This journal is©The Royal Society of Chemistry 2017 Soft Matter, 2017, 13, 8625--8635 | 8625

Cite this: SoftMatter, 2017,

13, 8625

The effect of hydrodynamic interactions onnanoparticle diffusion in polymer solutions:a multiparticle collision dynamics study

Anpu Chen, a Nanrong Zhao *a and Zhonghuai Hou *b

The diffusion of nanoparticles (NPs) in polymer solutions is studied by a combination of a mesoscale

simulation method, multiparticle collision dynamics (MPCD), and molecular dynamics (MD) simulations. We

investigate the long-time diffusion coefficient D as well as the subdiffusive behavior in the intermediate

time region. The dependencies of both D and subdiffusion factor a on NP size and polymer concentration,

respectively, are explicitly calculated. Particular attention is paid to the role of hydrodynamic interaction (HI)

in the NP diffusion dynamics. Our simulation results show that the long-time diffusion coefficients satisfy

perfectly the scaling relation found by experimental observations. Meanwhile, the subdiffusive factor

decreases with the increase in polymer concentration but is of little relevance to the NP size. By parallel

simulations with and without HI, we reveal that HI will generally enhance D, while the enhancement effect

is non-monotonous with increasing polymer concentration, and it becomes most pronounced at

semidilute concentrations. With the aid of a scaling law based on the diffusive activation energy model, we

understand that HI affects diffusion through decreasing the diffusive activation energy on the one hand

while increasing the effective diffusion size on the other. In addition, HI will certainly influence the

subdiffusive behavior of the NP, leading to a larger subdiffusion exponent.

1 Introduction

The transport property of proteins in complex fluids is a problemof broad importance in diverse fields ranging from materialsscience to cellular biophysics and even drug delivery. There isgrowing interest in understanding how biopolymers such asproteins move through crowded cytoplasmic environments.1–3

Since a typical cell is in general occupied by substantial macro-molecular content such as proteins, DNA, ribosomes andmembranes, up to 40% of the total mass and volume, the veryhigh total concentration of these molecules, or ‘macromolecularcrowding’ 4 plays an important role3,5–7 in the transport processesas well as the reaction stages, such as protein–protein association8

and self-assembly of various supramolecular structures.9,10

A semidilute polymer solution is a mimic system3,9 which helpsus to understand the crowding effect. Adding nanoparticles (NPs)to polymer liquids can result in novel electrical and photonicproperties of nanocomposite systems where the mobility of NPscan play a crucial role.2,11–13

In recent years, diffusion of NPs in polymer solutionshas received a lot of attention both experimentally14–18 andtheoretically.19–25 In experiments, fluctuation correlationspectroscopy (FCS),26–29 dynamic light scattering (DLS),27,30

and capillary viscosimetry are general tools to investigate thediffusion of a NP in complex fluids. It has been found that NPmotion in polymer solutions may exhibit typically non-Markoviancharacteristics of long-time memory,31,32 leading to anomaloussubdiffusive behavior, wherein the mean square displacement(MSD) scales with time as MSD(t) B ta with a o 1 (a = 1 corres-ponds to normal diffusion). As discussed in many literaturereports,3,21,22,24,25,33–37 the main mechanism leading to subdiffusionis the so-called cage effect the NP experiences exerted by thesurrounding crowding agents. Indeed, the subdiffusion of NPdiffusion has been even used as a probe of the crowding effect in acell.3,35,36 Besides, for complex polymer solutions, the long-timediffusion coefficient of a nano-scale probe has been found to deviateremarkably from the well-known Stokes–Einstein (SE) relation,which states that the translational diffusion coefficient D followsD = kBT/( fpZR), where R is the particle radius, kB is the Boltzmannconstant, T is the absolute temperature, Z is the solution viscosity,and f = 4 or 6 for slip or stick boundary conditions, respectively.Indeed, it has been found by experiments14,18,26,27,38–40 that thediffusion coefficient of a NP can deviate from SE, up to over severalorders of magnitude, in particular for small probe sizes.

a College of Chemistry, Sichuan University, Chengdu 610064, China.

E-mail: zhaonanr@scu.edu.cnb Hefei National Laboratory for Physical Sciences at the Microscale & Department of

Chemical Physics, University of Science and Technology of China, Hefei,

Anhui 230026, China. E-mail: hzhlj@ustc.edu.cn

Received 14th September 2017,Accepted 31st October 2017

DOI: 10.1039/c7sm01854a

rsc.li/soft-matter-journal

Soft Matter

PAPER

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Theoretically, great efforts have also been made to develop aquantitative strategy to understand the diffusion behavior of aparticle in complex polymer solutions. For instance, the well-known Phillies retardation formula41 D B exp(�bcd) where c isthe polymer concentration, and b and d are system parameters,describes the slowing down of a particle diffusion due to theaddition of polymer molecules. Besides the phenomenologicalstudies, from a microscopic viewpoint, mode-coupling theory(MCT) has been used to study a NP’s long-time diffusioncoefficient and subdiffusive dynamics in a semidilute polymersolution.21,22,24,25 In particular, very recently, Holyst et al. havestudied systematically the translational diffusion of NPs, pro-teins and dyes in semidilute polymer solutions and made a fewrather important discoveries. They introduced the length-scale-dependent nano-viscosity such that the SE relation can be validfor all length scales. More importantly, they established ascaling relationship27,42–46 for the effective viscosity which iswidely applicable for various length scales and polymer con-centrations, which suggested that the most relevant lengthscale is an effective radius Reff, which is determined by theparticle radius R and the hydrodynamic radius rh of the poly-mer molecule. Nevertheless, these inspiring scaling relationswere mainly obtained from phenomenological models. To thebest of our knowledge, direct simulation work at the micro-scopic level to study these scaling behaviors has not beenreported yet, which constitutes the first motivation of thepresent work.

