Is there any way to calculate the molecular weight of a polymer composite using DMA or rheology without knowing the K and alpha?
I need to calculate the molecular weight, or the polymer chain's weight between two crosslinks, in a polymer composite (polymer + non-polymer). Is there any way to estimate this number only using the mechanical/rheological properties of the composite? (and potentially knowing the mechanical properties of the pure polymer and the non-polymer parts)
I also know some equations exist to estimate the molecular weight from the storage and loss moduli. Yet, I have not seen them being applied to a composite material through, say, the volume fraction.
Thank you!
Dear Siavash,
you can calculate the molecular weight between cross-links according to the rubber elasticity theory from the rubbery plateau modulus. The procedure is described in detail here:
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It works for composites and nanocomposites too. See Fig. 4D in our recent paper:
Article Photoactivity, conversion kinetics, nanoreinforcement, post-...
Thank you so much Petr Lepcio for your answer.
So, in a composite material, the other phase potentially affects the composites' modulus both directly through its own modulus and indirectly through affecting the network crosslinks. I did not understand how you decomposed these two effects in your work.
The modulus contributions in composites are:
volume replacement (for all fillers stiffer than the matrix)
stress concentration (for anisotropic microfillers only)
nanoreinforcement (for nanocomposites only).
Using standard modulus-predicting equations, you must decouple the matrix modulus from the composite modulus to rule out the volume replacement and stress concentration. The exact equation depends on the filler type and orientation. The nanoreinforcement is usually considered to increase the effective crosslinking density. These are not covalent crosslinks, but only physical ones formed by polymer chains adsorbed on the nanoparticles.
Symmetrical nanoparticles do not allow stress transfer because they are too small, well below the critical length, and not anisotropic. Volume replacement is usually neglected because the replaced volume in most nanocomposites is very low, typically
Wow. That's an interesting approach Petr Lepcio .
Just a few follow-up questions on the side,
are you considering the finite strain, viscoelasticity (or loss modulus effect), and frequency dependency of the modulus when you consider the effect of the fillers, or do you ignore those to simplify comparison?
Sure, finite strain. How would you even consider infinite strain in solids? Anyway, we use a very low strain amplitude of 0.002%, so we can assume that the modulus is strain independent (see the stress-strain curve for rubber elasticity). The elastic modulus forms a plateau above Tg, its value is theoretically constant (according to rubber elasticity theory), but it practically varies slightly due to side processes (post-curing, degradation). Therefore, we pick a fixed temperature where we determine the E' for a whole series of samples. Ideally, this is done at the minimum of tan delta (or nearby) so that the E'' contribution to complex modulus is negligible. See the attached figure - it is Fig. 1 from the abovementioned paper (CC BY Open Access, feel free to share).
Article 3D printing and post-curing optimization of photopolymerized...
The temperature selection is material-specific, as various materials have rubbery plateaus at different temperatures. The frequency dependence is analogous to the temperature one according to the time-temperature superposition principle, giving you a flat plateau in the rubbery region. Thus, you do not have to consider the frequency effect either.
By the way, remember that if you deal with fibrous composites, you must consider the fiber-matrix adhesion strength as an efficiency factor for stress transfer.
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