# Composites, Volume fractions exceeding Max Packing Fractions

 P: 26 Hi, Sorry for the semi-book here I am working on a project where I am mixing h-BN nanoparticles into a polymer resin to try to tailor the thermal conductivity and dielectric strength of the resulting composite. Admittedly I am not very well versed when it comes to materials science, and I am having a bit of a problem with this concept of maximum packing fraction as determined by the geometry of the filler particles, and then the volume fraction of my filler particles to the whole composite volume. I am increasing my boron nitride concentration by weight percent (10% increments from 0-90%). In order to calculate the theoretical thermal conductivity of the resulting composite material after curing, I am using the Lewis-Nielsen model of thermal conductivity. The model incorporates both the maximum packing fraction and the volume fraction of the filler particles. Near 75% wt. percent BN/Resin I reach the theoretical maximum packing fraction of my nanoparticles (70nm random packed non-agglomerated spheres øm = 0.632 I guess I'm having trouble understanding exactly what the maximum packing fraction is defined as here. Is this the maximum packing fraction of the particles in an agglomerated sphere of the same particles over a set or defined volume, or is this the maximum packing fraction of the particles in "any matrix" or is this the packing fraction of the particles in an arbitrarily defined volume, not considering agglomeration? Additionally, with the Lewis-Nielsen model, I am running into issues with the ψ function. This function is defined as ψ = 1 + {[(1- øm) / øm2] * Vf } where I'm pretty sure Vf is the volume fraction of the filler particles in the composite. The problem I have with the ψ function is that with this definition, as I increase my volume fraction, the ψ value increases too. At some point, around 75% volume fraction (BN/polymer) the thermal conductivity explodes up to 160,000 W/mK, (which obviously doesn't make sense). I found another definition for ψ from McGee and McCullough (S. McGee and R L McCullough, Polymer Composites, 2, 149 (1981). That defines ψ as ψ = 1 + (ø1 / øm) * {øm*ø2 + (1 - øm)ø1} However this ψ was proposed for situations in which the modulus of the filler is much larger than that of the polymer (that is the case with Boron Nitride and just about any polymer or resin) and if the Einstein coefficient is much greater than 1.0. (With spherical particles the Einstein coefficient is presumably 2.5, and with another particle that I am using (500nm platelets), I have assumed an Einstein coefficient of roughly 5.5. The problem(s): First: my original ψ function increases as my Volume fraction increases. My 2nd ψ function decreases as my Volume fraction increases. Second: With the original ψ function my calculated thermal conductivity values explode around 77.8 weight % or 65.6 Vf This doesn't make sense, as it doesn't even correlate with the maximum packing fraction provided of (.632) Third: Using the second ψ function, my thermal conductivity values are calculated from 0% to 100% BN/Polymer and the maximum and minimum K-values are exactly what you'd expect, at 0% BN, the value is that of pure polymer, and at 100% the bulk K-value of BN. However if I change the Einstein coefficient in any way, my thermal conductivity values change to be unrealistic For Example: When I change my Einstein coefficient to match my new particles, the theoretical calculated thermal conductivity increases to just shy of double the possible thermal conductivity of the Boron Nitride particles. (100% BN in the composite, thereby making the K-value equal to that of the bulk material. Therefore making any calculated value higher than this impossible.) Does anyone have any experience with this, and if so, can you possibly clarify to me what I'm confused about? - (I have a volume fraction of BN that is higher than my theoretical maximum packing fraction. How does this make sense?)
 P: 26 Oh, and if anyone has any idea where I can find the original derivations for the equations in the Lewis-Nielson, or Halping-Tsai equations, I would sincerely appreciate it. I have been using Nielsen's book "Mechanical Properties of Polymers and Composites" for a lot of my research, but Nielsen fails to address the origins of the theory for his model, and as a Physicist, I find it significantly easier to understand what I am working with, when I can see the origin.
 P: 26 A related update on this. Following the Nielsen model, when I decrease the maximum packing fraction of the filler particles in the composite, the calculated thermal conductivity increases. To me, this was surprising, as I expected that a higher packing fraction (of a particle with significantly higher thermal conductivity) would improve the thermal conductivity value, not decrease it. To anyone who reads this and is familiar, am I right to be surprised by this result? To check this, I am learning some simulation software right now so that I can create a few simulations to run and see what happens based on the same type of physical scenario.

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