Thermal conductivity of emulsion with anisotropic microstructure induced by external field

Abstract

The structure formation influence on various macroscopic properties of disperse systems is poorly investigated in respect to emulsion systems. The present work deals with the experimental and theoretical study of the heat transfer in emulsions whose dispersed phase droplets are arranged into chain-like structures under the action of external force field. The studied emulsions are of water-in-oil and oil-in-water types; they are based on ferrofluid and contain dispersed phase droplets in a size range of the order of several tens of micrometers. It is demonstrated that the emulsion thermal conductivity grows notably under the effect of external magnetic field parallel to the heat flux and provoking structure formation. It is revealed that the maximal response of thermal conductivity on the magnetic field action takes place at the dispersed phase volume fraction of about 20–30%. The structure formation in magnetic field has been simulated, and the magnetic interactions of emulsion droplets with each other and with the sample boundaries have been considered and discussed. The macroscopic thermal conductivity of structured emulsions has been numerically calculated and compared with experimental data. The obtained results may be of interest in the development of potential applications of controlling the properties of colloids by magnetic field.

Appendices

Appendix 1. Finite-difference scheme

The finite-difference scheme used is illustrated in Fig. 13. The discrete approximation of Eq. (6) has the form

$$ {\displaystyle \begin{array}{r}-{a}_0{\psi}_{i,j,k}+{a}_1{\psi}_{i,j,k-1}+{a}_2{\psi}_{i-1,j,k}+{a}_3{\psi}_{i,j-1,k}+{a}_4{\psi}_{i,j+1,k}+\\ {}{a}_5{\psi}_{i+1,j,k}+{a}_6{\psi}_{i,j,k+1}=0,\end{array}} $$

(10)

where

$$ {\displaystyle \begin{array}{c}\begin{array}{c}{a}_0=3\left[{\mu}_{i,j,k}+{\mu}_{i,j,k-1}+{\mu}_{i-1,j,k}+{\mu}_{i-1,j,k-1}\right.+{\mu}_{i,j-1,k}+\\ {}\left.{\mu}_{i,j-1,k-1}+{\mu}_{i-1,j-1,k}+{\mu}_{i-1,j-1,k-1}\right],\end{array}\\ {}{a}_1={\mu}_{i,j,k-1}+{\mu}_{i,j-1,k-1}+{\mu}_{i-1,j-1,k-1}+{\mu}_{i-1,j,k-1},\\ {}{a}_2={\mu}_{i-1,j,k}+{\mu}_{i-1,j-1,k}+{\mu}_{i-1,j,k-1}+{\mu}_{i-1,j-1,k-1},\\ {}{a}_3={\mu}_{i,j-1,k}+{\mu}_{i-1,j-1,k}+{\mu}_{i,j-1,k-1}+{\mu}_{i-1,j-1,k-1},\\ {}{a}_4={\mu}_{i,j,k}+{\mu}_{i,j,k-1}+{\mu}_{i-1,j,k}+{\mu}_{i-1,j,k-1},\\ {}{a}_5={\mu}_{i,j,k}+{\mu}_{i,j-1,k}+{\mu}_{i,j,k-1}+{\mu}_{i,j-1,k-1},\\ {}{a}_6={\mu}_{i,j,k}+{\mu}_{i,j-1,k}+{\mu}_{i-1,j-1,k}+{\mu}_{i-1,j,k}.\end{array}} $$

(11)

Fig. 13

Finite-difference scheme for calculating the distribution of scalar magnetic potential and magnetic field strength

The discrete representation of the boundary conditions has the form:

$$ {\psi}_{\mathrm{bottom}}=\mathrm{B},\kern0.5em {\psi}_{\mathrm{top}}=\mathrm{T},\kern0.5em {\psi}_{\mathrm{boundary}}-{\psi}_{\mathrm{internal}}=0. $$

(12)

The solution of Eq. (10) has been obtained by means of successive over-relaxation method which consists in the iterative procedure:

$$ {\psi}_{i,j,k}^{\alpha +1}={\psi}_{i,j,k}^{\alpha }+\beta {R}_{i,j,k}^{\alpha } $$

(13)

where β is the relaxation factor; α, α + 1 denote two successive iteration steps; the residual R_{i, j, k} is defined by

$$ {\displaystyle \begin{array}{r}{R}_{i,j,k}=\frac{1}{a_0}\left({a}_1{\psi}_{i,j,k-1}+{a}_2{\psi}_{i-1,j,k}+{a}_3{\psi}_{i,j-1,k}+{a}_4{\psi}_{i,j+1,k}+\right.\\ {}\left.{a}_5{\psi}_{i+1,j,k}+{a}_6{\psi}_{i,j,k+1}\right)-{\psi}_{i,j,k}.\end{array}} $$

