Physics Letters A 355 (2006) 142–147

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Structural transition of sheared-liquid metal in quenching state

Y.N. Zhang, L. Wang, W.M. Wang ∗, J.K. Zhou

Key Laboratory of Liquid Structure and Heredity of Materials, Ministry of Education, Shandong University, Jinan 250061, People’s Republic of China

Received 2 August 2005; received in revised form 11 February 2006; accepted 13 February 2006

Available online 20 February 2006

Communicated by R. Wu

Abstract

In this Letter, the effects of shear rate on structural properties of liquid Al in quenching process were investigated via molecular dynamics(MD) simulations based on the EAM potential. Analyses in internal energy and pair correlation functions (PCF) reveal an increasing structuraltransition temperature as the shear rate is enhanced in the liquid. Results of pair analysis indicate that for liquid Al under normal condition, facecenter cubic (FCC) structure is clearly detected upon cooling; while in sheared liquid, structural transition from FCC to body center cubic (BCC)at temperature of 800 K is manifested, leading to the dominance of BCC structural order at low temperatures.© 2006 Elsevier B.V. All rights reserved.

Keywords: Structural transitions; Sheared-liquid; MD simulation

1. Introduction

Such extreme conditions as super-high pressure, super-low temperature, shearing action and magnetic field playan important role in the preparations of metastable phases,amorphous metals, nanocrystals and other new types of ma-terials [1,2]. Studies of solidification processes in extremeconditions revealed various new phenomena that did not ap-pear under normal conditions, giving prominence to theresearch of microstructural properties of nonequilibrium dy-namical systems. Although we have made great progress in theunderstanding of melt microstructures by capturing detailedliquid fragments [3–5], handful explorations have been donefor local structures of metallic liquids under extreme condi-tions due to experimental difficulties. Fortunately, computersimulations employing molecular dynamics or Monte Carlotechniques can provide important complementarities to experi-mental studies and make an important contribution to this topic[6–9].

* Corresponding author.E-mail address: weiminw@sdu.edu.cn (W.M. Wang).

0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2006.02.020

So for it is known that structural transitions as well as rheo-logical properties of the fluid will be influenced as shear ratesfunction. Recently, Fan and his co-workers investigated the evo-lution of solidification structures under different melt stirringconditions through Monte Carlo simulations and experimen-tal measures [10–13]. The nonequilibrium MD simulations fora flowing liquid at shear rate (γ < 0.01) were carried out byLacks to develop an inherent structural analysis of viscosity interms of potential energy landscapes [14]. Malandro focused onthe shear-induced changes of potential energy landscapes andthe consequences of such changes on the macroscopic mechan-ical behaviour of glasses [15]. Stankovic et al. used the ‘genericembedded-atom model’ (GEAM) with steady shear stressesand increased temperatures to study structural and mechanicalproperties of metals, and dramatically revealed structural tran-sitions in the system [16,17]. It is unquestionable that furtherresearches are needed to explore structural properties of shearedmetallic liquids as functions of shear rates and temperatures.

Here we will present results of internal energy, PCF and pairanalysis obtained by MD simulations with wide-range shearrates and various temperatures. The purpose of our project istrying to elucidate the effects of shear rate on structures ofliquid Al, and provide a reasonable discussion on the interdif-fusion mechanism in sheared liquids.

Y.N. Zhang et al. / Physics Letters A 355 (2006) 142–147 143

2. Simulation methods

2.1. Embedded-atom method (EAM)

EAM proposed by Daw and Baskes [18,19] has been widelyaccepted in computer simulations on metallic systems. Simula-tion results for aluminum [20–22] are known to lend themselvesto experimental descriptions accurately. In this Letter our dataare based on Mei’s EAM potential [23] by which the theoreti-cal PCFs agree well with the experimental profiles in terms ofWang’s results [24].

In the EAM potential, the total energy of a system is ex-pressed as:

(1)U =∑

i

Fi(ρi) + 1

2

∑i,j

(i �=j)

Φij (rij ),

where Fi is the embedding energy of placing an atom into thatelectron density ρi , and Φij (rij ) is a two-body central potentialbetween atoms i and j with the separation distance rij . In Mei’sempirical functions, ρi is total electron density at atom i due tothe rest of atoms in the system and it can be expressed as thesummation of local electron density:

(2)ρi =∑j (�=i)

f (rij ).