On the other hand, hydrodynamic interaction (HI), a long-range force which arises from the motion of one particle andacts on solvent and other particles (through the solventmedium), is an important factor which might affect thetransport property of a probe in solutions. HI has beenrevealed to have an important role in self-assembly,47–49 cellmobility,50,51 as well as the formation and folding ofproteins.52–54 The dynamical behavior of macromolecules insolution is strongly affected by HI.55–57 Generally, it is knownthat HI would accelerate the diffusion of simple sphericalcolloids as well as complex polymer chains. Nevertheless, therole of HI was usually implicitly considered, for instance, insome theoretical frameworks including MCT.21,24,25 For NPdiffusion in polymer solutions, the effect of HI was not explicitlystudied either. Moreover, despite the intuitive extra long-rangecoupling between NPs and polymer chains rendered by HI,how HI would practically influence the subdiffusive behaviorand the scaling relations for the long-time diffusion coefficienthas not been investigated before. This serves as another motiva-tion of the present work.

To investigate the effect of HI in an explicit way, a meso-scopic method, the so-called stochastic rotation dynamics(SRD) or multiple-particle collision dynamics (MPCD), hadbeen proposed recently.58 In MPCD, the solvent is representedby a large number of particles, which are ideal and move in acontinuous space, subject to Newton’s laws of motion. Atdiscrete time steps, a coarse-grained collision step allowsparticles to exchange momentum. The collision step locallyconserves mass, momentum, and energy, thus the correct

hydrodynamic Navier–Stokes equations are captured inthe continuum limit.58,59 With the coarse-grained form ofMPCD, enormous time and length scale differences betweenmesoscopic colloidal and microscopic solvent particles canpossibly be coupled at a reasonable resolution while retainingcomputational efficiency. Moreover, in MPCD, HI can beswitched off by regularly randomizing the absolute fluidparticle velocities. This makes it very convenient to investigatethe role of HI.59 MPCD can also be combined with standardmolecular dynamics (MD) simulation to account for theinteraction of the solute particles with the polymer molecules.This method has been applied to many systems ranging fromsimple flow systems60–62 to complex soft matter systemsembedded in a solvent.63–69 In particular, recently, we haveperformed a related study,70 where the MPCD method wasused to study the diffusion of a NP in polymer solutions.Therein, we mainly introduced the method and focused on thedependence of the long-time diffusion coefficient D on thepolymer concentration, with or without HI. We have usedthe phenomenological Phillies equation to fit the dependence ofD on concentration. The main finding there was that the scalingexponent in the equation is different if HI is off compared to thecase of HI-on. Nevertheless, the study there was very preliminaryand the analysis was rather thin. In particular, the effects of HI onthe subdiffusive behavior were not investigated. The physicalmeanings of the fitting parameters used in the Phillies equation,were not clear. In addition, the scaling of D with the nanoparticlesize Rn, which has been a very important topic in both experi-mental and theoretical studies in recent years, has not yet beenstudied carefully.

Motivated by the above considerations, in the present work,we will investigate the diffusion dynamics of spherical NPs insemidilute polymer solutions by using the combined MPCD–MD simulation method described above, in a more compre-hensive manner. Particular attention is paid to how HI influ-ences the subdiffusion as well as the long-time diffusionbehavior. Our results show that at medium concentrations,both subdiffusive and long-time behavior are evidently affectedby HI and the coupling of NP and polymer chain diffusiondynamics is strengthened by HI. Interestingly, the NP’s long-time diffusion coefficient recovers the scaling law of Holystboth with and without HI and implies a rather complicatedmechanism of HI effect. This complicated mechanism finallygives a nontrivial acceleration effect of HI as the polymerconcentration increases. The results in this paper indicate thatHI is important to the behavior of diffusing probes in polymersolutions.

The paper is organized as follows. Firstly, we introduceour model and method, including the interaction forcesand a brief introduction of the MPCD method includingthe basic idea and the switch of HI. Secondly, we ran simula-tions to calculate the translational diffusion coefficientsof NPs in a specific model of polymer solution. Comparisonbetween HI on and off will be addressed for subdiffusionand long-time diffusion of NPs. Finally, we concludethe paper.

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2 Model and method

We restrict our model to a single spherical NP of radius Rn

surrounded by polymer chains in solution, where the polymerchain is presented in the form of Nb connected beads withcharacteristic size s and mass M and the solvent is representedby a large number of identical fluid particles of mass m. Asnapshot of the model is given in Fig. 1.

