nanoprobe diffusion in entangled polymer solutions: linear vs ......nanoprobe diffusion in entangled...

Nanoprobe diffusion in entangled polymer solutions: Linear vs. unconcatenated ringchainsNegar Nahali, and Angelo Rosa

Citation: The Journal of Chemical Physics 148, 194902 (2018); doi: 10.1063/1.5022446View online: https://doi.org/10.1063/1.5022446View Table of Contents: http://aip.scitation.org/toc/jcp/148/19Published by the American Institute of Physics

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THE JOURNAL OF CHEMICAL PHYSICS 148, 194902 (2018)

Nanoprobe diffusion in entangled polymer solutions:Linear vs. unconcatenated ring chains

Negar Nahalia) and Angelo Rosab)Sissa (Scuola Internazionale Superiore di Studi Avanzati), Via Bonomea 265, 34136 Trieste, Italy

(Received 15 January 2018; accepted 1 May 2018; published online 17 May 2018)

We employ large-scale molecular dynamics computer simulations to study the problem of nanoprobediffusion in entangled solutions of linear polymers and unknotted and unconcatenated circular (ring)polymers. By tuning both the diameter of the nanoprobe and the density of the solution, we showthat nanoprobes of diameter smaller than the entanglement distance (tube diameter) of the solutiondisplay the same (Rouse-like) behavior in solutions of both polymer architectures. Instead, nanoprobeswith larger diameters appear to diffuse markedly faster in solutions of rings than in solutions oflinear chains. Finally, by analysing the distribution functions of spatial displacements, we find thatnanoprobe motion in rings’ solutions shows both Gaussian and ergodic behaviors, in all regimesconsidered, while, in solutions of linear chains, nanoprobes exceeding the size of the tube diametershow a transition to non-Gaussian and non-ergodic motion. Our results emphasize the role of chainarchitecture in the motion of nanoprobes dispersed in polymer solutions. Published by AIP Publishing.https://doi.org/10.1063/1.5022446

I. INTRODUCTION

The micro-mechanical and viscoelastic properties of com-plex polymer fluids and, more in general, of soft and biologicalmaterials can be efficiently explored owing to the advent ofmicrorheology,1–8 a versatile technique based on the directtracking of the diffusive motion of nanoprobes injected intothe medium. Through the analysis of the motion of the probes,one can measure the mechanical response of the materialand get quantitative information about its elastic and viscousproperties typically on larger time and length scales and athigher resolution than by traditional rheology. At the sametime, given amounts of nanoprobes embedded in polymermatrices may be employed to alter significantly their mechan-ical properties and thus contribute to the design of novelmaterials.9,10

The study of nanoprobe motion in semi-dilute and con-centrated solutions (melts) of polymer chains constitutes animportant and yet largely unexplored chapter in the long his-tory of microrheology.9,11–18 These systems appear in factparticularly challenging due to the complex interplay betweendifferent length scales such as nanoprobe size, mesh size ofthe polymer solution, and chain contour length.

More recently, it has been suggested15,17,18 that anotherfactor, namely, chain architecture, may play a quite sub-tle role when it comes to describe the detailed motion ofnanoprobes. In particular, by considering the two examples ofentangled solutions of linear chains vs. unknotted and uncon-catenated circular (ring) chains, Nahali and Rosa15 and Geet al.17 have shown that nanoprobes of linear sizes largerthan the mesh size of the solution move definitely faster

a)Electronic mail: [email protected])Electronic mail: [email protected]

in solutions of rings than in solutions of linear chains. Inthe same paper, Ge et al. described a theory where theobserved discrepancy is understood in terms of the differentspatial and temporal behaviors of linear chains vs. rings insolution.

Under the same environmental conditions, linear chainsand rings in solution do behave in fact quite differently: inparticular, because the unconcatenation constraint rings foldinto self-similar, compact conformations with non-Gaussianstatistics and because the lack of free ends do not performreptation,19–46 both features being instead well establishedtrademarks of linear chains in solution.47–49 Moreover, thepresence of threadings between close-by ring polymers in highdensity solutions has been held responsible for chain dynam-ics becoming glassy and heterogenous,44,46 a feature, onceagain, completely absent in linear chains. All these featuresare likely to influence the motion of nanoprobes embedded inthe solution.

