entropic barriers, activated hopping, and the glass ...imct and is often experimentally found to...

JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 2 8 JULY 2003

Entropic barriers, activated hopping, and the glass transitionin colloidal suspensions

Kenneth S. Schweizera) and Erica J. SaltzmanDepartments of Materials Science & Engineering and Chemistry, Frederick Seitz Materials ResearchLaboratory, University of Illinois, Urbana, Illinois 61801

�Received 5 February 2003; accepted 2 April 2003�

A microscopic kinetic description of single-particle transient localization and activated transport inglassy fluids is developed which combines elements of idealized mode-coupling theory, densityfunctional theory, and activated rate theory. Thermal fluctuations are included via a random forcewhich destroys the idealized glass transition and restores ergodicity through activated barrierhopping. The approach is predictive, containing no adjustable parameters or postulated underlyingdynamic or thermodynamic divergences. Detailed application to hard-sphere colloidal suspensionsreveals good agreement with experiment for the location of the kinetic glass transition volumefraction, the dynamic incoherent scattering relaxation time, apparent localization length, and lengthscale of maximum nongaussian behavior. Multiple connections are predicted betweenthermodynamics, short-time dynamics in the nearly localized state, and long-time relaxation byentropic barrier crossing. A critical comparison of the fluid volume fraction dependence of thehopping time with fit formulas which contain ideal divergences has been performed. Application ofthe derivative Stickel analysis suggests that the fit functions do not provide an accurate descriptionover a wide range of volume fractions. Generalization to treat the kinetic vitrification of morecomplex colloidal and nanoparticle suspensions, and thermal glass-forming liquids, is possible.© 2003 American Institute of Physics. �DOI: 10.1063/1.1578632�

I. INTRODUCTION

Slow dynamics of glass-forming materials is of majorscientific and technological importance and has recently beenthe subject of intense study.1 There are many different theo-retical approaches, the validity of which remain strongly de-bated. The relative importance of thermodynamics �includingpossible underlying phase transitions� versus kinetics in de-termining laboratory vitrification is particularly unclear.1,2

The microscopic ‘‘idealized’’ mode-coupling theory �IMCT�of Gotze and co-workers3–6 is a purely kinetic approachwhich focuses on the self-consistent description of local dy-namics due to particle caging. Nonlinear and non-Markoviancoupling of collective density fluctuations is proposed as theessential physical process which is treated using projectionoperator and factorization approximations. A primary resultis the prediction of an ‘‘ideal’’ nonergodicity or glass transi-tion. IMCT makes an impressive number of predictions fortransport coefficients and wave-vector-dependent collectiveand single-particle dynamic density fluctuations. These in-clude the bifurcation of correlators into �fast� beta and slow� relaxations, critical-like power-law decays at intermediatetimes, long-time stretched exponential relaxation, anomalousself-diffusion, and critical power-law divergences of all re-laxation times.3–5 IMCT is unique in being formulated at themicroscopic level of forces. Most explicit IMCT results havebeen obtained for fluids composed of spherical particles. Re-

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cently, extensions to simple molecular models such as di-atomics and ellipsoids7,8 and short-chain flexible polymers9

have been performed.So-called ‘‘schematic’’ formulations of IMCT have been

widely applied to interpret experimental measurements onthermal glass-forming molecular, ionic, and polymericliquids.3,4 A major problem is that the predicted criticalpower-law divergence of the � relaxation time is not ob-served and is generally agreed to be an unphysical conse-quence of the inherent approximations of IMCT. Based onfitting two or three parameters, IMCT typically describes�over an intermediate-temperature range� two or three ordersof magnitude variation of the � relaxation time and fails forthermal glass formers far above the laboratory transition.1,10

Analysis of Potts spin-glass models,11–13 experiments1 andcomputer simulations,14–18 all suggest that IMCT accuratelydescribes dynamical precursor phenomena in moderately su-percooled liquids, but does not account for �free� energylandscapes, barriers, or activated transport. The basic reasonfor the latter limitations appears to be that IMCT is of adynamic mean-field �infinite-dimensional� nature, with effec-tive barrier heights that scale with system size and hence areinsurmountable in the thermodynamic limit.11,13 The latteraspect is consistent with connections derived between IMCTand static density functional theory.12 In reality, particlesreach local minima in an energy landscape in a finite timeand long-time motion is controlled by activated dynamics.13

This insight has motivated the development of phenomeno-logical single-particle ‘‘trap models’’ which are fundamen-tally different than IMCT for the long-time � relaxationprocess.13,19

© 2003 American Institute of Physics

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1182 J. Chem. Phys., Vol. 119, No. 2, 8 July 2003 K. S. Schweizer and E. J. Saltzman

It is noteworthy that even above �but near� the apparentmode-coupling temperature Tc , computer simulations findevidence of a relatively sharp transition in single-particletransport from smooth, hydrodynamic-type motion to a dy-namics characterized by vibration about fixed positionscoupled with local activated hopping or jump diffusionevents.15–18 Failure of the Stokes–Einstein relation, or‘‘translation–rotational decoupling,’’ 1,2 is not predicted byIMCT and is often experimentally found to begin close to theestimated IMCT transition. This decoupling effect has beensuggested to be correlated with the emergence of activatedhopping as the primary mode of transport.17,18 Attempts toinclude activated transport within MCT have mainly in-volved low-order coupling of density fluctuations to currentmodes �phonons�,3,4 which destroys the IMCT nonergodicitytransition. This approach is largely phenomenological andnonpredictive. It is important to note that a number of IMCTresults �especially in the fast � relaxation regime� remainvalid even in the presence of hopping processes.

Incorporation of the activated hopping physics within apredictive approach formulated at the level of forces remainsa critical unsolved problem. An interesting dynamic densityfunctional theory has been proposed20 in an attempt to in-clude activated processes. This approach is numerically com-plex even for simple fluids, and results are rather limited.Alternative mesoscopic attempts to go beyond IMCT built onconcepts from spin-glass theory11 appear to have madeprogress in understanding some aspects of the low-temperature activated regime.12 The essential physics is tiedto an underlying ‘‘random first-order’’ thermodynamic phasetransition11 and ‘‘entropy catastrophe,’’ the existence and/orrelevance of which is the subject of vigorous debate.2,15,21

The goal of the present work is to propose a theory ofslow single-particle dynamics formulated at the level offorces. It is a kinetic approach which combines, and in someaspects extends, elementary ideas of mode-coupling, densityfunctional, and activated rate theory. We are motivated by thedesire2 to build a bridge between landscape and MCT ap-proaches at the simplest technical level. The theory does notaccount for the rich consequences of nonlinear and non-Markovian coupling of dynamic density fluctuations de-scribed by the full IMCT in the dynamic precursor regime.Rather, building on the ‘‘naive’’ single-particle version ofmode-coupling theory,22 a simpler approach is adopted sinceour primary goal is to go beyond IMCT in the sense of de-scribing free-energy barriers and activated hopping.