A computationally efficient hybrid algorithm is adapted herefor the dynamics of the polymers and NP in solution byexploiting the time-scale separation between the microscopicdynamics of the pair-wise interacting particles (NP and polymerbeads) and the much slower propagation of hydrodynamicmodes within the solvent.65

2.1 Dynamics of NP and polymer beads

Here we model the NP (n)–polymer bead (p) interaction potentialjnp(r) by steep repulsive interactions of the Lennard-Jones (L-J)form71 offset by the interaction range Rev:

jnpðrÞ ¼

1; r� Rev

es

r�Rev

� �12

� sr�Rev

� �6

þ1

4

" #; Revo roRevþ 2

16s

0; r� Rev þ 216s

8>>>>>><>>>>>>:

(1)

where Rev = Rn � s/2.Similarly, the polymer bead (p)–polymer bead (p) interaction

potential jpp(r) takes the L-J form:

jppðrÞ ¼e

sr

� �12� s

r

� �6þ 1

4

� �; 0o ro 2

16s

0; r � 216s

8>><>>: (2)

while bond effects among adjacent beads on the same chain aregiven by the Finite Extensible Nonlinear Elastic (FENE)potential:

jFENEðrÞ ¼ �kr02

2ln 1� r

r0

� �2" #

; ro r0 (3)

where k is the bond strength and r0 is used as the maximumbond length.

The dynamics of the NP and polymer beads is treated bymolecular dynamics (MD) simulations applying the velocityVerlet integration scheme72 with a time step DtMD:

Ri tþ DtMDð Þ ¼ RiðtÞ þ ViðtÞDtMD þFiðtÞ2Mi

DtMD2

Vi tþ DtMDð Þ ¼ ViðtÞ þFiðtÞ þ Fi tþ DtMDð Þ

2MiDtMD

(4)

where Ri, Vi, Fi and Mi are the position, velocity, total force andmass of particle i, respectively.

2.2 Dynamics of the solvent

The fluid particle–fluid particle interaction is simulated by theMPCD method.63 In this method, the dynamics of the fluidparticles is treated by alternating streaming and collisionprocedures. In the streaming procedure, the positions of thefluid particles are propagated synchronously with NP andpolymer beads applying a ballistic motion with the timestep DtMD:

Ri(t + DtMD) = Ri(t) + Vi(t)DtMD (5)

where Ri and Vi are the position and velocity of particle i,respectively.

After a total time DtMPC of ballistic motion, the collisionprocedure is operated. In the collision procedure, the fluidparticles are sorted into cubic cells of side length s and theirrelative velocities, with respect to the center-of-mass velocity ofthe cell, are rotated around a random orientation by a fixedangle a:

vinew = VCM + R(a)(vi

old � VCM) (6)

where viold and vi

new denote the velocity of fluid particle i beforeand after the collision procedure respectively, and R(a) is therotation matrix. In three dimensions, various schemes for therandom collisions are possible,58,60,73 the one employed hereconsists of choosing a random axis from xyz for each cell. VCM

is the center-of-mass velocity of the Nc fluid particles containedin the cell before the collision procedure:

VCM ¼1

Nc

XNc

i¼1viold (7)

2.3 The couplings with solvent

The solvent–polymer coupling is achieved by taking the polymerbeads into account in the collision procedure above; thiskind of coupling has been widely used in polymer solutions.Malevanets and Yeomans, and others, presented numericalresults describing the effect of HI on the dynamics of a shortpolymer chain in solution64,65,67,68,74 for its advantage of repro-ducing the expected scaling laws of polymer equilibriumproperties68 as well as avoiding spurious depletion forces.59

Fig. 1 A typical snapshot of the model at a semidilute concentration ofpolymer. The NP (golden sphere) is surrounded by connected polymerbeads (green beads) and fluid particles (blue point).

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The center-of-mass velocity of collision cells before the collisionprocedure is then written as:

VCM ¼

PNc

i¼1mviold þ

PNbeadc

j¼1Mv

jbead�old

mNc þMNbeadc

(8)

where Nbeadc is the number of polymer beads within the

collision cell and vjbead–old is the velocity of polymer bead j

before collision. At low temperatures or large collision frequencies,fluid particles could participate in successive collision proce-dures in one cell, which violates the assumption of molecularchaos and Galilean invariance. However, this anomaly is curedby a random shift of the collision lattice at every collisionprocedure.75,76

For NP–fluid particle coupling, we may be interested in howits diffusion behavior depends on its size while keeping com-putational efficiency. To this end, after each MD step DtMD, analternative bounce-back rule77 is adopted here by stochasticreflections where upon collision of the fluid particles and theNP surface, the fluid particles are given a random normalvelocity vN and tangential velocity vT in the NP-surface referenceframe taken from the following distributions

P vNð Þ ¼ mvN=kBTð Þ exp �mvN2�2kBT

; vN 4 0

P vTð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim=2pkBT

pexp �mvT

2�2kBT

(9)

Fluid particles interacting with the NP would typically havemultiple collisions, and stochastic boundary conditions cantherefore be viewed as a coarse-grained representation of theseprocesses.

