Controlling solvent quality by time: Self-avoiding sprints in nonequilibrium polymerization

A fundamental paradigm in polymer physics is that macromolecular conformations in equilibrium can be described by universal scaling laws, being key for structure, dynamics, and function of soft (biological) matter and in the materials sciences. Here we reveal that during diffusion-influenced, nonequilibrium chain-growth polymerization, scaling laws change qualitatively, in particular, the growing polymers exhibit a surprising self-avoiding walk behavior in poor and $\theta$ solvents. Our analysis, based on monomer-resolved, off-lattice reaction-diffusion computer simulations, demonstrates that this phenomenon is a result of (i) nonequilibrium monomer density depletion correlations around the active polymerization site, leading to a locally directed and self-avoiding growth, in conjunction with (ii) chain (Rouse) relaxation times larger than the competing polymerization reaction time. These intrinsic nonequilibrium mechanisms are facilitated by fast and persistent reaction-driven diffusion (“sprints”) of the active site, with analogies to pseudochemotactic active Brownian particles. Our findings have implications for time-controlled structure formation in polymer processing, as in, e.g., reactive self-assembly, photocrosslinking, and three-dimensional printing.

Figure 1

Simulation and kinetics of a growing polymer chain. (a) Schematic representation of the growth step of a polymer chain (blue) with an active center (AC, yellow) with reactive radius $R$ and reaction propensity $\lambda$. During a reaction a bond is formed between a free monomer of size $\sigma$ (green) and the AC to propagate the chain. The newly added monomer spans up an angle $\varphi$ by the terminal two bond vectors ${\vec{b}}_{n-1}$ and ${\vec{b}}_n$ during bond formation. The newly attached monomer becomes the new active center. (b) Concentration of free monomers $\rho \left(t\right)/{\rho }_0$ scaled by its initial value ${\rho }_0{\sigma }^3=0.3$ for fast (red; $\lambda ={10}^5{\tau }_B^{-1}$), intermediate (magenta;${10}^2{\tau }_B^{-1}$), and slow (blue; $\lambda ={10}^1{\tau }_B^{-1}$) reactions versus time $t$ (scaled by the Brownian timescale ${\tau }_B={\sigma }^2/D_0$) at $\theta$ conditions, $\varepsilon ={\varepsilon }_{\theta }$. Symbols represent simulated data, the lines show the corresponding fits to initial times according to the first order rate equation (2). (c) Snapshots of fast (red) and slowly (blue) growing chains. (d) Total diffusion coefficient $D(\lambda ,\rho )$ versus reaction propensity $\lambda$ for perfectly ideal ($\varepsilon =0$) and real chain $\theta$ conditions, $\varepsilon ={\varepsilon }_{\theta }$ (simulation: symbols; dash-dotted gray and orange line: Numerical solutions from the set of coupled Eqs. (3) and (4) for $D_{\mathrm{tb}}=D_m/N$ with $\alpha =1.0$ and $D_{\mathrm{tb}}=D_m$ with $\alpha =1.41$, respectively). (e) Scaled reaction rate $k(\lambda ,\rho )$ for the same parameters [simulation: Symbols; dash-dotted gray and orange line: Numerical solutions from the set of coupled Eqs. (3) and (4) for $D_{\mathrm{tb}}=D_m/N$ with $\alpha =1.0$ and $D_{\mathrm{tb}}=D_m$ with $\alpha =1.41$, respectively]; horizontal dashed black line: Smoluchowski bimolecular reaction rate $k_S=8\pi D_{0}R$.

Figure 2

Scaling behavior of growing polymer chains. [(a) and (b)] End-to-end distances for growing chains $R_{\mathrm{ee}}\left(N\right)$ in perfectly ideal ($\varepsilon =0$) and real-chain $\theta$ (${\varepsilon }_{\theta }$) solvents for slow ($\lambda {\tau }_B=1$) and fast ($\lambda {\tau }_B={10}^5$) rates and for densities ${\rho }_0=0.125$ and $0.3{\sigma }^{-3}$, compared with the expected equilibrium behavior for good solvents $\nu =3/5$ (black dashed), $\theta$ solvents $\nu =1/2$ (black dash-dotted), and poor solvents $\nu =1/3$ (black dotted). (c) Summary of scaling exponent $\nu (\lambda ,{\rho }_0)$ in dependence of the reaction propensity $\lambda$ for the two densities and the two solvent qualities as in (a) and (b).

