Universal size ratios of Gaussian polymers with complex architecture: radius of gyration vs hydrodynamic radius

Abstract

We study the impact of arm architecture of polymers with a single branch point on their structure in solvents. Many physical properties of polymer liquids strongly dependent on the size and shape measures of individual macromolecules, which in turn are determined by their topology. Here, we use combination of analytical theory, based on path integration method, and molecular dynamics simulations to study structural properties of complex Gaussian polymers containing $f^c$ linear branches and $f^r$ closed loops grafted to the central core. We determine size measures such as the gyration radius $R_g$ and the hydrodynamic radii $R_H$, and obtain the estimates for the size ratio $R_g /R_H$ with its dependence on the functionality $f=f^c+f^r$ of grafted polymers. In particular, we obtain the quantitative estimate of the degree of compactification of these polymers with increasing number of closed loops $f^r$ as compared to linear or star-shape molecules of the same total molecular weight. Numerical simulations corroborate theoretical prediction that $R_g /R_H$ decreases towards unity with increasing f. These findings provide qualitative description of polymers with complex architecture in $\theta$ solvents.

Introduction

Polymer macromolecules of complex branched structure attract considerable attention both from academical^1,2 and applied^3,4 perspective, being encountered as building blocks of materials like synthetic and biological gels⁵, thermoplastics⁶, melts and elastomers^7,8. High functionality of polymers provides novel properties with applications in diverse fields like drug delivery⁹, tissue engineering¹⁰, super-soft materials¹¹, and antibacterial surfaces¹² etc. On the other hand, multiple loop formation in macromolecules is often encountered and plays an important role in biological processes such as stabilization of globular proteins¹³ or transcriptional regularization of genes¹⁴. In this concern, it is of fundamental interests to study conformational properties of complex polymer architectures.

In statistical description of polymers, a considerable attention is paid to the universal quantities describing equilibrium size and shape of typical conformation adapted by individual macromolecule in a solvent^15,16. In particular, many physical properties are manifestations of the underlaying polymer conformation, including the hydrodynamic properties of polymer fluids¹⁷, the folding dynamics and catalytic activity of proteins¹⁸ etc. As a size measure of a single macromolecule one usually considers the mean square radius of gyration $R_g^2$, which is directly measurable in static scattering experiments^19,20. Denoting coordinates of the monomers along the polymer chain by $\vec {r}_n$, $n = 1, \ldots ,N$, this quantity is defined as:

$$\begin{aligned} \langle R_g^2 \rangle = \frac{1}{2N^2} \sum _{n, m}\langle (\vec {r}_n-\vec {r}_m)^2 \rangle , \end{aligned}$$

(1)

and is thus given by a trace of gyration tensor $\mathbf {Q}$²¹. Here and below, $\langle (\ldots ) \rangle$ denotes ensemble average over possible polymer conformations. Another important quantity that characterizes the size of a polymer coil is hydrodynamic radius $R_H$, which is directly obtained in dynamic light scattering experiments^22,23,24. This quantity was introduced based on the following motivation²⁵. According to the Stokes–Einstein equation, the diffusion coefficient D of a spherical particle of radius $R_s$ in a solvent of viscosity $\eta$ at temperature T is given by:

$$\begin{aligned} D=\frac{k_BT}{6\pi \eta R_s} \end{aligned}$$

(2)

where $k_B$ is Boltzmann constant. In order to generalize the above relation for the case of molecules of more complex shape, their center-of-mass diffusion coefficient D is given by Eq. (2) with $R_s$ replaced by $R_H$. The latter is given as the average of the reciprocal distances between all pairs of monomers²⁶:

$$\begin{aligned} \langle R_H^{-1} \rangle = \frac{1}{N^2} \sum _{n, m} \left\langle \frac{1}{ |\vec {r}_n-\vec {r}_m|} \right\rangle . \end{aligned}$$

(3)

