The presence of colloidal particles is known to increase the thermal conductivity of base fluids. The shape and structure of the solid particles are important in determining the magnitude of enhancement. Spherical particles—the only shape for which analytic theories exist—produce the smallest enhancement. Nonspherical shapes, including clusters formed by colloidal aggregation, provide substantially higher enhancements. We conduct a numerical study of the thermal conductivity of nonspherical structures dispersed in a liquid at fixed volume fraction in order to identify structural features that promote the conduction of heat. We find that elongated structures provide high enhancements, especially if they are long enough to create a solid network (colloidal gel). Cross-linking further enhances thermal transport by directing heat in multiple directions. The most efficient structure is the one formed by hollow spheres consisting of a solid shell and a core filled by the fluid. In both dispersed and aggregated forms, hollow spheres provide enhancements that approach the theoretical limit set by Maxwell’s theory.

Common fluids used in heating/cooling processes have very low thermal conductivity in comparison to solid materials [

Structural details of colloidal clusters are important in determining the conductivity of the dispersion. While Maxwell’s theory provides a baseline calculation for two idealized limits, that of fully dispersed and that of fully gelled states, for nearly all other cases theory is inadequate and one must resort to numerical simulations. To evaluate thermal conductivity of colloidal suspensions with different properties including particles shape and size along with investigating the effect of aggregation and cluster structure using model configurations, we need to go beyond experimental limitations and employ an accurate numerical model to avoid colloidal complications. In macroscopic simulations, large-scale structural effects can be captured. These simulations can be done by different standard models to solve the macroscopic conduction equation; however Monte Carlo method is both fast and accurate and especially well suited for complex geometries.

In this study we present a systematic investigation on the thermal conductivity of nonspherical particles with special interest in identifying structures that maximize conductivity at fixed volume fraction of the solid and approach the upper limit of the theory. We explore the range of validity of Maxwell’s theory for different particle shapes, evaluate thermal conductivity of solid particles, hollow particles as well as rods, and other nonspherical shapes, and identify the structures that produce maximum enhancement at fixed volume fraction of the dispersed phase.

Of the several methods available for the conductivity of heterogeneous structures, Monte Carlo is particularly useful because it allows the study of systems with arbitrarily complex geometries. In Monte Carlo we obtain the thermal conductivity of a two-phase system via the statistics of a biased random walk along sites with different thermal conductivities. The method as implemented here ignores the motion of the particle through the fluid medium as well as all fluid-mediated interactions between particles. We justify omitting these factors for two reasons. The first one is that Maxwell’s theory itself only considers conduction and neglects all other mechanisms of heat transfer. In this respect the simulation provides a direct comparison to the predictions of Maxwell’s theory. The second reason is based on the previous experimental and theoretical studies that show the enhancement of the thermal conductivity of clustered dispersions is fully captured by the conduction of heat along the solid backbone of the cluster and that other mechanisms, if present, make contributions that are at best within the error bounds of the experimental measurements [

The Monte Carlo method used here is based on the work of Van Siclen [

The dimensionless parameter that controls the conductivity of the suspension is the ratio of the solid-to-liquid thermal conductivity. Oxide materials in typical fluids have conductivity ratios in the range 2–50, and metallic particles can reach values of the order of 100. The ratio

An important element of the simulation is the discretization size of a unit element relative to the size of the spherical particle. In general, the smaller the discretization size, the more accurate the simulation but also more computationally intensive. It is common to represent primary solid particles by a single lattice site [

Effect of the number of lattice sites: a single spherical particle is composed of numerical simulations on thermal conductivity of the suspensions with

Comparison of thermal conductivity between simulations (points) and Maxwell’s theory (dashed line) for a well-dispersed suspension of spherical particles at different volume fractions with

Figure

Effect of particle size of spherical particles on thermal conductivity of the solution at fixed volume fraction of

Maxwell’s original derivation is based on a concentric core-shell model in which the core represents one phase (the dispersed solid) and the shell the other one (liquid) [

To test (

Thermal conductivity of hollow particles as the core radius increases while the volume fraction of the solid shell is constant at

In inhomogeneous two-phase systems, thermal conduction is dominated by the phase that provides the longest uninterrupted paths to heat transport. Given two materials of different conductivities layered in a core-shell arrangement, maximum conductivity is obtained when the most conductive material surrounds the least conductive phase. The smallest possible conductivity is achieved by reversing this order. In fact, if the less conductive phase is a perfect insulator, a core-shell particle with the insulator fully encapsulated will still conduct heat, whereas the inverted structure (insulator on the outside) is a perfect insulator. Generalizing this principle, structures that place the most conductive phase at the exterior while shielding the less conductive phase are expected to produce enhanced thermal conductivity relative to uniform dispersion of the more conductive phase. The hollow sphere is an exact analytic model that bridges the entire range of conductivities between the two bounds of Maxwell’s theory. At the lower limit we have a colloidal system of fully dispersed spheres. The upper limit represents an idealized system in which the entire fluid is found inside the core of a single hollow sphere. Such system cannot be made experimentally using primary spherical particles as its building blocks. Nonetheless, one does not need to reach this limit exactly to achieve high conductivity. With

Among the many other structures that have been studied, nanotubes and nanofibers are of special interest as model structures for nonspherical particles. They can be made out of materials with high thermal conductivity and their anisotropic shape makes them potentially excellent additives to thermal fluids [

For nonspherical particles (

The results in Figure

Thermal conductivity of suspensions of rods with aspect ratios in the range of

The general conclusion from these investigations is that Maxwell’s lower limit can be employed for suspension of monomers as long as we have symmetric shaped particles such as cubes and spheres, including core-shell structures. Anisotropic shapes, such as rods, cylinders, and ellipsoids, enhance conductivity above Maxwell’s lower limit. Nonetheless, the enhancement is always found to lie below the maximum limit of Maxwell’s theory.

Colloidal particles are susceptible to aggregation, especially in nanofluids prepared by dispersing dry particles in the fluid. Clusters are more complex in structure than the cases considered so far because of the random nature of contacts between the primary particles that make up the cluster. As a result, clusters with the same size (e.g., hydrodynamic radius) can have quite different structures, for example, linear, packed, fractal, and highly cross-linked. Although many studies have investigated different ways to create assemblies with controlled geometries [

Figure

Effect of aggregation state and cluster configuration on conductivity enhancement. Thermal conductivity of different model aggregates at

We have studied by systematic numerical simulation the effect of nanostructure on the thermal properties of nanofluids. At fixed volume fractions, spherical particles exhibit the lowest possible thermal conductivity. Structures formed by creating contacts between primary particles always have higher conductivity compared to fully dispersed spheres. Structural elements that contribute to enhanced conductivity are linear branches, which facilitate heat transport along uninterrupted paths, and cross-links, which help distribute the transport to multiple directions. Both elements are needed to produce large enhancements. Structures that encapsulate the fluid within regions that are more or less thermally isolated from the main fluid exhibit the maximum enhancement. The ideal structure is a solid spherical shell that is filled with the suspending fluid. Colloidal aggregates are reasonable approximations of this highly conducting structure. In all cases, the conductivity of the nanofluid is found to lie between the two limits of Maxwell’s theory. This adds further support to the suggestion that both the unusually large conductivities that have been reported in some studies and the inconsistencies as to the precise magnitude of the enhancement of the thermal conductivity of colloidal suspensions can be attributed to the presence of nonspherical particles/aggregates in the nanofluid.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Science Foundation under Grant no. CBET GOALI no. 1132220.