Theory of Localization and Activated Hopping of Nanoparticles inCross-Linked Networks and Entangled Polymer MeltsZachary E. Dell† and Kenneth S. Schweizer*,‡

†Department of Physics and ‡Department of Materials Science, University of Illinois, Urbana, Illinois 61801, United States

ABSTRACT: We develop a microscopic, force-level statistical dynamical theory forthe localization and activated hopping dynamics of dilute spherical particles, bothhard and soft, in polymer networks and entangled melts. The main factorcontrolling localization is the confinement a particle experiences from polymerentanglements and/or chemical cross-links as characterized by the ratio of theeffective nanoparticle diameter to the mechanical mesh length, 2Reff/dT. Dynamicallocalization occurs when the latter ratio is of order unity, and the activated hoppingmobility drastically decreases as the confinement parameter even modestly increasespast threshold. Local packing correlations slightly enhance the tendency to localizefor hard particles and dramatically enhance mobility for soft repulsive particles. Theconcept of an effective nanoparticle size largely, but not completely, collapses thediverse dynamical results for hard and soft repulsive particles. For hard spheres inentangled polymer melts we find the exponentially slow hopping diffusivity isgenerally much smaller than transport via reptation-driven entanglement networkdissolution, except for a narrow window of confinement parameters of 2R/dT ≈ 1.5−2 and long enough chains.

I. INTRODUCTIONNanoparticle motion in diverse crowded environments is ofbroad interest, with applications in physics, material science,cell biology, and medicine. Understanding the motion of atagged nanoparticle can provide length-scale-dependent in-formation about the properties of structurally and/orviscoelastically complex matrix materials.1−4 In polymernanocomposites and filled elastomers, nanoparticles canprovide massive reinforcement of elastic and other mechanicalproperties.5−7 In biology and medicine, molecular transportunderlies many cell functions and drug delivery applications.8,9

The problem has begun to be addressed theoretically, albeitlargely at a tentative and single particle diffusion level.10−16

Of specific interest in this article is how nanoparticles diffusein cross-linked polymer networks and entangled polymer meltsin the dilute tracer limit. Yamamoto and Schweizer havepreviously developed a microscopic statistical mechanicaltheory at the level of forces for spherical one10 and two11

particle diffusivity in unentangled and entangled polymer meltsbuilt on a combination of Brownian motion, polymer physics,and mode coupling theory (MCT) ideas. By adopting aconstraint release dynamical picture, appropriate when particlesare not too small compared to the entanglement tube diameter,single particle mobility is driven solely by length-scale-dependent collective relaxation of polymer density fluctuations.Their primary finding was nanoparticle diffusivity can bemassively enhanced relative to the continuum Stokes−Einsteinprediction due to a gradual coupling of the particle to thetransient entanglement network constraints as its size increases.A second, complementary advance extends the self-consistentgeneral Lengevin equation (GLE) framework developed in refs17 and 18 to the case of small nanoparticles which decouple

from the transient entanglement network. The resultant theoryis in good agreement with simulations17 of hard particlediffusion in both unentangled and lightly entangled melts.The present article builds on the above work but addresses a

different, third regime where nanoparticle motion is viathermally induced activated barrier hopping. Such a rareevent becomes the only option in chemically cross-linkedpolymer networks when the effective mechanical mesh sizebecomes smaller than the particle diameter. It also may beimportant over a narrow range of particle to entanglementmesh diameter ratios in melts. To address hopping transportrequires going beyond the dynamically Gaussian GLE approachpreviously developed.10,17 We employ the nonlinear Langevinequation (NLE) theory, widely utilized for colloidal glasses andgels and supercooled polymer melts.19−22 It is based on theconcept of a dynamic free energy and is a quantitative,microscopic, force-level description. Three questions areaddressed. First, under pure excluded volume conditions(athermal, nonadsorbing polymers), how does the onset oftransient hard sphere localization, the localization length,activated hopping time, and diffusivity depend on cross-linkednetwork and nanoparticle variables? Second, under what(subtle) conditions can hopping control particle diffusion inentangled polymer melts? Third, what is the influence ondiffusion of nonexcluded volume forces, specifically polymer−particle attractions and soft repulsions? The latter is a toymodel of sterically stabilized particles, nanogels, vesicles, andother soft objects.

Received: October 17, 2013Revised: November 25, 2013Published: December 17, 2013

Article

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© 2013 American Chemical Society 405 dx.doi.org/10.1021/ma4021455 | Macromolecules 2014, 47, 405−414

Section II presents the models and dynamical theoriesemployed. Analytic results for the athermal hard sphere systemare derived and discussed for networks and melts in sections IIIand IV, respectively. Section V establishes the role of variablepolymer−particle interactions and local packing correlations ondiffusivity. The paper concludes in section VI with a briefsummary and future outlook.

II. THEORY

A. Models. We consider spherical nanoparticles of coreradius Rcore larger than the polymer monomer size. Thepolymer is described at the lightly coarse-grained level of afreely jointed chain of hard sphere sites of diameter d (typically≲ nm).23,24 Monomers interact with the nanoparticle via a hardcore repulsion plus an exponential tail pair potential:

εα

=

∞ < +

−− +

≥ +

⎧⎨⎪⎪

⎩⎪⎪

⎛⎝⎜⎜

⎞⎠⎟⎟

U r

r R d

r R dr R d

( )

/2

exp( /2)

/2pn

pnpn

core

corecore

(1)

where r is the center-to-center separation of the particle andmonomer, εpn is the contact energy, and αpn is the spatial range.This potential models (i) athermal (εpn = 0), (ii) attractive (εpn< 0), and (iii) soft repulsive (εpn > 0) interactions. For thelatter, it is convenient to define an effective size using standardliquid state theory ideas:25

∫+ = − β∞

−R d r/2 d (1 e )U reff

0

( )pn

(2)