Our basic ideas are general and will likely be of mostvalue for thermal glass-forming materials. But for a varietyof reasons, we first develop and apply the theory for modelhard-sphere colloidal suspensions. Besides their molecularsimplicity, the microscopic structural information required asinput to the dynamical theory is readily available for hardspheres, and coarse graining of molecular complexities canbe avoided. The hard-sphere suspension is also the systemfor which IMCT has been most successful.3–5 IMCT predictsa literal nonergodicity glass transition at a packing fractionof �MCT�0.52, which is well below the kinetic experimentalvalue23–25 of �g�0.57– 0.58, which is well below classicrandom close packing �RCP�0.64 or other estimates26 of a


maximum jamming volume fraction �MAX . By empiricalrescaling �fitting� of �MCT , IMCT agrees nearly quantita-tively with a number of dynamic light scattering measure-ments on hard-sphere suspensions3–5,23,24 and also computersimulations for related model fluids which probe the dynami-cal precursor regime.15,16

A popular argument for the success of IMCT for hard-sphere suspensions is that since colloidal momentum is notconserved, phononlike excitations are absent �strongly over-damped�, which renders activated processes irrelevant.3,4,23,27

However, intrinsic entropic barriers of a suitably definedfree-energy surface must be present. Indeed, dissipativeBrownian dynamics simulations have shown that activatedhopping motions are present and may be the primary mecha-nism of the � relaxation even above the apparent IMCT idealglass transition.17 There are quantitative differences in theslow relaxation based on Newtonian and Brownian dynam-ics, the latter being slower by roughly an order ofmagnitude.16 But the qualitative aspects of the activated hop-ping processes are the same despite the lack of any collective‘‘phonon assistance.’’ 15–17 This strongly suggests that thelong-time processes that destroy the IMCT transition are, to afirst approximation, independent of the nature of the micro-scopic dynamics.15–17 Brownian dynamics simulations17 findthat the rather sharp crossover in transport mechanism doesnot result in any abrupt changes in the intermediate dynamicscattering function, and transient plateaus on the particle-sizelength scale only clearly emerge below the apparent mode-coupling transition temperature.15,17 These observationsagain suggest the importance of activated hopping processes.Experimentally, transient nonergodicity in hard-sphere sus-pensions as indicated by well formed plateaus in the collec-tive dynamic structure factor only becomes clearlyobservable23 above ��0.55–0.56.

From a practical perspective, the experimental study ofan activated dynamics regime in colloidal suspensions is se-verely hindered �relative to thermal glass formers� due to theslow elementary diffusive time scale for Brownian particles.Typically only a factor of 103 – 104 in time is accessible be-tween the freezing and kinetic vitrification volume frac-tions,23–25 thereby restricting experiments to a dynamic pre-cursor regime. The latter is also true of computer simula-tions, which typically probe �3 orders of magnitude ofdynamical slowing down.15,16 Hence, the existence of a di-vergent relaxation time can only be inferred via a large ex-trapolation. The reliability of the latter seems especiallyuncertain given recent light scattering,20 confocal micro-scopy,28–31 and microgravity experiments32 using model hardspheres of various sizes and chemical constitutions. All thesestudies find significant motion exists for ��g , which maypersist32 up to �MAX . Whether this motion corresponds tosmall-scale ‘‘aging’’ phenomena in a globally frozen solid orultraslow structural relaxation and flow is not well under-stood. Possible long-time diffusion in the ‘‘glass’’ due to ac-tivated processes has been suggested.23,27–31

The remainder of the paper is structured as follows: Thebasic theoretical ideas are developed in Sec. II. Section IIIpresents numerical results for an ‘‘effective free energy’’


1183J. Chem. Phys., Vol. 119, No. 2, 8 July 2003 Glass transition in colloidal suspensions

function and characteristic time, energy, and length scales.Comparisons between various features of this free-energyfunction and dynamic experiments are presented. Activatedbarrier hopping is the subject of Sec. IV, and quantitativecomparisons with incoherent dynamic light scattering mea-surements are given. Section V presents detailed compari-sons of the relaxation time results with various fit functions,all of which invoke a divergence. A summary and discussionis given in Sec. VI.

II. THEORY

We adopt a single-particle dynamical description, par-tially motivated by the seminal efforts of Wolynes andco-workers12,22,33 to connect ‘‘naive’’ IMCT and densityfunctional theory �DFT�. We are also motivated by the desireto quantify, and provide a more fundamental basis for, phe-nomenological ‘‘trap models.’’ 19 The latter aim to describesingle-particle motion in a cage via a static random potential.Although the restriction to single-particle dynamics carriesobvious limitations, it is important to note that many colloidexperiments23,24 and simulations15–18 have found a close cor-relation between local single-particle and collective cageslow dynamics.

A. Naive idealized MCT

The ‘‘naive’’ IMCT �Ref. 22� focuses on calculation ofthe force–force time correlation, or memory, function:

K� t ��F� �0 �"F� � t �

�1

3��2� dq�

�2��3 q2C2�q ��S�q � s�q ,t � c�q ,t �. �1�

F� (t) is the total force exerted on a tagged particle by thesurrounding fluid at time t , and � is the inverse thermalenergy. The second line is a Fourier decomposition whichfollows from the MCT approximation of projecting forcesonto a bilinear product of single-particle and collective den-sity variables, coupled with factorization of four-point corre-lations into products of pair correlations.3,34 The MCT ap-proximations are not correct at early times, but are intendedto capture the slow dynamics associated with many particlecaging. C(q) is the direct correlation function, S(q) the di-mensionless collective structure factor, and � the fluid num-ber density.34 The product q2�CSC is an effective mean-square force exerted by the surrounding liquid on a taggedparticle. The ‘‘propagator’’ s(q ,t) � c(q ,t)� is the t�0normalized single-particle �collective� dynamic structure fac-tor which decays to zero at long times in the fluid phase.

The propagators do not decay to zero at long times in aglass. The localized state enters Eq. �1� via Debye–Wallerfactors describing arrested single-particle and collective den-sity fluctuations. Adopting a harmonic model of the amor-phous solid state yields22

s�q ,t→��e�q2/4�, �2�

corresponding to a real-space Einstein oscillator descriptionof the single-particle density:

��r� ��/��3/2e��r2. �3�


The variable r is the particle displacement from a randomlylocated position. The Gaussian form is consistent with ex-periments on colloids,24 simulations,15–18 and IMCT.3–6 Theparameter � is the local order parameter related to a mean-square displacement, or localization length, as

rLOC2 �3/2� . �4�

In the naive IMCT analysis22 a ‘‘Vineyard’’ approximation34

was invoked corresponding to replacing the collective den-sity fluctuation propagator by its single-particle analog. Asemphasized by the original workers,22 this simplification isrigorously valid for very high wave vectors, but not for dy-namic fluctuations on the length scale of the cage or smallerwave vectors.34 Use of the Vineyard approximation in Eq. �1�corresponds to neglecting vertex corrections and certain col-lective dynamical effects present in the full IMCT.22 In thehypothetical infinite-dimensional limit, the vertex correctionshave been shown22 to vanish and naive IMCT is equivalentto IMCT.

We suggest a modest improvement of naive MCT con-cerning the Vineyard approximation motivated by the factthat on the cage length scale, single-particle dynamics, col-lective density fluctuations, and stress relaxation are dynami-cally strongly coupled.3,24 However, static collective correla-tions are always present which we propose to approximatelyaccount for in a mean-field fashion as

c�q ,t→��e�q2/4�S(q). �5�

This form is motivated by the presence in MCT, and theexact short-time analysis of S(q ,t), of the classic de Gennesnarrowing effect which correlates the relaxation rate of den-sity fluctuations of wave vector q with the inverse of thestructure factor.3,34,35 For example, in the overdampedBrownian limit at short times, c(q ,t)�e�q2t/�sS(q), where�s is a short-time friction constant.35 Within the context ofMCT, c(q ,t→�) is the collective ‘‘nonergodicity param-eter.’’ Based on both experiment23 and many Brownian andNewtonian computer simulations,15–18 it is known the wavevector dependence is of a damped oscillatory form which isin phase with the static structure factor. Hence, the adoptionof Eq. �5� is expected to result in a quantitatively more real-istic description than prior work.22 It also correctly sup-presses the small wave vector contributions in the integrandof Eq. �1� which are not important at high volume fractions.

