Flow-Induced Ring Polymer Translocation through Nanopores

Article pubs.acs.org/Macromolecules

Flow-Induced Ring Polymer Translocation through Nanopores Mingming Ding, Xiaozheng Duan, Yuyuan Lu,* and Tongfei Shi* State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, P.R. China ABSTRACT: We investigate the flow-induced linear and ring polymer translocation through nanopores using a hybrid simulation method that combines a lattice-Boltzmann approach for the fluid with a molecular dynamics model for the polymer chain. Our results illustrate that the critical velocity flux for the linear and ring chain exhibits characteristic differences: for the linear chain, the critical velocity flux is independent of the chain length, which is in agreement with previous work; whereas, for the ring chain, the critical velocity flux decreases slowly with the increase of the chain length and gets closer to the critical velocity flux of the linear chain. In addition, we find that the ring chain shows much faster translocation compared to the linear chain with the same chain length. Moreover, the ring chain and the folded linear chain display the similar configurational deformation during the translocation. These results above indicate that with the increase of the chain length, the separation between the linear and ring polymers in the flow-induced translocation through nanopores becomes increasingly difficult. have also deduced that the critical velocity flux of hyperbranched polymers (jc,h) depends on both polymerization degrees of the entire chain and the subchain.15,16 Recently, such predictions have been experimentally verified by Wu et al.7,8 They have showed that jc,s is only related to the number of forward arms ( f in), i.e., jc,s∼ jc,l(f + |f − 2f in|)/2, where f is the total number of arms.7 In addition, they have proposed that jc,h ∼ jc,lnγb, where nb is the number of branching points inside a hyperbranched polymer, and γ = 1/3 and 1/4 for the weak and strong confinements, respectively.8 These studies imply that one may separate and fractionate a mixture of linear polymers with other topological polymers if the corresponding critical velocity flux is known. Even if much work has been carried out to elucidate the mechanisms of the flow-induced translocation of linear, star, and branched polymers, there still exists the limitation in the understanding of the flow-induced ring polymer translocation through nanopores. Compared to other topological polymers, the ring polymers, which do not have free ends, can exhibit special statistical mechanics properties. Because the inherent difficulties in a systematic theoretical analysis of such objects constrained to the unique topology, the theory on the description of the ring polymers has not been well established.17 Although great process has been made in the polymer synthesis, the characterization and control of the experimental systems including polydispersity, concatenation, and linear contaminants is far from perfect.18 Therefore, the coarse-grained computational models can serve as an effective

1. INTRODUCTION The separation between polymers with different topologies but a similar hydrodynamic volume is essential in biological chemistry, material science, and many other fields of science and applications.1−3 Recently, experimental studies have shown that the flow-induced polymer translocation through nanopores can serve as a potential method for separating a mixture of polymers with different topologies.3−8 Especially, exploring the flow-induced polymer translocation through nanopores can be helpful for understanding the biological processes of DNA migration, gene delivery, and virus injection.9−11 On the basis of the polymer blob model, earlier theoretical studies have predicted that in a dilute solution, the linear polymers can pass through a nanopore only above a critical velocity flux, i.e., jc,l ∼ kBT/η, where kB is the Boltzmann constant, T denotes the temperature, and η represents the solvent viscosity.12,13 Surprisingly, jc,l is independent of the chain length and the nanopore size. Some studies have shown that jc,l is indeed independent of the chain length but unexpectedly decreases with the increase of the nanopore size.4−6,14 Using a scaling argument, Wu et al. have derived a scaling law, i.e., jc,l ∼ (kBT/η) D1−(1/α), where D is the nanopore size, and 1/2 ≤ α ≤ 3/5, depending on the solvent quality.5 Notably, it is almost impossible to separate linear polymers with different lengths in the flow-induced polymer translocation through nanopores.4 The flow-induced translocation of polymers with more free ends (such as star and branched polymers) exhibits more intricate behaviors.7,8,15,16 Theoretically, de Gennes et al. have demonstrated that the critical velocity flux of star polymers (jc,s) depends not only on the total number of arms, but also the number of forward arms squeezed into the nanopore.13 They © XXXX American Chemical Society

