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Dissipative Particle Dynamics Simulations of Polymer Chains: Scaling Laws
and Shearing Response Compared to DNA Experiments
Vasileios Symeonidis, 1,^ * George Em Karniadakis, 1,†^ and Bruce Caswell 2,‡ (^1) Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, USA (^2) Division of Engineering, Brown University, Providence, Rhode Island 02912, USA (Received 19 October 2004; revised manuscript received 28 June 2005; published 12 August 2005) Dissipative particle dynamics simulations of several bead-spring representations of polymer chains in dilute solution are used to demonstrate the correct static scaling laws for the radius of gyration. Shear flow results for the wormlike chain simulating single DNA molecules compare well with average extensions from experiments, irrespective of the number of beads. However, coarse graining with more than a few beads degrades the agreement of the autocorrelation of the extension.
DOI: 10.1103/PhysRevLett.95.076001 PACS numbers: 83.10.Pp, 83.10.Rs, 83.50.Ax, 83.80.Rs
Modeling realistic polymer motion in the microscopic or mesoscopic level in equilibrium and simple flow configu- rations has been a continuous challenge in terms of (a) the choice of appropriate polymeric interactions and (b) the simulation method itself. Brownian dynamics (BD) simu- lations [1] have shown a good comparison [2] with experi- ments on DNA molecules in shear flow [3]. Molecular dynamics (MD) has been used for comparison with worm- like chain (WLC) [4] and slip length measurements for sheared films [5] but the number of beads and the time scale interval is much shorter than the times for gathering experimental data (of the order of seconds [3]). Here we employ dissipative particle dynamics [(DPD), described below] to investigate the static scaling law for several model chains and the response of the WLC under shear. This mesoscopic method has already been used to model macromolecules in a variety of equilibrium and nonequi- librium configurations [6 –10]. In the motion of ideal chains, the bonds—characterized by linear elastic forces—are not restricted from passing through each other, crossings known as phantom collisions. For real chains in good solvents these unphysical collisions are eliminated by the constraint of self-avoiding walks on preset lattice paths. This dramatically affects their scaling properties. Our treatment of chains places no explicit con- straints on the interaction between chain segments. Since it is usually assumed that in physical systems self-avoidance is a consequence of excluded volume and restricted bond rotation, we attempt to exclude phantom collisions by appropriate choices of intrapolymer interactions. Our chain potentials contain no angle dependence and therefore only the excluded volume is available for the task of self- avoidance. Since the detection of phantom collisions is computationally complex, their frequency is measured in- directly by checking the scaling exponents of the radius of gyration, Rg , with respect to chain size. For a chain of M beads it is defined by h Rg^2 i h (^) M^1 Mi 1 Ri R cm^2 i, where Ri is the position vector of each bead, R cm the position
vector of the center of mass of the chain, and hi denotes time averaging. Statistical scaling arguments show that Rg / M 1 . The static exponent is 0.5 for ideal chains (in any space dimension d [11]), and for real chains is (^) d^32 0 : 6 (Flory’s formula), which has been veri- fied by light scattering experiments [11]. Previous DPD simulations involved linear chains [6,7] and manipulation of solvent characteristics [8] to obtain the 0.6 exponent without appropriate interbead forces. In 1992 Hoogerbrugge and Koelman [12] introduced the DPD method, which combines some of the detailed de- scription of the MD with the ability to describe larger time and length scales. The DPD method describes blocks of molecules moving together in a coherent fashion subject to soft potentials and governed by predefined collision rules. Hence, this method is very attractive for the computer simulation of polymer solutions, since by employing the bead-spring model of polymer chains we can formulate and compare a variety of realistic conservative interbead forces. In contrast to Langevin-equation methods, such as BD, the hydrodynamic resistance is accounted for implic- itly by the DPD solvent particles which behave as a Newtonian fluid [13]. As a particle-based mesoscopic method, DPD considers N particles, each having mass mi , whose momenta and position vectors are governed by Newton’s equations of motion. For a typical particle i , v i ddt r i , F i mi ddt v i , where v i its velocity, r i its position, and F i its net force. The interparticle force F ij exerted on particle i by particle j is composed of conservative ( F cij ), dissipative ( F dij ), and random ( F rij ) components. Hence, the total force on par- ticle i is given by F i i j F cij F dij F rij t p (^) , t being the simulation time step. The sum acts over all particles within a cutoff radius r (^) c beyond which the forces are considered negligible. We set the interaction radius to rc 1 , thus defining the length scale of the system. Denoting r ij r i r j , u ij u i u j , rij j r ij j, and the unit vector