2.4 The switch of HI

Hydrodynamic correlations arise due to local momentum con-servation, which is strictly guaranteed by the MPCD collisionstep. Therefore, hydrodynamic correlations in the system canbe switched off by randomizing and decorrelating the fluidparticle velocities,59,78 which causes the violation of momentumconservation. This idea was successfully applied to polymersin solution where the polymer beads were treated as pointparticles.62,65

The randomizing of fluid particle velocities can be achievedby re-sampling the velocities from a Maxwell–Boltzmann dis-tribution. Note that this velocity re-sampling method mayremove mechanisms responsible for the relaxation of thearising density modulations of the fluid particles surroundingthe massive NP, which causes highly non-trivial dynamics ofthe NP such as an extra caging effect.79 In order to relax fluidparticle density modulations correctly, a uniform re-samplingof the fluid particle position beyond NP volume isimplemented79 after the velocity re-sampling. The additionalposition re-sampling successfully restores Enskog behavior.59,80

With these ingredients in hand, simulations are performedwith the same simulation parameters for both HI switched onand off in order to compare the effect of hydrodynamics on theNP diffusion.

2.5 Simulation parameters

The collision cell size s, solvent particle mass m and kBT are setto be the unit of length, mass and energy, respectively. The

derived time unit t is then t ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffims2=kBT

p. The interaction

strength in the L-J potentials is e = 4.0kBT. The polymer chainlength is fixed at Nb = 50. Parameters for the FENE potential arebond strength k = 30e/s2 and maximum bond length r0 = 1.5s.For the MPCD method, the mean fluid particle number per cellis hNci = 10 and the rotation angle is a = 1301. M = hNcim and thedensity of the NP are equal to that of the polymer beads. MDtime step DtMD = 0.002t and MPCD collision time step DtMPC =0.1t. The selected MPCD parameters yield the solvent shear

viscosity59 Z ¼ 8:7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimkBT=s4

pand the Schmidt number Sc C 17,

which means a fluid is simulated81 rather than a gas. Thegyration radius of the polymer in pure solvent Rg as well as thehydrodynamic radius of the polymer rh are obtained fromsimulation, yielding Rg = 4.7s and rh = 4.2s respectively. Thepolymer overlap concentration (bead number density) c* is then

derived by c� ¼ Nb

4

3pRg

3

¼ 0:1208s�3. Note that the equilibrium

properties such as gyration radius of a polymer are not affectedby HI,74 thus the Rg and c* hold the same for the situationwhere HI is switched off. We couple a cell-level thermostat82 tothe fluid to guarantee a constant local temperature during asimulation and the periodic boundary condition is adapted.

We perform simulations at varied NP radii Rn and polymerconcentrations c by changing the number of polymer chains.We keep the box size L = 31.0s. Note that one should take intoaccount the finite-size effect to study the effect of HI, which is along-range interaction. For pure solvent, it has already beenshown that the diffusion coefficient D of a spherical particle inan infinite system is corrected from that in a L-length box DL by

D ¼ DL þ2:837kBT

6pZL, where the correction term scales as 1/L. As

suggested in some literature reports,83,84 generally the systemsize should be larger than 6Rg to make the finite-size effectnegligible. In the present work, as mentioned above, we dealwith a short chain with 50 beads, of which the gyration radius isRg C 4.7 in dimensionless units, and the largest particle size isRn = Rg. Therefore, the box size we used, i.e., L = 31.0 C 7Rg, issufficient. We performed extensive simulations with differentbox sizes, and we confirmed that a box with L = 31 cansuccessfully eliminate that finite-size effects. In addition, as abasic check, we computed the diffusion coefficient of bothMPCD fluid particles DMPCD and that of NP DNP with Rn = 4.7s

in pure solvent. As a result, we obtain DMPCD ’ 0:051sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikBT=m

p,

consistent with the theoretical estimation.73 Also, we have

DNP ’ 0:0012sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikBT=m

p, which apparently coincides with the

Stokes–Einstein prediction.All results are averaged over 30 independent runs for each

parameter set. The simulation programs are coded for NVIDIA-CUDA based GPU,85 and the independent runs for each para-meter set are performed simultaneously on the same GPU as anoptimization of instruction reuse. The term ‘‘concentration’’ in

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this paper refers to the number density of polymer beads if nototherwise emphasized.

Before the simulation of NP diffusion, simulations of asingle polymer with varied chain length in pure solvent (dilutesolution) are performed to obtain the scaling law of gyrationradius Rg against chain length Nb (shown in Fig. 2). For goodsolvent conditions, the Flory theory prediction86 and the resultof the self-avoiding walk model indicate that

Rg B Nnb (10)

where n C 0.59. Our simulation gives Rg B N0.6b , which agrees

well with the theoretical value for a good solvent. Therefore, oursimulation method can reproduce a good solvent condition forthe polymer chains.

Besides, it is also interesting for us to investigate the HIeffect on NP diffusion in pure solvent before going to polymersolutions. Fig. 3 depicts the velocity auto-correlation functions(VACF) as well as MSD (inset figure) with HI on and off, whereMSD(t) = h|r(t) � r(0)|2i with r(t) denoting the position vector ofNP at time t. Evidently, the long-time diffusion is faster forHI-on than HI-off implying that HI would accelerate the long-time diffusion of the NP even in pure solvent. In addition, theVACF shows long-time tails for HI-on, indicating a memoryeffect resulting from the HI, while it decays exponentially forHI-off. Note that Belushkin et al.79 adjusted the MPCD para-meter to recover the same long-time diffusion of a NP with HIon and off for the qualitative comparison of VACF behavior inboth situations while keeping the influence of the NP’s mobilitymagnitude excluded. In the present work, we aim to study the

NP diffusion in polymer solutions, and the effect of HI will bemanifested as a whole via its influence on both diffusioncoefficients and subdiffusive behavior. Therefore we do notparticularly isolate the role of NP–polymer interactionsmediated by the solvent.

