Modeling nucleation and growth of bubbles during foaming of molten aluminum with high initial gas supersaturation

Journal of Materials Processing Technology 214 (2014) 1–12

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Modeling nucleation and growth of bubbles during foaming of molten aluminum with high initial gas supersaturation S.N. Sahu a,∗ , A.A. Gokhale a , Anurag Mehra b a b

Defence Metallurgical Research Laboratory, Kanchanbagh, Hyderabad 500058, India Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

a r t i c l e

i n f o

Article history: Received 3 March 2013 Received in revised form 11 July 2013 Accepted 13 July 2013 Available online 24 July 2013 Keywords: Aluminum foam Modeling Bubble size distribution Nucleation Growth

a b s t r a c t An idealized nucleation and growth based model was used to predict bubble size distribution in liquid aluminum foam, based on the assumption that the entire quantity of hydrogen added as TiH2 was retained in solution initially. The model considered simultaneous nucleation and growth of bubbles in the first stage, and pure bubble growth in the second stage. Bubble nucleation was found to be feasible only heterogeneously within narrow crevices in non wetting substrates. Effects of initial gas supersaturation on total expansion, final bubble size distribution, total number of bubbles, and average bubble size were investigated. Model predictions of foam characteristics were compared with experimental observations on foams prepared by dissociating TiH2 foaming agent in liquid aluminium, and good match between the two was found with respect to average cell size and total number of bubbles. Differences between model predictions and experimental observations, especially in the nature of bubble size distribution, and limitations of the model were explained. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The characteristic features of closed cell aluminum (Al) foams include low apparent density, high stiffness to weight ratio and high energy absorption at low transmitted stress. Cell size and apparent density are known to influence both static and dynamic properties of foams, and their knowledge a priori is certainly advantageous to process engineers. Aluminum foam evolution was modeled by Körner et al. (2002). In their model, pre-existence of bubbles is assumed. Hydrogen is assumed to dissolve in molten Al due to dissociation of foaming agent TiH2 , and diffuse to pre-existing bubbles causing growth. They predicted bubble coalescence based on certain criteria and its effects on liquid drainage and net foam expansion. Deqing et al. (2006) predicted cell size in air-bubbled PVA solutions, and applied these predictions to aluminum foam. They studied the effects of air pressure, flow rate, liquid viscosity and orifice characteristics on the nature of air flow (continuous vs. discontinuous) and bubble size evolution during its travel from orifice to liquid surface. Cox et al. (2001) studied the simultaneous effects of liquid drainage and cooling on foam density profile in solidified foam. They suggested a criterion to predict uniformity of foam density. In another study, Li et al. (2009) predicted the

∗ Corresponding author. Tel.: +91 9490956742; fax: +91 4024342697. E-mail addresses: [email protected], sn [email protected] (S.N. Sahu). 0924-0136/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2013.07.009

evolution of bubble size distribution in Al–Si melt injected with compressed air by rotating impeller. In their study, an experimentally determined bubble size distribution is taken as the initial condition, and bubble coarsening due to inter-bubble gas diffusion and simultaneous liquid drainage is modeled. Bikard et al. (2005) studied expansion of polymer foams by assuming a pre-distribution of bubbles and then modeled the bubble growth resulting from gas generation in the melt. In all the above cited works, bubbles are assumed to exist before start of foaming. Banhart et al. (2001) studied early stages of evolution of bubbles in a Zn powder + TiH2 powder compact heated to 420 ◦ C. Two types of bubbles were identified. Type I bubbles, which envelope TiH2 particles, and type II bubbles which are free of the particles. The former seem to nucleate at the particle- matrix interface discontinuities, while the latter are believed to form by a series of phenomena such as TiH2 dissociation, hydrogen dissolution in bulk Zn, hydrogen diffusion along grain boundaries and eventual nucleation of bubbles at grain boundary triple points. Similar observations were made by Rack et al. (2009) in powder metallurgy aluminum based foams. Using X-ray micro tomography and quantitative image analysis, they showed that the pores form either as envelops around TiH2 particles (e.g. in 6061 alloy) or at weak points within the matrix (e.g. Al-7%Si and Al–Si–Cu alloys). Shafi et al. (1996) presented a homogeneous nucleation and growth model to predict bubble size distribution in LDPE-N2 system in which the polymer melt is assumed to be initially supersaturated with finite amount of gas. They assumed the presence of solute

