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A Free Volume Theory of Polymer Dynamics
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- Author / Creator
- Wong, Chi Pui Jeremy
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It is not trivial to understand the underlying principle of polymer dynamics due to its wide range of applications in our daily lives. This thesis mainly focuses on the theory and simulation of polyethylene melts, which are not only restricted to the classic linear structure, but also extended to ring and four-arm symmetrical star structures. The size dependence of the dynamic properties of these polyethylene molecules, such as diffusivity and viscosity, demonstrates a crossover from unentangled to entangled regimes. In the literature, there are two classic models, known as the Rouse model and reptation model, only for polyethylene with linear structure, which accounts for the crossover in the dynamics of linear polyethylene as a function of chain length ($N$) at a particular temperature.
The Rouse model is a simplified version of Brownian dynamics (BD) model for unentangled polymers, in which the inertia term is neglected. If such inertia term is included, it is then possible to compute the velocity time correlation function using the eigenvalue and eigenfunction method. The eigenfunctions can then be used to construct different time correlation functions. Such analysis can also be applied to ring and star polymers, by modifying the resultant matrix equation in such model. It was also found that by incorporating a finite equilibrium length in the harmonic bond stretching term in Rouse model, it becomes a nonlinear BD model, which makes the equation of motion nonlinear and the analytical derivation of eigenfunctions impossible. This is also true in MD simulation, which explicitly includes all the highly nonlinear atomic force as a result of the bonded and non-bonded potentials in the equation of motion. Proper orthogonal decomposition (POD), which is a method that can generate a reduced ordered model, was applied to the numerical data so as to obtain `eigenfunctions', which are later regarded as eigenmodes, for calculating relaxation times and viscosities. With the aid of POD analysis, it was observed that the viscosities calculated from nonlinear BD model and MD simulations concur with one another.
Another notable fact is that the diffusivity of linear polyethylene melt as a function of chain length as predicted by Rouse model is $D_{cm}\sim N^{-\nu}$ and $\eta\sim N^{\nu}$, with the exponent $\nu =1$, whereas in reality, $\nu$ is larger than $1$ and it becomes larger when temperature decreases with $\nu=2.6$ at 343.5 K for diffusivity. The same trend was also observed for viscosity with $\nu=1.8$.
In the same way as the Rouse model, the classic reptation model for linear polyethylene predicts that $D_{cm}\sim N^{-2}$ at $T=448$ K, whereas a stronger exponent was experimentally observed having a value of $-2.2$. Similarly, based upon such model, it can be calculated that $\eta\sim N^{3}$, whereas in reality, $\eta\sim N^{3.4}$ for linear polyethylene. The reptation model is based on the assumption that a polymer chain is reptating through a tube as defined by a number of fixed obstacles, known as entanglements. However, a foundation, i.e., how does the repulsive force among the chains give rise to the entanglements, leading to a reptative motion?, for the entanglement concept in such model is still lacking. In addition, to our knowledge, the quantification of the entanglements for identifying the primitive path is only possible in linear polymer, but not ring polymer.
With regard to this, an alternative free volume theory was proposed, for which many parameters can be theoretically obtained using the generic van der Waals' equation of state and the distribution function theory of polymer melts. Different radial distribution functions in this case can be obtained either by either MD simulation or more elegantly the Polymer Reference Interaction Site Model (PRISM). The free volume theory proposed in this work, which only considers the amount of free volume distributed within the macromolecules, can account for the crossover in the dynamics of both linear, ring and star polymers.
In summary, this work shed light on the theoretical approach of evaluating the dynamic properties of polymers with linear, ring as well as star structures. The proposed free volume theory also offers an alternative way of understanding the diffusivity and viscosity of both unentangled and entangled polymer melts within a particular range of molecular weight.
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- Graduation date
- Spring 2021
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- Type of Item
- Thesis
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- Degree
- Doctor of Philosophy
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- License
- This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.