Motivated by these considerations, in this paper, we(re)consider the problem of nanoprobe motion in entangledsolutions of linear vs. ring polymers. By employing largescale Molecular Dynamics (MD) computer simulations, weprovide a quantitative, in-depth description of nanoprobedynamics in terms of chain architecture and topological con-straints between close-by chains. Mainly, the present workextends the numerical analysis pioneered by Refs. 15 and17 along two directions: (1) we consider wider ranges forprobe sizes and solution densities and (2) we provide a sys-tematic characterization of nanoprobe dynamics in terms ofthe (van-Hove) probability distribution function of spatial dis-placements and other observables borrowed from the theory ofglassy systems50 which allow us to define the conditions underwhich nanoprobe dynamics shifts from Gaussian/ergodic tonon-Gaussian/non-ergodic behavior.

0021-9606/2018/148(19)/194902/11/$30.00 148, 194902-1 Published by AIP Publishing.

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194902-2 N. Nahali and A. Rosa J. Chem. Phys. 148, 194902 (2018)

The paper is structured as follows: In Sec. II, we give aconcise account of the state-of-the-art concerning single-chainconformations and nanoprobe diffusion in entangled solutionsof linear chains and rings. In Sec. III, we explain the com-putational model and the numerical methods employed in thearticle. In Sec. IV, we describe in detail our results and inter-pret some of them in the light of the theoretical framework ofSec. II. Finally, in Sec. V, we summarize the main conclusionswith an outlook for future work.

II. THEORY

In this section, we describe (Sec. II A) the relevant physicsof polymer conformations in entangled solutions and we high-light, in particular, the main differences between linear chainsand rings. This is followed (Secs. II B and II C) by a conciseaccount of the scaling theory by Ge et al.12,14,17 linking chainstatistics to nanoprobe diffusion.

A. Polymer conformations in entangled solutions:Linear vs. ring chains

The physics of polymer conformations in entangled solu-tions can be described in terms of a set of few relevant lengthscales:47–49 (1) The microscopic correlation length, ξ, definedas the average spatial distance from a monomer on one chainto the nearest monomer on another chain. (2) The Kuhnlength of the polymer fiber, `K , characterizing the crossoverfrom rigid rod to random coil behavior. (3) The entangle-ment length, Le, corresponding to the chain contour lengthspanning the tube-like region where topological constraintsfrom surrounding polymers confine each given chain. Thelength scale dT ≈

√Le`K corresponds to the tube diameter

(see Sec. III D), which is a measure for the mesh size of thesolution.

On chain contour lengths L . Le, topological constraintsplay no role and linear and ring chains display the same behav-ior, the mean-square end-to-end distances characterized by thefamiliar crossover from rigid rod to random coil behavior at`K

〈R2(L)〉lin, ring ≈

L2, L . `K

`K L, `K . L . Le. (1)

On larger contour lengths L & Le, linear chains remain idealbecause of the screening of excluded volume effects,47–49

while the additional unlinking constraint forces rings intospace-filling compact conformations.26,27,31,32 Summarizing

〈R2(L)〉lin ≈ `K Le(

LLe

)1, (2)

〈R2(L)〉ring ≈ `K Le(

LLe

)2/3. (3)

B. Nanoprobe motion: Solutions of linear polymers

The erratic motion of a single nanoprobe can be cap-tured51 by its mean-square displacement (MSD), ∆r2(T; τ),as a function of the lag-time τ and the measurement time T

∆r2(T; τ) ≡ 1T − τ

∫ T−τ0

(~r(t + τ) −~r(t))2dt, (4)

where ~r(t) is the nanoprobe spatial position at time t. Thetime-average displacement can be defined as

∆r2(τ) ≡ limT→∞∆r2(T; τ), (5)

while its ensemble average is defined as

〈∆r2(T; τ)〉 ≡ 1Nnp

Nnp∑i=1

∆r2i (T; τ) (6)

with the sum indicating that the average is performed overthe whole set of Nnp independent nanoprobes used to explorethe system. Accordingly, we indicate the time- and ensemble-average displacement as 〈∆r2(τ)〉.

According to nanoprobe diameter d, three regimes can bedistinguished:

(I) Small nanoprobes, d . ξ. In this case, nanoprobemotion is barely influenced by the surrounding polymers andits MSD is described by

〈∆r2(τ)〉lin ≈ Ds τ ≈κBTηs d

τ, (7)

where Ds is the diffusion coefficient, ηs is the viscosityof the solvent, κB is the Boltzmann constant, and T is thetemperature.