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depleted volume or solute boundary volume around growing bubbles, and a sub-volume of lower solute content within the boundary volume in which nucleation is assumed to be negligible (the latter entity is termed by them as ‘influence volume’). Published literature on modeling of aluminum foaming process excludes bubble nucleation stage. Models based on bubble nucleation and growth are important since they provide reference bubble size distribution for comparison with real processes and with other models which assume pre-existing bubble size distribution. In the present work, a transport phenomena based model applicable to the foaming of aluminum melt supersaturated with hydrogen is considered. Different nucleation mechanisms are examined and a suitable nucleation model relevant to the conditions of aluminum foaming is proposed. The nucleation model is coupled with the growth model described by Shafi et al. (1996) in arriving at the final bubble size distribution. Some of the features of their model, retained in the present work, include simultaneous nucleation and growth of bubbles, neglecting nucleation rate below 1% of its initial value and defining termination of foaming based on chemical equilibrium between bubbles and their corresponding ‘influence volumes’ in the melt. The above model is used to predict the final bubble size distribution, total number of bubbles and average bubble size at different levels of initial supersaturation. The initial values of hydrogen supersaturation are calculated from the quantities of TiH2 added in the experiments, corrected suitably for hydrogen losses. Subsequently, predicted foam characteristics are compared with experimental observations. 2. Experimental procedure Aluminum foam was produced by the ‘Alporas’ route as described by Miyoshi et al. (2000). 1.25 kg of Al-1.5 wt%Ca alloy was melted in a resistance furnace and stirred at 700 ◦ C for 5 min. Due to stirring, fresh surface of the Al–Ca melt is continuously created and brought in contact with the atmosphere, resulting in melt oxidation, and formation of oxides such as CaO and CaAl2 O4 (Banhart, 2000). The role of these oxides in aluminum foaming is multifold. Babcsan et al. (2004) stated that the oxides prevent drainage and assist in foam cell stabilization. Ma and Song (1998) reported that higher oxide content increases melt viscosity, which slows foam expansion. Thus, the oxide content needs to be optimized to balance foam cell stabilization with foam expansion kinetics. Subsequent to stirring, TiH2 powder (particle size <8 ␮m) was added to the melt which dissociates rapidly at the melt temperature expanding the melt. Two experiments were carried out for each of the TiH2 addition levels, viz. 0.5%, 1.0% and 1.5% (all TiH2 addition levels are expressed on weight basis). Subsequently, the liquid foam was cooled to room temperature by forced air, and solid foam heights were measured to calculate expansion ratios. Foams prepared using varying addition levels of TiH2 were sectioned for determining cell size distribution and total number of cells, the details of which are presented in a later section. It is important to note that actual foam expansions are less than predictions based on TiH2 additions. The reasons are: burning of TiH2 , loss of hydrogen to atmosphere during foam expansion and incomplete dissociation of TiH2 before the foam is removed from the crucible. Therefore, initial gas supersaturation values (expressed as equivalent pressures) are suitably corrected in the model.

considered. In the present model, it is assumed that the entire quantity of hydrogen needed to cause observed level of expansion was (D) dissolved in the melt initially under an external pressure P0 . When the external pressure is removed, the melt becomes supersaturated with hydrogen. Campbell (2003) estimated that the hydrogen supersaturation required for homogeneous nucleation of bubbles in aluminum melt was of the order of 31,000 bar. The critical radius of nucleation was assumed to be 0.29 nm and surface energy of aluminum melt to be 0.9 N m−1 . In the present case, the maximum initial hydrogen content in the aluminum melt, assumed to be equal to the observed volume expansion of foam made by adding 1.5 wt% TiH2 , is equivalent to maintaining hydrogen partial pressure of 200 bar (the calculation of this pressure value is shown later in Table 1). Since this value is much less than 31,000 bar, it discounts the possibility of homogeneous nucleation. Though homogeneous nucleation models have been applied successfully to polymer systems, it should be remembered that surface tension for polymer melts is much less than for metals. Next, heterogeneous nucleation of bubbles at either TiH2 particles or at oxides is considered. Oxides are known to be poorly wetted by molten Al, and are suggested to be preferred sites for heterogeneous nucleation of pores in Al castings. Tiwari and Beech (1978) carried out experiments to examine the formation of hydrogen bubbles in an aluminum melt by using the first bubble technique. The ease with which bubbles form in the castings at very low supersaturation pressures is suggestive of heterogeneous nucleation. They observed that the entrained alumina particles and bubbles remained in close association in the solidified ingots. The reason, they suggested, was the presence of cracks, holes and other defects on the surface of alumina particles, which make them suitable sites for heterogeneous nucleation. In another study related to aluminum foaming, Yang and Nakae (2003) mentioned the possibility of alumina particles acting as heterogeneous nucleation sites for bubbles. Due to the non-wetting nature of alumina particles, they would be the preferred heterogeneous nucleation sites unlike TiH2 particles which are wetted by molten aluminum. Due to similar experimental conditions between their work and our work, oxides were considered as nucleation sites for bubbles. Following assumptions are made while analyzing bubble nucleation and growth. • No pre-existing bubbles are present in the melt. • Bubbles are spherically symmetric and remain so over the entire period of growth. • Liquid drainage and bubble coalescence are neglected. (D)

Gas pressure in the bubble Pt is related to dissolved gas concentration at the bubble surface C(R, t) (mol/m3 ) by Henry’s law, (D)

C(R, t) = KH Pt

(1)

where KH is the solubility coefficient. • The solubility coefficient and the diffusion coefficient of hydrogen in liquid aluminum remain constant. • The viscosity of aluminum melt remains constant throughout the process. • The gas is considered to be ideal in nature. • The entire process occurs under isothermal condition.

