Quantification of interaction and topological parameters of polyisoprene star polymers under good solvent conditions

Mass fractal scaling, reflected in the mass fractal dimension $d_f$, is independently impacted by topology, reflected in the connectivity dimension $c$, and by tortuosity, reflected in the minimum dimension $d_{min}$. The mass fractal dimension is related to these other dimensions by $d_f=c{d}_{min}$. Branched fractal structures have a higher mass fractal dimension compared to linear structures due to a higher $c$, and extended structures have a lower dimension compared to convoluted self-avoiding and Gaussian walks due to a lower $d_{min}$. It is found, in this work, that macromolecules in thermodynamic equilibrium display a fixed mass fractal dimension $d_f$ under good solvent conditions, regardless of chain topology.  These equilibrium structures accommodate changes in chain topology such as branching $c$ by a decrease in chain tortuosity $d_{min}$. Symmetric star polymers are used to understand the structure of complex macromolecular topologies. A recently published hybrid Unified scattering function accounts for interarm correlations in symmetric star polymers along with polymer-solvent interaction for chains of arbitrary scaling dimension. Dilute solutions of linear, three-arm and six-arm polyisoprene stars are studied under good solvent conditions in deuterated $p$-xylene. Reduced chain tortuosity can be viewed as steric straightening of the arms. Steric effects for star topologies are quantified, and it is found that steric straightening of arms is more significant for lower-molecular-weight arms. The observation of constant $d_f$ is explained through a modification of Flory-Krigbaum theory for branched polymers.

Figure 1

Schematic of a six-arm PI star polymer of fractal dimension $d_f$ and composed of $z$ Kuhn units of length $l_k$. The structure can be decomposed into two sets of conjugate parameters describing connectivity $\left(s,c\right)$ and tortuosity $\left(p,d_{\min }\right)$. The connective path is composed of $s$ units and has its nascent fractal dimension called the connective dimension $c$; and describes the branching characteristics in the chain, shown in straight black dashed lines. Any two branches of a symmetric star form a minimum path across the whole structure composed of $p$ Kuhn units with a nascent fractal dimension called the minimum dimension of $d_{\min }$. $p$ describes the average topological tortuosity and is shown in units with dark borders.

Figure 2

(a) SANS data from $\sim 1\mathrm{wt}\% 38k$-arm star polyisoprene polymer solutions in xylene for linear, three-arm, four-arm, and six-arm polymers shown in black and dark gray dots, gray dash-dots, and light gray dashes, respectively. (b) SANS from a solution of $\sim 1\mathrm{wt}\%$ for four-arm star polyisoprene in xylene for $10.5k$, 38k, and $46k$ arms in black triangles, dark gray squares, and light gray circles with respective Hybrid Unified Fits [Eq. (19)] in solid contrast lines. The data and their respective fits are offset for visual clarification. Slopes of −5/3 and −1 are also shown for reference.

Figure 3

(a) Mass $z$, (b) radius of gyration, $R_g$, and (c) persistent lengths $l_p$ as functions of functionality $f$.

Figure 4

(a) Flory-Huggins interaction parameter $\chi$ and (b) second virial coefficient $A_2$ as functions of functionality $f$.

Figure 5

(a) Fractal dimension $d_f$, (b) connectivity dimension $c$, and (c) minimum dimension $d_{min}$ as functions of functionality $f$.

Figure 6

(a) Minimum path $p$, (b) connectivity path $s$, (c) $s/f$, and (d) meandering mole fraction $\left({\varphi }_M\right)$ as functions of functionality $f$.

Figure 7

(a) ${\varphi }_{\mathrm{Si}}$, and (b) $s/z$ as functions of functionality $f$.

Figure 8

Schematic demonstrating changes in scaling and thermodynamic parameters with functionality for the same arm length. The scaling model for the six-arm star polymer is compared with that of the Daoud-Cotton model [13]. $l$ and $v$ are the monomer length and excluded volume associated with each monomer.