The long-time mean-square displacement obeys the self-consistent relation22

��1

2�2K� t→��

1

6 � dq�

�2��3 �q2C2�q �S�q �

�exp� �q2

4��1�S�1�q �� . �6�

At sufficiently low volume fractions ��3/6, only the��0 fluid solution exists. Based on Percus–Yevick �PY�theory34 for the static correlations, Eq. �6� predicts a local-ization transition at �MCT�0.432, lower than the full IMCTresult5 of �MCT�0.515. The localization length is rLOC



�0.19� , comparable to the classic Lindemann length34 andin excellent agreement with the full IMCT result5 of rLOC

�0.183� .If the 1/S(k) correction in Eq. �5� is ignored, we find

�MCT�0.464 and rLOC�0.204� . All our results are quanti-tatively, but not qualitatively, dependent on whether the1/S(k) correction is included in Eq. �5�. Since we believethere is a good theoretical reason for including it, we shall doso in all the numerical results and figures presented below.Selected results based on ignoring this correction are pre-sented to illustrate the quantitative sensitivities.

B. Beyond idealized mode-coupling theory

We adopt the common interpretation that the IMCT non-ergodicity transition signals the emergence of finite barriersin a free-energy landscape and strong transient localizationand reequilibration in a cage which requires activated dy-namics to escape.11–15 To proceed we seek a stochastic equa-tion of motion �EOM� for the single-particle dynamic scalarorder parameter r(t) or �(t)�3/2r2(t). The purpose of thisEOM is restricted to describing the localization process andescape over a barrier, in the spirit of the Langevin approachto chemical reactions,36,37 and some applications of the so-called model B for the slow dynamics of a nonconservedorder parameter.38

The explicit construction of the EOM is guided by threeconsiderations: �i� For Brownian colloids, short-time motionis overdamped Fickian diffusion. �ii� In the absence of ther-mal fluctuations or noise, IMCT is assumed to correctly pre-dict the tendency to localize in a cage. Hence, in the deter-ministic limit the EOM description should recover the naiveIMCT localization condition. This idea guides constructionof an effective force ��F/�r , which at high density favorslocalization. We shall refer to F as an ‘‘effective free-energy’’ function, although it does not have a rigorous equi-librium meaning. �iii� Ergodicity-restoring thermal fluctua-tions are included which destroy the naive IMCT glasstransition and allow for activated hopping when ��MCT .

To address points �i� and �ii� we adopt a scalar, deter-ministic �no noise or random forces�, nonlinear, and Markov-ian EOM for the order parameter:

d

dtr��

�

�rF , �7�

with the initial condition r(t�0)�0. Here, � is a mobilitycoefficient that describes short-time frictional processes andsets the time scale for a coarse-grained description of particletrapping and escape dynamics. The form of Eq. �7� corre-sponds to assuming that the rate of change of particle dis-placement is linearly proportional to a driving force. It is thesimplest dynamical model that reduces at ‘‘equilibrium’’�long-time steady-state limit, dr/dt→0) to the condition ofvanishing effective force in the spirit of a steepest-descenttrajectory. Equation �7� can also be written as a simple forcebalance

��s

d

dtr�

�

�rF�0. �8�


This emphasizes its Brownian character and the descriptionof short-time motion by a dissipative drag term or corre-sponding short-time diffusion constant Ds�kBT/�s . In col-loidal suspensions25,39,40 �s��0G , where the single-particleStokes–Einstein �0�3��0 with solvent viscosity �0 andfactor G�1 is discussed in Sec. IV.

The effective free-energy function should describe thedifference between localized �nonzero �� and delocalized��0� states in the spirit of DFT.22,33,38 This motivates thefollowing explicit form for F �in units of thermal energy�:

F��3

2ln�� dq�

�2��3 �C2�q �S�q ��1�S�1�q ��1

�exp� �q2

4��1�S�1�q ��

�F0�FI , �9�

where irrelevant additive constants have been omitted. Thefirst ‘‘ideal’’ term favors the fluid state and is identical towhat is employed in DFT for a harmonic solid.32,38 Since ourinterest is restricted to motions on the particle size andsmaller scale, the strong localization form of F0 is a reason-able approximation. The second ‘‘interaction’’ contributioncorresponds to an entropic trapping potential favoring local-ization. Minimization of Eq. �9� with respect to the orderparameter, or solution of Eq. �8�, yields the naive IMCTlocalization condition of Eq. �6�. This is the physically moti-vated constraint which guided construction of Eq. (9). For��MCT , a barrier and local minimum appear in F(r).

We emphasize that Eq. �9� is not an equilibrium or ex-tensive free-energy functional as employed in the thermody-namic DFT of the glass transition32,33 and recent dynamicDFT.19 These approaches adopt a conventional equilibriumform for the interaction free energy. The thermodynamictheory22,33 predicts a fluid–glass phase transition and implic-itly corresponds to effective free-energy barriers of macro-scopic magnitude.13,22 In reality, a ‘‘mosaic’’ structure is be-lieved to emerge below the IMCT transition corresponding tofinite clusters of particles existing as local minimum free-energy configurations which are separated by finitebarriers.11–13 Our approach adopts a simple single-particledescription of intensive free-energy barriers.41

In general, white noise �thermal fluctuations� due torapid processes and, possibly, inertial effects �particle massM ) are important �point �iii��. These are accounted for by astandard fluctuating random force resulting in an EOM of theLangevin form:36,37

��s

d

dtr�

�

�rF�� f �M

d2

dt2 r ,

�10�� f �0 �� f � t ��6��1�s�� t �.

The fluctuating force is statistically uncorrelated with thetagged particle position and velocity. Equations �9� and �10�define the EOM description and are reminiscent of Kramers’approach36 for chemical reactions and activated processes.However, F(r) is not an equilibrium potential-of-meanforce, but rather is ‘‘self-generated’’ via collective liquidstructure. The Markovian form implies that a literal solution



of Eq. �10� does not automatically recover the long-time dif-fusion constant since the forces which define the cage con-straints in F(r) ultimately transform to an additional viscousfriction. Description of the latter process requires a self-consistent, non-Markovian formulation which is beyond thescope of this initial study. This problem is the analog ofconstructing a time-dependent random potential in the phe-nomenological trap models.13,19

C. Limiting behaviors and diffusion length

At early times or small displacements the interactioncontribution in Eq. �9� is irrelevant, and solving Eq. �10�correctly yields Fickian diffusion in the overdamped limit ofinterest. This follows explicitly from Eq. �9� by noting thatfor small displacements �large �� the ideal contribution to theeffective force is dominant since

�F0 /�r��3/r , �FI /�r�r . �11�

Equation �10� then simplifies to

�s

d

dtr�

3

r�� f , �12�

the solution of which is r2(t)�6Dst .A ‘‘diffusion length’’ RD , is defined as the displacement

beyond which Fickian motion applies. Without explicitlysolving the EOM, the latter displacement is not unambigu-ously defined. We consider two ideas, which are shown be-low to result in nearly identical results. One natural conditionis irrelevance of the interaction component of the effectiveforce, which is equivalent to

�r

3

�

�rF→1, at r→RD . �13a�

This condition is attained at ‘‘large enough’’ displacementssince the interaction force �FI /�r→0 exponentially fast at adisplacement ��. Equation �13a� is precisely quantified byrequiring the ratio T��FI /�r�/��F0 /�r��0.01. Calculateddisplacements are weakly sensitive to the chosen tolerance;for example, if a threshold of T�0.001 is used, we find thatRD changes by only �5%–20% over a wide range of volumefractions.

As a second estimate, we consider the displacementwhere the ‘‘rate of change of the work,’’ � (d/dr) (r dF/dr)�,has decayed to a fraction Q of its maximum value:

� d

dr � rdF

dr � ��Q� d

dr � rdF

dr � �MAX

. �13b�

As true for calculations based on Eq. �13a�, we find thatreduction of the tolerance from Q�0.01 to 0.001 results inonly �15% – 20% changes in RD , regardless of whether thetotal free energy or just its interaction part is employed inEq. �13b�.