The density function, pair interaction and embedding energyare given as follows:

(3)f (r) = ρe

5∑l=0

cl

12

[r0

r

]l

,

(4)Φ(r) = −Φ0

[1 + δ

[r

r0− 1

]]exp

[−γ ′

[r

r0− 1

]],

F (ρ) = −Ec

[1 − α

βln

[ρ

ρe

]][ρ

ρe

]α/β

+ 1

2Φ0

3∑m=1

sm exp[−(

√m − 1)γ ′]

(5)

×[

1 + (√

m − 1)δ − √m

α

βln

[ρ

ρe

]][ρ

ρe

]√mγ ′/β

,

where α, β , γ ′, δ, cl , sm are model parameters which are con-stants and can be found in Ref. [23].

2.2. Details of MD simulations

The liquid and solid aluminum are simulated by MD tech-niques based on the isothermal and isochoric (NVT) condition.The decline-force method is adopted to decrease the tempera-ture by forcing bath temperature to change linearly at every timeinterval. We simulate a small volume of bulk fluid implementedwith periodic boundary conditions that are devised by Leesand Edwards [25]. Hence boundary effects may be ignored andthe fluid is genuinely homogeneous isothermal Couette flow inwhich the fluid undergoes sheared flow due to boundary walls

Fig. 1. PCF at the temperature of 1000 K for two different system sizes, i.e.,N = 500 particles (solid line) and N = 2048 particles (dotted line).

that are in relative motion [26]. We assume a homogeneousfluid of atoms under planar shear flow in x direction, wherethe velocity gradient is constant in z direction. If the relativevelocity of walls is Vx , the shear rate γ of the fluid is equal tothe velocity gradient in the z direction, i.e., γ = ∂Vx/∂z. In oursimulations, the shear rate varies from 0 to 1.0 ps−1 (reducedunit, which is related to the lattice constant of solid metal) toshow the influence of shearing on the microstructure of the sys-tem.

Creating this shear flow is most readily done as one of initialconditions, which applies an impulse of the correct size anddirection to each atom when calculation starts (t = 0). A smallterm related to the shear rate γ is subtracted from the velocityterm in Lagrange multiplier to obtain the thermostat equationof motion and keep the temperature of sheared system constant.The detailed formulas are given by Rapaport [27].

Previous investigations implied that the calculated resultswere insensitive to system size [28,29]. Here Fig. 1 repre-sents consistent results through the calculations of PCF atT = 1000 K for normal and sheared liquids with 500 and 2048particles, respectively. Consequently in our simulations, 500particles are placed in a cubic box and the time step is 1.6 fs.First of all, internal energies for both the normal and shearedliquids are calculated at different cooling rates but with thesame shear rates. Then, we equilibrated the system for 10 000time steps at temperature of 1500 K. This is followed by a cool-ing process in which the system is cooled from 1500 to 400 Kin decrements of 100 K with a cooling rate of 6.25 K ps−1. Atthe selected temperature, another run of 5000 time steps is per-formed to determine the averaged energy and structure.

3. Structural analysis methods

3.1. Pair correlation function (PCF)

PCF is regarded as a common parameter used to describestructure characteristics of liquid and amorphous states [30]. It

144 Y.N. Zhang et al. / Physics Letters A 355 (2006) 142–147

is defined as

(6)g(r) = Ω〈ni(r, r + r)〉4πr2rN

,

where 〈ni(r, r + r)〉 is the averaged number of atoms aroundthe ith atom in the sphere shell ranging from r to r + r .Ω/N is the particle number density which is independent ofvolume of the simulation cell and total simulated particles. PCFshows the probability of finding another atom in a distance fromr to r + r and is adopted to detect the structural evolution inmelts.

3.2. Pair analysis (PA) technique

PA technique has been widely adopted in simulations tocharacterize local structures of system. In this technique, twoatoms are considered to be nearest neighbors if they are withina specified cutoff distance. The cutoff we usually use is thefirst minimum in the PCF of the system. Honeycutt and An-derson [31] used four integers i, j , m and n to distinguishthe bonded pairs. The first integer indicates whether or not theatoms are near neighbors, that is, if the atoms A–B form a bond,i = 1; otherwise, i = 2. The integer j denotes the number ofnear neighbors common to the two atoms. The m representsthe number of bonds formed among the shared neighbors. Thefourth integer n is added to differentiate between the cases whenthe first three integers are same but the bond geometries are dif-ferent. Each of various phases in a dense bulk system has itsown signature in the bonded pairs that characterizes its localstructures. The number of 1551, 1541 and 1431 bonded pairs inliquid is a direct measurement of degree of icosahedral order;1421 bonded pair is the typical pair for FCC crystal; 1661 and1441 bonded pairs are the characteristic pairs for BCC crystal;and 1421 and 1422 bonded pairs are related to the HCP crystal.

