Liquid phase separation in undercooled Cu–Co alloys under the influence of static magnetic fields

Abstract

Undercooling of Cu-based alloys often induces metastable liquid phase separation followed by rapid solidification of separated liquids. The rapid solidification can help freeze in the morphology of a higher-melting liquid and eases difficulties in studies of liquid phase separation kinetics. In the present work, the influence of static magnetic fields on liquid phase separation in bulk Cu₈₄Co₁₆ composition was investigated. Inductively melted samples were glass-fluxed, undercooled and solidified under uniform and non-uniform magnetic fields generated by a superconducting magnet. Solidification microstructure of the phase-separated samples was examined using an optical microscope. The imposition of the magnetic fields, both uniform and non-uniform, altered the morphology, segregation pattern and size distribution of Co-rich droplets due to liquid phase separation. The imposition of the non-uniform magnetic fields with positive and negative gradients brought about segregation of the Co-rich droplets at the top and the bottom side of the samples, respectively. Such influence of the static magnetic fields is interpreted by assuming intensification of convective flow and Kelvin force-controlled migration of the Co-rich droplets.

This article is part of the theme issue ‘Heterogeneous materials: metastable and non-ergodic internal structures’.

1. Introduction

Undercooling of a metallic liquid often leads to metastable solidification reactions and formation of technically interesting microstructure [1]. An undercooling-induced metastable liquid phase separation reaction followed by rapid solidification of two separated liquids is one of such examples. This kind of liquid-to-liquid reaction has been observed in a number of Cu-based alloys including Cu–Fe, Cu–Co and Cu–Cr [2–4]. What is common for those alloys is the existence of a metastable miscibility gap in the undercooled liquid [5–7]. When their homogeneous liquid is undercooled slightly below a binodal temperature of the miscibility gap [5,8], it will be separated into two liquids of different composition and properties. This liquid phase separation brings about either a dispersed structure or a bicontinuous structure depending on bulk composition [9–12]. In off-critical composition regions, the liquid phase separation proceeds via a nucleation mechanism [5,8,13–15]. Because of a small interfacial energy between two liquids, the nucleation rate of a minority phase is so high that a dispersion of fine droplets in the matrix of a majority liquid can be produced [8,9,14,15]. On the other hand, crystal nucleation is more difficult due to a higher interfacial energy between a liquid and a crystal. Thus, the liquid phase separation is often extended to a temperature well below the initial binodal temperature. In this extended separation, chemical composition of both liquids still follows the binodal of the miscibility gap as temperature declines. As a result, the undercooling of a higher melting liquid is promoted to such an extent that its diffusionless solidification into supersaturated solids can be kinetically favoured [16–20]. Technically, a fine dispersion of small-sized particles of a supersaturated solid solution in the matrix of another phase has been shown to be a key feature of microstructure for improvement of mechanical and electrical properties of Cu-based alloys [4,11,12]. For this reason, the metastable liquid phase separation in undercooled Cu-based alloys has aroused much interest in the past decades [20,21].