Received: April 23, 2015 Revised: June 26, 2015

A

DOI: 10.1021/acs.macromol.5b00857 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules method to investigate the flow-induced ring polymer translocation through nanopores, which is considerable significance for the understanding of the physical nature and the development of the corresponding application.19 Up to present, most simulation studies have only focused on the instance in which one end of a chain is pinned inside the nanopore before simulations.14,20−24 As we will show below, the capture process of the polymer chain is extremely important in determining the translocation dynamics. In this paper, we study the flow-induced ring polymer translocation through nanopores by means of a numerical method developed by Dünweg et al.25−27 This is a hybrid scheme that combines a lattice-Boltzmann approach for the fluid with a molecular dynamics model for the polymer chain. The two parts are coupled using a frictional coupling, which results in Oseen level hydrodynamics for the polymer chains.25 The paper is organized as follows: In section 2, we describe the hybrid model and the corresponding simulation details. In section 3, we systematically investigate the critical velocity flux, the translocation time, and the configurational deformation of the ring and linear polymers. In section 4, we present the conclusions of this work.

Solid boundaries in the simulation system are implemented using the well-known bounce-back rules.28 Just as verified by Ledesma-Aguilar et al., a linear relationship between j and Fext is found as Darcy’s Law, i.e., j = kFext, where k is a proportional coefficient.32 A linear fit to the data gives k ≃ 66.7 in our simulations. Molecular Dynamics Model for the Polymer Chain. In the simulations, we model the polymers as the Lennard-Jones (LJ) beads connected with the finite extension nonlinear elastic (FENE) potential.33,34 Both the excluded volume and van der Waals interactions between the beads are taken into account by a repulsive LJ potential: 12 ⎧ − (σ /r )6 ] + ϵ, r ≤ 21/6σ ⎪ 4ϵ[(σ / r ) ULJ(r ) = ⎨ ⎪ 0, r > 21/6σ ⎩

where r is the center-of-mass distance between two beads. ϵ and σ are the energy and length parameters, respectively. The FENE potential can be described as UFENE(r ) = −0.5κR 0 2 ln[1 − (r /R 0)2 ]

Ffl = −ζ(v − u(R)) + f

(5)

where ζ is the friction constant, v denotes the bead velocity, and u(R) represents the fluid velocity at the bead’s position R, which is estimated by a linear interpolation scheme.25,26 Here, f is a stochastic force of zero mean and ⟨fα (t )fβ (t ′)⟩ = 2kBTζδαβδ(t − t ′)

1 fi (r + ciτ , t + τ ) = fi (r, t ) − (fi (r, t ) − f ieq (r, t )) τ0

(6)

In order to complete the coupling between the bead and the fluid, we distribute the accumulated force on a bead back to the fluid according to the same interpolation.25,26 The equations of motion resulting from these potentials are integrated using the velocity−Verlet algorithm of the molecular dynamics with a time step Δt.35 The LJ parameters ϵ, σ, and the bead mass m define as the unit in the system.35 The parameters for the FENE potential are taken as κ = 30 and R0 = 1.5.34 We set Δt = 0.01, which is an optimal value for the polymer chain part.36 Following the procedure presented in ref25., the friction constant is ζ = 20 and the time step of the lattice−Boltzmann is τ = 5Δt. In addition, the lattice spacing is Δx = 1, and the fluid is characterized by the density (ρ = 1.0), the kinematic viscosity (ν = 2.8), and the temperature (kBT = 1.0), respectively.25 The velocity fluxes J are measured with respect to a reference value j0 = 2/3 × 10−3, which corresponds to Fext = 10−5. Simulation Details. Scheme 1a displays a schematic representation of the simulation, where a linear or a ring chain is put into a channel with dimensions of (Lx, Ly, Lz). In the middle of the channel we create a nanopore with dimensions of (D, D, D). As shown in Scheme 1a, the channel and the nanopore are formed by the same stationary particles within distance σ from one another which interact with the beads by the same repulsive LJ interaction. The size of the channel is fixed as Lx = 83, Ly = Lz = 25 and the size of the nanopore is D = 3. The coiled size of the polymer chain is smaller than the channel size but bigger than the nanopore size. Some simulation studies have found that the confinement can