3. Model description and formulation

3.1. Bubble nucleation

Due to the complexities of modeling the real process, and for benchmarking purposes, a simplified physical model, retaining some of the important features of the real process, was

In order to calculate the rate of heterogeneous nucleation of bubbles on oxide substrates, information on oxide contact angle and surface geometry is required. From nucleation point of view,

S.N. Sahu et al. / Journal of Materials Processing Technology 214 (2014) 1–12

higher contact angles (>90◦ ) are favorable for heterogeneous nucleation of bubbles due to lowering of activation energy barrier. But from bubble stability point of view, a contact angle around 90◦ is favored as it represents a good balance between particle adsorption at the liquid–gas interface and particle aggregation in the liquid, as postulated by Dickinson et al. (2004) for particle stabilized aqueous foams. In the absence of reliable experimental measurements, we have assumed a contact angle value of 100◦ , which is suitable from bubble stability and heterogeneous nucleation points of view. Equations for nucleation on flat oxide surfaces are presented first, followed by equations for nucleation in conical cavities. Fig. 1(a) presents a schematic diagram of nucleation of a gas bubble on a flat non-wetting oxide surface. Let c0 be the initial concentration (mol/m3 ) of hydrogen retained in the Al melt which, according to Eq. (1), can be converted to an equivalent partial pres(D) (D) sure P0 . With release of initial supersaturation pressure P0 at (C) t = 0, the melt is exposed to ambient pressure P , resulting in nucleation and growth of hydrogen bubbles. Eq. (2) gives the expression for heterogeneous nucleation rate J at oxide surface (expressed as number of nuclei per unit area of substrate per unit time) as given by Blander (1979),



2 J = N 2/3 Q (c ) mBF(c )



1/2 exp

−



16 3 F(c ) (D)

3kB T (Pt

− P (C) )

2

(2)

where N is the number of gas atoms per unit volume of liquid, B = 1 for chemical equilibrium at bubble surface, m is the mass of the gas molecule,  is surface tension, kB is the Boltzmann’s constant, T is the foaming temperature (K) and  c is the contact angle measured in the liquid phase. Among the two geometric factors, Q( c ) represents the ratio of the surface area of the liquid/gas interface to the surface

3

area of a spherical bubble of the same radius, while F( c ) represents the ratio of the volume of the nucleated bubble to the volume of a spherical bubble of the same radius, and are expressed as: Q (c ) =

(1 + cos c ) 2

(3)

F(c ) =

(2 + 3 cos c − cos3 c ) 4

(4)

Fig. 1(b) schematically shows the case of bubble nucleation in a conical cavity on an oxide surface. Here, ˇ is the cone semi-vertex angle. In this case, Q( c ) and F( c ) take the form of Q( c , ˇ) and F( c , ˇ), and are expressed as



Q (c , ˇ) =

 F(c , ˇ) =

1 − sin(c − ˇ) 2



2 − 2 sin(c − ˇ) + cos c cos2 (c − ˇ)/ sin ˇ 4

(5)

 (6)

Nucleation rate given in Eq. (2) is in the units of number of nuclei per unit surface area (or unit crevice surface area) per unit time. In order to obtain the volumetric nucleation rate, the area nucleation rate is multiplied by an appropriate ‘area per unit volume’ term given as: Jmod = JAT or Jmod = JAT

for nucleation on flat surfaces

A  C

AT

for nucleation in cavities

(7a)

(7b)

where AT is the total oxide surface area per unit volume and AC /AT is the area fraction for cavities suitable for nucleation. Eq. (2) shows that the nucleation rate is sensitive to the surface tension and the geometric factor F( c ) or F( c , ˇ) (as the case may be), and that it decreases exponentially with decreasing solute super saturation (expressed as pressure difference). Out of these, the surface tension and the contact angle are characteristics of the metal gas system under consideration, whereas solute supersaturation varies with the amount of TiH2 added and the amount of gas consumed in nucleation and growth. Another important factor is ˇ, the cone semi-vertex angle of rough oxide surface which can have a range of values significantly affecting the initial nucleation rate. Hence, it is worthwhile to investigate the effect of ˇ and the initial supersaturation pressure on nucleation rate, which will be presented in a later section. Critical radius for nucleation RC is given in Eq. (8). It represents the bubble radius at which the free energy of bubble formation is at its maximum and the bubbles have equal probability to grow or decay due to various perturbations. Once the bubble size is sufficiently beyond the critical size it spontaneously grows to form a stable bubble. Eq. (9) gives the critical momentum transfer time tC during which viscous transfer of momentum takes place over a characteristic length equal to the critical embryo size. Higher tC indicates that bubble growth is controlled by viscous resistance whereas lower tC indicates that pressure difference governs the kinetics of bubble growth. RC =

Fig. 1. (a) Flat oxide surface as a nucleating site for a bubble (Blander, 1979),  c is contact angle measured in the liquid phase. (b) Conical crevice on oxide surface acting as a nucleating site for a bubble (Blander, 1979),  c is contact angle measured in the liquid phase, ˇ is cone semi-vertex angle, r is the radius of nucleating bubble.

tc =

2 (D) P0

− P (C)

4 (D) P0

− P (C)

(8)

(9)

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displacement of surrounding liquid. As expected, bubble growth is proportional to the gas super saturation pressure. The initial condition for the above equation at t = 0 is given as R(0) = Rc

(11)

where Rc is critical radius of nucleation defined in Eq. (8). However, by definition, growth rate of bubble of radius Rc must be equal to zero, since it has equal probability to grow or decay. Once R(0) exceeds Rc it spontaneously grows to form a stable bubble. Based on perturbation theory, presented by Shafi et al. (1996), the initial radius of the bubble R(0) is taken to be slightly (∼ =7%) larger than Rc . The species mass balance in the radial direction in the surrounding melt, in spherical coordinates, gives ∂c + ∂t

 dR  R2 ∂c r 2 ∂r

dt

D ∂ r 2 ∂r

=

 r2

∂c ∂r

 (12)

with initial conditions denoted as, c(r, 0) = c0 (D)

(13)

(D)

Pt=0 = P0 Fig. 2. ‘Passive’ volume, Vs (t,t ) within solute depleted volume Vcb (t,t ) surrounding a growing bubble. cs represents the critical concentration below which nucleation is neglected. R(t,t ) is the bubble radius and S(t,t ) is the radius of the ‘passive’ volume (Shafi et al., 1996).