Namely, $R_H$ is related with the averaged components of the Oseen tensor $\mathbf{H}_{nm}$ characterizing the hydrodynamic interactions between monomers n and m²⁷. To compare $R^2_g$ and $R^{-1}_H$, it is convenient to introduce the universal size ratio

$$\begin{aligned} \rho =\sqrt{ R_g^2} / R_H , \end{aligned}$$

(4)

which does not depend on any details of chemical microstructure and is governed by polymer architecture. In the present paper we restrict our consideration to the ideal (Gaussian) polymers, i.e. monomers have no excluded volume. This to a certain extent corresponds to the behavior of flexible polymers in the so-called $\theta$-solvents. Note that our theoretical approach is not capable to correctly capture structural properties of more rigid branched polymers like dendrimers or molecular bottlebrushes. The intrinsic rigidity of these macromolecules is controlled by steric repulsions between connected branches or grafts. This approach allows to obtain the exact analytical results for the set of universal quantities characterizing conformational properties of macromolecules. In particular, for a linear Gaussian polymer chain the exact analytical result for the ratio (4) in $d=3$ dimensions reads^28,29,30:

$$\begin{aligned} \rho _{\mathrm{chain}}= \frac{8}{3\sqrt{\pi }}\approx 1.5045. \end{aligned}$$

(5)

The universal ratio of a Gaussian ring polymer was calculated in Refs. ^29,31,32 and is given by

$$\begin{aligned} \rho _{\mathrm{ring}} = \frac{\sqrt{2\pi }}{2}\approx 1.2533. \end{aligned}$$

(6)

The validity of theoretically derived ratios $\rho _{\mathrm{chain}}$ and $\rho _{\mathrm{ring}}$ was confirmed in several simulation studies ^30,32,33.

Figure 1

Schematic presentation of rosette polymer topology comprised $f^r=4$ rings (green) and $f^c=8$ linear chains (red) grafted to a central core (black).

Table 1 Literature data for the universal size ratio for different polymer topologies, derived using analytical theory $\rho _{\mathrm{theory}}$ and numerical simulations $\rho _{\mathrm{sim}}$. The theoretical values for tadpol and double ring architectures were calculated on the basis of our general analytical result, cf. Eq. (28).

The distinct example of branched macromolecule is the so-called rosette polymer³⁴, containing $f^c$ linear chains and $f^r$ closed loops (rings), radiating from the same branching point (see Fig. 1). Note that for $f^r=0$ one restores architecture of a star polymer with $f^c$ functionalized linear chains radiating from a central core, for which an exact analytical result is known for the size ratio (Ref.²⁶):

$$\begin{aligned} \rho _{\mathrm{star}}=\frac{8\sqrt{f(3f^c-2)}}{3(f^c)^2\sqrt{\pi }}(\sqrt{2}-1)(\sqrt{2}+f^c). \end{aligned}$$

(7)

The estimates for $\rho _{\mathrm{star}}$ have been also obtained by numerical Monte-Carlo simulations³⁵. Using molecular dynamics (MD) simulations, Uehara and Deguchi derived the universal size ratios for macromolecules such as single ring ($f^c=0$, $f^r=1$), tadpole ($f^r=1$, $f^c=1$) and double ring ($f^r=2$, $f^c=0$)³². The overview of existing literature data for universal size ratios obtained in analytical $\rho _{\mathrm{theory}}$ and numerical $\rho _{\mathrm{sim}}$ investigations are listed in Table 1. Note large discrepancy between previous numerical study of star polymers³⁵ and the theoretical result of Eq. (7). This significant difference between theory and simulations is due to too short chains that were used in Ref. ³⁵ with maximum degree of polymerization $N=150$. As it will be shown the finite-size effect of polymer chains strongly affects measured value of $\rho$. In our numerical study we calculate $\rho$ in the asymptotic limit. For this purpose we simulated long polymer chains with degree of polymerization equal to $N=6400$.

The aim of the present work is to extend the previous analysis of rosette-like polymers³⁴, by thoroughly studying their universal size characteristics. For this purpose we apply the analytical theory, based on path-integration method, and extensive numerical molecular dynamics simulations. The layout of the paper is as follows. In the next section, we introduce the continuous chain model and provide the details of analytical calculation of the universal size ratios $\rho$ for various polymer architectures applying path integration method. In Sect. 3 we describe the numerical model and details of MD simulations. In the same section we present numerical results and compare them with our theoretical predictions. We draw conclusions and remarks in Sect. 4.