III. EFFECTIVE FREE ENERGY

Examples of the effective free-energy function based onPY static input are shown in Fig. 1. There are three basicforms:42 �a� At relatively low packing fractions and/or smalldisplacements, F(r) decays monotonically with particle dis-

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placement. �b� As �→�MCT , an inflection or bifurcationpoint emerges at r�0.19� . This is the ‘‘precursor’’ regime,although the mathematical nature and physical consequencesof the naive bifurcation point are different, and far moreprimitive, than its full IMCT analog. �c� For larger �, strongtrapping can occur with relatively large intrinsic entropicbarriers.

The focus of this initial paper is to establish the mostbasic aspects of the theory in regime �c� where ��MCT andthere is a clean separation of time and length scales. Numeri-cal solution of the stochastic EOM is postponed for futurework. Our philosophy is that solution of the EOM will in-volve trajectories which continuously sample different partsof F(r), and the dynamics is largely controlled by the char-acteristic length and energy scales in F(r). The latter state-ment is the central physical idea underlying our approach. Inthis spirit, one can identify four dynamical processes corre-sponding to the displacement regimes indicated schemati-cally in Fig. 1:

�1� Crossover from short time Fickian diffusion to theearly stages of localization.

�2� Transient �near� localization at a displacement corre-lated with the location of the free energy minimum. We de-fine the latter as rLOC , although literal localization at pre-cisely this displacement is not expected. This regime isrelevant to a crude description of localized vibrational states,perhaps related to the so-called ‘‘boson peak,’’ 1 althoughsuch an excitation has been argued to not exist in over-damped colloidal suspensions.26 Other important features ofF(r) are the harmonic spring constant K0 , the correspondingfrequency �0�(K0 /M )1/2, and anharmonic aspects within�kBT of the minimum.

�3� Initial stages of ‘‘cage escape’’ �below the barrier�which are driven by thermal fluctuations. This is a strongly

FIG. 1. Effective ‘‘free energy’’ �units of thermal energy� as a function ofdimensionless displacement for �from top to bottom� ��0.3, 0.432, 0.50,0.57, and 0.6. Five characteristic lengths, the barrier height, and four re-gimes of dynamical behavior �1�–�4� are defined in the inset for the case of��0.55.

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non-Gaussian process. We expect that the strongest deviationfrom a diffusive or Gaussian process will be correlated withthe location of the maximum restoring force defined to occurat R*. Another length scale of possible interest is the dis-placement of maximum work, Rmw , defined by maximizingthe product r�F/�r .

�4� Activated hopping over an entropic barrier of mag-nitude FB located at the ‘‘transition state’’ displacementrB . The characteristic hopping time � is influenced by thelocal friction constant, barrier height, and attempt frequen-cy and driving force for barrier crossing via the well andbarrier harmonic �positive� curvature constants K0 and�KB , respectively. The elastic or spring constants enterthe determination of the ‘‘attempt frequency’’ �0 and thedynamical driving force near the top of the barrier via �B

�(�KB /M )1/2. A hopping diffusion constant follows fromdimensional analysis:

DHOP�LD2 /6� ,

�14�LD�RD�rLOC ,

where a mean ‘‘jump length’’ LD is estimated as the differ-ence between the displacement beyond which no interactioneffects are present �computed from Eq. �13�� and rLOC .

Throughout Secs. III–V, the discussion of numerical re-sults and figures is based on calculations employing the deGennes narrowing factor in Eq. �5� unless explicitly statedotherwise.

A. Characteristic length scales

Figure 2 shows the five characteristic length scales �inunits of the particle diameter� as a function of colloid volumefraction. The solid curves include the 1/S(q) correction inEq. �5�. The location of the minimum of F(r) decreasesstrongly with packing fraction and is well described by anexponential law rLOC�30e�12.2�. The displacement ofmaximum restoring force decreases exponentially as R*�3.3e�6.6�, more weakly than the localization length. Thedisplacement of maximum work also follows an exponentiallaw Rmw�0.92e�3.65� and is larger, and more weakly vary-ing, than R*. The barrier location �‘‘transition state’’� in-creases weakly with volume fraction and approaches �/2 atlarge �. The latter value is intuitively reasonable since itcorresponds to a displacement distance which maximally dis-rupts the strong local cage order. The mean diffusive jumplength is almost independent of volume fraction with LD

�0.8, independent of which version of Eq. �13� is employed.Thus, the displacement beyond which Fickian diffusion ap-plies is nearly constant and less than a particle diameter, astypically found in computer simulations.17,18 This value isalso consistent with both our restricted interest in small-length-scale dynamics and the idea that Fickian diffusioncommences once a particle escapes from its local cage.

Also shown in Fig. 2 as dashed curves are the corre-sponding results based on assuming the collective nonergod-icity factor equals its incoherent single-particle analog, c(q)� s(q). Neglect of the de Gennes narrowing factorresults in systematically larger values for all length scales, asmight be expected given the larger value of �MCT . However,

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FIG. 2. Log-linear plot of characteristic lengths as a function of volumefraction �solid curves�. The jump diffusion length is shown based on Eq.�13a� for T�0.01 �dashed curve� and 0.001 �dash-dotted curve� and Eq.�13b� with Q�0.01. The analogous results for the smallest four lengthscales based on ignoring the de Gennes narrowing factor � c(q)� s(q)�are shown as dashed lines. The solid �Ref. 28�, open �Ref. 24�, and partiallysolid �Ref. 31� circles are experimental data for the apparent localizationlength. The solid �Ref. 28� and partially solid �Ref. 31� squares indicateexperimental displacements of maximum non-Gaussian behavior. The soliddiamonds are the ‘‘apparent cage diameter’’ defined in Ref. 28.

c

the qualitative behavior is identical. The modest quantitativedifferences depend on property and vary from �3% to 35%.Except for Rmw , the differences decrease with increasingvolume fraction.

B. Experimental measurements of length scales

Incoherent scattering experiments and confocal micros-copy analyses of colloid trajectories28–31 have attempted toidentify important lengths in the transient localization andearly stage cage escape processes. We do not expect a quan-titative correspondence between the lengths deduced fromdynamic experiments and the characteristic length scales ofF(r). However, it is of interest to explore possible qualita-tive or semiquantitative correlations.

The dynamic mean-square displacement �MSD� in sev-eral different model hard-sphere suspensions has beenmeasured.24,28,31 The confocal microscopy experiments28–31

clearly find the apparent �transient� localization length is adecreasing function of volume fraction over the range��0.52–0.64. To be more specific, we first consider the ex-periments of Weeks and Weitz28 �WW� on ��2.36 �m col-loids. Perfectly horizontal �transient� plateaus in the MSDare never seen. Thus, we deduce a mean localization lengthas the displacement where the time dependence of the MSDpasses through an inflection point as quantified by when thenon-Fickian anomalous diffusion exponent attains aminimum.28 Experimental results are shown in Fig. 2 for twofluid states, and rather good agreement is found with thecalculated rLOC . An important caveat is that the ‘‘hard



spheres’’ studied in Ref. 28 carry a small amount of charge.This feature modifies the equilibrium fluid–crystal transitionand also introduces uncertainty in the precise determinationof suspension volume fraction.

The WW experiments28 have also measured a ‘‘cage re-arrangement time,’’ and corresponding displacement �‘‘cagesize’’�, deduced from the location of a maximum non-Gaussian parameter. Results are shown in Fig. 2 for two fluidsamples, and surprisingly good agreement with the calcu-lated R* is found. The experimental ratio R*/rLOC�1.8�0.2, compared with the theoretical result of �2.3�0.2. Analternative measure of ‘‘cage size’’ rcage has also been de-duced from single-particle trajectories as the displacementbeyond which a linear restoring force behavior fails.28 Thislength is shown in Fig. 2 for two fluid samples and is largerthan R*. Curiously, the experimental values of rcage are closeto the calculated ‘‘displacement of maximum work,’’ Rmw ,although a fundamental theoretical reason for this is notobvious.

