SCIENCE CHINA Physics, Mechanics & Astronomy • Article • August 2012 Vol.55 No.8: 1371–1375 doi: 10.1007/s11433-012-4804-8 Structure and property of metal melt IV—Evolution of titanium melt residual bond structure and its effect on dynamic viscosity MI GuangBao1,2*, CAO JingXia1, HUANG Xu1, CAO ChunXiao1, LI PeiJie2 & HE LiangJu2 1 Aviation Key Laboratory of Science and Technology on Advanced Titanium Alloys, Beijing Institute of Aeronautical Materials, Beijing 100095, China; 2 National Center of Novel Materials for International Research, Tsinghua University, Beijing 100084, China Received October 31, 2011; accepted December 13, 2011; published online June 20, 2012 Based on the concept of melt residual bonds, a calculating model quantitatively describing the evolution of the residual bond structure of titanium melt at the melting point or in a certain range above the melting point was established; i.e., both the size dS and the bond number n of the residual bond structure decrease monotonously with the increase of temperature. By mathematical deduction, a linear relationship between the residual bond structure size dS and the dynamic viscosity of Titanium melt was revealed, i.e., = 0.876 + 0.471·dS, which is of great significance to the investigation of the relationship between the melt microstructure and the macroscopic properties of metals with high melting temperature. titanium melt, residual bond structure, dynamic viscosity, calculating model PACS number(s): 61.25.Mv, 61.20.Gy, 36.40.-C, 66.20.+d Citation: Mi G B, Cao J X, Huang X, et al. Structure and property of metal melt IV—Evolution of titanium melt residual bond structure and its effect on dynamic viscosity. Sci China-Phys Mech Astron, 2012, 55: 13711375, doi: 10.1007/s11433-012-4804-8 Titanium has obtained increasing applications due to many excellent properties such as high specific strength and good corrosion resistance. Currently, people have acquired a clear understanding of the crystal structure of solid titanium [1]. Titanium is a transition metal with atomic number 22 and has two allotropic forms at room condition: -Ti below 1155.5 K, in hcp structure with the lattice constants a= 2.9504 Å and c/a=1.587; -Ti above 1155.5 K, in bcc structure with the lattice constant a=3.282 Å; the phase transformation from to results in a 0.17% change in volume. In addition, titanium is the metal with the highest melting point among light metals, whose melting point is 1943 K and the density is 4.50 g/cm3. However, due to the unstability and uncertainty of melt structure and the limitations of experimental conditions, experiments on the struc*Corresponding author (email: [email protected]; [email protected]. cn) © Science China Press and Springer-Verlag Berlin Heidelberg 2012 ture and properties of titanium melt are difficult to conduct, corresponding to a few research reports. Therefore, on the basis of earlier relevant researches [2–4], this paper theoretically studied the evolution of the titanium melt structure and its effect on viscosity at the melting point or in a certain range above the melting point, which laid a foundation for deeper investigations on the liquid structure of metals with high melting temperature and the microscopic nature of viscosity. 1 Evolution of the residual bond structure of titanium melt When the metal melts, the local atomic distribution still shows certain regularity; i.e., in the range of dozens, hundreds even thousands of atomic distance, the atomic arrangement is similar to that of solid and the change of orphys.scichina.com www.springerlink.com 1372 Mi G B, et al. Sci China-Phys Mech Astron dering is small. In melt, the region with such arrangement pattern is called local ordered structure or atomic cluster. Such facts have been confirmed by X-ray diffraction, neutron diffraction and other experiments [5–11]. Besides, with the development of cluster physics, it has been proved that the atomic cluster constructed by a certain quantity of atoms possesses a relatively stable structure [12–17]; e.g., the atomic cluster Al13 shows a stable structure, which provides additional evidence for the existence of local ordered structure in melt. When the liquid metal transforms into vapor, the mass spectrometry analysis on aluminium and copper showed that atomic clusters with a certain size exist in the vapor (Figure 1) [18]. As Figure 1 shows, the size of atomic clusters changes with inert gas environment; i.e., the cluster size increases gradually with the atomic size of inert gas and the cluster size increases obviously with the pressure of inert gas. The results above indicated that the bonds retained from the liquid still preserved in the metal vapor. When the atomic number of inert gas increases or the atomic size of inert gas enlarges, the chance of the interaction among the greatly weakened bonds in the vapor may grow from the point of probability. The bonds bind together and the atomic clusters form, while the inert gas atoms still move freely. To sum up, the local ordered structure of melt can be described from the view of chemical bond: assume that the atoms in free state with chemical bonds totally destructed are active atoms, and the atoms linked by the retained chemical bonds after phase transformation are defined as local ordered structure; i.e., the active atoms yield due to the destruction of a certain quantity of chemical bonds in the metal crystal, while large numbers of undestroyed chemical bonds retain in the local ordered region. From the point of statistical average, the retained chemical bonds bind the atoms other than the active atoms together in a certain arrangement pattern, and a time average of the space positions of all atoms composes local ordered structure. The retained chemical bonds in the local ordered structure are called melt residual bonds. In this sense, the local ordered structure constructed by melt residual bonds is called melt residual- Figure 1 Relationship between atomic cluster (particle) and inert gas pressure after the liquid to vapor transformation of liquid metals [18]. August (2012) Vol. 55 No. 8 Figure 2 Structure model of the