Encyclopedia of Surface and Colloid Science

Encyclopedia of Surface and Colloid Science Colloidal Thermal Fluids Themis Matsoukas∗ and Saba Lotfizadeh† Department of Chemical Engineering, Pennsylvania State University, University Park, PA 16802 (Dated: 02/26/2014) The thermal properties of colloidal dispersions have only recently come under the scope of investigators and several reviews have appeared on this topic [1–14]. The motivation is quite practical. The heat transfer fluids used in common heat exchangers (but all liquids in general) have low thermal conductivity when compared to solids (Fig. 1). Among common thermal transfer liquids, water (k = 0.608 W/m K) is about the most conductive. Ethylene glycol and most other organic liquids have conductivities that are lower by a factor of 2 or more. The thermal conductivity of solid materials is typically much higher. Silica, a poor conductor of heat, has twice the conductivity of water; the conductivity of alumina is an order of magnitude higher, while that of metals is larger by yet another order of magnitude. This large difference implies that a solid dispersed in a thermal fluid can lead to significantly higher thermal conductivity, even at very small amounts. It is for this reason that the thermal properties of colloidal dispersions are of interests. There are obvious advantages in using particles in the nanometer range for such applications. Most importantly, small particles remain well dispersed avoid difficulties associated with settling, a problem that becomes more severe as the material density of the particles increases. There are also difficulties. Producing colloidally stable dispersions can be a challenge, especially in non-aqueous systems. One of the most powerful tools for enhancing stability, pH control, is not of much practical use because most industrial heat transfer operations require near-neutral pH conditions. Despite these problems, a small number of commercial heat transfer fluids enhanced by colloidal additives are currently available [15]. Over the past 15 years there has been growing interest in thermal characterizations of nanoparticle suspensions and in the mechanisms that control their thermal behavior. Here we summarize the major experimental and computational developments in this area. II. MAXWELL’S THEORY Colloidal dispersions are inhomogeneous media consisting of solid phase dispersed within a continuous fluid. ∗ † 4 INTRODUCTION [email protected] [email protected] 10 thermal conductivity (W/m k) I. 3 10 2 10 1 10 0 10 -1 10 -2 10 C6F14 C6H14 C2H8O2 water SiO2 Al2O3 Al Cu CNT FIG. 1. Thermal conductivity of selected liquid and solid materials at 20 ◦ C. Their transport properties are quite complex and a general theory that is simple enough for calculations is not available. If the mobility of the phases is neglected, the system may modelled as a static inhomogeneous dispersion. The theory for this model was developed by Maxwell [16] in the context of electrical conduction, which translates directly into thermal conductivity. The theory applies to the dispersion of immobile spherical inclusions in a continuum medium at volume fractions sufficiently low that each particle may be treated independently of the rest. The assumption of immobility ignores the effect of Brownian motion but this effect is small. Particles in a temperature gradient develop a thermophoretic drift velocity, uT = DT ∇T , where DT is the thermal diffusion coefficient. Under no slip conditions, the thermophoretic velocity is essentially the characteristic velocity for the transport of heat via microconvection induced by the prepense of the particles. For typical colloids, DT ∼ 10−12 m2 /s K, yielding thermophoretic velocities that are far too low to have an appreciable effect [17]. Maxwell’s theory, therefore, may still apply and indeed represents the benchmark by which to analyze experimental results. Maxwell first addressed the problem of conduction in an inhomogeneous medium idealized in the form of two concentric spheres (Fig. 2a), an inner sphere with radius R1 and conductivity k1 , and outer one with radius R2 and conductivity k2 . Maxwell shows that this system is equivalent to a homogeneous sphere 2 2R1 10 kp / kf = 10 8 k1 k / kf k2 2R2 (a) 6 4 upper limit (b) lower limit 2 FIG. 2. Maxwell’s model for the conductivity of a dispersed phase. 0 0.0 0.2 0.4 0.6 volume fraction 0.8 1.0 of radius R2 whose conductivity is k1 + 2k2 + 2(R1 /R2 )3 (k1 − k2 ) . k = k2 k1 + 2k2 − (R1 /R2 )3 (k1 − k2 ) FIG. 3. Upper and lower bounds of Maxwell’s theory. (1) If the inner sphere is replaced by a collection of N smaller spheres of radius R1′ , Maxwell showed that the conductivity of the new arrangement can be obtained as an extension of the above result, if one assumes the distances between the spheres to be large enough such that “effects in disturbing the course of the currents may be taken as independent of each other” [16]. Maxwell’s result for this case is k = k2 k1 + 2k2 + 2N (R1′ /R2 )3 (k1 − k2 ) . k1 + 2k2 − N (R1′ /R2 )3 (k1 − k2 ) (2) This equation gives the conductivity of a homogenous dispersion of N spheres in a continuous medium. It is the standard model for the conductivity of a colloidal dispersion with particle conductivity kp = k1 and a fluid conductivity kf = k2 . This result is usually expressed as an enhancement ratio in the form k kp + 2kf + 2φ(kp − kf ) = , kf kp + 2kf − φ(kp − kf ) (3) where φ = N (R1′ /R2 )3 is the volume fraction of the dispersed phase. With kp > kf , as is the case with most solid dispersions, the conductivity of the dispersion is always higher than that of the fluid. To first-order in φ the above result becomes   k kp − kf , (4) ≈ 1 + 3φ kf 2kf + kp and clearly shows that the fractional enhancement is proportional to the difference between the conductivity of the two phases. Upon increasing the conductivity of the solid the conductivity of the dispersion increases but not indefinitely. Setting kp ≫ kf , Eq. (3) gives   1 + 2φ kf ≈ (1 + 3φ)kf . (5) kmax = 1−φ This gives the maximum possible conductivity in a system of well-dispersed spheres at fixed volume fraction. In this limit the enhancement ratio k/kf depends only on the volume fraction but not on the conductivities of the two phases. For example, at φ = 0.05 the maximum enhancement that can be expected is k/kf = 1.158. The result can be explained as follows. When the solid phase is infinitely conductive, the rate of heat transfer is limited by the conductivity of the less conductive phase, which occupies a fraction 1 − φ of the total volume. Thus the result depends only of kf and φ. This behavior is reached for relatively small ratios of kp /kf . For example, with kp /kf = 10, the actual enhancement of the thermal conductivity is 93% of the value predicted by Eq. (5). This means that even materials with conductivities in the midrange of Fig. 1 can deliver practically the same enhancement as materials with much higher conductivity. The upper limit in Maxwell’s theory Maxwell’s result makes two important predictions. The first one is that the enhancement is independent of the size of the dispersed phase and depends only on its volume fraction. A dispersion, regardless of particle size, reduces to the core-shell system of Fig. 2, in which the core represents the dispersed phase coalesced into a single sphere with the same volume fraction. A second, less obvious consequence is that the order of the layers in the core-shell model is important. In the basic model depicted in Fig. 2, the core represents the phase with the higher conductivity. If the order is switched such that the more conductive phase is in the outside, the conductivity of the new arrangement is obtained from Eq. (3) by interchanging kp and kf and replacing φ by 1 − φ: k = kf  kp kf  