The incoherent scattering measurements of van Megenand co-workers24 �VM� probe single-particle motion of a col-loidal system very close to the hard-sphere model ��400nm�. The time-dependent MSD was extracted. Since mea-surements of the non-Gaussian parameter were not reported,extraction of a localization length as described for the WWexperiments28,29 cannot be done. However, at a glassy vol-ume fraction of ��0.583 the plateau in the MSD is suffi-ciently flat to allow a direct estimate. The result is shown inFig. 2 and differs from the value of WW �Refs. 28 and 29�and the calculated rLOC .

Another set of confocal microscopy experiments in-volves ��900 nm cross-linked microgel fluids.27,31 Thesemodel hard spheres do not undergo a fluid–crystal transition,and calibration of the absolute magnitude of � is more un-certain. As in the WW study,28 a mean localization lengthand displacement of maximum non-Gaussian behavior canbe determined �see Fig. 2�. The localization length is roughlyconsistent with the VM experiments, while the experimentalR* is significantly larger than measured by WW. The experi-mental ratio R*/rLOC�1.8 in agreement with WW.

C. Vibrational features

The ‘‘vibrational’’ features of F(r) are summarized inthe inset of Fig. 3. The harmonic well curvature �in units ofkBT/�2) increases rapidly with packing fraction and is wellfit by an exponential law K0�10�3 e25.3�. A strong correla-tion with the ‘‘localization length’’ is found. The productK0rLOC

2 is nearly constant, and roughly equal to 1.4. Calcu-lations which ignore the de Gennes narrowing factor areshown as dashed curves. The elastic constant K0 is smaller,by a factor of �1.2–2 over the range ��0.52–0.62, withdeviations that decrease with volume fraction.

The harmonic well curvature also enters in the determi-nation of a characteristic oscillation frequency �0

�(K0 /M )1/2. For typical colloids such as polymethyl-methacrylate or silica the density �1 – 2 grams/cm3, andat ��0.58 a frequency in the range �0�105 – 1010 Hz ispredicted for diameters varying from 1 �m to 10 nm, respec-


FIG. 3. Mean-square vibrational amplitude about the F(r) minimum �solidcurve� and its harmonic analog �dashed curve�. Inset shows the dimension-less curvature constants �solid curves�; the indicated product �multiplied by20� is nearly independent of �. The analogous results for the curvatureconstants which ignore the de Gennes narrowing factor are shown as thedashed curves �with the indicated product multiplied by 20�.

c

tively. For a sphere of diameter �1 nm, a value relevantto small molecule glass formers, a frequency of �0/2��6�1011 Hz�20 cm�1 is obtained. Interestingly, this valuefalls in the typical frequency range of the so-called bosonpeak observed in the glassy state of small-moleculesystems43 such as orthoterphenyl and propylene. The pos-sible significance of this result is enhanced by theobservation44 for small-molecule organic glasses that thecharacteristic boson frequency scales as M �1/2, as in oursingle-particle description.

The mean-square fluctuation about the minimum, u2 ,has been computed using standard Boltzmann statistics. Re-sults are shown in Fig. 3 and are compared with the har-monic analog u20�3/K0 . Harmonic behavior is wellobeyed at very high �, but below ��0.61–0.62 increasinglystrong anharmonic effects are present resulting in larger fluc-tuations �‘‘softer vibrations’’�. We find that the deviation isempirically well described by a critical power law

u2�u20�225�0.64��3.7.

The trend of increasing anharmonicity as the glass transitionis approached from the solid state has been observed inquasielastic neutron scattering experiments on thermalglasses.1 Large deviations from harmonic behavior emerge asa precursor of glass melting, which �surprisingly� appear tobe closely related to long-time transport concepts such askinetic fragility.1 We are not aware of such measurementsbeing performed on hard-sphere colloidal suspensions.

The absolute magnitude of the barrier curvature is muchsmaller than its well analog �Fig. 3�. It is also weakly depen-dent on volume fraction, changing by only a factor of �2over the entire range studied. Calculations which ignore the



FIG. 4. Log-linear plot of the relative displacement probability as a functionof displacement normalized by the location of minimum of F(r). Resultsfor five values of suspension volume fraction are shown.

de Gennes narrowing factor � c(q)� s(q)� are shown asthe dashed curves, and again differences of only a factor of 2or smaller are found.

Figure 4 shows calculations of a probability distributionof displacements based on Boltzmann statistics, P(r)�exp�F(rLOC)�F(r)� , as a function of displacement. Nor-malization to the localization length rLOC �well minimum� isadopted. Only displacements below the barrier location areof interest, and the curves in Fig. 4 are terminated at thispoint. Our motivation for computing this quantity is twofold.First, for the quasilocalized dynamical regime P(r) may beclosely related to a ‘‘distribution of localization lengths.’’Second, confocal microscopy experiments have mea-sured27–30 the probability distribution of particle displace-ments in time windows where the colloids are nearly local-ized in a cage or in the early stages of escape. The experi-ments find non-Gaussian distributions, with long tails in thelarge displacement direction which roughly follow an expo-nential �not Gaussian� law.

We do not expect a rigorous correspondence betweenour calculations and the dynamic measurements. However,we speculate that qualitative agreement can occur if the dy-namics are largely controlled by a suitably defined effectivefree-energy function. The results in Fig. 4 are very similar tothe experimental findings reported in Fig. 3 of Ref. 28, es-pecially for ��0.56. For example, �i� the strong asymmetrytowards the large displacement direction roughly follows anexponential law, �ii� the distribution functions for differentvolume fractions nearly collapse down to the �1%–10%relative probability level, and �iii� the relatively large 1%probability of a particle displacement �5 times the mostprobable value.


D. Elastic shear modulus

Rigorous calculation of the elastic shear stress modulusG0 is not possible based on our single-particle approach.However, it can be estimated using simple ideas as recentlyapplied to interpret diffusing wave spectroscopy experimentson colloidal suspensions.45 Within a homogeneous con-tinuum elasticity description, which is consistent with theEinstein oscillator picture of the amorphous solid state, onehas45

G0�2kBT

��rLOC2 �

K0

�. �15�

The second equality is often employed in the analysis offractal networks.46 For the present problem, it is nearly anidentity since Fig. 3 shows K0rLOC

2 �kBT��1, correspondingto the very close connection between a mean localizationlength and the �harmonic� distribution of localization lengthsdiscussed in Sec. III A. For a 1 �m diameter colloid at roomtemperature one obtains G0�4�10�6e25.3� Pa. This corre-sponds to a modulus of �10 �1.5� Pa at ��0.58 �0.51�.Experimental measurements of the elastic shear modulus ofwell-characterized, nearly monodisperse glassy hard-spheresuspensions where � has been accurately determined by cali-bration with the fluid–crystal transition do not seem to exist.However, the predicted order of magnitude is reasonablecompared with the little data that are available.47,48 For a1 nm diameter sphere, crudely representative of molecularliquids, a shear modulus of �109 – 1010 Pa is predicted,which is in agreement with the typical experimental range.34

E. Entropic barrier

Calculations of the entropic barrier �in units of kBT) arepresented in Fig. 5. The barriers are low (FB�1) up to��0.49, followed by a rapid increase reaching a value of �8at ��0.58. The analogous results based on ignoring the deGennes narrowing factor are shown in the inset and arequalitatively identical. Quantitative differences of �2%–30% are found for ��0.52, which become monotonicallysmaller with increasing volume fraction. As the maximumpacking fraction is approached, we expect the barrier heightdiverges. Of course, based on PY theory input this does nothappen. Thus as �→�MAX our calculations may increas-ingly underestimate the barrier height. Lack of knowledge ofthe required correlation functions precludes quantitative ac-cessment.

