Brownian Dynamics (BD) simulations are a standard tool for understanding the dynamics of polymers in and out of equilibrium. Quantitative comparison can be made to rheological measurements of dilute polymer solutions, as well as direct visual observations of fluorescently labeled DNA. The primary computational challenge with BD is the expensive calculation of hydrodynamic interactions (HI), which are necessary to capture physically realistic dynamics. The full HI calculation, performed via a Cholesky decomposition every time step, scales with the length of the polymer as O(N3). This limits the calculation to a few hundred simulated particles. A number of approximations in the literature can lower this scaling to O(N2N2.25), and explicit solvent methods scale as O(N); however both incur a significant constant per-time step computational cost. Despite this progress, there remains a need for new or alternative methods of calculating hydrodynamic interactions; large polymer chains or semidilute polymer solutions remain computationally expensive. In this paper, we introduce an alternative method for calculating approximate hydrodynamic interactions. Our method relies on an iterative scheme to establish self-consistency between a hydrodynamic matrix that is averaged over simulation and the hydrodynamic matrix used to run the simulation. Comparison to standard BD simulation and polymer theory results demonstrates that this method quantitatively captures both equilibrium and steady-state dynamics after only a few iterations. The use of an averaged hydrodynamic matrix allows the computationally expensive Brownian noise calculation to be performed infrequently, so that it is no longer the bottleneck of the simulation calculations. We also investigate limitations of this conformational averaging approach in ring polymers.

1.
M.
Doi
and
S. F.
Edwards
,
The Theory of Polymer Dynamics
(
Clarendon
,
Oxford
,
1988
).
2.
R. B.
Bird
,
C. F.
Curtiss
,
R. C.
Armstrong
, and
O.
Hassager
,
Dynamics of Polymeric Liquids
(
Wiley
,
New York
,
1987
), Vol. 2.
3.
R. G.
Larson
, “
The rheology of dilute solutions of flexible polymers: Progress and problems
,”
J. Rheol.
49
,
1
(
2005
).
4.
E. S. G.
Shaqfeh
, “
The dynamics of single-molecule DNA in flow
,”
J. Non-Newtonian Fluid Mech.
130
,
1
(
2005
).
5.
T. T.
Perkins
,
S. R.
Quake
,
D. E.
Smith
, and
S.
Chu
, “
Relaxation of a single DNA molecule observed by optical microscopy
,”
Science
264
,
822
(
1994
).
6.
D. E.
Smith
,
T. T.
Perkins
, and
S.
Chu
, “
Dynamical scaling of DNA diffusion coefficients
,”
Macromolecules
29
,
1372
(
1996
).
7.
T. T.
Perkins
,
D. E.
Smith
, and
S.
Chu
, “
Single polymer dynamics in an elongational flow
,”
Science
276
,
2016
(
1997
).
8.
R. G.
Larson
,
T. T.
Perkins
,
D. E.
Smith
, and
S.
Chu
, “
Hydrodynamics of a DNA molecule in a flow field
,”
Phys. Rev. E
55
,
1794
(
1997
).
9.
D. E.
Smith
and
S.
Chu
, “
Response of flexible polymers to a sudden elongational flow
,”
Science
281
,
1335
(
1998
).
10.
D. E.
Smith
,
H. P.
Babcock
, and
S.
Chu
, “
Single-polymer dynamics in steady shear flow
,”
Science
283
,
1724
(
1999
).
11.
R. G.
Larson
,
H.
Hu
,
D. E.
Smith
, and
S.
Chu
, “
Brownian dynamics simulations of a DNA molecule in an extensional flow field
,”
J. Rheol.
43
,
267
(
1999
).
12.
C. M.
Schroeder
,
H. P.
Babcock
,
E. S. G.
Shaqfeh
, and
S.
Chu
, “
Observation of polymer conformation hysteresis in extensional flow
,”
Science
301
,
1515
(
2003
).
13.
D. J.
Mai
,
A. B.
Marciel
,
C. E.
Sing
, and
C. M.
Schroeder
, “
Topology-controlled relaxation dynamics of single branched polymers
,”
ACS Macro Lett.
