Swelling of asymmetric pom-pom polymers in dilute solutions

Within the continuous chain model [11], pom-pom polymer is described as a set of trajectories in continuous space. Each of the trajectories has the length $L$ and is parametrized by radius vector $\mathbf{r}(s)$ in $d$-dimensional space, with $s$ changing from $0$ to $L$. The Hamiltonian of this model can be written as:

where $F=f_{1}+f_{2}+1$ is the number of trajectories, the first term in the expression (2.1) represents a connectivity of each trajectory and the second term describes a two point interaction with a coupling constant $u$ known as excluded volume interaction [12].

It is a well known property of the model that all topologies are described by the same Hamiltonian [13, 14, 15] and differ only at the level of partition function, which for the pom-pom case reads:

Here, $\mathbf{r}_{0}$ denotes a backbone trajectory, the two sets of delta-functions place $f_{1}$ and $f_{2}$ trajectories of side stars, correspondingly, at the terminal points of the $0$-th trajectory, $H$ is Hamiltonian of the model given by equation (2.1), and $Z_{0}$ represents a partition function for a Gaussian case.

Observables calculated within the frames of continuous chain model depend on the chain length $L$ and diverge in the limit of $L\rightarrow\infty$. These divergences should be removed in order to receive universal values for the observables in question. In this work we use the des Cloiseaux’s direct renormalization scheme [12], where a set of renormalization factors are introduced, that are directly connected to the physical quantities for which the divergencies should be removed.

Different topologies are introduced into the model through partition function, and the Hamiltonian of the model contains only interactions. Therefore, the fixed points of the renormalization scheme, once calculated for a particular Hamiltonian, do not depend on the topology of the molecule under consideration. Consequently, it suffices to evaluate the fixed points of the model for the simplest case.

Starting with the calculation of the partition function of two interacting polymers $Z(L,L)$, the method then introduces the renormalization factors that are connected with the number of allowed trajectories (the partition function of a single chain) $[Z(L,u_{0})]^{-2}$, and with its characteristic size which is represented by the end-to-end distance $\langle R_{e}^{2}\rangle$ or, more precisely, a swelling factor $\chi_{0}(L,\{x_{0}\})$, defined as $\chi_{0}=\langle R_{e}^{2}\rangle/L$. Those functions allow one to introduce a renormalized coupling constant:

with $\epsilon=4-d$ being a deviation from the upper critical dimension for the coupling constant $u_{0}$.

In the limit of infinitely long chains, the dimensionless coupling constant $u_{0}=u(2\piup)^{-d/2}L^{2-d/2}$ diverges unlike the renormalized one $u_{R}$, which in the same limit reaches a fixed value:

In practice, the value $u_{R}^{*}$ is calculated from the equation:

For the model under consideration, those solutions are well known, and in the first order of the $\epsilon$-expansion read:[12]:

where the first one describes a Gaussian case, and the second one is a coupling constant for the model with excluded volume interaction.

Within the formalism of the random flight model [16], every linear sub-chain of $N$ monomers within a complex polymer structure is considered as a sequence of $N$ bond vectors $\sigma_{k}$, with $0\leqslant k\leqslant N$ with the origin denoted as $0$. We assume it to coincide with the center of the central (backbone) chain. Let us denote by ${\mathbf{R}_{\rm cl}}_{n}$ and ${\mathbf{R}_{\rm cr}}_{n}$ the position vectors of segments in the central chain to the right and to the left from the reference point (considered as $0$), correspondingly. They can be expressed as sums over the set of corresponding bond vectors:

Similarly, let $\mathbf{R}^{i}_{{\rm pl}_{n}}$ and $\mathbf{R}^{i}_{{\rm pr}_{n}}$ be the position vectors of monomers in the left and right side stars (poms), correspondingly, so that

All the bond vectors are assumed to have an equal length $l$ and are connected to each other by unrestricted joints. The direction of each bond is random and independent of the directions of its neighbours. Thus, performing the averaging over an ensemble of possible bond configurations, we have

with $\delta$ being the Kronecker delta symbol. In what follows, we take $l=1$.

Note that the polymer molecule in this model is assumed to capture Gaussian statistics, i.e., the excluded volume (self-avoidance of segments) is neglected.