3kf + 2φ(kp − kf ) 3kp − φ(kp − kf )  . (6) 3 (a) (b) (c) (d) FIG. 4. (a) Well-dispersed particles; (b) colloidal clusters; (c) colloidal gel; (d) fluid dispersed within solid matrix. Linearization of this result with respect to φ gives   k φ 2kp kf =1+ − − 1 + o[φ2 ] kf 3 kf kp (7) In the limit of high particle conductivity (kp ≫ kf ) Eq. (6) simplifies to   2φ ′ kmax = kp . (8) 3−φ The conductivity in this case depends on that of the solid (now the continuous phase), and in contrast to Eq. (5), it increases continuously with increasing kp . Equation (6) predicts conductivities that are higher than those from Eq. (3) at the same volume fraction. To understand why, we return to Fig. 2(a), which represents the dispersion as a core-shell structure. When the less conductive phase is placed in the shell, the system has lower overall conductivity because the exterior of the core-shell structure partially insulates the conductive core. In the limit that the conductivity of the shell goes to zero, the conductivity of the core-shell system goes to zero as well. On the other hand, if the more conductive phase is placed at the shell, the core-shell system will remain conductive even if the core is a perfect insulator. Therefore, given two phases with different conductivities, the most conductive core-shell structure is the one that places the more conductive material on the outside. Equations (3) and (6) are known as the lower and upper limits, respectively, of the Maxwell theory. They are often referred to as the Hashin-Shtrikman (H-S) bounds after the two authors who obtained them in the context of magnetic permeability [18]. The two bounds of Maxwell’s theory are shown in Fig. 3. Accordingly, the lower limit refers to a system of well-dispersed particles (the more conductive phase is dispersed within the less conductive phase) and the upper limit to a system in which the fluid is dispersed within the solid phase. The relevance of the Maxwell limits to colloidal systems has been elaborated in a series of papers by Eapen, Yi and coworkers [13, 17, 19]. The lower limit clearly represents a well-dispersed system of spherical particles at low volume fractions. The upper limit may be viewed as an idealized model for aggregated nanoparticles (Fig. 4). Colloidal clusters are typically fractal in structure that can be loosely modeled as interconnected chains. These provide a network of high-conductivity pathways that transfer heat over longer distances compared to well dispersed spheres. In the extreme case that the colloid forms a gel, the solid phase is truly continuous throughout the entire structure (Fig. 4c). This situation approximates the conditions of the upper limit in Maxwell’s theory. The analogy is not exact because the liquid forms also a continuous rather than a dispersed phase as Maxwell’s model assumes (Figure 4d) [20–23]. Nonetheless, and to the extent that the rate of heat transfer is dominated by the conductivity of the solid, continuous colloidal networks may be modelled by the upper limit of Maxwell’s theory. The two limits, shown in Fig. 3, represent the range of enhancement that can be expected from a colloidal dispersion and may be viewed as mixing rules for the conductivity of the two-phase system that depend on the degree of aggregation. Both Maxwell limits are below the diagonal that connects the conductivities of the pure phases and whose equation is k|| = (1 − φ)kf + φkp . (9) This expresses the conductivity of the system as a simple weighted average of the conductivities of the two phases and corresponds to a system of resistances in parallel. The upper limit of the theory comes close to the parallel resistance model but it still lies below it. Range of validity of Maxwell’s model The two Maxwell limits strictly apply to the core-shell structure of Fig. 2 with the more conductive phase placed in the inner core (lower limit) or in the shell (upper limit). In applying these to colloidal dispersions we must require the volume fraction to be small enough such that the effect of the dispersed phase may be treated as additive. For the lower limit this implies low volume fractions. This is usually interpreted to mean that Eq. (3) is exact to the first order in φ [13], however, direct calculation shows Maxwell’s result to hold up to surprising high volume fractions as long as particles are not touching [24]. Since most colloidal systems in thermal applications are at volume fractions below 10%, this requirement is normally met and Maxwell’s result is applicable. For the upper limit the requirement is φ → 1 because in this case the liquid forms the dispersed phase. This condition is never met in experimental systems. Equation (6) therefore must be viewed as a qualitative upper limit for the fully gelled colloid. Maxwell’s theory makes several other explicit or implicit assumptions. Particles are spherical and immobile, therefore, shape effects or the mobility of particles (or of the fluid for that matter) are not accounted for. The theory further assumes temperature profile at the fluid-solid interface is continuous, i.e., surface (Kapitza) resistance is not present. Finally, while the bounds of the theory provide a range of conductivities depending in the degree of aggregation, the conductivity of colloidal clusters is not predicted by the theory itself. These assumptions must 4 z 10 kp / kf = 10 R1 8 Wheatstone bridge k / kf 1 6 4 4 3 4 • L q 3 2 1 R3 2 Rw 1 2 rw r0 0.0 0.2 0.4 0.6 volume fraction 0.8 1.0 2 R2 FIG. 5. Predictions of he Hamilton-Crosser model for sphere, cube, cylinder with aspect ratio 10, and cylinder with aspect ratio 100 [25]. be taken into consideration when the theory is applied to experimental systems. FIG. 6. Schematic of the transient hot wire (THW) apparatus (adapted from Vadasz [27]). the same volume, the overall conductivity of the suspension increases. In the limit of high non-sphericity (e.g., a cylinder with aspect ratio that approaches ∞) the conductivity approaches the diagonal given by equation (9). Extension to non-spherical particles III. For non-spherical particles, the most common model is that of Hamilton and Crosser [25], which is based on the work of Fricke [26]. This model modifies Maxwell’s lower limit as follows: k= kp (1 + (n − 1)φ)) + kf (n − 1)(1 − φ) . kp (1 − φ) + kf (n − 1 − φ) (10) The shape of the particle is incorporated into the parameter n, whose general form is n = 3/ψ a , (11) where ψ is the sphericity of the particle, defined as the surface area of an equal volume sphere over the surface area of the particle. In Fricke [26] the exponent a is 1 for spheres, 2 for prolate ellipsoids, and 1.5 for oblate ellipsoids but the experiments of Hamilton and Crosser [25] are better described with a = 1 regardless of shape. With ψ = 1 the above reverts to Maxwell’s lower limit for spherical particles. For non-spherical particles (ψ < 1) Eq. (10) gives conductivities that are higher than that of spheres. Figure 5 shows the results of the HamiltonCrosser model for spheres, cubes, cylinders with aspect ratio 10, and cylinders with aspect ratio 100, using Eq. (11) with a = 1. Elongated particles such as cylinders and fibers facilitate heat transport along their primary axis. Upon increasing the aspect ratio at constant volume the enhancement along the backbone increases further and, even though transport along the perpendicular axis is decreased compared to the isotropic particle of EXPERIMENTAL MEASUREMENT OF CONDUCTIVITY The most direct way to measure conductivity is through application of Fourier’s law under steady state across a layer of fluid that is subjected to known heat flux. Steady-state methods, however, produce convective flows that interfere with the measurement and are difficult to control [28]. Transient methods avoid these