An effective model for the thermal conductivity of nanoparticle composites/polymers

Abstract

In this study, an effective model is proposed to predict the effect of nanoparticle agglomeration on the thermal conductivity of three-phase nanocomposites/polymers. In order to better describe this effect, the concept of agglomeration degree is introduced. The effect of particle volume fraction on thermal conductivity of composites is also studied by considering the interphase and agglomeration degree of particles. First, the relationship between agglomeration degree and particle volume fraction is discussed. Then, the effects of particle volume fraction, agglomeration degree and interphase thickness on thermal conductivity of composites are studied. The obtained results show that the agglomeration degree increases with increasing particle volume fraction. The thermal conductivity of composites increases first and then decreases with increasing particle agglomeration degree, and is also affected by the different thermal conductivity of particles and matrix, and the thickness of interphase.

Appendix: Aggregative growth

This appendix qualitatively explains why Eq. (26) can be set to power-law form. First, a simple kinetic mechanism is considered, which is called “aggregative growth” [34]. Accordingly, the agglomeration kinetics is controlled by the following equation:

$$\begin{aligned}&P+P\rightarrow A, \nonumber \\&P+A\rightarrow 1.5A, \end{aligned}$$

(A1)

where P stands for the nanoparticles and A for the agglomerate. The “nucleation rate,” defined as the rate of the first agglomerate formed from the initial nanoparticles, is given by

$$\begin{aligned} -\frac{1}{2}\frac{\mathrm{d}\left[ p \right] }{\mathrm{d}t}=k_\mathrm{n} \left[ p \right] ^{2} \end{aligned}$$

(A2)

with $k_\mathrm{n}$ the nucleation rate constant and square brackets indicating the molar concentration. The term $\frac{\mathrm{d}\left[ P \right] }{\mathrm{d}t}$ in (A2) stands for the consumption of P due to nucleation only. The “agglomerative growth rate” is given by:

$$\begin{aligned} \frac{\mathrm{d}\left[ p \right] }{\mathrm{d}t}=k_\mathrm{g} \left[ P \right] \left[ A\right] , \end{aligned}$$

(A3)

where $k_\mathrm{g}$ is the rate constant of agglomerate growth. The term $\frac{\mathrm{d}\left[ P \right] }{\mathrm{d}t}$ in (A3) stands for the consumption of P due to agglomerate growth only. From (A1):

$$\begin{aligned} \left[ P \right] =\left[ P \right] _{0} -2\left[ A \right] . \end{aligned}$$

(A4)

It follows from (A4) that $\frac{\mathrm{d}\left[ A \right] }{\mathrm{d}t}=-\frac{1}{2}\frac{\mathrm{d}\left[ P \right] }{\mathrm{d}t}$, so that the total rate law for the agglomeration mechanism is given by

$$\begin{aligned} \frac{\mathrm{d}\left[ A \right] }{\mathrm{d}t}=k_\mathrm{n} \left[ P \right] ^{2}+\frac{1}{2}k_\mathrm{g} \left[ P \right] \left[ A \right] . \end{aligned}$$

(A5)

Making use of the initial conditions $\left. {\left[ A \right] } \right| _{t=0} =\left[ A \right] _{0} =0$, $\left. {\left[ P \right] } \right| _{t=0} =\left[ P \right] _{0}$ and relation (A4), the solution of Eq. (A5) is given by

$$\begin{aligned} \left[ A \right] =\frac{2k_\mathrm{n} \left[ P \right] _{0} \left( {1-e^{\frac{k_\mathrm{g} \left[ P \right] _{0} t}{2}}} \right) }{4k_\mathrm{n} (1-e^{\frac{k_\mathrm{g} \left[ P \right] _{0} t}{2}})-k_\mathrm{g} }. \end{aligned}$$

(A6)

When $t\rightarrow \infty _{\mathrm { ,}}\left[ A \right] _{\infty } =\frac{\left[ P \right] _{0} }{2}$. This indicates that the concentration of agglomerates may be up to half of the initial particle concentration. It can also be seen from the formula that the final concentration of aggregates depends on the initial concentration of nanoparticles. This shows that the size of the agglomerate (and the degree of agglomeration) is a function of the initial volume fraction of nanoparticles. According to the law of proportion, the number of nanoparticles in agglomerates can be determined at a given time:

$$\begin{aligned} N_\mathrm{t} =\frac{2\left[ A \right] }{\left[ P \right] _{0} }N_{\infty } \end{aligned}$$

(A7)

with $N_{\infty }$ the final number of nanoparticles in the agglomerate. However, the limit $t\rightarrow \infty$ is not realistic and (A7) is not valuable at this value. Therefore, a middle value $N_{\mathrm{int}}$ is taken here:

$$\begin{aligned} N_\mathrm{t} =\frac{2\left[ A \right] ^{*}}{\left[ P \right] _{0} }N_{\mathrm{int}}, \end{aligned}$$

(A8)

where $\left[ A \right] ^{*}<\frac{\left[ P \right] _{0} }{2}$. The relation between the agglomerate size D and the number of nanoparticles in the agglomerate zone can be assumed as follows:

$$\begin{aligned} D\propto CN^{b}, \end{aligned}$$

(A9)

where C is a constant composed out of material constants and geometric data, and b is the fractal index. Coupling Eqs. (A8) and (A9) results in

$$\begin{aligned} \frac{D_{\mathrm{int}} }{D_{t} }\propto \frac{N_{\mathrm{int}} }{N_\mathrm{t} }\propto C\left( {\frac{\left[ P \right] _{0} }{2\left[ A \right] ^{*}}}\right) ^{b}, \end{aligned}$$

(A10)

where $\left[ P \right] _{0} \propto f_\mathrm{p}$. Therefore, Eq. (26) can be set to power-law form.