# Electrical Conductivity of Field-Structured Emulsions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{27}H

_{52}O

_{9}), a surface-active substance with the low hydrophilic-lipophilic balance equal to 3.5, was used. The surface-active substance was first dissolved in magnetic fluid (~3% vol), and then water was drop-by-drop added at continuous mechanical mixing with a homogenizer. This yielded an average ≈23 µm microdrop emulsion. Figure 1 shows the experimentally determined emulsion drop-size distribution obtained by the optical microscopy image processing using Matlab script. The dispersed phase drop sizes were by several orders of magnitude larger than those of the magnetic fluid magnetic nanoparticles (~10 nm); hence, within the framework of this study, the magnetic fluid can be considered as a continuous magnetizable medium. The obtained emulsions may be classified as water-in-oil.

^{5}–10

^{7}, which indicates the prevalence of magnetic interactions due to the large size of the droplets. At the low (~0.1) dispersed phase volume fractions, the emulsion drops are organized into separate chains, whose image obtained with an optical microscope in transmitted light is shown in Figure 2a. When the emulsion dispersed phase volume fraction grows, the drops arrange into dense column-shape structures instead of separate chains under action of magnetic field, as shown in Figure 2b. The transition from single chains to larger bundles has been previously observed in the suspensions of solid particles in magnetic fields [40,41].

^{−6}S/m for the magnetic fluid, 4.4 × 10

^{−2}S/m for the water, and 2.7 × 10

^{−8}S/m for the emulsifier. Due to the great specific conductivity of the dispersed phase compared with the dispersion medium, a significant change of the macroscopic electrical conductivity of such emulsions during structure formation can be expected. The dispersion frequency for the system under study can be assessed from the Wagner formula [17]:

_{i}, ε

_{e}are the dielectric permeabilities of the dispersed phase and dispersion medium correspondingly; λ

_{i}, λ

_{e}are the specific electrical conductivities of the dispersed phase and dispersion medium correspondingly; φ is the dispersed phase volume fraction. Substituting corresponding numerical values, one can ascertain that, here, the measuring frequency is much lower than the dispersion frequency for the system under study (7~9 × 10

^{6}Hz), hence, this measuring mode can be assumed quasistatic. Note that the physical parameters of a pure magnetic fluid can be considered constant and field-independent in a strength range used in the current experiments.

## 3. Experimental Results

## 4. Analysis and Discussion

#### 4.1. Conductivity Estimation

_{s}is the shell specific electrical conductivity; ν = [R/(R + d)]

^{3}; R is the dispersed phase droplet radius; d is the shell thickness. The length of a single surfactant molecule (polyglyceryl-3 polyricinoleate) is approximately 3.5 nm. The calculation according to Equation (4) gives a better agreement with the experimental data, as shown in Figure 9.

#### 4.2. Structure Simulation

**r**

_{i}is the Cartesian coordinates of ith droplet; η

_{i}, η

_{e}are the dynamic viscosities of the dispersed phased and dispersion medium correspondingly;

**F**

_{i}is the total force acting on a droplet excluding the hydrodynamic force. Equation (7) is the Hadamard–Rybchinsky formula, and it gives an algorithm of emulsion microstructure evolution simulation. The total force comprises the magnetic force and the force of droplets repulsion from each other and from bounding walls:

**r**

_{ji}=

**r**

_{j}−

**r**

_{i}; i, j are the droplets numbers;

**r**

_{iw}is the perpendicular vector dropped from the center of the ith drop to the wall; n = 2; A is the adjusting coefficient. The repulsive force has been attenuated in such a manner that the minimum spacing between the surfaces of the droplets and the surfaces of the droplets and walls was no more than 0.5 μm.

_{0}is the droplet volume;

**H**is the magnetic field strength related with the scalar magnetic potential ψ by the equation:

**r**is the coordinates of points at the boundary;

**n**is the outer normal.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**A horizontal layer of water in magnetic fluid emulsion under action of uniform constant magnetic field, directed horizontally along plane of figure. (

**a**) Dispersed phase volume fraction is φ = 0.1. (

**b**) Dispersed phase volume fraction is φ = 0.3, external magnetic field strength is H = 3.5 kA/m. The layer thickness is about droplets diameter.

**Figure 3.**Experimental setup for the electrical conductivity measurement: 1—Helmholtz coils; 2—measuring cell filled with sample under study; 3—alternating current impedance meter.

**Figure 4.**Experimental dependences of magnetic fluid emulsion specific electrical conductivity relative change on external magnetic field strength at different dispersed phase volume fractions. Magnetic and measuring electric fields are parallel-oriented.

**Figure 5.**Dependence of magnetic fluid emulsion electrical conductivity relative change on angle between directions of external magnetic and measuring electric fields. Dispersed phase volume fraction is 0.2; magnetic field strength is 1.1 kA/m. Dots are experiments; line is the experimental data approximation by Equation (5).

**Figure 6.**Experimental dependences of emulsion electrical conductivity relative change on dispersed phase volume fraction at different magnetic field strength values. External magnetic field is parallel to electric measuring field.

**Figure 7.**Experimental dependences of reduced relative change of emulsion conductivity on time of action of magnetic field (parallel to electric measuring field) at different strength values. Dispersed phase volume fraction is 0.4.

**Figure 8.**Experimental dependences of reduced relative change of emulsion conductivity on time of action of magnetic field at different dispersed phase volume fractions. Magnetic field strength is 3.3 kA/m.

**Figure 9.**Dependences of magnetic fluid emulsion specific electrical conductivity on dispersed phase volume fraction. Magnetic field is parallel to electric field. Dots are experiments; lines are calculations.

**Figure 10.**(

**a**) Magnetic field distribution in a computational domain containing the magnetic fluid emulsion sample enclosed in rectangular cell. External homogeneous magnetic field is vertical in image plane. (

**b**) Emulsion sample microstructure forming in magnetic field shown in (

**a**). Emulsion volume fraction is 0.2.

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**MDPI and ACS Style**

Zakinyan, A.R.; Kulgina, L.M.; Zakinyan, A.A.; Turkin, S.D.
Electrical Conductivity of Field-Structured Emulsions. *Fluids* **2020**, *5*, 74.
https://doi.org/10.3390/fluids5020074

**AMA Style**

Zakinyan AR, Kulgina LM, Zakinyan AA, Turkin SD.
Electrical Conductivity of Field-Structured Emulsions. *Fluids*. 2020; 5(2):74.
https://doi.org/10.3390/fluids5020074

**Chicago/Turabian Style**

Zakinyan, Arthur R., Ludmila M. Kulgina, Anastasia A. Zakinyan, and Sergey D. Turkin.
2020. "Electrical Conductivity of Field-Structured Emulsions" *Fluids* 5, no. 2: 74.
https://doi.org/10.3390/fluids5020074