(140a) Off-Lattice Monte Carlo Simulation for Heat Transfer through Carbon Nanotube Composites

The focus of this study is to develop a numerical methodology for the prediction of thermal properties of composite materials that are composed of a low thermal conductivity matrix and very small but very high conductivity inclusions. An example of such materials is Carbon Nanotube (CN) composites.

Theoretical predictions, such as Maxwell's equations for the calculation of the effective thermal conductivity of a composite, do not take into account the presence of thermal resistance at the inclusion-matrix interface. As a result, the effective thermal conductivity of some composites can be severely overpredicted. For example, using the modified Mawell equation for non-spherical inclusions, one would predict a 1% wt CN multi-wall composite with epoxy (for multi-wall CNs the thermal conductivity is about 3000 W/(K.m) and for epoxy it is about 0.2 W/(K.m)) to have an effective conductivity of 30 W/(K.m). Experiments have shown that instead it is 0.29 W/(Km) [1]. The reason is the heat resistance [2] that exists at the interface between different solids and between solid and liquid in contact. Since heat is transferred by phonons in insulator solids, mismatch of phonon frequencies (acoustic mismatch) causes this additional resistance that could be large even at room temperatures. We are using random walk simulations of thermal walkers to explore the effect of the interfacial resistance on heat flow in composites with inclusion of practically infinite conductivity. In the case of CN composites, the algorithm can calculate the effective conductivity as a function of nanotube length, orientation and percent composition.

The numerical approach is based on a parallelizable off-lattice Monte Carlo method. A large number of thermal markers (i.e., random walkers) travel in the computational cell with finite thermal conductivity, which includes cylindrical enclosures of very high thermal conductivity for a relatively long time. The heat markers move only due to molecular diffusion. The off-lattice Monte Carlo algorithm treats the matrix material with an effective medium approximation utilizing a Brownian motion for the thermal markers while they travel within the matrix. At the end of each time step, a marker can end up into an inclusion (i.e., into a CN). The probability that allows a thermal marker to enter a CN enclosure is proportional to the thermal resistance at the CN-matrix interface. If the move is allowed, and a marker enters an inclusion, it will be uniformly distributed anywhere within the inclusion, taking, thus, into account the fact that the CN thermal conductivity is orders of magnitude larger than that of the matrix material. Tomadakis and Sotirchos [3] have used a similar algorithm to investigate cylindrical enclosures with different properties than the matrix material, but did not take thermal resistance into account for their simulations.

The paper will discuss the development of the algorithm as well as an important (but subtle) point of simulating heat transfer in materials with different characteristic time and length scales. The assumption that a marker is distributed uniformly throughout the space occupied by an inclusion, once the marker crosses into the inclusion, is the mesoscopic result of the Brownian random movement of a thermal marker inside a CN. In other words, the random Brownian phonon transfer inside the CN appears as ballistic heat transport on the time scale for conduction in the matrix material. The determination of the correct value of the probability that a marker has to bounce back into the matrix (or back into the inclusion) when the random jump of the particle makes it cross the interface is an issue where scale considerations become important. The algorithm was developed so that once a walker in the matrix reached the interface between the matrix and a CN, the walker moved into the CN phase with a probability f, which represented the thermal resistance of the interface. Similarly, once a walker was inside a CN, the walker either re-distributed randomly within the CN at the end of a time step, or would cross into the matrix phase with a probability g. Even though it was assumed that the thermal resistance is the same for a heat walker traveling from the matrix to the enclosure and from an enclosure to the matrix phase, it was found that g ? f. The reason is that the assumption of a uniform distribution of a marker once inside a CN removed a length scale from the problem (that of the length of the thermal walker movement inside the CN). In order to maintain equilibrium, the two probabilities must be related as

f = c g

where c is a constant that depends on the shape of the high conductivity enclosures and on the properties of the matrix material. The paper will discuss methods to determine the value of c both theoretically and computationally (by conducting numerical experiments at thermal equilibrium).

Our algorithm is more efficient than a typical random walk algorithm, and much faster than a Molecular Dynamics algorithm. Even though it cannot provide results at the fundamental molecular level as Molecular Dynamics can, it can be used to model physico-chemical properties of randomly-dispersed high conductivity inclusions quite successfully.

REFERENCES: 1. M.J. Biercuk, M.C. Llaguno, M. Radosavljevic, J.K. Hyun, A.T. Johnson, Applied Physics Letters, 80, 2767-2779 (2002). 2. P.L. Kapitza, J. Phys. USSR, 4, 181 (1941). 3. M.M. Tomadakis, S.V. Sotirchos, J.Chem.Phys., 104, 6893-6900(1996).