Fractal aggregation kinetics contributions to thermal conductivity of nano-suspensions in unsteady thermal convection

Nano-suspensions (NS) exhibit unusual thermophysical behaviors once interparticle aggregations and the shear flows are imposed, which occur ubiquitously in applications but remain poorly understood, because existing theories have not paid these attentions but focused mainly on stationary NS. Here we report the critical role of time-dependent fractal aggregation in the unsteady thermal convection of NS systematically. Interestingly, a time ratio λ = tp/tm (tp is the aggregate characteristic time, tm the mean convection time) is introduced to characterize the slow and fast aggregations, which affect distinctly the thermal convection process over time. The increase of fractal dimension reduces both momentum and thermal boundary layers, meanwhile extends the time duration for the full development of thermal convection. We find a nonlinear growth relation of the momentum layer, but a linear one of the thermal layer, with the increase of primary volume fraction of nanoparticles for different fractal dimensions. We present two global fractal scaling formulas to describe these two distinct relations properly, respectively. Our theories and methods in this study provide new evidence for understanding shear-flow and anomalous heat transfer of NS associated non-equilibrium aggregation processes by fractal laws, moreover, applications in modern micro-flow technology in nanodevices.

Scientific RepoRts | 6:39446 | DOI: 10.1038/srep39446 where φ p is volume fraction of primary NP, and k Maxwell , k p and k f are the thermal conductivity of NS, NP and base fluid, respectively. The transmission electron microscopy (TEM) in Fig. 1 illustrates the typical aggregation in NS of Al 2 O 3 /water (left) and CuO/water (right), respectively 13 . In a recent paper 14 , we presented a multilevel equivalent agglomeration (MEA) thermal conductivity model for stationary NS, which have achieved successful theoretical predictions. The highly consistent predictions with classical data (by Lee et al. 13 ) are presented in that paper. It is common in NS that nanoparticles tend to aggregate with the time due to its high surface potential energy (surface-activity), in consequence, there is more difficulty in probing the thermal performance of NS with dispersion of fully irregular clusters. Weitz, et al. 15 found that the scaling behavior in diffusion-limited aggregation in colloidal solution depends on both the average cluster size and initial concentration with the evolution of time 15 , and the fractal dimension 1.75 ≤ d f ≤ 2.05 for gold aggregates was displayed 16 . Subsequently, scaling law in terms of volume fraction of aggregates, the contained number of nanoparticles per cluster and fractal dimensions of clusters size were reported in refs [17][18][19]. The scaling law relationships between the characteristic geometric size of aggregates and time was established by Hanus et al. 20 in which the aggregate characteristic time t p was proposed explicitly. Prasher et al. 21 investigated experimentally the distribution of t p for nano-alumina suspensions and a large time-domain independence of nanoparticles radius, temperature and pH was presented.
Unfortunately, up to now, most researchers have focused on the investigation of stationary NS. It is unclear how the non-equilibrium aggregation alters the heat transfer properties of kinetic NS induced via shear flow or thermal convection often encountered in modern nanotechnology applications 22 . In modern technology, nano-suspensions used in most engineering applications should meet the various thermal convection requirements, for instance, the shear flow and heat transfer of nano-suspensions in biological tissue, DNA replication and amplification via natural convection processes etc. 23,24 . Furthermore, enhanced heat transfer efficiency in other heat exchange devices using diluted NS, such as solar collectors, cooling systems etc., is often accompanied by the generation of thermal convections 25,26 . The aggregate morphology described by quasi-fractal has been modeled in altering the convection heat transfer performance in nanofluids, and some specialized models for convective heat transfer coefficient, dynamic viscosity etc. were proposed 27 , but there is no more exploration for unsteady convection heat transfer in nano-suspensions governed by N-S equations with adopting the perception on fractal aggregation kinetics.
