Metallic melts are of great importance in innovative applications.^[1–3] Viscosity is one of the important physical properties of metallic melts and is closely related to the melt structure.^[4–6] Electrical resistivity is another property that plays a prominent role in industrial applications such as mortar and concrete.^[7] Investigations of the viscosity and electrical resistivity of metallic melts remain an active domain of research in both technical and theoretical fields of condensed matter.^[8–10] But little research has been focused on the relationship between viscosity and electrical resistivity;^[11] there is no quantitative report about that relationship.

Studies of the viscosity of melts under traditional conditions have achieved much progress.^[12–15] But investigations of viscosity under extraordinary conditions are still challenging because of the limited experimental conditions, such as under external electrical field,^[16] pressure field,^[17] and external magnetic field.^[18] The external field strongly influences physical behaviors such as glass transition^[19] and dynamic behaviors including viscosity^[20] and diffusion.^[21] So much attention is still attracted by the application of different external fields in both technical and theoretical fields.

Magnetic field is one of the important external phenomena that effectively change the properties of metallic melts. In recent decades, the magnetic field has been considered to be a powerful tool in new-materials preparation^[22] and crystal growth.^[23] In addition, some investigations focus on the effects of magnetic field on controlling directional solidification.^[24,25] Meanwhile, the effects of high-intensity magnetic field on alloys have been studied, indicating that the magnetic field can increase the eutectic temperature and shift the eutectic point.^[26]

Investigating the viscosity of liquid melts under a magnetic field can help us understand the fundamental physical properties and the structures of the melts, and some investigations in viscosity under magnetic field have been done. Simultaneously, it should be noted that the dynamic viscosity under magnetic field show similar behaviors: a sharp increase in viscosity appears with increasing intensity of magnetic field, and researchers have attempted to explain this surprising phenomenon based on the activation energy, Lorentz force, valence bond, and so on.^[27–31] Unfortunately, up to now, the reason for this sharply increased dynamic viscosity in an external magnetic field remains disputable due to the limitations of experimental conditions and the absence of theoretical explanations. So a quantitative understanding of viscosity under a horizontal magnetic field still has not been provided.

In the present work, we have investigated the viscosity of pure Ga, Ga₈₀Ni₂₀, and Ga₈₀Cr₂₀ metallic melts under a horizontal magnetic field to elucidate how the magnetic field increases the viscosity. Furthermore, we have established a mathematical physical model to describe the viscosity under the magnetic field quantitatively, which is important for studying the effects of magnetic field on the solidification of metallic melts. The relationship between the viscosity and the electrical resistivity under the horizontal magnetic field has been studied, which provides a deep understanding for physicists.

The samples of Ga₈₀Ni₂₀ and Ga₈₀Cr₂₀ alloys used in the present study were prepared from pure Ga, Ni, and Cr of 99.99% purity in a high frequency induction furnace under Ar atmosphere.

Viscosity measurements were completed with a torsional OSVM oscillation viscometer. By measuring logarithmic decrement, the kinetic viscosity is calculated using the Shvidkovskiy equation^[32]

(1)

(2)

In these equations, Δ = δ/2π; I is the inertial momentum of the suspended system, δ is the logarithmic damping decrement, and δ₀ is the logarithmic damping decrement for an empty vessel. T is the period time of the oscillations, while T₀ is the period time of the oscillations for an empty vessel. M represents the mass of the liquid sample, R_v is the radius of the vessel, H is the height of the liquid sample in the vessel, a, b, c are constants, and n is the number of solid planes contacted horizontally by the liquid sample (i.e., in the case of a vessel having its lower end closed and its upper surface free, n = 1; if the vessel encloses the fluid at its top and bottom, n = 2).

After careful surface treatment of the specimens with a file, the specimens were put into an alumina crucible, 30 mm in diameter and 60 mm in height. The samples were measured during the cooling process from elevated temperature above liquidus to a supercooling temperature. Before the viscous experiment, the viscosity of the empty alumina crucible used in the experiment should be measured in the same condition during the process of the experiment in order to eliminate the influences of the atmosphere, ensuring the precision of the experiment. During the process of the viscosity experiment, the samples were homogenized for 1 h at the highest temperature and were held for 40 min at each temperature at which viscosity was measured. The data error of different measurements at the same temperature is less than 3%. The relative accuracy of the viscometer is better than 5%. The viscosity experiments were repeated four times and the values were averaged. The intensities of the magnetic field were set at 0 Gs, 95 Gs, 245 Gs, 475 Gs, 650 Gs, and 935 Gs respectively.