For hard and attractive particles we set Reff ≡ Rcore.Dynamically, the polymer network is described by Rouse

dynamics on length scales less than the entanglement ormechanical mesh diameter, dT, and an arrested soft solid onlarger length scales. We shall use the tube diameter jargon(literally relevant for polymer melts) to denote a mean totaldynamical confinement length scale including (in networks)both topological entanglement and chemical cross-link effects.Experimentally, this length scale can be defined in terms of therubbery plateau modulus23 Gx ∼ ρpkBT/Nx where dT ≡ Nx

1/2σand here is assumed to be unchanged in the presence of thenanoparticle. We note that the latter simplification is rigorousin the dilute particle limit at the ensemble-averaged level wework at. However, physically the polymer packing structurenear a nanoparticle will not be the same as in the bulk, andhence the rubbery modulus is expected to be locally perturbed.How to microscopically include this higher order effect is anunsolved problem at present, but we do not expect it toqualitatively modify any of our results. A schematic depiction ofthe model is given in Figure 1.B. Dynamical Theories. In ideal MCT, the onset of

(transient) nanoparticle localization is characterized by a finitevalue of the long time mean-square displacement20,26 (MSD),⟨(r(t → ∞) − r(0))2⟩ ≡ rL

2. We compute this quantity usingthe well-established naive mode coupling theory (NMCT) self-consistent equation:20,26

β= ⟨ → ∞ · ⟩r

F t F3

216

( ) (0)L

22

(3)

where β−1 is the thermal energy, F(t) is the total force on thenanoparticle at time t, and

∫β ρπ

⟨ · ⟩ =

Γ Γ −F t F k k C k S k k t k t( ) (0)d

(2 )( ) ( ) ( , ) ( , )p pn pp nn

sppc2

32 2

(4)

Here, ρp is the segment number density, Spp(k) is the polymerstatic (density fluctuation) structure factor, and Γnn

s (k,t)(Γpp

c (k,t)) is the normalized (at t = 0) dynamic structurefactor, or propagator, for the nanoparticle (polymer melt).Effective particle−monomer forces enter via Cpn(k) which is theFourier transform of the segment−nanoparticle directcorrelation function defined via the polymer referenceinteraction site model (PRISM) integral equation theoryas27−30

=h k C k S k( ) ( ) ( )pn pn pp (5)

where hpn(k) is the Fourier transform of hpn(r) = gpn(r) − 1, andgpn(r) is the monomer−particle pair correlation function.To dynamically close the theory for entangled polymer melts,

it was previously proposed10

π τ

Γ = −

+ − −

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

k tk D tS k

SN

k dS k

t

( , ) exp( )

exp3 ( )

exp

ppc

pp

x

T

pp

20

02 2

2rep (6)

where D0 is the segmental short time diffusion constant, τrep isthe reptation (disentanglement) time, and Spp(k = 0) ≡ S0 =ρpkBTκT = ρpkBT/KB is the dimensionless compressibility whichquantifies the amplitude of long wavelength density fluctua-tions; κT is the isothermal compressibility, and KB is the bulkmodulus.31 On short length scales (k−1 ≲ dT/2π), diffusiveRouse dynamics apply as encoded in the first term on the right-hand side of eq 6. The second term captures the longer (tubeand beyond) scale (k−1 ≳ dT/2π) physics where transientlocalization occurs due to entanglement effects. Three aspectsenter:10 (i) a Gaussian tube spatial localization (ala neutronspin-echo experiments32) including a de Gennes narrowingfactor, (ii) long time disentanglement via reptation, and (iii) anoverall amplitude, S0/Nx = Gx/KB ≪ 1, reflecting the couplingof density and stress fluctuations. Cross-linked networks aresolids at long times, and thus eq 6 becomes

Figure 1. A schematic depiction of a nanoparticle (hard core of radiusRcore) in a cross-linked polymer network. Soft particles are modeled bya repulsive polymer−particle tail potential (range αpn) from which onecan define an effective particle radius Reff via eq 2. Polymers arecomposed of segments of size d. Both chemical cross-linking andphysical entanglements are mapped to a single length scale, dT,characterizing monomer spatial confinement.

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πΓ → ∞ = −

⎛⎝⎜⎜

⎞⎠⎟⎟k t

SN

k dS k

( , ) exp3 ( )pp

c

x

T

pp

02 2

2(7)

Localization of the nanoparticle is modeled in the standardGaussian manner:20,26

Γ → ∞ = −⎛⎝⎜

⎞⎠⎟k t

k r( , ) exp

6nns L

2 2

(8)

Equations 3, 4, 7, and 8 define a self-consistent Gaussiantheory for rL

2. For particles small relative to dT we expectunbounded diffusion at long times, and hence rL

2 → ∞. Forlarge enough nanoparticles the cross-links should trap theparticle, and hence the localization length will be finite.NMCT describes localization but not activated hopping. The

latter is a strongly non-Gaussian process and is the focus of thenonlinear Langevin equation (NLE) theory19−22 for the scalardynamical displacement of the particle from its initial position,r(t), which obeys the stochastic evolution equation-of-motion:

ζ δ∂∂

= − ∂∂

+t

r tr

F r f t( ) ( ) ( )s sdyn (9)

where ζs is the nanoparticle short time friction constant, δfs(t) isthe corresponding thermal random force, and Fdyn(r) is thedynamic free energy. For networks the latter becomes

∫π

ρ

π

= −

× − +

⎜ ⎟⎛⎝

⎞⎠

⎡⎣⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟⎤⎦⎥⎥

F r

k T rk

C k S k

SN

kr d

S k

( ) 32

ln3

2d

(2 )( ) ( )

exp6 3 ( )

p pn pp

x

T

pp

dyn

B2 3

2

0 22 2

2(10)