problems. The technique most commonly used is transient the hot wire (THW) method, which applies a short heat pulse to a conductive wire immersed in the sample and extracts the conductivity from the transient response of the fluid. The transient nature of the experiment, its brief duration and small perturbation ensure that convection does not arise during the measurement. The technique has been proven highly accurate for both liquid and solid materials [28–31]. It is by far the most common method used for the conductivity of colloidal dispersions [27–66]. The Transient Hot Wire Method The basic setup of the THW apparatus is shown in Fig. 6. It consists of a thin metal wire, typically Pt, that runs along the axis of a cylindrical vessel that contains the liquid of interest. The wire is subjected to a step change of the applied voltage and its temperature rise is recorded as a function of time. The electrical circuit forms a Wheatstone bridge between the wire and three known resistances. By adjusting the resistance of the po- 5 tentiometer R3 such that no current runs between points 1 and 2, the resistance of the wire is calculated from the balance condition Rw = R1 R3 /R2 . This measurement produces both the resistive heat that is delivered through the wire as well as its temperature. The heat per unit length (W/m) is q̇L = i2 ρw /Aw , (12) where i is the current through the wire, and ρw , Aw , are the resistivity (Ω m) and cross sectional area (m2 ) of the wire. The temperature is obtained through the relationship between resistance and temperature, which is quadratic in T [57], Rw = a 0 + a 1 T + a 2 T 2 . (13) The calculation of conductivity requires a theoretical model for the temperature rise of the wire under a step change in the heat that is delivered through it. The model assumes a linear source of heat of infinite length, uniform temperature along the wire and within its cross section, and an infinite medium around the wire that transports heat by conduction only. These assumptions must be matched by the design of the apparatus. The characteristic time for establishing uni2 /αw , form temperature across the wire is of the order rw where αw is the thermal diffusivity of the wire. Using αw = 2.6 × 10−5 m2 /s [27], and rw = 50 µm, this time is of the order of 0.1 ms. Typical measurements last several s, therefore the above condition is well met. The conduction equation in the medium that surrounds the wire is [27] ∂T αf ∂ = ∂t r ∂r  r ∂T ∂r  , (14) where α = k/ρCp is the thermal diffusivity of the medium, ρ (kg/m) is its density and Cp (J/kg K) is its heat capacity. This is solved under the following conditions:   ∂T q̇L r =− , ∂r rw 2πk T (t = 0, r) = T0 . T (t, r → ∞) = T0 . The first of these is the boundary condition at the fluidwire interface and expresses the heat flux in terms of the temperature gradient on the fluid side of the interface. The second equation is the initial condition before the step change, and the third equation gives the far-field condition for temperature at all times. An analytic solu2 tion is obtained using the substitution x = rw /4αt [27]. The final form is  2  q̇L rw T − T0 = (15) Ei 4πk 4αt where Ei(x) is the exponential integral Ei = Z ∞ x e−x dx. x (16) 2 This equation can be expanded in terms of rw /4αt to produce a simple expression for experimental analysis,     4αt q̇L + ··· −γ + ln T − T0 = 2 4πk rw (17) where γ is Euler’s constant. The omitted terms are of the 2 order of rw /4αt and higher, an approximation that is acceptable over several s of the transient, provided that rw is sufficiently small. The conductivity is then calculated from the slope of a linear graph of T versus ln t: k= q̇L ∆(ln t) , 4π ∆T (18) which assumes that the physical properties of the fluid do not vary much with temperature. The basic setup described here, originally developed for gases, must be modified to accommodate liquids that are electrically conductive. The difficulty arises from partial flow of current through the liquid, polarization effects at the surface of the wire and poor signal-to-noise ratio [58]. These problems are generally avoided by applying a thin insulating layer on the wire. In one approach the wire is coated by a thin layer of polyester [58]. Other designs implement anodized tantalum wires in which a thin layer of metal oxide serves as the insulator [31, 45–47], and the use of a mercury capillary in which case mercury replaces the wire and the borosilicate glass offers the insulation [7, 40, 62]. Various corrections may be necessary to account for radiation losses, finite size of the apparatus, and other assumptions that are not matched by the experimental design. These can be found in the specialized literature (see for example references [7] and [58]) but essentially they apply corrections to the value of ∆T that is used in Eq. (18). Other methods The temperature oscillation method [67, 68] is an alternative technique for the measurement of conductivity. It applies an oscillating current that is introduced from the two ends of a cylindrical fluid volume and the thermal diffusivity is calculated from measurements of the amplitude and phase of these oscillations at various points. The oscillating input has certain advantages, for example, it helps erase concentration gradients that may develop by ionic species interacting with a charged wire but the method has not found widespread use in the study of colloidal systems yet. 6 IV. 2.0 (a) CONDUCTIVITY OF COLLOIDAL DISPERSIONS knf / kf 1.8 1.6 SiO2 Kang (2006) Eapen (2007) 1.4 1.2 1.0 0 5 10 15 20 % volume fraction 25 30 2.0 (b) knf / kf 1.8 1.6 Al2O3 Eastman (1996) Masuda (1993) Wang (1999) Das (2003) 1.4 1.2 1.0 0 5 10 15 20 % volume fraction 25 30 2.0 (c) knf / kf 1.8 1.6 1.4 CuO 1.2 Eastman (1996) Liu (2006) Xuan (2006) Zhang (2006) 1.0 0 5 10 15 20 % volume fraction 25 30 2.0 (d) knf / kf 1.8 1.6 1.4 Cu 1.2 Jna (2007) Xuan (2000) 1.0 0 5 10 15 20 % volume fraction 25 Since the mid 90’s there has been a rapidly growing interest in colloidal dispersions as thermal media for heat transfer. Alumina (Al2 O3 , kp = 40 W/m K) [33, 50, 52, 53, 60, 65, 69–76] and copper oxide (CuO, kp = 77 W/m K) [38, 60, 63, 64, 69, 71, 73, 74, 76, 77] are among the most widely used materials because they are rather inexpensive to obtain in nanometer-size particles. Moreover, even though their conductivity lies in the mid-range for solids (see Fig. 1), it is high enough that it can deliver the maximum enhancement predicted by Maxwell’s theory in Eq. (1). Colloidal silica, though not a good conductor, is often the subject of investigations primarily because of its availability in monodisperse form over a range of particle sizes [64, 78] but more important, it serves as a model colloid for well-controlled studies [17, 79]. Figure 7 summarizes results for silica, alumina, copper oxide and copper. Though silica offers a small advantage in thermal conductivity, enhancements of the order of 20% are possible but this requires volume fractions that are relatively high. As expected by Maxwell’s theory, the conductivity enhancement is more pronounced as the conductivity of the particles increases. Alumina and copper oxide, for example, produce enhancements of the order of 20% at volume fraction of about 5%. Even though copper has higher conductivity than any of the materials in Fig. 7, it does not lead to significantly better enhancements, due to the saturation effect expressed by Eq. (5). There are also practical difficulties associated with copper colloids: with a density of 8.96 g/cm3 , settling becomes a serious problem even for particles in the nanometer range. This explains in part the larger scatter in the experimental data when copper is compared to other lighter materials. With non-aqueous systems the enhancement of the thermal conductivity can be quite higher because of the larger solid-to-fluid ratio of conductivities. Among non-aqueous systems ethylene glycol is the most commonly used base fluid [34, 39, 53]. Some studies have considered less well-defined liquids such as engine oil and pump fluid, as a means of improving heat transfer in actual machinery. Nonetheless, the formation of stable suspensions in non-polar liquids remains a challenge that has limited both the practical application of colloidal thermal fluids, as well as the study of their thermal properties. 