A Direct Correlation between Viscosity and Liquid Structure in Cu-Sn Alloys

The viscosity and liquid structure of Cu100−xSnx (x = 10, 20, 30, 40, at.%) melts were investigated. Temperature dependence of viscosity η and correlation length D all shows an exponential decay function (Arrhenius-type equation), which is similar to our former studied Cu-Ag alloys. The correlation between viscosity and liquid structure had been studied. A simple relation between viscosity and correlation lengthD was found. The ratio ofD and η shows a linear relationship with temperature, which is different from our former studied results in Cu-Ag alloys. Among the four Cu-Sn alloys, Cu80Sn20 and Cu70Sn30 alloys have higher activation energy for viscous flow (Ea,V), activation energy for structural evolution (Ea,D), and the slope (γ) of the linear relationship of D/η with temperature due to the Cu3Sn clusters formation.


Introduction
Viscosity is one of the most important physical properties of melts.It can be seen as the reflection of interaction force in relative motion atoms or molecules, which is also the foundation of the liquid structure.The interatomic interactions change inevitably causes the change of the structure and vice versa.So, viscosity is seen as one of structure sensitive properties [1].The relationship between viscosity and liquid structure has attracted extensive interests and researches.Earlier in the 1940s, Born and Green [2] set up a correlation between the viscosity of liquid, the pair correlation function (), and the pair interatomic potential C().Successively in the 1950s and 1960s, Zwanzig et al. [3] and Rice and Allnatt [4] put forward a similar equation amended by the friction coefficient  and boundary condition function for finite differential equation ().Although some results calculated by these models are in good agreement with experiments in some unitary metals, some calculated results are not satisfied due to the difficulty of obtaining the accurate pair interatomic potential C() [5].This is especially true for alloys.Few studies on the relationship of viscosity and structure in alloys were reported.
This study is one of our series experimental studies on the correlation between viscosity and liquid structure in binary alloys.The viscosities of Cu 100− Sn  ( = 10, 20, 30, 40, at.%) melts were investigated by an oscillating viscometer.Based on the structural investigations through X-ray diffraction method [6], a direct correlation between viscosity and liquid structure in Cu-Sn alloys was found, which is different from our former studies in Cu-Ag alloys [7].

Material and Methods
Pure copper (99.99% mass) and tin (99.98% mass) were melted in a vacuum induction furnace to prepare the samples of Cu-Sn alloys used in this work When the samples cooled to room temperature, they were processed into the suitable size, a cuboid (25 mm×18 mm×8 mm) and a cylinder (Φ28 mm× 48 mm) for X-ray diffraction and viscosity experiments, respectively.
A high temperature torsional vibration viscometer was used for viscosity investigations.The details of measurement process can be seen in our former works [8,9].An - type X-ray diffractometer of high temperature melts was used for liquid structure experiments.MoK  radiation 2 Advances in Condensed Matter Physics  (wavelength  = 0.07089 nm) diffracted by sample was selected by a graphite monochromator.Firstly, Krogh-Moe-Norman method [10,11] was used for the coherent scattering intensity transformation.Secondly, the amended values given by Cromer and Mann [12] were used to correct the Compton scattering.At last, the structure factor () was obtained through Ashcroft-Langreth method [13,14].The detailed process can be seen in [15,16].The correlation length  used in this work was calculated as  = 2/Δ, where Δ is the half-height-width of the first peak of () [17,18].