If the thermal noise in eq 9 is dropped, then the ideal NMCTresult of strict localization of eq 3 is recovered. Examples of thedynamic free energy are shown in Figure 2, and the localizationlength, rL, barrier position, rB, and barrier height, FB, aredefined. The average hopping time is computed using thegeneral Kramers theory for stochastic processes:33

∫ ∫τ βζ= ′β β− ′r r2 d e d esr

rF r

rF r

hop( )

0

( )

L

Bdyn dyn

(11)

For small barriers, FB ≲ 2kBT, hopping events become less well-defined; however, eq 11 still accurately captures the averagetime it takes the particle to move a “jump distance”, rB − rL.When the barrier is large, FB ≳ 2kBT, eq 11 reduces to theclassic Kramers theory result:33

ττ

π=

β

K K

2e

s L

Fhop

B

B

(12)

where τs ≡ βζs(2Rcore)2 is the particle short time scale and Ki =

βKi (2Rcore)2 is the dimensionless absolute value of the

curvature of Fdyn.C. Structural Models. Equilibrium pair correlations are

input to the dynamical theories. We adopt two models: (i) ananalytically tractable structural (random) continuum model10

and (ii) numerical PRISM theory.27−30 The former is applicablewhen dT, 2Rcore ≫ d and all interactions are hard core, whichmotivates the two defining simplifications of model (i). First,monomers pack randomly around the nanoparticle:

=<

≥⎪

⎧⎨⎩g rr R

r R( )

0

1pncore

core (13)

π= −h k Rj kR

kR( ) 4

( )pn core

3 1 core

core (14)

where j1(x) is the spherical Bessel function of the first kind.Second, the polymer collective density fluctuations areapproximated by their long wavelength limit, Spp(k) ≈ Spp(0)≡ S0.To capture local packing effects and variable nanoparticle−

monomer interactions, we employ PRISM theory and modelthe polymer as a freely jointed chain of N hard spheres ofdiameter d and persistence (bond) length σ = 4d/3.10,27 Themonomer volume fraction is chosen to mimic melts or drynetworks, ηp = ρp4πd

3/3 = 0.4.27−29 For all correlations weadopt the Percus−Yevick site−site closure30,31 outside thedistance of closest approach:

= − βC r g r( ) (1 e ) ( )ijU r

ij( )ij

(15)

PRISM is a theory for melts. To the extent the pair correlationsin dry networks are melt-like it is applicable. There are multiplereasons this situation can hold. First, filled networks aretypically formed experimentally by melt mixing and then cross-linking. Second, we will show that particle localization occurson a relatively short length scale of the order of dT or much less.Third, at zeroth order one generally thinks of a network as amelt below the chemical cross-link scale.23 The latter is largelyrelevant in the lightly cross-linked systems of primary interesthere where the entanglement length is smaller than the meandistance between chemical cross-links. To mimic a rubbernetwork in numerical PRISM calculations, we set N = 1000since this is large enough such that the asymptotic limit isreached for the polymer−particle pair correlations.

III. ANALYTIC ATHERMAL RESULTS IN POLYMERNETWORKS

The random structure model considers only hard spheres andhence Reff = Rcore ≡ R. Given Cpn(k) is proportional to aspherical Bessel function (eqs 5 and 14), standard Gaussianintegration techniques can be used to analytically evaluate eq 10to obtain

π ρ σ

π

π

=

−

× − + − +

= +

− − − −

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

F r

k T rd

Rd

b b b b

bR d

R d r S

( ) 32

ln3

2 62

[ {2 3 (1 2 )e } erf( )]6(2 / )

(2 / ) 4/

p TT

b

T

T

dyn

B2

23

1/2 1 1

2

2 2 20 (16)

where r ≡ r/R. The first ideal-entropy-like term favors theliquid state, while the second term favors localization due topolymer−particle forces. This dynamic free energy isdetermined by three dimensionless parameters: (i) particlesize relative to tube diameter, 2R/dT, a measure of confinement,(ii) the tube diameter relative to the polymer packing length,dT/p, where p−1 ≡ (ρpσ

2), and (iii) the dimensionlesscompressibility, S0. At the analytic random structure level theparticle size enters only via the ratio 2R/dT.For polymer melts, the packing length and tube diameter are

proportional, and experiments find ρpσ2dT ≈ 18.34 In cross-

linked systems this value can be smaller (tighter effective tube).

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Here we perform calculations for ρpσ2dT = 18,9, and four values

of S0 = 0.1, 0.25, 0.4, and 1.0 where the first three span meltvalues27 and the last corresponds to ignoring the de Gennesnarrowing factor in eq 7. For fixed polymer parameters, particledynamics are studied as a function of the confinementparameter, 2R/dT.A. Localization Phenomena. NMCT predicts the onset of

localization, or within NLE theory the crossover of nanoparticlemotion from diffusive to activated hopping corresponding towhen the dynamic free energy first acquires a minimum.20

Figure 2 shows representative results for ρpσ2dT = 18 and S0 =

0.25. For 2R/dT ≤ 1.32, the nanoparticle is not trapped by theentanglement network, and beyond this value it is. The“critical” value for localization depends on polymer propertiesand is denoted as (2R/dT)c; numerical results are summarizedin Table 1. Note that in all cases the onset of localization is a

priori predicted (not assumed) to occur when 2R/dT ∼ 1,consistent with physical intuition. Variations in (2R/dT)c reflectspecific properties of the polymer network. One trend is that ahigher dimensionless compressibility reduces (2R/dT)c, aseemingly counterintuitive result. However, the latter deductionassumes that S0 can be changed independent of all otherpolymer properties. In real systems this is not true since thecompressibility is related to polymer chemistry and density27

and hence the tube diameter. To illustrate this point, we recallthe Gaussian thread “field theoretic” version of PRISM theory27

where a polymer is an uncrossable continuous ideal random

walk (space curve, d → 0). For long chains it is knownS0 ∝ (ρpσ

3)−2 ∝ (p/σ)2, and since in melts p ∝ dT, one has S0 ∝(dT/σ)