The analysis of inherent structures can be alternatively re-alized via the common neighbor analysis (CNA) method [32]which can provide direct interpretation of various features ofthe radial distribution function and analyse local atomic struc-tures [5].

4. Results and discussions

Temperature dependence of internal energy at several cool-ing rates is plotted in Fig. 2. In normal condition, energy–temperature (E–T ) curve shows less difference as the coolingrate changes from 12.5 to 4.17 K ps−1; but when the cool-ing rate decreases from 0.625 to 0.0625 K ps−1, a distinctdrop in energy arises and the transition temperature increaseswith the cooling rate, as shown in Fig. 2(a). Comparably,Fig. 2(b) displays that the final solid of the sheared systemwith γ = 1.0 ps−1 is a crystalline state, which is determinedby the sudden drop in each E–T curve. Both at high temper-atures (T > 1000 K) and at low temperatures (T < 600 K) theenergies of sheared system are hardly affected by the coolingrate, implying that the shear rate γ is a dominant factor in theenergy transition of sheared-liquid. Therefore, we employ thecooling rate of 6.25 K ps−1 as an example to explore the in-fluence of γ on the internal energy and transition temperature

Fig. 2. Internal energy as a function of temperature at several cooling ratesfor (a) the normal system and (b) the sheared liquid with γ = 1.0 ps−1. 500particles are contained in each system.

during the cooling process. From Fig. 3(a), we can easily findthat the energy of melt increases with shear rates and the distinctdrop occurs at a certain temperature upon cooling. The increas-ing shear rate makes the transition temperature Ts higher, up to800 K as shear rate γ = 1.0 ps−1 is imposed. The Ts as a func-tion of shear rate γ fits the exponent expression of Ts = Aeγ ,shown in Fig. 3(b), where A is a constant. Only the exponentialform is adopted in this Letter because of its simple form andsmall error.

The inherent structure formalism separates the dynamics ofliquids into vibrational motions within the local potential en-ergy minima and structural transitions between the different lo-cal minima [33,34]. In the temperature control mechanism, thevibrational contribution seems to be independent of shear rateand just dependent on the simulation temperature. Therefore,the change of overall energy originates solely from the struc-tural contribution. It is also found that shear rates will cause thedecrease of energy barrier for structural transitions [14,15], asa result of an easy shift to alternate local energy minima, whichis in favour of the structural transition in system. Therefore, wededuce that structural transition in melt takes place easily as the

Y.N. Zhang et al. / Physics Letters A 355 (2006) 142–147 145

Fig. 3. Internal energy and transition temperature as a function of shear rate:(a) the internal energy vs. temperature at the cooling rate of 6.25 K ps−1; (b) therelationship between the shear rate and transition temperature: Ts = Aeγ ,where the value of A is 288.

shear rate increases. However, as shown in Fig. 3(a), the distinctdrop in internal energy does not emerge in our simulations if theshear rate is below 0.25 ps−1. Due to the rapid cooling rate, aconsiderable supercooling exists in the system, which leads totoo low transition temperature in our simulations.

The microstructures of simulated liquids are convenientlydiscussed in terms of the PCF. We compare PCF curves forsheared liquid with various shear rates at temperature of 1400 Kin Fig. 4(a). With increasing shear rate, the first and secondpeaks in PCF curves become flatten in shape and the third peakalmost disappears in sheared liquid. Furthermore, the positionof the first peak shifts towards smaller r value, which suggeststhat the degree of disorder in liquids is strengthened and thenearest-neighbor distance between atoms becomes smaller asthe shearing is imposed in the liquid.

