Elsevier

Journal of Non-Crystalline Solids

Volume 363, 1 March 2013, Pages 103-115
Journal of Non-Crystalline Solids

Modelling of the conductive heat transfer through nano-structured porous silica materials

https://doi.org/10.1016/j.jnoncrysol.2012.11.053Get rights and content

Abstract

There is currently a growing interest in nano-structured silica based materials due to their remarkable thermal properties. These materials are notably used in Vacuum Insulating Panels (VIP). Their exceptional insulating performances have been demonstrated experimentally for a relatively long time. But the heat transfer mechanisms occurring in this kind of materials remain relatively badly known due to the nanometric dimensions and to the complexity of the porous structure. Therefore, the present study aims to develop a numerical model for estimating the magnitude of conductive heat transfer inside nano-structured silicas using a realistic representation of their complex porous structure. The model takes into account the special porous morphology of the materials at both the nanometric and microscopic scale. Moreover, the conduction heat transfer at the nanometric scale is treated using a numerical resolution of the Boltzmann equation since the validity of the macroscopic laws is then questionable. The computations are conducted using phonon properties of silica obtained in the literature. A parametric study allows us to analyse the influence of structural characteristics and thermo-physical properties on the insulating performances and thus to highlight the most important parameters.

Highlights

► Development of a numerical model for conductive transfer in nanostructured silica. ► Takes into account the realistic morphology at nanometric and microscopic scales. ► Resolves the Boltzmann equation since macroscopic laws are questionable. ► Computations conducted using phonon properties of silica obtained in literature. ► Influence of structural characteristics and thermo-physical properties analysed.

Introduction

Among different businesses, building domain is the most energy consuming one with more than 45% of the total energy, far more than the industry (28%) or transport fields. Most of this energy is devoted to the building heating. Therefore, the thermal insulation quality is of primary importance; and the improvement of thermal insulators' performances is of major issue in view of a significant reduction in the global energy consumption. Current researches are interested in the development of new insulating materials, notably nanoporous silica-based superinsulating materials. These materials are made of silica particles, forming aggregates several nanometres in diameter, which are themselves agglomerated in highly porous matrix (ε  90%). These nanostructured silica present a double-scale structure: (i) at the nanometric scale aggregates form chains of silica particles with undividable links; and (ii) at the microscopic scale, the structure is formed by aggregate agglomeration. Nano-structured silica structure is illustrated in Fig. 1, visualizing fumed silica formation process of when chlorosilane molecules are introduced inside a flame.

The good insulating properties of nanostructured silicas are due to the very small pore size where pores are of the same order of magnitude as the mean free path of air molecules. Indeed, according to Kaganer [1], the thermal conductivity of a confined gas can be expressed by:kf=kf.free1+2βkBσ0Ttp.P=kf.free1+2βLgtpwhere the mean free path of the gas molecules Lg and the constant β can be calculated by Lg=kBT2σ0P and β=5π322αα9γ5γ+1 with α being the accommodation coefficient and γ the adiabatic coefficient. For air, the adiabatic coefficient γ was set to 1.4 and for the accommodation coefficient α = 1 was applied.

Thus, under partial vacuum air thermal conductivity, which is the main part of the superinsulant's total thermal conductivity, gets reduced because of the “Knudsen effect” (non-confined air conductivity is kair  25 mW/m/K while for good insulators total conductivity is typically comprised between 30 and 40 mW/m/K). In such a case, heat transfer mainly occurs by thermal conduction through the solid phase and by thermal radiation propagation. Besides, the nanostructured silicas currently commercialised generally contain additional opacifing micrometric particles in order to reduce drastically the contribution of radiation heat transfer. Then, when a primary vacuum is applied and when opacifing particles are added, equivalent thermal conductivities lower than 10 mW/m/K can be achieved as reported by Quenard and Sallée [2] or the researchers of ZAE Bayern [3], [4], [5], [6]. This constitutes a substantial improvement of insulating properties compared to the most performing classical insulators.

Because of the growing interest for these materials, numerous recent studies have been devoted to experimental or theoretical characterization of the heat transfer in nano-structured silicas.

