Modeling intra- and intermolecular correlations for linear and branched polymers using a modified test-chain self-consistent field theory

A modified test-chain self-consistent field theory (SCFT) is presented to study the intra- and intermolecular correlations of linear and branched polymers in various solutions and melts. The key to the test-chain SCFT is to break the the translational symmetry by fixing a monomer at the origin of a coordinate. This theory successfully describes the crossover from self-avoiding walk at short distances to screened random walk at long distances in a semidilute solution or melt. The calculations indicated that branching enhances the swelling of polymers in melts and influences stretching at short distances. The test-chain SCFT calculations show good agreement with experiments and classic polymer theories. We highlight that the theory presented here provides a solution to interpret the polymer conformation and behavior under various conditions within the framework of one theory.

Figure 1

Schematic of linear and branched architectures [5].

Figure 2

Illustration to calculate the local density probability of a three-arm structure. In this example, the molecule is fixed at $t_{\mathrm{fix}}$. As $q3$ is solved from the end of the chain and reaches $t_{\mathrm{fix}}, q3$ is reset to an initial condition, $q3(r;t=t_{\mathrm{fix}};t_{\mathrm{fix}})=\delta \left(r\right)$.

Figure 3

Intramolecular density profiles of a linear single molecule for different values of excluded volume $\overline{v}$ on log-log scales. (a) Intramolecular density profiles of a linear molecule from the center. (b) Intramolecular density profiles of a linear molecule from the end. The length of the molecule $N=50$.

Figure 4

Intramolecular density profiles of a 4-star polymer with $N_{\mathrm{arm}}=25$ for different values of excluded volume $\overline{v}$ calculated (a) from the joint and (b) from the end.

Figure 5

Density distributions of $f$-star $N_{\mathrm{arm}}=25$ and $\overline{v}=0.1$ under a dilution solution condition calculating from (a) the joint and (b) the end, respectively. $N_{\mathrm{arm}}=25, N_{\mathrm{star}}=100$.

Figure 6

Single-chain end-to-end distribution function of (a) a linear molecule ($N=50$) and (b) a 4-star ($N=100$) by SCFT as a function of the reduced distance $x=\frac{r}{\sqrt{\langle R^2\rangle }}$. The Gaussian model is also provided for comparison.

Figure 7

Density profiles of a linear polymer with $N=50$ and ${\overline{\rho }}_{\mathrm{b}}=0.2$ in melts. The thin solid, dashed, and dash-dotted lines correspond to the total, intramolecular, and intermolecular density profiles with $\overline{v}=1$, respectively. The thick solid, dashed, and dash-dotted lines are the total, intramolecular, and intermolecular density profiles with $\overline{v}=100$, respectively.

Figure 8

Intramolecular density profiles for linear polymers in various solutions. (a) ${\overline{\rho }}_{\mathrm{b}}=0.2$; $\overline{v}$ varies from 0.01 to 100. The slope of the density profile on the log-log scales is between $-4/3$ and $-1$ at $\overline{r}\approx 0.1$. (b) $\overline{v}=1$; ${\overline{\rho }}_{\mathrm{b}}$ varies from 0.02 to 2.

Figure 9

Scaling analysis of the monomer density and correlation lengths versus the excluded volume in concentrated solutions and melts of linear polymers. (a) Scaling dependence of the scaled monomer density $\overline{\rho }\left(\overline{r}\right)/{\overline{r}}^{\alpha }$ on the excluded volume $\overline{v}$ at $\overline{r}=0.1$ in solutions with ${\overline{\rho }}_{\mathrm{b}}=0.2$ and $N=50$. (b) Scaling dependence of correlation lengths $\overline{\xi }$ on $\overline{v}$ in solutions. Circles are obtained from the intramolecular densities of solutions in Fig. 8. Crosses are obtained assuming that the intramolecular density profile in a melt is the same as in the dilute solution (Fig. 3). Inset in (b): The way to obtain $\overline{\xi }$, where $\rho \left(\overline{\xi }\right)={\overline{\rho }}_{\mathrm{b}}$, from the intramolecular density profiles. Dashed line: $\overline{v}=10, {\overline{\rho }}_{\mathrm{b}}=0.2$. Solid line: a dilute solution with $\overline{v}=10$.

Figure 10

Density profiles from the joint of 4-star chains with $N=100$. (a) ${\overline{\rho }}_{\mathrm{b}}=0.2$ and $\overline{v}$ varies. (b) $\overline{v}=1$ and ${\overline{\rho }}_{\mathrm{b}}$ varies.

Figure 11

Intramolecular $\omega \left(k\right)$ and intermolecular $h\left(k\right)$ structure factors of 4-star molecules ($N_{\mathrm{star}}=100$). The Gaussian chain is also shown in (a) for comparison.

Figure 12

Static structure factors of 4-star polymers with $N_{\mathrm{star}}=100$ in different solutions.