Simulation of the neck growth of non-isometric biosphere during initial sintering

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1 Acta Metall. Sin.(Engl. Lett.)Vol.22 No.4 pp August 29 Simulation of the neck growth of non-isometric biosphere during initial sintering Jixiang DU, Shuhua LIANG, Xianhui WANG and Zhikang FAN Department of Material Science & Engineering, Xi an University of Technology, Xi an 7148, China Manuscript received 6 September 28; in revised form 11 February 29 Because powders are mostly non-isometric during the sintering process, copper powders were chosen to study the effects of four material transport mechanisms, including surface diffusion, grain-boundary diffusion, volume diffusion, and multi-couplings. These material transport mechanisms were studied with respect to sintering neck growth of a non-isometric biosphere during initial sintering. The evolution of the neck growth in the four transport mechanisms was simulated by Visual C ++ as well based on the model of different particles. The results show that the increase of the sintering temperature, both the grain-boundary diffusion and volume diffusion play primary roles in neck growth, while surface diffusion gradually becomes the secondary mechanism. Both the sintered neck and the shrinkage of the two centers increase with increasing temperature by means of the coupling diffusion mechanism. The radius of the sintering neck decreased, and the shrinkage rate of the two centers increased with an increase of the diameter ratio of the two spheres. KEY WORDS Sintering simulation; Non-isometric biosphere model; Diffusion mechanism; Sintering neck 1 Introduction Sintering is a phenomenon or process in which powders or compacts are heated to a given temperature and in a particular atmosphere until they adhere to one another. Sintering causes interparticle bonding which results in a series of physics and chemical changes and causes the aggregation of powder particles. These changes cause the aggregation of the crystal grains and the sequential production of materials with often desirable physical and mechanical properties. However, it is difficult to study the sintering process and sintering mechanism, because the processing usually occurs at high temperatures and influences other process variables. Sintering models have been extensively studied and have resulted in many significant advances. Based on the double-globe model, Luis [1] established a new model, which can be used to analyze both activation energy and dynamics according to the expansion-curve, and the results had been verified by sintering. Su et al. [2,3] presented a classical sintering theory, i.e. the main sintering curve, which can describe the sintering of Corresponding author. Professor, PhD; Tel.: ; Fax: address: liangsh@xaut.edu.cn (Shuhua LIANG) DOI: 1.116/S (8)698-8

2 264 certain powders in atmosphere regardless of the sintering path. The establishment of the main sintering curve included the sintering models of volume diffusion and interface diffusion. Hassold et al. [4] adopted a two-dimensional sintering model to study the later stage of sintering and the changes in microstructure. Xu et al. [5,6] investigated the effect of the expansion and shrinkage system on the entire surface energy through dynamic analysis and mathematical computation. The results showed that the shrinking plays a dominant role during the sintering process. Jing et al. [7,8] studied the evolution of the coupled pore and grain growth at the later stage of sintering of ceramics by introducing a modified algorithm of a semi-implicit Fourier-spectral method into the phase-field model. The development of the sintering models based on computer simulation was very helpful in understanding the sintering process. At present, most studies involving simulating the sintering process are mainly focused on the diffusion growth of the sintering neck, the densification of powders, and the temperature and phase fields [9 11]. For convenience, almost all sintering models have assumed that the powders are isometric [12 15]. However, compact is usually described by the non-isometric-sintering model, because the billet is always composed of the powders with different dimensions [16,17]. Unfortunately, there is not a perfect quantitative description of the sintering model, and most researches have focused on a single diffusion mechanism [18 2], thereby ignoring the effective combination of many diffusion mechanisms. In order to fully understand the process of sintering, a non-isometric-sintering model is established and sintering via various diffusion mechanisms is studied in the present investigation. 2 Non-isometric biosphere model during initial sintering Fig.1 shows the non-isometric biosphere model that was developed by using the diffusion flux and the flux area of various diffusion mechanisms. In the figure, a 1 and a 2 are the radii of the two particles, x is the radius of the sintering neck, b is the intersected radius of two grains, dy is the intersected height of the two grains, and x 1 and y 1 are the parameters of the sintering neck of sphere a 1. The variables x 2 and y 2 are the parameters of the sintering neck of sphere a 2. Furthermore, ρ 1 and ρ 2 are the curvature radii of neck of the two spheres respectively. The following expressions can be obtained from a geometrical relation as Fig.1 Geometric model of two non-isometric particles.

