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Modelling of microstructure evolution in transient-liquid-phase diffusion bonding under temperature gradient
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Scripta Materialia 60 (2009) 780–782 www.elsevier.com/locate/scriptamat
Modelling of microstructure evolution in transient-liquid-phase diffusion bonding under temperature gradient M.A. Jabbareh and H. Assadi* Tarbiat Modares University, Department of Materials Engineering, Tehran, Iran Received 26 December 2008; revised 8 January 2009; accepted 9 January 2009 Available online 19 January 2009
A multiple-grain phase-field model is used to study formation of wavy bond lines in transient-liquid-phase diffusion bonding. Simulations show that the prime condition of unidirectional interface migration can be achieved under significantly smaller temperature gradients than those predicted by analytical models. They also show that morphological instabilities may form not only during solidification, but also during melting, i.e. on the retreating interface. Formation of melting perturbations is explained in light of a tailored version of the constitutional undercooling criterion. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Diffusion bonding; Phase field models; Solidification microstructure; Morphological instability
Transient-liquid-phase (TLP) diffusion bonding is a process for joining materials for which other bonding processes might not be viable. A main challenge in this process is that conventional methods may lead to a planar bond line with agglomerated oxide, and hence to degradation of the bond strength [1]. It has been shown that imposing a temperature gradient across the interface during bonding can alleviate this problem [2,3]. The basic aspects of TLP diffusion bonding under a temperature gradient (TG-TLP) have been explained in detail elsewhere [2]. Briefly, the presence of a temperature gradient across the liquid phase promotes unidirectional migration of both solid/liquid interfaces from the colder side towards the hotter side of the bond region, while the liquid phase shrinks and eventually vanishes due to depletion of solute from liquid. The common notion has so far been that during this migration, the advancing solid/liquid interface may become morphologically unstable, leading to a non-planar bond line. The problem of non-planar bond-line formation in TG-TLP diffusion bonding has already been tackled using analytical models [3]. These models are, nevertheless, based on a number of simplifying assumptions, and contain several adjustable parameters. Most importantly, previous models have focused only on the
* Corresponding author. Tel.: +98 21 82883305; fax: +98 21 88006544; e-mail: [email protected]
advancing solid/liquid interface, and almost completely disregarded the retreating interface. The present work, in contrast, aims to provide a basis for quantitative prediction of the final microstructure of the bonded region, by taking account of the entire system, including both the advancing and the retreating interfaces. For this purpose, a multiple-grain model is used for the simulation of microstructure evolution in TG-TLP bonding. Details of the modelling method are given elsewhere [4–6]. In brief, the model allows for a realistic simulation of polycrystalline solidification by combining the standard phase-field approach for solidification modelling [7] with a probabilistic algorithm for the evolution of crystal orientation. The simulation domain in the present analysis consists of two polycrystalline solid phases (pure aluminium) separated by a liquid layer (Al–30%Cu). The heat transfer is assumed to be quasi steady state, in that the temperature profile is taken as a straight line connecting the boundary values. The parameters used for the calculations are given in Table 1. Note that some of the thermodynamic parameters, e.g. the melting temperature of Cu, are deliberately adjusted to hypothetical values, with the sole aim of reproducing the Al-rich side of the phase diagram. Figure 1 shows the calculated bond-line displacement, Xb, as a function of temperature gradient, for an initial liquid thickness of 5 lm. According to the simulations, Xb increases with increasing temperature gradient, whereas it remains constant in the analytical solution [2]. This is so because the analytical model neglects spatial
1359-6462/$ - see front matter Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2009.01.013
M. A. Jabbareh, H. Assadi / Scripta Materialia 60 (2009) 780–782
781
Table 1. Input parameters used in the numerical modelling of TG-TLP diffusion bonding of pure aluminium using a Al–30%Cu interlayer. Parameter
Value
Melting temperature of pure Al Melting temperature of pure Cu Latent heat of Al Latent heat of Cu Initial liquid composition Solute diffusivity in the liquid layer Mobility of solid/liquid interface Energy of solid/liquid interface Thickness of solid/liquid interface Interaction parameter for solid Interaction parameter for liquid Mesh spacing Time increment
933 K 650 K 2333 MJ m3 1625 MJ m3 0.3 mole fraction 2 108 m2 s1 0.2 m s–1 K1 0.5 J m2 6 108 m 10 kJ mol1 2 kJ mol1 1 107 m 4 109 s Figure 2. Calculated microstructure selection map for TG-TLP diffusion bonding, showing three different regions of: (1) no cooling, leading to planar and unidirectional migration of the two solid/liquid interfaces; (2) low-gradient/high-cooling, leading to non-planar solidification from both sides; and (3) high-gradient/low-cooling, leading to the favourable condition for non-planar and unidirectional migration of both interfaces. The solid line signifies the boundary between the regions 2 and 3 according to the present simulations, whereas the dashed line corresponds to that obtained from the analytical model [3] in which the critical gradient is given as: Gcritical = qW0/2D.
