The simulation of polymer solutions often requires the development of methods that accurately include hydrodynamic interactions. Resolution on the atomistic scale is too computationally expensive to cover mesoscopic time and length scales on which the interesting polymer phenomena are observed. Therefore, coarse-graining methods have to be applied. In this work, the solvent is simulated using the well-established multi-particle collision dynamics scheme, and for the polymer, different coarse-graining methods are employed and compared against the monomer resolved Kremer–Grest model by their resulting diffusion coefficients. This research builds on previous work [Ruiz-Franco et al., J. Chem. Phys. 151, 074902 (2019)], in which star polymers and linear chains in a solvent were simulated and two different coarse-graining methods were developed, in order to increase computational efficiency. The present work extends this approach to ring polymers and seeks to refine one of the authors’ proposed model: the penetrable soft colloid model. It was found that both proposed models are not well suited to ring polymers; however, the introduction of a factor to the PSC model delivers satisfying results for the diffusion behavior by regulating the interaction intensity with the solvent.

Topics
Molecular dynamics, Monte Carlo methods, Coarse-grain model, Newtonian mechanics, Rotational dynamics, Atomic and molecular collisions, Polymers, Hydrodynamics simulations

In the last decades, computer simulations have become a powerful tool to understand physical principles of soft matter systems, such as colloidal suspensions and polymeric fluids. Conventional simulations of polymers at the atomistic level using full-atom molecular dynamics are limited in terms of length scales (typically nanometer) as well as time scales (typically microseconds). In order to overcome such limitations, several coarse-grained methods have been developed to capture the dynamics at mesoscopic scales while neglecting the behavior at the atomistic level. For example, for polymers, these strategies involve the substitution of interactions on the atomistic level by effective interaction potentials between coarse-grained segments of a polymer. A nowadays widely used approach approximates the polymer by a chain of spherical beads connected by springs.1 An important example is the well-known Kremer–Grest model2 that connects adjacent beads by nonlinear springs and bead-bead repulsion is modeled as a truncated Lennard-Jones potential, also known as the Weeks–Chandler–Anderson (WCA) potential.3 In these approaches, the solvent is often taken into account in an effective way. In the simplest case, it acts as a local heat bath that transmits uncorrelated stochastic forces onto the beads where the strength of these forces depends not only on temperature but on the bead size and the fluid viscosity.

However, such a simplified procedure neglects solvent-induced non-local effects that appear as hydrodynamic interactions between different beads. While equilibrium properties do not depend on capturing hydrodynamic interactions, for example, the dynamics of polymers under non-equilibrium conditions often involve fluid flow, such as in phoretic transport,4 polymers under shear,5 or in the presence of external forces.6 In order to keep the computational expenses of resolving hydrodynamic interactions reasonable, the hydrodynamic interactions can be treated by another level of coarse-graining, such as by Stokesian dynamics,7 the Lattice Boltzmann (LB) method,8 Dissipative Particle Dynamics (DPD),9 or Multiparticle collision dynamics (MPCD).10,11

In particular, MPCD, which had first been introduced by Malevanets and Kapral more than 20 years ago,12 has since been widely used for the numerical simulation of complex fluids at the nano- and the micrometer scale. Examples include simulations of polymers,13–19 colloids,20–22 microswimmers,23–25 and biological cells.26 The MPCD method models the solvent by effective coarse-grained fluid particles that perform alternating streaming and collision steps, where momentum is conserved locally, and it thus approximates solutions of the Navier–Stokes equations and includes thermal fluctuations.11 Polymer dynamics is usually efficiently coupled to the solvent in the collision step.

Performing simulations of the bead-resolved polymer dynamics, with or without the presence of the explicit MPCD solvent, can become very time-consuming, in particular, when modeling large semidilute suspensions or polymer melts. Therefore, there has been great interest in finding even stronger levels of coarse-graining of polymers. In these approaches typically star polymers or dendrimers are treated as effective soft colloids,27 where the multitude of configurations is averaged out and individual polymers interact with each other only via an effective pair potential,28 instead of the monomer-resolved interactions.

While the aforementioned approaches using effective potentials are usually applied in equilibrium conditions, where fluid flow and hydrodynamic interactions can be neglected, recently Ruiz-Franco et al. investigated how linear and star polymers can be coarse grained as effective colloids and how they can be coupled to an MPCD fluid,29 allowing for incorporating hydrodynamic effects in the system. The authors introduced two different methods for describing linear polymers and star polymers as soft colloids in an MPCD fluid, which they validate against the fully monomer-resolved model (MRM) for polymers. The first method models the polymer as a rigidly connected distribution of monomers, which captures the real polymer density distribution obtained from the MRM. This method, also termed effective monomer model (EMM), had been proven to work well for linear polymers. The second method considers the polymer as a single penetrable soft colloid (PSC), where the solvent particles are allowed to penetrate the colloid with a certain probability, which is again related to the monomer density obtained from the MRM. This methods works particularly well for star polymers.29 

In this work, we extend the works from Ruiz-Franco et al.29 to ring polymers. Recently, there has been a great interest in understanding the dynamics and structural properties of ring polymer solutions,30–32 in particular, because of their importance in biological systems, such as chromatin organization.33 In previous studies, monomer-resolved models of ring polymers had been coupled to an MPCD fluid,34–36 and in particular the importance of hydrodynamic flows for ring polymers out of equilibrium had been demonstrated.36 We first introduce the methods, such as MPCD, the MRM, and the coarse-grained models developed in Ref. 29 (see Sec. II). We then introduce two modifications of the PSC model that we apply, in addition to the other model, to ring polymers. In Sec. III, we show our results on comparing the diffusive motion obtained of the effective descriptions of ring polymers for the different coarse-grained soft colloid models. We provide summary and conclusion in Sec. IV.