The PCFs for both normal and sheared liquids are shown inFig. 4(b). Compared with the normal liquid, PCF for shearedsystem shows following characteristics: (1) the PCF retains anoverall shape of liquid phase as the temperature is higher than800 K which is identical with the transition temperature de-duced from E–T curves (Fig. 3(a)); (2) the crystal-like peak

Fig. 4. PCF as functions of shear rates and temperatures: (a) at temperature of1400 K with various shear rates; (b) at the cooling rate of 6.25 K ps−1 withγ = 0 (solid line) and with γ = 1.0 ps−1 (dotted line).

appears in PCF curves below 800 K and becomes more obvi-ous with the decrease of temperature; (3) the position of thefirst peak moves towards large r value and the amplitude be-comes larger at low temperatures. At the temperature of 400 K,these two liquids solidify to imperfect crystals, which is simi-lar to results in the literature [35]. The difference in shapes ofPCF curves implies different microstructures for both systemsat low temperatures.

Results of pair analysis for the systems with γ = 0 and γ =1.0 ps−1 are given in Fig. 5. For the normal liquid, the num-ber of bonded pairs changes mostly at temperature of 500 K.With the decreases of temperature from 1400 to 500 K, the1551, 1541 and 1431 bonded pairs corresponding to icosahedralstructure as well as 1661 and 1441 bonded pairs represent-ing BCC structure increase slightly; while the 1421 and 1422bonded pairs keeps unchanged. When the temperature is below500 K, the 1421 and 1422 bonded pairs increase rapidly and the1421 bonded pairs reaches to 20.3% of all bonded pairs at lowtemperatures, indicating that the crystallized sample has mostly

146 Y.N. Zhang et al. / Physics Letters A 355 (2006) 142–147

Fig. 5. Temperature dependence of the relative numbers of various bondedpairs: (a) 1551, 1541 and 1431; (b) 1421 and 1422; (c) 1661 and 1441.Solid lines are simulation data for normal liquid and the transition tempera-ture Ts0 = 500 K; dotted lines are simulation results for sheared liquid withγ = 1.0 ps−1 and the transition temperature Ts1 = 800 K.

FCC crystal phase after a quenching process. As for the case ofγ = 1.0 ps−1, structural transition temperature is around 800 K.We will see that all the bonded pairs keep unchanged as thetemperature is higher than 800 K. But there is a substantial in-crease in the number of 1551, 1661 and 1441 bond pairs justbelow 800 K; on the contrary, the number of 1421 and 1422bond pairs drops quickly at low temperatures. Differing fromFCC crystal phase obtained in normal liquid, at low tempera-tures the number of 1661 and 1441 bonded pairs representingBCC structure is in a dominant position in the sheared systemas shown in Fig. 5(c).

The phenomenon that the structural transition from FCC toBCC arises as a high shear rate is imposed is similar to thatdiscussed by Stankovic et al. [16,17]. In their studies, the ini-tial configuration of atoms which was placed as FCC latticesites did not survive under the shear-flow and the traces ofBCC structure appeared via transformation of a part of FCC.For BCC structure, it has a smaller packing factor than theinitial configuration of Al, i.e., FCC close packed structure(KFCC = 0.74, while KBCC = 0.68); also, the diffusion coeffi-cients of dissolved hydrogen, oxygen and carbon in BCC-liquidiron are larger than those in FCC-liquid iron [36]. Therefore,structure transition from FCC to BCC is reasonable to min-imize the resistance to shear. The vacancy in atoms of BCCstructure is also larger than that in FCC structure, which allowsother atoms to penetrate into this vacancy and the interdiffusionprocess occurs easily.

5. Conclusion

We discuss the features of structural transition as functionsof shear rates and temperatures for liquid Al through MD simu-lations based on EAM potential. Analyzing the results of in-ternal energy and PCF, we deduce that the system size andcooling rate have less effect on the structure and energy ofsheared system. The increasing transition temperature whichcan be expressed as: Ts = Aeγ , implies the enhanced crystal-lization ability in sheared liquid. The structural transition fromFCC to BCC can be observed via PA technique when a highshear rate γ = 1.0 ps−1 is imposed on the liquid. MD simu-lations display that an appropriate shear rate would accelerateatomic interdiffusion processes.

Acknowledgements

The authors give grateful thanks for the supports of NationalNatural Science Foundation of China (grant Nos. 50301008,50501012), the Natural Science Foundation of Shandong Pro-vince (No. 2004BS04016) and Shandong Young ScientistFoundation, respectively.

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