As a result, the aggregate formation in unsteady flowing NS should change drastically relative to stationary ones. With such an understanding, nevertheless, existing investigations have not yet demonstrated how important a role non-equilibrium aggregates play on the thermal transport of NS. In this study, we report a novel physical process that time-dependent fractal aggregation kinetics affects the unsteady thermal convection boundary layer of NS. The analysis both of convection flow and temperature distributions of NS in the boundary layer are considered to obtain what we regard as interesting results by incorporating non-equilibrium fractal aggregation mechanisms.

Theoretical description and formula
According to natural convection boundary layer (NCBL) theories 28 (Zheng and coauthors have carried out some studies [29][30][31], there are large velocity gradient (shear flow) and temperature gradient (heat transfer) in NCBL and they are both coupled. Consider a two-dimension unsteady laminar natural convection flow and heat transfer of the nano-suspensions on a heated vertical plate as shown in Fig. 2. The velocity and temperature boundary layers both develop over time near the plate along the x-axis. Herein, T w denotes the constant temperature of the plate surface and T ∞ the ambient temperature of NS. It is assumed that the homogeneous NS is at rest at the initial time t = 0 and no mass transfer occurs over all the time. More importantly, the base fluid has large magnitude in altering the rheology properties of NS, and the consensus is that the rheology properties have little changes from Newtonian characteristics with particles loading φ p ≤ 0.1 for water base fluid with a certain range of shear rate, in spite of the dynamic viscosity and effective thermal conductivity of NS are dependent on the particles volume fraction apparently. The NS will exhibit very complex performances, perhaps shear-thinning or shear-thicken, once the particles loading is higher (> 0.1).
It is applicable that the rheology and heat conducting constitutive models of NS with maximum particle loading 0.1 are addressed as Newtonian models within the circumstance of natural convection boundary layer [29][30][31] . Then, the governing equations of this system can be written as ns ns ns where V(u, v) is the two-dimension velocity vector with u along the x-axis and v along the y-axis, respectively, g the gravitational acceleration, T the temperature and t the time. ρ ns and c ns are density and specific heat of NS, respectively.
is the material derivative. ∇ and ∇ 2 are the gradient and the Laplace operator, respectively. The pressure gradient is ρ ∇ = = − ∞ P dp dx g / ns with ρ ∞ ns the density of the ambient NS. By invoking the Boussinesq approximation, we have ρ

ns ns ns ns
, with β ns the thermal expansion coefficient of NS. The modified Fourier heat conduction law for NS is employed as = − ∂ ∂ q k T y / ns with k ns the thermal conductivity of NS, subject to the boundary conditions We now investigate the time-dependent fractal aggregation kinetic model by introducing two important scaling laws as 20,21 where r p is the radius of primary nanoparticle, R a an average gyration radius of clusters (shown in Fig. 2  Furthermore, if φ p is the primary volume fraction of NP, φ int denotes the primary NP volume fraction inside aggregates and φ a the volume fraction of aggregates in the entire NS, we have the relation φ p = φ int φ a 20,21 . With the hard-sphere and homogenization assumptions that the aggregates formed by primary nanoparticles are in uniform size (a sphere with radius R a ), one can derive another important scaling law according to Eq. (7) as Note that φ a = φ p /φ int signifies the formation process of time-dependent fractal aggregation with time evolution. Subsequently, we obtain the density, effective heat capacity and effective thermal conductivity of fractal aggregates by considering the percolation effects in clusters, respectively: where, subscript "f" is for the based fluid, "a" for aggregates and "p" for nanoparticles. Non-equilibrium aggregation occurs in time 0 ≤ t ≤ t p , otherwise an equilibrium state of aggregates form if t > t p . Consequently, we obtain the renovated Maxwell model by considering fractal aggregation effects, once Eq. (8) is introduced, as Here the enhanced thermal conductivity of dynamic NS is described by means of this modified model with the focus on time-dependent fractal laws prominently, which is regarded as underlying mechanism in thermal convection flow rather than other factors commonly are involved for stationary NS 14 . Based on Eq. (9), we derive the effective physical parameters of flowing nano-suspensions as ρ ns ns a f f a a a and dynamic viscosity is suggested by the Brinkman model 33 . Furthermore, we seek similarity solutions of Eqs (2)(3)(4) to simplify the process of solving partial differential equations systems [29][30][31] . The elaborated similarity transformation for unsteady boundary layer thermal convection including the stream function ψ, similarity variables and dimensionless temperature function are shown as where ν f = μ f /ρ f is kinematic viscosity of base fluid, f(η) the dimensionless stream function and τ = t/t p the dimensionless time, respectively. Substituting Eq. (11) into Eqs (2-6), in view of (u, v) = (ψ y , − ψ x ), we derive the following coupled nonlinear equations as where Pr is the Prandtl number of the base fluid (water), and λ = is average velocity at a reference length L in NCBL 29,30 . Obviously, S → 0 as the dimensionless time τ → ∞ , i.e., the unsteady model will be reduced to the steady model.