In this section, we introduce several methods to simulate a polymer in an MPCD solvent. Each method uses a different level of coarse-graining.

The MPCD algorithm is a particle based approach, where a number of n point-like solvent particles each with mass m are initialized. Their initial positions are random, and their velocities follow a Boltzmann distribution with temperature T. The dynamics consist of two alternating steps: streaming and collision.11,12,37 During the streaming step, the particles move ballistically for the duration of one time step h with
(1)
where r⃗i(t) represents the position of solvent particle i at time t and v⃗i(t) represents its velocity. For the collision step, the simulation box is divided into cubic collision cells of side length a. Particles that share a collision cell interact with each other. The collision exchange of momentum is realized by rotating the particles’ relative velocity vectors v⃗i−v⃗cm with respect to the center-of-mass velocity of the collision cell v⃗cm⁠, which is defined as
(2)
with nc being the number of solvent particles inside the collision cell. The velocities after collision can then be calculated using a standard universal rotation matrix D̂(R⃗,α)⁠, which rotates the solvent particle relative velocities about a randomly chosen unit vector R⃗⁠, i.e., the rotation axis, by a fixed angle α,
(3)
This ensures that the center-of-mass velocity of each collision cell remains unaffected by the collision. Only the relative velocities are rotated, and as a result, linear momentum is conserved locally. While the side length a of the collision cell remains fixed during the entire simulation, their center points have to be shifted by a uniformly picked random vector with components ∼U(−a/2, a/2) before each collision step. This is done as to not violate Galilean invariance. Galilean invariance and the local conservation of linear momentum are necessary to reproduce hydrodynamic interactions correctly.11 
The kinematic viscosity ν = νkin + νcoll of the solvent can be tuned by the choice of parameters for collision cell side length a, average number of solvent particles per collision cell nc⁠, rotation angle α, and time step size h.38 It can be separated into a kinetic part νkin, in which momentum is transported mainly through particle displacements, and a collisional part νcoll, in which momentum is transported mainly through particle collisions11,
(4)
(5)
where kB represents the Boltzmann constant and nc refers to the mean number of particles per cell. The temperature T is controlled using cell-level Maxwell–Boltzmann scaling.37 The reference units for length, mass, energy, and time are a, m, kBT, and τ=ma2/kBT⁠, respectively, and they are generally set to a = 1, m = 1, and kBT = 1, following τ = 1. We chose the remaining parameters as follows: h = 0.1τ and α = 3π/5. Furthermore, the cubic simulation box side length l = 46a and solvent particle number n = 106, which results in nc=n/l3=10.3⁠. These are all within the usual range for an MPCD solvent,14,39 and they result in a kinematic viscosity (with unit ν0=a2kBT/m⁠) of
νkin/ν0=0.039andνcoll/ν0=0.656.
The monomer-resolved ring polymer consists of N monomers each with mass M. The monomer–monomer interactions are simulated using the purely repulsive Lennard-Jones potential ULJ(R) for excluded-volume interactions as well as the finitely extensible nonlinear elastic (FENE) potential UFENE(R) for bonds.40 This combination of potentials to simulate polymers is a well-established model developed by Grest and Kremer.2 The purely repulsive Lennard-Jones potential3,
(6)
is applied between all monomer pairs. Here, θ(x) denotes the heavyside step function, and R is the absolute distance between two monomers. We set the energy and length scale to ɛ = kBT and σ = a, respectively. The FENE potential
(7)
is only applied between neighboring monomers. The constants in the FENE-potential are generally chosen as K = 30ɛ/σ2 for the spring constant and Rmax = 1.5σ for the maximum allowed distance between two neighboring monomers.2 The linear combination of the two potentials, Eqs. (6) and (7), then leads to an equilibrium monomer–monomer distance of ≈σ⁠. In the streaming step the solute monomers do not interact with the fluid particles. A molecular dynamics of the polymer beads is performed by integrating Newton’s equations of motion using Eqs. (6) and (7) numerically with the velocity Verlet algorithm41 at discrete time steps hMD.
A simple yet effective way to couple solute monomers to the MPCD-solvent is to let them participate in the multi-particle collision.11,13 Analogous to the solvent, monomers are treated as point-like particles for the collision step with according mass M and velocity V⃗i⁠. The center-of-mass velocity of each cell with nc solvent particles and Nc monomers is then calculated as
(8)
replacing Eq. (2) in that regard. Equation (3) to calculate the new velocities after collision is not modified when adding solute particles to the collision, except for using the expression for v⃗cm from Eq. (8), and applies to both solvent particles and monomers. We decided against including the polymer in the thermostating step (for all proposed models), but rather let thermostating happen indirectly through the solvent.

The parameters of the polymer and for the MD should be in accordance with the solvent and MPCD specific values. For this work, a time step of hMD = 0.002τ was selected. MD steps and MPCD steps have to be alternated correctly such that the time is synchronized. Hence, the MD algorithm has to be performed h/hMD = 50 times for each MPCD step. Also, the monomer mass should be roughly equal to the total solvent mass per cell;38 hence, we set M = 10m. This ensures that both the embedded object and the solvent influence each other equally. In this work, a ring polymer with N = 100 monomers in a cubic box using periodic boundary conditions was chosen for all simulations.

We initialize the ring polymer as a regular polygon in 2D. The results of the MD-MPCD simulations are averaged over 400 independent runs, where each run consists of 5 × 105 MD steps of pre-equilibration without solvent, by which a typical equilibrium conformation for the polymer is produced, and 5 × 104 MPCD steps both for equilibration and for the actual simulation. Note: Only the MRM needs time for equilibration. The subsequent coarse-grained models do not need equilibration due to their inherent coarse-grained nature.