3 265 shown in Fig.1 [21] : x 1 = ρ 1 cosϕ 1 (1) y 1 = ρ 1 sinϕ 1 (2) x 2 = ρ 2 cosϕ 2 (3) y 2 = ρ 2 sinϕ 2 (4) tgϕ 1 = a 1 dy 2 ρ 1 + x tgϕ 2 = a 2 dy 1 ρ 2 + x ρ 1 = x2 2a 1 dy 2 + dy 2 2 2(a 1 x) (5) (6) (7) b = The total surface area of the sintering neck is ρ 2 = x2 2a 2 dy 1 + dy1 2 (8) 2(a 2 x) a 2 1 (a 1 dy 2 ) 2 dy = dy 1 + dy 2 (9) A = A 1 + A 2 = 2πρ 1 [ϕ 1 (a 1 + ρ 1 )cosϕ 1 ρ 1 sinϕ 1 ] + 2πρ 2 [ϕ 2 (a 2 + ρ 2 )cosϕ 2 ρ 2 sinϕ 2 ] (1) The total volume of sintering neck is [ ( V = πa 2 1cos 2 ϕ 1 y 1 2πϕ 1 ρ 2 1 x + ρ 1 2y ) ( 1 2πy 1 x 1 3ϕ 1 x + ρ 1 2x 1 3 πy [ ( 1 6 (3a2 1cos 2 ϕ 1 + 2b 2 + y1) 2 + πa 2 2cos 2 ϕ 2 y 2 πϕ 2 ρ 2 2 x + ρ 2 2y ) 2 3ϕ 2 πx 2 y 2 (x + ρ 2 2 )] 3 x 2 πy 2 6 (y b 2 + 3a 2 2cos 2 ϕ 2 ) (11) Copper powders were used for these simulations. Some physical properties of the copper are shown in the Table 1. Table 1 Physical parameters of the copper D os D ob D ov γ s γ b γ v cm 2 /s cm 2 /s cm 2 /s J/cm 2 J/cm 2 J/cm Q os Q ob Q ov δ s δ b Ω kj/mol kj/mol kj/mol cm cm cm )] 2.1 Surface diffusion mechanism During surface diffusion [22], the cavities on the surface of the particles are gradually filled during the process of sintering, due to the diffusion of the surface atoms and the increase in surface bonding. According to Mullins [18], during sintering with surface diffusion

4 266 the surface chemical gradient is proportional to the surface curvature, and the surface diffusion flux is in proportion to the chemical gradient. The surface diffusion flux in the non-isometric biosphere model can be expressed as follows: γ s Ω ( 1 j s = D s 1 2kT ρ 1 ρ 1 x + 2 ) a 1 + D s γ s Ω 2kT ρ 2 ( 1 ρ 2 1 x + 2 a 2 ) where, relevant variables are defined as follows: D s is the surface diffusion coefficient; γ s is the surface free energy per unit area; Ω is the atomic volume; k is Boltzmann s constant; T is absolute temperature. The flux area of surface diffusion is (12) A s = 2πxδ s (13) where δ s is the surface diffusive width. Therefore, the volume increment of the sintering neck for surface diffusion is the diffusion flux multiplied by the flux area. dv dt = j sa s (14) The surface diffusion coefficient D s can be deduced from Eq.(15) [23]. D s = D os exp( Q os /RT ) (15) where, D os is the constant of surface diffusion; Q os is the activation energy of surface diffusion; R is the constant of gas. From the above equations, these parameters, such as the time to reach the pre-setting value, the volume of the sintering neck, the radii of curvature, and the surface area, can be accurately calculated using VC ++ only for the surface diffusion mechanism. The dynamic simulation result is shown in Fig.2. In Fig.2 it can be seen that the surface diffusion is caused by the flow along the surface of particles, which cause the configuration of the neck to changes with surface morphology. Fig.2 Simulated result of the neck growth at surface diffusion. 2.2 Grain-boundary diffusion mechanism From the grain-boundary diffusion mechanism [24], the migration of matter is mainly a diffusion process from a grain-boundary to a pore. The main driving force of the interface diffusion is a result of chemical and stress gradients along the interface. A steady grain boundary can easily be formed on the contact surface of grains. The activation energy of grain boundary diffusion is only half of the volume diffusion, but its diffusion coefficient is 1 times larger than that of volume diffusion. The grain-boundary diffusion flux can be expressed as shown: γ b Ω j b = 4D b (x + ρ) (16) kt ρx2