30 Non-spherical particles FIG. 7. Experimental conductivities of selected aqueous colloidal systems. Solid lines are the lower and upper bounds of Maxwell’s theory. Among the many other materials that have been studied, carbon nanotubes (CNT) and nanofibers are of special interest. CNTs consist of graphitic sheets that form multiwall nanotubes with diameter 20-500 nm and length that can be several micrometers. They are characterized by very high thermal conductivity and their anisotropic 7 1.4 1.25 preparation method A B C f 1.20 = 22% 1.15 k f 1.2 / k / kf 1.3 desrepsid ylluf k 1.10 1.1 1.05 1.0 f = 11% aggregation 1.00 0 100 200 300 400 homogenization time (min) 500 FIG. 8. Thermal conductivity of 0.6 vol. % carbon nanotube suspensions under various preparation conditions (adapted from [47]). shape makes them potentially excellent additives to thermal fluids. Several studies have reported on the thermal properties of these systems [21, 41, 46, 47, 49, 51, 52, 59, 80–83]. Figure 8 summarizes the results of one such study by Assael et al. [47] under various preparation conditions. A maximum enhancement of 38% is obtained at 0.1 vol. %, which represents a significant improvement over the base fluid (water) with very small amount of solid material. However, maintaining stable suspensions over extended periods of time is not easy. A dispersant is necessary (in that study SDS) in combination with extended sonication. However, conductivity decreases with sonication time and reaches a plateau of 10% above the conductivity of the fluid. This is partly attributable to breakage of the nanotubes, which results in smaller aspect ratio and thus lower enhancement (see Fig. 5), but it also reflects the fact that improved dispersibility comes at the expense of destroying the interconnected network of fibers that contributes to the large enhancement seen at short sonication times. While the thermal performance of these systems is superior to those involving colloidal particles, the difficulties associated with the preparation of stable dispersions with acceptable flow characteristics represent challenges that must be overcome before CNT suspensions can fulfill their potential in thermal applications involving liquid media. Beyond Maxwell’s Limit The measured conductivity of colloids is often found to exceed the lower limit of Maxwell’s theory, as seen in Fig. 7. This has led to various speculations about possible nanoscale mechanisms that could explain effects that are stronger than those predicted by classical continuum theories [1, 32, 34, 84–86]. It is now understood that this behavior can be fully accounted for by colloidal aggregation within the context of Maxwell’s theory [13, 19]. As 0 50 100 150 200 250 300 size (nm) FIG. 9. Conductivity of colloidal silica (39 nm) as a function of cluster size: the conductivity at fixed volume fraction increases with increasing cluster size [101]. discussed in the theory section, clustered colloids are predicted to exhibit conductivity that is higher than that of the well-dispersed system. Maxwell’s upper limit in Eq. (6) offers an upper bound, albeit approximate, of the maximum effect due to clustering. Figure 7 demonstrates that the experimental data lie indeed with the two classical limits. An extensive review of literature given by [13] shows this to be invariably the case. Several studies, both experimental and computational, have looked into the effect of aggregation [2, 13, 19, 35, 50, 61, 87–100]. The most direct evidence is shown in Fig. 9. In this study, modified silica by surface silanization produced a colloid whose degree of aggregation can be controlled reversibly by pH in the full range, from well-dispersed particles to a colloidal gel [101]. This allows the determination of the conductivity of the colloidal system at fixed volume fraction as a function of the degree of clustering, as measured by the size of the clusters. As theory predicts, conductivity increases with size, an effect that can be attributed to increased rate of heat transfer along the solid network formed by the aggregated particles. Although colloidal clusters are more efficient media for heat transfer than well-dispersed particles, the practical difficulties associated with the handling and stability of aggregated colloids makes it difficult to implement in as a thermal fluid. It is possible, however, to take advantage of the increased transport along a percolated network to produce thermal switches that capitalize on reversible formation of such networks [99]. V. SIMULATION Maxwell’s theory provides a baseline calculation for an idealized system of well dispersed spheres in the limit of low volume fraction. For nearly all other cases, theory is inadequate and one must resort to numerical simulation. Generally, simulations may be performed at one of three 8 levels: 1. Macroscopic 2. Atomistic (molecular dynamics) 3. Coarse grained (dissipative particle dynamics) Each level offers advantages and has its own limitations. Macroscopic simulations are capable of capturing large scale structural effects, in particular clustering. The system is essentially modelled as a macroscopic object of interconnected spheres (clusters of primary particles) by solving the macroscopic conduction equation. This can be done by any standard method (for example, finite differences, finite elements) but Monte Carlo methods are especially suited to handle complex geometries and one such algorithm will be discussed in greater detail below. Molecular dynamics (MD) focuses on atomistic-level effects with pico-second time resolution and makes it possible to study the effect of molecular interactions as well as spatial effects at the nm range. Molecular dynamics simulations provide atomistic resolution at the expense of computational cost and are thus highly restricted by the number of atoms and the required simulation time to estimate properties. Large aggregates such as those seen in experiments cannot be modeled at the atomistic scale. Dissipative particle dynamics (DPD) focuses at a scale intermediate between atomistic and macroscopic. Particles are treated as objects without internal structure and the fluid is modelled as a continuum. This level of simulation focuses on particle-fluid interactions and their effect on heat transfer. All of the above methods have been applied to study the thermal properties of colloids, and they will be described below. Macroscopic simulations by Monte Carlo Monte Carlo (MC) methods take advantage of the mathematical similarity between diffusion and heat conduction to calculate the thermal conductivity of a composite phase using random walks. Compared to classical numerical methods that solve the steady-state conduction equation, MC is much simpler to implement, straightforward to program, and does not require numerical libraries beyond access to a random number generator. This makes MC particularly appealing for the study of complex structures such as aggregated colloids, colloidal gels, nanofiber networks [92, 102–106] and twophase systems in general [24, 107–113]. The theoretical foundation of the algorithm was given by DeW. Van Siclen [108] and its implementation is as follows. The simulation volume is discretized in cubic