Their outstanding thermal properties have been demonstrated experimentally for a relatively long time. For example, Quenard and Sallée [2] have analysed samples with porosities higher than 90% composed of fumed silicas combined with micometric fibres insuring the mechanical strength and opacifing particles. At ambient pressure and temperature, the equivalent conductivity was near 20 mW/m/K and can be reduced to 6 mW/m/K when a primary vacuum such as 1–10 hPa is applied. The noticeable increase of this conductivity with temperature reported by the authors suggests that a significant part of the heat transfer is due to thermal radiation. Quenard and Sallée also pointed out experimentally the detrimental influence of adsorbed water since the equivalent conductivity increases almost linearly with the water content with a rate of 1.5–2 mW/m/K per % of adsorbed water. Similarly, the researchers of ZAE Bayern [3], [4], [5], [6] have measured equivalent thermal conductivities close to 3 mW/m/K for evacuated and opacified nanoporous materials based on fumed, precipitated and fumed silicas at ambient temperature. They also show that their thermal performances are strongly affected by water content.

In parallel, several theoretical studies have been initiated in order to model the thermal transfer in silica-based super-insulating materials (fumed or precipitated). Coquard and Quenard [7] have proposed a model of computation of their equivalent conductivity taking into account the radiation–conduction coupling. The effective thermal conductivity (conductive transfer alone) was estimated using a Finite Volume Model considering the heat conduction through gas phase, through solid phase and through water possibly adsorbed and forming meniscus at the contact area between two nanoparticles. The authors assumed that the macroscopic laws of heat conduction are valid at the nanoscale. They depicted the structure of nano-structured silicas as chains of silica grains distributed at the edges of a cubic network. They studied the influence of air pressure and of the contact area between two particles touching each other. The results of their model highlighted the influence of the air pressure and of the water content. The effective conductivities computed were in good agreement with the values measured at the CSTB [2] or ZAE Bayern [3], [4], [5], [6] for different air pressures and different water content. The numerical results also highlighted the influence of the particle diameter and the contact area between neighbouring particles. Rochais, Domingues and Enguehard [8] have also developed a model of computation of the effective conductivity of silica-based nanoporous super-insulating materials. They applied Finite Volume computations to different 2-D or 3-D fractal structures to evaluate the influence of the solid fraction on the conductivity. Finally, during her Ph.D., Spagnol [9] had used a Finite Difference Method to solve numerically the heat conduction equation through various fractal geometries. Like in preceding studies [7], [8], the author considered the macroscopic heat conduction laws to be valid and used Eq. (1) for estimating the gas conductivity. Computations were applied to simple fractal geometries like brick walls, Von Koch flakes and to random fractal geometries obtained by diffusion limited aggregation process.

These first theoretical studies permitted to highlight the main mechanisms responsible for the conduction heat transfer in nano-structured silica-based materials and to point out the parameters that mostly affect the effective conductivity. However, all these studies considered the macroscopic Fourier law of heat conduction ϕc=KgradT as valid at nanoscale. However, according to Kittel [10], the phonon mean free path in silica is about one to several nm and is therefore non-negligible compared to the particle size. Thus, the validity of the macroscopic laws is questionable. In view of the models improvement, it seems indispensable to use another approach for treating conduction heat transfer at the nanometric scale.

Besides, recent studies are initiated to account for the “nanoscopic effects”, using notably Molecular Dynamics (MD) simulations. However, the works which are interested specifically in the heat transfer between two silica nano-particles remain relatively rare and have been conducted essentially by Domingues et al. [11], [12] and Mahajan et al. [13]. Domingues et al. have estimated the thermal conductance between two particles in contact [12] or separated by a submicronic distance [11]. For this purpose, they used the fluctuation-dissipation theorem and applied a MD computation. For silica particles in contact with diameters comprised between 1.5 and 5 nm, they showed that the thermal conductance is proportional to the number of atoms in contact. Then, they managed to compute an “atomic” conductance which is comprised between 0.5 and 3 nW/K. This leads to a total heat exchange between two particles of about 1 to 2 μW per nm2 of contact area. Similarly, Mahajan, Subbarayan and Sammakia [13] have estimated the thermal conductivity of silica nanoparticles by MD simulations using the Green–Kubo relations. They applied their computations to silica particle composed of 600 atoms (several nm in diameter) and obtained a thermal conductivity of 0.59 W/m/K corresponding to approximately 40–50% of the silica at the macroscopic scale. These results show the significant influence of the particle size at the nanometric scale. MD simulations are promising and give rise to a growing interest. However, the two preceding studies have highlighted their major drawback: they require high computation costs and time when dimensions of several nm or tenth of nm are considered. This is prohibitive in view of parametric studies.