Figure 1. Calculated bond-line displacement as a function of temperature gradient in TLP diffusion bonding. The inset shows schematic composition profiles across the bonded region.
variations in the equilibrium compositions, which is not necessarily a valid assumption, especially when the temperature gradient, G, the slope of liquid, m, or both G and m are large. The equilibrium compositions would in fact vary to match the respective interfacial temperature, which would naturally increase as the interface moves forward under a positive temperature gradient. As shown in the inset of Figure 1, for a binary system of negative m, such variations lead inevitably to an extension of the bond-line displacement. A further outcome of the previous analysis has been that temperature gradient alone does not lead to morphological instabilities. Simulations also confirm this. As shown in Figure 2, in situations without cooling (q = 0, region 1), the solid/liquid interface remains planar, regardless of the magnitude of the temperature gradient. By increasing the cooling rate, in contrast, the solid/liquid interface becomes unstable and thus— depending on the thermal conditions—wavy or dendritic patterns emerge. Also in agreement with the previous analysis, simulations show that higher cooling rates lead to larger deviations from planar growth. They further demonstrate that unidirectional migration of the solid/ liquid interfaces occurs only beyond a critical temperature gradient (region 3). Below this critical G, which scales with cooling rate, the interlayer solidifies from
both sides, leading to unfavourable segregation at the bond line (region 2). It is interesting to note that, despite the qualitative agreement between simulations and previous analysis, the analytical model overestimates the critical gradient by a factor of more than two. The reason for this discrepancy lies in the fact that the analytical model ignores the velocity term in the differential equation of diffusion, i.e. Eq. (10) in Ref. [3]. As a result of this oversimplification, diffusive fluxes towards the hotter interface are underestimated, and hence, the threshold of melting at this interface shifts towards higher temperature gradients. A most remarkable feature in the present simulations is the formation of what appear to be morphological instabilities on the retreating interface. This is shown
Figure 3. Simulated microstructure evolution during TLP diffusion bonding subject to temperature gradient and cooling, showing formation of morphological instabilities on the retreating interface. The indicated times are normalized with respect to the final process time.
782
M. A. Jabbareh, H. Assadi / Scripta Materialia 60 (2009) 780–782
Figure 4. Schematic representation of the conditions for morphological instabilities during solidification and melting in TLP diffusion bonding.