3 Results and discussion3.1 Subdiffusion

To begin, we first study the subdiffusive behavior of the NP bycalculating the ensemble-averaged MSD. The MSDs of a NPwith Rn = 0.5Rg as a function of time for different values ofpolymer concentration are shown in Fig. 4(a) for illustration.For all the curves, MSD(t) scales as t2 at very short time due tothe ballistic motion with inertia, while it scales as t in the long-time limit corresponding to normal diffusion. If the polymerconcentration is low (c o c*), one can see that MSD B t over alltime scales after the ballistic motion. For higher polymerconcentrations, the NP takes a subdiffusive behavior in theintermediate time range, where MSD(t) B ta with a o 1. Asalready discussed in many literature reports, this subdiffusivebehavior is a consequence of the cage effect that the surround-ing polymer chains exert on the NP before the polymer influ-ence has sufficient time to relax.87 Obviously, along with theincrease in concentration, the onset time for subdiffusiondecreases. Accordingly, the length scale over which the sub-diffusive behavior is present also decreases, indicating a strongercage effect. Besides, Fig. 4(b) is plotted for the MSD under typicalnanoparticle sizes, i.e., Rn = 0.33, 0.5, 0.8, and 1.0Rg at a certainconcentration c = 4c*. It shows that NP size also influences thesubdiffusive behavior as expected. For smaller NP size, the lengthscale for the onset of subdiffusive behaviour tends to be larger.This means that a smaller NP will experience a broader cage andis subject to a weaker cage effect.

In order to reveal the effect of HI, we have performed thesimulation under the same conditions but with HI switched off.For instance, the MSDs with HI on and off are shown in Fig. 4(c)for Rn = 0.5Rg and c = 4c*. Clearly, HI has a strong effect on theNP’s MSD, except for very short time scales where the NP takesballistic motion. With HI off, the subdiffusive behaviorbecomes stronger with an apparently smaller a. In the long-time limit, the NP also shows normal diffusion behavior withHI off, but with a much smaller diffusion coefficient comparedto the value with HI on.

Clearly, the exponent a plays a key role in characterizing thesubdiffusive behavior. In Fig. 5(a), a as a function of polymerconcentration for Rn = 0.5Rg is shown for both HI on and off.For all the concentrations, aon is always larger than aoff.In general, the long-range HI provides fluid-mediated coupleddynamics between polymer beads, and that between polymerbeads and the NP, as depicted in Fig. 6 (left). For a betterunderstanding, we also calculated the MSD of the polymercenter-of-mass (CM) to compare with that of the NP. InFig. 5(a), the exponent a for the polymer CM under severaltypical different polymer concentrations with HI-on is shown

Fig. 2 Log–log plot of gyration radius of the polymer chain as a functionof chain length Nb. The red solid lines are fitted via eqn (10).

Fig. 3 VACF of NPs with different sizes in pure solvent. The inset showsthe corresponding MSD for Rn = 0.5Rg with HI on and off.

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(black dots). Interestingly, the a value for polymer CM withHI-on is nearly identical to that of the NP, indicating that HIleads to nearly synchronized subdiffusive behavior of the NPwith the whole polymer chain. This is further demonstrated inFig. 5(c), as an example for c = 3c*, where the MSD of polymerCM is depicted, compared with that of a NP with Rn = Rg. For HIon, we can see that the MSDs for both the polymer CM and NPenter the subdiffusion regime at nearly the same time scale andthe values of a are nearly equal. Such a synchronization of theNP subdiffusive behavior with the polymer CM indicates thatthe NP is coupled to the whole polymer chain due to the long-range HI. For HI-off, the situation becomes quite different.As shown in Fig. 5(c) (the inset), with an adequate polymerconcentration (3c*), the subdiffusive behavior of the NPdeviates remarkably from that of the polymer CM which

exhibits a larger subdiffusion exponent, while it approaches thatof the polymer beads. This indicates that the coupling betweenthe NP and polymer is dominated by the direct interactionswhich are local and short-ranged, and thus the NP tends to becoupled with a small section of beads instead of the whole chain.

Another feature shown in Fig. 5(a) is that the discrepancybetween aon and aoff is most pronounced for an intermediateconcentration. For low or high polymer concentration, one cansee that the HI effects are almost suppressed in terms of thedifferences in the a values. For very low concentration, the cageeffect is negligible and the subdiffusive behavior is not sig-nificant such that a is close to 1 whether HI is on or off. For veryhigh concentration, on the other hand, it is known that the HIeffect is screened88 by the very crowded polymer moleculessuch that aon also nearly equals aoff.