Analytical approach

The model

Within the frame of continuous chain model³⁶, a single Gaussian polymer chain of length L is represented as a path $\vec {r}(s)$, parameterized by $0<s<L$. We adapt this model to more complicated branched polymer topologies, containing in general $f^c$ linear branches and $f^r$ closed rings (see Fig. 1). In the following, let us use notation $f=f^c+f^r$ for total functionality of such structure. The weight of each ith path ($i=1,\ldots ,f$) is given by

$$\begin{aligned} W_i={\mathrm{e}}^{-\frac{1}{2}\int \limits _0^L ds \left( \frac{{\mathrm{d}} {\vec{r}_{i}}}{\mathrm{d}s}\right) ^2}. \end{aligned}$$

(8)

The corresponding partition function of rosette polymer is thus:

$$\begin{aligned} Z_{f^c,f^r} = \frac{\int \mathcal{D}\{\vec {r}\} \prod \limits _{j=1}^{f^r} \delta (\vec {r}_j(L){-}\vec {r}_j(0))\prod \limits _{i=1}^{f} \delta (\vec {r}_i(0))\, W_i}{\int \mathcal{D}\{\vec {r}\} \prod \limits _{i=1}^{f} \delta (\vec {r}_i(0)) \, W_i}, \end{aligned}$$

(9)

where $\mathcal{D}\,\{\vec {r}\}$ denotes multiple path integration over trajectories $\vec {r}_i(s)$ ($i=1,\ldots ,f$) assumed to be of equal length $L_i=L$, the first product of $\delta$-functions reflects the fact that all $f^c+f^r$ trajectories start at the same point (central core), and the second $\delta$-functions product up to $f^r$ describes the closed ring structures of $f^r$ trajectories (their starting and end points coincide). Note that (9) is normalised in such a way that the partition function of the system consisting of $f^c+f^r$ open linear Gaussian chains (star-like structure) is unity. The expression for partition function of rosette-like polymer architecture have been evaluated in Ref. ³⁴ and in Gaussian approximation reads:

$$\begin{aligned} Z_{f^c,f^r} =(2\pi L)^{-df^r/2}. \end{aligned}$$

(10)

where d denotes spatial dimensionality. Within the frame of presented model, the expression for the mean square gyration radius from Eq. (1) can be rewritten as

$$\begin{aligned} \langle R_g^2 \rangle = \frac{1}{2(fL)^2} \sum _{i,j=1}^{f}\int _0^L\int _0^{L}\,ds_2\,ds_1 \langle (\vec {r}_i(s_2)-\vec {r}_j(s_1))^2 \rangle , \end{aligned}$$

(11)

whereas the expression (3) for hydrodynamic radius reads:

$$\begin{aligned} \langle R_H^{-1} \rangle = \frac{1}{(fL)^2} \sum _{i,j=1}^{f} \int _0^L\int _0^{L}\,ds_2\,ds_1 \langle |\vec {r}_i(s_2)-\vec {r}_j(s_1)|^{-1} \rangle , \end{aligned}$$

(12)

where $\langle (\ldots ) \rangle$ denotes averaging over an ensemble of all possible configurations defined as:

$$\begin{aligned}&\langle (\ldots ) \rangle = \frac{1}{Z_{f_c,f_r}} \nonumber \\&\quad \times \frac{\int \mathcal{D}\{\vec {r}\} \prod \limits _{j=1}^{f^r} \delta (\vec {r}_j(L){-}\vec {r}_j(0))\prod \limits _{i=1}^{f} \delta (\vec {r}_i(0))(\ldots \,) W_i}{\int \mathcal{D}\{\vec {r}\} \prod \limits _{i=1}^{f} \delta (\vec {r}_i(0)) \, W_i}. \end{aligned}$$

(13)

Calculation of hydrodynamic radius and universal size ratio

The crucial point in the calculation of the hydrodynamic radius is utilization of the following equality³⁷:

$$\begin{aligned} |\vec {r}|^{-1}{ =} (2\pi )^{-d} \int {\mathrm{d}}\vec {k} \,2^{d-1} \pi ^{\frac{d-1}{2}} \Gamma \left( \frac{d-1}{2}\right) k^{1-d}{\mathrm{e}}^{i\vec {r}\vec {k}}. \end{aligned}$$

