Thermal conductivity of carbon nanotubes and their polymer nanocomposites: A review

This paper reviews the crystal chemistry, synthesis, densification, microstructure, mechanical properties, and oxidation behavior of zirconium diboride (ZrB2) and hafnium diboride (HfB2) ceramics. The refractory diborides exhibit partial or complete solid solution with other transition metal diborides, which allows compositional tailoring of properties such as thermal expansion coefficient and hardness. Carbothermal reduction is the typical synthesis route, but reactive processes, solution methods, and pre-ceramic polymers can also be used. Typically, diborides are densified by hot pressing, but recently solid state and liquid phase sintering routes have been developed. Fine-grained ZrB2 and HfB2 have strengths of a few hundred MPa, which can increase to over 1 GPa with the addition of SiC. Pure diborides exhibit parabolic oxidation kinetics at temperatures below 1100°C, but B2O3 volatility leads to rapid, linear oxidation kinetics above that temperature. The addition of silica scale formers such as SiC or MoSi2 improves the oxidation behavior above 1100°C. Based on their unique combination of properties, ZrB2 and HfB2 ceramics are candidates for use in the extreme environments associated with hypersonic flight, atmospheric re-entry, and rocket propulsion.

Refractory Diborides of Zirconium and Hafnium

A methodology is introduced for predicting the effective thermal conductivity of arbitrary particulate composites with interfacial thermal resistance in terms of an effective medium approach combined with the essential concept of Kapitza thermal contact resistance. Results of the present model are compared to existing models and available experimental results. The proposed approach rediscovers the existing theoretical results for simple limiting cases. The comparisons between the predicted and experimental results of particulate diamond reinforced ZnS matrix and cordierite matrix composites and the particulate SiC reinforced Al matrix composite show good agreement. Numerical calculations of these different sets of composites show very interesting predictions concerning the effects of the particle shape and size and the interfacial thermal resistance.

Effective thermal conductivity of particulate composites with interfacial thermal resistance

Nanofluids, a new class of solid/liquid suspensions, offer scientific challenges because their measured thermal conductivity is one order of magnitude greater than predictions. It has long been known that liquid molecules close to a solid surface form layered solid-like structures, but little is known about the connection between this nanolayer and the thermal properties of the suspensions. Here, we have modified the Maxwell equation for the effective thermal conductivity of solid/liquid suspensions to include the effect of this ordered nanolayer. Because this ordered nanolayer has a major impact on nanofluid thermal conductivity when the particle diameter is less than 10 nm, a new direction is indicated for development of next-generation coolants.

The Role of Interfacial Layers in the Enhanced Thermal Conductivity of Nanofluids: A Renovated Maxwell Model

Theoretical formulation, Navier's solutions of rectangular plates, and finite element models based on the third-order shear deformation plate theory are presented for the analysis of through-thickness functionally graded plates. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The formulation accounts for the thermomechanical coupling, time dependency, and the von Karman-type geometric non-linearity. Numerical results of the linear third-order theory and non-linear first-order theory are presented to show the effect of the material distribution on the deflections and stresses. Copyright © 2000 John Wiley & Sons, Ltd.

Analysis of functionally graded plates

Evaluation ofKIc of brittle solids by the indentation method with low crack-to-indent ratios

A modification of the original theories of Rayleigh and Maxwell permitted the deriva tion of expressions for the effective thermal conductivity of composites consisting of a continuous matrix phase with dilute concentrations of dispersions with spherical, cylin drical and flat plate geometry with a thermal barrier resistance at the interface between the components.

Effective Thermal Conductivity of Composites with Interfacial Thermal Barrier Resistance

A fracture-mechanical theory is presented for crack propagation in brittle ceramics subjected to thermal shock. The criteria of crack stability are derived for a brittle solid uniformly cooled with triaxially constrained external boundaries. Thermal stress crack instability occurs between two values of critical crack length. For short initial crack length, crack propagation occurs kinetically, with the total area of crack propagation proportional to the factor St2 (1-2v)/EG, where St is tensile strength, v is Poisson's ratio, E is Young's modulus, and G is surface fracture energy. Under these conditions the newly formed crack is subcritical and requires a finite increase in temperature difference before propagation will proceed. For long initial crack length, crack propagation occurs in a quasi-static manner and can be minimized by maximizing the thermal stress crack stability parameter Rst= [G/α2E]1/2, where α is the coefficient of thermal expansion. For heterogeneous brittle solids, such as porous refractories, the concept of an “effective flaw length” is introduced and illustrated on the basis of experimental data in the literature. The relative change in strength of a brittle solid as a function of increasing severity of thermal shock is estimated. Good qualitative agreement with literature data is found.

Unified Theory of Thermal Shock Fracture Initiation and Crack Propagation in Brittle Ceramics

We have observed that the thernal conductivity of zincsulphide is increased by adding large particles of highly conducting diamond, but lowered by the addition of sub-micron size particles of diamond. This effect is explained in terms of the interfacial thermal resistance which becomes increasingly dominant as the particles becomes smaller (because that increases their surface to volume ratio). A phenomonological model in which the interface resistance is expressed as an effective Kapitza radius, ak, is presented. The conductivity of the composite is analyzed for different values of α, which is defined to be equal to the Kapitza radius divided by the particle radius. If α = 1, that is, the actual particle radius is equal to ak then the effective thermal conductivity of the particles is equal to that of the matrix. If α > 1, that is the particles are very small, then the contribution of the particles to the thermal conductivity of the composite is dominated by interfaces; if α < 1 then the bulk property of the particles is important. Our measurements yield ak ≈ 1.5 μm for the ZnS-diamond interface.

The effect of particle size on the thermal conductivity of ZnS/diamond composites

The physical properties which affect the propagation of cracks in specimens fractured by thermal shock are discussed. The driving force for crack propagation is provided by the elastic energy stored at fracture. The mechanism of energy dissipation which will tend to arrest the propagating cracks is provided by the “effective surface energy” required to produce the newly formed crack surfaces. An expression is derived applicable to a body of spherical shape for the mean area traversed by cracks nucleated by thermal shock. Three numerical examples are given for materials with widely different physical properties, and their fracture behavior is predicted. Good agreement with experiment was obtained. Thermal shock damage resistance parameters suitable for the relative comparison of the “degree of damage” to be expected in materials fractured by thermal shock are proposed. The criteria for a low degree of damage are high values of Young's modulus of elasticity, Poisson's ratio, and effective surface energy and low values of strength. Recommendations are made for the selection of materials for severe thermal shock, where the best materials available are known to fail.

D. P. H. Hasselman

Papers

Evaluation ofKIc of brittle solids by the indentation method with low crack-to-indent ratios

Effective Thermal Conductivity of Composites with Interfacial Thermal Barrier Resistance

Unified Theory of Thermal Shock Fracture Initiation and Crack Propagation in Brittle Ceramics

The effect of particle size on the thermal conductivity of ZnS/diamond composites

Elastic Energy at Fracture and Surface Energy as Design Criteria for Thermal Shock