2. Now, from Table 1 for ρpσ2dT = 18 we numerically

find (2R/dT)c ∝ S0−1/6. Since dT ∝ S0

1/2, the particle size atlocalization scales as Rc ∝ S0

1/3, restoring the intuition thatparticles are harder to localize in more compressible polymericmedia.A related trend in Table 1 is that with decreasing ρpσ

2dT thelocalization onset shifts to larger confinement values, againseemingly counterintuitive. However, this also is a manifes-tation of assuming polymer properties are independentlyadjustable. In real cross-linked systems, the ratio p/dT isreduced compared to the melt because the effective tubediameter is smaller. If this is taken into account, dT will decreaseby a factor of ∼2 for ρpσ

2dT varying from 18 to 9, implying theparticle size 2Rc at localization is ∼0.6 times smaller for ρpσ

2dT= 9 compared to 18. Thus, we do predict nanoparticles localizemore easily as ρpσ

2dT is decreased, as expected physically.The tightness of particle trapping is quantified by the

localization length, rL. Figure 3 shows the two extrema lengths

of the dynamic free energy as a function of reduced particlesize; the lower (upper) branches of these curves are rL (barrierlocation, rB). In all cases near the localization transition rL ≈ dT≈ 2R, but as the confinement parameter grows, the localizationlength significantly shrinks. For small increases of 2R/dT >(2R/dT)c, it decreases approximately as a critical power law:

−∼ −

±⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

r r

RR

dR

d2 2L c L

T T c

,0.24 0.09

(17)

The exponent variability reflects both a weak dependence on S0and uncertainties in fitting our numerical calculations to theform of eq 17. Concerning the former aspect, the apparentexponent slightly decreases as S0 increases, implying localizationtightens more slowly for more compressible systems. Equation17 remains a reasonable representation only up to 2R/dT ≈(2R/dT)c + 0.3. From 2R/dT ≈ (2R/dT)c + 0.3 to ≈ (2R/dT)c +3, there is a transition regime with no discernible simpleanalytic behavior. For very strong confinement, 2R/dT ≥(2R/dT)c + 3, an asymptotic limit is reached that can be derived

Figure 2. Dynamic free energy profiles, in units of the thermal energy,for various nanoparticle sizes based on the analytic hard sphere modelwith ρpσ

2dT = 18 and S0 = 0.25. The important length and energyscales are indicated: rL, rB, FB.

Table 1. Localization Transition for the Analytic AthermalStructure Modela

ρpσ2dT S0 (2R/dT)c

18 0.10 1.580.25 1.320.40 1.221.00 1.08

9 0.25 1.53aValues of the dimensionless nanoparticle confinement ratio at theonset of localization (2R/dT)c are given for several polymer systems.Here ρpσ

2dT is the ratio of the tube diameter to the polymer packinglength, and S0 is the dimensionless compressibility of the polymersystem.

Figure 3. The two extrema length scales (units of particle radius) ofthe dynamic free energy as a function of the confinement parameterfor several different dimensionless compressibilities in the analytic hardcore model. ρpσ

2dT = 18 for all curves. The lower (upper) branchrepresents the localization length rL (barrier position rB). The bluecrosses are results using the PRISM theory structural input and shouldbe compared to the S0 = 0.25 curve.

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analytically from eq 16 for 2R/dT ≫ 1 and rL/R ≪ 1. Thisresult is

ρ σπ= −

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

rR d

Sd

R72

(3 )2

L

p T

T2

1/2

30

1/42

(18)

The localization length is related to dT as expected, since thepolymer network is driving nanoparticle confinement; however,for large particles rL is much smaller than the tube diameter.Even under strong localization conditions, where hopping willnot be physically relevant in real systems (see section III.B), thetrends in eq 18 may be experimentally observable.B. Dynamic Barrier and Activated Hopping. Three

quantities mainly characterize the particle hopping process: (i)the jump distance rB − rL, (ii) barrier height FB, and (iii) meanhopping time τhop. Figure 3 implicitly shows results for thejump distance which one sees grows as confinement increases.Near threshold, 2R/dT ≈ (2R/dT)c, an approximate criticalpower law fits our calculations:

−∼ −

±⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

r rR

Rd

Rd

2 2B L

T T c

0.48 0.03

(19)

which is robust to variations in the dimensionless compressi-bility except for S0 = 1.0. Equation 19 applies until 2R/dT ≈(2R/dT)c + 1. For 2R/dT ≫ (2R/dT)c, an analytic result canagain be derived. Since rL → 0, the hopping distance isdominated by the barrier location which obeys rB ≫ dT, and inthis limit we find

π ρ σ−

≈ ≈ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

r rd

rd

dR

d(6 )

148

2B L

T

B

Tp T

T

1/6 21/3 2

(20)

The hopping distance grows very rapidly as the confinementparameter is increased, independent of the compressibility. As acaveat, NLE theory is less reliable when the jump distance islarge.19,20 Moreover, as a practical matter hopping isexperimentally irrelevant in the large confinement parameterregime since barriers are enormous as shown below.Figure 4 shows results for the barrier height (inset) and mean

hopping time (main) for the ρpσ2dT = 18 systems. Here the

confinement parameter is expressed as the distance from itsvalue at the onset of localization, a format that results in a nearcollapse of all the curves with the exception of S0 = 0.1. It isevident that very small changes in the degree of confinementinduce a dramatic growth of the barrier and hopping time. Formodest values of 2R/dT ≈ (2R/dT)c, a power law fit to thebarrier calculations works well:

∼ −±⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

Fk T

Rd

Rd

2 2B

T T cB

1.7 0.1

(21)