Figure 3

Structural nonequilibrium properties of growing polymer chains for ${\rho }_0{\sigma }^3=0.125$. (a) Number of nonbonded chain beads within a distance of $1.5\sigma$ as measure for the overlap $N_{\mathrm{overlap}}\left(N\right)$. The nonequilibrium results for fast and slow growth are compared with the degree of overlap for equilibrium ideal chains (black, dash-dotted line) and equilibrium chains with essentially repulsive, SAW-like, pairwise interactions of $\varepsilon =0.1k_{B}T$ (black, dotted line). (b) Normalized radial densities profiles $\rho (r,t)/\rho \left(t\right)$ around the active center for different reaction conditions compared with the low-density equilibrium approximation of $g\left(r\right)$ for ${\varepsilon }_{\theta }$ (black, dashed line) and the theoretical profile in the steady-state diffusion-controlled (“Smoluchowski”) limit $\rho (r,t)=\rho \left(t\right)(1-R/r)$ (black, dash-dotted line). (c) Schematic representation of directed growth of the (yellow) AC of a polymer chain away from the monomer-depleted zones (dashed-line circles) around the reacted blue polymer beads. The green spheres represent free monomers. (d) Probability distribution of bond angles $P\left(\varphi \right)/sin\varphi$ between the new ${\vec{b}}_n$ and the previously formed bond vectors ${\vec{b}}_{n-1}$. The distribution is normalized by $sin\varphi$ to account for the isotropic azimuthal distributions of vectors in 3D space. The black, dash-dotted line represents a homogeneous distribution among all possible angles.

Figure 4

Summary of conformational scaling and state diagram. (a) Scaling exponent $\nu$ as a function of the solvent quality (expressed by interaction paramer $\varepsilon$) for fast growing (red square), slowly growing (blue diamond), and nongrowing polymer chains in equilibrium (gray circle), all at ${\rho }_0{\sigma }^3=0.125$. The black dash-dotted, dashed, and dotted lines serve as orientation for reference exponents $\nu =3/5$, $1/2$, and $1/3$, respectively. (b) State diagram for growing ideal polymer chains ($\varepsilon =0$). The circles, crosses, and squares represent exponents $\nu <0.525$ ($\theta$ solvent), $0.525\le \nu <0.55$ (intermediate), and $\nu \ge 0.55$ (good solvent), respectively. The smooth contours in the back represent the ratio between reaction timescale ${\tau }_{\mathrm{react}}$ and the segmental relaxation ${\tau }_{\mathrm{relax}}$ time of a segment with $N/p^{*}=s^{*}=5$. Red regions illustrate chain relaxation times ${\tau }_{\mathrm{relax}}\gg {\tau }_{\mathrm{react}}$, while blue regions depict ${\tau }_{\mathrm{relax}}\ll {\tau }_{\mathrm{react}}$. The bright-white thin region in between, together with the black dashed line corresponds to the boundary ${\tau }_{\mathrm{relax}}={\tau }_{\mathrm{react}}$. The horizontal black dotted line represents the critical Smoluchowski percolation density ${\rho }_{\mathrm{crit}}=0.43{\sigma }^{-3}$, as predicted from the self-consistent Smoluchowski approach, Eq. (5).

Figure 5

Determination of the $\theta$ point for different $\varepsilon$ for different densities ${\rho }_0$. Panels (a) and (c) show the mean simulated end-to-end distances $R_{\mathrm{ee}}\left(N\right)\propto N^{\nu }$ and the squared radius gyration $R_{\mathrm{gyr}}^2\left(N\right)\propto N^{2\nu }$ for ${\rho }_0{\sigma }^3=0.125$ and 0.3, respectively. Symbols represent simulation results for a given chain length $N$ and interaction parameter $\varepsilon$. Lines represent the corresponding fits following Eq. (A1), respectively. Dotted, dashed, and dash-dotted line represent theoretical scaling behavior for bad ($\nu =1/3$), $\theta$ ($\nu =1/2$), and good solvent conditions ($\nu =3/5$), respectively. Panels (b) and (d) show the $\varepsilon$ dependency of the scaling exponent. Circles, squares, and crosses represent fitted scaling exponents $\nu \left(\varepsilon \right)$ [Eq. (A1)] from simulated data for $R_{\mathrm{ee}}$ (red), $R_{\mathrm{gyr}}^2$ (gray), and the mean of the two (blue), respectively. The lines represent the corresponding sigmoidal fits using Eq. (A2). Values for ${\varepsilon }_{\theta }$ are represented by the brown dashed lines. Black dotted, dashed, and dash-dotted line represent theoretical scaling behavior for bad ($\nu =1/3$), $\theta$ ($\nu =1/2$), and good solvent conditions ($\nu =3/5$), respectively.

