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Based on a recent simulation study [Masubuchi et al., Macromolecules, 56, 9359 (2023)], the cycle rank plays a significant role in determining the fracture characteristics of network polymers. However, the study only considered energy-minimized networks without the effects of thermal agitation. We conducted Brownian dynamics simulations at various stretch rates to address this gap. The results showed that even with Brownian motion, the strain and stress at the break obtained for different node functionalities and conversion ratios exhibited master curves if plotted against cycle rank. These master curves were dependent on the strain rate, with the curves tending to approach those observed in energy-minimized simulations as the strain rate decreased, even though the fracture process was affected by the competition against Brownian motion, elongation, and bond degradation., Comment: 12 pages, 4 figures

Despite many efforts, the fracture of network polymers under extension is yet to be elucidated. This study investigated the effect of strand molecular weight on the fracture characteristics of the networks made from star-branched prepolymers with various node functionalities $3 \le f \le 8$ and conversion ratios $0.6 \le {\phi}_c \le 0.95$ via molecular simulations with phantom chains. The networks were created via end-linking reactions of star polymers with the arm molecular weight $N_a$ = 2, 5, and 10, simulated by a Brownian dynamics scheme. The cycle rank of the percolated network ${\xi}$ was entirely consistent with the mean-field theory, implying that the reaction occurred independently and the network connectivity was statistically fair. Afterward, the networks were alternatively subjected to energy minimization and uniaxial stretch until the break. From the stress-strain curves recorded during the stretch, the stretch and stress at the break, ${\lambda}_b$ and ${\sigma}_b$, and work for fracture $W_b$ were obtained. For these values, the following relations were found: ${\lambda}_b \sim N_a^{(1/2)} {\xi}^{(-1/3)}$, ${\sigma}_b \sim {\nu}_{br} N_a^{(4/3)} {\xi}^{(2/3)}$, and $W_b \sim {\nu}_{br} N_a^{(4/3)} {\xi}^{(2/3)}$, where ${\nu}_{br}$ is the branch point density., Comment: 21 pages, 10 figures

Although the reverse non-equilibrium molecular dynamics (RNEMD) simulation method has been widely employed, the range of applicability is yet to be discussed. In this study, for the first time, we systematically examine the method against an unentangled melt of the Kremer-Grest type chain. The simulation results indicate that as the shear rate increases, the temperature and density become inhomogeneous. However, the average viscosity remains consistent with the results obtained using the SLLOD method under homogeneous temperature and density. We also confirm that the temperature-density inhomogeneity does not significantly affect polymer conformation.

Despite many attempts, the relation between fracture and structure of polymer networks is yet to be clarified. For this problem, a recent study for phantom chain simulations [Macromolecules, 56, 9359 (2023)] has demonstrated that the fracture characteristics obtained for polymer networks with various node functionalities and conversion ratios lie on master curves if they are plotted against cycle rank. In this study, we extended the simulation to the effect of prepolymer concentration on the relationships between cycle rank and fracture characteristics within the concentration range of 1

A few experiments have reported that the time development of shear stress under fast start-up shear deformations exhibits double peaks before reaching the steady state for bimodal blends of entangled linear polymers in specific conditions. To analyze the molecular origin of this phenomenon, multi-chain slip-link simulations based on the primitive chain network model were conducted for the data of a bimodal polystyrene solution reported by Osaki et al. [J Pol Sci B Pol Phys, 38, 2043 (2000)] Owing to the reasonable agreement with the data and the simulation results, the stress was decomposed into contributions from long and short-chain components and decoupled into segment number, stretch, and orientation. The analysis revealed that the first and second peaks correspond to the short-chain orientation and the long-chain stretch, respectively. The results also imply that the peak positions are not affected by the mixing of short and long chains, whereas the second peak intensity depends on the mixing conditions in a complicated manner, dominating the emergence of double peaks., Comment: 16 pages, 5 figures