This result holds to 2R/dT ≈ (2R/dT)c + 0.4, at which point thebarrier is already rather high, FB ∼ 8−10kBT. Likewise, thehopping time can be fit as an exponential law:

ττ

= −⎡

⎣⎢⎢

⎛⎝⎜⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟⎤

⎦⎥⎥

Rd

Rd

0.08 exp 162 2

s T T c

hop

(22)

as shown in Figure 4. This works until 2R/dT ≈ (2R/dT)c + 0.3,corresponding to FB ≤ 3kBT. We emphasize that when barriersare small, hopping may become difficult to distinguish from

other nonactivated motions. However, we can clearly concludethat in experiments, hopping will only be possible to observe inthe smaller confinement regime.To explicitly illustrate the massive reduction in mobility, the

S0 = 0.25 system was analyzed further. By choosing dT = 8 nmto mimic a polystyrene melt,34 and taking τs = 1 ms as a typicalnanoparticle short time constant, we find for 2R ≈ 2Rc = 10 nmthat τhop ≈ 1 ms ≈ τs. As the particle size is increased by amodest amount to 2R = 18 nm, the hopping time increases to100 000 years! Thus, there is a massive reduction in hoppingmobility with small changes in particle size.In between 2R/dT ≈ (2R/dT)c + 0.4 and 2R/dT ≈ (2R/dT)c +

10 a transition regime is observed, where no simple analyticbehavior is found. Above 2R/dT ≈ (2R/dT)c + 10, we find anasymptotic limit can be derived

π ρ σ≈ =⎛⎝⎜

⎞⎠⎟F R k T

dV G

16

(2 )B pT

n x3

B

2

2(23)

where Vn is the particle volume and Gx is the network shearplateau modulus. Equation 23 is reminiscent of elastic modelsfor flow and relaxation in deeply supercooled liquids.35,36 Here,it reflects that in order to move, the particle must isotropicallydeform the surrounding network on a scale of order its size. Apractical caveat is that at the onset of the asymptotic scaling thebarrier is already FB ≈ 104kBT, and thus in experiment orsimulation such a R3 scaling will never be observed.Finally, one can ask how hopping transport is affected by

polymer properties? As mentioned above, when confinement isviewed relative to the onset of localization, the dimensionlesscompressibility of the polymer liquid, S0, has little effect onhopping (e.g., see Figure 4). Decreasing ρpσ

2dT weakens therate of growth of the barrier and hopping time. However, evenwith this downward shift, the conclusion is the same: smallincreases in the ratio 2R/dT above its critical value results in adrastic decrease of particle mobility.

IV. ACTIVATED HOPPING IN POLYMER MELTSIn cross-linked systems, once the particle becomes localized theonly channel for motion is hopping. On the other hand, in

Figure 4. Mean barrier hopping time in units of the particle short timescale, τs ≡ βζs(2R)

2, as a function of the distance of the confinementparameter from its critical value for the analytic hard sphere model.ρpσ

2dT = 18 for all curves. The black, dot-dot-dashed curve is a fit ofthe low confinement regime to the exponential form of eq 21 . (Inset)Barriers for the same systems. The blue crosses are the barriers usingthe PRISM structural model input and should be compared to the S0 =0.25 curve.

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entangled polymer melts the tube constraints have a finitelifetime, and two classes of competing, system-specific channelsof motion can be envisioned: (i) nanoparticle “active” motionto escape dynamical constraints and (ii) “passive” motion viapolymer constraint release. A recent theory17,18 has combinedthese two mechanisms by solving the GLE self-consistently forthe particle mean square displacement as a function of time.However, the GLE theory was dynamically closed at theGaussian level, which does not account for large amplitudeactivated hopping.Hence, there remains the question of under what conditions

hopping contributes to nanoparticle diffusion in entangledpolymer melts? Intuitively, we expect two conditions musthold. First, reptation must be slow enough for hopping to berelevant; otherwise, constraint release would dominate. Thiswill only be true for “long enough” chains. Second, the particleto tube diameter ratio must fall in a narrow window near unityto ensure that the particle becomes transiently localized butthat the barrier height is sufficiently low so that hopping canoccur quickly enough. At zeroth order, we will quantify theseconditions based on comparing the time scales for hopping andconstraint release. A caveat is that calculating hopping andconstraint release mobilities independently will be quantita-

tively accurate only if the time scales are sufficiently separatedso the two processes can be distinguished.Given the reptation time sets the time scale for constraint

release motion, and the particle short time scale follows from itsRouse friction constant, the hopping time relative to theconstraint release time is

ττ

π ρ σ π=

β−⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢⎢

⎤⎦⎥⎥d

NN

Rd K K18

( )2 2

ep Tx T L B

Fhop

rep

32

3 5

B

(24)

When τhop/τrep ≪ 1, hopping will dominate, and when τhop/τrep≫ 1, constraint release will dominate. Figure 5 showscalculations using ρpσ

2dT = 18 and S0 = 0.25 for severaldegrees of entanglement. Results are shown for FB = 2−20kBTsince for low barriers hopping is not well-defined, and for FB >20kBT hopping is astronomically slow. One sees from Figure 5that if N/Nx = 5, then particle hopping is slower than polymerdisentanglement, suggesting it is not important, consistent withrecent simulations of lightly entangled melts.17 As the degree ofentanglement increases, the window of relevant confinementratios, 2R/dT, grows. However, for experimentally accessiblevalues of N/Nx ≲ 200, the window remains very small, and weestimate 2R/dT ≈ 1.5−1.8.One can also phrase the above question as for a given particle

size and tube diameter, what is the minimum chain lengthnecessary to see hopping? By defining the threshold as whenτhop = τrep, one can solve eq 24 for Nc:

πρ σ π=

β

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢⎢

⎤⎦⎥⎥N N

d Rd K K182 2

ec xp T

T L B

F2 1/3 5/3 1/3

B

(25)