(14)

where $\Gamma (x)$ is Gamma function. Applying the above expression to Eq. (12) allows to rewrite the mean reciprocal distance from the definition of $R_H$ as

$$\begin{aligned} \langle |\vec {r}_i(s_2)-\vec {r}_j(s_1)|^{-1}\rangle \,&{ =}\, (2\pi )^{-d} \int {\mathrm{d}}\vec {k}\, 2^{d-1} \pi ^{\frac{d-1}{2}}\times \nonumber \\&\quad \times \Gamma \left( \frac{d-1}{2}\right) \, k^{1-d} \langle \xi (s_1,s_2) \rangle \end{aligned}$$

(15)

with notation

$$\begin{aligned} \xi (s_1,s_2) \equiv {\mathrm{e}}^{i\vec {k}(\vec {r}_i(s_2)-\vec {r}_j(s_1))}. \end{aligned}$$

(16)

Below we will apply path integration approach to calculate the mean reciprocal distances.

Exploiting the Fourier-transform of the $\delta$-functions in definition (13)

$$\begin{aligned} \delta (\vec {r}_j(L)-\vec {r}_j(0)) =(2\pi )^{-d}\int {\mathrm{d}}{\vec{q}}_j\, {\mathrm{e}}^{-i\vec {q}_j(\vec {r}_j(L)-\vec {r}_j(0))} \end{aligned}$$

(17)

we get a set of wave vectors $\vec {q}_j$ with $j=1,\ldots ,f^r$ associated with $f^r$ closed loop trajectories, which is an important point in following evaluation. To visualize different contributions into $\langle |\vec {r}_i(s_2)-\vec {r}_j(s_1)|^{-1}\rangle$, it is convenient to use the diagrammatic technique (see Fig. 2). Taking into account the general rules of diagram calculations¹⁵, each segment between any two restriction points $s_a$ and $s_b$ is oriented and bears a wave vector $\vec {p}_{ab}$ given by a sum of incoming and outcoming wave vectors injected at restriction points and end points. At these points, the flow of wave vectors is conserved. A factor $\exp \left( -{{p}_{ab}}^{\,\,2}(s_b-s_a)/2\right)$ is associated with each segment. An integration is to be made over all independent segment areas and over wave vectors injected at the end points.

Figure 2

Diagrammatic presentation of contributions into $\langle R_H^{-1} \rangle$ according to (12). Solid lines are schematic presentation of polymer paths, arrows denote point $s_1$, $s_2$.

To make these rules more clear, let us start with diagram (1), corresponding to the case when both points $s_1$ and $s_2$ are located along any linear arm of rosette polymer. The vector $\vec {k}$ is injected at restriction point $s_1$ and the segment $s_2-s_1$ is associated with factor $\exp \left( -{{k}}^{\,\,2}(s_2-s_1)/2\right)$. Next step is performing integration over k. Passing to d-dimensional spherical coordinates, we have:

$$\begin{aligned} \int {\mathrm{d}}{\vec {k}}\, k^{1-d} f(k^2)= \frac{2\pi ^{d/2}}{\Gamma \left( \frac{d}{2}\right) }\int \mathrm{d} {k}\, f(k^2), \end{aligned}$$

(18)

and thus integration over k can be easily performed

$$\begin{aligned} \int _0^{\infty }\mathrm{d} k\, \mathrm{e}^{-\frac{k^2(s_2-s_1)}{2}}=\sqrt{ \frac{\pi }{2} }(s_2-s_1)^{-1/2}. \end{aligned}$$

(19)

The analytic expression corresponding to contribution from diagram (1) thus reads

$$\begin{aligned} \langle \xi (s_1,s_2)\rangle _{(1)} {=}\frac{(2\pi ^{d+1})^{\frac{1}{2}}}{\Gamma \left( \frac{d}{2}\right) }\int _0^L\mathrm{d}s_2\int _0^{s_2}\mathrm{d}s_1\left( s_2{-}s_1\right) ^{-\frac{1}{2}}. \end{aligned}$$

(20)