(1)

where i represents a lattice vector, ci is the discrete velocity, τ denotes the time step, and τ0 characterizes the relaxation time scales of the fluid, which is defined as τ0 = 3ν + 1/2 (ν is the fluid kinematic viscosity).31 f eq i represents the local equilibrium distribution function and Fext is the external force density, which i can model a pressure-driven field.28 The fluctuation is taken into account by the random terms in the nonequilibrium stresses during the collision process.29 Accordingly, the fluid density, the fluid velocity, and the velocity flux can be calculated via ρ = ∑i f i, u = 1/ρ∑i f ici, and j = ∑u, respectively. The D3Q19 model is used and f eq i is defined as 2 ⎡ c ·u 9 (c i · u ) 3 u2 ⎤ ⎥ f ieq (r) = wciρ(r)⎢1 + 3 i 2 + − 4 2 c 2 c2 ⎦ c ⎣

(4)

where κ is the spring constant and R0 is the maximum allowed separation between connected beads. The hydrodynamic force Ffl on the bead exerted by the fluid through a frictional coupling is

2. MODEL AND METHODS Lattice−Boltzmann Approach for the Fluid. In the simulations, we model the fluid with the lattice−Boltzmann method, which is widely used to simulate complex fluids because of the possibility of straightforward implementation of complex boundaries.28,29 In this method, the single-particle distribution function f i(r,t) constitutes the fundamental quantity. According to the Bhatnagar-Gross-Krook approximation,30 the time evolution of f i(r,t) at the lattice node r and the time t can be described as

+ Fiext (r, t )

(3)

(2)

where wci is the weight factor, which depends on the sublattice i and c = Δx/ 3 τ is the sound speed. The lattice spacing Δx is uniform along the lattice axes, and the set of ci consists of stationary particles and 18 velocities corresponding to the nearest ([100]) and next nearest ([110]) neighbor directions of a simple cubic lattice. The magnitudes of the velocities corresponding to these three sets of velocities are {|ci|} = (0,1, 2 )Δx/τ. The weight factors for the D3Q19 model are w0 = 1/3, w1 = 1/18, and w 2 = 1/36. B


Article

Macromolecules

3. RESULTS AND DISCUSSION Figure 1 depicts the translocation probability P(J) as a function of the velocity flux J for the linear and ring chain with the chain

Scheme 1. (a) Schematic Representation of the Initial Conformation of a Linear or a Ring Chainaand (b−d) Schematic Representations of the Unfolded Translocation (b), the Folded Translocation (c) for the Linear Chain, and the “Folded Translocation” for the Ring Chain (d)

Figure 1. Translocation probability P(J) as a function of the velocity flux J for the linear and ring chain with the chain length N = 80.

length N = 80. In our simulations, we find that the polymer chain mainly exhibits random motion on the left-hand side of the nanopore under the lowest velocity flux during the all simulation times. With the increase of the velocity flux, the polymer chain can pass through the nanopore in some events. By further increasing the velocity flux, we find that the polymers always present successful translocation. The results show that the translocation probability crosses over smoothly from 0 to 1 both for the linear and ring chain. However, the translocation probability for the ring chain is always lower than that for the linear chain with the same velocity flux. In Figure 2, we display the critical velocity flux Jc as a function of the chain length for the linear and ring chain, where the critical velocity flux is defined as the velocity flux when P(J) = 0.5.8 For the linear chain, the critical velocity flux is independent of the chain length, which is in agreement with previous results;4−6,12−14 However, for the ring chain, the

a

This is put into a channel with dimensions of (Lx, Ly, Lz). A nanopore with dimensions of (D, D, D) is created in the middle of the channel.