The main frame of Figure 6 shows hopping is relevant only for2R/dT ≈ 1.5−2.0 based on this criterion, and N/Nx ≳ 10 isnecessary to observe hopping.As an alternative, but related, measure of the relevance of

hopping, we compare the particle diffusivity achievable basedon activation to that of the constraint release mechanism. Asimple and natural choice for the hopping diffusion constant is

τ=

−D

r r( )6

B Lhop

2

hop (26)

In the analytic model the short time diffusivity, Ds, sets themobility scale. Since hopping is associated with escape fromentanglement constraints, we assume all relevant Rouse modesfirst relax, and thus obtain the N-independent result10

ζπ η= =

−⎡⎣⎢⎢

⎤⎦⎥⎥D

k Tk T

RR

83s R

B

RouseB

3

g2

1

(27)

where the Rouse viscosity is ηR = η0N and η0 is the segmentalmelt viscosity. Expressing the constraint release diffusivity fromref 10 in terms of the same units, one finds

ζζ

ζζ π

π

π π

= +

=

× + + −π

−

−

⎡⎣⎢

⎤⎦⎥

⎛⎝⎜

⎞⎠⎟

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

DD

NN

Sd

R

S R d S R d

1

112

32

e 12

3 ( / )1

23 ( / )

CR

s

x

T

S R d

T T

ent

Rouse

1

ent

Rouse

33

0

3 ( / )2

02 2

02

T2

02

(28)

Figure 5. Ratio of the mean hopping time to the reptation time inmelts of various degrees of entanglement N/Nx, with ρpσ

2dT = 18 andS0 = 0.25.

Figure 6. Minimum value of N/Nx for which hopping is important inmelts based on the four indicated criteria. Here ρpσ

2dT = 18 and S0 =0.25 are chosen to mimic melt conditions. (Inset) Hopping diffusivity(purple, solid) compared to its constraint release analogue in melts.The horizontal lines correspond to (from top to bottom) N/Nx = 5,10, 25, 100, and 200.

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A direct comparison of the diffusivities can now be made, as afunction of N/Nx, 2R/dT, and S0. The inset of Figure 6 showsresults for barriers again spanning the range FB ∼ 2−20kBT.Note that for N/Nx = 5, the polymer is only slightly entangled,and we expect (and find) hopping is not relevant. As N/Nxgrows, the window in 2R/dT where hopping is faster thanconstraint release diffusion does widen. But, even for the mostheavily entangled N/Nx = 200 system, hopping is importantonly for 2R/dT ≈ 1.5−1.8, consistent with our conclusion basedon the time scale comparisons. Figure 6 also shows the value ofN/Nx where Dhop = DCR, along with lower and upper boundson this threshold corresponding to when Dhop = DCR/10 andDhop = 10 DCR, respectively. We again conclude that in all caseslong chains are necessary to see hopping (N/Nx ≳ 10), and it isonly relevant for a narrow range of confinement parameters oforder unity. Our conclusion that hopping is the primary modeof transport only for strongly entangled polymers over a smallwindow of confinement parameter appears to be consistentwith the qualitative arguments of Cai and Rubinstein.12,13

V. ROLE OF VARIABLE NANOPARTICLE−POLYMERINTERACTIONS

To explicitly study the influence of equilibrium local structureand interactions on our dynamical results, we employ PRISMtheory as described in section II. Typical monomer−particlepair correlations functions, gpn(r), are shown in Figure 7 for

2Rcore = 10d with a tail potential range of αpn = d for three typesof systems: (i) hard spheres, (ii) attractive spheres, and (iii) softrepulsive spheres. Compared to the hard sphere case,attractions enhance local correlations of monomers aroundthe tagged particle, while repulsions decrease the correlations.For the highly repulsive systems, a significant correlation holedevelops, ala a dewetting of the polymer−particle interface.A. Hard Spheres. By comparing the dynamic free energy

extrema length scales and barrier for hard spheres based onPRISM theory structural input to that deduced based on theanalytic structural continuum model, the role of localnonrandom packing correlations can be investigated. In order

to make a fair comparison, the finite size of monomers isincorporated into the analytic model by taking R → R + d/2 ineq 16, which ensures that the contact separations in gpn(r)match. Additionally, we match the polymer parameters of bothmodels by choosing N = 1000 and ηp = 0.4, which fixes S0 =0.25. Finally, the packing parameter for the analytic model wasρpσ

2dT = 18, which implies dT = 27πd/16ηp = 13.25d.Results for the extrema lengths are shown in Figure 3 as

crosses. Comparing to the S0 = 0.25 case, the major differencebetween the two results is the precise onset of localization; thePRISM-based calculations predict (2R/dT)c = 1.05, modestlysmaller than the analytic result of (2R/dT)c = 1.32. Thisdifference is consistent with the intuitive idea that the confiningforces on the nanoparticle are enhanced when local packingcorrelations (layering) is included. When viewed relative to thelocalization transition onset, the behavior of the extrema lengthscales are essentially identical in the two calculations. Barrierheight calculations are shown in Figure 4 (inset), expressedrelative to (2R/dT)c. The PRISM-based calculations growmodestly more quickly with increasing particle size, and henceso will the hopping time, consistent with the idea that localpacking correlations quantitatively reduce mobility.Overall, the dynamical results based on the continuum model

agree remarkably well with the fully numerical results. Thisseems understandable as a consequence of the short-rangenature of the structural correlations and the fact they areoscillatory in a roughly symmetric manner about the randomvalue of unity. As a result, their dynamical consequences arelargely averaged out, as a priori done by the analytic structuremodel.