Two interesting connections between the entropic barrierheight and other properties have been discovered. First, FB islinearly correlated with a thermodynamic property: the in-verse dimensionless isothermal compressibility,34 S(q�0)�(1��)4/(1�2�)2, which quantifies the amplitude ofstatic density fluctuations. This is a somewhat surprising andnontrivial result since the theoretical input involves stronglywave-vector-dependent correlation functions with localstructural information on the q*�7/� cage scale making thedominant contribution to the integrals in Eq. �9�. Hence,some type of ‘‘self-averaging’’ of the local cage structure issuggested.



The barrier height is also strongly correlated with twofeatures of the effective free-energy minimum: FB�K0

1/2

�1/rLOC . Since �as discussed below� ��exp(FB), the firstrelation connects the long-time barrier hopping process andthe localized-state oscillation �attempt� frequency as is ex-plicitly demonstrated in Fig. 5. This is reminiscent of therecent connections found experimentally in thermal glassformers between short-time vibration dynamics �boson peak�and the � relaxation flow process.1 The relation FB

�1/rLOC�1/�u20 also follows from Figs. 3 and 4. Neu-tron scattering measurements49 on thermal glass formershave found an empirical correlation between the viscosity or� relaxation time and the inverse of the mean-square vibra-tion amplitude ��exp(b/u2). An analogous relation hasbeen theoretically suggested50 and observed in molecular dy-namics simulations for a specific model system.51 A similarrelation is found from our approach although the power towhich u2 is raised differs.

IV. BARRIER HOPPING TIME

The mean barrier hopping time � is expected to beclosely correlated with the � relaxation time. In this sectionwe present model calculations for � and a comparison withexperimental measurements of a single-particle relaxationtime.

A. Model calculations

The conditions for validity of the Kramers first-passagetime analysis are FB�1 and Markovian local friction,36,37 forwhich

FIG. 5. Entropic barrier height �solid circles� as a function of volume frac-tion. Linear fits to the inverse compressibility �solid curve, 0.08S�1(0;�)�3.51] and the well elastic constant �dashed curve, 0.216K0

1/2�2.437) areshown. The inset compares the barrier height with �solid curve� and without�dashed curve� the de Gennes narrowing factor.


��1��0

2� ��s

2M�B�� s

2M�B� 2

�1�e�FB. �16�

Kramers theory ignores the possible importance of correlatedthermal force fluctuations and lack of a time-scale separationbetween single-particle displacements and collective cagedynamics �non-Markovian effect�. However, within this sim-plified picture Eq. �16� does apply for general degrees of therelative importance of diffusive versus inertial motion nearthe barrier top. For a concentrated colloidal suspension withthe broad barriers found in Sec. III B, the high friction limitis expected to apply corresponding to diffusive motion overthe barrier. Formally, this requires the inequality

�s/2�MKB�1. �17�

In this limit, Eq. �16� simplifies to

��1��K0KB

2��se�FB. �18�

Note that the particle mass drops out. To explicitly evaluatethe applicability of Eq. �17�, it is rewritten using �0

�3��0 �solvent viscosity �0), �s��0G with G�1, KB

�K̃BkBT��2, M��m��3/6, where �m is the colloid massdensity. The result is

�s/2�MKB�3�G��0

2�K̃BkBT��m�/6�1. �19�

Using the numerical results for KB from Sec. II B, ��1 �m,a typical solvent viscosity of 10�2 P, T�300 K, a typicalcolloid mass density of 1 – 2 g/cm3, and G�5 – 10 at highvolume fractions �see below�, yields a value for this ratio of�105. Even for particles of diameter �1–10 nm the ratio islarge.

The hopping time of Eq. �18�, nondimensionalized bythe elementary Brownian diffusion time �0��2�0 /kBT , isthus given by

�

�0�

2�G

�K̃0K̃B

eFB. �20�

The quantity G represents the short-time dynamical correc-tion to the single-sphere Stokes–Einstein friction factor.There are alternative theories which ascribe the primary ori-gin of this correction as either independent binarycollisions39 �IBCs� or solvent-mediated high-frequency hy-drodynamic interactions associated with two-particle lubrica-tion forces.25,40 In practice, the different approaches yieldnearly identical results. For the IBC or Enskog approach, Gis the contact value of the hard-sphere radial distributionfunction.34,39 An accurate analytic expression for g(�) isgiven by the Carnahan–Starling formula34 below ��0.5 andsmoothly crosses over to a different form at higher packingfractions which diverges at random close packing:52

g��1��/2

�1��3 ,

�21�

g��0.84

0.64��, ��0.5.



FIG. 6. Log-linear plot of the dimensionless barrier hopping time �solidcurve� and incoherent relaxation time �short-dashed curve�. Analogous re-sults which ignore the de Gennes narrowing factor are shown as the longdashed and dash-dotted curves, respectively. The end points of the horizon-tal bars indicate the experimental data �Ref. 24� which includes the uncer-tainties in absolute volume fraction discussed in footnote 22 of Ref. 24.

In the hydrodynamic approach, G is the ratio of the high-frequency viscosity to the solvent viscosity. An accurate in-terpolation formula for this ratio is given by40

��

�0�

1�1.5��1��1��2.3�2�

1��1��1��2.3�2�, ��0.56�

��15.78 ln�1�1.16�1/3��42.47, ��0.6�. �22�

To within a factor of less than 2, G is insensitive to which ofthese expressions are used for all volume fractions up to��0.62. Equation �21� is employed in our calculations.

Results for the dimensionless barrier hopping time arepresented in Fig. 6. Interestingly, the prefactor of the expo-nential term in Eq. �20� is found to change by only a factorof �2 over the range of � ��0.5–0.62� where � increases by�1.5�106. This is a nontrivial result since the well andbarrier curvatures and local friction constant in Eq. �20� varysignificantly with �. Apparently, for hard-sphere suspensionsthe increase of friction with � is nearly perfectly compen-sated by the increase in attempt frequency. Calculationswhich ignore the de Gennes narrowing factor � c(q)� s(q)� are also shown and differ by only �3% – 35%over the entire range of volume fractions of interest.

B. Comparison with experiment

The mean barrier hopping time is of fundamental inter-est, but its quantitative relation to experiment is propertyspecific.1 For example, an � relaxation time can be deducedfrom macroscopic (q�0) stress or dielectric relaxation mea-surements. Alternatively, length-scale-dependent probes of


single-particle, F(q ,t), or collective, S(q ,t), density fluctua-tions can be employed to extract a wave-vector-dependentrelaxation time.

The kinetic glass transition depends on experimentalmethod. Typically �expt�102 – 104 s is the operationaldefinition.1 Nonuniversal short-time dynamics enters via �0

which for colloidal suspensions can vary significantly de-pending on particle diameter and solvent viscosity. This vari-ability implies a kinetically defined glass transition volumefraction does depend on particle size and other factors whichinfluence short-time dynamics. Reduction of particle sizeand/or solvent viscosity results in an increase in the predictedvolume fraction of the hard-sphere kinetic glass transition.This aspect is in contrast with the IMCT scenario3 based oninterpretation that �g��MCT , but is in accordance with theexperimental behavior of thermal glass formers where it in-fluences the dynamic ‘‘fragility.’’ 1 We are unaware of ex-periments or simulations for colloidal suspensions which ad-equately address this point. Uncharged hard-sphereexperiments23–25 where the volume fraction has been quan-titatively calibrated against the fluid–crystal transition haveemployed colloids of nearly identical diameters of ��350–400 nm.