Results and Discussion
The measured viscosities (the scattered points) of the four Cu-Sn alloys are shown in Figure 1.The changing trend of viscosity with temperature can be fitted well by an Arrhenius equation: where  0 is a constant, which is decided by the nature of the material,  , the activation energy for viscous flow,  the gas constant, and  the Kelvin temperature.
From Figure 1, it can be seen that the viscosity of Cu-Sn alloys decreases with the increasing of Sn content as a whole.But the viscosities of Cu 80 Sn 20 and Cu 70 Sn 30 alloys change faster with temperature than the other two alloys.We believe this is associated with the formation of Cu 3 Sn clusters with temperature decreasing in Cu-Sn alloys [19].The values of  0 and  , / for the four Cu-Sn melts can be gotten through fitting the viscosity data in Figure 1 by (1).The detailed fitting results are shown in The correlation length  (the scattered points) of the studied four Cu-Sn alloys is shown in Figure 2. It can be seen that the data of correlation length  also show an exponential function change with temperature, which can be fitted by an Arrhenius-type equation: where  0 is a constant, which is decided by the nature of the material, and  , is the activation energy for structural evolution.Similar to the temperature dependence of viscosity, the correlation length  decreases with increasing of Sn content, and the correlation length  of Cu 80 Sn 20 and Cu 70 Sn 30 alloys changes faster with temperature than the other two alloys also.Through fitting the correlation length  in Figure 2 by (2),  0 and  , / of the four Cu-Sn alloys can be gotten.The detailed fitting results are shown in Table 1.The results show that Cu 80 Sn 20 alloy has the minimum value of  0 (0.3296 nm) and the maximal value of  , / (1058.307K).
The viscosity  and correlation length  for the four studied Cu-Sn alloys all show Arrhenius-type relations with temperature, that is, an exponential decay function with temperature.This is similar to the Cu-Ag alloys we studied earlier [7].Does this mean that the correlations between viscosity  and correlation length  in Cu-Ag alloys are also fit for Cu-Sn alloys?For Cu-Ag alloys, the correlations between viscosity  and correlation length  were found as [7]  −  =  exp ( ) , , ,  = 1. ( By calculation, the correlations between viscosity and correlation length in Cu-Ag alloys as shown above are not fit for Cu-Sn alloys.As one of the simple eutectic binary alloys, Cu-Ag melt is homogeneous; the viscosity and liquid structure with temperature change synchronously.It can be seen in our earlier study that the values of activation energy for viscous flow ( , ) and structural evolution ( , ) are nearly equal to each other, which means the temperature dependence characteristics of viscosity and correlation length (the exponential changing rules) are similar.But, for the Cu-Sn alloy, its structure is inhomogeneous due to the formation of Cu 3 Sn and Sn-Sn clusters [19].It can be seen from Table 1 that the values of activation energy for viscous flow ( , ) and structural evolution ( , ) have a greater difference.So the viscosity and liquid structure with temperature change asynchronously in Cu-Sn alloy.
In consideration of instability of atomic clusters in high temperature, they have less influence on atomic transport properties, such as the viscosity.But, in low temperature area, atomic clusters are relatively stable, and interaction between them is reinforced, so they have greater influence on viscosity.Namely, relative to the structure change, viscosity changes slowly with temperature in high temperature and changes quickly in low temperature.So the relationship between viscosity and structure should also be related to the temperature.By the study on the ratio of correlation length and viscosity, we found that it is proportional to the temperature, which is shown in Figure 3.The lines are fitted by a linear equation: The slopes () of the fitted lines are shown in Table 1.It can be seen from Figure 3

Conclusion
To summarize, the viscosity and liquid structure of four Cu-Sn melts were investigated by an oscillating viscometer and X-ray diffraction method.The correlation between viscosity and liquid structure was studied.The results show that temperature dependence of viscosity  and correlation length  can be all fitted well by an Arrhenius-type equation.The ratio of correlation length  and viscosity  shows a linear relationship with temperature, which is different from our earlier studied results in Cu-Ag alloys.Among the four Cu-Sn alloys, Cu 80 Sn 20 and Cu 70 Sn 30 alloys have higher  , (the activation energy for viscous flow),  , (the activation energy for structural evolution), and  (the slope of the linear relationship of / with temperature) due to the more Cu 3 Sn clusters formation in Cu 80 Sn 20 and Cu 70 Sn 30 melts than the other two Cu-Sn alloys.The results of our work will enhance the understanding of the correlations between dynamic properties and liquid structure of alloy melts.

Figure 1 :
Figure 1: Temperature dependence of viscosities of Cu-Sn alloys (the lines are fitted by Arrhenius equation).

Figure 2 :
Figure 2: Temperature dependence of correlation length of Cu-Sn alloys (the lines are fitted by the Arrhenius equation).

Figure 3 :
Figure 3: Temperature dependence of the ratio of correlation length and viscosity of Cu-Sn alloys (the lines are fitted by a linear equation).

Table 1 .
The results show that Cu 90 Sn 10 and Cu 60 Sn 40 alloys have smaller value of  0 and larger value of  , /.And Cu 80 Sn 20 alloy has the minimum value of  0 (0.297 mPa⋅s) and the maximal value of  , / (2604.146K).
and Table 1 that the slopes of Cu 80 Sn 20 and Cu 70 Sn 30 alloys are larger than the other two Cu-Sn alloys.We believe that this is associated with more Cu 3 Sn clusters forming in Cu 80 Sn 20 and Cu 70 Sn 30 alloys than the other two.CＯ 90 SＨ 10 CＯ 80 SＨ 20 CＯ 70 SＨ 30 CＯ 60 SＨ 40