Diagram (2) describes the situation when restriction points $s_1$ and $s_2$ are located along two different linear arms of rosette polymer. We thus have a segment of length $(s_2+s_1)$ between them, associated with factor $\exp \left( -{{k}}^{\,\,2}(s_2+s_1)/2\right)$. After performing integration over k we receive

$$\begin{aligned} \langle \xi (s_1,s_2)\rangle _{(2)} { =}\frac{(2\pi ^{d+1})^{\frac{1}{2}}}{\Gamma \left( \frac{d}{2}\right) }\int _0^L\mathrm{d}s_2\int _0^{L}\mathrm{d}s_1\left( s_2{+}s_1\right) ^{-\frac{1}{2}}. \end{aligned}$$

(21)

In the case (3), both $s_1$ and $s_2$ are located on the closed loop, let it be the loop with $j=1$. Here, we need to take into account the wave vector $\vec {q}_1$, “circulating” along this loop, so that three segments should be taken into account with lengths $s_1$, $s_2-s_1$, and $L-s_2$, correspondingly, with associated factors $\exp \left( -{{q_1}}^{\,\,2}s_1/2\right)$, $\exp \left( -(q_1+k)^{\,\,2}(s_2-s_1)/2\right)$, $\exp \left( -{{q}}^{\,\,2}(L-s_2)/2\right)$. Integration over the wave vector $q_1$ gives

$$\begin{aligned}&(2\pi )^{-d}\int {\mathrm{d}}\vec {q}_1\, {\mathrm{e}}^{-\frac{q_1^2L}{2}-\vec {q}\vec {k}(s_2-s_1)}=\nonumber \\&\quad =(2\pi L)^{-d/2}(s_2-s_1)^{-\frac{1}{2}}{\mathrm{e}}^{\frac{k^2(s_2-s_1)^2}{2L}}. \end{aligned}$$

(22)

After performing final integration over k we receive

$$\begin{aligned} \langle \xi (s_1,s_2)\rangle _{(3)} &=\frac{(2\pi ^{d+1})^{\frac{1}{2}}}{\Gamma \left( \frac{d}{2}\right) }\int _0^Lds_2\int _0^{s_2}ds_1 \nonumber \\&\times \left( s_2{-}s_1-\frac{(s_2{-}s_1)^2}{L}\right) ^{-\frac{1}{2}}. \end{aligned}$$

(23)

Following the same scheme, we receive analytic expressions, corresponding to diagrams (4) and (5) on Fig. 2:

$$\begin{aligned} \langle \xi (s_1,s_2)\rangle _{(4)} &=\frac{(2\pi ^{d+1})^{\frac{1}{2}}}{\Gamma \left( \frac{d}{2}\right) }\int _0^Lds_2\int _0^{L}ds_1\nonumber \\&\times \left( s_2{+}s_1- \frac{s_2^2}{L}{-}\frac{s_1^2}{L}\right) ^{-\frac{1}{2}}, \end{aligned}$$

(24)

$$\begin{aligned} \langle \xi (s_1,s_2)\rangle _{(5)} &=\frac{(2\pi ^{d+1})^{\frac{1}{2}}}{\Gamma \left( \frac{d}{2}\right) }\int _0^Lds_2\int _0^{L}ds_1\nonumber \\&\times \left( s_2{+}s_1-\frac{s_1^2}{L}\right) ^{-\frac{1}{2}}. \end{aligned}$$

(25)

Note that each diagram in Fig. 2 is associated with the corresponding combinatorial factor. Namely, the contribution (1) in above expressions is taken with the pre-factor $f^c$, contribution (2) with $\frac{f^c(f^c-1)}{2}$, (3) with $f^r$, (4) with $\frac{f^r(f^r-1)}{2}$ and the last contribution (5) with the pre-factor $f^r f^c$. Summing up all contributions from Eq. (25) with taking into account corresponding pre-factors, on the base of Eq. (15) we finally obtain the expression for the hydrodynamic radius of a rosette structure:

$$\begin{aligned} \langle R_{h,{\mathrm{rosette}}}\rangle= & {} \frac{\Gamma \left( \frac{d-1}{2}\right) }{\Gamma \left( \frac{d}{2}\right) \sqrt{2}}12(f^c+f^r)^2\sqrt{L}\nonumber \\&\times \left[ -6 f_r\pi \left( \sqrt{2}(f^r-1)-2 f^r+1\right) \right. \nonumber \\&+16 \left( \sqrt{2}-1 \right) f^c \left( \sqrt{2}+ f^c \right) \nonumber \\&+\left. 3f^cf^r \left( 10 \arcsin \left( \frac{\sqrt{5}}{5} \right) -\pi +4 \right) \right] ^{-1}. \end{aligned}$$