affect the conformations of the polymer chain and thus the translocation dynamics,23,24,37,38 whereas, in our simulations, the polymer conformations are independent of the effect of the channel size. Initially, we put the polymer chain on the left-hand side of the nanopore and the beads of the polymer chain are under thermal collisions to obtain an equilibrium conformation. During this process, the beads are controlled to stay outside the nanopore by a soft virtual wall near the entrance of the nanopore,14 which can avoid the chain being close to the nanopore in the equilibration. The wall is then removed and a fluid with certain velocity flux is applied. A successful translocation denotes the polymer ending up on the righthand side of the nanopore in the maximum simulation time. Accordingly, an event is deemed as a failure translocation if the polymer chain always stays on the left-hand side of the nanopore in the maximum simulation time. Recently, both unfolded translocation and folded translocation in the linear polymer translocation through nanopores have been experimentally observed,39 which are also found in our simulations as shown in Scheme 1b and Scheme 1c, respectively. In the present study, if any of the four endmost beads on the linear chain can first enter the nanopore and then the translocation is completed, it is defined as the unfolded translocation. The others are defined as the folded translocation. Naturally, there is only “folded translocation” for the ring chain as shown in Scheme 1d. We obtain the final results by averaging our data over 300 independent simulations. The translocation probability is calculated as the fraction of runs leading to successful translocation. Our simulation time has been tested long enough for the stability of the translocation probability.

Figure 2. Critical velocity flux Jc as a function of the chain length N for the linear and ring chain. C


Article

Macromolecules critical velocity flux decreases slowly with the increase of the chain length and gets closer to the critical velocity flux of the linear chain. This finding illustrates that the critical velocity flux of the ring chain depends on the chain length. For the short chain, it is possible to separate the ring and linear chains with the same chain length in the flow-induced polymer translocation through nanopores; whereas, for the long chain, it might be impossible. In fact, in order to pass through a nanopore where the number of allowed conformations is drastically reduced, the polymer chain needs to cross an energy barrier.40 Importantly, the hydrodynamics force from the flow field is increased with the increase of the chain length. Thus, the beads of entering the nanopore can obtain more hydrodynamics force from the beads outside the nanopore due to the effect of the hydrodynamic interactions. This makes the critical velocity flux of the ring chain decrease with the increase of the chain length. However, for the linear chain, Storm et al. inferred the chain is more likely to be transported with an unfolded translocation because of the lower energy cost,41 which implies that the energy barrier for the unfolded translocation is lower than that for the folded translocation. According to the physical model proposed by Stein et al.,39 we can deduce that the relationship between the probability of the unfolded translocation P0(N) and the chain length for the linear chain is P0(N) ∼ N−0.8, which has been tested in our simulations. This indicates that the probability of the unfolded translocation increases with the decrease of the chain length, meaning that the energy barrier is reduced. Even if the hydrodynamics force decreases with the decrease of the chain length, the lower energy barrier reduces the corresponding effect, which results in the independence of the chain length for the critical velocity flux of the linear chain. Because of the increase of the folded translocation for the linear chain, the translocation difference between the linear chain and ring chain tends to become obscure with the increase of the chain length, which indicates the critical velocity flux of the ring chain approaches that of the linear chain. However, the unfolded translocation always exists in the linear polymer translocation, meaning that the critical velocity flux of the linear chain is always lower than that of the ring chain. In addition, even if one arm is longer than the other for the folded translocation of the linear chain, we find that the polymer chain can be transported as long as a small part of the polymer chain has passed through the nanopore, which indicates that the effect of the chain conformations outside the nanopore may be very faint. Therefore, the critical velocity flux of the ring chain decreases slowly with the increase of the chain length and gets closer to the critical velocity flux of the linear chain, which is independent of the chain length. In order to investigate our analysis in detail, we display the translocation probability as a function of the velocity flux for the linear and ring chain with the chain length N = 40 and N = 160 in Figure 3, parts a and b, respectively, where the translocation probability of the linear chain is divided into two parts with the unfolded translocation contribution P-Ends and the folded translocation contribution P-Middle. As shown in Figure 3a, when the translocation probability is less than 0.5, the unfolded translocation contribution holds a strong majority for the linear chain, whereas the ring chain can hardly pass through the nanopore. With the increase of the velocity flux, the folded translocation contribution emerges and exhibits an increase. At the same time, the translocation probability of the ring chain starts to increase. However, as shown in Figure 3b,