During the MRM simulation, we quantify the conformation of the polymer using the instantaneous gyration tensor36 
(9)
where Si,α(t) = Ri,α(t) − Rcm,α(t) is the α-coordinate of the relative position vector of monomer i with respect to the polymer center-of-mass R⃗cm⁠. The eigenvalues of the individual instantaneous gyration tensors are sorted ({Λ1(t) > Λ2(t) > Λ3(t)}) and then averaged over time and ensemble. Of interest are especially the square-roots of the sorted, averaged eigenvalues {Λ1(t),Λ2(t),Λ3(t)}={λ1,λ2,λ3}⁠, which represent the three principal radii of gyration of the polymer along the corresponding three principal axes.42 We further use them to calculate the radius of gyration RG via
(10)
The results for a ring polymer with N = 100 in an MPCD-solvent are
λ1/σ=4.22±0.12,λ2/σ=2.87±0.15,λ3/σ=1.52±0.05,RG/σ=5.35±0.15.
The uncertainty indication refers to one standard deviation of the mean (s̄=s/n)⁠.

The effective monomer model (EMM) is a coarse-grained model for polymers, which was introduced by Ruiz-Franco et al.29 The principle behind this model is to keep the polymer as monomer resolved for the interaction with the solvent, but remove internal interactions between the monomers. By integrating out the fluctuation degrees of freedom, the EMM describes a rigid but soft object and can be categorized between a polymer and a colloid. Therefore, we will refer to the coarse-grained polymers also using the term “colloid.”

To initialize the effective polymer, the first step is to divide the simulation box into cells each with side length equal to the monomer diameter a = σ. From simulations using the monomer resolved model (Sec. II B), we obtain the monomer distribution relative to the center-of-mass as a radial density function ρmon(r), with r being the absolute distance to the center-of-mass. The radial monomer density fulfills the normalization condition43 
(11)
where Nmon(r) is the cumulative particle number from the center-of-mass at r′ = 0 up to r′ = r. With that, we also define a cutoff radius Rcoll, on which we impose the condition ρmon(Rcoll)σ3 < 10−3, to limit the polymer’s extension. Then, accordingly, Nmon(Rcoll) = N is the total monomer number.

To continue with initializing the effective polymer, a uniform random number R∼U(0,1) is drawn for each of the cells and compared to the value of the volume density ρmon(rcell)σ3 at the distance rcell of the cell’s center to the simulation box center. If R≤ρmon(rcell)σ3⁠, we insert a monomer at the center of the respective cell. At the end, we have a number of monomers Neff equal to the number of successful insertions with expectation value Neff=N⁠. Figure 1 shows an example configuration generated through this process.

FIG. 1.
Example configuration after initialization for the effective monomer model (EMM). The figure shows a projection on the xy-plane of the 3D-colloid.
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Example configuration after initialization for the effective monomer model (EMM). The figure shows a projection on the xy-plane of the 3D-colloid.

FIG. 1.
Example configuration after initialization for the effective monomer model (EMM). The figure shows a projection on the xy-plane of the 3D-colloid.
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Example configuration after initialization for the effective monomer model (EMM). The figure shows a projection on the xy-plane of the 3D-colloid.

Close modal

Instead of undergoing molecular dynamics following the intermolecular potentials as described for the MRM (Sec. II B), the colloid moves as a rigid body and keeps its monomer configuration fixed over the course of the simulation. Consequently, the EMM neglects the angular momentum of the polymer. The colloid displacement, and with it all effective monomers, is ballistic in the same way as for the solvent particles following Eq. (1) and uses its center-of-mass velocity V⃗cm for the ballistic movement.

As for the interaction with the solvent, we can simply let the effective monomers participate in the collision as described for the MRM. After the collision, the center-of-mass velocity V⃗cm needs to be updated using the new effective monomer velocities
(12)

Ruiz-Franco et al.29 suggested that after calculating the new center-of-mass velocity, the effective monomer velocities should follow such that each monomer has the same velocity, which is the center-of-mass velocity (V⃗j(t+h)=V⃗cm(t+h))⁠, before the next collision step. If we observe the colloid temperature, however, we find that this balancing makes the colloid cooler compared to the thermostated solvent. This is because the momentum exchange occurs through the effective monomers with mass Mmon as opposed to the entire colloid with mass M = NeffMmon. Through matching the velocities of the monomers with the center-of-mass velocity, however, they are no longer Maxwell–Boltzmann distributed and their temperature is on average comparably lower. Even after a collision with the thermostated solvent particles, the temperature has not adjusted sufficiently fast before the next averaging of the velocities occurs. For this reason, after a collision, the individual monomer velocities are kept for the next collision in this work. The average velocity is only used for the colloid displacement. Figure 2 shows the difference in temperature for both methods.

FIG. 2.
Temperature development of the EMM polymer, with initial velocity V⃗cm(0)=0. (a) Temperature development of the EMM polymer if the individual effective monomer velocities V⃗j are averaged with each other before each collision step. The temperature fluctuates around an average value far lower than 1 (kBT < 1). (b) Temperature development of the EMM polymer if the individual effective monomer velocities V⃗j are kept between collision steps. The temperature fluctuates well around 1 (kBT = 1).
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Temperature development of the EMM polymer, with initial velocity V⃗cm(0)=0⁠. (a) Temperature development of the EMM polymer if the individual effective monomer velocities V⃗j are averaged with each other before each collision step. The temperature fluctuates around an average value far lower than 1 (kBT < 1). (b) Temperature development of the EMM polymer if the individual effective monomer velocities V⃗j are kept between collision steps. The temperature fluctuates well around 1 (kBT = 1).

