(T) for Al and Si,respectively. The validity of this assumption should be further verified.

Albeit such limitations, it has been presumed that the approach to quantify thevariables in Equation (28) is fairly reliable as confirmed by the validation of thequantified viscosity of pure Al by that evaluated using the Born–Green theory usingthe pair potential functions.

7. Conclusions

The liquid structure of Al–Si hypoeutectic alloy has been extensively studied by high-energy X-ray diffraction experiments at various melt temperatures. Liquid structureparameters such as S(Q), g(r) and RDF have been evaluated as a function of melttemperature. The disorder in these liquid alloys increased with increasing melttemperature and silicon content as reflected by the decrease in coordination numberand packing density with increasing melt temperature for all Al–Si alloy composi-tions. At any given temperature, the coordination number decreased with increasingsilicon content. In order to better understand the liquid structure, the partial paircorrelations of Al–Al, Al–Si and Si–Si atom pairs obtained by RMC analysis havebeen evaluated using structural data obtained from high-energy X-ray diffractionexperiments. Number density values used in this study were matched with thenumber density values used by Waseda [17], but quite different from the values usedby Bian et al. [15] and Wang et al. [14]. The partial pair correlations obtained fromrunning the newly modified procedure for the RMC analysis with 10,000 atomsresulted in well-defined curves contrary to the results of earlier researchers [14,29].Also, the peak heights of partial pair distribution functions gAl-Al(r), gsi-Al(r) andgsi-si(r) decreased with increase in melt temperature. The coordination numbers ofAl–Al, Al–Si and Si–Si decreased with increase in melt temperature. Further, theviscosity of Al and Al–Si alloys has been determined using the atomic structureinformation obtained from diffraction experiments and RMC analysis coupled with


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a semi-empirical method. The viscosity of Al–Si alloys decreased with increasing melttemperature; further, the viscosity of aluminum melt decreased with increase insilicon content. It is believed that the results of this work could lead to furtherinvestigation of the structure information of these commercially important alloysand lead to further refinement in evaluating physical properties of these alloys, suchas viscosity, surface tension, free energy, and thermal/electrical conductivity fromfundamental experimental diffraction data. Further, it is believed that the criticaltotal and partial structure parameters and the empirical equations for thecoordination numbers as a function of Si content and melt temperature developedin this study would prove valuable to researchers modeling this alloy system usingmolecular dynamics.

Acknowledgements

This work was performed with the financial support of the Natural Science and EngineeringResearch Council (NSERC) of Canada and US Department of Energy (DOE) under contractnumber W-7405-Eng-82. The high-energy X-ray work at the MUCAT sector of the APS wassupported by the US Department of Energy, Office of Science, Basic Energy Sciences undercontract number W-31-109-Eng-38. Special acknowledgements are extended to Dr. DouglasRobinson, beam line scientist at Sector 6-ID-D, Advanced Photon Source, Argonne, IL, USA,for his cooperation and support in performing the liquid diffraction experiments. The authorsgratefully acknowledge the financial support of General Motors Corporation, specifically thecontribution from Dr. Michael J Walker and Dr. Carlton Fuerst.

Notes

1. http://www.isis.rl.ac.uk/RMC/downloads/useful.htm2. RMCA v.3.14 program, http://wwwisis2.isis.rl.ac.uk/RMC/rmc.htm3. General Electric Corporation, Schenectady, New York, USA.

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