B. Repulsive Interactions. A largely open question is therole of particle−monomer interactions on nanoparticlediffusion. Recently, there has been much interest in compositesystems of soft nanoparticles, such as nanogels, microgels, hardcolloids with grafted polymer layers, vesicles, and micelles.37,38

To crudely model the effects of particle softness, we studyparticles that interact with the polymers via a hard core plus softrepulsive exponential tail as in eq 1. Using the PRISM structuralcorrelations as input, the effects of a soft interfacial interactionon particle dynamics in cross-linked polymers are explored.Consider first the influence of repulsion strength εpn at

constant 2Rcore = αpn = 5d for tube diameters in the range dT =2−20d. System parameters are summarized in the first sectionof Table 2 along with the localization transition tube diameterratio dT,c/d for each case. Note that as βεpn increases, thelocalization transition tube diameter, and hence (dT/2Rcore)c,increases. This effect is dramatic since dT,c varies from ∼d forthe hard core case to ∼20d for the very soft sphere case of βεpn= 5. There is no seemingly obvious relationship betweenparticle core size and the transition value. However, sinceincreasing εpn results in an effectively larger particle diameter asit appears to the monomers, this trend is qualitatively expected.In an attempt to collapse the above calculations, we scale dT

by the effective particle size 2Reff of eq 2. The resulting values of2Reff/d and (2Reff/dT)c are given in Table 2. This normalizationapproximately collapses the localization onset to (2Reff/dT)c ≈1−2. To further investigate this, extrema length scale profileswere calculated as a function of both dT/2Rcore and dT/2Reff, asshown in Figure 8. When expressed as a function of dT/2Rcore(inset), there is a huge variance in the curves mainly associatedwith the shifts to higher tube diameters. This variance can belargely collapsed if dT is scaled by the effective particle size. Theextrema length scales can then be compared to Reff as suggested

Figure 7. Equilibrium monomer−particle pair correlation functions forattractive (dashed red), soft repulsive (dotted black, dash-dotted green,dash-double dotted yellow), and hard core (solid blue) systems. Theparticle core diameter is 2Rcore = 10d, and the attraction/repulsionrange is αpn = d. (Inset) Fourier transform of the nonrandom part ofthe pair distribution function, hpn(k), for the same conditions of themain frame and attractive (solid red and dashed blue), hard sphere(dotted black), and soft repulsive (dot-dashed green) systems.

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by the analytic model. Doing so, we find the massive (but notperfect) collapse shown in the main panel of Figure 8.We now investigate the role of softness on hopping

dynamics. In order to accomplish this while holding everythingelse constant, a collection of systems of fixed 2Reff = 10d werestudied. By choosing specific values of 2Rcore and εpn, therepulsion range αpn was then appropriately determined. Varyingthe core size from 2Rcore = 0.1−10d spans cases from a nearlyfully soft particle to a hard sphere. The system parameters andlocalization transition values for these systems are summarizedin the center section of Table 2. First note the onset oflocalization is shifted to smaller dT,c values for softer particles.This is an intuitive trend and also applies to the extrema lengthscale profiles in Figure 9. Additionally, it is evident from thelatter that as 2Rcore → 0, the results tend toward the same limitof a fully soft particle. Barrier calculations are shown in the insetof Figure 9. In all cases, small changes of the tube diameteryield large changes of the barrier height. The main effect of

particle softness is slowing down how quickly the barrier grows.For softer fillers, more confinement relative to the transitionvalue is necessary to produce the same barrier (and hencehopping time). All these results are intuitive as softer particlesshould be able to squeeze through the network more easily.The above studies show that the main effect driving

localization in cross-linked nanocomposites is the confinementa (soft) particle experiences from the network as characterizedby the value of 2Reff/dT. When scaled in this manner, the largestvariations in dynamics are collapsed, and all localizationtransitions occur at 2Reff/dT ≈ 1. There are smaller, second-order perturbations due to particle softness. We find that fortwo particles with the same effective size the softer the particleis, the more difficult it becomes to induce localization (smallerdT,c). The nanoparticle mobility drastically decreases in all casesas 2Reff/dT grows; however, the decrease is slower as particlesbecome softer.

Table 2. Localization Transition for PRISM Structural Modelsa

system 2Rcore/d βεpn αpn/d 2Reff/d dT,c/d (2Reff/dT)c

repulsive: constant 5 0 5 5.00 4.25 1.18core size 1 5 12.97 7.25 1.79

2 5 18.29 12.25 1.484 5 24.67 19.25 1.28

repulsive: constant 0.1 5 2.26 10 6.25 1.60effective size 1 5 2.06 6.45 1.55

2.5 5 1.71 6.75 1.485 5 1.14 7.60 1.3110 0 0.00 9.10 1.10

attractive 10 0 0.0 10 9.10 1.10−2 0.5 8.25 1.21−4 0.5 7.60 1.32−2 1.0 7.75 1.29−4 1.0 7.10 1.41

aCalculations of the tube diameter at the onset of localization, dT,c, are shown for several different nanoparticle diameters, 2Rcore, and monomer−nanoparticle interfacial interaction strengths (ranges), εpn (αpn) (defined in eq 1). It is convenient to define an effective particle size (via eq 2) andcompare it to the transition tube diameter (2Reff/dT)c. All length scales are expressed in terms of the polymer segment diameter d.

Figure 8. The two extrema lengths of the dynamic free energy for softrepulsive systems with 2Rcore = 5d and αpn = 5d and several differentrepulsion strengths εpn. Both the tube diameter and extrema lengthsare normalized by the effective particle size. (Inset) To exhibit theextent of data collapse in the main frame, the uncollapsed results areshown where dT is normalized by the particle core size and the extremalengths are normalized by the monomer diameter.

Figure 9. The two extrema lengths (main) and barrier height (inset)of the dynamic free energy for soft particles of constant effective size2Reff = 10d with various core sizes. For all cases βεpn = 5 and αpn/d isdetermined by the constant effective size condition of eq 2. The2Rcore/d = 10 case represents a hard particle, and as the core size isdecreased the “softness” of the filler is increased. The barrier heightresults are shown as a function of the inverse mesh length dT

−1

rendered dimensionless by the mesh length at the onset of localizationdT,c.