(14)

Boundary conditions are (D)

c(R, t) = cR (t) = KH Pt

(15)

c(∞, t) = c0

(16)

3.2. Bubble growth For the purpose of the present analysis, bubble growth is divided into two stages. In stage I, bubble growth is accompanied by nucleation, while stage II represents continued bubble growth after termination of nucleation. In stage I analysis, bubble nucleation and growth equations are solved simultaneously to estimate the bubble size distribution as well as the remaining solute concentration of the melt as functions of time. The bubble size distribution arrived at the end of Stage I serves as the initial condition for stage II analysis in which the bubbles are allowed to grow by consuming excess solute in the melt till chemical equilibrium is achieved between the bubbles and the melt. The mathematical representations of stages I and II will be presented in the subsequent sections. 3.2.1. Stage I bubble growth The stage I bubble growth model presented here, and shown schematically in Fig. 2, is based on the work of Shafi et al. (1996) and, hence, only essential features will be presented. When bubbles nucleate, gas starts to diffuse into them through the bubble-melt interface. A concentration gradient propagates radially into the melt (Fig. 2) forming a solute boundary layer. Beyond the boundary layer, solute concentration ca is uniform but reduces with time from its initial value c0 due to formation and growth of gas bubbles. Various important volumes and radii terms are explained in Fig. 2. At any time t, the radius of the bubble nucleated at time t is denoted as R(t,t ). Based on mass and momentum balance, the bubble growth rate is expressed as a function of bubble pressure and radius R. 1 dR = 2 dt



(D)

(Pt

− P (C) )R − 2



(10)

where  is the melt viscosity. According to Eq. (10), bubble growth rate is inversely proportional to , implying that viscous forces retard bubble growth, since bubble growth has to involve

The stage I bubble growth analysis presented above is carried out till nucleation rate drops to 1% of its initial value, which occurs when melt solute concentration reduces by 2% of c0 . Thus, bubble nucleation is neglected within a volume Vs (t,t ) around the bubble in which hydrogen concentration is less than this critical concentration for nucleation (marked as gray area in Fig. 2). This region is called as ‘passive’ volume (indicating the lack of nucleation activity). All bubbles are assumed to have their own ‘passive’ volumes around them. The remaining melt can be termed as the ‘active volume’ VL (in which nucleation occurs). The nucleation rate Jmod is calculated only in the ‘active’ volume. The mass balance at the bubble surface gives d dt



(D)

4 Pt R3 3 Zg T





∂c

= 4R D ∂r

2

(17) r=R

where Z is the compressibility factor of hydrogen inside the bubble, g is the universal gas constant, and D is the diffusion rate of H in liquid Al. Solution of Eqs. (10)–(17) is described later in the paper. Simultaneous nucleation and growth of the bubbles consumes the solute in the ‘active’ volume, the average concentration of which is calculated as:

 ca VL = c0 VL,0 −



0

t

t

Jmod (t  )VL (t  )

− 0

(D)

P (t, t  )R3 (t, t  )  4 Jmod (t  )VL (t  ) t dt 3 g T



S(t,t  )

4r 2 c(r, t, t  )drdt 

(18)

R(t,t  )

The first, the second and the third terms on the right hand side of Eq. (18) represent the total solute present in the system, the solute in the bubbles and the solute in the ‘passive’ volumes, respectively. As nucleation and growth of bubbles progress, ca diminishes and approaches the critical concentration cs below which nucleation can be neglected.

S.N. Sahu et al. / Journal of Materials Processing Technology 214 (2014) 1–12

3.2.2. Stage II bubble growth and final bubble size distribution When nucleation stops, the entire melt becomes the ‘passive’ volume, and the existing bubbles continue to grow, driven by the concentration gradient between the bubble interface and the surrounding volumes. This stage II growth continues till a chemical equilibrium is established (at t = tf ) between the bubbles and their boundary volumes, as represented by Eq. (19): c0 Vs (tf , t  ) = KH P (D) (tf , t  )Vs (tf , t  ) +

4 P (D) (tf , t  )R3 (tf , t  ) g T 3

schemes cannot be used to solve the growth equations, as they introduce large numerical error. Using this method, the equations were descritized in to non-linear algebraic equations. These algebraic equations with the initial and the boundary conditions were solved simultaneously and iteratively to determine the bubble size as a function of time. In the second step of the calculation, nucleation rate for the base line conditions was determined to find the total number of bubbles. Solute content in these nucleated bubbles and in their passive volumes is determined using the expressions of Eq. (18). Subsequently, from Eq. (18) the new (depleted) average concentration of the melt was determined. This new average concentration was used to calculate the nucleation rate for the next time step. Now two generations of bubbles with different sizes (one freshly nucleated and the other a single time step old) were taken into considerations while calculating the solute consumption. These steps were repeated for several time steps till nucleation rate decreased to below 1% of its initial value. At this point nucleation was assumed to have ceased. Afterwards, the total number of bubbles remained constant but the bubbles formed at different time steps were different in their number and size. In the third step of calculation, final bubble size distribution was obtained. It was important to note that even after nucleation stops, a large amount of solute still remains in the passive volumes of the bubbles. This makes the bubbles grow further by the transfer of solute from the melt to the bubbles. In order to determine the final bubble size distribution, Eq. (19) was solved using Newton–Raphson iterative technique. As Eq. (19) is based on mass balance, stage II bubble growth was not tracked as a function of time.