Numerical simulations for a pom-pom polymer are performed within the lattice model of self-avoiding walk (SAW) on a simple cubic lattice via pivot algorithm [17, 18]. Each trajectory is forbidden to visit the site occupied by itself or by the other trajectory. In total, we consider $f_{1}+f_{2}+1$ walks, so that $f_{1}+1$ of them start at the origin and $f_{2}$ — at the end of the walk numbered as $0$-th. We consider both branching parameters $f_{1}$ and $f_{2}$ to vary from $1$ to $13$ with all walks being of the same length top bounded by $N=100$ steps, making a total amount of steps in the pom-pom to be $N(f_{1}+f_{2}+1)$. Then, the pivot algorithm is applied again to receive a new configuration. We start from the initial configuration as a set of straight lines. Then, a randomly chosen transformation is applied to the part of a walk. If the new trajectory is acceptable under the self-avoidance rules, the new conformation is accepted; if not — it is rejected and the previous one is accounted for one more time. Then, another pivot operation is performed at a new randomly chosen site. The first $20N(f_{1}+f_{2}+1)$ operations are rejected for the equilibration, and the next $10^{5}$ are used for calculation of the observables. All observables are evaluated as the mean arithmetic average over the set of structures obtained by means of simulations. Numerical modelling is performed for $N$ varying from $10$ to $100$ with the step of $10$. Final values are received by application of the finite size approximation $g_{x}=g_{x,\infty}+B/N$, with $B$ being a constant. The snapshots of the obtained conformations are illustrated in figure 1.

Another numeric approach used in this study is the dissipative particle dynamics (DPD) [19]. In this method, the polymer and solvent molecules are modelled as soft beads of equal size, each representing a group of atoms. For this reason, the interaction potential between two beads are, in general, density and temperature dependent. Different parametrization strategies are possible and in this paper we closely followed the original parametrization by Groot and Warren based on matching the model compressibility to that of water [19]. The length-scale is given by the diameter of a soft bead, and the energy unit is $k_{\rm B}T=1$, where $k_{\rm B}$ is the Boltzmann constant and $T$ is the temperature. Monomers in a polymer chain are bonded via harmonic springs resulting in a force:

where $k$ is the spring constant, and $\mathbf{x}_{ij}=\mathbf{x}_{i}-\mathbf{x}_{j}$, $\mathbf{x}_{i}$ and $\mathbf{x}_{j}$ are the coordinates of $i$-th and $j$-th bead, respectively. The non-bonding force $\mathbf{F}_{ij}$ acting on the $i$-th bead as a result of its interaction with its $j$-th counterpart is expressed as a sum of three contributions

where $\mathbf{F}^{\mathrm{C}}_{ij}$ is the conservative force responsible for the repulsion between the beads, $\mathbf{F}^{\mathrm{D}}_{ij}$ is the dissipative force mimicking the friction between them and the random force $\mathbf{F}^{\mathrm{R}}_{ij}$ works in pair with a dissipative force to thermostat the system. The expression for all these three contributions are given below [19]

where $a$ is the amplitude for the conservative repulsive force, $x_{ij}=|\mathbf{x}_{ij}|$, $\mathbf{v}_{ij}=\mathbf{v}_{i}-\mathbf{v}_{j}$, $\mathbf{v}_{i}$ is the velocity of the $i$-th bead. The dissipative force has an amplitude $\gamma$ and decays with distance according to the weight function $w^{\mathrm{D}}(x_{ij})$. The amplitude for the random force is $\sigma$ and the respective weight function is $w^{\mathrm{R}}(x_{ij})$. $\theta_{ij}$ is the Gaussian random variable. As was shown by Español and Warren [20], to satisfy the detailed balance requirement, the amplitudes and weight functions for the dissipative and random forces should be interrelated: $\sigma^{2}=2\gamma$ (we choose $\gamma=6.75$, $\sigma=\sqrt{2\gamma}=3.67$ here) and $w^{\mathrm{D}}(x_{ij})=\left[w^{\mathrm{R}}(x_{ij})\right]^{2}$. Here, we use the weight functions quadratically decaying with the distance:

Following parametrization of Groot and Warren [19], the reduced number density of the system is set at $\rho^{*}=N/V=3$, where $N$ is the total number of beads (pom-pom polymer and a solvent) in a system and $V$ is the volume of a simulated system. We consider the case of a single pom-pom polymer within a simulation box, which reproduces the conditions of an infinite dilution. To avoid self-interaction between periodic images of a pom-pom molecule, the linear dimension of a cubic simulation box $L$ was chosen accordingly. In particular, we used the expression: $L\approx 1.75R^{*}_{g}$, where $R^{*}_{g}=(3N_{f}-1)^{\nu}$ estimates the radius of gyration of the longest linear sub-chain of a pom-pom of length $3N_{f}$ and bond length of $1$ in a good solvent regime. Here $N_{f}$ is the number of beads in a single arm and $\nu=0.59$ is the Flory exponent.