FIG. 2.
Temperature development of the EMM polymer, with initial velocity V⃗cm(0)=0. (a) Temperature development of the EMM polymer if the individual effective monomer velocities V⃗j are averaged with each other before each collision step. The temperature fluctuates around an average value far lower than 1 (kBT < 1). (b) Temperature development of the EMM polymer if the individual effective monomer velocities V⃗j are kept between collision steps. The temperature fluctuates well around 1 (kBT = 1).
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Temperature development of the EMM polymer, with initial velocity V⃗cm(0)=0⁠. (a) Temperature development of the EMM polymer if the individual effective monomer velocities V⃗j are averaged with each other before each collision step. The temperature fluctuates around an average value far lower than 1 (kBT < 1). (b) Temperature development of the EMM polymer if the individual effective monomer velocities V⃗j are kept between collision steps. The temperature fluctuates well around 1 (kBT = 1).

Close modal

The monomer configuration is not redrawn during the simulation process, but colloid dynamics are evaluated by averaging over a sample size of 300 fixed rigid body configurations instead, with each run consisting of 5 × 104 MPCD steps.

The penetrable soft colloid model (PSC) was also introduced by Ruiz-Franco et al.29 In this coarse-grained model, the monomer resolution is removed, and instead the polymer is depicted as one single spherical colloid that is to a certain extent penetrable for solvent molecules. Movement of the solvent particles inside the colloid leads to interaction between the two, where the colloid itself acts as one rigid particle with no internal interaction.

The soft colloid cannot be viewed as a point-like particle in this model. Therefore, to govern angular momentum exchange with the solvent, we need the moment of inertia I of the colloid in addition to its total mass M = NMmon. For a radially symmetric object with axis of rotation through the center-of-mass, the moment of inertia tensor can be simplified to a scalar. In this case, it is calculated as
(13)
with the radial monomer density function ρmon(r) and cutoff radius Rcoll as defined in Sec. II C.
The coupling of the PSC to the solvent is implemented during the streaming step of the solvent particles, contrary to the previous models where the coupling is during the multi-particle collision step. During the streaming step, the PSC moves ballistically, which works analogously to the solvent
(14)
using the position vector R⃗(t) of the colloid and its velocity vector V⃗(t)⁠.
The solvent particles each have to pass a Monte Carlo trial move in order to follow through with their ballistic movement according to Eq. (1). If the trial move is rejected, they are scattered by the PSC instead. For this step, it is convenient to define the solvent particles position in reference to the PSC such that r denotes the distance of the solvent particle relative to the colloid center R⃗⁠. Trial moves for which both the initial and the final position lie outside the colloid are always accepted. If either one of the solvent particle’s initial relative position r1=|r⃗1|=|r⃗i(t)−R⃗(t+h)| or trial end relative position r2=|r⃗2|=|r⃗i(t+h)−R⃗(t+h)| are inside the colloid (r1 ≤ Rcoll or r2 ≤ Rcoll), the acceptance probability is calculated as follows:
(15)
The Monte Carlo acceptance probability utilizes the solvent volume density ρsol(r), which is the local volume fraction that is populated by the solvent after the monomer excluded volume is subtracted. Since monomers are assumed to be spherical the solvent density is calculated via
(16)
There are very few instances where the solvent density calculated this way is negative. In this case, it is set to 0.
If the Monte Carlo trial move is rejected, the solvent particle is scattered by the PSC. For that, we first define the scattering point s⃗⁠. Dependent on whether both a trial initial position and trial end position of the particle lie inside the colloid, or one of them lies outside, the particles are either scattered at the halfway point of that movement or at the surface of the colloid, respectively. In other words,
(17)
To show this more clearly, the two different cases are demonstrated in Fig. 3.
FIG. 3.
Scattering of the solvent particle by the PSC. Case 1: The solvent trial movement would be only partially inside the colloid, hence the particle is scattered at the colloid surface. Case 2: The solvent trial movement would be fully inside the colloid; therefore, scattering happens at the halfway point.
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Scattering of the solvent particle by the PSC. Case 1: The solvent trial movement would be only partially inside the colloid, hence the particle is scattered at the colloid surface. Case 2: The solvent trial movement would be fully inside the colloid; therefore, scattering happens at the halfway point.

FIG. 3.
Scattering of the solvent particle by the PSC. Case 1: The solvent trial movement would be only partially inside the colloid, hence the particle is scattered at the colloid surface. Case 2: The solvent trial movement would be fully inside the colloid; therefore, scattering happens at the halfway point.
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Scattering of the solvent particle by the PSC. Case 1: The solvent trial movement would be only partially inside the colloid, hence the particle is scattered at the colloid surface. Case 2: The solvent trial movement would be fully inside the colloid; therefore, scattering happens at the halfway point.

Close modal
Naturally, on elastic collision, we would impose energy conservation, linear momentum conservation, and angular momentum conservation. However, the PSC is technically not a continuous rigid body with a smooth surface, but more a construct of monomers, each with their individual momentum. Therefore, instead of imposing energy conservation, the PSC should be viewed as a thermostated heat bath and the new solvent relative velocities are assigned randomly according to the distributions21 
(18)
and
(19)
with the velocity component νn that is perpendicular to the colloid surface and the velocity component νt that is tangential to it. Linear and angular momentum conservation remain upheld. The new absolute velocity of the solvent particle then calculates as
(20)
using the PSC’s angular velocity vector Ω⃗(t)⁠, and both the unit vector normal to the collision plane ên and the unit vector tangential to it êt⁠. The normal unit vector ên is defined such that it points away from the colloid center, independent of whether the solvent trial move constitutes penetration or escape, with
(21)
By choosing the normal unit vector ên as such, detailed balance is restored for the major part of the colloid. As a result, the desired solvent density profile ρsol(r) can be achieved in the end. The tangential unit vector êt is chosen randomly under the condition that it is orthogonal to the normal unit vector: êt⊥ên⁠.
After collision with particle i, the velocity and angular velocity of the PSC also have to be updated,
(22)
(23)
The collisions are resolved sub-sequentially for all solvent particles, while instantaneously updating the colloid linear and angular velocity in between.
The positions of solvent particles with rejected Monte Carlo trial moves are updated according to their new velocities as
(24)
The last step, which is the multi-particle collision among solvent particles, is performed in the same way as in the absence of a solute. The dynamics of the PSC is evaluated by averaging over 400 independent runs, where each run consists of 5 × 104 MPCD steps.