(19)

In Eq. (19), the left hand side represents the initial gas mass in the passive volume, while the first and second terms on the right hand side represent gas mass within the bubble and its ‘passive’ volume, respectively. Based on the assumption of chemical equilibrium between the bubble and its ‘passive’ volume, concentration in the ‘passive’ volume is represented in terms of bubble pressure in the first term on the right hand side of Eq. (19). The final size of bubbles nucleated at different times can be determined from this equation. The above equations for nucleation, growth and final bubble size distribution are required to be non-dimensionalized for solving numerically. The methodology of non-dimensionalization was discussed in detail by Shafi et al. (1996) and not discussed here. 3.3. Final expansion ratio Total volume of bubbles Vb can be calculated as: Vb =

i=n 4

3

3 nb,i Rf,i

5

(20)

i=1

5. Results and discussion

Total volume of foam VT : VT = Vb + VL,0

(21)

5.1. Experimental foams

Expansion ratio (E.R.): E.R. =

VT VL,0

Fig. 3 shows the variation of the expansion ratio as a function of the amount of TiH2 added into the melt. For a given TiH2 addition, the difference in the expansion values of the two experiments is less than 10%. The foam expansion increases linearly with TiH2 content, as seen in Fig. 3. Out of the two experiments carried out for each TiH2 content, one having higher expansion value is selected (see Table 1) for validation of simulated cell size distribution. Table 1 gives further information on these experiments regarding the predicted and the actual expansion ratios, and the estimated values of hydrogen supersaturation and the equivalent gas pressure after making corrections for losses. These values of the concentration (D) and the gas pressures were taken as initial values c0 and P0 in the model. The thermo-physical properties of the Al–H system used in the calculations are shown in Table 2.

(22)

where n is the total number of size classes present in the bubble size distribution (this is equal to the total number of time steps in which nucleation event was simulated), Rf,i is the final bubble size in the ith class after reaching chemical equilibrium as calculated from Eq. (19) and nb,i is the number of bubbles present in the ith size class. 4. Solution methodology Solution methodology to arrive at the final bubble size distribution can be divided into three steps. In the first step, Eqs. (10)–(17) were solved which represent the stage I bubble growth. These equations are coupled, highly stiff and non-linear in nature. In order to obtain their solutions, implicit Gears method for stiff differential equations (Boundary Difference Formula 3), suggested by Shafi et al. (1996) was used. Other methods such as explicit numerical

5.2. Bubble nucleation Eq. (2) was used to determine the nucleation rates corresponding to material properties and process variables given in Table 2. As

Table 1 Details of the foam expansion and the calculated partial pressures of hydrogen. Out of the two experiments carried out at each % TiH2 content, the higher expansion value is shown in the table. Expt. No.

wt% TiH2 added

Equivalent hydrogen concentration (mol/m3 )

Equivalent hydrogen partial pressure (bar)

1 2 3

1.5 1 0.5

774 516 258

548 366 183

Predicted expansion ratio

18 13 7

Actual expansion ratio

6.8 5.2 3.7

Corrected initial hydrogen concentration (c0 moles/m3 )

Corrected hydrogen partial pressure (bar)

280 190 122

200 135 87

6

S.N. Sahu et al. / Journal of Materials Processing Technology 214 (2014) 1–12

Fig. 3. Foam expansion ratio as a function of TiH2 content for initial melt height of 64 mm. The experiments were carried out in a graphite crucible of diameter 100 mm with 1.25 kg melt.

mentioned earlier, for the given gas super saturation levels, homogeneous nucleation is not feasible. Next, heterogeneous nucleation on oxide surfaces was assumed. The maximum contact angle for oxides in aluminum melt is 160◦ as reported by Campbell (2003). Even for such highly non-wetting oxides, heterogeneous nucleation rate calculated using Eqs. (2)–(4) remained negligibly small (i.e. Jmod ≈ 10−500 m−3 s−1 ), indicating that the heterogeneous nucleation on flat oxides can be neglected. Consequently, in the present case, the oxide contact angle was assumed to be 100◦ (see Section 3.1). Oxides are known to form multiple folds during melting operation and, hence, possibility of cavity nucleation can be considered. Keeping with the assumption of no preexisting bubbles, presence of air pockets within the oxide folds was not considered. Next, cavity nucleation rate was calculated using Eqs. (2), (5) and (6) for the geometry shown in Fig. 1(b). In order to determine the volumetric nucleation rate Jmod in Eq. (7), it is necessary to determine the oxide interfacial area per unit volume of the melt (AT ). It may be mentioned that oxide particles have poor metallographic contrast with aluminum, and their characterization is not trivial. Based on the limited data available, effective particle diameter of 2 ␮m and effective volume fraction of 0.03 were assumed (Babcsan et al., 2004) to calculate oxide interfacial area to volume ratio (AT ).

Fig. 4. Effect of ˇ on geometric factors F( c , ˇ) and Q( c , ˇ) plotted at  c equal to 100◦ .

