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Multiscale modeling of the solidification microstructure evolution in the presence of ultrasonic

stirring

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2012 IOP Conf. Ser.: Mater. Sci. Eng. 33 012079

(http://iopscience.iop.org/1757-899X/33/1/012079)

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Multiscale modeling of the solidification microstructure

evolution in the presence of ultrasonic stirring

Laurentiu Nastac

THE UNIVERSITY OF ALABAMA, Department of Metallurgical and Materials

Engineering, P. O. Box 870202, Tuscaloosa, AL, 35487-0202, USA

E-mail: lnastac@eng.ua.edu

Abstract. Ultrasonic treatment (UST) was studied to improve the quality of cast ingots as well

as to control the solidification microstructure evolution. Ultrasonically-induced cavitation

consists of the formation of small cavities (bubbles) in the molten metal followed by their

growth, pulsation and collapse. These cavities are created by the tensile stresses that are

produced by acoustic waves in the rarefaction phase. The cavitation threshold pressure for

nucleation of the bubbles may decrease with increasing the amount of dissolved gases and

especially with the amount of inclusions in the melt. A UST model was developed to predict

the ultrasonic cavitation and acoustic streaming. The developed UST modeling approach is

based on the numerical solution of Lilley model (that is founded on Lighthills’s acoustic

analogy), fluid flow, and heat transfer equations, and mesoscopic modeling of the grain

structure. The UST model was applied to study the solidification of Al-based alloys) under the

presence of ultrasound. It is found that the predicted ultrasonic cavitation region is relatively

small, the acoustic streaming is strong and thus the created/survived bubbles/nuclei are

transported into the bulk liquid quickly. The predicted grain size under UST condition is at

least one order of magnitude lower than that without UST, which is in excellent agreement

with the experimental data.

1. Introduction

Ultrasound technology is a powerful technological tool to affect a material. Ultrasonic vibrations can

significantly affect heat and mass transfer in liquids, modify the structure and properties of solids and

interfere with the interactions of solids and liquids. The primary causes are due to ultrasonic

cavitation, acoustic streaming and movement of dislocations associated with propagation of ultrasound

waves in media. Ultrasonically-induced cavitation consists of the formation of small cavities

(bubbles) in the molten metal followed by their growth, pulsation and collapse. These cavities are

created by the tensile stresses that are produced by acoustic waves in the rarefaction phase [1].

Relevant applications include structural refinements of solidifying metals with applications to

continuous casting as well as to remelting processes (vacuum arc remelting (VAR), electroslag

remelting (ESR), electron beam remelting (EBR) and plasma arc melting (PAM)) for Ti alloys,

specialty steels and superalloys. Additional potential applications include die-casting, semi-solid

metal processing, and forging of high fractions of fine spherical grains. Some critical technologies

that have immediate application to manufacturing industry are shown in [1-7].

MCWASP XIII IOP PublishingIOP Conf. Series: Materials Science and Engineering 33 (2012) 012079 doi:10.1088/1757-899X/33/1/012079

Published under licence by IOP Publishing Ltd 1

The objective of this work is to develop an ultrasonic modeling tool to assist in the scale up of the

following processes: (i) An ultrasonic technology to significantly refine the solidification

microstructure of castings and ingots and (ii) A nanotechnology-assisted by ultrasonic cavitation to

manufacture metal-matrix-nano-composite (MMNC) castings with enhanced mechanical properties.

This paper discusses the ultrasonic model development effort and simulation results for an ingot

casting process.

2. Modeling approach for analysis of grain refinement with ultrasound

Fully couple energy, mass, acoustic, and compressible fluid flow models were used for this ultrasonic

analysis. Several acoustic models [8] were investigated including Fows Williams and Hawkings

Model, Broadband Noise Source Model (Proudman’s and more recently Lilley model based on

Lighthills’s acoustic analogy), Jet Noise Source Model (by Goldstein and Ribner) and The Boundary

Layer noise source Model. It was determined that Lilley model based on Lighthills’s acoustic analogy

is the most adequate for this work. Figure 1 shows the geometry of the liquid pool. Since time-

domain computations were used for resolving the acoustic field, a very small time step of the order of

1e-07 s was used. The ultrasonic probe has a diameter of 20 mm, an amplitude, A = 10 microns, and a

resonant frequency, f = 17.5 kHz.

Figure 1. A356 liquid pool (the ultrasound probe is shown in the middle of the pool).