It turns out that neither of the two models developed by Ruiz-Franco et al.29 deliver satisfying diffusion coefficients compared to the MRM for a ring polymer (see Table I in Sec. III B for results). While the EMM works well for linear chains and the PSC is able to capture star polymers, ring polymers require the development of a different model. Therefore, we introduce two new coarse-graining models to test against the MRM for ring polymers: the PSC model with ellipsoidal shape (EPSC) and the PSC model with increased interaction factor (PSC-f).

TABLE I.

The summarized results regarding polymer diffusion for all employed models. In this table, nsim expresses the number of independent simulations that were used to form the average value. The standard deviation of the sample mean is used as an uncertainty measure.

ModelnsimDl/D0RH/aD/D0
MRM 400 0.001 28 ± 0.000 06 4.29 0.001 73 
EMM 300 0.001 15 ± 0.000 06 4.64 0.001 60 
PSC 400 0.007 02 ± 0.000 32 0.99 0.007 48 
EPSC 300 0.006 30 ± 0.000 30 1.10 0.006 56 
PSC-f (th.)a 400 0.001 91 ± 0.000 09 3.14 0.002 36 
PSC-f (emp.)b 400 0.001 32 ± 0.000 06 4.19 0.001 77 
ModelnsimDl/D0RH/aD/D0
MRM 400 0.001 28 ± 0.000 06 4.29 0.001 73 
EMM 300 0.001 15 ± 0.000 06 4.64 0.001 60 
PSC 400 0.007 02 ± 0.000 32 0.99 0.007 48 
EPSC 300 0.006 30 ± 0.000 30 1.10 0.006 56 
PSC-f (th.)a 400 0.001 91 ± 0.000 09 3.14 0.002 36 
PSC-f (emp.)b 400 0.001 32 ± 0.000 06 4.19 0.001 77 
a

PSC-f whose factor was calculated from theoretical considerations.

b

PSC-f whose factor was empirically approximated.

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Within the standard PSC model we have approximated the ring polymer to a spherical shape. While this may be reasonable for star polymers or polymers in a collapsed state due to bad solvent conditions, for ring polymers or linear chain polymers in good solvent conditions, an ellipsoidal shape poses a much more appropriate choice. Considering that the gyration tensor obtained from the MRM suggests an ellipsoid with an aspect ratio of λ1/λ3 ≈ 3 for our ring polymer, this can hardly be considered an approximately spherical shape. The EPSC is a refined version of the PSC discussed in Sec. II D, which does not assume spherical shapes but allows the polymer to be approximated by an ellipsoid with any set of three length parameters {a, b, c} that represent its extension along the three principal axes. It is known that the diffusion coefficient of ellipsoids is smaller than that of spheres with the same volume.44 This indicates that an ellipsoidal colloid could close the gap between discrepancies in the measured diffusion coefficients between the MRM and the standard PSC model.

1. Parameterizing the EPSC

Because of the anisotropy of the colloid, its orientation is tracked over time. Hence, we define a rotation matrix Q̂(t) that we use to track the orientation of the colloid. Q̂(t) is updated every step using its angular velocity Ω⃗(t)⁠.

We define the frame of reference of the colloid as the frame whose axes are aligned with the principal axes of the colloid, its origin is located at the center-of-mass of the colloid, and it moves and rotates together with the colloid. In other words, for all times t in the frame of reference of the colloid (denoted by primed variables), we have R⃗′(t)=0⁠, V⃗′(t)=0⁠, Ω⃗′(t)=0⁠, and Q̂′(t)=1⁠. In the frame of reference of the colloid, we parameterize the EPSC with the parameters {a, b, c} and resulting volume VE,
(25)
To facilitate the calculations involving the EPSC, we use stretched dimensionless spherical coordinates (r′, θ′, ϕ′) instead of Cartesian coordinates (x, y, z),
(26)
The EPSC has a moment of inertia Î (defined below), and total mass M = NMmon. For the monomer density function, we use the radially symmetric density function evaluated previously, but transform it according to the stretched spherical coordinates. That means, we transform r → r′ and then normalize again the monomer density ρmon(r′),
(27)
(28)
where dV′ = abc r′2 sin θ dr′dθdϕ and N = Nmon(Rcoll). We can hereby calculate all elements to calculate the moment of inertia tensor for the ellipsoid Î and can write it in diagonal form45 
(29)
with
(30a)
(30b)
(30c)
By transitioning from a sphere to an ellipsoid, it makes sense to impose the condition that the volume of the colloid does not change,
(31)
The square-roots of the gyration tensor’s eigenvalues {λ1, λ2, λ3}, obtained from the MRM simulation, are equal to the semi-axis lengths of the ellipsoid of inertia.46 In order to match a physical ellipsoid, they can simply be scaled with a factor. Using Eq. (31) then allows us to determine {a, b, c},
(32)

2. EPSC movement and coupling to the solvent

In principal, the EPSC is simulated in the same way as the spherical PSC. Each time step consists of three steps: EPSC ballistic movement, solvent particles ballistic movement with colloid interaction, and the multi-particle collision between the solvent particles.

