On Clustering Impurities by Liquid Density Fluctuations

Recent developments in liquid technology have created a new class of fluids called “nanofluids” which are two-phase mixtures of a non-metal-liquid matrix and addon particles usually less than 100 nm in size. It is reputed that such liquids have a great potential for application. Indeed, many tests have shown that their thermal conductivity can be increased by almost 20% compared to that of the base fluids for a relatively low particle loading (of 1 up to 5% in volume). It is confirmed by experimental data and simulation results. In this study, the author considers an effect of impurity clustering by liquid density fluctuations as a natural mechanism for stabilizing microstructure of the colloidal solution and estimates the effect of fractal structure of colloidal particles on thermal conductivity of water. The results of this study may be useful for motivating choosing the composition of heat-transfer suspension and developing technology for making the appropriate nanofluid.

Presently, nanofluids are produced by two techniques [8][9][10][11].A two-step technique starts with microparticles as a dry powder and then is dispersed into the base liquid.This method may result in a large degree of particle agglomeration.The single-step one simultaneously makes and disperses the particles directly into the base liquid.Nanofluids containing oxides (Al 2 O 3 , CuO, and TiO 2 ), nitrides (AlN and SiN), and carbides (SiC and TiC) are produced by the two-step process, and the ones containing metals (Ag, Au, Cu, and Fe) are produced by the single-step process.
Obviously, it is necessary to study any water nanofluid composition [12], since a key factor in understanding thermal properties of nanofluids is clustering effects that provide paths for rapid heat transport [5].It is very important to consider density fluctuation of water as normal modes in the low energy dynamics and the most stable atomic configurations, where the system spends most of its time [13].
Furthermore, the observed enhancement of thermal nanofluid conductivity is larger than predicted by wellestablished theories.Other perplexing results in this rapidly evolving field include a surprisingly strong temperature dependence of the thermal conductivity and a threefold higher critical heat flux compared with the base fluids [6].
At the same time, it is known [14] that a position of any impurity in the liquid matrix has dual character due to density fluctuations of liquid.They consist of almost regular tetrahedrons connected in pairs by faces in ramified chains [15,16].Impurity atoms can be placed on the external faces of these chains not changing the microstructure of the dense liquid part forming an "introduction" solution [14,16].In opposite, they can add tetrahedral clusters to the dense part of the basic component.Then, the mixture becomes diphase for the impurity component and single phase for the basic one, that is, it forms an "addition" solution [17].
The last as a two-phase state for impurity is the very interesting way for stabilizing the nanofluid structure and to improve its thermal properties.For this, an effect of ramified fractal clusters as a solid-state part of the addition solution is considered within the framework of the heat-transfer theory.A subject of investigations in the given work is the thermal conductivity correction for such the fractal particles in the liquid.

Possible States of Impurities in Liquids
It is found [18,19] that cations and anions in liquid water reside on the surface of clusters H + (H 2 O) 100 , Na + (H 2 O) 100 , Na + (H 2 O) 20 , and Cl − (H 2 O) 17 below the cluster melting temperature, but they are solvated into the interior of the cluster above the melting temperature.It should be noted that the exclusion of the Cl − ion from the cluster surface was far less dramatic than that for the Na + clusters studied.In increasing temperature, the Cl − maxima transitions occur gradually from near surface to interior displaying a significant interior bias only at the highest temperatures considered.
On the other hand, impurity atoms (at a low concentration) are placed on the external faces of tetrahedral clusters of the dense liquid part not changing its microstructure, that is forming the introduction solution [14,16].In increasing the concentration, the impurity adds its tetrahedral clusters to the dense part of the basic liquid, so that the solution becomes nonhomogeneous in addition of the solution [17].
In two-structured water, the low-density tetrahedralordered ice-regions are divided by higher density clusters with an asymmetrical structure [20].A typical sample of ramified tetrahedral cluster of water density fluctuations simulated by molecular-dynamics method [13] is shown in Figure 1.This tetrahedral cluster is complicated due to hydrogen bonds, but the frame of them as a broken red line connecting the centers of tetrahedrons is also ramified as the one of liquid metal which is presented in Figure 2.
At the same time, a parameterization of the energy functional cannot be arbitrary since the condensate structure at the fixed density weakly depends on temperature for a wide class of matter: metals and ionic melts, liquid semiconductors, and nonconductors [14].Such a model can  explain many properties of the liquid state, in particular, fractal properties of impurity clusters in any liquid which can be formed at changing the composition of binary mixture.So, in Figure 3, one can see such a microstructure behavior of any solution at changing the correlation radii of fluctuations of the solution density, R 1 , and its composition, R 2 [14].
One can see a sharp falling R 2 at the some value of impurity concentration that specifies a disintegration of the composition fluctuations in molecular clusters against a background of long-wave fluctuations of the solution density.Monotonous increasing of the correlation radius of the composition fluctuations in some interval of the impurity concentration should be interpreted as a region of cluster existence in the phase diagram, and then the allocation of these particles from the solution when the correlation radius R 2 increases sharply [14].
This impurity clustering differs from the first-order phase transition in the solution when the excessive phase of an impurity compound precipitates.It may be considered as an analogue of such transition only in microregions of liquid As illustration of such density fluctuations clustering of any impurity, Figure 4 shows potassium quasioxide cluster (K 2 O) n received by molecular-dynamic simulation of nonsaturated alloy K-O [14].
As one can see in Figure 3, the low bound of impurity concentration for its clustering in liquid is ∼ 0.1C s , where C s is the saturation concentration of the impurity.It is clear that this range for clustering any impurity in water is the effective way for stabilizing the microstructure of water nanofluid.
Thus, in order to prepare stable nanofluids, it is important to form nanoparticles in water directly from the dissolved impurity.Such ramified fractal cluster is shown in Figure 5 as a natural state of solution when the density fluctuations in water induce clustering the impurities.In this connection, one can offer a fractal model for nanoparticles in water as a percolation cluster of solid-like filaments shown in a diagram form of Figure 6 or a micelle presented in Figure 7.  Here, is assumed that fractal-cluster mass, M, enclosed in a sphere of radius, r, satisfies the scaling law [21] where D is Hausdorff 's dimension of particle fractal equal to ∼2.5.
In real objects, the fractal structure is observed on scales, r, satisfied the condition a < r < d/2, where d is the size of fractal nanoparticle.Then, one can easily obtain the volume fraction, ϕ f , of a particle material in the fractal as Here, a is the thickness of fractal filament.
Physics Research International