5 267 where, ρ= 1 2 (ρ 1+ρ 2 ), D b is the grain boundary diffusion coefficient, γ b is the grain-boundary free energy per unit area, flux area of grain boundary diffusion is A b = 2πxδ b (17) where δ b is diffusion width of the grain boundary. Therefore, the volume increment of the sintering neck for grain boundary diffusion is the diffusion flux multiplied by flux area, i.e. dy dt = j ba b (18) The grain boundary can cause the center shrinkage between the two spheres [24]. The shrinkage rate between the two spheres is The grain-boundary diffusion coefficient D b can be calculated from the Eq.(2) [23]. D b = D ob exp( Q ob /RT ) (2) dl dt = 8D bγ b Ωδ b kt ρx 3 (x + ρ) (19) where, the key variables are defined as follows: D ob is the constant of grain-boundary diffusion; Q ob is the activation energy of grain-boundary diffusion. Using a method similar to that used in Section 2.1, the dynamic simulation of the non-isometric particles was conducted only at the site of grain boundary diffusion, and the result is presented in Fig.3. The sintering neck grows linearly, and the center distance of the two particles is contracted only at the location grain-boundary diffusion. Fig.3 Simulated result of the neck growth at the grain-boundary diffusion. 2.3 Volume diffusion mechanism For the volume diffusion mechanism [25], it is believed that the intergranular material compression between the two particles migrates to the neck by volume diffusion. The diffusion is a consequence of the differential vacancy concentration between the compressed region and the neck. The volume diffusion flux can be expressed by the following equation: j v = j vs + j vb = D v γ v Ω kt ρ 1 ( 1 ρ 1 1 x + 2 a 1 ) + D v γ v Ω kt ρ 2 ( 1 ρ 2 1 x + 2 a 2 ) + γ v Ω 4D v kt ρ 1 x 2 (x + ρ γ v Ω 1) + 4D v kt ρ 2 x 2 (x + ρ 2) (21) where, j vs is the volume diffusion flux via surface; j vb is the volume diffusion flux via grain-boundary; γ v is volume free energy per unit area; D v is volume diffusion coefficient.

6 268 Flux area of volume diffusion is represented as follows: A v = A v1 + A v2 A v1 = 2πρ 1 [ϕ 1 (a 1 + ρ 1 )cosϕ 1 ρ 1 sinϕ 1 ] A v2 = 2πρ 2 [ϕ 2 (a 2 + ρ 2 )cosϕ 2 ρ 2 sinϕ 2 ] (22) Therefore, the volume increment of the sintering neck for volume diffusion is the diffusion flux multiplied by flux area, i.e. dv dt = j va v (23) The shrinkage rate of the distance between the two centers is dl dt = 4D vγ v ΩA v1 πkt ρ 1 x 4 (x + ρ 1) + 4D vγ v ΩA v2 πkt ρ 2 x 4 (x + ρ 2) (24) The volume diffusion coefficient can be calculated from the equation [23]. where, D ov is the constant of volume diffusion; Q ov is activation energy of volume diffusion; The dynamic simulation of volume diffusion is given in Fig.4, and suggests that volume diffusion can also cause the shrinkage of the two centers. D v = D ov exp( Q ov /RT ) (25) Fig.4 Simulated result of neck growth at the volume diffusion. 2.4 Coupling effect of the surface diffusion, grain-boundary diffusion and volume diffusion Each diffusion mechanism should be considered during sintering. Under the coupling mechanism of surface diffusion, grainboundary diffusion and volume diffusion, the total mass increments at the neck should be the sum of mass increments for surface diffusion, grain-boundary diffusion, and volume diffusion, i.e. dv dt = j sa s + j b A b + j v A v (26) The shrinkage rate of the distant between the two centers is dl dt = 8D bγ b Ωδ b kt ρx 3 (x + ρ) + 4D vγ v ΩA v1 πkt ρ 1 x 4 (x + ρ 1) + 4D vγ v ΩA v2 πkt ρ 2 x 4 (x + ρ 2) (27) The result of the dynamic simulation is shown in Fig.5. It can be seen that the sintering neck at the coupling mechanism is composed of two arcs with different radii at the ends. Here, the shrinkage of the two centers occurs as well.