The ballistic movement of the EPSC in this model encompasses both translation and rotation. The translational ballistic movement is the same as for the spherical PSC according to (14). Additionally, in this model, we also track colloid orientation Q̂(t)⁠. The new colloid orientation matrix Q̂(t+h) is calculated with the help of a standard rotation matrix D̂(R⃗(t),α(t))⁠,
(33)
The rotation matrix D̂(R⃗(t),α(t)) is defined through the rotation angle α(t) and the rotation axis R⃗(t)⁠, both of which can be calculated from the angular velocity Ω⃗(t) following
(34)
(35)
The ballistic movement of the solvent particles [Eq. (1)] is a trial move, which will be performed if accepted. In case of rejection, the solvent particle is scattered by the EPSC. This works analogously to the spherical PSC, but now using stretched spherical coordinates. The calculation of the normal unit vector ên changes for the EPSC. It is not simply parallel to the scattering vector anymore (see Fig. 4). The normal unit vector ên for the EPSC is calculated by
(36)
where (xs, ys⁠, zs) define the scattering point coordinates in the colloid frame of reference. The new colloid angular velocity is calculated using the inverse moment of inertia tensor Î−1 such that
(37)
FIG. 4.
For the ellipsoid, the normal unit vector ên at the scattering point (xs, ys, zs) is generally not parallel to the scattering vector s⃗=(xs,ys,zs).
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For the ellipsoid, the normal unit vector ên at the scattering point (xs, ys⁠, zs) is generally not parallel to the scattering vector s⃗=(xs,ys,zs)⁠.

FIG. 4.
For the ellipsoid, the normal unit vector ên at the scattering point (xs, ys, zs) is generally not parallel to the scattering vector s⃗=(xs,ys,zs).
View largeDownload slide

For the ellipsoid, the normal unit vector ên at the scattering point (xs, ys⁠, zs) is generally not parallel to the scattering vector s⃗=(xs,ys,zs)⁠.

Close modal

The third step, which is the multi-particle collision among solvent particles, is performed the same way as if there was no solute present. The EPSC dynamics is evaluated by averaging over a sample size of 300 independent runs of 5 × 104 MPCD steps each.

If the theoretical diffusion coefficient is known, or it can be acquired by a simulation following the MRM algorithm (as described in Sec. II B), it is possible to adjust the acceptance probability pacc(r1 → r2) of the PSC algorithm in such a way that the desired diffusion coefficient is enforced. The algorithm of the PSC-f is based on the standard PSC algorithm. The only difference is in the calculation of the Monte Carlo acceptance probability pacc(r1 → r2) and consequences arising from that.

If the generation probability is symmetric, the acceptance probability in the Metropolis algorithm is usually defined as47 
(38)
where ρ(ri) describes the probability density to find state ri. If we introduce a factor 0 < f ≤ 1, detailed balance would be upheld for
(39)
as well. That means for 0 < f ≤ 1 the ensemble will converge to its equilibrium state. The factor f only lowers the rate of this convergence. The PSC algorithm does not use the usual Metropolis algorithm, but the acceptance probability is conditional. Tailored to the PSC algorithm, it results in
(40)
In order to follow detailed balance in this model also, the orientation of the normal unit vector ên needs to be adapted. The modified acceptance probability ensures more interaction on average, but the ratio of solvent particles that are reflected to those that are only deflected or pass unhindered has to be the same regardless of the factor f. One way to realize this is by distinguishing between four cases (as opposed to the two cases in the original PSC model) in order to determine the orientation of the normal unit vector ên⁠:

  • r2 ≤ r1 and ρsol(r2)ρsol(r1)≥1⇒ên points outwards,

  • r2 ≤ r1 and ρsol(r2)ρsol(r1)<1⇒ên points outwards,

  • r1 < r2 and ρsol(r1)ρsol(r2)≥1⇒ên points inwards,

  • r1 < r2 and ρsol(r1)ρsol(r2)<1⇒ên points inwards, with probability: (1−f)1−fρsol(r1)ρsol(r2)−1⁠.

This algorithm facilitates comparison between the original PSC and the PSC-f. For future applications of this model, we recommend to adapt the PSC-f model by using the Metropolis algorithm with factor [Eq. (A3)] unconditionally everywhere and have the normal unit vector ên always point backward compared to the trial direction. This simplifies the model, but naturally requires a different numerical value for the factor in order to obtain the same diffusion coefficient.
It is possible to estimate the factor f from theoretical considerations, which is shown in the  Appendix. However, it turns out that the resulting diffusion constant does not resemble the desired diffusion constant sufficiently well (see Table I). Therefore, the factor has to be empirically estimated, keeping in mind that decreasing the factor increases the interaction rate and therefore decreases the diffusion coefficient. For the PSC-f, we tuned f to fit the desired outcome regarding the diffusion coefficient and obtained a value of
f=0.995.
This value is somewhat smaller than the theoretical estimate (see the  Appendix). Colloid dynamics are evaluated by averaging over 400 independent runs of 5 × 104 MPCD steps each.

In this section, we present results for the monomer and solvent density profiles and calculate the diffusion coefficient of the ring polymer within the different models.

The radial monomer density ρmon(r) is assessed through simulation using the MRM. In order to obtain a smooth analytical function, the discrete data are approximated using an empirically created fit function. Figure 5 shows the simulation data for a ring polymer organized into a histogram and their empirical fit, namely,
(41)
with fit parameters A, b, d, s. The fit parameters for the ring polymer with N = 100 result in A = 2.246 × 10−4σ−8, b = 66.16σ3, d = 365.2σ5, and s = 2.128σ.
FIG. 5.
Radial monomer density distribution from the MRM with MPCD, and fit function according to Eq. (41) of a ring polymer embedded in an MPCD solvent.
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Radial monomer density distribution from the MRM with MPCD, and fit function according to Eq. (41) of a ring polymer embedded in an MPCD solvent.