Despite numerous studies, the relationship between network structure and fracture remains unclear. In this study, the fracture properties of end-linking networks were compared with those of loop-free analogs made from star prepolymers by performing phantom chain simulations. The networks were created from equilibrated sols of stoichiometric mixtures of linear prepolymers and f-functional linkers through end-linking reactions using Brownian dynamics schemes. The examined networks, with various f values (between 3 and 8) and strand-connection rates ({\phi}_s), were evaluated in terms of the primary loop fraction and the cycle rank {\xi}. These structural characteristics were consistent with mean-field theories that assume independent reactions. Energy minimization and uniaxial stretch were applied to the networks until they broke without Brownian motion. The fracture characteristics, including strain ({\epsilon}_b), stress ({\sigma}_b), and work for fracture (W_b), were obtained from stress-strain curves. The end-linking networks exhibited larger {\epsilon}_b and smaller {\sigma}_b and W_b than those for star networks due to primary loops, at the same set of f and {\phi}_s. However, {\epsilon}_b, {\sigma}_b/{\nu}_br and W_b/{\nu}_br (with {\nu}_br being the branch point density) lie on the same master curves as those for star networks if they are plotted against {\xi}. This result implies that the fracture of end-linking networks is essentially the same as that for star analogs, and the effects of primary loops are embedded in {\xi}., Comment: 20 pages, 8 figures

Slip-spring models are valuable tools for simulating entangled polymers, bridging the gap between bead-spring models with excluded volume and network models with presumed reptation motion. This study focuses on the DPD-SS (Dissipative Particle Dynamics - Slip-Spring) model, which introduces slip-springs into the standard DPD polymer model with soft-core interactions. By systematically adjusting the fugacity of slip-springs, the density of slip-springs within the system is varied. Simulation results demonstrate the compatibility of models with different slip-spring densities in terms of diffusion and linear relaxation modulus when the average number of slip-springs per chain is the same. The conversion between DPD-SS models concerning length and time is achieved through Rouse scaling, which utilizes the average number of DPD beads between consecutive anchoring points of slip-springs. Additionally, the modulus conversion is accomplished through the plateau modulus that takes account of fluctuations around entanglement. Notably, diffusion and relaxation modulus from the DPD-SS model align with those reported for standard Kremer-Grest and DPD models featuring strong repulsive interactions., Comment: 19 pages, 9 figures

Anisotropic shear relaxation is an interesting but rarely discussed issue in polymer dynamics under confinement [Abberton et al., Macromolecules, 48, 7631, 2015]. According to the earlier study of bead spring simulations for an unentangled polymer melt confined between two flat plates, the shear relaxation modulus taken perpendicular to the interface is accelerated by decreasing the distance between plates, whereas the parallel component is unchanged. This study observed similar anisotropic shear relaxation for entangled polymer melts under confinement in multi-chain slip-link simulations (primitive chain network simulations). The analysis demonstrated that the accelerated relaxation in the perpendicular component reflects the Rouse-type constraint release dynamics, for which the coarsening is upper-limited by the geometry. This result suggests a novel mechanism for anisotropic shear relaxation different from the modified chain statistics under confinement considered for unentangled systems., Comment: 12 pages, 6 figures

Fujiyabu et al. have experimentally reported that mixing of 3-arm star prepolymers into 4-arm analog improves the toughness of the resultant polymer networks compared to the base network composed of 4-arm star polymers only. For the mechanism of this phenomenon, this study conducted phantom chain simulations for polymer networks composed of mixtures of star branch prepolymers with equal arm length and different arm numbers, (f_1,f_2 )=(3,4), (3,6) and (3,8), for various f_1=3 prepolymer fractions {\phi}_3. The networks were created via end-linking reactions between prepolymers traced by a Brownian dynamics scheme, and the network structure was stored at different conversion ratios {\phi}_c, ranging from 0.6 and 0.9. The cycle rank of the gelated networks {\xi} is fully consistent with the mean field theory, demonstrating that the examined network structure is statistically fair. The networks were stretched with energy minimization until the break, and fracture characteristics including strain at break {\epsilon}_b, stress at break {\sigma}_b, work for fracture W_b, and the ratio of broken strands {\phi}_bb, were obtained. {\epsilon}_b, {\sigma}_b/{\phi}_bb, and W_b/{\phi}_bb data plotted against {\xi} roughly follow the master curves reported for the base networks without mixing, implying that the change of fracture properties by the mixing of f_1=3 mainly corresponds to a decrease of {\xi}. The mixing slightly suppresses {\sigma}_b/{\phi}_bb and W_b/{\phi}_bb for large f_2 cases compared to the base networks because of a biased breakage at the network strands without extenders, which are prepolymers with only two reacted arms. The analysis for broken strands exhibited a new master curve for the {\xi}-dependence of the molecular weight of broken strands., Comment: 17 pages, 8 figures