4
,
446
(
2015
).
14.
D. J.
Mai
and
C. M.
Schroeder
, “
Single polymer dynamics of topologically complex DNA
,”
Curr. Opin. Colloid Interface Sci.
26
,
28
(
2016
).
15.
K.-W.
Hsiao
,
C. M.
Schroeder
, and
C. E.
Sing
, “
Ring polymer dynamics are governed by a coupling between architecture and hydrodynamic interactions
,”
Macromolecules
49
,
1961
(
2016
).
16.
J. S.
Hur
,
E. S. G.
Shaqfeh
,
H. P.
Babcock
, and
S.
Chu
, “
Dynamics and configurational fluctuations of single DNA molecules in linear mixed flows
,”
Phys. Rev. E
66
,
011915
(
2002
).
17.
N. J.
Woo
and
E. S. G.
Shaqfeh
, “
The configurational phase transitions of flexible polymers in planar mixed flows near simple shear
,”
J. Chem. Phys.
119
,
2908
(
2003
).
18.
H.
Ma
and
M. D.
Graham
, “
Theory of shear-induced migration in dilute polymer solutions near solid boundaries
,”
Phys. Fluids
17
,
083103
(
2005
).
19.
N.
Watari
,
M.
Makino
,
N.
Kikuchi
,
R. G.
Larson
, and
M.
Doi
, “
Simulation of DNA motion in a microchannel using stochastic rotation dynamics
,”
J. Chem. Phys.
126
,
094902
(
2007
).
20.
A.
Alexander-Katz
and
R. R.
Netz
, “
Dynamics and instabilities of collapsed polymers in shear flow
,”
Macromolecules
41
,
3363
(
2008
).
21.
C. E.
Sing
and
A.
Alexander-Katz
, “
Globule-stretch transitions of collapsed polymers in elongational flow
,”
Macromolecules
43
,
3532
(
2010
).
22.
S.
Somani
,
E. S. G.
Shaqfeh
, and
J. R.
Prakash
, “
Effect of solvent quality on the coil-stretch transition
,”
Macromolecules
43
,
10679
(
2010
).
23.
C. E.
Sing
and
A.
Alexander-Katz
, “
Dynamics of collapsed polymers under the simultaneous influence of elongational and shear flows
,”
J. Chem. Phys.
135
,
014902
(
2011
).
24.
C. E.
Sing
and
A.
Alexander-Katz
, “
Elongational flow induces the unfolding of von Willebrand factor at physiological flow rates
,”
Biophys. J.
98
,
L35
(
2010
).
25.
A.
Alexander-Katz
, “
Toward novel polymer-based materials inspired in blood clotting
,”
Macromolecules
47
,
1503
(
2014
).
26.
N.
Laachi
,
M.
Kenward
,
E.
Yariv
, and
K. D.
Dorfman
, “
Force-driven transport through periodic entropy barriers
,”
Europhys. Lett.
80
,
50009
(
2007
).
27.
Y.
Zhang
,
J. J.
de Pablo
, and
M. D.
Graham
, “
An immersed boundary method for Brownian dynamics simulation of polymers in complex geometries: Application to DNA flowing through a nanoslit with embedded nanopits
,”
J. Chem. Phys.
136
,
014901
(
2012
).
28.
M. P.
Allen
and
D. J.
Tildesley
,
Computer Simulation of Liquids
(
Clarendon
,
Oxford
,
1987
).
29.
B. H.
Zimm
, “
Dynamics of polymer molecules in dilute solution: Viscoelasticity, flow birefringence and dielectric loss
,”
J. Chem. Phys.
24
,
269
(
1956
).
30.
A.
Saadat
and
B.
Khomami
, “
Computationally efficient algorithms for incorporation of hydrodynamic and excluded volume interactions in Brownian dynamics simulations: A comparative study of the Krylov subspace and Chebyshev based techniques
,”
J. Chem. Phys.
140
,
184903
(
2014
).
31.
R.
RodriguezSchmidt
,
J. G.
HernandezCifre
, and
J.
Garciade la Torre
, “
Comparison of Brownian dynamics algorithms with hydrodynamic interaction
,”
J. Chem. Phys.
135
,
084116
(
2011
).