(II) Intermediate nanoprobes, ξ . d . dT . Now, nano-probe motion is affected by the polymers and its MSD displaysthree regimes

〈∆r2(τ)〉lin ≈

Ds τ, τ < τξ ≈ ηsξ3

κBT, (a)

Dsτξ(ττξ

)1/2, τξ < τ < τd ≈ τξ

(dξ

)4, (b)

Ds(ξd

)2τ, τ > τd . (c)

(8)

At short times [Eq. (8a)], nanoprobe motion is driven onlyby random collisions with the solvent, as in I. This regimestops at τξ , the relaxation time of a polymer strand of spatialsize ξ. Then [Eq. (8b)], the nanoprobe experiences a time-

dependent viscosity η(τ) ≈ ηs nstr(τ) ≡ ηs(ττξ

)1/2, where

nstr(τ) is the number of strands which have relaxed at timeτ. This regime stops at time τd , the relaxation time of a largerpolymer strand of spatial size = d = ξ

√nstr(τd). Above τd

[Eq. (8c)], nanoprobe motion becomes diffusive again witheffective viscosity ≈ηs·nstr(τd), which is ≈ (d/ξ)2 times largerthan the value in pure solvent.

(III) Large nanoprobes, d & dT . The regime describedby Eq. (8a) still holds, while the regime of Eq. (8b) stops

at τd=dT = τξ(

dTξ

)4. Above τdT , nanoprobes are trapped by

entanglements and the MSD becomes plateau-like: 〈∆r2(τ)〉lin= 〈∆r2(τdT )〉lin ≈ Ds

(ξdT

)2τdT ≈

ξd d

2T . Interestingly, this last

expression depends on all three relevant time scales of the prob-lem. Neglecting hopping14 between close-by entanglements,the trapping regime persists up to complete chain reptation48

at τ ≈ τrep ≈ τe(

LcLe

)3, where τe (the entanglement time,

see Sec. III D) corresponds to the relaxation time scale ofpolymer strands of contour length = Le. With bulk viscosity


ηbulk ≈ κBTd2T ξτrep, at larger times nanoprobe motion becomes

diffusive again

〈∆r2(τ)〉lin ≈κBTηbulk d

τ ≈ ξd

d2T

(LcLe

)−3τ

τe. (9)

C. Nanoprobe motion: Solutions of ring polymers

On length scales smaller than the tube diameterdT ≈ (`K Le)1/2, ring and linear polymers behave similarly [seeEq. (1)], so dynamics of nanoprobes of small and intermediatesizes d . dT is described by the same expressions derived fornanoprobes immersed in linear chains, Eqs. (7) and (8).

Conversely, for nanoprobes with diameters &dT , dynam-ics at time scales & τdT reflects the different spatial organiza-tion of rings compared to linear polymers (Sec. II A). In thiscase, the time-dependent friction of the solution49 η = η(τ)≈ τG(τ). G(τ) ≈ κBT

b2Le

(ττe

)−αis the power-law stress relax-

ation modulus expected for rings in entangled solutions and bis the monomer linear size,25 and it corresponds to the simul-taneous rearrangement of polymer strands of contour lengthL(τ) ≈ Le

(ττe

)α. The time MSD of the nanoprobe is thus given

by

〈∆r2(τ)〉ring ≈κBTη(τ)d

τ ≈ b2Led

(τ

τe

)α. (10)

The regime breaks down at τ′d ≈ τe(

ddT

)3/α, namely, at the

relaxation time of a polymer strand of spatial extension ≈din the compact regime [Eq. (3)]. Above τ′d , the nanoprobe isdiffusive with MSD

〈∆r2(τ)〉ring ≈κBTη(τ′d)d

τ ≈ b2Led

(ddT

)3(1−1/α)τ

τe. (11)

With α ≈ 0.4,25 Eqs. (9) and (11) for diffusional motions insolutions of linear vs. ring polymers suggest that, already atmoderate ratios Lc/Le > 1, nanoprobes diffuse much faster inthe latter case than in the former.

III. MODEL AND METHODSA. The model

Polymer model—To model solutions of dilute nanoprobesand linear and ring polymers at various monomer densities, weemploy the same numerical framework considered in our pre-vious studies15,46 consisting of a variant of the known Kremerand Grest52 polymer model.

Excluded volume interactions between beads (includingconsecutive ones along the contour of the chains) are modelledby the shifted and truncated Lennard-Jones (LJ) potential

ULJ(r) =

4�[(σr

)12 − (σr )6 + 14 ] r ≤ rc0 r > rc

, (12)

where r denotes the separation between the bead centers. Thecutoff distance rc = 21/6σ is chosen so that only the repulsivepart of the Lennard-Jones is used. The energy scale is set by� = κBT and the length scale is set by σ, both of which are setto unity in our simulations. Consistent with that, in this workall quantities are reported in reduced LJ units.