By phantom chain simulations, it has been recently discovered that the fracture characteristics of star polymer networks with different node functionality and conversion ratios can be described by the cycle rank of the networks [Masubuchi et al., Macromolecules, doi: 10.1021/acs.macromol.3c01291]. However, due to the employed simplifications and idealizations of the examined model networks, the results cannot be cast into realistic systems straightforwardly. For instance, the equimolar reaction was assumed in a limited volume for the binary mixture of star prepolymers. For this issue, the present study investigated the effects of stoichiometry by phantom chain simulations. Examined polymer networks were created from binary mixtures of star prepolymers with various mixing ratios by the end-linking reaction via Brownian dynamics simulations. The networks were stretched with energy minimization until the break. From the mechanical response, strain and stress at break and work for fracture were obtained. These fracture characteristics slightly decrease with increasing the contrast of volume fractions of the binary prepolymer blends when the node functionality is small and the conversion ratio is large. For the other cases, the stoichiometry does not impact the fracture behavior. The number ratio of broken bonds and the cycle rank exhibit similar stoichiometry dependence. Consequently, stoichiometry of prepolymer blends does not disturb the previously reported relationships between the fracture characteristics and the cycle rank., Comment: 13 pages, 6 figures

This study presents a coarse-grained molecular dynamics simulation model to investigate the process of oxidative aging in polymers. The chemical aging effect is attributed to the auto-oxidation mechanism, which is initiated by radicals, leading to the chain scission and crosslinking of polymer chains. In this study, we integrate a thermal oxidation kinetic model in the oxygen excess regime into the Kremer-Grest model, thereby enabling a reactive coarse-grained molecular dynamics simulation to capture the process of oxidative degradation. Our simulation reveals that when the timescale of the characteristic reaction step of oxidative degradation is shorter than the longest relaxation time of polymer chains, the scission sites exhibit spatial heterogeneity. This innovative simulation model possesses the potential to enhance our comprehension of polymer aging phenomena, thus making noteworthy contributions to the realm of polymer science and degradation chemistry., Comment: 32 pages, 16 figures

The influence of node functionality (f) on the fracture of polymer networks remains unclear. While many studies have focused on multi-functional nodes with f>4, recent research suggests that networks with f=3 exhibit superior fracture properties compared to those with f=4. To clarify this discrepancy, we conducted phantom chain simulations for star-polymer networks varying f between 3 and 8. Our simulations utilized equimolar binary mixtures of star branch prepolymers with a uniform arm length. We employed a Brownian dynamics scheme to equilibrate sols and induce gelation through end-linking reactions. We prevented the formation of odd-order loops owing to the binary reaction and second-order loops algorithmically. We stored network structures at various conversion ratios ({\phi}_c) and minimized energy to reduce computation costs induced by structural relaxation. We subjected the networks to stretching until fracture to determine stress and strain at break and work for fracture, {\epsilon}_b, {\sigma}_b, and W_b. These fracture characteristics are highly dependent on {\phi}_c for networks with small f but relatively insensitive for those with large f. Thus, the networks with small f exhibit greater fracture properties than those with large f at high {\phi}_c, whereas the opposite relationship occurs at low {\phi}_c. We analyzed {\epsilon}_b, {\sigma}_b, and W_b concerning the cycle rank {\xi} and the broken strand fraction {\phi}_bb. We found {\epsilon}_b, {\sigma}_b/{\phi}_bb, and W_b/{\phi}_bb monotonically decrease with increasing {\xi}, and the data for various f and {\phi}_c superpose with each other to draw master curves. These results imply that the mechanical superiority of the networks with small f comes from their smaller {\xi} that gives higher {\epsilon}_b, {\sigma}_b/{\phi}_bb, and W_b/{\phi}_bb than the networks with large f., Comment: 21 pages including 12 figures

We develop a novel Moving Particle Simulation (MPS) method to accurately reproduce the motion of fibers floating in sheared liquids. In conventional MPS schemes, if a fiber suspended in a liquid is represented by a one-dimensional array of MPS particles, it is entirely aligned to the flow direction due to the lack of shear stress difference between fiber-liquid interfaces. To address this problem, we employ the micropolar fluid model to introduce rotational degrees of freedom into the MPS particles. The translational motion of liquid and solid particles and the rotation of solid particles are calculated with the explicit MPS algorithm. The fiber is modeled as an array of micropolar fluid particles bonded with stretching and bending potentials. The motion of a single rigid fiber is simulated in a three-dimensional shear flow generated between two moving solid walls. We show that the proposed method is capable of reproducing the fiber motion predicted by Jeffery's theory being different from the conventional MPS simulations., Comment: 10 pages, 8 Postscript figures