The liquid phase separation in undercooled Cu–Co alloys has been extensively studied compared to that in other Cu-based alloys. In earlier studies, different methods were used to determine the binodal and spinodal of the metastable miscibility gap below a flat and long liquidus of Co-rich solid solution. The existence of such a miscibility gap was first suggested by Nakagawa in terms of his measurements of magnetic susceptibility of undercooled alloys [2]. Later, Munitz & Abbaschian [18] and Li et al. [19] proposed a complete binodal of the miscibility gap by energy-dispersive X-ray spectrometric measurements of local composition of phase constituents in containerlessly solidified alloys and by pyrometric studies of thermal events in continuous cooling of glass-fluxed alloys, respectively. However, there were serious discrepancies between their studies, especially in the location of the critical point and a low-temperature part of the binodal at the Co-rich side. Cao et al. [22] resolved those discrepancies reasonably. They determined the binodal and spinodal of the miscibility gap by combining differential thermal analyses of glass-fluxed alloys with thermodynamic calculations. The currently accepted metastable miscibility gap of the Cu–Co system was established by Curiotto and colleagues [6,23]. Those researchers performed differential scanning calorimetric studies and thermodynamic assessment of stable and metastable phase equilibria at elevated temperatures. Thus, the determined miscibility gap has high accuracy. Microstructure formation in phase-separated Cu–Co alloys has been well documented. It is very sensitive to processing conditions such as liquid undercoolings and cooling rates [10–12,14,15,17–35]. Electromagnetic levitation and glass-fluxing were used to undercool bulk Cu–Co alloys [8,15,17–19,22,24–30]. Accessed undercoolings often exceeded a critical value for triggering the metastable liquid phase separation. Then, large undercoolings brought about an extension of liquid phase separation down to low temperatures and promoted coarsening of droplets significantly. Quantitative studies showed that the mean or the largest droplet sizes of a minority phase increased with rising undercooling generally [25,30]. High cooling rates were achieved by drop tube processing, gas atomization, melt spinning and splat quenching [10–12,14,31–35]. They not only promoted undercoolings of small drops or thin liquid layers (typically less than 1 mm at least in one dimension), but also reduced the total time available for coarsening of an initially fine structure. As a result, the formation of an egg-like coarse structure was avoided and the mean droplet size was reduced. However, undercoolings of individual drops or quenched liquids were left unknown and had to be estimated in terms of observed solidification microstructure or by numerical modelling of thermal history [14].

Apart from undercoolings and cooling rates, liquid flow was often quoted as a critical factor that affects microstructure evolution in phase-separated Cu–Co and other Cu-based alloys. As reviewed elsewhere [13], a density difference between separated liquids and a thermal gradient in the bulk liquid can induce Stokes motion and Marangoni migration, respectively. While the Stokes motion causes sedimentation of droplets or the surrounding matrix, the Marangoni migration due to a thermally or chemically induced gradient of the liquid/liquid interfacial energy brings droplets from a cold end or shell towards a warm end or core. Both of them can promote the formation of an egg-like core–shell structure [2,6,15,19,24], which is quite detrimental to applications of Cu-based alloys. Therefore, efforts were made to suppress or reduce such flow effects. An effect of forced convection in electromagnetically levitated and phase-separated Cu–Co alloys was noticed by Munitz & Abbaschian [18] and Cao and colleagues [26,36]. Those researchers suggested that strong electromagnetic stirring in the levitated samples deformed droplets of a minority phase into a worm-like morphology. They also suggested that the convection either broke large-sized droplets into smaller ones or brought about collision and coagulation of small-sized droplets. The effect of forced convection on microstructure formation of phase-separated alloys was quantitatively revealed by electromagnetic levitation experiments under different gravity conditions [37,38]. It was found that reduced convection related to microgravity established during parabolic flights lowered the possibility of coagulation of the droplets of the minority phase and produced a monomodal droplet size distribution in contrast to a bimodal one under the normal gravity condition [37–39]. Recent studies by Zhang and colleagues [40–42] showed that imposition of static magnetic fields on electromagnetically levitated Cu–Co alloys damped forced convection and brought about a microgravity-like effect on the droplet size distribution. Such a magnetic field effect was observed also by Sugioka and colleagues [43,44]. Compared to parabolic flights or space experiments, ground-based experiments with the use of static magnetic fields are cheap and offer a new opportunity for studies of liquid phase separation in undercooled Cu–Co and other Cu-based alloys. Yasuda et al. [45] showed that the use of static magnetic fields of high intensities suppressed Stokes motion-induced sedimentation of Pb droplets in mould-cast ingots of hypermonotectic Cu–Pb alloys and reduced macrosegregation. Inspired by such progress, we investigated the influence of static magnetic fields on liquid phase separation in glass-fluxed Cu–Co alloys in the present work. Unlike the electromagnetic levitation experiments, the glass-fluxing experiments allowed for a switch-off of heating power. Thus, the convection in the undercooled bulk liquid was already weak even if no static magnetic fields were imposed [46]. Apart from high intensities, large gradients of static magnetic fields can be generated by superconducting magnets. They impose a so-called Kelvin force (also known as magnetic buoyancy force) onto a chemically inhomogeneous fluid or a phase mixture providing an additional way for control of melt flow or motion of floating particles [47–49]. The effect of the Kelvin force on liquid phase separation in undercooled Cu-based alloys is little explored in literature. The present experiments showed that both the high intensities and large gradients of the static magnetic fields had significant influence on liquid phase separation in glass-fluxed and phase-separated Cu–Co alloys.