(26)

The expression for the mean square gyration radius of a rosette architecture is³⁴:

$$\begin{aligned} \langle R^2_{g,{\mathrm{rosette}}}\rangle &=\frac{Ld}{12(f^r+f^c)^2}[f^r(2f^r-1)\nonumber \\&+2f^c(3f^c-2)+8f^rf^c]. \end{aligned}$$

(27)

Finally, using Eqs. (26) and (27), we calculate the the universal size ratio (4) of rosette-like polymer architecture in Gaussian approximation:

$$\begin{aligned} \rho _{{\mathrm{rosette}}}= & {} \frac{\sqrt{6\,d}\,\Gamma \left( \frac{d-1}{2}\right) }{72(f^r+f^c)^3\Gamma \left( \frac{d}{2}\right) }\nonumber \\&\times \sqrt{6(f^c)^2+8f^cf^r+2(f^r)^2-4f^c-f^r}\nonumber \\&\times \left[ -6 f_r\pi \left( \sqrt{2}(f^r-1)-2 f^r+1\right) \right. \nonumber \\&+16 \left( \sqrt{2}-1 \right) f^c \left( \sqrt{2}+ f^c \right) \nonumber \\&+\left. 3f^cf^r \left( 10 \arcsin \left( \frac{\sqrt{5}}{5} \right) -\pi +4 \right) \right] . \end{aligned}$$

(28)

Substituting $d=3$ in expression (28), for $f^r=0$, both at $f^c=1$ and $f^c=2$ we restore the universal size ratio of a linear polymer (5), whereas $f^c>2$ and $f^r=0$ gives the expression for a star polymer (7). For $f^c=0$ and $f^r=1$ we reproduce the known analytical expression of a single ring from Eq. (6). Consequently $f^c=0$ and $f^r=2$ Eq. (28) provides the formula for universal size ratio of a star comprised of two ring polymers:

$$\begin{aligned} \rho _{{\mathrm{double\,ring}}}=\frac{\sqrt{3\pi }}{4}(3-\sqrt{2})\approx 1.217. \end{aligned}$$

(29)

For $f^c=1$ and $f^r=1$ we find analytic expression for the so-called tadpole architecture:

$$\begin{aligned} \rho _{{\mathrm{tadpole}}}= & {} \frac{\sqrt{22}}{96\sqrt{\pi }}\left[ 3\pi +28+30\arcsin \left( \frac{\sqrt{5}}{5}\right) \right] \nonumber \\\approx & {} 1.415. \end{aligned}$$

(30)

Figure 3

Summary of theoretical results for universal size ratio $\rho$ as given by (28) vs functionality $f=f^c+f^r$ for different polymer topologies. Data for architectures containing: only linear chains (star-like polymer with $f^r=0$) as function of $f=f^c$ (red symbols), only ring polymer (with $f^c=0$) as function of $f=f^r$ (blue symbols) and “symmetric” rosette structure with equal number of rings and linear branches $f=f^r+f^c$ (purple symbols).

In Fig. 3 we plot calculated theoretical values of the universal size ratio vs number of functionalized chains for stars comprised of linear polymers with $f^c>0$, $f^r=0$ (red symbols) and ring polymers $f^r>0$, $f^c=0$ (blue) as well as rosette polymers with equal number of grafted linear chains and rings $f^r=f^c>0$ (purple). For all architectures we observe decrease in $\rho$ with increasing functionality. In the next subsection we compare our theoretical predictions with the result of MD simulations.

Numerical approach

The method

Numerical data in this work have been obtained from MD simulations. We consider simple three-dimensional numerical model of a rosette polymer consisting of arms which are $f^c$ linear chains and/or $f^r$ ring polymers. Each arm is composed of N sizeless particles of equal mass m connected by bonds. We study ideal (Gaussian) conformations of rosette polymers corresponding to a certain extent to the conformations of real rosette polymers at dilute $\theta$ solvent conditions. In our numerical model the connectivity along the polymer chain backbone is assured via harmonic potential

$$\begin{aligned} V(r)=\frac{k}{2}(r-r_0)^2, \end{aligned}$$

(31)

where $k=200$ $k_BT/b^2$ is the interaction strength measured in units of thermal energy $k_BT$ and and the equilibrium bond distance $r_0=b$.