elements of equal size, each element representing either fluid or particle (Fig. 10). A walker is launched to perform a random walk of unit steps that is biased by the conductivity in each phase. Starting at a site, the walker on the discretized 3-d lattice can move in one of 6 directions. A FIG. 10. Schematic of Monte Carlo simulation of thermal conductivity. The volume is discretized into cubic elements and the conductivity is calculated by analyzing the trajectory of random walkers whose steps are biased by the conductivity of the phases. direction is chosen at random and the walker is advanced to a neighbor site with probability pi→j = kj ki + kj where ki is the conductivity of the current site, and kj the conductivity of the neighbor. If the target site is of the same material (ki = kj ) the walker has a 50% chance to make the move. For dissimilar materials, the walker always has higher probability to remain in the more conductive phase. Time is advanced by 1/ki and the process is repeated to produce a trajectory in time, r(t). The thermal diffusivity D of the composite material is obtained from the Einstein relationship [92], 1 |r(t + τ )r(t)|2 , τ →∞ 6t D = lim (19) and the conductivity is finally calculated from its relationship to the thermal diffusivity, k = ρCP D. A large number of trajectories is generally required, especially if the difference in the conductivities of the two phases is large, in order to obtain results of sufficient accuracy. In practice, a large number of independent walkers is launched and the results are averaged to obtain the mean squared displacement. The method is very simple to implement and is capable of handling complicated structures such as non-spherical particles and colloidal aggregates. Using this method, it was shown that the conductivity of fractal colloidal clusters increases well beyond that for well-dispersed spheres. Such studies highlight the fact that the structure of the clusters is an important factor, for example, clusters that contain a higher percentage of their particles along the backbone of the network exhibit higher conductivity than those composed of more compact arrangements [88]. This further emphasizes the importance of characterizing the colloidal state of a system when interpreting its thermal properties. 9 Molecular Dynamics (MD) The MD simulation is a computational tool that can provide detailed information on the motion of atoms during the evolution of materials under applied thermal and stress conditions. To estimate the thermal properties of materials from MD simulations, the two most popular methods are equilibrium Green-Kubo method, and nonequilibrium molecular dynamics (NEMD) simulations. In equilibrium Molecular Dynamics (EMD), transport coefficients are obtained from fluctuations in the vicinity of the equilibrium state using the Green-Kubo equation [114, 115]: Z V hJ(0) · J(t)i dt (20) k= 3kB T 2 where V is the volume of system, T is the system temperature, and J is the heat flux:   X 1 X J(t) = (Ei − hEi i) vi + (Fij · rij )vj  . V i j,j6=i Here, Ei = (Ek + Ep )i ) is the total energy (kinetic plus potential) of atom i; Fij is the force exerted on ith atom from j th atom; rij is the distance between ith and j th atoms; vi is the velocity of atom i. The conductivity is then calculated by the autocorrelation of the heat flux J in Eq. (20), which can also be expressed in discrete form suitable for numerical calculations, k= M NX −m X 1 ∆T Jm+n Jn . 3V kB T 2 m=1 N − m n=1 (21) The method is rigorous but depends on weak gradients produced by thermal fluctuations in order to determine the rate of heat transfer. This introduces noise in the results and requires fairly long simulations to reach acceptable accuracy. Additionally, finite size effects can have be significant and must be thoroughly analyzed when using this method [116]. Non-equilibrium Molecular Dynamics (NEMD) has the ability to study transport coefficients and rheological properties of fluids mimicking real experiments [118]. What makes it different from equilibrium molecular dynamics (EMD) is the presence of an external force field that drives the system arbitrarily away from equilibrium. The basic principles are common to these algorithms, namely, they solve numerically Newton’s equations of motion and compute the physical properties of the system as a function of its phase-space variables. However, while the theoretical framework for equilibrium systems is well established by equilibrium Statistical Mechanics, for non-equilibrium systems it is still developing [119]. It nonetheless offers a very useful tool to recreate and test real experiments or otherwise situations impossible or very difficult to obtain experimentally. Besides being very simple to implement, the scheme also offers several FIG. 11. MD simulation of colloidal dimer in a fluid [117]. advantages such as compatibility with often used periodic boundary conditions, conservation of total energy and total linear momentum, and the sampling of a rapidly converging quantity (temperature gradient) rather than a slowly converging one (EMD). A classical non-equilibrium Molecular Dynamics method for computing the thermal conductivity is developed by Muller [120]. The method establishes a temperature gradient and the conductivity is calculated by application of Fourier’s law, q̇ = −k∇T (22) where q̇ is heat flux. To establish the temperature gradient, the method divides the simulation volume into two slabs, one hot and one cold. At predetermined intervals, the energy of the hottest atom in the cold region is exchanged with that of the coldest atom of the hot region. This practically “pumps” heat from the cold slab to the hot slab and establishes a gradient that reaches steady state. The corresponding flux in the direction of the gradient is P ij ∆Eij (23) Jz = − 2tA where ∆Eij the energy difference between the hot and cold atoms that are exchanged, t is time and A is the cross sectional area of the simulation volume in the direction of the gradient. In practice this implemented by dividing the simulation volume into three slabs, one hot at the middle, two cold ones at the side, which produces two symmetric gradients. As a numerical method, molecular dynamics is computationally intensive. For example, to simulate a volume fraction φ = 4% of a simple solid in a simple liquid (i.e., each phase consists of a single type atom), requires approximately 7000 atoms in the liquid and 1200 in the solid to represent a volume that contains a single particle with a diameter of 2.54 nm. It is, however, is ideally suited to probe phenomenal at the microscale. 