Figure 3. Translocation probability P(J) as a function of the velocity flux J for the linear and ring chain with the chain length N = 40 (a) and N = 160 (b), where the translocation probability of the linear chain is divided into two parts with the unfolded translocation contribution PEnds and the folded translocation contribution P-Middle.

the contribution of the unfolded translocation is faint, as a result, the difference of the translocation probability between the linear and ring chain is not obvious. These results further verify that the chain length is a key factor to affect the contribution of the unfolded translocation for the linear chain, which can result in differential critical velocity fluxes for the linear and ring chains. In addition, the translocation time t is defined as the time interval between the entrance of the first bead into the nanopore and the exit of the last bead, which can dynamically reflect the translocation process. The scaling relationships between the translocation time and the chain length for the linear and ring chain are shown in Figure 4. An exponent of 0.99 is found for the linear chain, which is in agreement with the linear dependence of the experimental value.1 Using a dissipative particle dynamics simulation, Liang et al. have found an exponent of 1.152 for the fluid flow driven translocation, which seems to be closer to the linear dependence.42 However, an exponent of 1.4 has also been obtained for the translocation driven by an external field, where one end of the linear chain is pinned inside the nanopore before simulations.22 This assumption neglects the capture process, which is very different from the experiments. Thus, the capture process is extremely D


Article

Macromolecules

the gyration radius ratio ⟨Rg/Rg0⟩ as a function of the distance ΔXc between the chain’s center of mass and the center of the nanopore for the linear and ring chain, where the linear chain is divided into two cases: the unfolded translocation L-ChainEnds and the folded translocation L-Chain-Middle. The changing process for the ring chain is very similar to the linear chain for the folded translocation, even if there exists a little bit of shrink before starting the nanopore because of the unique topology. However, the configurational deformation of the unfolded translocation for the linear chain is bigger than that for the other cases. Because the folded translocation is preferred, the overall behavior of the linear chain is dominated by the folded translocation. In addition, for the linear chain, the configurational deformation achieves a maximum stretch before the chain’s center of mass reaches the center of the nanopore; whereas for the ring chain, it achieves a maximum stretch near the center of the nanopore. These results explain the mechanism of the polymer translocation, where the ring and linear chain share a similar configurational deformation. Therefore, as the probability of the unfolded translocation of the linear chain decreases with the increase of the chain length, the separation between the linear and ring polymers with the same polymerization degree in the flow-induced polymer translocation through nanopores becomes increasingly difficult.