In the current multi-phase ultrasonic cavitation modeling approach, a basic two-phase cavitation

model is used that consists of using the standard viscous flow equations governing the transport of

phases (Eulerian multi-phase) and the k- turbulence model. In cavitation, the liquid-bubble mass

transfer is governed by the cavity (bubble) transport equation:

CGbbbbb

RRVfft

(1)

where b subscript denotes the cavitation bubble phase, fb is the bubble volume fraction, b is the

bubble density, b

V

is the bubble phase velocity, RG and RC are the mass transfer source terms related

to the growth and collapse of the cavitation bubbles, respectively.

In equation (1), the interphase mass transfer rates per unit volume (RG and RC) account for the

liquid and bubble phases in cavitation. They are calculated using the growth of a single bubble based

on the Rayleigh-Pleasset model [9]. The model assumes no barrier for nucleation; thus, the bubble

dynamics can be obtained from the general Rayleigh-Plesset equation as follows [9]:

dt

dR

RR

pp

dt

dR

dt

RdR

b

b

L

bL

L

L

bbb

b

42

2

32

2

2

(2)

where Rb is the bubble radius, L is the surface tension coefficient of the liquid phase, L is the liquid

density, L is the kinematic viscosity of the liquid phase, pb is the bubble surface pressure, and p is the

local far-field pressure.

MCWASP XIII IOP PublishingIOP Conf. Series: Materials Science and Engineering 33 (2012) 012079 doi:10.1088/1757-899X/33/1/012079

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3. Simulation results

3.1. Cavitation

The predictions of the flow, pressure and ultrasonic cavitation are presented in figures 2-7 for an A356

alloy at two time sequences: 8.8e-06 s and 2e-05 s. As illustrated in figure 7a, time t = 8.8e-06 s

corresponds to the beginning of the cavitation event. The cavitation region will increase with time

(see figure 7b, t = 2e-05s). Though the predicted ultrasonic cavitation region in figure 7 is relatively

small, the acoustic streaming is strong (see figures 2-5) and thus the created/survived bubbles/nuclei

are transported into the bulk liquid quickly.

Figure 2. Flow induced by ultrasound: Flow field (streamline, in kg/s).

t = 8.8e-06 s V, m/s t = 2.0e-05 s V, m/s

Figure 3. Flow induced by ultrasound: Flow field (velocity vector).

MCWASP XIII IOP PublishingIOP Conf. Series: Materials Science and Engineering 33 (2012) 012079 doi:10.1088/1757-899X/33/1/012079

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t = 8.8e-06 s V, m/s t = 2.0e-05 s V, m/s

Figure 4. Flow induced by ultrasound: Flow field (velocity contour).

t = 8.8e-06 s. (%) t = 2.0e-05 s (%)

Figure 5. Flow induced by ultrasound: turbulent intensity (%).

Also, the pressure for nucleation of the bubbles (cavitation threshold pressure or Pc) in A356 alloy

(see fiigure 6) is of the same order of magnitude as the experimentally determined pressures to

nucleate gas bubbles in other metallic systems (see reference [1], p. 123 and reference [10]). Note also

that the cavitation threshold may decrease with increasing the amount of dissolved gases (such as

Hydrogen in A356) and especially with the amount of inclusions in the melt (such as Al203 in A356)

(see figure 2.39 in reference [1]).

MCWASP XIII IOP PublishingIOP Conf. Series: Materials Science and Engineering 33 (2012) 012079 doi:10.1088/1757-899X/33/1/012079

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Figure 6. Flow induced by ultrasound: Pressure contour (in Pa).

(a) t = 8.8e-06 s (b) t = 2.0e-05 s

Figure 7. Cavitation induced by ultrasound. Legends show the volume fraction of

cavities/bubbles/potential nuclei (cavitation region).

MCWASP XIII IOP PublishingIOP Conf. Series: Materials Science and Engineering 33 (2012) 012079 doi:10.1088/1757-899X/33/1/012079

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3.2. Prediction of the solidification structure in the presence of ultrasound

The mesoscopic model developed previously in [11, 12] is adapted to include the effect of UST on

solidification structure evolution and therefore on grain refinement of alloys. The mesoscopic model

includes computations of the grain size and columnar-to-equiaxed transition (CET), as well as of

segregation. It also includes computations at the dendrite tip length scale (mesoscopic scale), for

prediction of dendritic morphology and microsegregation patterns.

Table 1 presents the modeling data used in the microstructure modeling of alloy A356. Equation 4

in reference [13] was used to estimate the number of nuclei in the presence of UST. It was assumed in

equation 4 [13] that the cavitation-induced heterogeneous nucleation is the main mechanism for grain

refinement. This is also in accordance with the work reported in reference [6], that is, the cavitation-

induced heterogeneous nucleation plays a more important role than the “sluggish” dendrite

fragmentation process in the formation of equiaxed grains during solidification of A356 alloy castings.