The viscosities of Ga₈₀Cr₂₀ and Ga₈₀Ni₂₀ alloy melts under different magnetic fields during the cooling progress are shown in Fig. 1. Some researchers have discovered that the temperature-dependence of viscosity of melts under magnetic fields of different intensities in a wide temperature range follows the Arrhenius law^[27,28]

(3)

Fig. 1.

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	Fig. 1. (color online) The viscosities of (a) Ga80Cr20 and (b) Ga80Ni20 alloy melts under magnetic fields of intensities 0 Gs, 95 Gs, 245 Gs, 475 Gs, 650 Gs, and 935 Gs. The solid lines are the fitting curves according to the Arrhenius law.

Table 1.

Table 1.

Table 1. The viscous parameters of Ga80Cr20 and Ga80Ni20 melts under magnetic fields of different intensities. . Sample Parameter Gs 95 Gs 245 Gs 475 Gs 650 Gs 935 Gs Ga80Cr20 A 0.4712 0.8954 0.5471 0.6499 0.7619 1.6721 Ga80Cr20 E 1573 1251 1429 1949 1967 2170 Ga80Ni20 A 0.6001 0.8295 0.8961 1.0599 1.0048 1.4621 Ga80Ni20 E 1089 1266 1474 1566 1556 1987

Table 1. The viscous parameters of Ga80Cr20 and Ga80Ni20 melts under magnetic fields of different intensities. .

Figures 2(a) and 2(b) are schematics of the torsional oscillation viscometer and the simplified optical measuring system. At first, the driving motor activates the lever, which makes the sample rotate with the lever. A mirror is placed on the lever; when the lever rotates, the beam transmitted by the laser transmitter is reflected by the mirror. The reflected beam will pass receivers A and B because of the rotation of the mirror. When the motor stops working, the velocity of rotation decreases because the viscosity of the melt impedes the rotation of the samples. When measuring, the shape of the samples is shown as Fig. 2(c). Besides the external effects, which have been ruled out by measuring the empty alumina crucible, the viscous force whose direction is opposite to the direction of the velocity is the only factor resulting in the decrease of the speed of rotation. In this ideal state, the stress analysis can be simplified as Fig. 2(d). Based on this mathematical physical model, we can obtain the equation

(4)

Fig. 2.

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	Fig. 2. (color online) (a) Schematic of a torsional oscillation viscometer. (b) Schematic of a simple measuring system. (c) The physical sample. (d) Cross section of the sample.

In the above equation, J is the inertial momentum of the sample, which can be described as

(5)

In Eq. (4), α is the angular acceleration, r is the radius, and F is the force applied to the sample resulting from the wall of the alumina crucible, which can be described as

(6)

According to Eqs. (4)–(6), we can solve the angular acceleration α,

(7)

(8)

If the angular velocity is measured by the optical system, we can obtain the viscosity of the sample in the ideal state.

When a magnetic field is applied to the sample, the causes of the decrease of the angular velocity can be divided into two aspects – the viscosity of the sample and the force caused by the magnetic field, as shown in Fig. 2(d). In the macroscopic view, the magnetic force results from the Ampere Force. The samples rotate during the measuring process. When the samples rotate in the magnetic field, the samples cut the magnetic induction line. So an induction current is produced, and the Ampere force is generated. In our experiments, the magnetic induction line is horizontal. For the cylindrical sample, the velocity is not always vertical to the magnetic induction line. The induction current can be obtained as

(9)

For the whole cylindrical sample, the induction current can be described as

Then the induced magnetic force F_M can be obtained from

Then, the forces under magnetic field can be described as

(12)

According to Eq. (12), we can easily obtain the equation

(13)

Figure 3 shows the dependence of the viscosity of pure Ga metallic melt, Ga₈₀Cr₂₀ metallic melt, and Ga₈₀Ni₂₀ metallic melt upon the intensity of the magnetic field at various temperatures. The solid lines are the fitting curves according to Eq. (13). It is obvious that the experimental viscous data fit well with Eq. (13). The fitting parameters of β which can be described as 2H/πΩ, are shown in Fig. 4. It can be seen that β decreases with increasing temperature. During the measuring process, the temperature is changed, which results in a change of the electrical resistivity of the liquid melt, but the height of the sample is unchanged, so we can conclude that the change of β is the same as the reciprocal electrical resistivity. The electrical resistivity increases with increasing temperature,^[33] indicating that equation (13) conforms to the well known fact that electrical resistivity increases with increasing temperature. According to Eq. (13), we can test electrical resistivity by measuring the viscosity under horizontal magnetic field, and vice versa.