Figure 4. Translocation time t as a function of the chain length N for the linear and ring chain.

important in determining the translocation dynamics for the flow-induced polymer translocation. In addition, we also find that the exponent is almost independent of the velocity flux when it is above the critical velocity flux, which has also been verified in the previous simulation.21 Importantly, the results show that the scaling behavior for the ring chain is very similar to that of the linear chain. However, for the same chain length, the ring chain displays faster translocation than the linear chain, meaning that the translocation process of the ring chain is faster compared with that of the linear chain with the identical length. Even if the ring chain is more difficult to be captured by the nanopore, its contour distance from the initial entering bead to the last entering bead is always less than or equal to that of the linear chain. This effect is not beneficial for entering of the ring chain into the nanopore, but can promote the translocation process if the chain enters the nanopore. In order to elucidate how the flow-induced linear and ring chain pass through nanopores, we explore the configurational deformation during the translocation process. Figure 5 shows

4. CONCLUSIONS In this work, we study the flow-induced linear and ring polymer translocation through nanopores using a hybrid simulation method that combines a lattice−Boltzmann approach for the fluid with a molecular dynamics model for the polymer chain. The critical velocity flux for the ring and linear chain exhibits characteristic differences: for the linear chain, the critical velocity flux is independent of the chain length, which is in agreement with previous results; whereas, for the ring chain, the critical velocity flux decreases slowly with the increase of the chain length and gets closer to the critical velocity flux of the linear chain. Compared to the unfolded translocation and the folded translocation of the linear chain, only the symmetricalfolded translocation is available for the ring chain. Thus, as the probability of the unfolded translocation of the linear chain decreases with the increase of the chain length, the difference of the translocation conformation between the linear and ring chain tends to become obscure. As a result, the linear chain and ring chain gradually display the similar critical velocity flux. In addition, we also find that the translocation process of the ring chain is faster than that of the linear chain with the same chain length. The ring chain and the folded translocation of the linear chain share a similar configurational deformation during the translocation process. Our results indicate that with the increase of the chain length, the separation between the linear and ring polymers with the same chain length in the flow-induced polymer translocation through nanopores becomes increasingly difficult. We hope this work can be helpful for understanding the microscopic dynamics of the flow-induced ring polymer translocation through nanopores, and provide theoretical guidance for the efficient separation and purification of the polymer chains with different topologies.

■

Figure 5. Gyration radius ratio ⟨Rg/Rg0⟩ as a function of the distance ΔXc between the chain’s center of mass and the center of the nanopore for the linear and ring chain, where the linear chain is divided into two cases: the unfolded translocation L-Chain-Ends and the folded translocation L-Chain-Middle.

AUTHOR INFORMATION

Corresponding Authors

*(Y.L.) E-mail: [email protected]. *(T.S.) E-mail: [email protected]. E


Article

Macromolecules Notes

(36) Kremer, K.; Grest, G. S. J. Chem. Phys. 1990, 92, 5057. (37) Polson, J. M. J. Chem. Phys. 2015, 142, 174903. (38) Sean, D.; de Haan, H. W.; Slater, G. W. Electrophoresis 2015, 36, 682. (39) Mihovilovic, M.; Hagerty, N.; Stein, D. Phys. Rev. Lett. 2013, 110, 028102. (40) Muthukumar, M. J. Chem. Phys. 1999, 111, 10371. (41) Storm, A.; Chen, J.; Zandbergen, H.; Dekker, C. Phys. Rev. E 2005, 71, 051903. (42) Guo, J.; Li, X.; Liu, Y.; Liang, H. J. Chem. Phys. 2011, 134, 134906.

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank Burkhard Dünweg of Max Planck Institute for Polymer Research and Ulf Schiller of University College London for discussions and sharing the details of the simulation method. This work is supported by the National Natural Science Foundation of China (Nos. 21234007, 21304097, 21404103 and 51473168).