The major engineering parameters for this problem are the skin friction coefficient are the surface shear stress and heat flux along the vertical plate. In view of the similarity variables above, we define the local skin friction and local Nusselt number, respectively as

Discussion
Slow and Fast aggregation. The characteristic time ratio parameter λ = t p /t m is closely related to the aggregates formation processes and thermal convection flow, which can be used to characterize the slow aggregation (SA) and fast aggregation (FA) processes relative to thermal convection processes. As reported by Prasher et al. 21 , the aggregate characteristic time in a broad range 10 < t p < 10 5 seconds is corresponding to the primary nanoparticles size around 5~10 nm at 55 °C, or larger for aqueous nano-alumina suspension with hard sphere shape nanoparticles, which is also chosen as the working NS in this study. Prandtl number Pr ≈ 3.2 (water at 55 °C), initial volume fraction φ p = 0.05 and d f = 1.8 and other important thermophysical properties presented in Table 1 are suitable for calculations. Eqs (12)(13)(14) are solved numerically by employing a shooting technique with a high-efficiency fourth order Runge-Kutta algorithm.
Here, we may set an empirical average convection time t m = 66.67 seconds (about one minute, generally determined by the temperature difference of the vertical plate and nano-suspensions). For slow aggregation (λ  1), set t p = 10 3 seconds (16.67 min) and gives λ = 15.00. For fast aggregation (λ < 1), set t p = 27.00 seconds (0.45 min), yield λ = 0.40.
We compute local skin friction Cf local and the local Nusselt number Nu local to confirm the time to reach full development of natural convection. Figure 3(a) shows the developing slow aggregation process over the time till τ = 1, and the local skin friction reaches a maximum (magnification figure inset Fig. 3(a)) at about τ = 0.35, i.e., t = 350 s (5.83 min), which manifests a full development of a convection flow. So the coupled development of both convection and aggregation together occurred in 0 ≤ τ ≤ 0.35. Subsequently, the dynamic aggregation plays its role continuously in duration 0.35 ≤ τ ≤ 1 to result in the decrease of local skin friction. Figure 3(b) shows that the fast aggregation processes are in confinement 0 ≤ τ ≤ 1 with the initial development of natural convection. Subsequently, the convection, with an equilibrium aggregation (thermophysical properties will not alter any more), is ongoing to reach a steady state at around τ = 12, i.e., t = 324 s (5.40 min) and Cf local and Nu local are kept invariant. These results indicate that the unsteady thermal convection of a dynamic NS is apparently dependent on aggregation formation duration. SA makes the convection strong first (large Cf local ) and then weaken, but convection is always growing to a steady state without weakening in FA.