32.
M.
Fixman
, “
Construction of Langevin forces in the simulation of hydrodynamic interaction
,”
Macromolecules
19
,
1204
(
1986
).
33.
A.
Saadat
and
B.
Khomami
, “
Matrix-free Brownian dynamics simulation technique for semidilute polymeric solutions
,”
Phys. Rev. E.
92
,
033307
(
2015
).
34.
W.
Jiang
,
J.
Huang
,
Y.
Wang
, and
M.
Laradji
, “
Hydrodynamic interaction in polymer solutions simulated with dissipative particle dynamics
,”
J. Chem. Phys.
126
,
044901
(
2007
).
35.
L.
Jiang
,
N.
Watari
, and
R. G.
Larson
, “
How accurate are stochastic rotation dynamics simulations of polymer dynamics
,”
J. Rheol.
57
,
1177
(
2013
).
36.
P.
Ahlrichs
and
B.
Dunweg
, “
Simulation of a single polymer chain in solution by combining lattice Boltzmann and molecular dynamics
,”
J. Chem. Phys.
111
,
8225
(
1999
).
37.
T.
Geyer
and
U.
Winter
, “
An O(N2) approximation for hydrodynamic interactions in Brownian dynamics simulations
,”
J. Chem. Phys.
130
,
114905
(
2009
).
38.
P. G.
de Gennes
, “
Coil-stretch transition of dilute flexible polymers under ultrahigh velocity gradients
,”
J. Chem. Phys.
60
,
5030
(
1974
).
39.
H. C.
Öttinger
, “
Consistently averaged hydrodynamic interaction for Rouse dumbbells in steady shear flow
,”
J. Chem. Phys.
83
,
6535
(
1985
).
40.
H. C.
Öttinger
, “
Generalized Zimm model for dilute polymer solutions under theta conditions
,”
J. Chem. Phys.
86
,
3731
(
1987
).
41.
H. C.
Öttinger
, “
Gaussian approximation for Rouse chains with hydrodynamic interaction
,”
J. Chem. Phys.
90
,
463
(
1989
).
42.
J. J.
Magda
,
R. G.
Larson
, and
M. E.
Mackay
, “
Deformation-dependent hydrodynamic interaction in flows of dilute polymer solutions
,”
J. Chem. Phys.
86
,
2504
(
1988
).
43.
C. M.
Schroeder
,
E. S. G.
Shaqfeh
, and
S.
Chu
, “
Effect of hydrodynamic interactions on DNA dynamics in extensional flow: Simulation and single molecule experiment
,”
Macromolecules
37
,
9242
(
2004
).
44.
Y.
von Hansen
,
M.
Hinczewski
, and
R. R.
Netz
, “
Hydrodynamic screening near planar boundaries: Effects on semiflexible polymer dynamics
,”
J. Chem. Phys.
134
,
235102
(
2011
).
45.
M.
Hinczewski
,
X.
Schlagberger
,
M.
Rubinstein
,
O.
Krichevsky
, and
R. R.
Netz
, “
End-monomer dynamics in semiflexible polymers
,”
Macromolecules
42
,
860
(
2009
).
46.
J.
Rotne
and
S.
Prager
, “
Variational treatment of hydrodynamic interaction in polymers
,”
J. Chem. Phys.
50
,
4831
(
1969
).
47.
H.
Yamakawa
, “
Transport properties of polymer chains in dilute solution: Hydrodynamic interaction
,”
J. Chem. Phys.
53
,
436
(
1970
).
48.
T.
Ando
,
E.
Chow
,
Y.
Saad
, and
J.
Skolnick
, “
Krylov subspace methods for computing hydrodynamic interactions in Brownian dynamics simulations
,”
J. Chem. Phys.
137
,
064106
(
2012
).
49.
P. E.
Rouse
, “
A theory of the linear viscoelastic properties of dilute solutions of coiling polymers
,”
J. Chem. Phys.
21
,
1272
(
1953
).
50.
C. M.
Schroeder
,
R. E.
Teixeira
,
E. S. G.
Shaqfeh
, and
S.