■

REFERENCES

(1) Kasianowicz, J.; Brandin, E.; Branton, D.; Deamer, D. Proc. Natl. Acad. Sci. U. S. A. 1996, 93, 13770. (2) Matthews, R.; Louis, A.; Yeomans, J. Phys. Rev. Lett. 2009, 102, 088101. (3) Ge, H.; Wu, C. Macromolecules 2010, 43, 8711. (4) Jin, F.; Wu, C. Phys. Rev. Lett. 2006, 96, 237801. (5) Ge, H.; Jin, F.; Li, J.; Wu, C. Macromolecules 2009, 42, 4400. (6) Béguin, L.; Grassl, B.; Brochard-Wyart, F.; Rakib, M.; Duval, H. Soft Matter 2011, 7, 96. (7) Ge, H.; Pispas, S.; Wu, C. Polym. Chem. 2011, 2, 1071. (8) Li, L.; He, C.; He, W.; Wu, C. Macromolecules 2012, 45, 7583. (9) Obukhov, S. P.; Rubinstein, M.; Duke, T. Phys. Rev. Lett. 1994, 73, 1263. (10) Jun, S.; Mulder, B. Proc. Natl. Acad. Sci. U. S. A. 2006, 103, 12388. (11) Benková, Z.; Cifra, P. Macromolecules 2012, 45, 2597. (12) De Gennes, P. J. Chem. Phys. 1974, 60, 5030. (13) De Gennes, P. Adv. Polym. Sci. 1999, 38, 91. (14) Markesteijn, A.; Usta, O.; Ali, I.; Balazs, A.; Yeomans, J. Soft Matter 2009, 5, 4575. (15) Gay, C.; De Gennes, P.; Raphaël, E.; Brochard-Wyart, F. Macromolecules 1996, 29, 8379. (16) Sakaue, T.; Raphaël, E.; De Gennes, P.; Brochard-Wyart, F. Europhys. Lett. 2005, 72, 83. (17) Sheng, J.; Luo, K. Soft Matter 2012, 8, 367. (18) Halverson, J. D.; Grest, G. S.; Grosberg, A. Y.; Kremer, K. Phys. Rev. Lett. 2012, 108, 038301. (19) Slater, G. W.; Holm, C.; Chubynsky, M. V.; de Haan, H. W.; Dube, A.; Grass, K.; Hickey, O. A.; Kingsburry, C.; Sean, D.; Shendruk, T. N.; Zhan, L. Electrophoresis 2009, 30, 792. (20) Matysiak, S.; Montesi, A.; Pasquali, M.; Kolomeisky, A.; Clementi, C. Phys. Rev. Lett. 2006, 96, 118103. (21) Luo, K.; Metzler, R. J. Chem. Phys. 2011, 134, 135102. (22) Luo, K.; Ollila, S.; Huopaniemi, I.; Ala-Nissila, T.; Pomorski, P.; Karttunen, M.; Ying, S.; Bhattacharya, A. Phys. Rev. E 2008, 78, 050901. (23) Zhang, K.; Luo, K. J. Chem. Phys. 2014, 140, 094902. (24) Zhang, K.; Luo, K. Soft Matter 2013, 9, 2069. (25) Ahlrichs, P.; Dünweg, B. J. Chem. Phys. 1999, 111, 8225. (26) Pham, T. T.; Schiller, U. D.; Prakash, J. R.; Dünweg, B. J. Chem. Phys. 2009, 131, 164114. (27) Schiller, U. D. Comput. Phys. Commun. 2014, 185, 2586. (28) Ladd, A.; Verberg, R. J. Stat. Phys. 2001, 104, 1191. (29) Ladd, A. J. Fluid Mech. 1994, 271, 285. (30) Bhatnagar, P. L.; Gross, E. P.; Krook, M. Phys. Rev. 1954, 94, 511. (31) Qian, Y.; D’Humières, D.; Lallemand, P. Europhys. Lett. 1992, 17, 479. (32) Ledesma-Aguilar, R.; Sakaue, T.; Yeomans, J. M. Soft Matter 2012, 8, 1884. (33) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237. (34) Grest, G. S.; Kremer, K. Phys. Rev. A: At., Mol., Opt. Phys. 1986, 33, 3628. (35) Frenkel, D.; Smit, B. Understanding molecular simulation: from algorithms to applications; Academic Press: San Diego, CA, 2001. F


Flow-Induced Ring Polymer Translocation through Nanopores

Recommend Documents