The predicted solidification structure is shown in figure 8, where GE is the equiaxed grain density

and the legend illustrates the grain crystallographic orientation. The simulation of the microstructure

shown in figure 8 was performed assuming a steel mold of 25 mm wall thickness. In these

computations, the pouring temperature of the molten A356 alloy was 625°C and the initial temperature

of the mold was 20 °C. As it can be seen from figure 8, the predicted grain size under UST condition

is approximately one order of magnitude lower than that without UST. The predicted average grain

size in figure 8b, which is about 200 m, is in excellent agreement with the experimental results

presented in reference [6]. This validates the mesoscopic solidification model.

Table 1. Data Used in Microstructure Modeling of Alloy A356

Solidification Kinetics Property A356

Liquid diffusivity, DL [m2 s-1] 3x10-9

Solid diffusivity, DS [m2 s-1] 1x10-12

Liquidus slope [K wt. %-1] mL = -6.5

Initial Si concentration [wt. %] Co = 7.0

Si partition coefficient k = 0.14

Eutectic [wt. %] 12.6

Gibbs-Thomson coefficient, [K m] 0.9 x 10-7

4. Concluding remarks and future work

An approach for modeling the effects of ultrasonic vibration on the ingot structure was developed.

This modeling approach was implemented into Ansys’s Fluent, a commercial CFD software [14]. It

was shown that the UST is a promising technology that can be used to improve the solidification

structure and consequently to increase the mechanical properties of ingots and castings made of steel,

Ti, Al, Mg, superalloys, etc.

Well-defined experiments on ultrasonic stirring of A356 alloys are being conducted at the

University of Alabama, Solidification Laboratory, to advance the current understanding of the

influence of ultrasonic cavitation and acoustic streaming on the microstructure evolution of solidifying

alloys. Also, the present UST model is currently used to assist in the development and optimization of

cast MMNCs. The UST modeling approach for the A356-MMNC system was introduced in [15].

The developed UST modeling approach will be used to study the globular/dendritic transition in

alloys that are solidifying in the presence of ultrasonic stirring. Also, additional R&D work will be

performed at the University of Alabama to further develop, refine and integrate the developed UST

and microstructure models into commercial modeling and simulation software packages.

MCWASP XIII IOP PublishingIOP Conf. Series: Materials Science and Engineering 33 (2012) 012079 doi:10.1088/1757-899X/33/1/012079

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(a) without UST (b) with UST color index

Figure 8. Ultrasonic treatment: prediction of the solidification structure for an A356 alloy

(2 mm x 9 mm test geometry): (a) GE = 5 x 107 nuclei/m2 and (b) GE = 5 x 109 nuclei/m2.

MCWASP XIII IOP PublishingIOP Conf. Series: Materials Science and Engineering 33 (2012) 012079 doi:10.1088/1757-899X/33/1/012079

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References

[1] Abramov O V 1998 High-Intensity Ultrasonics Theory and Industrial Applications Gordon and

Breach Science Publishers

[2] Eskin G I 1998 Ultrasonic Treatment of Light alloy Melts Gordon and Breach Science

Publishers

[3] Rosenberg L D ed. 1969 Sources of high-intensity ultrasound vols. 1-2 New York: Plenum

Press

[4] Campbell J 1981 International Metals Reviews 26 71-108

[5] Jian X, Meek T T, and Han Q 2006 Scripta Materialia 54 893-896

[6] Jian X, Xu H, Meek T T, and Han Q 2005 Material Letters 59 190-193

[7] Meek T T and Han Q 2006 Ultrasonic Processing of Materials Final DOE Technical report,

Oak Ridge National Laboratory

[8] Fluent 6.3: 2006 User’s Guide Manual Fluent Inc.

[9] Brennen C E 1995 Cavitation and Bubble Dynamics Oxford University Press 51

[10] El Kaddah N H and Robertson D G C 1977 Journal of Coloids and Interface Science 60 2 349-

360

[11] Nastac L 2004 Modeling and Simulation of Microstructure Evolution in Solidifying Alloys

Springer: http://www.springer.com/materials/special+types/book/978-1-4020-7831-6

[12] Nastac L 1999 Acta Materialia 47 17 4253–4262

[13] Nastac L 2011 Metallurgical and Materials Transactions B 24 1297-1305

[14] Fluent: http://ansys.com/Products/Simulation+Technology/Fluid+Dynamics/ANSYS+FLUENT

[15] Nastac L and Dai Y 2008 Ultrasonic Model Development and Applications to Ingot Casting

Processes Proc. 2008 International Ansys Conference (Pittsburgh, PA, August 25-28, 2008)

MCWASP XIII IOP PublishingIOP Conf. Series: Materials Science and Engineering 33 (2012) 012079 doi:10.1088/1757-899X/33/1/012079

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