2. Methods

Alloys of bulk composition of Cu₈₄Co₁₆ were prepared by arc-melting of elemental Cu (99.9999% purity) and elemental Co (99.99% purity) in an argon atmosphere. Each alloy had a mass of about 1.0 g. Mass losses of individual alloys during arc-melting were typically less than 0.1 wt.%. Glass fluxing experiments were carried out using a radio frequency induction heater installed in the bore of a superconducting magnet (figure 1). In each experiment, a single sample was seated on a sample holder together with a small amount of soda lime glass and sealed in a vacuum chamber. The vacuum chamber includes a metal part and a quartz part. The quartz part has a smaller outer diameter than that of the metal part. It was inserted and fixed in the coil of the induction heater [50]. The vacuum chamber was evacuated to a pressure of 2–5 Pa and back-filled with high purity argon to a pressure of 20 kPa. This evacuation and back-filling procedure was repeated at least twice to lower the oxygen concentration of the atmosphere. The position of the sample and the coil was adjustable depending on the choice of the magnetic fields. In experiments with uniform magnetic fields, the position of the coil was adjusted to allow the sample to rest at a site, where the intensity of the magnetic field reached a maximum for a given current of the superconducting magnet. In experiments with non-uniform magnetic fields, the position of the sample was lifted or lowered by 100 mm along the central axis of the magnet with respect to that in the experiments with the uniform magnetic fields. In such a way, the sample rested at a site where the gradient of the magnetic fields had a positive or a negative maximum. For convenience, the gradients of the magnetic fields are given below in the form of B · dB/dz, where B is the magnetic field intensity and dB/dz represents the gradients of the magnetic fields. Before the experiments, a magnetic field of intensity ranging from B = 1 T to B = 6 T was excited by supplying a given current in the superconducting coil of the magnet. By adjusting the position of the sample and the coil, a non-uniform magnetic field with a positive or a negative gradient ranging from B · dB/dz = 2.26 T² m⁻¹ to B · dB/dz = 81.4 T² m⁻¹ (absolute magnitude) was established by supplying the same current in the superconducting coil of the magnet. After the uniform or the non-uniform magnetic field was excited at the location of the sample, the sample was inductively heated, melted and overheated. Then, the power of the coil of the induction heater was switched off. The sample was cooled predominantly by radiation to the surrounding atmosphere and the vacuum chamber. After an undercooling was achieved, the sample started to solidify. In undercooled solidification, the sample often showed a rapid rise of temperature, termed recalescence. The sample was usually melted and solidified several times in order to attain a large undercooling. The intensity and gradient of the magnetic field was kept constant through such melting-solidification cycles of individual samples. In all cycles, the temperature of the top surface of the sample was measured using a single-wavelength pyrometer at a sampling rate of about 100 Hz and recorded transiently using a computer. Measured temperature of the sample surface was calibrated by reference to the maximum temperature reached through a second recalescence event associated with solidification of a Cu-rich majority liquid. In terms of the calibrated temperature-time profile, undercooling of the sample was defined as the difference between the equilibrium melting temperature of the bulk composition and the onset temperature of solidification. After the experiments, all samples were cross-sectioned along a plane parallel to the direction of the magnetic field (antiparallel to the gravity). Their microstructure was examined using an optical microscope. More than one cross-sections of few samples were examined for a better statistic analysis.

Figure 1.