7 269 3 Simulation Results and Discussion 3.1 Role of different diffusion mechanisms in the neck growth during sintering Based on the non-isometric biosphere model presented above, the dominant mechanism is determined by comparing the time needed to form the same sintering neck for different diffusion mechanisms. Fig.6a Fig.6e are the growth rates of the neck at 75 C, 8 C, 85 C, 9 C and 1 C for the surface diffusion mechanism, the grain-boundary diffusion mechanism, the Fig.5 Simulated result of the neck growth at coupling diffusion. volume diffusion mechanism, and the coupling diffusion mechanism, respectively when the diameter ratio of the two copper spheres is.5 (1 µm/2 µm). It can be seen from Fig.6a and Fig.6b that the formation time of a sintering neck with the same size is the smallest for surface diffusion. Also, the time for the grain-boundary diffusion is smaller, while the time for volume diffusion is the longest. Furthermore, the time required for the coupling diffusion mechanism is shorter than of volume diffusion, but much longer than for surface diffusion and grain-boundary diffusion. The neck growth rate at 85 C is shown in Fig.6c. In comparison with that at 75 C and at 8 C, it can be seen that the grain-boundary diffusion rather than the surface diffusion is the dominant mechanism. The formation time for grain-boundary diffusion is the shortest, while the t / s (a) Surface diffusion Grain-boundary diffusion Volume diffusion Couping diffusion t / s (b) Surface diffusion Grain-boundary diffusion Volume diffusion Couping diffusion t / s (c) Surface diffusion Grain-boundary diffusion Volume diffusion Coupling diffusion t / s (d) Surface diffusion Grain-boundary diffdusion Volume diffusion Coupling diffusion t / s (e) Surface diffusion Grain-boundary diffdusion Volume diffusion Coupling diffusion Fig.6 Comparison of neck growth rates of diffusion mechanism at 75 C (a), 8 C (b), 85 C (c), 9 C (d) and 1 C (e).

8 27 surface diffusion mechanism is shorter, and the changes of volume diffusion mechanism and the coupling diffusion mechanism are least. It can be seen from the Fig.6d and Fig.6e that at 9 1 C the effect of surface diffusion and volume diffusion mechanisms are pronounced. The formation time for the coupling diffusion mechanism is the longest, but is far below that at 75 C and 85 C. As discussed above, it can be found that, all diffusion mechanisms affect the neck growth during sintering, but the effects of each diffusion mechanisms varies at different temperatures and times. The general trend is that the surface diffusion gradually becomes the secondary mechanism from the dominant mechanism below 85 C, while the grainboundary diffusion and the volume diffusion gradually contribute more to the neck growth and become the dominant mechanisms above 85 C. The trend is closely related to all diffusion mechanisms. Other studies [26] have shown that the characterization of copper could be expressed by Eqs.(28) (3) during surface diffusion, grain-boundary diffusion and volume diffusion. When the activation energy of copper is 8.7, 21 and 42 times larger than the melting point, surface diffusion, grainboundary diffusion and volume diffusion will easily occur, respectively. H surface /T melting point = 8.7 (28) H grain boundary /T melting point = 2.1 (29) H volume /T melting point = 42 (3) Therefore, the effect of surface diffusion on the neck growth is very obvious at lower temperatures, while the volume diffusion and the grain-boundary diffusion will act on the neck growth at higher temperatures. When the grain-boundary diffusion coefficient of copper is 1 times larger than its volume diffusion coefficient, grain-boundary diffusion has a larger contribution to the neck growth than the volume diffusion at elevated temperatures. 3.2 Effect of the diffusion mechanism on the shrinkage during sintering During sintering, the shrinkage of two centers occurs in parallel with neck growth. Both grain-boundary diffusion and volume diffusion can cause the shrinkage of two centers, which cannot happen with just surface diffusion. The rate of shrinkage at 1 C for the four diffusion mechanisms is shown in the Fig.7 for copper sphere diameter ratio of.5 (1 µm/2 µm). The shrinkage of two centers during volume diffusion and grainboundary diffusion increases with time, but the shrinkage during grain-boundary diffusion is larger than during volume diffusion. The shrinkage during the coupling diffusion mechanism is smaller than for grainboundary diffusion, but is larger than that for the volume diffusion mechanism. It is clear that surface diffusion slightly decreases dl / 1-8 m Grain-boundary diffdusion Volume diffusion Coupling diffusion t / s Fig.7 Shrinkage of two centers at different diffusion mechanisms at 1 C