Other numerical methods have been developed to model the heat transfer at nanometric scale when macroscopic laws are invalid. The most common one consists in solving the Boltzmann equation governing the phonon transport inside nano-structured materials. This approach can be used for dielectric or semiconductor materials for which phonons are the unique heat carriers (the contribution of electron is negligible). This method is rigorously applicable only when the smallest dimension is more important than the phonon wavelength so that the phonons can be considered as massless particles by neglecting the interference between phonon waves. It has been largely used for estimating the thermal conductivity of silicon-based nanostructures notably by Mazumder and Majumdar [14] Lacroix et al. [15], [16] or Randrianalisoa et Baillis [17], [18], [19], [20]. These authors resorted to Monte Carlo procedure for the resolution of the Boltzmann Equation. The principle of this approach is to follow the path of a great number of phonons emitted by the materials and carrying thermal energy. Each phonon has its own wavelength and polarisation. During their path, the phonons can undergo phonon–phonon scattering, scattering by impurities or defaults, can be reflected at the interface between the phases or be absorbed at the boundary surfaces of the system or inside the material. The phonon properties required are: (1) the phonon mean free path before interaction with other phonons or with impurities/defaults, (2) the phase velocities, (3) the group velocities, (4) the density of states, (5) the heat capacity, and (6) dispersion curves. Each of these properties varies with the frequency and the polarisation of the phonon. Due to the numerous applications of silicon in microelectronics where the size of the components might be submicronic, its phonon properties have been extensively characterized. Mazumder and Majumdar [14], Lacroix et al. [15], [16] and Randrianalisoa and Baillis [17], [18] have computed the thermal conductivity of very thin silicon films and have shown that it gets lower than the macroscopic conductivity when the thickness of the film is from the same order of magnitude as the phonon mean free paths. Lacroix [16] and Randrianalisoa and Baillis [18] have also investigated the conductivity of silicon nano-wires while Randrianalisoa and Baillis [20] have applied the method to nanoporous silicon films of various thicknesses. They managed to estimate the influence of the porous characteristics (porosity, pore size, shape of pores) on their conductivity.

The Boltzmann equation numerical resolution via Monte Carlo procedures is very interesting for the present study problem as it permits to take into account accurately the morphology of the porous structure as well as is well-adapted to the dimensions of the nano-structured silica materials particles. That is the reason why, this approach is used in the present study for evaluating the heat exchange between two neighbouring particles. For simplicity purpose, the particles are assumed spherical with homogeneous diameter.

Once the heat exchange between nano-particles has been evaluated, the proposed model should also take into account the morphology of the nano-structured silica at the microscopic scale. This has been done by assimilating the material to a random agglomeration of aggregates composed of silica particles chains. According to previous studies [21], [22], [23], these aggregates present a fractal like structure. Thereafter, the effective conductivity of this agglomeration of aggregates is evaluated by simulating a 1-D heat transfer across a slab of material. The heat transfer problem is solved using a Finite Volume Scheme. The obtained numerical results allow us to study the evolution of the silica-based nano-structured materials effective conductivity with air pressure, structural parameters (such as diameter of particles, contact area between particles, fractal characteristics, size of aggregates) and material thermo-physical properties.

In the present study, one focuses on heat transfer in fumed silicas, as they are raw materials in commercial VIPs.

Section snippets

Representation of the nano-porous structure

As explained in the introduction, the nano-structured silicas consist of randomly organized agglomerates of aggregates (see Fig. 2). The aggregates are made of a chain of silica particles related among them by undividable connections. The silica particles are assumed spherical with a unique diameter denoted by Dp. The contact area between two neighbouring particles is characterized by a contact diameter denoted Dc.

Each aggregate, composed of Np particles, is assumed to have a fractal like

Nanometric scale — variations of the particle–particle conductivity

We have applied the resolution of the Boltzmann equation using Monte Carlo simulations for particle–particle junctions with varying geometrical characteristics Dp and Dc/Dp using the phonon properties given by the model of Goodson et al. at a temperature of 290 K. The main results are illustrated on the following figures for two different values of contact diameters: Dc  0.44 × Dp and Dc  0.66 × Dp. On the same figure, we have also illustrated the particle–particle thermal conductivity that would be

Conclusions

We have developed a parametric model for estimating the thermal behaviour of nano-structured silica-based materials; that model takes into account a faithful representation of the complex porous structure. Several conclusions on heat transfer in nano-structures silica-based materials can be drawn from the analysis of our numerical results.

At nanometric scale:

  • The “nanoscopic effects” due to the very small dimensions of the silica particles remain relatively limited for silicas in commercial VIPs

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