10 Here, Fi is the force on particle i and the summation on the right-hand side goes over all other particles. The conservative force is given by the interaction between particles (hard sphere, DLVO, or other) while the dissipative (FD ) and stochastic (FS ) terms are given by [123, 124] 2.0 triplets doublets 1.8 k f 1.6 dispersed spheres Upper k / fn Maxwell limit 1.4 Lower D FD ij = −γω (rij · vij )rij (25) FSij (26) S = −σω ζij rij , Maxwell and are coupled via the conditions limit 1.2 1.0 0.00 0.02 0.04 0.06 0.08 0.10 volume fraction FIG. 12. Molecular dynamics simulation of dispersed spheres, spheres forming doublets, and spheres forming triplets. The conductivity of the dispersion increases as spheres form longer chains connected via narrow necks [117]. Using MD, Eapen et al. [121] were able to identify self correlations of the potential flux as a major contributor to the enhancement of the conductivity. These correlations arise from the strong interaction between solid and liquid molecules. Molecular dynamics makes it possible to simulate realistic clusters by heating two touching clusters until they form a connecting neck, as shown in Fig. 11. When such clusters are placed in vacuum their thermal conductivity is about 100 lower than that of the bulk material, due to the thermal resistance of the nanosized constriction [97]. But when they are placed in a fluid, they enhance the conductivity of the dispersion and the enhancement is larger than that of non-touching spheres with the same volume fraction [117]. Figure Fig. 12 shows MD calculations of the conductivity of dispersed spheres compared to doublets and triplets formed by joining spheres via a narrow neck at the same volume fraction of primary spheres. The conductivity increases with increasing chain length, a result that offers atomistic-level support for the hypothesis that clustered nanoparticles are more efficient conductors of heat. Dissipative Particle Dynamics (DPD) In DPD, simulation particles represent a cluster of molecules or fluid regions, rather than single atoms, and atomistic details are tied in through the interactions between these particles [122–125]. In this coarsened picture, the motion of a particle obeys the usual equations of motion, under a force that includes conservative, dissipative and stochastic contributions [122]: Fi = X j6=i D S FC ij + Fij + Fij . (24) σ 2 = 2γkB T  2 ω D = ω S = function of rij , (27) (28) where rij is the interparticle distance, vij = vi −vj , is the velocity difference between particles i and j, kB is Boltzmann’s constant and T is temperature. Such simulations have been performed before for individually dispersed particles but not for clusters [86, 125, 126]. Local momentum conservation requires the application of a random force between two interacting particles, thus distinguishing DPD technique from Brownian dynamics where each particle experiences a random force. The thermal conductivity is estimated using the non-equilibrium MullerPlathe method. The key challenge in these simulations is the coarse graining of the atomistic features in a nanoparticle to form a DPD particle and accurately representing the bonding between two DPD particles by a Hookean spring. DPD provides a level of modeling that lies between atomistic and macroscopic. It is most suitable for probing effects due to fluid motion and particle diffusion, none of which are captured by the other two methods. With respect to particle diffusion in particular, DPD studies have dispelled the notion that microconvection due to Brownian motion has any significant effect on thermal conductivity [125], as it had been hypothesized in the literature [127]. VI. CONCLUSIONS The use of colloidal particles as additives for thermal fluids is an attractive way to improve the heat transfer characteristics of the base fluid. The most significant observation is that improvements between 10% and 40% can be achieved with less that 10% by volume of the solid phase. Many challenges remain, however. While it has been established that large enhancements are possible, it has also become clear that a substantial part of it is due to clustering, an effect that is undesirable in practical settings. Indeed, providing sufficient stability under temperature swings and flow conditions in aqueous and organic solvents remains the key obstacle to commercial development. On the other hand, the potential benefits, particularly with respect to more efficient utilization of energy, are quite substantial. In this respect, this relatively new area provides colloidal scientists with new opportunities. 11 ACKNOWLEDGMENTS This work was supported by the National Science Foundation under Grant no. CBET GOALI #1132220. [1] S. K. Das, S. U. S. Choi, and H. E. Patel, Heat Transfer Engineering 27, 3 (2006), ISSN 01457632, URL http://ezaccess.libraries.psu.edu/login?url= http://search.ebscohost.com/login.aspx?direct= true&db=a9h&AN=22483223&site=ehost-live. [2] J. Buongiorno, D. C. Venerus, N. Prabhat, T. McKrell, J. Townsend, R. Christianson, Y. V. Tolmachev, P. Keblinski, L. wen Hu, J. L. Alvarado, et al., Journal of Applied Physics 106, 094312 (pages 14) (2009), URL http://link.aip.org/link/?JAP/106/094312/1. [3] X.-Q. Wang and A. S. Mujumdar, International Journal of Thermal Sciences 46, 1 (2007), ISSN 1290-0729, URL http://www.sciencedirect. com/science/article/B6VT1-4KM4722-1/2/ e062c698f2b3d392ac7b1d721f569470. [4] V. Trisaksri and S. Wongwises, Renewable and Sustainable Energy Reviews 11, 512 (2007), ISSN 1364-0321, URL http://www.sciencedirect. com/science/article/B6VMY-4FW7K7H-1/2/ 75b617fe4d027d23c1c06caa47eb82ab. [5] W. Yu, D. France, S. U. S. Choi, and J. L. Routbort, Tech. Rep., Argonne National Laboratory, Argonne, Illinois (2007), URL http: //www.osti.gov/bridge/product.biblio.jsp?query_ id=0&page=0&osti_id=919327. [6] W. Yu, D. M. France, J. L. Routbort, and S. U. S. Choi, Heat Transfer Engineering 29, 432 (2008), URL http://ezaccess.libraries.psu.edu/login?url= http://search.ebscohost.com/login.aspx?direct= true&db=a9h&AN=30009383&site=ehost-live. [7] M. Beck, Ph.D. thesis, Georgia Institute of Technology (2008). [8] S. Das, S. S. Choi, W. Yu, and T. Pradeep, Nanofluids Science and Technology (Wiley Interscience, New Jersey, 2008). [9] M. Pantzali, A. Mouza, and S. Paras, Chemical Engineering Science 64, 3290 (2009), ISSN 0009-2509, URL http://www.sciencedirect. com/science/article/B6TFK-4W1JW36-2/2/ b904aa07a38470a0e7117476adf1b112. [10] P. Keblinski, Thermal Nanosystems and Nanomaterials (Springer Berlin Heidelberg, 2009), vol. 118, chap. 8. Thermal Conductivity of Nanofluids, pp. 213–221, ISBN 978-3-642-04257-7, URL http://dx.doi.org/10. 1007/978-3-642-04258-4_8. [11] K. V. Wong and O. D. Leon, Advances in Mechanical Engineering 210, 519659 (2010). [12] K. V. Wong and M. J. Castillo, Advances in Mechanical Engineering 2010, Article ID 795478 (2010). [13] J. Eapen, R. Rusconi, R. Piazza, and S. Yip, Journal of Heat Transfer 132, 102402 (pages 14) (2010), URL http://link.aip.org/link/?JHR/132/102402/1. [14] R. Taylor, S. Coulombe, T. Otanicar, P. Phelan, A. Gunawan, W. Lv, G. Rosengarten, R. Prasher, and H. Tyagi, Journal of Applied Physics 113, 011301 [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] (pages 19) (2013), URL http://link.aip.org/link/ ?JAP/113/011301/1. The following companies advertise commercial nanofluids: Cool-X (http://cool-x.com); Ice Dragon Cooling (http://www.icedragoncooling.com); Mayhems (http://www.mayhems.co.uk/). J. K. Maxwell, Treatise on Electricity and Magnetism, vol. 1 (Dover, 1954 (reprint of the 1891 edition)). J. Eapen, W. C. Williams, J. Buongiorno, L.-w. Hu, S. Yip, R. Rusconi, and R. Piazza, Phys. Rev. Lett. 99, 095901 (2007). Z. Hashin and S. Shtrikman, Journal of Applied Physics 33, 3125 (1962), URL http://link.aip.org/link/ ?JAP/33/3125/1. P. Keblinski, R. Prasher, and J. Eapen, Journal of Nanoparticle Research 10, 1089 (2008), URL http: //dx.doi.org/10.1007/s11051-007-9352-1. J. Tobochnik, D. Laing, and G. Wilson, Phys. Rev. A 41, 3052 (1990), URL http://link.aps.org/doi/10. 1103/PhysRevA.41.3052. S. Kumar, J. Y. Murthy, and M. A. Alam, Phys. Rev. Lett. 95, 066802 (2005), URL http://link.aps.org/ doi/10.1103/PhysRevLett.95.066802. J. Eapen, J. Li, and S. Yip, Phys. Rev. E 76, 062501 (2007), URL http://link.aps.org/doi/10. 1103/PhysRevE.76.062501. S. I. White, B. A. DiDonna, M. Mu, T. C. Lubensky, and K. I. Winey, Phys. Rev. B 79, 024301 (2009), URL http://link.aps.org/doi/10. 