A schematic diagram illustrating the experimental set-up. In the experiments, samples were placed at different positions, where either the magnetic field intensity, B or its gradient along the vertical axis, dB/dz had a maximum. (Online version in colour.)

3. Results

Most of the samples reached a large undercooling in their last solidification cycles. Those undercoolings are distributed in a narrow range from 177 to 200 K. They exceeded a critical undercooling of 112 K that is required for triggering of the metastable liquid phase separation. Thus, the solidification microstructure of those samples showed distinct evidence for liquid phase separation. Details of their microstructure are presented below. There were a few exceptions. The sample solidified under a uniform magnetic field of B = 6 T reached an undercooling slightly smaller than the critical undercooling. Thus, its solidification microstructure consisted of coarse primary dendrites of Co-rich solid solution due to a near-equilibrium solidification path. In addition, two samples melted in non-uniform magnetic fields of high and positive gradients were lost because of their splitting into two parts in a phase-separated state. As displayed in figure 2, the phase-separated samples showed similar recalescence behaviour in solidification under the uniform magnetic fields. Two recalescence events were observed at a higher and a lower temperature, respectively due to sequential solidification of two separated liquids. The first one corresponded to rapid solidification of the high-melting Co-rich liquid, whereas the second one corresponded to rapid solidification of the Cu-rich liquid, which had a larger volume fraction. This is also the case for the samples solidified under the non-uniform magnetic fields of negative gradients. The recalescence behaviour of the samples solidified under the non-uniform magnetic fields of positive gradients showed a dependence on the positive gradients of the non-uniform magnetic fields. As shown by the curves 13–15 of figure 2, their first recalescence events showed an overshooting of the sample temperature compared to those observed under other magnetic field conditions. This overshooting of the sample temperature became more and more pronounced as the positive gradient of the magnetic fields was increased. The splitting of the two samples under magnetic fields of higher positive gradients suggested that the overshooting of the sample temperature during the first recalescence event might not represent a true rise of the sample temperature. Rather, it is more likely to be a consequence of the sample motion or a change of the emissivity of the sample surface. Cooling rates of all samples were similar, about 30 K s⁻¹, regardless of the intensities and gradients of the magnetic fields. Such similar undercoolings and cooling rates of the samples allowed us to identify the influence of the uniform and non-uniform magnetic fields on liquid phase separation readily.

Figure 2.

Temperature–time profiles of samples undercooled and solidified under (a) uniform magnetic fields and non-uniform magnetic fields with (b) negative and (c) positive gradients. Undercoolings of the samples (ΔT) as well as intensities (B) and gradients (B · dB/dz) of the magnetic fields are shown below each curve.

Optical micrographs of figure 3 illustrate solidification microstructure of phase-separated samples at low magnifications. Both the intensity and the gradients of the static magnetic fields were found to have significant influence on the morphology and segregation patterns of Co-rich droplets due to the metastable liquid phase separation. As shown in figure 3a, the Co-rich droplets were frozen into a round morphology in the sample solidified under a uniform magnetic field intensity of B = 1 T. Many of them have radii of less than 10 µm and can be discerned in high-magnification micrographs only (not shown here). The Co-rich droplets showed a deformed morphology as the intensity of the uniform magnetic fields was increased. They appeared to be elongated along the height of the samples. The width of the elongated droplets declined steadily with rising intensity of the magnetic fields. These changes reflect that the uniform magnetic fields of higher intensities promoted coagulation of Co-rich droplets in contrast to an opposite influence at a lower intensity. The Co-rich droplets were also frozen into a sphere-like morphology under the non-uniform magnetic field of a small and negative gradient of B · dB/dz = ‒2.26 T² m⁻¹ (figure 3b). Droplets of relatively large sizes gathered in a zone slightly above the middle height of the sample. As the magnitude of the negative gradients was increased, the Co-rich droplets showed a deformed morphology as in figure 3a. Compared to those frozen in under the uniform magnetic fields, the deformed droplets were less elongated along the height of the samples. They tended to segregate at the bottom side of the samples. Deformation of the Co-rich droplets is also evident in the samples solidified under the non-uniform magnetic fields of positive and medium gradients (figure 3c). Surprisingly, the droplets of large sizes tended to gather at the top side of the samples, suggesting that a reversal of the sign of the gradients brought about opposite segregation patterns. The deformed droplets are not seen in the micrographs of the samples solidified under the non-uniform magnetic fields of positive gradients of B · dB/dz = 20.3 T² m⁻¹ and 36.1 T² m⁻¹. They were likely to lie out of the cross-sections investigated. On the other hand, many small-sized droplets were observed in the two samples as in other samples. Because their radii are generally smaller than 10 µm, they are hardly discerned in the low-magnification micrographs of figure 3c.