The velocity distributions over time for the slow and the fast aggregation process with respect to convection are displayed in Fig. 4(a) and (b), respectively. As time goes on, the convection flow strengthens rapidly due to the thermal buoyancy force. For slow aggregation with fraction dimension d f = 1.8 in Fig. 4(a), there are apparent intersection points (magnification figure inset Fig. 4(a)) between velocity profiles after the convection flow fully develops but before equilibrium aggregation (0.35 ≤ τ ≤ 1), i.e., the maximum of velocity do not change but its boundary layer thickens. Nevertheless, the convection velocity increase rapidly until the full development without any intersection points for fast aggregation in Fig. 4(b). The coupling development of velocity and temperature fields as an inherent feature in thermal convection system signifies the susceptible temperature field by different aggregation cases. Figure 4(c) and (d) illustrate the temperature distributions for slow and fast aggregation cases, respectively. The temperature increases rapidly over time, and the thickened temperature boundary layer shows    enhancement heat transfer (the increased temperature) as shown in Fig. 5(b), as well as a strengthen convection flow as shown in Fig. 5(a). In addition, results show that significant influences on both of F and θ with variation of d f appear at high concentration 0.05 ≤ φ p ≤ 0.1; in contrary, the effects in small magnitude are for dilute nano-suspensions φ p = 0.01. As is well known, the larger fractal dimension the more random the aggregates is 34 , that is to say, for aggregates, the increased d f manifests the more complicated aggregates in morphology. For concentrated NS, the increased d f , i.e., high complexity aggregation, not only can improve convection flow, but enhance heat transfer capability due to the thinner thermal boundary layer thickness shown in Fig. 5(b). Practically, one should adopt some measures to control aggregation in concentrated NS in order to maintain a system optimum working state. In general, some physical and chemical approaches, such as surfactant, ultrasonic vibration, temperature & pH control and particle surface charge modification, etc., can weaken aggregate processes 3,4 , namely, d f is changeable for a certain concentration φ p of NS. Obviously, it requires manpower and financial resources. Furthermore, if we define the energy efficiency (heat transfer efficiency) as the heat current absorbed by NS per unit area on the plate q w , then the increasing percentage of energy efficiency in micro devices using NS and base fluid is presented as κ = q q ( ) /( ) . For slow aggregation case, the enhanced energy efficiency κ using Al 2 O 3 /water nano-suspensions relative to the base fluid water for different primary volume fraction is shown in Table 2. The energy efficiency enhancement is prominent with the increase of φ p , particularly after the fully developed convection. However, the increased d f can reduce κ, which we realize is important for the control of the aggregation processes. Although, the high volume fraction of NS can enhance energy efficiency, such NF can often cause the abrasion and blockage in micro devices due to the presence of numerous nanoparticles and clusters. On the other hand, a very dilute NS also can't meet the requirements of energy efficiency enhancement. Consequently, the optimal options we suggest are at φ p ~ 0.05 and we suggest to hold the diffusion-limited aggregation at d f ~ 1.8.

The variable momentum and enthalpy boundary layer thickness.
Based on the theory of natural convection boundary layer (NCBL), momentum layer thickness (MLT) and enthalpy layer thickness (ELT) are primary characteristic values, whose expressions are given as 28 for simplicity 35 , yielding dimensionless momentum layer thickness δ m and enthalpy layer thickness δ e , respectively: ns ns w f 3 2 is Grashof number of nano-suspensions. In this approach, time-dependent fractal aggregation kinetics plays a critical role in unsteady NCBL. Therefore, δ m and δ e are evidently affected by time τ, volume fraction φ p and fractal dimension d f altogether.
In Fig. 6, one can readily observe the rapid growth of two layers in coupling development area, e.g., 0.01 ≤ τ ≤ 0.35 for d f = 1.8. The increased d f (in a proper range) makes both of the two layers thinner, but such effects are dominant after full development of convection, in particular, at a later stage of aggregation. Furthermore, the time τ con for the full development of thermal convection can be extended with the increase of d f in SA, i.e., τ τ τ < < . . . con c on con  In addition to time τ, δ m and δ e are also evidently affected by φ p and d f , which are displayed in Fig. 7(a-f). The dimensionless momentum layer thickness δ m and enthalpy layer thickness δ e increase largely with the increase of φ p , but the decrease of d f . Theoretically, the thinner boundary layer, the larger local Nusselt number Nu local , which signifies the enhanced convective heat transfer at the liquid-solid surface. For the fixed concentration φ p , the magnitude of the effects of d f is gradually increased over time from τ = 0.2 to τ = 1 in the NCBL as shown in Fig. 7(a-f).