Further, area fraction of the crevice surface to the oxide surface (AC /AT ) was assumed to be about a tenth of the total oxide surface area, though detailed characterization of oxides is required to ascertain this value. Fig. 4 shows that increase in ˇ increases both F( c , ˇ) and Q( c , ˇ). Among the two factors, F( c , ˇ) has more dominant effect on Jmod than Q( c , ˇ) does, since F( c , ˇ) appears in the exponential term in Eq. (7). Due to this, increasing ˇ, which causes both F( c , ˇ) and Q( c , ˇ) to increase, results in a sharp decrease in Jmod till 12◦ and gradually thereafter (Fig. 5). Beyond ˇ > 12◦ , the nucleation rate becomes negligible. The effect of supersaturation pressures on Jmod is shown in Fig. 6. These pressure values are taken from Table 1 which represents the three experimental conditions. The variation of Jmod for different pressures is qualitatively similar to that of Fig. 5. However, for lower pressures, the asymptote shifts to lower values of ˇ. Considering the sensitivity of nucleation rate with respect to ˇ, observable nucleation rate (assumed as log10 Jmod > 2) was possible for the three pressure values when ˇ remained within a narrow range of 11–12◦ . This observation is qualitatively similar to that of Wilt (1986) where he showed observable nucleation of bubbles in water/CO2

Table 2 Thermo-physical parameters for the aluminum–hydrogen system. Surface tension Viscosity Solubility Coefficient Diffusivity (hydrogen in liquid aluminum)

Temperature Initial supersaturation pressures

Pressure at continuous phase Compressibility factor Dimensionless number Solubility number Nsl = 0.114 Peclet number (for 1.5 wt% TiH2 ) Npe = 7.13 × 102

 = 0.866 N m−1 (Szekely, 1979)  = 1.5 × 10−3 Nsm−2 (Szekely, 1979) KH = 1.41 × 10−5 mol N−1 m−1 (Talbot, 2004) D = 4.475 × 10−8 m2 s−1 (Eichenauer and Markopoulos, 1974) T = 700 ◦ C (D) P0 = 200 bar for 1.5 wt% TiH2 (D) P0 = 135 bar for 1.0 wt% TiH2 (D) P0 = 87 bar for 0.5 wt% TiH2 P(C) = 1.01 bar Z=1

Fig. 5. Effect of ˇ on nucleation rate Jmod for  c equal to 100◦ and supersaturation pressure of 200 bar.

S.N. Sahu et al. / Journal of Materials Processing Technology 214 (2014) 1–12

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Fig. 6. This is an important result, since a range of cavity angles may be present in real systems, and the kinetically favored ones will be active, making the initial nucleation rate relatively independent of the supersaturation pressure. 5.3. Stage I bubble growth

Fig. 6. Effect of ˇ on the nucleation rate as a function of supersaturation pressures, and different pressure values. Contact angle  c is maintained at 100◦ . The shaded portion in the figure shows the observable nucleation rates i.e. log10 Jmod > 2 for different combinations of ˇ and pressure values.

solution for ˇ value of 4.7◦ with  c = 94◦ and  (surface tension) as 0.065 N m−1 for a supersaturation as low as 5 times the atmospheric pressure. The present system, due to the high supersaturation (the lowest value corresponding to 0.5% TiH2 addition is 87 times the ambient pressure) does not require very sharp cavities for nucleation to occur, as in aqueous systems. For a given supersaturation pressure, a cavity of suitable ˇ (in the range 11–12◦ ) will be the most active to generate observable nucleation as shown in the band in

Bubble growth behavior during nucleation regime was determined with reference to 1.5 wt% TiH2 , (the most commonly reported addition level corresponds to a partial pressure of 200 bar). Since the average solute concentration in the melt reduced only by about 2% of the initial value during stage I bubble growth, it was assumed to remain constant with respect to bubble growth kinetics during stage I. Therefore, bubbles nucleated at different times were assumed to follow the same growth kinetics during stage I. In other words, bubbles of the same age were assumed to attain the same size, internal pressure and interface solute concentration, irrespective of their birth time. Fig. 7(a) and (b) depicts the history of bubble growth and the accompanying bubble pressure drop with respect to bubble age, t − t’ till the end of nucleation, i.e. over a period of 5 ms. Fig. 7(a) shows that bubble growth rate is very low initially, increases sharply to the maximum value, and reduces thereafter at decreasing rate. Fig. 7(b) shows that initially, bubble pressure drops marginally when bubble growth is slow, followed by sharp drop, and then remains constant near the ambient value. Eq. (10) shows that bubble growth rate is proportional to the product of bubble radius R and bubble excess pressure. The initial small values of bubble growth rates are clearly related to small bubble sizes existing soon after nucleation, while the low values of bubble growth rate during later times are related to near zero excess bubble pressure. At intermediate values of bubble radius and pressure, growth rate attains the maximum value.

Fig. 7. (a) History of bubble growth till completion of nucleation. (b) Bubble pressure variation shown up to the end of nucleation. (c) Hydrogen concentration in the active volume of the melt. (d) Variation of nucleation rate with time.

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Fig. 10. Nucleation rate as a function of time and initial supersaturation. Fig. 8. Discrete bubble size distribution obtained in the simulation for 1.5 wt% TiH2 content (200 bar supersaturation pressure).