Fig. 4 (a) Time-dependent MSD of the NP with Rn = 0.5Rg and variednormalized polymer concentration c/c*. (b) Time-dependent MSD of theNP with polymer concentration c = 4c* and NP size Rn = 0.33Rg, 0.5Rg,0.8Rg, and 1.0Rg respectively. Solid lines denote the subdiffusive scalings.(c) A comparison of the MSD with HI switched on and off for polymerconcentration c = 4c*. Dashed lines are linear fits at long-time scales, andsolid lines denote the subdiffusive scalings.

Fig. 5 (a) Subdiffusion factor a of the NP and polymer CM as a function ofnormalized polymer concentration c/c* with Rn = 0.5Rg. (b) Subdiffusionfactor a as a function of normalized NP size Rn/Rg with polymer concen-tration c/c* = 1 and 6 respectively when HI is switched on (full shapes) andoff (open shapes). (c) MSDs of the NP (Rn = Rg) and polymer CM with HI onat c = 3c*. Inset: MSDs of the NP, polymer CM and polymer bead with HIoff. Dashed lines denote the subdiffusive scalings.

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For further information, we have also investigated thedependence of a on the NP size Rn, as shown in Fig. 5(b) fora typical low concentration c = c* and another high concen-tration c = 6c*. For the low concentration, one can see that aremains nearly constant when HI is on, while it decreasesmonotonically with Rn if HI is turned off. When HI is switchedoff, a larger volume is taken by the NP with increasing Rn andthe influence of local interactions between the NP and beads onthe NP is more evident leading to a decreasing a. When HI isswitched on, the NP’s subdiffusion dynamics is only deter-mined by that of the polymer chains, which results in asuppressed dependency of a on NP size. Nevertheless, at highconcentration (6c*), the NP is totally trapped even at a smallsize and the long-range HI is screened by local direct inter-actions between the NP and polymer beads as depicted in Fig. 6(right). In this case, we see that both aon and aoff show very weakdependencies on the particle size, indicating that the cageeffect at high concentrations is mainly determined by thecharacteristic length scales of the polymer solution rather thanthe particle size.

3.2 Scaling law of the diffusion coefficient

Now we turn to the long-time diffusion coefficient D, payingparticular attention to its scaling relations with the particle size.Note that the well-known Phillies equation D/D0 B exp(�bfd)provides an inspiring scaling relation for the diffusion coeffi-cient. As mentioned above, in our previous work,70 the long-timediffusion coefficient can be well fitted by this relation where thedependence of D on the concentration c is accounted for. Here inthe present work, we mainly focus on the dependence of D onboth particle size and concentration. Therefore, alternatively, wechoose the Holyst relation where the relevant length scaleinvolves the particle size and the polymer concentration. In fact,the Holyst model has been demonstrated to be successful inexplaining many experimental observations regarding NP diffu-sion in complex solutions. Notice that besides the Holystmodel, there is another interesting theory,89 which performeda very detailed scaling analysis for the mobility of nonsticky

nanoparticles in polymer liquids, including solutions andmelts. Scaling relations for different size regimes were proposedbased on an equivalent freely jointed chain model of the polymer.However, such a model seems not applicable to our present studyof a short chain with only 50 beads, for which the chain-end-effect is not negligible. Based on the above considerations, wewill investigate the HI effect on the long-time diffusion coefficientwithin the theoretical framework of the Holyst model. Accord-ingly, we interpret our numerical results based on the followingscaling law:42,46

D ¼ D0 exp �g

RT

Reff

x

� �a� �(11)

where x is the correlation length of the polymer solution. g(g4 0)and a(a 4 0) are system-dependent parameters (but not depen-

dent on concentration) in terms of DEa ¼ gReff

x

� �a

which corre-

spond to an effective excess diffusion activation energy45 overthat observed in a pure solvent. In the last section, we have shownthat the gyration radius Rg scales with the number of beads Nb asin a good solvent, for which

x ¼ Rgc

c�

� ��0:75(12)

holds according to Flory.86 Eqn (11) involves an important lengthscale, the effective radius Reff which is defined as

Reff�2 = rh

�2 + Rn�2 (13)

If the particle size Rn is much larger than rh, Reff would beapproximately rh, such that the ratio D/D0 will not depend on itssize as the conventional SE relation would tell. On the otherhand, if Rn { rh corresponding to a small probe, Reff C Rn,such that D will show strong dependence on the particle size Rn

as shown by eqn (11). It has been shown that such a scalingrelation can fit rather well many experimental observations oftranslational diffusion coefficients.27,42,43,45,46

In our present work, we have performed extensive simula-tions to obtain D for various particle sizes as well as polymerconcentrations. Interestingly, we find that the data can be wellfitted by the scaling relation given by eqn (11). This is demon-

strated in Fig. 7(a), where ln(D/D0) is plotted againstReff

x

� �a

with two fitting parameters g and a when setting temperature Tin eqn (11) as T = 298 K. For each particle size, all the results fellvery well along a straight line on the scaling plot, stronglysupporting the validity of the scaling relation. Furthermore,a asymptotically approaches a C 1.2 (inset of Fig. 8(b)) which isquite close to the macroscopic experimental value a C 1.29.46