(14)

After having the magnetic potential obtained, the field strength distribution can be calculated according to the discrete analog of Eq. (5):

$$ {\displaystyle \begin{array}{c}\begin{array}{c}{\left({H}_x\right)}_{i,j,k}=\frac{1}{4h}\left({\psi}_{i,j,k}-{\psi}_{i,j+1,k}+{\psi}_{i+1,j,k}-{\psi}_{i+1,j+1,k}+\right.\\ {}\left.+{\psi}_{i,j,k+1}-{\psi}_{i,j+1,k+1}+{\psi}_{i+1,j,k+1}-{\psi}_{i+1,j+1,k+1}\right),\end{array}\\ {}\begin{array}{c}{\left({H}_y\right)}_{i,j,k}=\frac{1}{4h}\left({\psi}_{i,j,k}-{\psi}_{i+1,j,k}+{\psi}_{i,j+1,k}-{\psi}_{i+1,j+1,k}+\right.\\ {}\left.+{\psi}_{i,j,k+1}-{\psi}_{i+1,j,k+1}+{\psi}_{i,j+1,k+1}-{\psi}_{i+1,j+1,k+1}\right),\end{array}\\ {}\begin{array}{c}{\left({H}_z\right)}_{i,j,k}=\frac{1}{4h}\left({\psi}_{i,j,k}-{\psi}_{i,j,k+1}+{\psi}_{i+1,j,k}-{\psi}_{i+1,j,k+1}+\right.\\ {}\left.+{\psi}_{i,j+1,k}-{\psi}_{i,j+1,k+1}+{\psi}_{i+1,j+1,k}-{\psi}_{i+1,j+1,k+1}\right),\end{array}\end{array}} $$

(15)

where h is the step of grid equal in all directions.

Finally, the magnetic force acting on a single droplet can be obtained by the equations of the form:

$$ {\displaystyle \begin{array}{c}{\left({F}_m\right)}_x=\frac{\mu_0\left(\frac{\mu_i}{\mu_e}-1\right){V}_0}{2h{N}_i}\sum \limits_{i,j,k}\left\{{\mathbf{H}}_{i,j,k}\cdot \left[\left({\left({H}_x\right)}_{i,j+1,k}-{\left({H}_x\right)}_{i,j-1,k}\right){e}_x,\right.\right.\\ {}\left.\left.\left({\left({H}_x\right)}_{i+1,j,k}-{\left({H}_x\right)}_{i-1,j,k}\right){e}_y,\left({\left({H}_x\right)}_{i,j,k+1}-{\left({H}_x\right)}_{i,j,k-1}\right){e}_z\right]\right\},i,j,k\in \varOmega \end{array}} $$

(16)

Here e_x,y,z are the unit vectors of corresponding coordinate axes; Ω is the region of the finite-difference grid for the field strength occupied by the drop; N_i is the number of mesh points inside Ω; dot · denotes the scalar product. Analogous expressions hold for the other components of the magnetic force.

Equation (9) for the effective conductivity in a discrete representation has the form:

$$ {\lambda}_{eff}=\frac{\sum \limits_{i,j,k}{\lambda}_{i,j,k}{\left(\frac{\partial T}{\partial y}\right)}_{i,j,k}\cdot {h}^2{L}_y}{\Delta {TL}_x{L}_z\left({N}_y-1\right)}, $$

(17)

where L_x,y,z are the dimensions of the sample (computational domain) along the corresponding coordinate axes, N_y is the number of grid nodes for temperature along the y axis.

Appendix 2. Magnetic interactions

In many existing studies, the magnetic force acting on particles in a suspension under the uniform magnetic field is calculated using point-dipole approximation:

$$ {\displaystyle \begin{array}{c}{\mathbf{F}}_{\mathrm{m},i}=\frac{3{\mu}_0}{4\pi}\sum \limits_{j\ne i}\frac{1}{r_{ij}^5}\left[\left({\mathbf{p}}_j\cdot {\mathbf{r}}_{ij}\right){\mathbf{p}}_i+\left({\mathbf{p}}_i\cdot {\mathbf{r}}_{ij}\right){\mathbf{p}}_j+\left({\mathbf{p}}_i\cdot {\mathbf{p}}_j\right){\mathbf{r}}_{ij}-\right.\\ {}\left.\frac{5\left({\mathbf{p}}_j\cdot {\mathbf{r}}_{ij}\right)\left({\mathbf{p}}_i\cdot {\mathbf{r}}_{ij}\right)}{r_{ij}^2}{\mathbf{r}}_{ij}\right],\end{array}} $$