The stage I bubble growth regime can be divided into (a) ‘free expansion’ stage, when bubble pressure is higher than the ambient value and (b) ‘limited expansion’ stage, when bubble pressure remains near ambient value. During free expansion, momentum transfer limits bubble growth, while during limited expansion, solute transfer controls bubble growth. A similar situation is reported in polymer foam expansion by Mao et al. (2006). However, for similar solute super saturation values, critical momentum transfer time (Eq. (9)) for polymers is about 8 orders of magnitude higher than for metals, resulting in delayed initial stage of bubble expansion in case of polymer foaming, and relatively slow pressure drop thereafter till end of nucleation. In metal foams, bubble pressure drops to near ambient values in much shorter times due to very small values of critical momentum transfer times. Fig. 7(c) shows the variation of average solute concentration in the ‘active’ volume as a function of time. Average hydrogen concentration in the ‘active’ volume shows a marginal decrease initially followed by a sharp fall thereafter. In the beginning, bubbles are small and few in number. At this stage, the amount of gas consumed in the bubbles and their surrounding ‘passive’ volumes is less, thus not affecting average solute content in the ‘active’ volume.

Fig. 9. Individual bubble growth as a function of initial supersaturation pressure.

However, with time the size and the number of bubbles increases due to growth of existing bubbles and also due to fresh nucleation, leading to rapid depletion of solute in the melt. The variation of nucleation rate with time can be seen in Fig. 7(d). With decrease in average concentration in the ‘active’ volume, nucleation rate decreases appreciably. As a result of continuous nucleation and growth of bubbles, and their expanding ‘passive’ volumes, the available volume for nucleation (‘active’ volume) decreases with time (figure not shown). 5.4. Stage II bubble growth and final bubble size distribution As explained in Section 3.2.2, after nucleation stops bubble growth continues till chemical equilibrium is reached between the bubbles and the surrounding melt. Eq. (19) can be used to calculate the final bubble size distribution. Fig. 8 shows the predicted discrete bubble size distribution for foam made with 1.5 wt% TiH2 addition. It can be seen that larger bubbles, having nucleated earlier at higher nucleation rates, are more abundant than their smaller counterparts. The small plateau near the maximum bubble sizes is most likely due to the fact that nucleation rate does not vary appreciably

Fig. 11. Cumulative bubble size distribution as a function of initial supersaturation pressure.

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Fig. 12. (a) Vertical section of a typical foam ingot; lines show the locations where the sections were taken in each experimental foam for cell size measurements. (b) Section 1 of the foam prepared using 1.5 wt% of TiH2 content, cell walls are seen as bright and interiors as dark. (c) Section 1 of the foam prepared using 1.5 wt% of TiH2 content, after inversion cell walls are seen as dark and interiors as bright. (d) Processed image in ‘ImageJ’. Each cell has been taken into account during the measurements.

with time in the beginning (Fig. 7(d)), producing similar number of large sizes bubble. For the base line condition, predicted bubble diameter varies from 0.05 to 2.8 mm. 5.5. Effect of initial solute supersaturation on bubble growth Fig. 9 shows stage I bubble growth for three initial supersaturation levels (expressed as equivalent pressures). With increase in supersaturation, bubble growth becomes faster, and nucleation stops earlier. Overall, the bubble size for the same age bubbles and the maximum bubble size at the end of stage I growth were larger for higher super saturations. 5.6. Effect of initial supersaturation pressure on the nucleation rate Variation of bubble nucleation rate with time for three different supersaturation pressure conditions is presented in Fig. 10.

Initial nucleation rate remained the same for all the three cases. This has been explained in Section 5.2 based on the assumption of availability of cavities of a range of angles each being suitable to act as substrate for heterogeneous nucleation for given super saturation pressures. With nucleation and growth, gas supersaturation decreases, reducing the nucleation rate. The duration of stage I growth (limited by the end of nucleation) is observed to be the shortest for 200 bar supersaturation pressure due to the higher growth rates which quickly deplete the melt of the solute.

5.7. Effect of initial supersaturation pressure on final bubble size distribution Cumulative bubble size distributions for three different hydrogen supersaturation pressures are shown in Fig. 11. Any point in the curve represents total number of bubbles of all sizes present up to the corresponding diameter. Thus, the number of bubbles on the curve corresponding to the largest bubble diameter (i.e. number of

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Fig. 14. Comparison of experimental and theoretical bubble size distributions. Fig. 13. Measured cell size distribution at different sections of aluminum foam prepared using 1.5 wt% TiH2 content. Average of these distributions is also shown.

bubbles of size less than the maximum bubble size) is equal to the total number of bubbles in the system. With increase in supersaturation pressure total numbers of bubbles are less and the maximum bubble size at the end of stage II is higher. The latter is the result of higher growth rates occurring under greater supersaturation pressure (Fig. 9). As a result of the shorter duration of stage I under high supersaturation condition (Fig. 10), total number of bubbles is less. 5.8. Final expansion ratio Final foam expansion ratio was calculated using Eqs. (20)–(22) for three supersaturation pressure conditions mentioned in Table 1, and other process conditions mentioned in Table 2. The difference in the foam volume and melt volume found experimentally (2950 cm3 for 1.5 wt% TiH2 content, 2100 cm3 for 1 wt% TiH2 content, and 1350 cm3 for 0.5 wt% TiH2 content) matched the volume of hydrogen input into the system confirming the numerical scheme with respect to mass balance. 5.9. Comparison of the bubble size distributions Fig. 12(a) schematically shows typical locations where the foams were sectioned. The sections were brush coated with graphite emulsion and polished. Fig. 12(b) shows the image of typical section of foam prepared with 1.5 wt% TiH2 addition in the as polished condition. Light regions represent cross sections of cell walls, while dark regions represent graphite coated cell walls. Fig. 12(c) is an inverted image of Fig. 12(b) in which the contrast is reversed to give more clarity to the cell boundaries. Finally, ‘ImageJ’ software was used to process the image further (Fig. 12(d)) to determine individual cell areas from which equivalent diameters were calculated. Fig. 12(d) is the processed image of the software which shows the cell boundaries after evaluation. It clearly shows the absence of any unintended continuities between the cells. Cells with diameter less than 0.5 mm were neglected owing to poor resolution at 1:1 magnification. The equivalent diameters of all the cells were classified into equal size intervals of 0.5 mm width and the normalized number density distribution was determined. Similar procedures were undertaken for Sections 2 and 3 of the foam to determine the cell size distribution in each section. Fig. 13 shows discrete cell size distributions corresponding to the three sections and also the ‘average’ distribution combining data from all the three sections.