Rod-shaped particles embedded in certain matrices have been reported to exhibit an increase in their center of mass diffusivity upon increasing the matrix density. This increase has been considered to be caused by a kinetic constraint in analogy with tube models. We investigate a mobile rod-like particle in a sea of immobile point obstacles using a kinetic Monte Carlo scheme equipped with a Markovian process, that generates gas-like collision statistics, so that such kinetic constraints do essentially not exist. Even in such a system, provided the particle's aspect ratio exceeds a threshold value of about 24, the unusual increase in the rod diffusivity emerges. This result implies that the kinetic constraint is not a necessary condition for the increase in the diffusivity., Comment: 9 pages, 10 figures

Diffusivity in some soft matter and biological systems changes with time, called the fluctuating diffusivity. In this work, we propose a novel origin for fluctuating diffusivity based on stochastic simulations of binary gas mixtures. In this system, the fraction of one component is significantly small, and the mass of the minor component molecule is different from that of the major component. The minor component exhibits fluctuating diffusivity when its mass is sufficiently smaller than that of the major component. We elucidate that this fluctuating diffusivity is caused by the time scale separation between the relaxation of the velocity direction and the speed of the minor component molecule., Comment: 20 pages, 12 figures

We derive the stress tensor of a rigid dumbbell by using the virtual work method. In the virtual work method, we virtually apply a small deformation to the system, and relate the change of the energy to the work done by the stress tensor. A rigid dumbbell consists of two particles connected by a rigid bond of which length is constant (the rigid constraint). The energy of the rigid dumbbell consists only on the kinetic energy. Also, only the deformations which do not violate the rigid constraint are allowed. Thus we need the dynamic equations which is consistent with the rigid constraint to apply the virtual deformation. We rewrite the dynamic equations for the underdamped SLLOD-type dynamic equations into the forms which are consistent with the rigid constraint. Then we apply the virtual deformation to a rigid dumbbell based on the obtained dynamic equations. We derive the stress tensor for the rigid dumbbell model from the change of the kinetic energy. Finally, we take the overdamped limit and derive the stress tensor and the dynamic equation for the overdamped rigid dumbbell. We show that the Green-Kubo type linear response formula can be reproduced by combining the stress tensor and the dynamic equation at the overdamped limit., Comment: 14 pages,to appear in Nihon Reoroji Gakkaishi (J. Soc. Rheol. Jpn.)

In some binary fluids, mass transport is observed under a temperature gradient. This phenomenon is called the Soret effect. In this study, we discuss the influence of inter-particle interaction. We considered equimolar binary Lennard-Jones fluids with a mass contrast, whereas the interaction was common for all the particle pairs with various cut-off lengths. We performed molecular dynamics simulations of such fluids under equilibrium to obtain the cross-transport coefficients L1q between the fluxes of mass and heat. The simulation revealed that this quantity strongly depends on the cut-off length. Further, we decomposed the heat flux into kinetic and potential contributions and calculated the cross-correlations between decomposed fluxes and the mass flux. The result indicates that the potential contribution dominates L1q, implying that the Soret coefficient is altered by the inter-particle interaction., Comment: submitted Mol. Sim., 12 pages, 4 figures

Ianniruberto and Marrucci developed a theory whereby entangled branched polymers behave like linear ones in fast elongational flows. In order to test such prediction, Huang et al. performed elongational measurements on star polymer melts, indeed revealing that, in fast flows, the elongational viscosity is insensitive to the molecular structure, provided the molecular weight of the backbone is the same. Inspired by these studies, we here report on results obtained with multi-chain slip-link simulations for symmetric and asymmetric star polymer melts, as well as calculations of the Rouse time of the examined branched structures. The simulations semi-quantitatively reproduce the experimental data if the Kuhn-segment orientation-induced reduction of friction (SORF) is accounted for. The observed insensitivity of the nonlinear elongational viscosity to the molecular structure for the same span molar mass may be due to several factors. In the symmetric case, the calculated Rouse time of the star marginally differs from that of the linear molecule, so that possible differences in the observed stress fall within the experimental uncertainty. Secondly, it is possible that the flow-induced formation of hooked star pairs makes the effective Rouse time of the aggregate even closer to that of the linear polymer because the friction center moves towards the branchpoint of the star molecule. In the asymmetric case, it is shown that the stress contributed by the short arms is negligible with respect to that of the long ones. However, such stress-reduction is balanced by a dilution effect whereby the unstretched arms reduce SORF as they decrease the Kuhn-segment order parameter of the system. As a result of that dilution, the stress contributed by the backbone is larger. The two effects compensate one another so that the overall stress is virtually the same as the other architectures., Comment: 19 pages, 6 figures