9 271 shrinkage due to material migration to the neck by surface diffusion, which increases the radius of curvature of the sintering neck. The shrinkages for grain-boundary diffusion are larger than for the volume diffusion. On the one hand, in comparison with the grain-boundary diffusion coefficient, the volume diffusion coefficient is quite small, usually 1 times smaller than the grain-boundary diffusion coefficient. On the other hand, the concentration difference of the same material is small along the direction of volume diffusion. As a result, volume diffusion has a smaller driving force than that grain-boundary diffusion. It can be deduced from above analyses that the conditions and time should be specified when determining a dominant mechanism. The following discussions are based on the coupling diffusion mechanism. 3.3 Effect of the diameter ratio of two spheres on neck growth The diameter ratio of two spheres is an important parameter in the non-isometric biosphere model. This falls within the range of Copper was chosen as the sintering body. The sintering temperature was 1 C, the ratio changed from.5 to.75, 1, 1.25 and 1.5 when a 1 was 1 µm. The neck growth at different diameter ratios is shown in Fig.8. It can be seen that the radius of the sintering neck decreases with an increasing diameter ratio of two spheres. When the diameter ratio increases, the contact radius of the two spheres decreases, which causes a decrease in the sintering neck. It can be seen that the size of the contact radius can affect the sintering neck in the non-isometric biosphere model (as shown in Fig.1). Fig.9 shows the shrinkage of two centers at different sphere diameter ratios. It can be seen from Fig.9 that a smaller ratio yields a smaller shrinkage of the two centers. When the diameter is smaller, the contact radius is bigger. Shrinkage occurs first from the contact region of two particles. The shrinkage rate depends on the extents of contact. The bigger the contact radius, the more difficult it is to shrinkage the two centers. 3.4 Effect of the sintering temperature on the neck The sintering simulation was performed at 75 C, 85 C, and 95 C with a diameter ratio of.5 (ratio=a 1 /a 2 =1 µm/2 µm=.5). From the Fig.1, it can be seen that the Ratio=.5 Ratio=.75 Ratio=1 Ratio=1.25 Ratio=1.5 l / 1-7 m Ratio=.5 Ratio=.75 Ratio=1 Ratio=1.25 Ratio= lg(t/s) lg(t/s) Fig.8 Curves of the grain growth in the different diameter ratios. Fig.9 Shrinkage curves of two grains in the different diameter rating.