In order to determine total number of bubbles present in the foam and average cell size, the curves in Fig. 13 are not sufficient as they only give the number of cell sections present per unit area of the foam, whereas number of cells per unit volume of foam is required for model validation. Hence, a method developed by Schwartz–Saltykov (DeHoff and Rhines, 1968) was used to modify the area normalized distribution curve into volume normalized distribution curve assuming the cells to be spherical. The detailed mathematical procedure is described in the referred literature and not discussed here. Using the above technique, volume normalized distribution curves were determined for all the foams. The volume normalized cell size distributions averaged over three sections and theoretically predicted bubble size distributions for all the foams are shown in Fig. 14. The simplicity of the assumptions in the model allows only trend comparisons at this stage. The predicted discrete bubble size distributions are monotonically increasing since higher nucleation rates prevail initially due to high initial supersaturation levels, resulting in largest number of bubbles getting longest durations to grow to the largest sizes. On the contrary, experimental bubble size distributions show a peak at intermediate cell sizes. This is attributed to the fact that, in real experiments, supersaturation builds over a period of time and then decays affecting the nucleation rates similarly. Secondly, the maximum bubble size of experimental foam is higher than theoretical estimations. This is because in real foams TiH2 dissociates over a period of time, thus letting initially nucleated cells to grow over longer periods of time (i.e. to larger sizes) than theoretically predicted values. Also, bubble coarsening and coalescence remain important possibilities which have not been considered in the model. In spite of these, the differences in each size class are within the same order of magnitude. Next, total number of bubbles and average cell size were calculated from the experimental curves and compared with the predicted values (see Fig. 15). It can be seen from the figure that the total number of bubbles predicted by the model is higher than the number estimated from the experiments, though both are of the same order of magnitude. This can be related to the extremely high initial nucleation rates predicted by the model, and the fact that mesoscale bubbles were not counted in experimental foams. The average bubble sizes predicted by the model for foams with 0.5 and 1.0% TiH2 additions were comparable to the experimentally observed values, while for the 1.5% TiH2 addition level, larger deviation between the two is observed. This is because the predicted distribution curves become wider with increase in TiH2 quantity

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model is insensitive to such microscale phenomena. However, the model is sensitive to material and process parameters relevant to industrial practices justifying its utility. 6. Conclusions • Bubble size distribution was numerically predicted in liquid aluminum foam prepared under high initial gas super saturation in Al–Ca melt without preexisting bubbles. • The assumed value of hydrogen partial pressure is insufficient to cause homogeneous nucleation of bubbles, and also heterogeneous nucleation on flat surfaces of non-wetting oxides. Observable nucleation rates are obtained only when nucleation in cavities of non-wetting oxides is allowed. • Experimental cell size distributions for foams made with three different TiH2 addition levels were compared with model predictions. Predicted cell size distributions are observed to increase monotonically to the maximum size whereas experimental cell size distribution shows an intermediate peak. • Predicted cell size range and total number of bubbles was of the same order of magnitude as experiments. The range of cell sizes was narrower and the total number of bubbles was higher than the observed values due to instantaneous supersaturation assumption and neglecting coalescence in the model. Acknowledgements The authors wish to thank the Defence Research & Development Organization for financial assistance and colleagues from Light Alloy Casting Group, DMRL for experimental assistance. References

Fig. 15. (a) Variation of total number of bubbles with TiH2 content. (b) Variation of average cell size with quantity of TiH2 content.

as seen in Fig. 14. Given the uncertainties in assumptions made with regard to oxide size, volume fraction and cavity geometry, the match between predicted and observed values of the cell size is considered very good, and encourages to explore the model further by incorporating TiH2 dissociation kinetics and possibly cell coarsening phenomena. 5.10. Limitations of the model The model is expected to be limited to predicting foam characteristics in a certain range of supersaturations. For example, simulations were carried out under conditions of up to five times higher solubility number than the base line value given in Table 2, representing a condition in which five times higher solute is dissolved for the same overpressure. The predicted foam expansion ratio in this case was 32:1. Such high expansions are not feasible since they require cell walls which are too thin to resist coalescence and collapse under self weight at the foaming temperatures. Secondly, at lower super saturations, cavities on the oxide surface are required to be too sharp for nucleation to be feasible. Total number of such fine cavities may be too small in number or may not be sustained in real experiments, as they may tend to open up or rupture under high stirring speeds prevailing during processing. The

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