Polymers in solution exposed to shear flow exhibit large conformational changes due to tumbling motion,i.e.,they stretch and collapse with a characteristic frequency depending on shear rate. This behavior has intensively been studied experimentally and numerically ^{[1, 2, 3, 4, 5, 6, 7]}. In contrast,the flowinduced dynamical behavior of polymers absorbed on a surface has received far less attention ^{[8]}. Insight into the behavior of such systems is of fundamental importance in a wide spectrum of systems ranging from the separation of macromolecules using microcircuit devices to the gene therapy in which DNA is transported to a desired target site ^{[9, 10, 11, 12, 13]}.
The dynamic behavior of an absorbed polymer in shear flow is governed by various parameters; aside from the shear rate,the properties of surfaces are of major importance. Since the tumbling dynamics is usually observed at high shear rates close to the critical shear rate at which chains desorb,hydrodynamic interactions (HI) are also believed to play an important role in this behavior.
In this letter,we investigate the tumbling dynamics of individual absorbed polymers by a hybrid mesoscale simulation approach,which combining molecular dynamics for the polymer molecules with the multiparticle collision dynamics describing the solvent ^{[14]}. As has been shown,this hybrid method is very well suited to study the properties of polymer solutions,where both thermal fluctuations and hydrodynamics are important ^{[15, 16]}.
2. ExperimentalIn our model,a flexible polymer chain in an explicit solvent consistsofN = 20monomersofmassMeach.Theexcludedvolume interactions between monomers are described by the truncated shift purely repulsive LennardJones (LJ) potential with the parameter s characterizing the monomer size and e the energy. A finite extensible nonlinear elastic potential (FENE),with the maximum bond length 1.5 s and the spring constant 30 e/s^{2},is used to model the connectivity between adjacent monomers of the chain. The monomer dynamics is determined by Newton’s equations of motion,which are integrated by the velocity Verlet algorithm with time step h_{p}. The multiparticle collision dynamics method is used to describe the solvent. It is composed of Nspoint like particles of mass m. The algorithm consists of alternating streaming and collision steps. In the streaming step,the solvent particles move ballistically for a time h. In the collision steps, particles are sorted into cubic cells of side length and their relative velocities,with respect to the centerofmass velocity of their cell, are rotated around a randomly oriented axis by a fixed anglea. The solventpolymer coupling is achieved by taking the monomers into account in the collision step. The system is confined between two parallel walls as shown in Fig. 1. Shear flow is imposed by sliding the upper wall relative to the lower wall at a constant rate and periodic boundary conditions are applied parallel to the walls.We impose noslip boundary conditions at walls for solvent particles. The chain is absorbed on the lower wall by the LJ potential with the interaction strength ε_{w} and the cutoff distance 2.5 s,Uadsorb. Two types of walls are considered. The first is a smoothsurface for monomers; the second is atomically corrugated and the interactions between monomers and the surface are U = U_{adsorb} [1 + A cos(2px/q) sin(2py/q)] with q = 1 and A = 0.2 consistent with previous works ^{[17]}. The energy of interaction of the chain with the lower wall includes repulsive and attractive parts,the attractivepart makesthechainadsorb onthesurfaceand the repulsive part avoids monomer moving outside the wall. Periodic boundary conditions are applied in the directions parallel to the walls. The shear flow with shear rate = V_{wall}/L_{z} is introduced into the system by the motion of the upper wall and bounceback boundary condition being applied simultaneously,which means the solvent particles reverse their direction of velocity butkeep the magnitude constant. The system is in the canonical ensemble (NVT) and the viscous friction of particles with the walls of the simulation box creates heat that result in temperature rising, which can influence the velocity of particles,therefore thermo stattingisrequiredinthenonequilibriumMPCDsimulation.Alocal Maxwellian thermostat is used to keep the temperature at the desired value ^{[18]}.
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Fig. 1. Schematic depiction of an absorbed polymer chain under shear flow. 
All simulations are performed with σ = ε = α = 1,α = 1308,the average number of solvent particles per cell ρ = 10,M = rm, h = 0.1s(m/k_{B}T)^{1/2} (where k_{B} is the Boltzmann constant and T the temperature),h_{p}/h = 20,m = 1,and k_{B}T = 1. A simulation box with size 30 × 20 × 20 is applied. More than 50 parallel samples with different initial conditions for each data to improve the quality of the results.