Figure 3.

Optical micrographs showing cross-sectional microstructure of samples undercooled and solidified under (a) uniform magnetic fields and non-uniform magnetic fields with (b) negative and (c) positive gradients. All cross-sections were chosen to be parallel to the direction of the magnetic fields. The minor phase with dark contrast is Co-rich solid solution, and the matrix phase with bright contrast is Cu-rich solid solution. (Online version in colour.)

As shown in figure 4, the static magnetic fields also had significant influence on the size distribution of the Co-rich droplets (the raw data of droplet sizes can be found in the electronic supplementary material). At low intensities or smaller gradients of the magnetic fields, the droplet size distribution showed a single peak with a long tail on the side of larger sizes. This feature represents a singular-modal distribution and has a high resemblance to that observed for the electromagnetic levitated samples [40–42]. However, the droplet size distribution became bimodal at higher intensities or larger gradients of the magnetic fields. Even a multimodal distribution was observed at the intensity of B = 5 T of the uniform magnetic field.

Figure 4.

Semi-logarithmic plots of size distribution of Co-rich droplets resulting from liquid phase separation under (a) uniform magnetic fields and non-uniform magnetic fields of (b) negative and (c) positive gradients. Curves shown in each panel are only guides for the eyes. Sizes of small droplets refer to their averaged radii, whereas those of large droplets refer to their equivalent circle radii. (Online version in colour.)

Attention was also paid to the droplets having a maximum frequency or a maximum size in the cross-sections of each sample. As plotted in figure 5, the sizes of the droplets having the maximum frequency showed very little changes with increasing intensity or gradient generally. An exception is noticed for the droplets in the sample solidified under the uniform magnetic field of B = 5 T. The sizes of the droplets having the maximum frequency are almost doubled relative to those in other samples. On the other hand, the sizes of the largest droplets showed different dependences on the intensity and gradient of the magnetic fields. The largest droplet size in the sample solidified under the uniform magnetic field of B = 2 T is by a factor of four larger than that of the samples solidified under uniform magnetic fields of lower or higher intensities. Unlike small variations of the size of the largest droplets in the samples solidified under the non-uniform magnetic fields of negative gradients, a clearly declining tendency was observed for the size of the largest droplets in the samples solidified under the non-uniform magnetic fields of positive gradients. The reasons for such changes are discussed in §4.

Figure 5.

Magnetic field dependence of (a) the droplet size having the maximum frequency and (b) the size of the largest droplets in samples solidified different magnetic field conditions. Curves shown in each panel are guides for the eyes only. The sizes of the largest droplets refer to the radii of their equivalent circles. (Online version in colour.)

4. Discussion

The influence of the static magnetic fields on the metastable liquid phase separation in the glass-fluxed Cu–Co samples exhibits similarities and differences compared to those observed for electromagnetically levitated samples [40–44]. The similarities are seen in the sphere-like morphology and the monomodal size distribution of the Co-rich droplets frozen in at lower intensities or smaller gradients of the magnetic fields. They can be understood by taking into account a similar damping effect on liquid flows by the magnetic fields [40–42]. On the other hand, the differences are seen in the elongated morphology, biased segregations and the bimodal or multimodal size distributions of the Co-rich droplets at higher intensities or larger gradients of the magnetic fields. These differences gave a hint at enhanced liquid flow. The underlying mechanisms, however, are dependent on the intensities and gradients of the magnetic fields as discussed below.