metal crystal after melting [3]. bond structure, as shown in Figure 2. With atoms aggregating and dispersing randomly, the melt residual bond structure fluctuates under the statistical law. The main parameters describing the melt residual bond structure are the statistical average size ds and the residual bond number n in the structure. Based on the concept of melt residual bonds, the evolution of residual bond structure can be described quantitatively by the melt structure information calculating model [3,4]; i.e., ds and n are: Z 1 C Q 1 1 0 d (T ) 2r 1 exp 1 1 , k T T 2 C0 m 3 (1) Q 1 1 Z 3 1 C0 exp 1 , n(T ) 1 k T T 8 C0 m (Tm T TC ), where k is the Boltzmann constant; α, the geometrical morphology factor (0<1); Z1, the coordination number of metal before melting; r, the half atomic distance in the residual bond structure, which is assumed to be equal to the single-bond radius of atom; Q, the activation energy; C0, the relative concentration of active atoms at the melting point; Tm, the melting point; TC, the temperature when the first transformation of melt structure happens during the process from liquid to vapor. The thermophysical and structure parameters of titanium are [19,20]: Tm =1943 K, Hb=425.0 kJ/mol, Hm=14.15 kJ/mol, r=1.467 Å. Substitute the parameters into eq. (1) and obtain the formulas for dS and n: 1 1 d (T ) 79.218exp 5615.025 16.137, T 1943 3 (2) 1 1 n ( T ) 216 6 exp 5615.025 1 T 1943 , (Tm T TC ). Mi G B, et al. Sci China-Phys Mech Astron Using eq. (2), the residual bond structure parameters of titanium melt between the melting point and 200 K above the melting point were calculated, as shown in Figure 3. As shown in Figure 3, the average size of the titanium melt residual bond structure and the bond number in the structure decrease monotonously with the increase of temperature. 2 Relationship between the dynamic viscosity and the residual bond structure parameters of titanium melt Using function transformation on the model to calculate the residual bond structure parameters of melt, the relationship between the size of the residual bond structure and melt temperature is obtained: T (ds ) 1 , (Tm T TC ). (3) C0 ds 2r 1 k ln 1 Tm Q 1 C0 z1r Substituting eq. (3) into the Arrhenius equation [21] we obtain the melt kinematic viscosity: H C d 2r H v(ds ) A exp v v ln 0 s 1 RTm R 1 C0 z1r (Tm T TC ), k Q , (4) v(ds ) H C0 2 1 A exp v 1 C0 z1 RTm H C0 1 1 A exp v ds , (Tm T TC ), (5) 1 C0 z1 r RTm Substituting the relation between kinematic viscosity and dynamic viscosity v Figure 3 H C0 2 1 A exp v 1 C0 z1 RTm H C0 1 A exp v ds , (Tm T TC ). (6) 1 C0 z1 r RTm (d s ) According to the dynamic viscosity at the melting point H m A exp v , the dynamic viscosity above the RTm melting point is: (ds ) C0 1 C0 2 C0 m 1 1 m ds , 1 C0 z1 r z1 (Tm T TC ). into eq. (5) we obtain: (7) Define 0 C0 2 C0 m 1 . 1m , K1 1 C0 z1 1 C0 z1 r Thus, eq. (7) can be expressed as: 0 K1 ds , (8) where ds is the average size of the residual bond structure; 0 and K1 are the constants. Based on the semi-empirical formula of viscosity derived by Andrade, which considers Lindemann melting law [21], the dynamic viscosity at the melting point is obtained: m 1.8 107 After simplification: 1373 August (2012) Vol. 55 No. 8 ( MTm ) 2 Vm 3 1 2 , (9) where M is the atomic mass (kg), Tm is the absolute temperature of the melting point (K) and Vm is the atomic volume at the melting point (m3). Combining eq. (7), eqs. (8) and (9), substituting the basic parameters of titanium (M=0.04788 kg, Tm=1943 K, Vm= 1.165×105 m3) into the equations, we obtain the relationship between the dynamic viscosity and the residual bond structure size of titanium melt: Evolution of the residual bond structure size dS of Titanium melt and the bond number n in the structure. (a) dS; (b) n. 1374 Mi G B, et al. 0.876 0.471 ds . Sci China-Phys Mech Astron (10) According to eq. (10), the dependence of the dynamic viscosity of titanium melt on the residual bond structure size in the temperature range from 1943 K to 2143 K was calculated, as shown in Figure 4. At 1943 K, the dynamic viscosity = 3.850×103 Pa·s, corresponding to the earlier work = 4.42×103 Pa·s with deviation smaller than 15% [22]. Thus, on the one hand, the relationship between the macroscopic viscosity and the microstructure parameters of titanium melt is revealed by eq. (10); i.e., the viscosity increases linearly with the residual bond structure size of the melt, which provides a new way to calculate the viscosity of metal melt. On the other hand, the microscopic mechanism of the dependence of the titanium melt dynamic viscosity on temperature is reflected by eq. (10); i.e., the microscopic nature of viscosity can be described as the size variation of the viscous flow unit (residual bond structure), which provides direct theoretic evidence for views such as the concentration of atoms in the crystal-like atomic clusters of tin melt is only a function of temperature [23], aluminium atomic cluster is an independent unit of viscous flow [24], the kinematic viscosity of magnesium and aluminium has a one-to-one correspondence with the microscopic structure parameters [25–27] and the increase of viscosity reflects the improvement of ordering [28]. +0.471·ds, which provides a new way to calculate the viscosity of titanium melt. (3) The microscopic mechanism of the dependence of viscosity on temperature is attributed to the evolution of the residual bond structure size, which provides direct theoretic evidence for experiment results about the viscosity and the atomic cluster size of tin and aluminium melts, etc. This work was supported by the National Basic Research Program of China (Grant Nos. 2007CB613803 and 2007CB613702). 1 2 3 4 5 6 7 8 3 Conclusions 9 (1) Based on the concept of melt residual bonds, a model quantitatively describing the evolution of the residual bond structure of titanium melt at the melting point or in a certain range above the melting point was established; i.e., the residual bond structure parameters of titanium melt (ds, n) decrease monotonously with the increase of temperature. 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