## 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.

This is a preview of subscription content, log in to check access.

## References

- 1.
Chiew YC, Glandt ED (1983) The effect of structure on the conductivity of a dispersion. J Colloid Interface Sci 94:90–104. https://doi.org/10.1016/0021-9797(83)90238-2

- 2.
Wu C, Cho TJ, Xu J, Lee D, Yang B, Zachariah MR (2010) Effect of nanoparticle clustering on the effective thermal conductivity of concentrated silica colloids. Phys Rev E 81:011406. https://doi.org/10.1103/PhysRevE.81.011406

- 3.
Ammar A, Chinesta F, Heyd R (2017) Thermal conductivity of suspension of aggregating nanometric rods. Entropy 19:19. https://doi.org/10.3390/e19010019

- 4.
Timofeeva EV, Gavrilov AN, McCloskey JM, Tolmachev YV, Sprunt S, Lopatina LM, Selinger JV (2007) Thermal conductivity and particle agglomeration in alumina nanofluids: experiment and theory. Phys Rev E 76:061203. https://doi.org/10.1103/PhysRevE.76.061203

- 5.
Bica I, Liu YD, Choi HJ (2013) Physical characteristics of magnetorheological suspensions and their applications. J Ind Eng Chem 19:394–406. https://doi.org/10.1016/j.jiec.2012.10.008

- 6.
Ruan X, Wang Y, Xuan S, Gong X (2017) Magnetic field dependent electric conductivity of the magnetorheological fluids: the influence of oscillatory shear. Smart Mater Struct 26:035067. https://doi.org/10.1088/1361-665X/aa5fe5

- 7.
Forero-Sandoval IY, Vega-Flick A, Alvarado-Gil JJ, Medina-Esquivel RA (2017) Study of thermal conductivity of magnetorheological fluids using the thermal-wave resonant cavity and its relationship with the viscosity. Smart Mater Struct 26:025010. https://doi.org/10.1088/1361-665X/26/2/025010

- 8.
Heine MC, de Vicente J, Klingenberg DJ (2006) Thermal transport in sheared electro- and magnetorheological fluids. Phys Fluids 18:023301. https://doi.org/10.1063/1.2171442

- 9.
Shima PD, Philip J (2011) Tuning of thermal conductivity and rheology of nanofluids using an external stimulus. J Phys Chem C 115:20097–20104. https://doi.org/10.1021/jp204827q

- 10.
Katiyar A, Dhar P, Nandi T, Das SK (2016) Magnetic field induced augmented thermal conduction phenomenon in magneto-nanocolloids. J Magn Magn Mater 419:588–599. https://doi.org/10.1016/j.jmmm.2016.06.065

- 11.
Mousavi NSS, Kumar S (2018) Effective in-field thermal conductivity of ferrofluids. J Appl Phys 123:043902. https://doi.org/10.1063/1.5010275

- 12.
Song D-X, Ma W-G, Zhang X (2019) Anisotropic thermal conductivity in ferrofluids induced by uniform cluster orientation and anisotropic phonon mean free path. Int J Heat Mass Transf 138:1228–1237. https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.120

- 13.
Vinod S, Philip J (2020) Impact of field ramp rate on magnetic field assisted thermal transport in ferrofluids. J Mol Liq 298:112047. https://doi.org/10.1016/j.molliq.2019.112047

- 14.
Doganay S, Turgut A, Cetin L (2019) Magnetic field dependent thermal conductivity measurements of magnetic nanofluids by 3ω method. J Magn Magn Mater 474:199–206. https://doi.org/10.1016/j.jmmm.2018.10.142

- 15.
Doganay S, Alsangur R, Turgut A (2019) Effect of external magnetic field on thermal conductivity and viscosity of magnetic nanofluids: a review. Mater Res Express 6:112003. https://doi.org/10.1088/2053-1591/ab44e9

- 16.
Popplewell J, Rosensweig RE (1996) Magnetorheological fluid composites. J Phys D Appl Phys 29:2297–2303. https://doi.org/10.1088/0022-3727/29/9/011

- 17.
Ramos J, Klingenberg DJ, Hidalgo-Alvarez R, de Vicente J (2011) Steady shear magnetorheology of inverse ferrofluids. J Rheol 55:127–152. https://doi.org/10.1122/1.3523481

- 18.
Ortiz-Salazar M, Pech-May NW, Vales-Pinzon C, Medina-Esquivel R, Alvarado-Gil JJ (2018) Magnetic field induced tunability on the thermal conductivity of ferrofluids loaded with carbon nanofibers. J Phys D Appl Phys 51:075003. https://doi.org/10.1088/1361-6463/aaa5cb