Fig. 3.

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	Fig. 3. (color online) Dependence of the viscosity of (a) pure Ga melt, (b) Ga80Cr20 melt, and (c) Ga80Ni20 melt upon the intensity of the magnetic field at various temperatures. The red solid lines are the fitting curves.

Fig. 4.

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	Fig. 4. (color online) β of Ga80Cr20 and Ga80Ni20 melts dependence upon temperature.

Sun et al. measured the viscosity of Sb–Bi melts under magnetic fields of different intensities at various temperatures.^[27] Now we use their data to fit with Eq. (13) as shown in Fig. 5. We can find that the viscosities of Sb–Bi melts under magnetic fields fit well with Eq. (13). It is easily obtained that β increases with decreasing temperature, which means that the electrical resistivity increases with increasing temperature according to Eq. (13), consistent with the well established fact that the electrical resistivity increases with increasing temperature.

Fig. 5.

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	Fig. 5. (color online) Dependence of the viscosity of (a) Bi, (b) Sb, (c) Bi20Sb80, (d) Bi40Sb60, (e) Bi60Sb40, (f) Bi80Sb20 melts upon the intensity of the magnetic field at various temperatures.[27] The solid lines are the fitting curves.

Mao et al. measured the viscosities of Cu–10% Sn and pure Cu melts under magnetic fields of different intensities at various temperatures.^[28] Yang measured the viscosities of some single liquid melts, such as aluminum, indium, tin, and bismuth.^[29] Zhang measured the viscosity of Al–5 at% Si liquid melt under a horizontal magnetic field.^[30] We can see that the above viscous data all fit well with Eq. (13). The change of coefficient β with the temperature accords with the law that the electrical resistivity increases with increasing temperature.

Based on so many viscous data under different intensities of magnetic fields, such as the same components with different atomic percents, different single metallic melts, and different alloy systems, we can conclude that all the above viscosities are in accordance with Eq. (13).

However, there is no liquid–liquid transition during the cooling process for the above melts. So we find the viscous data of Cd₇₀Ga₃₀, which include the liquid–liquid transition.^[31] We know that when the liquid–liquid transition happens, the viscosity presents a turning point. We turn these data into a diagram that shows the relationship between the viscosity and the intensity of the magnetic field, as shown in Figs. 6(a) and 6(b). Figure 6(a) shows the viscosity’s dependence upon the intensity of the magnetic field before the liquid–liquid transition appears, and it indicates that the viscosity follows Eq. (13). Figure 6(b) shows the viscosity’s dependence upon the intensity of the magnetic field during the liquid–liquid transition process. The fitting curves show that when the liquid–liquid transition starts, the coefficient β decreases sharply, as shown in Table 2. This sharp change may be related to the liquid structure transition. Otherwise, during the transition process, when the intensity of the magnetic field is 420 Gs, the viscosity is much smaller than the fitting data. During the transition process, the metallic melt is metastable, the external magnetic field has a more obvious effect on the liquid structure, which results in deviation of the viscosity. In a nutshell, the liquid–liquid transition results in a sharp change of the electrical resistivity. The viscosity of single and binary metallic melts under magnetic field may deviate from Eq. (13) because of the metastable liquid structure during the transition process.

Fig. 6.

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	Fig. 6. (color online) (a) Viscosity’s dependence upon the intensity of the magnetic field before the liquid–liquid transition appears. (b) Viscosity’s dependence upon the intensity of the magnetic field during the liquid–liquid transition process.[31]

Table 2.

Table 2.

Table 2. The values of fitting β during the liquid–liquid transition process. . T/K β/10−6 mPa·s·Gs−2 568 9.489 562 8.926 561 8.8816 560 8.7536 558 8.7436

Table 2. The values of fitting β during the liquid–liquid transition process. .

In this work, we have studied the viscosities of pure Ga melt, Ga₈₀Ni₂₀ melt, and Ga₈₀Cr₂₀ melt under a horizontal magnetic field. A quantitative description of this dynamic behavior is given by an explicit analytical equation based on a mathematical physical model that allows us to predict the viscosity of single and binary metallic melts and to control the viscosity under a magnetic field. We can show that the magnetic viscosity is proportional to the square of the intensity of the magnetic field. The relationship between the magnetic viscosity and the electrical resistivity has been established. The change of the proportionality coefficient with temperature in our equation, which is related to the electrical resistivity, shows that our equation is correct. We also use some other data to fit with the equation, which indicates the correctness of our equation when there is no liquid–liquid transition.