3. Results and discussionWhen polymer fluids are subjected to a simple shear flow,the velocity across the chain is substantial,and hence it may undergo large conformational changes; i.e.,a polymer chain stretches and recoilsinthecourseoftime^{[1, 2, 3, 4, 5]}.Theinstantaneousconformation of polymers can be quantitatively characterized by the gyration tensor,which is defined as Gab¼PN i¼1Dri;aDri;b=N,where Dri,ais the position of monomer i in the center of mass reference frame of the polymer anda,b 2 x,y,z denote Cartesian components^{[16]}. The average gyration tensor is denoted as 〈G_{αβ}〉,where h?i denotes ensemble average. It is noticed that Gabcan be directly accessed in single molecule experiments and 〈G_{αβ}〉 in scattering experiments. At infinite dilution,the flow strength is characterized by the Weissenberg number Wi ¼ · t,where is the shear rate and t_{0} the longest relaxation time obtained at equilibrium.
The tumbling dynamics of polymers absorbed on surfaces is observed for Wi ? 1 where the chains are not able to relax back to the equilibrium conformation. The time traces of relative deformation of chains along the flow and gradient directions are shown in Fig. 2. It is clear that in the tumbling process,the extension of a chain along the flow direction always follows the shrinkage of its gradient thickness,or a chain stretching along the gradient direction follows the retraction in the flow direction. Furthermore,the maximum extension in the flow (gradient) direction always relates to the shrinkage in the gradient (flow) direction.
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Fig. 2. Time trajectories of the extensions of the radius of gyration along the flow and gradient directions for Wi = 617.7 with ε_{w}= 4.0. 
In order to obtain the characteristic time for the tumbling dynamics,we determine the crosscorrelation function between the conformational changes in flow and gradient directions, calculated as ^{[3, 6, 15, 19]}
Here δGaa= Gaa? hGaai. Fig. 3 shows crosscorrelation func tions for several Weissenberg numbers for ε_{w}= 4.0. Similarly to unconstrained linear polymers, the tumbling dynamics of absorbed polymers is not perfectly periodic because the autocor relation,Cxz,decays to zero at large timelags. Cxzexhibits the maximum at t?,indicating that positive values of dGxx are correlated with positive values dGzz,or a collapsed state along the xdirection is correlated with a previous collapsed sated in ydirection. The minimum of Cxzat t+reveals that positive values of dGaaare linked with negative ones of the orthogonal directions;i.e.,polymer compresses in the zdirection is linked with its extension in xdirection. Hence,the difference t+? t?is related to conformational changes of polymers due to tumbling. Here,the characteristicoftumblingmotionisdefinedastt= 2(t+? t?),where the factor two is attributed to two nonequivalent conformations leading to a maximum and a minimum,respectively.
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Fig. 3. Crosscorrelation functions for a polymer absorbed on the smooth surface with ε_{w}= 4.0 for various Weissenberg numbers as indicated. 
Normalized tumbling frequencies f_{t}= t_{0}/tt,scaled by the relaxation time t_{0},as a function of the Weissenberg number are shown in Fig. 4. Data for various absorption strengthscollapse onto a universal curve,which can be described by a power law f_{t}? Wim with m = 0.80. Lo¨wen and his coworkers reported a larger exponent m = 0.90 obtained by Brownian dynamics simulations ^{[8]}. In their work,the characteristic time is calculated from the Fourier transform of the autocorrelation of the radius of gyration ^{[8]}. However,DelgadoBuscalionipointedout that the morenature measure of the tumbling motion is the cross correlation between the chain extension along flow and gradient directions,namely, Eq. (1) ^{[19]}. This scaling behavioris practically consistent with that of a polymer with one end tethered on a surface,for which ft? Wi0.78is observed ^{[19]}. For unconstrained individual polymers in dilute solution,a somewhat smaller exponent m = 0.67 was obtained in Refs. ^{[2, 3, 6]}. Furthermore,Fig. 4 also shows that the corrugation of surfaces has no effect on the shear dependence of f_{t}.
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Fig. 4. The normalized tumbling frequencies f_{t},scaled withtherelaxationtimet_{0}for various ewas indicated. 
In summary,we study the tumbling dynamics of individual absorbed polymer chains in shear flow. The crosscorrelation betweenthechainextensionalongflowandgradientdirectionsare used to characterize the tumbling motion. The results show that the normalized tumbling frequencies follow the power law ft ≈ Wi^{0.80}. We find that both adsorptive strength and surface corrugation have no influence on the scaling behavior of tumbling dynamics.
AcknowledgmentsThis work is supported by the National Natural Science FoundationofChina(No.21274153).WearegratefultoComputing Center of Jilin Province for essential support.
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