More importantly, we find a nonlinear growth of δ m , but a linear growth of δ e , with increased φ p in the whole development process. The momentum boundary layer generated by the viscosity of fluids strongly depends on the enhanced effective viscosity of nano-suspensions in low-shear flow, e.g., natural thermal convection 29,30 . Numerous direct measurements of viscosity enhancement of hard sphere NS as the function of primary particles volume fraction exhibit a jamming transition near a random close packing (RCP) φ rcp of hard sphere in low-shear flow, as a result, we logically infer an divergence packing friction in which the asymptotical infinite of momentum layer thickness is expected. The pioneering models for viscosity enhancement of NS suggested the divergence φ rcp ≈ 0.64 with the optimal fitted values for data [36][37][38] (also some model 39 presented a value φ rcp = 0.637 ± 0.0015). Here, we also take the RCP φ rcp = 0.64 as a most probably divergence of δ m , while we know the possible maximum packing fraction is 0.7405 for general cases 40 .
Consequently, we establish the nonlinear and linear fractal scaling relations for δ m and δ e , respectively, as evidenced by the high-precision least square data fitting (R 2 > 0.999) in Fig. 7(a-f): and γ ≅ . ± . 0 05893 0 00002 e at fully developed convection regime (0.4 ≤ τ ≤ 1) for different fractal dimensions with almost no change over time, thus, these two scaling formulas can be reduced to simple expressions. Nevertheless, both M(τ, d f ) and E(τ, d f ) increase with time, in particular, M(τ, d f ) is largely reduced in magnitude comparing to E(τ, d f ) as d f increases. It is a remarkable fact that the correlation parameters M(τ, d f ) and E(τ, d f ) as mentioned form earlier are failed to fit the data for the stage of unsteady convection (τ < 0.4, the primary stage of aggregation) by our repeated simulations. Therefore, an in-depth study is still essential in the future to derive the functions M(τ, d f ) and E(τ, d f ) to present the scaling laws more explicitly. At present, the fractal scaling relations Eq. (18) is more applicable for fully developed convection regime τ ≥ 0.4.
In conclusion, non-equilibrium aggregation as highlighted by its time-dependent and fractal behaviors play a critical role in the unsteady thermal convection boundary layer of the nano-suspensions. It can be categorized into two regimes, i.e., the slow and fast aggregation in terms of the thermal convection process using λ = t p /t m . Technically, we transform the original physical governing equations to the corresponding similarity equations by unique similarity variables which simplify solving complexity significantly and make the clear discussions consequently. For the slow aggregation, the aggregation (with certain fractal dimension value) reduces thermal convection due to the decrease of Cf local and Nu local after the flow fully develops. The increase of fractal dimension not only extends the time for fully developed convection, it but also reduces the thickness of the momentum and enthalpy boundary layers. In particular, the fractal scaling laws proposed by us on the basis of the data can well describe the dependence of δ m and δ e on both φ p and d f mathematically for the stage after a fully developed convection, in spite of the unsettled probe of fitting unsteady convection stage with primary stage of aggregation. Our results provide the theoretical perspective on the regimes how the dynamic NS induced by unsteady convection flow (shear-flow) exhibit the anomalous thermal conduction with effects of non-equilibrium aggregations, which is the convincing evidences in terms of phenomenological relations. In addition, leave unsettled whether the enthalpy boundary layers, unlike the momentum layer corresponding to the critical packing fraction due to the viscosity, are also divergent near the critical packing fraction of hard sphere particles in NS, which is indeed further research.