wwceramics. org/ACT Composites of 0-3 Connectivity Alu G SCAN B=√3V/2,C=-4√2/3Ⅵ By solving Laplace's equation, Maxwell of randomly distributed noninteracting particles homo- geneous in the filled polymers. This model assumes that the particles are well dispersed and the potential around each sphere will not be influenced by the presence of each particle(most parts are not touc each other) These conditions are fulfilled only for low filler loadings kr + 2km +2Vr(lr-km Filler Volume Percentage 灯+2km-Ⅵ(k-km) ig. 2. Thermal conductivity nction of volume loading for different fillers(afier Wong and Bollampally) Nielson derived a semitheoretical model by introducing the shape, size, and orientation of the par ticles into the composite and modifying the halpi Tsai equation for thermal transfer, which can be expressed kr -ket 1-B (12) km+ kerr kr+ 2kem kr/km-1 where Vm is the volume fraction of the polymer matrix B一k/km+Aψ=1+ The constant A is a function of the geometry of the Based on Tsao's model, which gives the thermal filler particles and m is the maximum filler content conductivity of a two-phase solid mixture, possible while still maintaining a continuous matrix and Vachon assumed a parabolic distribution of the phase. B is a constant related to the relative conductiv discontinuous phase in ty of the components. The values of A and pm for constants of this parabolic distribution are determined many geometric shapes and orientations are reported by and presented as a function of the discontinuous-phase Kumlutas et al. volume fraction. Thus, the equivalent thermal conduc 6. Agari-Uno model tivity of the two-phase solid mixture is derived in terms Agari and Uno developed a model that takes of the distribution function and the thermal conductiv- into account the parallel and series conducting mecha ity of the constituents nism in the particle-filled polymer composites For k>k log ke= VrC2 log kr+(l-vr)log(Cikm)(13) k√C(km-k)km+B(k一k and crystal size of the polymer. C2 is a measure of the ease with which the particles form conductive chains of xlmy{如a+B-k)+√C(如-k the fillers /km+B(kr-km) C(km-kr The ceramics in general have a relatively high thermal conductivity as compared with the polymers Hence, the thermal conductivity of the composite ases with the increase in the fille r content 40,98,112,113 Figure 10 shows the variation of the thermal conduc-for thermal transfer, which can be expressed as102,103 Vm km keff km þ 2keff þ Vf kf keff kf þ 2keff ¼ 0 ð9Þ where Vm is the volume fraction of the polymer matrix in the composite. 3. Cheng–Vachon model: Based on Tsao’s model, which gives the thermal conductivity of a two-phase solid mixture,104 Cheng and Vachon assumed a parabolic distribution of the discontinuous phase in the continuous phase.110 The constants of this parabolic distribution are determined and presented as a function of the discontinuous-phase volume fraction. Thus, the equivalent thermal conduc￾tivity of the two-phase solid mixture is derived in terms of the distribution function and the thermal conductiv￾ity of the constituents. For kf4km 1 kc ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cðkm kfÞ½km þ Bðkf kmÞ p  ln ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½km þ Bðkf kmÞ p þ B 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cðkf kmÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½km þ Bðkf kmÞ p B 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cðkm kfÞ p þ 1 B km ð10Þ where B ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 3Vf=2; p C ¼ 4 ffiffiffiffiffiffiffiffiffiffiffiffi 2=3Vf p 4. Maxwell model: By solving Laplace’s equation, Maxwell100,111 ob￾tained a simple relationship for the thermal conductivity of randomly distributed noninteracting particles homo￾geneous in the filled polymers. This model assumes that the particles are well dispersed and the potential around each sphere will not be influenced by the presence of each particle (most parts are not touching each other). These conditions are fulfilled only for low filler loadings kc ¼ km kf þ 2km þ 2Vfð Þ kf km kf þ 2km Vfð Þ kf km   ð11Þ 5. Neilson model: Nielson derived106,107 a semitheoretical model by introducing the shape, size, and orientation of the par￾ticles into the composite and modifying the Halpin– Tsai equation108 kc ¼ km 1 þ ABVf 1 BVfc   ð12Þ B ¼ kf=km 1 kf=km þ A c ¼ 1 þ 1 fm f2 m !Vf The constant A is a function of the geometry of the filler particles and fm is the maximum filler content possible while still maintaining a continuous matrix phase. B is a constant related to the relative conductiv￾ity of the components. The values of A and Fm for many geometric shapes and orientations are reported by Kumlutas et al. 109 6. Agari–Uno model: Agari and Uno105 developed a model that takes into account the parallel and series conducting mecha￾nism in the particle-filled polymer composites log kc ¼ VfC2 log kf þ ð Þ 1 Vf logðC1kmÞ ð13Þ where C1 is the coefficient of the effect on crystallinity and crystal size of the polymer. C2 is a measure of the ease with which the particles form conductive chains of the fillers. The ceramics in general have a relatively higher thermal conductivity as compared with the polymers. Hence, the thermal conductivity of the composite in￾creases with the increase in the filler content.40,98,112,113 Figure 10 shows the variation of the thermal conduc- 0 10 20 30 40 50 0.0 0.5 1.0 1.5 2.0 2.5 Thermal Conductivity (W/mK) Filler Volume Percentage Alumina SCAN Silica Fig. 9. Thermal conductivity as a function of volume loading for different fillers (after Wong and Bollampally98). www.ceramics.org/ACT Polymer–Ceramic Composites of 0–3 Connectivity 425