To make quantitative comparisons with experiment weconsider the incoherent scattering measurements of Ref. 24where ��400 nm, �0�0.5 s, and the maximum observationtime �104 s. The most extensive results were obtained for awave vector of q��2.6, corresponding to a length scale2�/q�2.42� . Since the latter is significantly larger than theparticle diameter, nearly pure Fickian diffusion is expected.This is consistent with the observations of a Gaussian wavevector dependence and nearly single-exponential decay ofF(q ,t) at this wave vector. A slow, single-particle relaxationtime � inc was measured. The product of the self-diffusionconstant and relaxation time, D� inc , was found to be nearlyconstant over the range ��0.47–0.58, with the dimension-less ratio D� inc /(D0�0)�0.38�0.05. Interestingly, the mea-sured single-particle relaxation time and the wave-vector-independent collective density fluctuation relaxation time onthe cage length scales q��4.5– 9 are numerically veryclose, over �3 orders of magnitude in time variation.23,24

This again suggests the central role played by single-particlemotion on the cage scale.

The kinetic glass transition defined as when � inc�104 swas experimentally determined24 to occur at �g�0.573 �or�0.59; see footnote 22 in Ref. 24�. However, motion wasstill detected beyond this volume fraction, as indicated byupturns �downturns� in the dynamic mean-square displace-ment �incoherent structure factor�. Hence, the reported valueof �g does not appear to represent a literal nonergodicity orlocalization transition. Rather, in analogy with the behaviorof thermal glass formers, it may indicate when the system‘‘falls out of equilibrium’’ in the sense that the characteristicdynamic time scale exceeds the experimental time scale.1

To make contact with this experiment we employ Fick’slaw and Eq. �14� to compute the relevant relaxation time:



� inc��2�/q �2

6DHOP, �23�

with 2�/q�2.42� . Adopting the experimental definition ofthe glass transition of � inc�104 s, the results in Fig. 6 predict�g�0.579. For multiple reasons the very good agreementwith experiment may be fortuitous. However, we do believethe agreement is significant since all prior attempts �e.g,IMCT, free-volume model� to confront experiment require anempirical rescaling of � and/or fitting of a hypotheticalmaximum volume fraction where there is a dynamicalsingularity.1,3–5,53 Calculations which ignore the de Gennesnarrowing factor yield �g�0.582.

The experimental relaxation times24 as a function of �are also shown in Fig. 6. Good agreement with the calcula-tions is found, although some tendency for the theory to varymore slowly with volume fraction than experiment seemsapparent at the highest colloid concentrations studied. Thehorizontal error bars are due to the mixture nature of thesample which is required to perform the incoherent scatter-ing measurement. This aspect introduces uncertainties in thedetermination of the absolute magnitude of � using thefluid–crystal phase transition as an internal calibration �seefootnote 22 in Ref. 24�. Beyond the kinetic glass transition,particle motion has been detected experimentally,24,28–31 andthe higher-� theoretical results serve as testable predictions.

Other experiments on weakly charged ��2 �m suspen-sions28,29 and ��900 nm cross-linked microgel fluids27,31

have reported kinetic glass transitions in the range �g

�0.56–0.6. As discussed in Sec. III B, calibration of the ab-solute magnitude of � is more uncertain for these systems.Moreover, the kinetic �g is determined in Refs. 28 and 29 bywhen the time-dependent non-Gaussian parameter undergoesa dramatic change, while the experiments of Refs. 27 and 31employ a different kinetic glass transition criterion, and bothdiffer from the � inc�104 s definition of van Megen et al.24

These differences render impossible any definitive evaluationof the existence of a nonuniversal dependence of the kinetic�g on colloid size. Variation of the colloid diameter by afactor of �5 results in changes in the elementary Browniantime of �102. From Fig. 6 this implies a variability of�0.02–0.03 in the predicted kinetic �g .

All three experimental studies24,27,28,31 detect significantmotion, and perhaps long-time diffusion, beyond the esti-mated kinetic glass transition volume fraction. This feature isconsistent with the absence of any zero-mobility solid statein our approach.

V. COMPARISON WITH CRITICAL SCALING,FREE VOLUME, AND ADAMS–GIBBS APPROACHES

In this section we treat our theoretical results as ‘‘data’’and explore the fitting accuracy of both classic empiricaland, more recently proposed, functional forms. Historically,the search for an understanding of glassy dynamics has beenintimately connected with the ability of empirical functionsto fit relaxation time or viscosity data.1 The fit functionsgenerally involve at least three adjustable parameters, oftenof unclear physical meaning. They almost all invoke under-lying divergences, which greatly increases their flexibility in


providing adequate fits over limited ranges of controlvariables.1 The plotting format adopted for making compari-sons with experiment can significantly change the relativeweight of different dynamical regimes and can strongly in-fluence the conclusions drawn. The recent analyses of Stickeland co-workers of relaxation in thermal glass-forming liq-uids clearly documents such vagaries.10 We employ their‘‘derivative analysis’’ to more critically test the adequacy ofmodel fit functions. The lack of ‘‘noise’’ in the theoreticalresults enhances the usefulness of this procedure. The resultsgiven in this section employ the de Gennes narrowing cor-rection in Eq. �5�; all conclusions and results are virtuallyidentical if this correction is ignored.

A. Critical power laws

Critical power-law behavior of the relaxation time hasbeen proposed in various very different contexts. These in-clude the microscopic IMCT �Refs. 3–5� and phenomeno-logical approaches such as the critical slowing down ideas ofSouletie54 and a cluster percolation model of Colby.55 For acolloidal suspension, all these approaches would claim that‘‘close enough’’ to the glass transition the dimensionless re-laxation time scales as

�

�0�A��c��, �24�

where A is a numerical factor, � is a critical exponent, and�c is a critical volume fraction.

For hard spheres IMCT predicts5 ��2.6 and �c�0.52.The accuracy of this critical power law can be tested in sev-eral ways. In Fig. 7 the relaxation time raised to the 1/2.6

FIG. 7. Dimensionless relaxation rate raised to the inverse of the criticalexponent � as a function of volume fraction for ��2.6 �solid circles� and��5.46 �squares�. The lines drawn are linear fits to the lower volume frac-tion region of prime experimental relevance and extrapolate to an apparentnonergodic state at ��0.570 ��2.6� and 0.616 ��5.46�. The inset showsthe volume fraction dependence of the effective exponent obtained by ap-plying the Stickel derivative analysis to the theoretical results.



power is plotted. This is a format for which linear behavior isexpected, which is accurately found in the interval ��0.5–0.56. The apparent ideal glass volume fraction deduced bylinear extrapolation is �c�0.57, very close to the experi-mental analysis.24 At higher volume fractions the theoreticalresults strongly deviate from linearity, which is interpretedwithin the IMCT scenario as signaling the emergence of ac-tivated processes. Of course, the present calculations arebased on a theory where transport is controlled by activatedprocesses for all the volume fractions shown. The apparentlinear regime is associated with hopping over modest barri-ers �roughly �6kBT), and deviations from linearity continu-ously emerge with increasing barrier heights. Thus, thephysical interpretation of the shape of the relaxation plot inFig. 7 is fundamentally different than offered by IMCT. Ex-periments on suspensions composed of cross-linked networkparticles find a plot of the same shape as Fig. 7, includingsignificant long-time diffusion beyond �MCT which does notfollow a critical power law.27 However, the deformability ofthese model hard spheres may introduce complications.

One can adopt a more empirical approach and treat � asan adjustable parameter to optimize the fit. An example ofthis procedure is shown in Fig. 7. The choice of a largercritical exponent of ��5.46 �and corresponding larger appar-ent �c�0.616) results in a quite accurate fit up to volumefractions of �0.60.