Nearest-neighbour monomers along the contour of thechains are connected by the finitely extensible nonlinear elastic(FENE) potential, given by

UFENE(r) =

−0.5kR20 ln(1 − (r/R0)2

)r ≤ R0

∞ r > R0, (13)

where k = 30� /σ2 is the spring constant and R0 = 1.5σ is themaximum extension of the elastic FENE bond.

In order to maximize mutual chain interpenetration atrelatively moderate chain length23 and hence reduce the com-putational effort, we have introduced an additional bendingenergy penalty between consecutive triplets of neighbouringbeads along the chains in order to control polymer stiffness

Ubend(θ) = kθ(1 − cos θ

). (14)

Here, θ is the angle formed between adjacent bonds andkθ = 5κBT is the bending constant. With this choice, thepolymer is equivalent to a worm-like chain with Kuhn length`K = 10σ.53

Nanoprobe model—Nanoprobe-monomer and nanoprobe-nanoprobe interactions are described by the model potentialsintroduced by Everaers and Ejtehadi.54

The total interaction energy between probes at center-to-center distance r can be represented as the sum of two functions

Ucc(r) =

UAcc(r) + URcc(r) r ≤ rcc

0 r > rcc. (15)

UAcc(r) is the attractive component and it is given by

UAcc(r) = −Acc6

[2a2

r2 − 4a2+

2a2

r2+ ln

(r2 − 4a2

r2

)]. (16)

The repulsive component of the interaction, URpp(r), is

URcc(r) =Acc

37800σ6

r

[r2 − 14ar + 54a2

(r − 2a)7

+r2 + 14ar + 54a2

(r + 2a)7− 2 r

2 − 30a2r7

], (17)

where54 Acc = 39.478 kBT. We have considered non-sticky,athermal probe particles with diameters d/σ ≡ 2a/σ = 2.5,5.0, 7.5 which correspond to truncating the interaction Ucc(r)to rcc/σ = 3.08, 5.60, 8.08. The smallest probe is comparable toor larger than the smallest polymer correlation length ξ . 2.3σ(Table I), meaning that all probes are on or above length scaleswhere polymer effects become dominant (Sec. II).

Finally, the interaction potential, Umc, between a singlemonomer and a nanoprobe with center-to-center distance r isgiven by

Umc(r) =

2a3σ3Amc9(a2−r2)3

[1− (5a

6+45a4r2+63a2r4+15r6)σ6

15(a−r)6(a+r)6

]r ≤ rmc

0 r ≥ rmc,

(18)

where54 Amc = 75.358 kBT. According to our choices of probediameters, the interaction Umc(r) is truncated to rmc/σ = 2.11,3.36, 4.61.


TABLE I. Summary of the physical properties of polymer solutions and diffusing nanoprobes as functions of solution density expressed in monomer (ρ) orKuhn segment (ρK ) units: (a) Le, entanglement length. (b) dT , tube diameter. (c) τe, entanglement time. (d) d, nanoprobe diameter. (e) ξ , correlation length.(f) D, terminal diffusion coefficient of dispersed nanoprobes obtained by averaging the results of linear fits to single-nanoprobe displacements for lag-timesτ > 100τe. Error bars correspond to standard deviations.

Linear polymers Ring polymers

ρσ3 ρK`3K Le/σ [Eq. (19)] dT /σ [Eq. (20)] τe/τMD d/σ ξ /σ D [σ

2/τMD] ξ /σ D [σ2/τMD]

2.5 (4.73± 2.33)× 10�1 (4.91± 2.36)× 10�10.1 10 40.0 8.2 ≈1600 5.0 2.1 (1.28± 0.72)× 10�1 2.3 (1.23± 0.63)× 10�1

7.5 (3.18± 1.59)× 10�2 (3.37± 1.37)× 10�2

2.5 (1.51± 0.84)× 10�1 (1.60± 0.75)× 10�10.2 20 16.2 5.2 ≈570 5.0 1.6 (2.46± 1.28)× 10�2 1.7 (3.22± 1.60)× 10�2

7.5 (2.82± 1.32)× 10�3 (7.91± 3.83)× 10�3

2.5 (6.24± 3.13)× 10�2 (6.62± 3.28)× 10�20.3 30 11.0 4.3 ≈490 5.0 1.4 (6.24± 3.04)× 10�3 1.4 (1.25± 0.56)× 10�2