First of all, the enhanced flow is most likely to be related to a vertical thermal gradient in the glass-fluxed samples with a cooler top surface. This thermal gradient was revealed by numerical modelling of the thermal field of a spherical Ni sample, of which the lower part is immersed in a glass flux [46]. It represents a temperature difference of 10 K over a height of 6 mm of the spherical sample. Although it has a small magnitude, it may excite Marangoni convection in the phase-separated samples by introduction of a gradient of interfacial tension along the surface of the Co-rich droplets. The cooler surface of the samples provides better conditions for nucleation of the Co-rich droplets than those offered by other parts of the samples. Then, the Marangoni convection tends to move the Co-rich droplets towards the bottom side of the samples. It was shown elsewhere that a Co-rich liquid has a larger mass density than that of a Cu-rich liquid at high temperatures [51]. For this difference, the Stokes motion is supposed to move the Co-rich droplets from the upper side towards the bottom side of the samples as well. However, any liquid flow along a direction antiparallel to the magnetic field cannot be suppressed by electromagnetic damping. Thus, both the Marangoni convection and Stokes motion are allowed under the uniform magnetic fields. In the light of the Young's theory [52], the Marangoni migration velocity and Stokes sedimentation velocity of the droplets are functions of droplet radii. Droplets moving faster may catch those moving slower leading to single or multiple coagulation. Such coagulation is likely to be promoted with rising intensity of the uniform magnetic fields in terms of the changes of the droplet size distribution shown in figure 4a. The promoted coagulation of the droplets may be due to a stabilizing effect of the vertical magnetic fields on Marangoni convection [53]. The coagulated droplets usually restore a spherical-like morphology very quickly. However, it is not the case for the Co-rich droplets in the samples solidified under the uniform magnetic field intensities of B = 2–5 T. The reason is that any transverse fluid flow due to droplet coagulation can be damped by the vertical magnetic fields of a few Tesla on a scale of the order of 1 mm. For this damping effect, the transverse stretching of the coagulated droplets might be restricted whereas their stretching along the magnetic field direction is permitted. The assumption of such anisotropic stretching can account for the elongated morphology and the deformation of the large-sized droplets in the present samples. Unlike the glass-fluxed samples, electromagnetically levitated samples were shown to have a relatively uniform temperature distribution in their surfaces due to a skin effect of eddy currents [54,55]. Thus, thermal gradients in the surface of the electromagnetically levitated samples are supposed to be smaller than that of the glass-fluxed samples. This difference explains why the droplets with the elongated morphology were not observed by Zhang and colleagues [40–42]. However, recent studies by Sugioka and colleagues [43,44] showed the formation of elongated Co-rich droplets in a Co₈₀Co₂₀ sample electromagnetically levitated under a magnetic field of B = 4 T. Those elongated droplets may also be interpreted by assuming suppression of the transverse stretching of the coagulated droplets. Because of smaller thermal gradients, the Marangoni convection in the levitated bulk samples are supposed to be weaker than in the glass-fluxed samples. Nevertheless, Marangoni convection may be introduced by asynchronous crystallization of the Co-rich droplets, which was observed both in the glass-fluxed and in the electromagnetically levitated samples [17,55]. Owing to release of latent heat, a solidifying Co-rich droplet may produce a localized thermal gradient and excites Marangoni convection in local liquid. This local Marangoni convection may also be intensified by the magnetic fields of high intensities. Then, the elongated morphology of the Co-rich droplets in the levitated samples can be understood. Apart from Marangoni convection, liquid flow can be excited via another mechanism. That is, a difference in Seebeck coefficient arises naturally between the Cu-rich liquid and the Co-rich droplets in phase-separated Cu–Co alloys. This difference gives rise to thermoelectric currents in the Cu-rich liquid. Then, the magnetic fields act Lorentz forces onto the currents exciting thermoelectric magnetohydrodynamic convection in local liquid [50]. We assume that this thermoelectric magnetohydrodynamic convection may account for the rapid increase of the size of the Co-rich droplets having the maximum frequency at the uniform magnetic field intensity of B = 5 T (figure 5a). Under this assumption, the formation of the elongated droplet of an equal circle radius of more than 1500 µm in the sample solidified under the magnetic field intensity of B = 2 T (figure 5b) is more likely to be induced by locally intensified Marangoni convection than by the thermoelectric magnetohydrodynamic convection. Such explanations and assumptions await validation by new experiments or by numerical simulations.