We calculate the plateau moduli of several single-chain slip-link and slip-spring models for entangled polymers. In these models, the entanglement effects are phenomenologically modeled by introducing topological constraints such as slip-links and slip-springs. The average number of segments between two neighboring slip-links or slip-springs, $N_{0}$, is an input parameter in these models. To analyze experimental data, the characteristic number of segments in entangled polymers $N_{e}$ estimated from the plateau modulus is used instead. Both $N_{0}$ and $N_{e}$ characterize the topological constraints in entangled polymers, and naively $N_{0}$ is considered to be the same as $N_{e}$. However, earlier studies showed that $N_{0}$ and $N_{e}$ (or the plateau modulus) should be considered as independent parameters. In this work, we show that due to the fluctuations at the short time scale, $N_{e}$ deviates from $N_{0}$. This means that the relation between $N_{0}$ and the plateau modulus is not simple as naively expected. The plateau modulus (or $N_{e}$) depends on the subchain-scale details of the employed model, as well as the average number of segments $N_{0}$. This is due to the fact that the subchain-scale fluctuation mechanisms depend on the model rather strongly. We theoretically calculate the plateau moduli for several single-chain slip-link and slip-spring models. Our results explicitly show that the relation between $N_{0}$ and $N_{e}$ is model-dependent. We compare theoretical results with various simulation data in the literature, and show that our theoretical expressions reasonably explain the simulation results., Comment: 26 pages, 3 figures, to appear in Macromolecules

In some recent experiments on entangled polymers of stress growth in startup of fast shear flows an undershoot in the shear stress is observed following the overshoot, i.e., before approaching the steady state. Whereas tumbling of the entangled chain was proposed to be at its origin, here we investigate another possible cause for the stress undershoot, i.e., slippage at the interface between polymer and solid wall. To this end, we extend the primitive chain network model to include slip at the interface between entangled polymeric liquids and solid walls with grafted polymers. We determine the slip velocity at the wall, and the shear rate in the bulk, by imposing that the shear stress in the bulk polymers is equal to that resulting from the polymers grafted at the wall. After confirming that the predicted results for the steady state are reasonable, we examine the transient behavior. The simulations confirm that slippage weakens the magnitude of the stress overshoot, as reported earlier. The undershoot is also weakened, or even disappears, because of a reduced coherence in molecular tumbling. In other words, the disentanglement between grafted and bulk chains, occurring throughout the stress overshoot region, does not contribute to the stress undershoot., Comment: 38 pages and 9 figures

Although it has been established that the multi-chain slip-spring (MCSS) model can reproduce entangled polymer dynamics, the effects of model parameters have not been fully elucidated yet. In this study, we systematically investigated the effects of slip-spring density. For the diffusion and the linear viscoelasticity, the simulation results exhibited universality. Namely, the results from the simulations with various slip-spring densities can be superposed with each other by the conversion factors for the bead number per chain, unit of length, unit of time, and modulus. The diffusion and the viscoelasticity were in good agreement with the literature data for the standard bead-spring simulations, including the molecular weight dependence. The universality among the MCSS simulations with various slip-spring density also held under mild shear if the slip-spring density was not significantly high. The results imply that the level of coarse-graining for the MCSS model can be arbitrarily chosen as in the Rouse model., Comment: 22 pages, 12 figures

Primitive chain network simulations for randomly cross-linked slip-link networks were performed. For the percolated networks, the stress-strain relationship was compared to the theories by Ball et al. [Polymer, 22, 1010 (1981)] and Rubinstein and Panyukov [Macromolecules, 35, 6670 (2002)]. The simulation results were reasonably reproduced by both theories, given that the contributions from cross-links and slip-links were used as fitting parameters. However, these parameters were model dependent. Besides, the theories cannot describe the simulation results if the parameters were determined from the number of active links involved in the percolated networks. These results reveal that the fitting of experimental data to the theories does not provide a fraction of entanglements in the system unless the network only consists of Gaussian strands and it correctly reaches the state of free-energy minimum., Comment: 20 pages, 6 figures