PRL 95, 076001 (2005) P H Y S I C A L^ R E V I E W^ L E T T E R S^
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0031-9007 = 05 = 95(7) = 076001(4)$23.00 076001-1 2005 The American Physical Society
e ij r ij r (^) ij , the forces are
F cij F c ^ rij e ij ; F dij ! d rij u ij e ij e ij ;
F rij !r r (^) ij (^) ij e ij ;
where the (^) ij are symmetric Gaussian random variables with zero mean and unit variance and ; are coupled by ^2 2 kB T , kB being the Boltzmann constant and T the temperature of the system [14]. A common choice for the conservative force is a soft repulsion given by F c ^ rij
a (^) ij maxf 1 r (^) ij rc ;^^0 g. The dissipative and random forces, on the other hand, are characterized by strengths_! d_ rij and ! r rij coupled by_! d_ rij ! r r (^) ij ^2 maxf 1 r (^) ij rc
(^2) ; 0 g. The above relation is necessary for thermodynamic
equilibrium. The dissipative forces represent friction be- tween the particles and account for energy loss, while the random ones compensate for lost degrees of freedom due to coarse graining and heat up the system. The conservative forces present in the DPD equations can be tailored to describe a variety of interactions F c ^ rij r V r (^) ij , for a potential V. In this work, poly- mers are chains of beads (DPD particles) subject to the standard DPD forces: soft repulsive (conservative), dissi- pative, and random. In addition , they are subject to intra- polymer forces arising from different combinations of the following types: Lennard-Jones.— The force for each pair of bead parti- cles is given by the shifted (to avoid numerical instabili- ties) LJ potential U LJ 4 (^) rLij ^12 (^) rL (^) ij ^6 14 truncated to
act only for pairs with r (^) ij < r (^) c. We pick kB T , L 2 ^1 =^6 , and r (^) c L 2 ^1 =^6 1. We note that the LJ po- tential used here is defined at the mesoscopic level to improve polymeric self-avoidance —softer repulsion rules [15] are an alternative. Hookean and Fraenkel. — The interbead force is derived from a pairwise potential with equilibrium length r eq. Since the force is proportional to j r i r i 1 j r eq, it is attractive or repulsive depending if j r i r i 1 j > or <r eq. As r eq! 0 we recover the Hookean spring. FENE.— The finitely extensible nonlinear elastic (FENE) spring has a maximum bond extension r max be- yond which the force becomes infinite, and hence any length greater than r max is not allowed. The potential for
M beads is described by U FENE 2 r^2 max log 1 j r i r i 1 j^2 r^2 max^ . Wormlike chain (WLC).— The Marko-Siggia (M-S) [16] force expression kBp^ T (^4) 1 ^1 R 2 14 R , where (^) p is the
persistence length, L sp is the maximum length of the
spring, and R j r~^ i L^ ~r sp i 1 j. For chains with more than 2
beads, (^) p was adjusted using the analysis and results presented in [17].
In the above, is the spring constant, i 2 ;... ; M , and the interbead force in each case is F p^ r U. Single chains immersed in ‘‘an ocean’’ of DPD particles constitutes a mesoscopic model of a dilute polymer solu- tion. Hence, the dynamics of a single flexible polymer chain is of great importance for validation and physical understanding of the method. Our work introduces combi- nations of forces aiming to illustrate excluded volume effects and phantom collision minimization (well docu- mented with other methods), not through immiscibility of the solvent but through different bead-spring representa- tions. Figure 1 summarizes results for different spring laws with and without bead-bead repulsions. The corresponding static exponent values ( ) of the radius of gyration are computed for each case, using 5-, 10-, 20-, 50-, and 100- bead chains. The LJ repulsion seems to be mostly respon- sible for capturing self-avoidance while the underlying spring force (Hookean or FENE) appears to have a sec- ondary effect on the scaling exponent, when coupled with hard repulsions. However, FENE forces alone scale close to the Flory exponent, rendering the model realistic with- out any additional repulsions. The FENE parameter r max was also varied, with the values 2 r (^) c and 3 rc giving very similar scaling laws (Fig. 1). The parameters for the WLC were consistent with the rest of the models [18]. DNA molecules under steady shear have been exten- sively studied in experimental [3] and computational [2,15] works. Using DPD we investigated the dynamics of a single WLC. The moving boundaries at y 0 , y L (^) y are modeled using Lees-Edwards boundary conditions [19]: particles leaving the domain at y 0 , Ly are ad- vanced or retarded by an increment of r U (^) x t , U (^) x t , respectively, in the x direction, where t is the time elapsed from an appropriate origin of times and U (^) x denotes twice
100 101 102
100
101
102
Static exponent scaling (ν) for different polymer spring models
M−1 (=number of beads−1)
<R gyratio
n 2
Stiff:FENE 2rc: ν = 0.49719ν = 0. FENE 2rc,LFENE 3rc,L−−Jones:Jones:νν = 0.61669= 0. Hookean,LStiff, L−Jones:−Jones:ν = 0.5736ν = 0. Wormlike:ν = 0.