Figure 6

Growth at good ($\varepsilon =0.1k_{B}T$) and poor solvent conditions ($\varepsilon =1.0k_{B}T$) for densities ${\rho }_0{\sigma }^3=0.125$ and 0.4. (a) Simulated diffusion coefficients $D$ (symbols) for different reaction rates $k$ and initial densities ${\rho }_0$. The orange line and the dash-dotted gray line represent the fastest and slowest theoretical limit for ${\rho }_0{\sigma }^3=0.4$ with $D_{\mathrm{tb}}=D_m$ and $\alpha =1.41$ and for ${\rho }_0{\sigma }^3=0.125$ with $D_{\mathrm{tb}}=D_m/N$ and $\alpha =1.0$, respectively. (b) Simulated reaction rate constants $k$ (symbols) for different reaction propensities $\lambda$ and initial densities ${\rho }_0$. The orange line and the dash-dotted gray line represents the fastest and slowest theoretical limit for ${\rho }_0{\sigma }^3=0.4$ with $D_{\mathrm{tb}}=D_m$ and $\alpha =1.41$ and for ${\rho }_0{\sigma }^3=0.125$ with $D_{\mathrm{tb}}=D_m/N$ and $\alpha =1.0$, respectively, and the dashed black line the fastest Smoluchowksi reaction rate $k_S=8\pi D_{0}R$. (c) End-to-end distance $R_{\mathrm{ee}}\left(N\right)$ at a density ${\rho }_0{\sigma }^3=0.125$ for good and poor solvent conditions during fast and slow reactions. Symbols show selected simulated data points, colored dashed lines represent corresponding fits with $R_{\mathrm{ee}}\left(N\right)\propto N^{\nu }$. Black dash-dotted, dashed, and dotted lines represent theoretical curves for $\nu =3/5$ (good solvent), $\nu =1/2$ ($\theta$ solvent), and $\nu =1/3$ (poor solvent). (d) Size scaling exponents $\nu$ for different propensities $\lambda$. (e) Normalized densities profiles $\rho (r,t)/\rho \left(t\right)$ around the active site for different reaction conditions compared with the theoretical profile in the diffusion-controlled (“Smoluchowski”) limit $\rho (r,t)=\rho \left(t\right)(1-R/r)$ (black, dash-dotted line) [47, 65] and the low-density equilibrium approximation $g\left(r\right)$ for ${\varepsilon }_{\theta }/\left(k_{B}T\right)=1.0$ (black, dashed line). (f) snapshots of chains at different solvent conditions for a density ${\rho }_0{\sigma }^3=0.125$ at fast ($\lambda {\tau }_B={10}^5$) and slow ($\lambda {\tau }_B={10}^0$) reaction conditions with yellow-colored active sites.

Figure 7

Growth in an ideal solvent at different initial densities. (a) Diffusion coefficients $D$ (symbols) for different reaction rates $k$ and initial densities ${\rho }_0$. The orange line and the dash-dotted gray line represent the fastest and slowest theoretical limit for ${\rho }_0{\sigma }^3=1.0$ with $D_{\mathrm{tb}}=D_m$ and $\alpha =1.41$ and for ${\rho }_0{\sigma }^3=0.01$ with $D_{\mathrm{tb}}=D_m/N$ and $\alpha =1.0$, respectively. (b) Simulated reaction rate constants $k$ (symbols) for different reaction propensities $\lambda$ and initial densities ${\rho }_0$. The orange line and the dash-dotted gray line represents the fastest and slowest theoretical limit for ${\rho }_0{\sigma }^3=1.0$ with $D_{\mathrm{tb}}=D_m$ and $\alpha =1.41$ and for ${\rho }_0{\sigma }^3=0.01$ with $D_{\mathrm{tb}}=D_m/N$ and $\alpha =1.0$, respectively, and the dashed black line the fastest Smoluchowksi reaction rate $k_S=8\pi D_{0}R$. (c) End-to-end distance at different densities for an intermediate reaction frequency $\lambda {\tau }_B={10}^3$. Symbols show selected simulated data points, colored dashed lines represent corresponding fits with $R_{\mathrm{ee}}\left(N\right)\propto N^{\nu }$. Black dash-dotted, dashed, and dotted lines represent theoretical curves for $\nu =3/5$ (good solvent), $\nu =1/2$ ($\theta$ solvent), and $\nu =1/3$ (poor solvent). (d) Size scaling exponents $\nu$ for different propensities $\lambda$. (e) Normalized densities profiles $\rho (r,t)/\rho \left(t\right)$ around the active site for different reaction conditions compared with the theoretical profile in the diffusion-controlled (“Smoluchowski”) limit $\rho (r,t)=\rho \left(t\right)(1-R/r)$ (black, dash-dotted line) [47, 65]. (f) Snapshots of fast growing chains ($\lambda {\tau }_B={10}^5$) with yellow-colored active sites.