in Figure 3. In analogy with solidification instabilities, which emerge on the advancing solid/liquid interface, ‘‘melting instabilities” appear to form only when there is thermal gradient and cooling at the same time. The latter perturbations, however, emerge at later stages of the process, and grow to smaller amplitudes, as compared to those on the solidifying front. This phenomenon can be explained as follows. Similar to the wellknown constitutional undercooling criterion [8], one may link the driving force for the growth of melting perturbations to the following parameter: /m ¼ G mGc ;
ð1Þ
which signifies the difference between the temperature gradient, G, and the gradient of the liquidus, mGc, at the retreating interface. This parameter can be conceived to represent the degree of ‘‘constitutional superheating”. Note that both G and Gc are evaluated for a case where the solid domain is to the right of the liquid domain, so that melting occurs when the liquidus gradient, mGc, is positive. Figure 4 summarizes schematically the conditions leading to solidification and melting morphological instabilities in TLP bonding. Following the analytical approximation described in Ref. [3], /m also can be shown to scale with the cooling rate according to: /m ¼
qW 0 ; 2D
ð2Þ
where W0 is the initial liquid thickness and D is the solute diffusivity in liquid. This relation is identical to that obtained for the degree of constitutional undercooling at the advancing interface. Despite these similarities, there are fundamental differences between the two types of morphological instabilities. On the retreating interface, the perturbations ‘‘grow” into the phase of the significantly lower solute diffusivity, i.e. into the solid. As a result, the depth into which they penetrate is controlled not by the solute profile within the solid, but rather by the solute influx delivered to the interface from the adjacent liquid. In fact, the solute influx at the retreating interface can be conceived to carve the roots of what would appear as solidification perturbations on an advancing interface. These findings appear to be in general agreement with previous experimental observations [3,9]. Careful examination of these observations shows evidence of melting perturbations, presumably caused by effects other than non-uniform diffusion ahead of the solidification perturbations. Overall, the present analysis illustrates microstructural features beyond what can be shown by the existing analytical models alone. The findings provide a basis for understanding the sequence of microstructure formation in TLP diffusion bonding, as well as in processes—such as fusion welding—where melting constitutes a major stage of the process. [1] A.A. Shirzadi, H. Assadi, E.R. Wallach, Surface and Interface Analysis 31 (2001) 609. [2] A.A. Shirzadi, E.R. Wallach, Acta Materialia 47 (1999) 3551. [3] H. Assadi, A.A. Shirzadi, E.R. Wallach, Acta Materialia 49 (2001) 31. [4] H. Assadi, A phase-field model for crystallization into multiple grain structures, in: D.M. Herlach (Ed.), Solidification and Crystallization, Wiley–VCH, Weinheim, 2004, pp. 17–26. [5] H. Assadi, Acta Materialia 55 (2007) 5225. [6] H. Assadi, M. Oghabi, D.M. Herlach, Acta Materialia (2009), doi:10.1016/j.actamat.2008.12.004. [7] W.J. Boettinger, J.A. Warren, C. Beckermann, A. Karma, Annual Reviews in Material Research 32 (2002) 163. [8] W. Kurz, D.J. Fisher, Fundamentals of Solidification, Trans Tech Publications, Switzerland, 1989. [9] A.A. Shirzadi, E.R. Wallach, Science and Technology of Welding and Joining 2 (1997) 89.
Scripta Materialia 60 (2009) 780–782 www.elsevier.com/locate/scriptamat
Modelling of microstructure evolution in transient-liquid-phase diffusion bonding under temperature gradient M.A. Jabbareh and H. Assadi* Tarbiat Modares University, Department of Materials Engineering, Tehran, Iran Received 26 December 2008; revised 8 January 2009; accepted 9 January 2009 Available online 19 January 2009
A multiple-grain phase-field model is used to study formation of wavy bond lines in transient-liquid-phase diffusion bonding. Simulations show that the prime condition of unidirectional interface migration can be achieved under significantly smaller temperature gradients than those predicted by analytical models. They also show that morphological instabilities may form not only during solidification, but also during melting, i.e. on the retreating interface. Formation of melting perturbations is explained in light of a tailored version of the constitutional undercooling criterion. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Diffusion bonding; Phase field models; Solidification microstructure; Morphological instability
Transient-liquid-phase (TLP) diffusion bonding is a process for joining materials for which other bonding processes might not be viable. A main challenge in this process is that conventional methods may lead to a planar bond line with agglomerated oxide, and hence to degradation of the bond strength [1]. It has been shown that imposing a temperature gradient across the interface during bonding can alleviate this problem [2,3]. The basic aspects of TLP diffusion bonding under a temperature gradient (TG-TLP) have been explained in detail elsewhere [2]. Briefly, the presence of a temperature gradient across the liquid phase promotes unidirectional migration of both solid/liquid interfaces from the colder side towards the hotter side of the bond region, while the liquid phase shrinks and eventually vanishes due to depletion of solute from liquid. The common notion has so far been that during this migration, the advancing solid/liquid interface may become morphologically unstable, leading to a non-planar bond line. The problem of non-planar bond-line formation in TG-TLP diffusion bonding has already been tackled using analytical models [3]. These models are, nevertheless, based on a number of simplifying assumptions, and contain several adjustable parameters. Most importantly, previous models have focused only on the