- 19.
Wang RH, Knudsen JG (1958) Thermal conductivity of liquid-liquid emulsions. Ind Eng Chem 50:1667–1670. https://doi.org/10.1021/ie50587a042

- 20.
Chiesa M, Garg J, Kang YT, Chen G (2008) Thermal conductivity and viscosity of water-in-oil nanoemulsions. Colloids Surf A Physicochem Eng Asp 326:67–72. https://doi.org/10.1016/j.colsurfa.2008.05.028

- 21.
Yadegari M, Seifi M, Sabbaghzadeh J (2012) Examination of thermal properties of paraffin/water emulsion with implications for heat transfer. J Thermophys Heat Transf 26:535–539. https://doi.org/10.2514/1.T3842

- 22.
Gamot T, Bhattacharyya AR, Sridhar T, Fulcher AJ, Beach F, Tabor RF, Majumder M (2019) Enhanced thermal conductivity of high internal phase emulsions with ultra-low volume fraction of graphene oxide. Langmuir 35:2738–2746. https://doi.org/10.1021/acs.langmuir.8b04116

- 23.
Puri PS, deMan JM (1977) Thermal conductivity of liquid-liquid emulsions. Can Inst Food Sci Technol J 10:49–52. https://doi.org/10.1016/S0315-5463(77)73436-4

- 24.
Liu J, Lawrence EM, Wu A, Ivey ML, Flores GA, Javier K, Bibette J, Richard J (1995) Field-induced structures in ferrofluid emulsions. Phys Rev Lett 74:2828–2831. https://doi.org/10.1103/PhysRevLett.74.2828

- 25.
Brojabasi S, Lahiri BB, Philip J (2014) External magnetic field dependent light transmission and scattered speckle pattern in a magnetically polarizable oil-in-water nanoemulsion. Physica B 454:272–278. https://doi.org/10.1016/j.physb.2014.08.003

- 26.
Mohapatra DK, Laskar JM, Philip J (2020) Temporal evolution of equilibrium and non-equilibrium magnetic field driven microstructures in a magnetic fluid. J Mol Liq 304:112737. https://doi.org/10.1016/j.molliq.2020.112737

- 27.
Zakinyan A, Dikansky Y, Bedzhanyan M (2014) Electrical properties of chain microstructure magnetic emulsions in magnetic field. J Dispers Sci Technol 35:111–119. https://doi.org/10.1080/01932691.2013.769109

- 28.
Yang R-J, Hou H-H, Wang Y-N, Fu L-M (2016) Micro-magnetofluidics in microfluidic systems: a review. Sensors Actuators B Chem 224:1–15. https://doi.org/10.1016/j.snb.2015.10.053

- 29.
Pankhurst QA, Thanh NTK, Jones SK, Dobson J (2009) Progress in applications of magnetic nanoparticles in biomedicine. J Phys D Appl Phys 42:224001. https://doi.org/10.1088/0022-3727/42/22/224001

- 30.
Mefford OT, Woodward RC, Goff JD, Vadala TP, Pierre TGS, Dailey JP, Riffle JS (2007) Field-induced motion of ferrofluids through immiscible viscous media: testbed for restorative treatment of retinal detachment. J Magn Magn Mater 311:347–353. https://doi.org/10.1016/j.jmmm.2006.10.1174

- 31.
Elmore WC (1938) Ferromagnetic colloid for studying magnetic structures. Phys Rev 54:309–310. https://doi.org/10.1103/PhysRev.54.309

- 32.
Popplewell J, Al-Qenaie A, Charles SW, Moskowitz R, Raj K (1982) Thermal conductivity measurements on ferrofluids. Colloid Polym Sci 260:333–338. https://doi.org/10.1007/BF01447973

- 33.
Xu L, Yang S, Huang J (2019) Thermal illusion with the concept of equivalent thermal dipole. Eur Phys J B 92:264. https://doi.org/10.1140/epjb/e2019-100377-5

- 34.
Nagel JR (2014) Numerical solutions to Poisson equations using the finite-difference method. IEEE Antennas Propag Mag 56:209–224. https://doi.org/10.1109/MAP.2014.6931698

- 35.
Asami K (2005) Simulation of dielectric relaxation in periodic binary systems of complex geometry. J Colloid Interface Sci 292:228–235. https://doi.org/10.1016/j.jcis.2005.05.076