Chu
, “
Characteristic periodic motion of polymers in shear flow
,”
Phys. Rev. Lett.
95
,
018301
(
2005
).
51.
B.
Dünweg
and
K.
Kremer
, “
Microscopic verification of dynamic scaling in dilute polymer solutions: A molecular-dynamics simulation
,”
Phys. Rev. Lett.
66
,
2996
(
1991
).
52.
D. J.
Mai
,
C.
Brockman
, and
C. M.
Schroeder
, “
Microfluidic systems for single DNA dynamics
,”
Soft Matter
8
,
10560
(
2012
).
53.
P. G.
de Gennes
,
Scaling Concepts in Polymer Physics
(
Cornell University Press
,
Ithaca
,
1979
).
54.
Y.
Zhang
,
A.
Donev
,
T.
Weisgraber
,
B. J.
Alder
,
M. D.
Graham
, and
J. J.
de Pablo
, “
Tethered DNA dynamics in shear flow
,”
J. Chem. Phys.
130
,
234902
(
2009
).
55.
C.
Sendner
and
R. R.
Netz
, “
Single flexible and semiflexible polymers at high shear: Non-monotonic and non-universal stretching response
,”
Eur. Phys. J. E
30
,
75
81
(
2009
).
56.
C. E.
Sing
and
A.
Alexander-Katz
, “
Giant nonmonotonic stretching response of a self-associating polymer in shear flow
,”
Phys. Rev. Lett.
107
,
198302
(
2011
).
57.
I. S.
Dalal
,
A.
Albaugh
,
N.
Hoda
, and
R. G.
Larson
, “
Tumbling and deformation of isolated polymer chains in shearing flow
,”
Macromolecules
45
,
9493
9499
(
2012
).
58.
I. S.
Dalal
,
N.
Hoda
, and
R. G.
Larson
, “
Multiple regimes of deformation in shearing flow of isolated polymers
,”
J. Rheol.
56
,
305
332
(
2012
).
59.
R. M.
Robertson
,
S.
Laib
, and
D. E.
Smith
, “
Diffusion of isolated DNA molecules: Dependence on length and topology
,”
Proc. Natl. Acad. Sci. U. S. A.
103
,
7310
(
2006
).
60.
Y.
Li
,
K.-W.
Hsiao
,
C. A.
Brockman
,
D. Y.
Yates
,
R. M.
Robertson-Anderson
,
J. A.
Kornfield
,
M. J.
San Francisco
,
C. M.
Schroeder
, and
G. B.
McKenna
, “
When ends meet: Circular DNA stretches differently in elongational flows
,”
Macromolecules
48
,
5997
(
2015
).
61.
V.
Bloomfield
and
B. H.
Zimm
, “
Viscosity, sedimentation, et cetera, of ring- and straight-chain polymers in dilute solution
,”
J. Chem. Phys.
44
,
315
(
1966
).
62.
B. H.
Zimm
and
W. H.
Stockmayer
, “
The dimensions of chain molecules containing branches and rings
,”
J. Chem. Phys.
17
,
1301
(
1949
).
63.
D. R.
Tree
,
A.
Muralidhar
,
P. S.
Doyle
, and
K. D.
Dorfman
, “
Is DNA a good model polymer?
,”
Macromolecules
46
,
8369
(
2013
).
64.
C.
Sasmal
,
K.-W.
Hsiao
,
C. M.
Schroeder
, and
J. R.
Prakash
, “
Parameter-free prediction of DNA dynamics in planar extensional flow of semidilute solutions
,”
J. Rheol.
61
,
169
(
2017
).
65.
A.
Jain
,
P.
Sunthar
,
B.
Dünweg
, and
J. R.
Prakash
, “
Optimization of a Brownian-dynamics algorithm for semidilute polymer solutions
,”
Phys. Rev. E
85
,
066703
(
2012
).
66.
C.
Stoltz
,
J. J.
de Pablo
, and
M. D.
Graham
, “
Concentration dependence of shear and extension rheology of polymer solutions: Brownian dynamics simulations
,”
J. Rheol.
50
,
137
(
2006
).
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