* Corresponding author. Tel.: +98 21 82883305; fax: +98 21 88006544; e-mail: [email protected]
advancing solid/liquid interface, and almost completely disregarded the retreating interface. The present work, in contrast, aims to provide a basis for quantitative prediction of the final microstructure of the bonded region, by taking account of the entire system, including both the advancing and the retreating interfaces. For this purpose, a multiple-grain model is used for the simulation of microstructure evolution in TG-TLP bonding. Details of the modelling method are given elsewhere [4–6]. In brief, the model allows for a realistic simulation of polycrystalline solidification by combining the standard phase-field approach for solidification modelling [7] with a probabilistic algorithm for the evolution of crystal orientation. The simulation domain in the present analysis consists of two polycrystalline solid phases (pure aluminium) separated by a liquid layer (Al–30%Cu). The heat transfer is assumed to be quasi steady state, in that the temperature profile is taken as a straight line connecting the boundary values. The parameters used for the calculations are given in Table 1. Note that some of the thermodynamic parameters, e.g. the melting temperature of Cu, are deliberately adjusted to hypothetical values, with the sole aim of reproducing the Al-rich side of the phase diagram. Figure 1 shows the calculated bond-line displacement, Xb, as a function of temperature gradient, for an initial liquid thickness of 5 lm. According to the simulations, Xb increases with increasing temperature gradient, whereas it remains constant in the analytical solution [2]. This is so because the analytical model neglects spatial
1359-6462/$ - see front matter Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2009.01.013
M. A. Jabbareh, H. Assadi / Scripta Materialia 60 (2009) 780–782
781
Table 1. Input parameters used in the numerical modelling of TG-TLP diffusion bonding of pure aluminium using a Al–30%Cu interlayer. Parameter
Value
Melting temperature of pure Al Melting temperature of pure Cu Latent heat of Al Latent heat of Cu Initial liquid composition Solute diffusivity in the liquid layer Mobility of solid/liquid interface Energy of solid/liquid interface Thickness of solid/liquid interface Interaction parameter for solid Interaction parameter for liquid Mesh spacing Time increment
933 K 650 K 2333 MJ m3 1625 MJ m3 0.3 mole fraction 2 108 m2 s1 0.2 m s–1 K1 0.5 J m2 6 108 m 10 kJ mol1 2 kJ mol1 1 107 m 4 109 s Figure 2. Calculated microstructure selection map for TG-TLP diffusion bonding, showing three different regions of: (1) no cooling, leading to planar and unidirectional migration of the two solid/liquid interfaces; (2) low-gradient/high-cooling, leading to non-planar solidification from both sides; and (3) high-gradient/low-cooling, leading to the favourable condition for non-planar and unidirectional migration of both interfaces. The solid line signifies the boundary between the regions 2 and 3 according to the present simulations, whereas the dashed line corresponds to that obtained from the analytical model [3] in which the critical gradient is given as: Gcritical = qW0/2D.
Figure 1. Calculated bond-line displacement as a function of temperature gradient in TLP diffusion bonding. The inset shows schematic composition profiles across the bonded region.
variations in the equilibrium compositions, which is not necessarily a valid assumption, especially when the temperature gradient, G, the slope of liquid, m, or both G and m are large. The equilibrium compositions would in fact vary to match the respective interfacial temperature, which would naturally increase as the interface moves forward under a positive temperature gradient. As shown in the inset of Figure 1, for a binary system of negative m, such variations lead inevitably to an extension of the bond-line displacement. A further outcome of the previous analysis has been that temperature gradient alone does not lead to morphological instabilities. Simulations also confirm this. As shown in Figure 2, in situations without cooling (q = 0, region 1), the solid/liquid interface remains planar, regardless of the magnitude of the temperature gradient. By increasing the cooling rate, in contrast, the solid/liquid interface becomes unstable and thus— depending on the thermal conditions—wavy or dendritic patterns emerge. Also in agreement with the previous analysis, simulations show that higher cooling rates lead to larger deviations from planar growth. They further demonstrate that unidirectional migration of the solid/ liquid interfaces occurs only beyond a critical temperature gradient (region 3). Below this critical G, which scales with cooling rate, the interlayer solidifies from
both sides, leading to unfavourable segregation at the bond line (region 2). It is interesting to note that, despite the qualitative agreement between simulations and previous analysis, the analytical model overestimates the critical gradient by a factor of more than two. The reason for this discrepancy lies in the fact that the analytical model ignores the velocity term in the differential equation of diffusion, i.e. Eq. (10) in Ref. [3]. As a result of this oversimplification, diffusive fluxes towards the hotter interface are underestimated, and hence, the threshold of melting at this interface shifts towards higher temperature gradients. A most remarkable feature in the present simulations is the formation of what appear to be morphological instabilities on the retreating interface. This is shown
Figure 3. Simulated microstructure evolution during TLP diffusion bonding subject to temperature gradient and cooling, showing formation of morphological instabilities on the retreating interface. The indicated times are normalized with respect to the final process time.