The molecular dynamics simulations were performed by solving the Langevin equation of motion for the position $\vec {r}_i=[x_i,y_i,z_i]$ of each monomer,

$$\begin{aligned} m\ddot{\vec {r}}_i = \vec {F}_i -\zeta \dot{\vec {r}}_i + \vec {F}_i^{\mathrm{R}}, \,\,\, i=1,\ldots ,fN, \end{aligned}$$

(32)

which describes the motion of bonded monomers. Forces $\vec {F}_i$ in Eq. (32) above are obtained from the harmonic interaction potential between (Eq. 31). The second and third term on the right hand side of Eq. (32) is a slowly evolving viscous force $-\zeta \dot{\vec {r}}_i$ and a rapidly fluctuating stochastic force $\vec {F}_i^{\mathrm{R}}$ respectively. This random force $\vec {F}_i^{\mathrm{R}}$ is related to the friction coefficient $\zeta$ by the fluctuation-dissipation theorem $\langle \vec {F}_i^{\mathrm{R}}(t) \vec {F}_j^{\mathrm{R}}(t')\rangle = k_BT\zeta \delta _{ij}\delta (t-t')$. The friction coefficient used in simulations was $\zeta =0.5\,m\tau ^{-1}$ where $\tau = [mb^2/(k_BT)]^{1/2}$ is the unit of time. A Langevin thermostat was used to keep the temperature constant. The integration step employed to solve the equations of motions was taken to be $\Delta t=0.0025\tau$. All simulations were performed in a cubic box with periodic boundary conditions imposed in all spatial dimensions. We used Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)³⁸ to perform simulations. Simulation snapshots were rendered using Visual Molecular Dynamics (VMD)³⁹.

Figure 4

Molecular dynamics data for the universal size ratio $\rho$ of linear chains (red symbols) and ring polymers (blue symbols) plotted as a function of correction-to-scaling variable $N^{-1/2}$ with corresponding simulation snapshots for polymer architectures with degree of polymerization $N=6400$. Solid lines represent fitting functions of the general form given in Eq. (33). Horizontal dotted lines correspond to asymptotic values $\rho _{\infty }$ predicted by theory, cf. Eqs. (5) and (6).

Results

Simulations of rosette polymers were performed for the following number of monomer beads per arm $N=100,200,400,800,1600$ and 6400. The number of arms for star polymers composed of solely linear chains (i.e. with $f^r$=0) and ring polymers (i.e. with $f^c=0$) were varied in the range between 1 to 4. In the case of rosette polymers which are hybrid polymer architectures comprised of linear chains and ring polymers we considered two arm functionalities with $f^c=f^r=1$ and 2. To increase conformational sampling each simulation was carried out with 50 identical molecules in a simulation box. In the course of simulations the universal size ratio was measured, cf. Eq. (4). In the numerical calculation of quantities like $\rho$ a crucial aspect is finite degree of polymerization N that we are dealing with in simulations, while theoretically obtained values of $\rho$ hold in the asymptotic limit $N\rightarrow \infty$. Thus, the finite-size effects (or corrections to scaling) should be appropriately taken into account. For the size ratio of an ideal linear chain, this correction is given by

$$\begin{aligned} \rho =\rho _{\infty }(1+aN^{-\Delta }), \end{aligned}$$

(33)

where $\rho _{\infty }$ is the asymptotic value obtained at $N\rightarrow \infty$, a is non-universal amplitude, $\Delta$ is the correction-to-scaling exponent for $\theta$-solvent is $\Delta =1/2$³⁰ whereas for good solvent conditions is $\simeq 0.53$³³. In our numerical analysis we use Eq. (33) to obtain the universal size ratio in the asymptotic limit for all considered architectures. For this purpose we plot $\rho$ vs correction-to-scaling term $N^{-1/2}$ and get $\rho =\rho _{\infty }$ for $N\rightarrow \infty$.