A small tagged particle immersed in a fluid exhibits the Brownian motion and diffuses at the long-time scale. Meanwhile, at the short-time scale, the dynamics of the tagged particle cannot be simply described by the usual generalized Langevin equation with the Gaussian noise, since the number of collisions between the tagged particle and fluid particles is rather small. At such a time scale, we should explicitly consider individual collision events between the tagged particle and the surrounding fluid particles. In this study, we analyzed the short-time dynamics of the tagged particle in an ideal gas, where we do not have static nor hydrodynamic correlations between fluid particles. We performed event-driven hard sphere simulations and show that the short-time dynamics of the tagged particle is correlated even under such an idealized situation. Namely, the velocity autocorrelation function becomes negative when the tagged particle is relatively light and the fluid density is relatively high. This result can be attributed to the dynamical correlation between collision events. To investigate the physical mechanism, which causes the dynamical correlation, we analyzed the correlation between successive collision events. We found that the tagged particle can collide with the same ideal gas particle several times, and such collisions cause the strong dynamical correlation for the velocity., Comment: 23 pages, 11 figures, to appear in Phys. Rev. E

The overdamped (inertialess) dumbbell model is widely utilized to study rheological properties of polymers or other soft matters. In most cases, the effect of inertia is merely neglected because the momentum relaxation is much faster than the bond relaxation. We theoretically analyze the effect of inertia on the linear viscoelasticity of the harmonic dumbbell model. We show that the momentum and bond relaxation modes are kinetically coupled and the inertia can affect the bond relaxation if the momentum relaxation is not sufficiently fast. We derive an overdamped Langevin equation for the dumbbell model, which incorporates the weak inertia effect. Our model predicts the bond relaxation dynamics with the weak inertia effect correctly. We discuss how the weak inertia affects the linear viscoelasticity of a simple harmonic dumbbell model and the Rouse model., Comment: 17 pages, 3 figures, to appear in Nihon Reoroji Gakkaishi (J. Soc. Rheol. Jpn.)

It has been established that the entangled polymer dynamics can be reasonably described by single chain models such as tube and slip-link models. Although the entanglement effect is a result of the hard-core interaction between chains, linkage between the single chain models and the real multi-chain system has not been established yet. In this study, we propose a multi-chain slip-spring model where bead-spring chains are dispersed in space and connected by slip-springs inspired by the single chain slip-spring model [A. E. Likhtman, Macromolecules 38, 6128 (2005)]. In this model the entanglement effect is replaced by the slip-springs, not by the hard-core interaction between beads so that this model is located in the niche between conventional multi-chain simulations and single chain models. The set of state variables are the position of beads and the connectivity (indices) of the slip-springs between beads. The dynamics of the system is described by the time evolution equation and stochastic transition dynamics for these variables. We propose a simple model which is based on the well-defined total free-energy and the detailed balance condition. The free energy in our model contains a repulsive interaction between beads, which compensate the attractive interaction artificially generated by the slip-springs. The explicit expression of linear relaxation modulus is also derived by the linear response theory. We also propose a possible numerical scheme to perform simulations. Simulations reproduced expected bead number dependence in transitional regime between Rouse and entangled dynamics for the chain structure, the central bead diffusion, and the linear relaxation modulus., Comment: 25 pages, 10 figures, to appear in J. Chem. Phys

We consider statistical mechanical properties of the primitive chain network (PCN) model for entangled polymers from its dynamic equations. We show that the dynamic equation for the segment number of the PCN model does not reduce to the standard Langevin equation which satisfies the detailed balance condition. We propose heuristic modifications for the PCN dynamic equation for the segment number, to make it reduce to the standard Langevin equation. We analyse some equilibrium statistical properties of the modified PCN model, by using the effective free energy obtained from the modified PCN dynamic equations. The PCN effective free energy can be interpreted as the sum of the ideal Gaussian chain free energy and the repulsive interaction energy between slip-links. By using the single chain approximation, we calculate several distribution functions of the PCN model. The obtained distribution functions are qualitatively different from ones for the simple slip-link model without any direct interactions between slip-links., Comment: 29 pages, 7 figures, to appeare in J. Chem. Phys