7.5 (7.67± 6.17)× 10�5 (2.66± 1.12)× 10�3

2.5 (2.57± 1.21)× 10�2 (2.93± 1.58)× 10�20.4 40 8.8 3.8 ≈540 5.0 1.3 (1.94± 1.03)× 10�3 1.3 (4.92± 2.49)× 10�3

7.5 (1.75± 1.55)× 10�6 (7.13± 3.64)× 10�4

B. Simulation details

As in our former work,15 we consider polymer solutionsconsisting of M = 80 circular or linear chains made of Nm = 500monomers each. The total number of monomers is then fixedto 40 000 monomer units. Each polymer solution includesalso Nnp = 100 nanoprobes of diameters d/σ = 2.5, 5.0, 7.5.These values were suitably chosen to cover the range frombelow to above the tube diameters of the polymer solutions(see Sec. III D). The volumes of the simulation box accessibleto chain monomers have been chosen in order to exploremonomer densities ρσ3 = 0.1, 0.2, 0.3, 0.4, corresponding,respectively, to Kuhn segment densities ρK`3K = 10, 20, 30, 40.

The static and kinetic properties of chains and nanoprobesare studied using fixed-volume and constant-temperatureMolecular Dynamics (MD) simulations with implicit sol-vent and periodic boundary conditions. MD simulations areperformed using the LAMMPS package.55 The equationsof motion are integrated using a velocity Verlet algorithm,in which all beads are weakly coupled to a Langevin heatbath with a local damping constant Γ = 0.5τ−1MD, whereτMD = σ(m/�)1/2 is the Lennard-Jones time and m = 1 is theconventional mass unit for monomer and colloidal probes. Theintegration time step is set to ∆t = 0.012τMD.

The length of each MD run is equal to 2.4 × 107τMD,during which (see the supplementary material of Ref. 15) eachpolymer moves on average of a spatial distance larger than itsown average size or gyration radius Rg.

C. Choice of initial configurations and checkfor equilibration

At any given ρ, we started from the equilibrated poly-mer solutions made of M × Nm = 80 × 500 monomers andNnp = 100 dilute nanoprobes of diameter d/σ = 5.0 describedin our previous publication.15 With no modifications, thoseconfigurations have been used as the starting configurations forMD runs with the same probe diameter. For simulations with

probe diameter of d/σ = 2.5, we have simply replaced the for-mer probes by the corresponding smaller ones and proceededto compress the simulation box slightly, in order to work atthe same monomer densities. Sticking into the same logic, forsimulations with larger probe diameter d/σ = 7.5, we have pro-ceeded to inflate simultaneously the probes and the simulationbox. We have accomplished this task by performing short (ofthe order of a few tens of τMD’s) MD runs with a soft (i.e.,non-diverging), capped repulsive interaction between chainmonomers and probes. At the end of these preparatory runs, wehave checked the perturbation of chains’ conformations afterthe deflation/inflation steps by measuring53 the mean-squareinternal distances, 〈R2(L)〉, between pairs of monomers at con-tour length separation L, see Fig. 1. The perfect agreementbetween different curves at each density demonstrates that theprobes have perturbed in no sensible manner the overall chains’conformations.

D. Estimating entanglement length (Le), tube diameter(dT ), and entanglement time (τe)

As briefly mentioned in Sec. II A, topological constraintsconfine the motion of any given linear chain to a tube-likeregion of diameter dT . The contour length of sub-chains span-ning a region of linear size = dT and the associated diffusiontime define, respectively, the entanglement length, Le, and theentanglement time, τe.

As in our previous studies,15,32,56 all these quantities canbe estimated by employing the results by Uchida et al.57 Theentanglement length, Le, is given by the following functionof the polymer Kuhn length, `K , and the density of Kuhnsegments, ρK :

Le`K=

1

(0.06(ρK`3K ))2/5

+1

(0.06(ρK`3K ))2

. (19)

Accordingly, the tube diameter is defined48,58 as the averagegyration radius of a linear chain of contour length Le


FIG. 1. Mean-square internal dis-tances, 〈R2(L)〉, between pair ofmonomers at contour length separationL: results for linear chains (solid lines)and ring polymers (dashed lines).Averages have been calculated on thefirst parts of the corresponding MDtrajectories, immediately after theintroduction of the nanoprobes.

dT =

√`K Le

6. (20)

Finally, the entanglement time (τe) was estimated from themonomer mean-square displacement, 〈g1(τ)〉, at lag-time τby using48,58 〈g1(τe)〉 = 2d2T . As for nanoprobe displacement〈δr2(τ)〉 (Sec. II B), 〈g1(τ)〉 is both time- and ensemble-averaged

〈g1(τ)〉 ≡1

M × Nm

M×Nm∑i=1

limT→∞

1T − τ

×∫ T−τ

0(~ri(t + τ) −~ri(t))2dt (21)

with ~ri(t) being the spatial coordinate of monomer i at time t.Le, dT , and τe have been calculated for our solutions of

linear chains at different Kuhn densities ρK and summarizedin Table I. For the sake of comparison, the same values havebeen applied to ring polymers.