1103/PhysRevB.79.024301. I. Belova and G. Murch, Journal of Materials Processing Technology 153–154, 741 (2004), ISSN 09240136, ¡ce:title¿Proceedings of the International Conference in Advances in Materials and Processing Technologies¡/ce:title¿, URL http://www.sciencedirect.com/ science/article/pii/S0924013604005825. R. L. Hamilton and O. K. Crosser, Industrial & Engineering Chemistry Fundamentals 1, 187 (1962), http://pubs.acs.org/doi/pdf/10.1021/i160003a005, URL http://pubs.acs.org/doi/abs/10.1021/ i160003a005. H. Fricke, Phys. Rev. 24, 575 (1924), URL http:// link.aps.org/doi/10.1103/PhysRev.24.575. P. Vadasz, Journal of Heat Transfer 132, 081601 (2010), URL http://dx.doi.org/10.1115/1.4001314. J. Kestin and W. A. Wakeham, Physica A: Statistical Mechanics and its Applications 92, 102 (1978), URL http://www.sciencedirect.com/science/article/ pii/0378437178900237. J. J. De Groot, J. Kestin, and H. Sookiazian, Physica 75, 454 (1974), URL http://www.sciencedirect.com/ science/article/pii/0031891474903413. J. J. Healy, J. J. de Groot, and J. Kestin, Physica B+C 82, 392 (1976), URL http://www.sciencedirect.com/ science/article/pii/0378436376902035. 12 [31] M. Assael, M. Dix, K. Gialou, L. Vozar, and W. Wakeham, International Journal of Thermophysics 23, 615 (2002), ISSN 0195-928X, URL http://dx.doi.org/10. 1023/A%3A1015494802462. [32] S. U. S. Choi, Z. G. Zhang, W. Yu, F. E. Lockwood, and E. A. Grulke, Applied Physics Letters 79, 2252 (2001), URL http://scitation.aip.org/content/ aip/journal/apl/79/14/10.1063/1.1408272. [33] C. H. Chon, K. D. Kihm, S. P. Lee, and S. U. S. Choi, Applied Physics Letters 87, 153107 (2005), URL http://scitation.aip.org/content/ aip/journal/apl/87/15/10.1063/1.2093936. [34] J. A. Eastman, S. U. S. Choi, S. Li, W. Yu, and L. J. Thompson, Applied Physics Letters 78, 718 (2001), URL http://scitation.aip.org/content/ aip/journal/apl/78/6/10.1063/1.1341218. [35] K. S. Hong, T.-K. Hong, and H.-S. Yang, Applied Physics Letters 88, 031901 (pages 3) (2006), URL http: //link.aip.org/link/?APL/88/031901/1. [36] H. Zhu, C. Zhang, S. Liu, Y. Tang, and Y. Yin, Applied Physics Letters 89, 023123 (2006), URL http://scitation.aip.org/content/aip/journal/ apl/89/2/10.1063/1.2221905. [37] H. Zhu, C. Zhang, Y. Tang, J. Wang, B. Ren, and Y. Yin, Carbon 45, 226 (2007), URL http://www.sciencedirect.com/science/article/ pii/S0008622306003757. [38] M.-S. Liu, M. C.-C. Lin, I.-T. Huang, and C.-C. Wang, Chemical Engineering & Technology 29, 72 (2006), URL http://dx.doi.org/10.1002/ceat.200500184. [39] Y. J. Hwang, Y. C. Ahn, H. S. Shin, C. G. Lee, G. T. Kim, H. S. Park, and J. K. Lee, Current Applied Physics 6, 1068 (2006), URL http://www.sciencedirect. com/science/article/B6W7T-4GX64X1-1/2/ a44ea2bc3ac99c98e260dfa41964e949. [40] R. M. DiGuilio and A. S. Teja, Industrial & Engineering Chemistry Research 31, 1081 (1992), URL http://dx. doi.org/10.1021/ie00004a016. [41] M.-S. Liu, M. Ching-Cheng Lin, I.-T. Huang, and C.-C. Wang, International Communications in Heat and Mass Transfer 32, 1202 (2005), URL http://www.sciencedirect. com/science/article/B6V3J-4GCWY6R-1/2/ bb427f7fd9f6273fe2f4a776c530e0a5. [42] Y. Xuan and Q. Li, International Journal of Heat and Fluid Flow 21, 58 (2000), ISSN 0142-727X, URL http://www.sciencedirect. com/science/article/B6V3G-3YGDCXV-6/2/ 7a59877c774f6db5ed3967bbb8d07d41. [43] M.-S. Liu, M. C.-C. Lin, C. Y. Tsai, and C.-C. Wang, International Journal of Heat and Mass Transfer 49, 3028 (2006), URL http://www.sciencedirect.com/ science/article/pii/S0017931006001347. [44] S. M. S. Murshed, K. C. Leong, and C. Yang, International Journal of Thermal Sciences 44, 367 (2005), URL http://www.sciencedirect.com/science/article/ pii/S129007290500013X. [45] M. J. Assael, I. N. Metaxa, K. Kakosimos, and D. Constantinou, International Journal of Thermophysics 27, 999 (2006), URL http://dx.doi.org/10. 1007/s10765-006-0078-6. [46] M. Assael, I. Metaxa, J. Arvanitidis, D. Christofilos, and C. Lioutas, International Journal of Thermophysics 26, 647 (2005), ISSN 0195-928X, URL http://dx.doi. org/10.1007/s10765-005-5569-3. [47] M. Assael, C.-F. Chen, I. Metaxa, and W. Wakeham, International Journal of Thermophysics 25, 971 (2004), ISSN 0195-928X, URL http://dx.doi.org/10.1023/ B%3AIJOT.0000038494.22494.04. [48] T.-K. Hong, H.-S. Yang, and C. J. Choi, Journal of Applied Physics 97, 064311 (2005), URL http://scitation.aip.org/content/aip/journal/ jap/97/6/10.1063/1.1861145. [49] H. Xie, H. Lee, W. Youn, and M. Choi, Journal of Applied Physics 94, 4967 (2003), URL http://scitation.aip.org/content/aip/journal/ jap/94/8/10.1063/1.1613374. [50] H. Xie, J. Wang, T. Xi, Y. Liu, F. Ai, and Q. Wu, Journal of Applied Physics 91, 4568 (2002), URL http: //link.aip.org/link/?JAP/91/4568/1. [51] Y. Yang, E. A. Grulke, Z. G. Zhang, and G. Wu, Journal of Applied Physics 99, 114307 (2006), URL http://scitation.aip.org/content/ aip/journal/jap/99/11/10.1063/1.2193161. [52] X. Zhang, H. Gu, and M. Fujii, Journal of Applied Physics 100, 044325 (2006), URL http://scitation.aip.org/content/aip/journal/ jap/100/4/10.1063/1.2259789. [53] S. H. Kim, S. R. Choi, and D. Kim, Journal of Heat Transfer 129, 298 (2006), URL http://dx.doi.org/ 10.1115/1.2427071. [54] S. Lee, S. U.-S. Choi, S. Li, and J. A. Eastman, Journal of Heat Transfer 121, 280 (1999), URL http://link. aip.org/link/?JHR/121/280/1. [55] H. Xie, J. Wang, T. Xi, Y. Liu, and F. Ai, Journal of Materials Science Letters 21, 193 (2002), URL http: //dx.doi.org/10.1023/A%3A1014742722343. [56] H. Xie, J. Wang, T. Xi, Y. Liu, and F. Ai, Journal of Materials Science Letters 21, 1469 (2002), ISSN 0261-8028, URL http://dx.doi.org/10.1023/A% 3A1020060324472. [57] J. P. Bentley, Journal of Physics E: Scientific Instruments 17, 430 (1984), URL http://stacks.iop.org/ 0022-3735/17/i=6/a=002. [58] Y. Nagasaka and A. Nagashima, Journal of Physics E: Scientific Instruments 14, 1435 (1981), URL http:// stacks.iop.org/0022-3735/14/i=12/a=020. [59] D. Wen and Y. Ding, Journal of Thermophysics and Heat Transfer 18, 481 (2004), URL http://dx.doi. org/10.2514/1.9934. [60] J. A. Eastman, U. S. Choi, S. Li, L. J. Thompson, and S. Lee, MRS Online Proceedings Library 457 (1996), ISSN null, URL http://journals.cambridge. org/article_S1946427400215996. [61] E. V. Timofeeva, A. N. Gavrilov, J. M. McCloskey, Y. V. Tolmachev, S. Sprunt, L. M. Lopatina, and J. V. Selinger, Phys. Rev. E 76, 061203 (2007). [62] M. Hoshi, T. Omotani, and A. Nagashima, Review of Scientific Instruments 52, 755 (1981), URL http://scitation.aip.org/content/aip/journal/ rsi/52/5/10.1063/1.1136672. [63] D. Lee, J.-W. Kim, and B. G. Kim, The Journal of Physical Chemistry B 110, 4323 (2006), pMID: 16509730, http://pubs.acs.org/doi/pdf/10.1021/jp057225m, URL http://pubs.acs.org/doi/abs/10.1021/jp057225m. [64] Y. Hwang, J. K. Lee, C. H. Lee, Y. M. Jung, S. I. Cheong, C. G. Lee, B. C. Ku, and S. P. Jang, Thermochimica Acta 455, 70 (2007), URL 13 [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] http://www.sciencedirect.com/science/article/ pii/S0040603106005995. D.-H. Yoo, K. S. Hong, and H.-S. Yang, Thermochimica Acta 455, 66 (2007), URL http://www.sciencedirect.com/science/article/ pii/S0040603106006162. W. Wakeham, A. Nagashima, and J. Sengers, eds., Measurement of the Transport Properties of Fluids. Experimental Thermodynamics. (Oxford, 1991). P. Bhattacharya, S. Nara, P. Vijayan, T. Tang, W. Lai, P. E. Phelan, R. S. Prasher, D. W. Song, and J. Wang, International Journal of Heat and Mass Transfer 49, 2950 (2006), URL http://www.sciencedirect.com/ science/article/pii/S001793100600144X. W. Czarnetzki and W. Roetzel, Int J Thermophys 16, 413 (1995), URL http://dx.doi.org/10.1007/ BF01441907. R. Gowda, H. Sun, P. Wang, M. Charmchi, F. Gao, Z. Gu, and B. Budhlall, Advances in Mechanical Engineering 2010 (2010), URL http://dx.doi.org/10. 1155/2010/807610. E. Abu-Nada, International Journal of Heat and Fluid Flow 30, 679 (2009), URL http: //www.sciencedirect.com/science/article/pii/ S0142727X09000265. X. Zhang, H. Gu, and M. Fujii, International Journal of Thermophysics 27, 569 (2006), URL http://dx.doi. org/10.1007/s10765-006-0054-1. C. H. Li and G. Peterson, Journal of Applied Physics 101, 044312 (2007), ISSN 0021-8979. C. H. Li and G. P. Peterson, Journal of Applied Physics 99, 084314 (2006), URL http://scitation.aip.org/ content/aip/journal/jap/99/8/10.1063/1.2191571. S. K. Das, N. Putra, P. Thiesen, and W. Roetzel, Journal of Heat Transfer 125, 567 (2003), URL http: //dx.doi.org/10.1115/1.1571080. D.