(18)

where r_ij = r_i − r_j, and the dipole moments are calculated within the isolated particle approximation:

$$ {\mathbf{p}}_i=\frac{3{\mu}_e\left({\mu}_i-{\mu}_e\right){V}_0{\mathbf{H}}_0}{2{\mu}_e+{\mu}_i} $$

(19)

where H₀ is the external field strength. The correction of Eq. (19) can be made by taking into account the dipolar fields of all particles in the system when calculating the magnetization of each single particle. In this case, the dipole moments can be obtained from the solution of the following system of linear equations:

$$ {\mathbf{p}}_i=\frac{3{\mu}_e\left({\mu}_i-{\mu}_e\right){V}_0}{2{\mu}_e+{\mu}_i}\left({\mathbf{H}}_0+\frac{1}{4\pi}\sum \limits_{j\ne i}\left[\frac{3{\mathbf{r}}_{ij}\left({\mathbf{p}}_j\cdot {\mathbf{r}}_{ij}\right)}{r_{ij}^5}-\frac{{\mathbf{p}}_j}{r_{ij}^3}\right]\right),\kern0.5em i=1,\dots, N $$

(20)

where N is the number of particles in a system. Figure 14 demonstrates the comparison of the calculation results for the magnetic interaction force between two magnetic spheres in a uniform magnetic field. The forces were calculated in a simple point-dipole approximation (Eq. (19)), in a corrected point-dipole approximation (Eq. (20)), and by using the exact procedure (Eq. (4)). The spheres are identical and immersed in a nonmagnetic environment; magnetic permeability of the sphere’s material is equal to 2; diameter of the spheres is 23 μm; external magnetic field of the strength 6.45 kA/m is directed along the line of centers of the spheres. As is seen, the mutual influence of the spheres in the current problem geometry leads to the enhancement of their magnetization and results in the increase of the interaction force. Moreover, the point-dipole approximation gives inexact outcome for the interaction force between the closely spaced particles; in the same time, for the particles separated by the distance larger than two particle diameters, the difference between the point-dipole approximation and the exact force calculation becomes negligible. It should be noted that some discussion of the impotence of the correct calculation of interaction force between magnetizable particles in an external field has been previously presented in several studies (see, e.g., [44, 45]).

Fig. 14

Magnetic interaction force between two magnetic spheres in a uniform magnetic field calculated by different methods

The problem is rather complicated for the nonmagnetic particles immersed in a finite volume of ferrofluid subjected to the external magnetic field. Such situation takes place for the inverse emulsions described above. In this case, the particles magnetically interact not only with each other but also with the sample boundaries. It should be noted that the structure formation processes in systems of such type have not been previously investigated nether experimentally nor theoretically (only unbounded volumes or infinite layers of magnetic holes were considered previously, e.g., [46, 47]). The point-dipole approximation cannot be applied in such a case on principle, and the only way is a direct calculation of magnetic force according to Eq. (4). To illustrate this point, Fig. 15 shows the calculated force experienced by the sole nonmagnetic sphere in dependence of its position inside the rectangular volume of ferrofluid subjected to the external magnetic field. The sphere diameter is 23 μm; the ferrofluid magnetic permeability is 2; the space outside the ferrofluid volume is also nonmagnetic; the dimensions of ferrofluid volume are height 0.25 mm, width and length 0.2 mm; the external uniform magnetic field of the strength 6.45 kA/m is directed along the height dimension. The sphere was moved along the volume centerline (parallel to the external field direction) from the bottom to the top and the magnetic force was calculated as a function of the distance from the volume bottom. As is seen, the notable magnetic interaction between the nonmagnetic particle and the boundary of magnetic and nonmagnetic media is taking place. This interaction has an effect on the process of structure formation in a suspension of nonmagnetic particles in a bounded volume of ferrofluid. It should be noted that the effect of space restriction on the structure formation in a suspension of magnetic particles in a nonmagnetic liquid has been theoretically analyzed previously in [48]. However, the case considered in the current paper is more complex because there is not only a mechanical interaction of particles with the sample boundaries but also a magnetic interaction. Also note that the force experienced by the single nonmagnetic body immersed in a finite volume of ferrofluid under the action of external magnetic field has been previously considered analytically in [49] and experimentally in [50].

Fig. 15

Force experienced by the sole nonmagnetic sphere in dependence of its position inside the rectangular volume of ferrofluid subjected to the external uniform magnetic field