FIG. 1. Scaling of the radius of gyration of a single polymer chain governed by linear, WLC, and FENE forces and the effect of hard LJ potentials. The chain sizes vary from 5 to 100 beads, with r max 2 rc ; 3 rc , (^1) p 7 , r eq rc , L sp 2 rc. Inte- gration time is 10 000 units with time step t 0 : 01.
PRL 95, 076001 (2005) P H Y S I C A L^ R E V I E W^ L E T T E R S^
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- is taken to be 4 throughout, we compute the density scaling (for water solvents) to be - 14 997 kg = m 3 249 : 25 kg = m 3 at T 25 C. Hence, the corresponding mass of each particle (each cell contains 4 particles on average) is given by m 249 : 25 7 : 05 10 ^19 kg 1 : 76 10 ^16 kg. Next, we use the thermal velocity given by V rms
3 kB T= m
p , where kB 1 : 38 10 ^23 J = K and T 298 K to find the time units based on the solvent
motion: V s 0 : 0084 ms and therefore t s rV^ c 0 : 89 10 ^6 m 0 : 0084 m = s ^1 :^06 ^10
(^4) s. Our DPD time step is t
0 : 02 , a physical time of t 2 : 12 10 ^6 s. For reaching a solution time of 10 000 units, our total integrating time is 500 000 t 1 : 06 s. The solvent-based computational time compares well with DNA experimental statistics gathered over seconds. However, the slower dynamics of the polymer units yield markedly larger time scales. We take a typical polymer relaxation time + 51 : 8 (20 beads) which gives t p 516 :^3 : 8 s 0 : 12 s. The close agreement with BD simulations [2] and the experimental data [3] suggests these scales to be appropriate. Moreover, the shear velocity range for a length Ly 20 r (^) c spans the interval 1 ; 32 in DPD units, which scales to rc 1 ; 32 = t p 7 ; 237 ) sm , within reasonable agreement with the values 10 ; 200 ) sm used in [3]. The authors would like to acknowledge the support of NSF/CTS and NSF/ITR.
*Electronic address: sjoh0341@dam.brown.edu †Electronic address: gk@dam.brown.edu ‡Electronic address: caswell@dam.brown.edu [1] P. S. Doyle and E. S. G. Shaqfeh, J. Non-Newtonian Fluid Mech. 76 , 43 (1998). [2] J. S. Hur, E. S. G. Shaqfeh, and R. G. Larson, J. Rheol. (N.Y.) 44 , 713 (2000).
[3] D. E. Smith, H. P. Babcock, and S. Chu, Science 283 , 1724 (1999). [4] M. Cheon, I. Chang, J. Koplik, and J. R. Banavar, Europhys. Lett. 58 , 215 (2002). [5] N. V. Priezjev and S. M. Troian, Phys. Rev. Lett. 92 , 018302 (2004). [6] A. G. Schlijper, P. J. Hoogerbrugge, and C. W. Manke, J. Rheol. (N.Y.) 39 , 567 (1995). [7] N. A. Spenley, Europhys. Lett. 49 , 534 (2000). [8] Y. Kong, C. W. Manke, W. G. Maddenand, and A. G. Schlijper, J. Chem. Phys. 107 , 592 (1997). [9] C. M. Wijmans and B. Smit, Macromolecules 35 , 7138 (2002). [10] S. Chen, N. Phan-Thien, X.-J. Fan, and B. C. Khoo, J. Non-Newtonian Fluid Mech. 118 , 65 (2004). [11] P.-G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979). [12] P. J. Hoogerbrugge and J. M. V. A. Koelman, Europhys. Lett. 19 , 155 (1992). [13] R. D. Groot, T. J. Madden, and D. J. Tildesley, J. Chem. Phys. 110 , 9739 (1999). [14] P. Espan˜ol and P. Warren, Europhys. Lett. 30 , 191 (1995). [15] R. M. Jendrejack, J. J. de Pablo, and M. D. Graham, J. Chem. Phys. 116 , 7752 (2002). [16] J. F. Marko and E. D. Siggia, Macromolecules 28 , 8759 (1995). [17] P. T. Underhill and P. S. Doyle, J. Non-Newtonian Fluid Mech. 122 , 3 (2004). [18] Different parameters, taken from DNA molecules, pro- duced 12 —indicative that the DNA-wormlike model in equilibrium operates mostly in the linear regime. [19] A. W. Lees and S. F. Edwards, J. Phys. C 5 , 1921 (1972). [20] S. F. Sun, Physical Chemistry of Macromolecules (Wiley, New York, 1994). [21] C. P. Lowe, Europhys. Lett. 47 , 145 (1999). [22] The products for each pair were collected from single- chain simulations in time bins with + + DPDEXP (^1) t sample points, + DPD being the longest relaxation time in DPD units for each case, + EXP typically fixed to 6.3, and t the simulation time step, as done in [3].
PRL 95, 076001 (2005) P H Y S I C A L^ R E V I E W^ L E T T E R S^
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