In Fig. 4 we display the results of our MD simulations for two “benchmark” systems which are Gaussian linear chain (red circles) and Gaussian ring (blue circles). For both architectures systematic increase in the size ratio is observed with increasing value of N. In the asymptotic limit $N\rightarrow \infty$ we obtain $\rho _{\mathrm{chain}}=1.499\pm 0.005$ and $\rho _{\mathrm{ring}}=1.244\pm 0.004$. These numerical values with very good accuracy reproduce known theoretical results. The latter are given by Eq. (5) for linear chains and by (6) for rings. The complete list of numerically derived universal size ratios and their comparison to theoretical values can be found in Table 2.

Figure 5

Molecular dynamics data for the universal size ratio $\rho$ of star polymers comprised of a) linear, b) ring polymers and c) rosette polymers plotted as a function of correction-to-scaling variable $N^{-1/2}$. Data displayed for different amount of arms $f^c$ and $f^r$ as indicated. For rosette polymers data are for symmetric number of arms $f^c=f^r$. Solid lines represent fitting functions according to Eq. (33). Horizontal dotted lines correspond to asymptotic values obtained from analytical theory, see Table 2. Insets show simulation snapshots for with $N=6400$ and: (a) $f^c=3$, (b) $f^r=2$ and (c) $f^c=f^r=1$.

Table 2 Summary of theoretical results for the size ratio $\rho _{\mathrm{theory}}$ calculated using Eq. (28) and asymptotic values $\rho _{\mathrm{sim}}$ obtained from MD simulations for rosette polymer architectures comprised of different number of $f^c$ linear chains and $f^r$ ring polymers.

In Fig. 5 we show numerically derived universal size ratios as a function of $N^{-1/2}$ for more complex architectures. We investigated conformations of stars comprised of linear chains, stars of ring polymers and rosette polymers with equal number of grafted linear and ring chains. For all architectures we observe systematic approaching to asymptotic values predicted by theory with increasing value of N per arm. For stars of linear chains with functionality $f^c=3$ and 4 (cf. Fig. 5a) simulations provide the following universal size ratios: $1.395\pm 0.006$ and $1.336\pm 0.006$. Both values are with very good agreement to the theoretical prediction given by Eq. (7). Note that the values of $\rho$ calculated in the course of our simulations are much closer to the analytical theory results as compared to existing numerical data³⁵. For stars comprised of cyclic macromolecules (cf. Fig. 5b) we reproduce the theoretical value of Eq. (29) for double ring architecture ($f^r=2$) as well as for stars with larger number of grafted rings, cf. Eq. (28) with $f^c=0$ and $f^r=3$ or 4. Namely, we get $1.204\pm 0.010$ for $f^r=2$, $1.165\pm 0.011$ for $f^r=3$ and $1.135\pm 0.012$ for $f^r=4$. For the tadpole architecture, the simplest rosette polymer which is comprised of $f^c=1$ and $f^r=1$ arms (see snapshot in Fig. 5c), we obtain the size ratio of $1.401\pm 0.008$ which matches theoretically predicted value for this type of polymer from Eq. (30). For rosette polymers with $f^c=2$ and $f^r=2$ our simulations provide $1.295\pm 0.018$ which is comparable with the corresponding value calculated from the formula given in Eq. (28). The full list of calculated values of $\rho$ is in Table 2.

Conclusions

We have studied by combination of analytical theory and molecular dynamics simulations conformational properties of rosette polymers which are complex macromolecules consisting of $f^c$ linear chains (branches) and $f^r$ closed loops (rings) radiating from the central branching point. Our focus was on characterizing structure of ideal polymer conformation with no excluded volume interactions. For this purpose we investigated basic structural quantities such as the mean square radius of gyration $R_g^2$, the hydrodynamic radius $R^{-1}_H$ and most importantly the universal size ratio $\rho \equiv \sqrt{R_g^2}/R_H$. Our calculations demonstrated gradual decrease in $\rho$ with increasing functionality $f=f^c+f^r$ of grafted polymers. The analytical results are in perfect agreement with our numerical simulations data. Since both quantities $R_g^2$ and $R_H$ are directly accessible via correspondingly static and dynamic scattering techniques we hope that our results will stimulate further experimental studies on the behavior of complex polymer architectures in solutions.