10 272 sintering temperature has a significant effect on the growth of the sintering neck. The neck grows rapidly with an increase in sintering temperature, because the diffusion coefficient increases sharply with an increasing in temperature. The results of this sharp increase in diffusion coefficient are as follows: (1) there is a greater probability of diffusion for the atoms to gain enough energy to overcome the potential barrier caused by the thermal fluctuation associated with increasing temperature; (2) the vacancy assists in an increase in diffusion with increasing temperature. As shown in Fig.11, a higher temperature corresponds to a greater shrinkage of the two centers. At low temperature, the dominant diffusion mechanism is surface diffusion, but it cannot cause the shrinkage of two centers. However, at elevated temperatures both the grain-boundary diffusion and the volume diffusion become the dominant mechanisms that could cause the shrinkage of two centers. Particularly because of the grain growth, the grain boundary migrates across pores, decreasing a large number of pores and increasing the shrinkage of the two centers o C 85 o C 75 o C o C 85 o C 75 o C l / 1-7 m lg(t/s) lg(t/s) Fig.1 Curves of the grain growth at the different temperatures. Fig.11 Shrinkage of two centers at different temperatures. 3.5 Effect of sintering time on the neck Fig.12 shows the shrinkages of the centers of two spheres at 95 C over a range of times for a diameter ratio of.5 (ratio=a 1 /a 2 =1 µm/2 µm=.5). The curve was obtained by polynomial fitting. It can be seen that the shrinkage can be divided into two stages during isothermal sintering. The center of two grains shrinks quickly at the primary stage and then shrinks slowly until the last moment when shrinkage almost ceases. This result is in accord with the practice of obtaining higher density of compact by prolonging sintering time. The change l / 1-7 m Simulate data Polynomial fit lg(t/s) Fig.12 Polynomial fitting curve of the shrinkage of the two grains with the time. of the porosity depends on the pore that is interconnected with the surface. Some pores are fully filled while some pores are separated or closed. At the initial stage of sintering, the interconnected pores can become to be fully

11 273 filled quickly, so that the shrinkage of the two centers occurs rapidly. When the remaining pores are closed, the decrease in the quantity of pores results in a decrease in the rate of shrinkage. 4 Experimental Verification In order to validate the results of simulating, copper powders with diameters of 3 µm and 38 µm were chosen to sinter at 15 C for 3 min. The fracture of the samples was observed by scanning electron microscopy (SEM). The SEM micrograph of the sintering neck is shown in Fig.13. It can Fig.13 SEM micrograph of the sintering neck of copper powders. be seen from Fig.13 that the diameter of the sintering neck is approximately 21 µm. The diameter of the sintering neck (d) was calculated using the coupling diffusion model and was found to be µm. The calculated result is almost in accordance with the measured result. 5 Conclusions With increasing temperature, the surface diffusion gradually becomes the secondary mechanism from the originally dominant mechanism below 85 C, while grain-boundary diffusion and volume diffusion play dominant roles in growth of neck above 85 C. The increase in the diameter ratio of two spheres cause the decrease in the sintering neck growth and the increase in the shrinkage of the two centers. For coupling diffusion, both the sintering neck and the shrinkage of two centers increase with sintering temperature. For the process of the isothermal sintering over time, the center of two grains shrinks quickly during the primary stage, then shrinks slowly and the almost ceases during later stages. In order to obtain a dense compact, the sintering temperature should be increase quickly and the dominant diffusion mechanisms should be volume diffusion and grain-boundary diffusion. Acknowledgements The present work was supported by the National Natural Science Foundation of China (No ), New Century Excellent Talents in University (NCET-5-873) and Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP26711 and 4JC22). REFERENCES [1] A.P.M. Luis, M.C. José and R. Concepción, J Am Ceram Soc 85(22) 76. [2] H.H. Su and D.L. Johnson, J Am Ceram Soc 79 (1996) [3] H.H. Su and D.L. Johnson, J Am Ceram Soc 79 (1996) [4] N.H. Gregory, I.W. Chen and J.S. David, J Am Ceram Soc 73 (199) [5] S.M. Xu, H.D. Ding and Y. Li, The Chinese J Nonferrous Metal 11 (21) 176. [6] S.M. Xu, H.D. Ding and Y. Li, J Arm Force Eng Institute 15 (21) 22. [7] X.N. Jing, Y. Ni and L.L. He, J Inorganic Mater 17 (22) 178. [8] X.N. Jing, Y. Ni and L.L. He, Mater Sci Eng 21 (23) 17. [9] J. Anand and R.D. Raul, J Am Ceram Soc 73 (199) 173.

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