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C. Attractive Polymer−Filler Interactions. Chemistrycan be chosen to favor monomer adsorption, creating anattractive interfacial potential. We study this case for hardparticles with short-range attractions modeled via eq 1 with εpn< 0, 2Rcore/d = 10, and αpn/d = 0.5 and 1.0. The attractionstrength εpnwas varied from zero to the strong value of −5kBT.The bottom panel of Table 2 summarizes the system

parameter choices and the computed localization onset values.Short-range attractions shift the localization transition tosmaller tube diameters for both attraction ranges studied; thesame trend is found in the full extrema length scale profiles inFigure 10. Based on a local dynamical perspective, this result is

counterintuitive; as attractions are added, one might expect itshould be more difficult for the particle to move. On the otherhand, we are concerned with localization driven by physicaleffects on a mesoscopic length scale, the tube diameter, not thevery short local scale characteristic of the attraction range. Theprior work of ref 10 observed a similar trend for nanoparticlediffusion based on the polymer constraint release mechanism.For highly entangled systems, they found that nanoparticlediffusivity increases as the interfacial attraction was increased, atrend, however, that is reversed in unentangled melts where thephysics is more local.Hence, our results are consistent with the prior study10 and

can be explained, in the context of our approximate theory, in asimilar manner. Specifically, the effective forces on the particleare quantified by the structural correlations on the relevantlength scales dictated by dynamical relaxation. In Figure 7, onesees that as the attraction is increased the local correlationsgrow (adsorption), indicating stronger forces locally. But,considering hpn(k) on the larger mesoscopic length scale, thereis a reduction of correlation (and hence forces) since |hpn(k)|decreases with attraction strength. This raises the question as towhether the local or global structure is more relevant tonanoparticle motion. Given that dT ≈ 2Rcore near the transitionand that the cutoff wavevector in the polymer propagator (eq7) is kcut = (3π2S0)

1/2/dT, we find kcutd ≈ 0.27. Hence, the forcecorrelations are controlled to leading order by the mesoscopicregime, thereby mathematically explaining our results.A physical caveat to the above discussion is that while the

theory is internally consistent, the seemingly counterintuitive

trends could be a consequence of the core approximation ofMCT and NLE theory which replace real forces by effectiveforces determined by pair structure. This approach works wellfor repulsive systems. However, when there is a combination ofattractive and repulsive forces, concerns have been raised in thecontext of supercooled liquids as to whether such a MCT-likeforce renormalization accurately captures the dynamicalconsequences of competing forces.39,40 Our results for theattractive systems may be related to this issue, and futureresearch and simulations are needed to systematically addressthis open issue.

VI. DISCUSSIONWe have developed a microscopic, force-level statisticaldynamical theory for the localization and activated hoppingdynamics of dilute spherical particles, both hard and soft, incross-linked networks and entangled polymer melts. Ourprimary findings and predictions are as follows. First andforemost, the main factor controlling localization is theconfinement the particle experiences from polymer entangle-ments and cross-links as characterized by the ratio of theeffective nanoparticle diameter to the mechanical mesh length,2Reff/dT. Dynamic localization occurs when 2Reff ≳ dT, and thehopping mobility drastically decreases as the confinementparameter even modestly increases past threshold. In additionto this main effect, local packing correlations modestly enhancethe tendency to localize for hard particles, and for repulsiveparticles increasing softness, even at fixed effective particlediameter, can enhance mobility. While the effects of softinterfacial repulsions are small compared to those of particlesize (confinement), they still have dramatic consequences forthe onset of localization and in turn the nanoparticle mobility.For entangled melts, we find the exponentially slow activatedhopping diffusivity is ineffective relative to particle transportcontrolled by entanglement network dissolution for most cases.The exception appears to be a narrow window of confinementparameters of 2R/dT ≈ 1.5−2 for sufficiently long chains.Future experiments and simulations to test the regimes wherehopping is relevant for networks and melts would be veryvaluable. Additionally, experimental evidence for the scaling ofthe localization length and the hopping time with theconfinement ratio would be an important test of ourpredictions.Our approach relies on two key theoretical assumptions

inherent to both MCT and NLE theory. First, we assumeparticle hopping and polymer network fluctuations areisotropic. This, of course, is true in the context of ensembleaveraged results, but how much error it might incur at the levelof an individual hopping event is unknown. Second, we do notallow the nanoparticle to actively (and perhaps nonlinearly)modify the local network structure as part of the activatedhopping motion.13 Relaxing this assumption involves takinginto account higher order (greater than two body) static anddynamical correlations in MCT and NLE theory, an openproblem for microscopic force-level approaches.The theory presented here can be straightforwardly

generalized to treat particles in semidilute polymer solutions.Nonspherical particles can also be studied, including the role oftranslational versus rotational degrees of freedom in thelocalization and hopping process. The present work providesa foundation for attacking the very difficult nondilutenanoparticle regime of high interest for polymer nano-composites. However, this requires addressing several compli-

Figure 10. The two extrema lengths (main) and barrier height (inset)of the dynamic free energy for particles with core 2Rcore = 10d andvarious attraction ranges and strengths. The hard sphere case (solidred) is shown for reference. The barrier heights are plotted as afunction of the inverse mesh length dT

−1 rendered dimensionless bythe mesh length at the onset of localization dT,c.

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cations, a primary one being the assumption that polymerdynamics is unperturbed by the presence of nanoparticles.

■ AUTHOR INFORMATIONCorresponding Author*E-mail kschweiz@illinois.edu (K.S.S.).NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work was funded by Michelin-France. We thank UmiYamamoto and Sanat Kumar for helpful discussions.

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