-H. Yoo, K. S. Hong, T. E. Hong, J. A. Eastman, and H.-S. Yang, Journal of the Korean Physical Society 51 (2007). X. Wang, X. Xu, and S. U. S. Choi, Journal of Thermophysics and Heat Transfer 13, 474 (1999), URL http://dx.doi.org/10.2514/2.6486. L. Zhou, B.-X. Wang, X.-F. Peng, X.-Z. Du, and Y.-P. Yang, Advances in Mechanical Engineering 2010, Article ID 172085 (2010). H. U. Kang, S. H. Kim, and J. M. Oh, Experimental Heat Transfer 19, 181 (2006), URL http://dx.doi. org/10.1080/08916150600619281. X. Yang and Z.-h. Liu, Nanoscale Research Letters 5, 1324 (2010), ISSN 1931-7573, 10.1007/s11671010-9646-6, URL http://dx.doi.org/10.1007/ s11671-010-9646-6. B. Yang and Z. H. Han, Applied Physics Letters 89, 083111 (pages 3) (2006), URL http://link.aip.org/ link/?APL/89/083111/1. S. Jana, A. Salehi-Khojin, and W.-H. Zhong, Thermochimica Acta 462, 45 (2007), URL http://www.sciencedirect.com/science/article/ pii/S0040603107002675. L. Chen, H. Xie, Y. Li, and W. Yu, Thermochimica Acta 477, 21 (2008), URL http://www.sciencedirect.com/ science/article/pii/S0040603108002311. R. Zheng, J. Gao, J. Wang, S.-P. Feng, H. Ohtani, J. Wang, and G. Chen, Nano Letters 12, 188 (2011), URL http://dx.doi.org/10.1021/nl203276y. [84] H. E. Patel, S. K. Das, T. Sundararajan, A. S. Nair, B. George, and T. Pradeep, Applied Physics Letters 83, 2931 (2003), URL http://link.aip.org/link/?APL/ 83/2931/1. [85] Y. Feng, B. Yu, P. Xu, and M. Zou, Journal of Physics D: Applied Physics 40, 3164 (2007), URL http:// stacks.iop.org/0022-3727/40/i=10/a=020. [86] S. Jain, H. Patel, and S. Das, Journal of Nanoparticle Research 11, 767 (2009), URL http://dx.doi.org/10. 1007/s11051-008-9454-4. [87] B.-X. Wang, L.-P. Zhou, and X.-F. Peng, International Journal of Heat and Mass Transfer 46, 2665 (2003), ISSN 0017-9310, URL http://www.sciencedirect. com/science/article/B6V3H-47YPF5N-2/2/ 5fbe086f006c9c3091dc8f83349e60e9. [88] R. Prasher, W. Evans, P. Meakin, J. Fish, P. Phelan, and P. Keblinski, Applied Physics Letters 89, 143119 (pages 3) (2006), URL http://link.aip.org/link/ ?APL/89/143119/1. [89] S. Kim, I. Bang, J. Buongiorno, and L. Hu, International Journal of Heat and Mass Transfer 50, 4105 (2007), ISSN 00179310, URL http://www.sciencedirect. com/science/article/B6V3H-4NBRFWF-4/2/ 5874db11124e15c8cd3693424ce54af5. [90] A. R. Studart, E. Amstad, and L. J. Gauckler, Langmuir 23, 1081 (2007), http://pubs.acs.org/doi/pdf/10.1021/la062042s, URL http://pubs.acs.org/doi/abs/10.1021/la062042s. [91] J. Philip, P. D. Shima, and B. Raj, Applied Physics Letters 92, 043108 (pages 3) (2008), URL http://link. aip.org/link/?APL/92/043108/1. [92] W. Evans, R. Prasher, J. Fish, P. Meakin, P. Phelan, and P. Keblinski, International Journal of Heat and Mass Transfer 51, 1431 (2008), ISSN 00179310, URL http://www.sciencedirect.com/science/ article/pii/S0017931007006278. [93] P. Kanninen, C. Johans, J. Merta, and K. Kontturi, Journal of Colloid and Interface Science 318, 88 (2008), ISSN 0021-9797, URL http://www.sciencedirect. com/science/article/B6WHR-4R05BRV-1/2/ 7ab34e86907269208eaf490ef51d2f15. [94] G. Chen, W. Yu, D. Singh, D. Cookson, and J. Routbort, Journal of Nanoparticle Research 10, 1109 (2008), ISSN 1388-0764, 10.1007/s11051-007-9347-y, URL http://dx.doi.org/10.1007/s11051-007-9347-y. [95] J. W. Gao, R. T. Zheng, H. Ohtani, D. S. Zhu, and G. Chen, Nano Letters 9, 4128 (2009), pMID: 19995084, http://pubs.acs.org/doi/pdf/10.1021/nl902358m, URL http://pubs.acs.org/doi/abs/10.1021/nl902358m. [96] C. Wu, T. J. Cho, J. Xu, D. Lee, B. Yang, and M. R. Zachariah, Phys. Rev. E 81, 011406 (2010), URL http: //link.aps.org/doi/10.1103/PhysRevE.81.011406. [97] T. G. Desai, Applied Physics Letters 98, 193107 (pages 3) (2011), URL http://link.aip.org/link/ ?APL/98/193107/1. [98] D. Venerus and Y. Jiang, Journal of Nanoparticle Research 13, 3075 (2011-07-01), URL http://dx.doi. org/10.1007/s11051-010-0207-9. [99] R. Zheng, J. Gao, J. Wang, and G. Chen, Nat Commun 2, 289 (2011), URL http://dx.doi.org/10.1038/ ncomms1288. 14 [100] J. Hong and D. Kim, Thermochimica Acta 542, 28 (2012), URL http://www.sciencedirect.com/ science/article/pii/S0040603111006253. [101] S. Lotfizadeh and T. Matsoukas, Nano Letters (2014). [102] P. E. Gharagozloo, J. Eaton, and K. E. Goodson, Applied Physics Letters 93, 103110 (2008), ISSN 00036951. [103] Y. Feng, B. Yu, K. Feng, P. Xu, and M. Zou, Journal of Nanoparticle Research 10, 1319 (2008), ISSN 1388-0764, URL http://dx.doi.org/10.1007/ s11051-008-9363-6. [104] P. E. Gharagozloo and K. E. Goodson, Journal of Applied Physics 108, 074309 (2010), ISSN 0021-8979. [105] P. E. Gharagozloo and K. E. Goodson, International Journal of Heat and Mass Transfer 54, 797 (2011), ISSN 0017-9310, URL http://www.sciencedirect. com/science/article/pii/S0017931010005880. [106] N. Pelević and T. van der Meer, International Journal of Thermal Sciences 62, 154 (2012), ISSN 1290-0729, ¡ce:title¿Thermal and Materials Nanoscience and Nanotechnology¡/ce:title¿, URL http://www.sciencedirect.com/science/article/ pii/S1290072911002912. [107] D. C. Hong, H. E. Stanley, A. Coniglio, and A. Bunde, Phys. Rev. B 33, 4564 (1986), URL http://link.aps. org/doi/10.1103/PhysRevB.33.4564. [108] C. DeW. Van Siclen, Phys. Rev. E 59, 2804 (1999), URL http://link.aps.org/doi/10.1103/PhysRevE. 59.2804. [109] Y. Shoshany, D. Prialnik, and M. Podolak, Icarus 157, 219 (2002), ISSN 0019-1035, URL http://www.sciencedirect.com/science/article/ pii/S0019103502968156. [110] I. Belova and G. Murch, Journal of Physics and Chemistry of Solids 66, 722 (2005), ISSN 00223697, URL http://www.sciencedirect.com/science/ article/pii/S0022369704003579. [111] T. Fiedler, A. Ochsner, N. Muthubandara, I. Belova, and G. E. Murch, Materials Science Forum 553, 51 (2007), URL http://www.scientific.net/MSF.553. 39. [112] I. V. Belova, G. E. Murch, A. Öchsner, and T. Fiedler, Defect and Diffusion Forum 279, 13 (2008), URL http: //www.scientific.net/DDF.279.13. [113] P. C. Millett, D. Wolf, T. Desai, S. Rokkam, and A. El-Azab, Journal of Applied Physics 104, 033512 (pages 6) (2008), URL http://link.aip.org/link/ ?JAP/104/033512/1. [114] M. S. Green, The Journal of Chemical Physics 22, 398 (1954). [115] R. Kubo, Journal of the Physical Society of Japan 12, 570 (1957). [116] P. K. Schelling, S. R. Phillpot, and P. Keblinski, Phys. Rev. B 65, 144306 (2002). [117] Y.-H. Chen, M.S. thesis, Pennsylvania State Univesity, University Park (2013). [118] A. Berker, S. Chynoweth, U. C. Klomp, and Y. Michopoulos, J. Chem. Soc., Faraday Trans. 88, 1719 (1992). [119] S. R. De Groot and P. Mazur, Non-equilibrium thermodynamics (Courier Dover Publications, 2013). [120] F. Muller-Plathe, The Journal of Chemical Physics 106, 6082 (1997), URL http://link.aip.org/link/?JCP/ 106/6082/1. [121] J. Eapen, J. Li, and S. Yip, Phys. Rev. Lett. 98, 028302 (2007), URL http://link.aps.org/doi/10. 1103/PhysRevLett.98.028302. [122] P. J. Hoogerbrugge and J. M. V. A. Koelman, EPL (Europhysics Letters) 19, 155 (1992), URL http://stacks. iop.org/0295-5075/19/i=3/a=001. [123] P. Español and P. Warren, EPL (Europhysics Letters) 30, 191 (1995), URL http://stacks.iop.org/ 0295-5075/30/i=4/a=001. [124] P. Español, Phys. Rev. E 52, 1734 (1995). [125] P. He and R. Qiao, Journal of Applied Physics 103, 094305 (pages 6) (2008), URL http://link.aip.org/ link/?JAP/103/094305/1. [126] P. Bhattacharya, S. K. Saha, A. Yadav, P. E. Phelan, and R. S. Prasher, Journal of Applied Physics 95, 6492 (2004), URL http://link.aip.org/link/?JAP/ 95/6492/1. [127] R. Prasher, P. Bhattacharya, and P. E. Phelan, Phys. Rev. Lett. 94, 025901 (2005).