Model of Heat Transfer by Fractal Particles in Water
It is evident that thermal and physical properties of fiber (filaments) composites are locally anisotropic.However, these properties are isotropic in a clew of filaments as a whole.
Many theoretical and numerical studies exist about thermal conductivity of nanometer-sized particles in liquids [2][3][4][5][6].They are proposed to predict the effective thermal conductivity of nanofluids.This approach was developed in a previous paper [13].
A thermal conductivity, λ f , of fractal mater as a percolation cluster filled with liquid is the same as the solid particle with respect to a heat flux in the fluid where λ m is the thermal conductivity of dispersed material as spherical fractal particles (see Figure 6) which have developed interface of the solid/liquid contact.We offer that its heat-variable resistor is negligible.Now, for well-dispersed fractal particles, we can use the potential theory of Maxwell [22] giving a simple relationship for the reduced conductivity of randomly distributed and noninteracting spherical particles in the liquid where λ is the thermal conductivity of basic liquid and λ n is the obtained thermal conductivity of nanofluid.Since ϕ p as the volume fraction of particles in liquid is equal to ϕ/ϕ f = (D/3)(2a/d) D−3 ϕ and ϕ ≡ ϕ f ϕ p is the volume fraction of their material in the liquid, we will finally obtain Here, α = λ/λ m 1.The function (5) of three parameters: α, 2a/d, and D can be easily calculated for estimating the effect of fractal structure of particles on the thermal conductivity of nanofluid.

Results
Many of the obtained experimental data for a high fraction of particles can be understood if one allows clustering particles into fractal aggregates (see Figure 7).Since the fractal matter of particle occupies less space than the liquid into any individual particle, the volume fraction, ϕ f , of the fractal particle is substantially smaller than unity (ϕ f 1).Using (5) for different values of α, 2a/d, and D, one can estimate the effect of fractal structure of nanoparticles on the thermal conductivity of water nanofluid (see Table 1)

Discussion
We discuss the case of spherical fractal particles in which Hausdorff 's dimension is equal to ∼2.5 when the fine filaments of 0.5-1.0nm in diameter constitute the fractal cluster of 10-50 nm in size.The obtained effect is substantial.
In the limit of the small volume fraction of particles and the high thermal conductivity of particles, our version of the effective medium theory converges to the prediction that the thermal conductivity of colloidal solution can enhance up to 18ϕ.
Thus, the significant property of fractal colloidal particles is an explanation of observed enhancement obtained by several research groups of nanofluids.
At the same time, it is important to understand that the fractal particles can be produced in water solution as a result of complex chemical reactions between a chosen impurity and the liquid.Therefore, it is necessary to develop a special technology for getting them in the medium directly.

Conclusions
The theoretical studies show how one can provide the stable formation of particles in water solution.It is important to form clusters in water directly from impurities which are dissolved there.Then, ramified fractal clusters as natural solidstate part of addition solution can be stable constituents of nanofluid on water base, and its thermal conductivity can be enhanced up to 18ϕ with the volume fraction of fractal particles, ϕ.
The results of microscopic investigation may be useful for motivating choosing a composition of a heat-transfer suspension and developing technology for making the appropriate nanofluid.
Thus, the fractal structure of colloidal particles can give the sixfold enhancement of usual nanofluid contribution to the thermal conductivity of water.

Figure 1 :
Figure 1: Dense-part cluster in MD model of water at 300 K and its frame (broken red line); blue points are the molecules and red points are the centers of the cluster tetrahedrons.

Figure 2 :
Figure 2: Typical tetrahedral cluster in MD model of the dense part of liquid potassium and the frame connecting the centers of tetrahedrons (broken red line with its three projections as blue lines).

Figure 3 :
Figure 3: Logarithm of correlation radius, R i , for fluctuations of the solution density (1) and the one, R 2 , for its composition fluctuations (2) as a function of the impurity concentration, x.

Figure 4 :Figure 5 :
Figure 4: The quasioxide cluster in oxygen-nonsaturated melt of potassium, Big circles are oxygen anions and small circles are potassium cations.

Figure 6 :
Figure 6: The scheme of percolation fractal cluster of solid-like filaments.

Figure 7 :
Figure 7: The fractal cluster as a solid-like micelle.