IV. RESULTSA. Nanoprobe diffusion in entangled polymersolutions: Average properties

In order to study the influence of chain architecture onthe diffusion of nanoprobes dispersed in polymer solutions,we consider their time- and ensemble-averaged displacement〈∆r2(τ)〉 at lag-time τ (see Sec. II B for definition). Weemploy the notation 〈∆r2(τ)〉lin (respectively, 〈∆r2(τ)〉ring) forsolutions of linear (respectively, ring) polymers.

Results are shown in Fig. 2, with the horizontal and ver-tical axes scaled down by units of entanglement times τeand tube diameters dT (Table I), respectively. Insets showthe corresponding ratios of MSDs for unconcatenated rings’solutions vs. linear chains’ solutions. Figure 3 shows resultsfor the corresponding time local exponents, α(τ), defined

as α(τ) ≡ d log〈∆r2(τ)〉

d logτ , which provides a direct visualiza-tion of those regimes where diffusion markedly deviates fromstandard Brownian motion (i.e., α = 1).

By comparing the results for different densities ρ (or,tube diameters dT ) and probe diameters d, we may clearlydistinguish two regimes: (1) for d < dT , entanglementshave no effect on probe diffusion which looks almostidentical for solutions of linear and ring polymers. More-over, at short times 〈∆r2(τ)〉∼ τα with α approaching 1/2suggesting coupling of nanoprobe motion to the Rouse modesof the chains; (2) conversely, for d > dT , nanoprobe diffusionis markedly slower in solutions of linear chains compared torings. This is particularly evident in the “ρσ3 = 0.4”-panel inFig. 2 and nanoprobe diameter d = 7.5σ. Here, nanoprobes inrings’ solutions shift from anomalous to normal diffusion atabout τe, while the same probes in solutions of linear chainsshow a longer subdiffusive behavior with a smaller scalingexponent (see the corresponding panel in Fig. 3). Then, nor-mal terminal diffusion is characterized by a ≈ 400× smallerdiffusion coefficient, D (Table I).

In general, these results are in qualitative agreement withthe theory by Ge et al. summarized in Secs. II B and II C.More quantitatively, it is instructive to compare the ratio ofterminal diffusion coefficient for linear chains and rings pre-dicted by Eqs. (9) and (11), respectively. We consider againthe case of ρσ3 = 0.4 and nanoprobe diameter d = 7.5σ.By using data reported in Table I with polymer diameterb/σ ≈ 1, we get limτ→∞〈∆r2(τ)〉ring/〈∆r2(τ)〉lin ≈ 4 × 103.This is ≈10× our numerical estimate. Several factors may beadvocated, which may explain this discrepancy. First, Eqs. (9)and (11) are not exact but systematically neglect unknownprefactors. Second (and equally important), as it was pointedout by Ge et al.14 “hopping” between close-by entanglementsmay enhance nanoprobe diffusion in solutions of linear chains,which, in turn, may lead to better agreement with numericalsimulations.


FIG. 2. Mean-square displacements asfunctions of lag-time τ for nanoprobesof diameters d/σ = 2.5, 5.0, 7.5:results for entangled solutions of linearpolymers (〈∆r2(τ)〉lin, solid lines)and unconcatenated ring polymers(〈∆r2(τ)〉ring, dashed lines). Insets:corresponding ratios 〈∆r2(τ)〉ring/〈∆r2(τ)〉lin. Probes of diameter dlarger than the tube diameter dT of thecorresponding polymer solution diffusemarkedly faster in rings’ systems. Colorcode and symbols are as in Fig. 1.

On the other hand, there is more to nanoprobe motion thatcannot be captured by just its mean-square displacement: inSec. IV B, we will discuss the statistical properties of distribu-tion functions for several selected observables which providea far more vivid description of nanoprobe behavior.