- 36.
Löwen H (2001) Colloidal soft matter under external control. J Phys Condens Matter 13:R415–R432. https://doi.org/10.1088/0953-8984/13/24/201

- 37.
Chiolerio A, Quadrelli MB (2017) Smart fluid systems: the advent of autonomous liquid robotics. Adv Sci 4:1700036. https://doi.org/10.1002/advs.201700036

- 38.
Ge J, He L, Hu Y, Yin Y (2011) Magnetically induced colloidal assembly into field-responsive photonic structures. Nanoscale 3:177–183. https://doi.org/10.1039/c0nr00487a

- 39.
Torres-Díaz I, Rinaldi C (2014) Recent progress in ferrofluids research: novel applications of magnetically controllable and tunable fluids. Soft Matter 10:8584–8602. https://doi.org/10.1039/c4sm01308e

- 40.
Leal-Calderon F, Thivilliers F, Schmitt V (2007) Structured emulsions. Curr Opin Colloid Interface Sci 12:206–212. https://doi.org/10.1016/j.cocis.2007.07.003

- 41.
McClements DJ (2012) Advances in fabrication of emulsions with enhanced functionality using structural design principles. Curr Opin Colloid Interface Sci 17:235–245. https://doi.org/10.1016/j.cocis.2012.06.002

- 42.
Kawanami T, Togashi K, Fumoto K, Hirano S, Zhang P, Shirai K, Hirasawa S (2016) Thermophysical properties and thermal characteristics of phase change emulsion for thermal energy storage media. Energy 117:562–568. https://doi.org/10.1016/j.energy.2016.04.021

- 43.
Shao J, Darkwa J, Kokogiannakis G (2016) Development of a novel phase change material emulsion for cooling systems. Renew Energy 87:509–516. https://doi.org/10.1016/j.renene.2015.10.050

- 44.
Wang M, He L, Yin Y (2013) Magnetic field guided colloidal assembly. Mater Today 16:110–116. https://doi.org/10.1016/j.mattod.2013.04.008

- 45.
Derks RJS, Dietzel A, Wimberger-Friedl R, Prins MWJ (2007) Magnetic bead manipulation in a sub-microliter fluid volume applicable for biosensing. Microfluid Nanofluid 3:141–149. https://doi.org/10.1007/s10404-006-0112-9

- 46.
Toussaint R, Akselvoll J, Helgesen G, Flekkøy EG, Skjeltorp AT (2004) Interactions of magnetic holes in ferrofluid layers. Progr Colloid Polym Sci 128:151–155. https://doi.org/10.1007/b97120

- 47.
de Lange KK, Helgesen G, Skjeltorp A (2006) Braid theory and Zipf-Mandelbrot relation used in microparticle dynamics. Eur Phys J B 51:363–371. https://doi.org/10.1140/epjb/e2006-00241-7

- 48.
Bossis G, Lançon P, Meunier A, Iskakova L, Kostenko V, Zubarev A (2013) Kinetics of internal structures growth in magnetic suspensions. Physica A 392:1567–1576. https://doi.org/10.1016/j.physa.2012.11.029

- 49.
Rosensweig RE (1985) Ferrohydrodynamics. Cambridge University Press, Cambridge

- 50.
Khokhryakova CA, Pshenichnikov AF, Lebedev AV (2019) Determination of the weight of a non-magnetic body immersed in magnetic fluid exposed to uniform magnetic field. Magnetohydrodynamics 55:73–78. https://doi.org/10.22364/mhd.55.1-2.9

## Funding

The reported study was funded by Russian Foundation for Basic Research (RFBR) according to the research project no. 18-33-00796.

## Author information

### Affiliations

### Corresponding author

## Ethics declarations

### Conflict of interest

The authors declare that they have no conflicts of interest.

### Availability of data and material

Not applicable.

### Code availability

Not applicable.

## Additional information

### Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## 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

where

The discrete representation of the boundary conditions has the form:

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

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

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

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:

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:

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:

where **r**_{ij} = **r**_{i} − **r**_{j}, and the dipole moments are calculated within the isolated particle approximation:

where **H**_{0} 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:

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]).

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].

## Rights and permissions

## About this article

### Cite this article

Zakinyan, A., Arefyev, I. Thermal conductivity of emulsion with anisotropic microstructure induced by external field.
*Colloid Polym Sci* **298, **1063–1076 (2020). https://doi.org/10.1007/s00396-020-04672-x

Received:

Revised:

Accepted:

Published:

Issue Date:

### Keywords

- Emulsion
- Structure formation
- Thermal conductivity
- Ferrofluid