FIG. 5.
Radial monomer density distribution from the MRM with MPCD, and fit function according to Eq. (41) of a ring polymer embedded in an MPCD solvent.
View largeDownload slide

Radial monomer density distribution from the MRM with MPCD, and fit function according to Eq. (41) of a ring polymer embedded in an MPCD solvent.

Close modal

From the monomer density function and using Eq. (16), we obtain the theoretical solvent density profile. Considering time for equilibration, the actual radial distribution of the solvent particles is compared with the theoretical distribution for all soft colloid models. The solvent density function in the original PSC and the PSC-f can be seen in Fig. 6. By introducing the Monte Carlo acceptance probability for solvent movement in the PSC models, the colloid creates an excluded volume, which enables a more realistic solvent density distribution. This is a feature that is not inherent to the MRM or the EMM because, in the standard MPCD model, there is no excluded volume for solvent particles.1 As shown in Fig. 6, the desired solvent density distribution is achieved with the PSC. Introducing an increased interaction factor in the PSC-f keeps this feature if the acceptance rules are properly executed.

FIG. 6.
The actual radial solvent density distribution (green histogram) compared to the theoretical function (black line) in (a) the PSC model and (b) the PSC-f model. At small r/σ, there are higher fluctuations due to the smaller volume elements dV. At r/σ = 8, the solvent is almost at bulk density, which is depicted by the dashed line at ρsol(r)σ3 = 1.
View largeDownload slide

The actual radial solvent density distribution (green histogram) compared to the theoretical function (black line) in (a) the PSC model and (b) the PSC-f model. At small r/σ, there are higher fluctuations due to the smaller volume elements dV. At r/σ = 8, the solvent is almost at bulk density, which is depicted by the dashed line at ρsol(r)σ3 = 1.

FIG. 6.
The actual radial solvent density distribution (green histogram) compared to the theoretical function (black line) in (a) the PSC model and (b) the PSC-f model. At small r/σ, there are higher fluctuations due to the smaller volume elements dV. At r/σ = 8, the solvent is almost at bulk density, which is depicted by the dashed line at ρsol(r)σ3 = 1.
View largeDownload slide

The actual radial solvent density distribution (green histogram) compared to the theoretical function (black line) in (a) the PSC model and (b) the PSC-f model. At small r/σ, there are higher fluctuations due to the smaller volume elements dV. At r/σ = 8, the solvent is almost at bulk density, which is depicted by the dashed line at ρsol(r)σ3 = 1.

Close modal
The diffusion coefficient D can be estimated from the mean squared-displacement of the center-of-mass over time (MSD = Δr⃗cm2(t)⁠) according to
(42)
by fitting the MSD with a linear function. The MSD of all employed models is shown in Fig. 7. Compared to the monomer resolved model (MRM), only the PSC-f model with empirically estimated factor [PSC-f (emp.)] exhibits a similar diffusion behavior for ring polymers. In all other models, the diffusion coefficient is either slightly underestimated (EMM) or significantly overestimated (PSC and EPSC).
FIG. 7.
The mean squared-displacement (MSD = Δr⃗cm2(t)) of all presented models over time. The gray continuous line shows a linear curve to indicate the diffusive regime. All lines that are parallel to that curve are of diffusive behavior. The PSC-f is shown with both, the theoretically calculated factor [PSC-f (th.)], and the factor we tuned empirically [PSC-f (emp.)].
View largeDownload slide

The mean squared-displacement (MSD = Δr⃗cm2(t)⁠) of all presented models over time. The gray continuous line shows a linear curve to indicate the diffusive regime. All lines that are parallel to that curve are of diffusive behavior. The PSC-f is shown with both, the theoretically calculated factor [PSC-f (th.)], and the factor we tuned empirically [PSC-f (emp.)].

FIG. 7.
The mean squared-displacement (MSD = Δr⃗cm2(t)) of all presented models over time. The gray continuous line shows a linear curve to indicate the diffusive regime. All lines that are parallel to that curve are of diffusive behavior. The PSC-f is shown with both, the theoretically calculated factor [PSC-f (th.)], and the factor we tuned empirically [PSC-f (emp.)].
View largeDownload slide

The mean squared-displacement (MSD = Δr⃗cm2(t)⁠) of all presented models over time. The gray continuous line shows a linear curve to indicate the diffusive regime. All lines that are parallel to that curve are of diffusive behavior. The PSC-f is shown with both, the theoretically calculated factor [PSC-f (th.)], and the factor we tuned empirically [PSC-f (emp.)].

Close modal
Hydrodynamic interactions are of long-range nature and velocity correlations persist over a long time.48 Consequently, for a box with periodic boundaries, the diffusion coefficient is dependent on the system size. For this reason, the diffusion coefficient obtained by the simulation is only the finite-size diffusion coefficient Dl of system size l. The infinite-system-size diffusion coefficient D can be obtained from the finite-size diffusion coefficient Dl by solving for the hydrodynamic radius RH in49 
(43)
with the system size l and the dynamic viscosity of the solvent ηsol = ϱν. The infinite-system-size diffusion coefficient can subsequently be calculated using the Stokes–Einstein relation
(44)
The unit of the diffusion coefficient is D0=a2/τ=kBTa2/m⁠. Considering the system size l = 46a and a solvent dynamic viscosity ηsol/η0 = 7.143 (η0=kBTm/a2)⁠, the results regarding polymer diffusion for all employed models are summarized in Table I.