B. Nanoprobe diffusion in entangled polymersolutions: Distribution functions

In order to characterize nanoprobe dynamics beyondthe MSD, we introduce the probability distribution functions

P(τ; ∆x) ≡ 〈(∆x � (x(t + τ) � x(t)))〉 of one-dimensionaldisplacements ∆x for a given lag-time τ. P(τ; ∆x) correspondsto the self-part of the so-called van-Hove function,59 and itmeasures the probability that a probe reaches the spatial posi-tion x(t + τ) from x(t) after time τ. Thus, while 〈∆r2(τ)〉= 3〈∆x2(τ)〉 is just proportional to the second moment ofP(τ; ∆x), the detailed knowledge of P(τ; ∆x) gives deeperinsight for the whole diffusion process.

To fix the ideas, we have considered the three represen-tative lag-times τ/τe = 10�1, 100, 103. Results are illustrated

FIG. 3. Time-dependent differentialexponents of nanoprobe mean square

displacements, α(τ)≡ d log〈∆r2(τ)〉

d log τ .Color code and symbols are as in Fig. 1.


FIG. 4. Probability distribution functions of one-dimensional nanoprobe displacements, P(τ; ∆x), for lag-time τ. Representative curves for given polymersolution density ρ, nanoprobe diameter d, and lag-time τ. The black solid line in each panel corresponds to the Gaussian distribution. Color code is as in Fig. 1.

in Fig. 4 where, in order to ease visualization between differ-ent lag-times, horizontal axes have been scaled to the squareroot of the corresponding second moment,

√〈∆x2(τ)〉 and the

curves compared to the universal Gaussian curve P(τ;∆x)

= P(∆x) =√

12π〈∆x2〉 exp(−

∆x2

2〈∆x2〉 ) (black solid lines).

Interpretation of results appears thus straightforward: withthe notable exception of nanoprobes of diameter = 7.5σ, distri-butions are always or almost Gaussian. Tiny deviations fromGaussian behavior can be seen at the shortest lag-time andshould be ascribed to the coupling of nanoprobes with theRouse modes of the chains. Unperturbed Gaussian behavior isalso observed for large nanoprobes in rings’ solutions at anydensity.

Intriguingly, the case of large (d = 7.5σ, bottom row)nanoprobes in solutions of linear chains appears far from triv-ial. For low-density solutions, nanoprobes remain Gaussian.Conversely, for the highest densities ρσ3 = 0.3, 0.4, we observe(1) a more pronounced peak around ∆x = 0 and (2) the appear-ance of “fat”, exponential tails which become increasinglyevident at large lag-times.

The rationale behind the highest peak is the follow-ing: nanoprobe dynamics is hindered by the presence ofentanglements arising from surrounding polymers whichact as “cages”. In order to quantify this further, we con-sider6 the distribution function P(τ; θ) ≡ 〈θ − cos−1(

(~r(t+τ)−~r(t))·(~r(t+2τ)−~r(t+τ))|~r(t+τ)−~r(t)| |~r(t+2τ)−~r(t+τ)|

)〉 of angles θ between spatial

FIG. 5. Probability distribution function, P(τ; θ), of angles θ between spatial orientations of pairs of nanoprobe displacements separated by lag-time τ.Representative curves for given polymer solution density ρ, nanoprobe diameter d and lag-time τ (as in Fig. 4). The black solid line in each panel correspondsto the “random” distribution P(θ) = sin(θ)/2. Color code is as in Fig. 1.


orientations of pairs of nanoprobe displacements separated bylag-time τ. For random displacements, we expect that P(τ; θ)= sin(θ)/2. Caged particles, instead, are expected to be skewedtoward θ > π/2. These predictions get confirmed by the plotsshown in Fig. 5 representing distribution functions P(τ; θ)corresponding to the same lag-times as in Fig. 4. Particularlypronounced is the difference between distributions observedin solutions of linear chains vs. rings with large nanoprobes(bottom row), a vivid signature of the different role of entan-glements in the two systems. Deviations from the “random”distribution are also observable in rings’ solutions; however,they appear overall “less persistent” in time than for solutionsof linear chains.

The presence of exponential tails in the van-Hove functionhas been reported long ago50 in connection to single-particledynamics in glassy and jammed systems. Recently, we haveshown46 that this property characterizes also the dynamics ofunconcatenated rings in high-density solutions, while linearchains remain Gaussian. Simultaneously, in that work we havedemonstrated that topological constraints between close-byrings tend to cluster the polymers into distinct sub-populationswith definite diffusivities, thus triggering non-ergodic hetero-geneous dynamics.