The total number of branches was fixed at $F=f_{1}+f_{2}+1=15$ in all cases. This number of branches was chosen to show the effect of asymmetry and to avoid the effect of crowdedness which has been studied in our previous work [21]. Two cases of the arm length, $N_{f}=8$ and $N_{f}=16$ were examined, similarly to the case of a symmetric pom-pom polymer [8], the results were found to agree well. In work [19] it was shown that the reduced density is coupled to the amplitude $a$ of a repulsive force in equation (3.3). Following this paper, we set time-step $\Delta t=0.04$ and $a=25$ for all combinations of the interacting beads: polymer-polymer, solvent-solvent and polymer-solvent.

The first $2\cdot 10^{6}$ steps are allowed for the system equilibration and are skipped from the analysis. The productive runs span from the step $2\cdot 10^{6}$ up to the maximal simulation step in each case (typically $8\cdot 10^{6}$). The error estimates are made by splitting the whole productive run in four equal pieces and by evaluating the partial averages $A_{i}$, $i=1-4$ for a given property $A$ in each of them. The final average is $\langle A\rangle=\frac{1}{4}{\sum_{i=1}^{4}A_{i}}$, whereas the conservative estimate for the standard error is given by

We start with the evaluation of the expression for the partition function (2.2) as a series in a coupling constant $u$, where the zero order term corresponds to the unperturbed Gaussian case. We restrict the series to the linear term in $u$, the so-called one loop approximation:

where

is the Gaussian result for the partition function of a pom-pom polymer, in which case the excluded volume interactions are neglected. In order to calculate the contribution from the second term in (4.1), we consider a set of diagrams presented in figure 2. It is important to note that the diagram $Z_{1}$ describes the excluded volume interaction within a single trajectory and needs to be taken into account with the pre-factor $f_{1}+f_{2}+1$, diagram $Z_{2}$ corresponds to the contributions from all the interactions between two chains that are separated by one branching point, and contains a pre-factor $f_{1}+f_{2}+[f_{1}(f_{1}-1)+f_{2}(f_{2}-1)]/2$, and the last diagram corresponds to the interactions between the chains separated by both branching points and has a pre-factor $f_{1}f_{2}$.

Using a Fourier transform presentation of $\delta$-function:

and calculating the contributions of all diagrams, we obtain the following expressions:

As already mentioned in the previous section, we consider all the observables in the first order of the $\epsilon$-expansion. Therefore, the expressions for the diagrams read:

Getting all terms together, we obtain the expression for the partition function of a pom-pom molecule in one-loop approximation

where $u_{0}=u(2\piup)^{-d/2}L^{2-d/2}$ is a dimensionless coupling constant.

All the observables calculated below are averaged over an ensemble of all possible conformations with the partition function (4.9):

A set of specific size characteristics can be defined for the pom-pom polymer. Some of them were introduced for a symmetric case in our previous study [8]. Here, we add a number of new characteristics that allow us to describe the asymmetry of pom-pom molecule when $f_{1}\neq f_{2}$. The gyration radius of the structure is defined as:

where $s_{2}$ and $s_{1}$ are the “monomer numbers” along the chain, such that the average square distance is calculated between them. They are often called restriction points. In order to calculate this quantity we use the identity:

and evaluate $\xi(\mathbf{k})$ in the path integration approach. In calculations of the contributions into $\xi(\mathbf{k})$, it is convenient to use the diagrammatic presentation, as given in figure 3 for the Gaussian chain.

It is interesting to note that diagrams in figures 2 and 3 look similar with the only difference: in the former we consider them with interactions and in the latter one with the restriction points, though the pre-factor will be the same and the expression for the gyration radius in the Gaussian approximation reads:

Considering all possible combinations for diagrams in figures 2 and 3, a contribution to $\xi(\mathbf{k})$ the one-loop approximation is obtained:

An expression for $C_{g}$ is provided in the Appendix in expression A.1. In the case of $f_{1}=f_{2}$ we recover our previous result for the symmetric case [8].