When the glass-fluxed samples were solidified under the non-uniform magnetic fields, the intensities of the fields at the location of the samples were small compared to those under the uniform magnetic fields. From this point of view, the coagulation of the Co-rich droplets is supposed to be weakened. Nevertheless, the magnetic field intensities are still high and can damp flow along any transverse directions. Then, the mechanism for the formation of the elongated droplets is supposed to be operative also in the samples solidified under the non-uniform magnetic fields. What is more influential is that the gradients of the magnetic fields can impose a Kelvin force, F_K, onto the Co-rich droplets. In addition to the Kelvin force, other forces were also acted upon the Co-rich droplets. Those forces included gravity (F_g), the driving force for Stokes motion (F_S), and the Marangoni force (F_M). The magnitude of such forces can be calculated using the following equations [13,49,56,57]:

$$\begin{array}{rl}{F}_{\mathrm{G}}& =\frac{4}{3}\pi r^3{\rho }_{\mathrm{d}}g,\end{array}$$ 4.1

$$\begin{array}{rl}{F}_{\mathrm{S}}& =\frac{4}{3}\pi r^3({\rho }_{\mathrm{d}}-{\rho }_{\mathrm{m}})g,\end{array}$$ 4.2

$$\begin{array}{rl}{F}_{\mathrm{M}}& =4\pi r^2\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }T}\frac{\mathrm{\partial }T}{\mathrm{\partial }r}\end{array}$$ 4.3

$$\begin{array}{rl}\text{and}{F}_{\mathrm{K}}& =\frac{({\chi }_{\mathrm{d}}-{\chi }_{\mathrm{m}})}{{\mu }_0}B\frac{\text{d}B}{\text{d}z}\frac{4}{3}\pi r^3,\end{array}$$ 4.4

where r is the droplet radius, ρ_d is the mass density of the Co-rich droplets, ρ_m is the mass density of the Cu-rich liquid, g is the gravitational acceleration on the Earth, ∂σ/∂T is the temperature coefficient of the liquid–liquid interfacial energy, ∂T/∂r is the thermal gradient along the height of the glass-fluxed samples, χ_d is the volumetric susceptibility of the Co-rich droplets, χ_m is the volumetric susceptibility of the Cu-rich liquid, μ₀ is the magnetic permeability in vacuum, B is the intensity of the magnetic field, dB/dz is the gradient of the magnetic field. The calculations were done using the parameters listed in table 1 [51,58,59]. As shown in figure 6, the driving force for Stokes motion of droplets of r < 1000 µm is generally smaller than the Marangoni force acting on the droplets of the same sizes. This large difference suggested that the Marangoni migration is dominating in the coagulation of the Co-rich droplets when no magnetic fields are applied. However, the Kelvin force acted upon the droplets by the non-uniform magnetic fields of the smallest gradient of B · dB/dz = 2.26 T m⁻² can be comparable to the gravity (figure 6a). As a result, the Marangoni migration is influential on the droplets of small sizes (r < 40 µm) only. In contrast, the migration of the larger droplets is led by the Kelvin force. As the gradients of the non-uniform magnetic fields are increased to B · dB/dz = 36.2 T m⁻², the Kelvin force acted upon most of the droplets is larger than the Marangoni force (figure 6c). At the gradient of B · dB/dz = 81.4 T m⁻², the Kelvin force on the droplets becomes generally larger than the Marangoni force (figure 6d). Thus, the migration of the droplets is determined overwhelmingly by the Kelvin force. The overwhelming role of the Kelvin force explains why the segregation patterns of the Co-rich droplets of the present samples were reversed when the sign of the gradients of the non-uniform magnetic fields was changed from negative to positive. The Kelvin force is supposed to accelerate both the coagulation and the segregation of the Co-rich droplets when the gradients are negative. In contrast, it counterbalances the effect of the Marangoni force and the Stokes motion when the gradients are positive. As a result, the droplet coagulation is reduced significantly. This effect can account for the continuous declining of the largest droplet size with rising gradient of the non-uniform magnetic fields (figure 5b). More critically, the upward Kelvin force reverses the segregation patterns of the droplets upon a change of the sign of the gradients. In this sense, the large gradients of the non-uniform magnetic fields are a more powerful tool for control of coagulation and segregation of the droplets in the phase-separated Cu–Co samples than the high intensities of the uniform magnetic fields. Such effects of the Kelvin force are supposed to be effective also in other phase-separated systems and thus, may find extensive applications in sophisticated studies and industries.

