Non-Gaussian polymers described by alpha-stable chain statistics: Model, effective interactions in binary mixtures, and application to on-surface separation

The Gaussian chain model is the classical description of a polymeric chain, which provides analytical results regarding end-to-end distance, the distribution of segments around the mass center of a chain, coarse-grained interactions between two chains and effective interactions in binary mixtures. This hierarchy of results can be calculated thanks to the $\alpha$ stability of the Gaussian distribution. In this paper we show that it is possible to generalize the model of Gaussian chain to the entire class of $\alpha$-stable distributions, obtaining the analogous hierarchy of results expressed by the analytical closed-form formulas in the Fourier space. This allows us to establish the $\alpha$-stable chain model. We begin with reviewing the applications of Levy flights in the context of polymer sciences, which include: chains described by the heavy-tailed distributions of persistence length; polymers adsorbed to the surface; and the chains driven by a noise with power-law spatial correlations. Further, we derive the distribution of segments around the mass center of the $\alpha$-stable chain and construct the coarse-grained interaction potential between two chains. These results are employed to discuss the model of binary mixture consisting of the $\alpha$-stable chains. In what follows, we establish the spinodal decomposition condition generalized to the mixtures of the $\alpha$-stable polymers. This condition is further applied to compare the on-surface phase separation of adsorbed polymers (which are known to be described with heavy-tailed statistics) with the phase separation condition in the bulk. Finally, we predict the four different scenarios of simultaneous mixing and demixing in the two- and three-dimensional systems.

Figure 1

Top: schematic representation of a polymer adsorbed to the surface, dots indicate the adsorbed segments. Bottom: planar trajectories connecting the subsequent adsorbed segments. (a) In the strong adsorption regime freely-diffusing loops are short and subsequent adsorbed nodes are found close to each other. The radius of gyration parallel to the surface scale as $R_{g,\left|\right|}\propto N^{3/4}$ [25]. (b) In the weak adsorption limit the long, freely diffusing loops are numerous and introduce Levy flights into the planar trajectory. $R_{g,\left|\right|}$ scales as $\propto N^{3/5}$ [25].

Figure 2

The probability distribution $S_n$ of finding the $n$-segment-long linearized fragment in a chain driven by the spatially correlated noise. Each plot represents the results for noise amplitude $2kT$ and correlation length $\lambda$. Triangles: exponential noise correlation function $\langle \mathit{}\mathbf{\xi }\left(\mathbf{r}\right)\mathit{}\mathbf{\xi }(\mathbf{r}+\Delta \mathbf{r})\rangle \propto exp(-|\Delta \mathbf{r}|/\lambda )$; dots: Cauchy noise correlation function $\langle \mathit{}\mathbf{\xi }\left(\mathbf{r}\right)\mathit{}\mathbf{\xi }(\mathbf{r}+\Delta \mathbf{r})\rangle \propto {1/[1+(\left|\Delta \mathbf{r}\right|/\lambda )}^2]$. The tail behavior ($n\ge 17$) has been fitted with exponential decay model $S_n=c{e}^{-n/b_i}$ (dashed lines) with $b_i$ given on the plot. For $\lambda \ge 40$ the data has been also fitted with power law $S_n=c{n}^{-(\alpha +1)}$ for $n\ge 17$, indicating the power-law asymptotic behavior.

Figure 3

The spinodal decomposition condition (27) for the binary systems of particles described with the $\alpha$-stable statistics as a function of gyration radii ratio $g=R_{g,1}/R_{g,2}$ and the distribution exponent $\alpha$ for $D=3$ dimensional system. $\tilde{\epsilon }={\epsilon }_{12}/\sqrt{{\epsilon }_{11}{\epsilon }_{22}}$ is the common energy scale. In the region above the plotted surface the binary system undergoes demixing due to the prevalence of the attractive effective interactions, in the region below the surface the effective interaction is repulsive and the system is homogeneous.

Figure 4

Schematic representation of four mixing and demixing scenarios in the binary system consisting of a solution and an adsorbing surface. In the bulk polymers follow the Gaussian statistics (${\alpha }_b=2, D_b=3$). On the surface ($D_s=2$) polymers are described either by ${\alpha }_{ss}=2$ in the strong adsorption limit or ${\alpha }_{ws}=1$ for the weak adsorption.

Figure 5

The exemplary comparison of the phase separation conditions on the surface in the weak adsorption limit ($f_{ws,s}$, solid line) and in the bulk ($f_{b,b}$, dashed line) for the binary system of Gaussian polymers characterized by the persistence length ratio $b_1/b_2=0.1$. $f_{ws,s}$ and $f_{b,b}$ are defined by (30). $f_{ws,s}$ and $f_{b,b}$ divide the plot into four regions: (a) simultaneous mixing on the surface and in the bulk, (b) simultaneous demixing on the surface and in the bulk, (c) mixing on the surface, demixing in the bulk, (d) demixing on the surface, mixing in the bulk.

Figure 6

The density plot of the difference $f_{ws,s}-f_{b,b}$ as a function of persistence length ratio $b_1/b_2$ and the ratio of monomer numbers $N_1/N_2$. $f_{ws,s}-f_{b,b}$ indicates which surface/bulk demixing scenarios are allowed (see Sec. 7 and Fig. 5 for explanation). White meshed region: $f_{ws,s}-f_{b,b}<0$ indicates scenario (d) allowed (demixing on the surface, mixing in the bulk), complement region: $f_{ws,s-}{f}_{b,b}>0$ scenario (c) allowed (mixing on the surface, demixing in the bulk).