We introduce two other characteristics that quantify the asymmetry of the molecular conformation: the average squared distances from a monomer to the first and to the second branching point, respectively:

In the process of evaluating these parameters, we again use the diagrammatic technique and consider a set of diagrams similar to those for the gyration radius but with an additional restriction: points are permanently fixed at the branching point. The expressions in the Gaussian approximation read:

Similarly, the expressions in the one loop approximation can be presented in a general form:

with $C_{1}(f_{1},f_{2},d)$ and $C_{2}(f_{1},f_{2},d)$ being the functions of the space dimension and topology. $\epsilon$-expansions for $\langle{r_{1}^{2}}\rangle$ and $\langle{r_{2}^{2}}\rangle$ are provided in the Appendix in expressions (A.3) and (A.4), respectively.

To describe the impact of the effects of asymmetry on the size and shape of pom-pom structure, we consider a set of size ratios. We start by considering the ratios that compare the size of the asymmetric pom-pom to the other molecular topologies: a linear polymer chain, a star, and a symmetric pom-pom of the same total molecular weight, defined as:

respectively. Within the renormalization group approach, these quantities are presented as the series in $\epsilon=4-d$

with the expressions for $A_{g}(0,0),A_{g}(f,f),A_{g}(f_{1},f_{2})$ provided in the Appendix [expression (A.2)]. Note that the first two expressions above reproduce the well known results for the gyration radius of a linear chain [12] and of a star [14].

Thus, the final expressions for the size ratios (4.20, 4.21) and (4.22) read:

where $u_{0}$ is replaced by the fixed point (2.7).

Let us note that the expressions (4.27, 4.28) and (4.29) contain the first order terms of the perturbation theory only and, therefore, provide rather qualitative than the quantitative information. It is, however, known that the renormalization group approach requires at least the second order terms in $\epsilon$ to get the quantitatively sound values. This requires the evaluation of the terms up to $u_{0}^{2}$, since $u_{0}\sim\epsilon$. These calculations for the case of branched structures are very challenging, in particular, since the total number of diagrams grows exponentially and leads to cumbersome expressions that are difficult to handle analytically. Because of this, it is practical to use the approximation proposed in reference [22], based on a general idea of presenting the gyration radius in a form:

with $N$ being the number of monomers, $\Lambda$ is a coarse-grained length scale. Parameter $\eta$ here defines the transition from the Gaussian case at $\eta=0$ [$\nu(\eta)=1/2$ and $f_{p}(\eta)=1$] to the case of a polymer with an excluded volume interaction at $\eta\rightarrow\infty$ [$\nu(\eta)=\nu$ and $f_{p}(\eta)=1+a$]. Since only the factors $\langle R_{g}^{2}\rangle_{0}$ and $f_{p}(\eta)$ depend on the polymer topology, the size ratio can be presented as:

with ${\langle R_{g,1}^{2}\rangle_{0}}/{\langle R_{g,2}^{2}\rangle_{0}}$ being the ratio between topologies “$1$” and “$2$” in Gaussian approximation. And the parameters $a_{1}$, $a_{2}$ are calculated for these topologies using an expression:

This approximation allows us to partially account for the higher orders in $\epsilon$ in the $C_{g}$ expansion, resulting in such terms to be added to equations (4.27, 4.28) and (4.29).

The ratios (4.20, 4.21) and (4.22) allow us to examine how the asymmetricity of a pom-pom structure affects its overall shape as compared to the cases of a linear chain, star-polymer and a symmetric pom-pom structure. It would also be informative to introduce some specific characteristic, which addresses the shape asymmetry explicitly. For that purpose, we suggest the following combination, which can be termed as the asymmetry factor:

with $\langle r_{1}^{2}\rangle$ and $\langle r_{2}^{2}\rangle$ given by equations (4.14) and (4.15). It is strictly zero for the symmetric case, $f_{1}=f_{2}$, and its maximum value is reached for the most asymmetric cases, $f_{1}=1$ and $f_{2}=F-2$, or vice versa. In the Gaussian approximation, the corresponding expression reads:

Hydrodynamic properties of the polymer solutions are often studied by replacing each polymer molecule by an effective soft convex body (e.g., the sphere or equivalent ellipsoid), and, consequently, the center-center effective interactions between such bodies are examined [23, 24, 25]. Thus, the center of mass (CM) of each polymer appears in a natural way, as well as the distribution of monomers around it [26, 27]. For the evaluation of the quantities presented in this subsection, the random-flight model (section 2.3) approach is the most convenient. The CM of a molecule is defined, therefore, as:

Expressing the position vectors as sums of bond vectors [equations (2.8)–(2.3)], we have

Taking into account relations (2.10) and (2.11), we obtain an expression for the mean-squared location of the center of mass:

For the center of mass of a single chain of the same total length with a reference point in the middle of a chain, we have:

Thus, we can introduce the ratio:

which describes the impact of the presence of side stars (see figure 4) on the shift of the CM position of complex polymer structure as compared with that of linear chain as given by equation (4.38). This quantity may serve as yet another characteristic of the asymmetry of the pom-pom structure. In the Gaussian approximation, we have:

Let us introduce the distance of each $n$-th monomer from the CM

Applying the scheme introduced above, we obtain the analytic expressions for these distances for the monomers belonging to a central chain and these from the left and right branches, respectively:

The main focus of this study are the theoretic results for the shape characteristics of asymmetric pom-pom polymers, whereas the numeric studies, performed by means of the DPD and MC simulations, were conducted to support these theoretical findings. To this end, a set of size ratios, defined above, were obtained by means of the above mentioned numeric approaches. This set includes: the ratio between the gyration radius of asymmetrical pom-pom to that of a linear chain $g_{c}$ (4.20), of a star-like polymer (4.21) and of a symmetrical pom-pom polymer (4.22). These results, alongside with the theoretical findings, are shown in figures 5 and 6. Red open discs and magenta squares represent the DPD simulations with $N_{f}=8$ and $N_{f}=16$, respectively, green squares represent the MC study using a pivot algorithm. Yellow dash line displays the Douglas–Freed approximation, blue dash line: the first order (one loop) renormalization group results, and a black dot-dash line: these obtained within the Gaussian model.

The results for the $g_{c}$ ratio (4.20) are shown in the top frame of figure 5. First observation is that $g_{c}$ is always below $1$. This is a natural consequence of the pom-pom connectivity, which organizes monomers into two star-like bunches, which cannot stretch as much as a single linear chain of the same molecular weight. Hence, the gyration radius of a pom-pom is always smaller than that of a linear chain. Second observation is that the $g_{c}$ curve has a convex shape and $g_{c}$ is minimal for the highly asymmetric pom-pom structures, $f_{1}=1$ and $f_{1}=13$, since in this case one of the branching points holds a maximum possible number of arms and thus maximizes the spatial confinement effect. Third observation is that there is a reasonable agreement between the results of the DPD and MC simulations, despite great difference between soft potential and off-lattice nature of the former and hard potentials and lattice arrangement of monomers in the latter. The Douglas–Freed approximation is found to be in a good agreement with the simulation data, coinciding perfectly with the DPD results obtained at the arms length of $N_{f}=16$. This shows that such arms length is long enough to mimic the “long arms limit” assumed in the theory, and that $g_{c}$ clearly demonstrates a strong universality feature. Fourth observation is about the Gaussian result for $g_{c}$, which is found below both simulation results. Switching off the excluded volume effects (the case of the Gaussian model), reduces the denominator of $g_{c}$: $R^{2}_{g,{\rm chain}}\sim N^{0.5}$ as compared to its value $R^{2}_{g,{\rm chain}}\sim N^{0.59}$ for the case with excluded volume effects (as it is in both simulations here). As far as the ratio $g_{c}$ decreases in the case of the Gaussian model, the nominator $R^{2}_{g,{\rm pom-pom}}$ in the Gaussian approximation must decrease even more compared to the case of simulations, indicating a stronger effect of “compactization” of a pom-pom structure compared to the linear chain when the excluded volume effects are switched off. Finally, the fifth observation is that the first order renormalization group result overestimates $g_{c}$ essentially in the whole interval of $f_{1}$. This situation is somewhat typical of the renormalization group findings for other properties of interest, and a much better qualitative agreement is typically expected when the second order terms are included.