In my previous post, I developed a GPC post-processing program that uses the continuous wavelet transform (CWT) to detect and correct baselines, and the MH equation to convert GPC data into a molecular weight distribution. In this follow-up work, I determine the molecular weight distributions in a two-step synthesis process. In the first step, polymers […]

Consider a polymer chain dissolved in a solvent, where its behavior is influenced by the Flory parameter, denoted by $\chi$. As the interaction parameter $\chi$ varies, the chain undergoes several conformational transitions: • For $\chi > \chi_c$, the polymer collapses into a dense, globular structure with a radius of gyration scaling as $R_g \sim N^{1/3}$. • At […]

This analysis focuses on the probability distribution function, $P_i(\mathbf{r}_i-\mathbf{r}_\mathrm{cm})$, of the $i$th segment relative to the center of mass in an ideal chain. An ideal chain is modeled as a multidimensional random walk with independent steps. The distribution of each step, with a mean length of $b$, is assumed to be Gaussian: $P(\mathbf{r})\sim\mathcal{N}(0,b^2)$. Let $\mathbf{b}_i=\mathbf{r}_{i+1}-\mathbf{r}_{i}$ […]

Introduction Saddle point methods are widely used to estimate integrals of the form: $I = \int \exp\left(-f(x)\right)\mathrm{d}x$ where the function $f(x)$ can be approximated by the first two terms of its Taylor series expansion around a point $x_0$: $f(x)\approx f(x_0) + f^\prime (x_0)(x-x_0) + \frac{1}{2}f^{\prime\prime}(x_0)(x-x_0)^2$ The integral is then approximated by its value at the […]

The Boltzmann superposition principle states that the strain response of a viscoelastic material is a superposition of the responses to all previous stress histories. Assuming no stress before time $t=0$, the constitutive equation can be written as: $\begin{equation} \gamma(t)=\int_0^t J(t-t^\prime)\dot{\sigma}(t^\prime)\mathrm{d}t^\prime \end{equation}$ Applying the Laplace transform to this equation yields: $\begin{align} \hat{\gamma}(s)&=\hat{J}(s)\hat{\dot{\sigma}}(s)\\ &=\hat{J}(s)\left(s\hat{\sigma}(s)-\sigma(0^{-})\right) \\ &=\hat{J}(s)\left(s\hat{G}(s)\hat{\dot{\gamma}}(s)-\sigma(0^{-})\right)\\ &=\hat{J}(s)\left(s\hat{G}(s)(s\hat{\gamma}(s)-\gamma(0^-))-\sigma(0^{-})\right)\\ […]