For different particle sizes, however, the parameter g is differentas shown in Fig. 7(b). Clearly, g decreases with the particle sizeand finally reaches a plateau when the particle is large enough.Remarkably, for small NP size (Rn = 0.33Rg) which tends toexperience a microscopic viscosity contributed by a short sectionof a long polymer chain, our results of g obtained from simula-tions agree quite well with the experimental ones for the PEGsolution.45 For large NP sizes, the effective viscosity experienced

Fig. 6 A snapshot of the NP (golden sphere, Rn = 0.8Rg) diffusing in apolymer (green beads) solution of concentration c = c* (left) and c = 6c*(right) respectively. At low concentrations, the volume between the NP andpolymer chains is filled with a considerable quantity of fluid particles. Athigh concentrations (right), the space for the mediation of HI is fullyoccupied by the NP and surrounding polymer beads, which results in adominance of direct local collision between the NP and polymer beadswhether HI is on or off.

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by the NP approaches the macroscopic value. Note that thismacroscopic viscosity is contributed by complete polymer chainsand more possibly, entanglement, which has been widelyobserved in experiments but is not the case in our work, thusa smaller g in comparison to the experiment is obtained here.

Now it is instructive to ask how HI would influence thisscaling relation. Quite interestingly, we find that the scalingrelation (eqn (11)) still holds rather well as shown in Fig. 8(a),but with different values of g and a. As NP size increases,a approaches a C 0.75 (inset of Fig. 8(b)), but its value issmaller than 1.2 for HI-on. The values of g for HI-off as afunction of particle size are plotted in Fig. 8(b). Similarly to thecase of HI-on, g also decreases with Rn and reaches a constantvalue in the large size limit. Nevertheless, the values of g forHI-off are all about 1.0 kJ mol�1 smaller than those for HI-on.

The system parameters g and a provide a relatively moremicroscopic view to understand how HI enhances the diffusionof NPs in polymer solutions. Note that goff is larger than gon,while aoff is smaller than aon. With HI included, the diffusionactivation energy is lowered by approximately 1.0 kJ mol�1 per

effective relative sizeReff

x

� �a

. Since g physically corresponds to

an effective activation energy related to the hopping motion ofthe particle,45,90,91 the inclusion of HI seems to favor this

hopping process. On the other hand, the termReff

x

� �a

refers

to an effective relative diffusion size of the particle in thesolution. The increment of a with HI-on implies a larger

effective particle, which may arise from the dragging of theneighboring fluid particles by the NP. Overall, the decrease inthe activation energy g would dominate the increase of relative

sizeReff

x

� �a

, thus leading to an increase of D with HI-on.

Finally, the dependence of D on concentration c with HI-onand off for different particle sizes is shown in Fig. 9(a).Obviously, diffusion with HI on is faster than that with HI offover all ranges of polymer concentrations. Note that Don

amounts to several times as large as Doff, indicating that HI isan important factor that should be considered for NP diffusionin polymer solutions. Both Don and Doff results show that theNP mobility decreases with increasing polymer concentration.Nevertheless, as polymer concentration increases, Doff

decreases more slowly than Don and gradually approximatesDon, indicating that HI takes no effect at high concentration asdiscussed before.

In Fig. 9(b), the acceleration effect of HI on the diffusivity ofNP, Don/Doff, as a function of polymer concentration is shownwith varied NP size. Two features can be observed from thisfigure. Firstly, the acceleration effect is more significant forlarger particles. This may be understood based on the investi-gation of the friction z exerted on the NP related to the MPCDmethod. As discussed in a recent paper, there are two sources offriction92,93 for the NP in a pure MPCD solvent, namely,

1

z¼ 1

zSþ 1

zE(14)

Fig. 7 (a) Relative diffusivity of the NP fitted (solid lines) according to theproposed scaling eqn (11) with HI switched on, NP size Rn = 0.33Rg, 0.5Rg,0.8Rg, and 1.0Rg. (b) Dependence of g [eqn (11)] on the size ratio Rn/rh inthe situation when HI is on. The two dashed lines denote the range of gvalues obtained from the macroscale experiments.

Fig. 8 (a) Relative diffusivity of the NP fitted (solid lines) according to theproposed scaling eqn (11) with HI switched off, NP size Rn = 0.33Rg, 0.5Rg,0.8Rg, and 1.0Rg. (b) Dependence of g [eqn (11)] on the size ratio Rn/rh inthe situation with HI on and off. The inset shows the a value of thecorresponding size ratio Rn/rh.

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where zE denotes the Enskog term resulting from direct localcollisions with the fluid particles94 and zS denotes the hydro-dynamic term obtained by integrating the Stokes solution to thehydrodynamic field over the surface of the NP.59 It is known thatzE B Rn

2 and zS B Rn. If HI is switched on, the contributions of

zS and zE add in parallel such that Don ¼ kBT1

zSþ 1

zE

� �. While if

HI is switched off, the friction would be entirely dominated by

the Enskog contribution zE, and Doff ¼kBT

zE. Therefore,

Don

Doff¼

1þ zEzS

withzEzS� Rn, which will increase with the particle size Rn.

In polymer solutions, the friction exerted on the particle mayhave another source, namely, that from interaction with thepolymer beads. How this third source depends on the particlesize is not easy to obtain. Nevertheless, we expect that the aboveanalysis would also apply to a polymer solution.