782
M. A. Jabbareh, H. Assadi / Scripta Materialia 60 (2009) 780–782
Figure 4. Schematic representation of the conditions for morphological instabilities during solidification and melting in TLP diffusion bonding.
in Figure 3. In analogy with solidification instabilities, which emerge on the advancing solid/liquid interface, ‘‘melting instabilities” appear to form only when there is thermal gradient and cooling at the same time. The latter perturbations, however, emerge at later stages of the process, and grow to smaller amplitudes, as compared to those on the solidifying front. This phenomenon can be explained as follows. Similar to the wellknown constitutional undercooling criterion [8], one may link the driving force for the growth of melting perturbations to the following parameter: /m ¼ G mGc ;
ð1Þ
which signifies the difference between the temperature gradient, G, and the gradient of the liquidus, mGc, at the retreating interface. This parameter can be conceived to represent the degree of ‘‘constitutional superheating”. Note that both G and Gc are evaluated for a case where the solid domain is to the right of the liquid domain, so that melting occurs when the liquidus gradient, mGc, is positive. Figure 4 summarizes schematically the conditions leading to solidification and melting morphological instabilities in TLP bonding. Following the analytical approximation described in Ref. [3], /m also can be shown to scale with the cooling rate according to: /m ¼
qW 0 ; 2D
ð2Þ
where W0 is the initial liquid thickness and D is the solute diffusivity in liquid. This relation is identical to that obtained for the degree of constitutional undercooling at the advancing interface. Despite these similarities, there are fundamental differences between the two types of morphological instabilities. On the retreating interface, the perturbations ‘‘grow” into the phase of the significantly lower solute diffusivity, i.e. into the solid. As a result, the depth into which they penetrate is controlled not by the solute profile within the solid, but rather by the solute influx delivered to the interface from the adjacent liquid. In fact, the solute influx at the retreating interface can be conceived to carve the roots of what would appear as solidification perturbations on an advancing interface. These findings appear to be in general agreement with previous experimental observations [3,9]. Careful examination of these observations shows evidence of melting perturbations, presumably caused by effects other than non-uniform diffusion ahead of the solidification perturbations. Overall, the present analysis illustrates microstructural features beyond what can be shown by the existing analytical models alone. The findings provide a basis for understanding the sequence of microstructure formation in TLP diffusion bonding, as well as in processes—such as fusion welding—where melting constitutes a major stage of the process. [1] A.A. Shirzadi, H. Assadi, E.R. Wallach, Surface and Interface Analysis 31 (2001) 609. [2] A.A. Shirzadi, E.R. Wallach, Acta Materialia 47 (1999) 3551. [3] H. Assadi, A.A. Shirzadi, E.R. Wallach, Acta Materialia 49 (2001) 31. [4] H. Assadi, A phase-field model for crystallization into multiple grain structures, in: D.M. Herlach (Ed.), Solidification and Crystallization, Wiley–VCH, Weinheim, 2004, pp. 17–26. [5] H. Assadi, Acta Materialia 55 (2007) 5225. [6] H. Assadi, M. Oghabi, D.M. Herlach, Acta Materialia (2009), doi:10.1016/j.actamat.2008.12.004. [7] W.J. Boettinger, J.A. Warren, C. Beckermann, A. Karma, Annual Reviews in Material Research 32 (2002) 163. [8] W. Kurz, D.J. Fisher, Fundamentals of Solidification, Trans Tech Publications, Switzerland, 1989. [9] A.A. Shirzadi, E.R. Wallach, Science and Technology of Welding and Joining 2 (1997) 89.