Table 1.

List of literature data of physical parameters used in calculations of different forces.

parameter	symbol (unit)	value	reference
mass density of Co-rich droplets	ρd (kg m−3)	8132	[51]
mass density of Cu-rich liquid	ρm (kg m−3)	7856	[51]
gravitational acceleration	g (m s−2)	9.8	—
temperature coefficient of interfacial energy	∂σ/∂T (J m−2 K−1)	–5.947 × 10–4	[58]
thermal gradient	∂T/∂r (K m−1)	1619	present work
volumetric susceptibility of Co-rich droplets	χd (–)	0.0385a	[59]
volumetric susceptibility of Cu-rich droplets	χm (–)	–0.0088a	[59]
magnetic permeability in vacuum	µ0 (H m−1)	4π × 10–7	—

^aDerived from the measured data of mass susceptibility of phase-separated Cu₇₀Co₃₀ and Cu₅₄Co₁₆ composition.

Figure 6.

Comparison of absolute magnitudes of gravity, driving force for Stokes motion, Marangoni force and Kelvin force acting on Co-rich droplets under different gradients of the non-uniform magnetic fields. The first three forces always point from the top to the bottom of the samples, whereas the Kelvin force is parallel or antiparallel to them depending on the sign of B · dB/dz. (a) B · dB/dz = ±2.26 T² m⁻¹; (b) B · dB/dz = ±9.05 T² m⁻¹; (c) B · dB/dz = ±36.2 T² m⁻¹; (d) B · dB/dz = ±81.4 T² m⁻¹. (Online version in colour.)

5. Conclusion

The present results have shown that the imposition of the static magnetic fields of high intensities or larger gradients has multiple effects on the liquid phase separation in glass-fluxed Cu–Co samples. Firstly, the magnetic fields bring about deformation of the Co-rich droplets. Secondly, the magnetic fields promotes coagulation of the Co-rich droplets leading to a bimodal or a multimodal droplet size distribution. Last but not least, a change of the sign of the gradients of the magnetic fields brings about a reversal of the segregation patterns of the Co-rich droplets. Such effects have been attributed to intensification of Marangoni convection, intensification of thermoelectric magnetohydrodynamic convection and an overwhelming role of the Kelvin force in determining the migration of the Co-rich droplets. The large gradients of the magnetic fields have been suggested to be more powerful for control of droplet motion in a phase-separated mixture than the high intensities.

Data accessibility

The datasets supporting this article have been uploaded as part of the electronic supplementary material.

Authors' contributions

D.Z. performed experiments and metallographic studies, collected and interpreted data and drafted the paper. J.G. conceived and designed the experiments, interpreted the data and helped draft the paper.

Competing interests

We declare we have no competing interests.

Funding

J.G. and D.Z. are grateful to financial support by the National Natural Science Foundation of China (51271055). D.Z. is grateful to the China Scholarship Council for a visiting PhD studentship.