More interestingly, Fig. 9(b) indicates that the accelerationeffect of HI on long-time diffusion Don/Doff is non-monotonouswith increasing polymer concentration, which reaches its maxi-mum in semidilute regimes for all NP sizes. This is quitesimilar to the effect of HI on the subdiffusion exponent a aspresented in Fig. 5(a). As already discussed above, both Don andDoff satisfy the scaling relation. We note that in the semidiluteregime, the concentration dependence of Don/Doff is introducedthrough the correlation length x of the polymer solution as afunction of concentration c. Therefore, one has

Don

Doff¼ exp � gon

kBT

Reff

x

� �aon

þ goffkBT

Reff

x

� �aoff� �

(15)

wherein only x depends on c via x B Rg(c/c*)�3/4(c 4 c*). From

the analysis of dDon

Doff

� ��dc and d2

Don

Doff

� ��dc2 according to

eqn (15), one can show that the non-monotonic tendency is relatedto the relative magnitude of aon and aoff. When aon = aoff, Don/Doff ismonotonic and the tendency is determined by the relative magni-tude of gon and goff, Don/Doff increases with c when goff 4 gon and viceversa. When aon a aoff, Don/Doff has its extremum at c = cex where cex

satisfies lncex

c�

� �¼ 4

1

aon � aoffln

aoffgoffaongon

� �� ln

Reff

Rg

� �� ��3.

When aon o aoff, cex serves as the minimum position while whenaon 4 aoff, cex denotes the maximum position of Don/Doff, which isessentially our case.

As demonstrated above, we apply the Holyst relation (11) tounderstand the long-time diffusion behavior of the NP inpolymer solutions. The effect of HI has been manifested basedon the activated diffusion picture. We would like to point outthat the two parameters g and a should be dependent on thedetailed interactions at the microscopic level, and thus shouldbe correlated. If we change interaction parameters such asnanoparticle–polymer bead interaction strength e and polymerchain potential strength k, we can systematically study thedependence of g and a on microscopic interactions as well asthe correlation of these two parameters. As a primary work, forsimplicity, we fix interaction parameters based on generalvalues in order to focus on the effect of polymer concentrationand nanoparticle size. Further work on the dependence of g anda on microscopic interaction models as well as their correlationis worthy of study in the future.

4 Conclusions

We have presented a study of the diffusion dynamics of NPs ina polymer solution with a hybrid mesoscopic MD–MPCD simu-lation method. The solvent hydrodynamics is modeled byMPCD. The couplings of the NP and polymer beads with thesolvent are efficiently taken into account in coarse-grainedcollisions. We investigated the dependence of the NP’s long-time diffusion coefficient D and subdiffusive behavior on NPsize Rn and polymer concentration c. Our simulation shows anexcellent agreement with the Holyst’s scaling behavior found byexperiments, which serves as a validation of the MD–MPCDmethod. Also, the NP’s subdiffusion factor is found to decreasewith polymer concentration but has little relevance to NP size.

Particular attention is paid to the role of HI by parallelsimulations with and without HI. For the NP’s subdiffusivebehavior, particularly at low polymer concentrations, HI givesan extra long-range correlation between the NP and polymerbeads mediated by fluid particles. The weak dependence of thesubdiffusion factor on NP size with HI on reveals that this long-range correlation gives a strong coupling of the NP and polymerchains’ diffusion dynamics. For the NP’s long-time diffusiondynamics, when HI is switched off, fitting of the Holyst’sscaling law gives a larger diffusion activation energy and lowereffective diffusion size. The fitting results indicate that the

Fig. 9 (a) Long-time diffusion coefficient D of the NP (Rn = 0.5Rg, 0.8Rg,1.0Rg) as a function of polymer concentration for HI switched on (fullshapes) and off (open shapes) respectively. (b) Long-time diffusion coeffi-cient ratio with NP size Rn = 0.5Rg, 0.8Rg, and 1.0Rg.

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influence of HI on NP diffusion is rather complicated. On theone hand, HI reduces the diffusive activation energy g, resultingin a larger diffusion rate. On the other hand, HI enhances theeffective diffusion size, which leads to the retardation of NPdiffusion. However, the decrease of g plays the most importantrole thus the overall effect of HI is the enhancement of the NPdiffusion coefficient. For NPs of larger size, the accelerationeffect of HI is more significant. While just due to the competi-tion between the two above mentioned aspects, we find a veryinteresting phenomenon, that the acceleration effect of HI isnon-monotonous with increasing polymer concentration, andis most pronounced at the semidilute concentration. Ourresults suggest that careful attention should be paid to the HIeffect in the study of diffusion dynamics of NPs in semidilutepolymer solutions.

The MD–MPCD simulation used in the present paper helpsus to investigate the effect of HI explicitly. Although theNP–polymer solution system considered here is rather simple,our study brings deep insights into the significant role of HI inthe diffusion relevant dynamics. We hope that this work willprovide a useful starting point to study diffusion as well as otherimportant processes, such as protein folding, self-assembly anddrug transportation in complex solutions or crowding environ-ments, with particular attention paid to the role of HI.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Founda-tion of China (Grant No. 21373141, 21673145 and 21673212).

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