To verify if anything similar applies to nanoprobe dynam-ics as well and in order to better understand their devi-ations from Gaussian behavior, we consider46 the ratio

FIG. 6. Spatial heterogeneity ofdisplacements ∆r2(T;τ)/〈∆r2(τ)〉 ofnanoprobes of diameter d = 7.5σ as afunction of measurement time T andlag-time τ/τe = 10. At high densi-ties, nanoprobe dynamics becomesheterogeneous alternating slow to fasttrajectories. Color code and symbolsare as in Fig. 1.


FIG. 7. Ergodicity-breaking (EB) para-meter [Eq. (22)] as a function of mea-surement time T and lag-time τ/τe = 10.EB(T ) ∼ T −1 marks standard diffusiveprocesses, whereas EB(T ) ∼ T 0 is asignature of ergodicity breaking. Colorcode and symbols are as in Fig. 1.

∆r2(T; τ)/〈∆r2(τ)〉 as a function of measurement time T andfixed lag-time τ. Figure 6 shows representative curves forthe largest nanoprobes (d = 7.5σ) in solutions of linear andring polymers. Thus, while Gaussian behavior implies thatthe curves converge to the corresponding average value, non-Gaussian nanoprobes in high-density solutions of linear chains(see Fig. 4) alternate between slow to fast trajectories andare thus characterized by heterogeneous dynamics. Then, toquantify relaxation to equilibrium and ergodicity breaking(EB), we introduce the so-called ergodicity-breaking (EB)parameter46,60

EB(T ) = 〈(∆r2(T; τ))2〉

〈∆r2(T; τ)〉2− 1, (22)

which captures how fast the single-nanoprobe trajectories∆r2(T; τ) narrow around the mean 〈∆r2(τ)〉 for some specificlag-time τ. For standard diffusive solutions, EB(T ) decays as60T −1, whereas in nonergodic systems61 EB(T ) ∼ T 0. Indeed,Fig. 7 shows that topological constraints in systems of lin-ear chains trigger ergodicity breaking of diffusive nanoprobeswhen the size of those exceeds the tube diameter of the solu-tion. Inversely, we highlight that solutions of ring polymers donot induce anything similar and, thus, probe dynamics remainsergodic.

V. CONCLUSIONS AND OUTLOOK

In this work, we have employed large-scale computer sim-ulations to investigate the physics of nanoprobe motion inpolymer solutions of linear chains vs. unconcatenated ringpolymers which represent the most simple, yet not trivial,examples of complex polymer systems where entanglementsbetween close-by chains operate on polymer conformationsin “distinct” manners: linear chains follow Gaussian statisticsand strongly overlap with each other,48 while ring polymerstend to crumple as space-filling objects.31,32

By tuning physical parameters like nanoprobe diam-eter and solution density, we have focused, in partic-ular, on nanoprobe diffusion at the transition from thenon-entangled to the entangled regime corresponding tonanoprobes smaller/larger than the tube diameter of the solu-tion. We have thus confirmed and extended (Sec. IV A)recent15,17 numerical results demonstrating that nanoprobesin rings’ solutions diffuse faster than in the correspondingsolutions of linear chains whenever nanoprobe size exceedsthe tube diameter of the embedding solution, in agreementwith the theoretical picture proposed recently by Ge et al.(Secs. II B and II C). Then, we have shown (Sec. IV B) that thecorresponding distribution functions for spatial displacementsmarkedly deviate from Gaussian statistics only in solutionsof linear chains with nanoprobe motion turning to be non-ergodic. Conversely, nanoprobe motion in solutions of ringsremains both Gaussian and ergodic.

In a recent publication,46 we have demonstrated that “non-Gaussian/non-ergodic” dynamics is a characteristic trademarkof self-diffusing rings in sufficiently dense solutions while lin-ear chains follow conventional dynamics: this strikes as theopposite of the trend reported here for dispersed nanoprobesin the same media. At the same time, a consistent amountof theoretical and experimental studies (see, for instance, therecent studies10,62) have demonstrated that doping polymersolutions through the insertion of nanoprobes may enhance toa significant extent the mechanical property of the solution.As a follow-up of this work, it would be then interesting toquantify to which extent the presence of nanoprobes is able toalter the dynamical properties of the chains, especially ringsin high-density solutions which appear to be characterized byheterogeneous dynamics. Second, future investigations shouldnot be bound only to homogeneous solutions: notoriously,even a minimal amount of linear contaminants in solutions ofring polymers is able to enhance their mechanical response in


comparison to pure samples.25 It would then be especiallyinteresting to address nanoprobe diffusion in such mixedlinear/ring polymer solutions.

ACKNOWLEDGMENTS

The authors acknowledge computational resources fromSISSA HPC-facilities.

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