When we calculate the ratio between radius of gyration and hydrodynamic radius, we can further validate the accuracy of our methods. Literature suggests a ratio of RG/RH = 1.2533 for Gaussian ring polymers.50 Using the MRM, we find a ratio of RG/RH = 1.25, which is in excellent agreement with the theoretical value.

Neither the EMM nor the PSC model from Ruiz-Franco et al.29 are able to adequately reproduce the long-time diffusion constant of the ring polymer. Adapting the PSC model by enabling not only spherically shaped colloids but general ellipsoids (EPSC) does not significantly improve the diffusion behavior for ring polymers either, despite the fact that they have rather large aspect ratios a/c ≈ 3. On the other hand, introducing a factor to the PSC’s Monte Carlo acceptance probability (PSC-f), thus freely increasing interaction, provides the desired results and slows down the diffusion coefficient such that it matches the MRM. An advantage of this method is that it offers a lot of freedom since the factor can be chosen anywhere between 0 < f ≤ 1. As a result, the diffusion coefficient can be decreased arbitrarily within range while maintaining the desired solvent density profile. A disadvantage of this method is that the target diffusion coefficient has to be known, and the factor has to be approximated empirically.

The reason why the PSC model as proposed by Ruiz-Franco et al.29 does not work well for ring polymers lies most likely in the fact that it relies on the solvent density gradient to control the probability of interaction. Compared to that, interaction in the monomer-resolved model is better resembling the dependence on absolute solvent density. The reason why this model works for star polymers but not for linear chains and rings is probably that star polymers have a steeper monomer density function, and hence a higher solvent density gradient overall. As a result they possess higher interaction probabilities and consequently a slower diffusion.

For future work, it would be interesting to investigate the different coarse-grained models using varying degrees of polymerization N. Furthermore, testing the several coarse-grained models on many-polymer-systems would be the natural next step in this analysis.

We would like to acknowledge the Vienna Scientific Cluster (VSC) for providing us with the computational resources that were necessary to perform our simulations.

The authors have no conflicts to disclose.

Lisa Sappl: Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (lead). Christos N. Likos: Conceptualization (lead); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Project administration (lead); Resources (lead); Supervision (equal); Validation (supporting); Writing – review & editing (lead). Andreas Zöttl: Formal analysis (equal); Investigation (equal); Methodology (supporting); Supervision (equal); Writing – original draft (supporting); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The finite-size diffusion coefficient Dl is inversely proportional to the solvent’s viscosity [Eq. (43)]. For the solvents that are of interest to these kind of simulations, usually the collisional viscosity dominates over the kinetic viscosity νkin ≪ νcoll, and therefore, the kinetic viscosity may be neglected and the viscosity is approximately inversely proportional to the time step h. This leads to an inverse relationship between the diffusion coefficient and the expected Monte Carlo rejection probability prej(r1 → r2) = 1 − pacc(r1 → r2),
(A1)
Suppose that the diffusion coefficient obtained using the PSC model is DPSC, then the theoretically desired value Dtheo can be obtained using the MRM. The two differ by a factor fD, which can be expressed in terms of the expected Monte Carlo rejection probabilities as
(A2)
The increased interaction factor f used in the Monte Carlo scheme directly changes the acceptance probability
(A3)
By combining Eqs. (A2) and (A3), we obtain the relationship between the two factors
(A4)
The expected Monte Carlo rejection probability prej(r1→r2) can be tracked empirically during a simulation using the original PSC model. It can also be derived theoretically using the joint probability density f(r1, r2) for r1 and r2 in
(A5)
In equilibrium, the initial positions r1 are distributed following ρsol(r) as provided by the Monte Carlo scheme. The trial end position r2=|r⃗1+hv⃗| is dependent on the initial position and its probability density is also a function of r1. The variable that connects r1 and r2, which is Δr=|hv⃗|⁠, on the other hand, is independent of r1 and follows only the Maxwell–Boltzmann distribution. The joint probability density f(r1, Δr) = f1(r1) · fΔ(Δr) is therefore separable and simply a product of the individual probability densities
(A6)
(A7)
Both quantities are expressed using spherical coordinates, and for Δr, they are oriented such that θΔ denotes the angle between the vectors r⃗1 and Δr⃗ (see Fig. 8). The rejection probability prej(r1 → r2) uses the quantities r1 and r2. From geometrical considerations, it is possible to derive the transformation (Δr, θΔ, ϕΔ⁠) → (r2, θ2, ϕ2) with
(A8)
resulting into the Jacobian J,
(A9)
so that
(A10)
where θ2 denotes the angle between the vectors r⃗1 and r⃗2⁠. The Jacobi determinant J helps to transform the volume element. The joint probability density for r1 and r2 can now be calculated by transforming and subsequent integration over the other variables,
(A11)
FIG. 8.
Visualization of the relationship between r⃗1, r⃗2, and Δr⃗.
View largeDownload slide

Visualization of the relationship between r⃗1⁠, r⃗2⁠, and Δr⃗⁠.

FIG. 8.
Visualization of the relationship between r⃗1, r⃗2, and Δr⃗.
View largeDownload slide

Visualization of the relationship between r⃗1⁠, r⃗2⁠, and Δr⃗⁠.

Close modal
For a general solvent density function ρsol(r), the expression in Eq. (A5) can be calculated numerically, although it is wise to rearrange Eq. (A11) for that by joining the hyperbolic sine and exponential function, as otherwise the hyperbolic sine increases too quickly. The factor we obtained using this theoretical approach was
f=0.99757,
which did not lead to satisfying results for the diffusion constant. Also, by not neglecting the contributions from the kinetic viscosity in the theoretical approach, the factor did not improve significantly (f = 0.997 38).
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