An unbiased test of the appropriateness of a criticalpower law follows from the Stickel derivative analysis.10 Theproper logarithmic derivative of the relaxation time shouldresult in a constant equal to the inverse apparent scaling ex-ponent:

�d

d� � d

d�ln��/�0� � �1

��1. �25�

The inset in Fig. 7 shows the effective critical exponent ob-tained from applying Eq. �25� is not constant and increasesmonotonically with volume fraction. One could argue thelack of accuracy of a fit which employs the IMCT criticalexponent is not unexpected since it describes precursor dy-namical phenomena and our calculations involve activatedhopping. But we do find the apparent exponent equals theIMCT value of ��2.6 at ��0.51, which, remarkably, is veryclose to the a priori IMCT prediction5 of �c�0.515 basedon PY input.

We have also considered Eq. �24� with a critical volumefraction being the random close packing- �RCP-� like valueof 0.64 as suggested by others.26,32,40 Figure 8 shows theoptimized fit occurs for a critical exponent of ��8.2. Curi-ously this much higher exponent is close to what has beenfound ��9� from analyzing polymer melt relaxation data.55

Globally, a better representation of the theoretical results isobtained with the literal hard sphere IMCT form, althoughthe inset of Fig. 7 shows no critical power law is truly anaccurate representation. This conclusion is inevitable sinceour theoretical results do not contain a critical divergence.Minor variation of the assumed �c �e.g., 0.62, 0.63� does notchange our conclusions although the best-fit exponent canchange by 10%–20%.


B. Free-volume forms

The classic free-volume theory, or Williams–Landel–Ferry �WLF� expression, for the relaxation time also in-volves three adjustable parameters and is of an essential sin-gularity form1,40

�

�0�A exp� B�x�c

�c�� , x�0,1, �26�

where A and B are constants. The common choice is x�0,although some authors employ x�1. The value of x makeslittle difference due to the dominance of the essential singu-larity. Figure 9 shows the best fit to our results with x�0 and�c�0.64. Rather good agreement is found over roughly fourorders of magnitude variation in time which is comparableto, or exceeds, the entire range of volume fractions typicallyprobed experimentally.19,24,32,40 Our results remain virtuallyunchanged if x�1 is employed in Eq. �26�.

Systematic deviations between our calculations and theWLF formula are present in Fig. 9. The differences are quan-tified in the inset by applying the Stickel derivative method10

appropriate for Eq. �26� which states the following quantityis a linear function of volume fraction if the WLF expressionis exact:

�� d

d�ln��/�0� � �1

——→x�0

��c��B�1/2. �27�

Significant deviations are present at low and high volumefractions. The deviations are in the direction of weaker thanWLF dependence. The latter trend is qualitatively identicalto that found in recent studies of polymer melts56 where thetendency at high and low �below Tg but in equilibrium� tem-peratures is towards an Arrenhius law. For colloids, the ana-

FIG. 8. Log-linear plot of the dimensionless relaxation time �solid circles�as a function of volume fraction. The two fits of Fig. 7 are reproduced in thisformat as the dashed ��2.6� and dash-dotted ��5.46� curves. The solidcurve is the best critical power law fit based on a critical volume fraction of0.64; the corresponding exponent is ��8.23.




log of inverse temperature is volume fraction. The calcula-tions in Fig. 9 show that above the kinetic glass transitionvolume fraction the logarithm of the predicted relaxationtime is reasonably linear over roughly three orders of mag-nitude in time, which is the hard-sphere suspension analog ofan Arrenhius dependence.

C. Adams–Gibbs forms

The classic Adams–Gibbs �AG� approach1 argues therelaxation time is exponentially related to a barrier whichvaries inversely with a ‘‘configurational entropy,’’

�

�0�A exp� C

T�Sc� , �28�

where A and C are numerical factors. Considerable disagree-ment surrounds how the ‘‘configurational entropy’’ �Sc

should be defined.1 We consider two different choices em-ployed in the literature. For hard spheres, the free energy isof a purely entropic origin.

The classic idea of Kauzmann is that the configurationalentropy is the difference between the fluid and crystalentropies.1 Following Ref. 2, this is given in units of kB by

�Sc�Sfluid�Scrystal

�ln��e3/6��ln��2�1��1��1��2�2.8776

�2.3033 ln�z ��3 ln�1�z ��0.10463 ln�z�0.601�,

�29�

FIG. 9. Log-linear plot of the dimensionless hopping time �circles� as afunction of volume fraction. The solid curve is a WLF fit based on thechoice �c�0.64: ln(�/�0)�0.65/(0.64��)�3.718. The dashed curve is alinear fit �simple Arrenhius� for volume fractions starting just above thekinetic glass transition: ln �/�0�109.5��61.2. The inset shows the deriva-tive Stickel analysis quantity which should be a linear function of volumefraction if the WLF form is exact. The line through the data �circles� hasbeen optimized in the volume fraction region relevant to experiments and isgiven by �sqrt�d(ln �/�0)/d��1��0.78��0.54.

where z is a volume-fraction-dependent function chosen suchthat the crystal is at the same pressure as the fluid. Analyticequations for the fluid and crystal pressures required to en-force this constraint are given in Ref. 2. Note that �Sc→0 at��0.635.

An alternative approach is to assume that �Sc is thedifference in entropy between the fluid and some ultrahigh-volume-fraction fluid reference state.57 For hard-sphere sus-pensions, one ‘‘natural’’ choice is the maximally jammedstate. Without judging its correctness, we adopt a RCP-likevalue of �MAX�0.64 as the reference state. The relevantentropy is chosen to follow from Eq. �9� which controls the‘‘localization driving force’’ in our kinetic description.Hence,

T�Sc�Fm��Fm��MAX�, �30�

where Fm is the minimum value of the ‘‘effective’’ �nonequi-librium� free energy of Eq. �9�.

The results of fitting Eq. �28� with Eq. �29� or �30� to ourtheoretical data are shown in Fig. 10, and the entropy differ-ences are plotted in the inset. The volume fraction depen-dence of the latter are surprisingly similiar. Adjustment ofthe constant C in Eq. �28� has been performed to optimizethe fit in the experimentally relevant range of ��0.50–0.58.Reasonably good fits are obtained over this range where therelaxation time increases by �3 orders of magnitude. The fitbased on Eq. �30� performs slightly better, but only at lowervalues of � where the appropriateness of Eq. �28� �and ourcalculation of the hopping time� is most questionable. Bothfits fail badly as volume fraction increases above �0.58,

FIG. 10. Fits of the Adams–Gibbs form to the theoretical relaxation timeresults based on two choices for the configurational entropy. The solid�dashed� curve employs Eq. �30� �Eq. �29��. The best fit parameters inEq. �28� are A�exp(�11.31) and C�240.14 based on Eq. �30� and A�exp(�10.93) and C�20.3 based on Eq. �29�. The inset shows the corre-sponding entropy difference functions based on Eq. �29� �dashed curve,multiplied by 12� and Eq. �30� �solid curve�.



even though the divergences of both expressions are at��0.63. The quality of the AG fits are nearly identical tothat of the WLF results of Fig. 9.

VI. SUMMARY AND DISCUSSION

We have presented a simple kinetic approach for el-ementary aspects of slow single-particle dynamics whichcombines elements of naive idealized mode-couplingtheory,22 density functional theory, and activated rate theory.The primary new idea is that the slow dynamics are stronglyinfluenced by an underlying ‘‘effective �nonequilibrium� freeenergy’’ which is determined by the dynamical consequencesof structural cage correlations. Quantification of this ideaemploys the naive version of IMCT to construct the effectivefree energy which describes, within a Langevin equation-of-motion framework, both the driving force favoring particlelocalization and entropic barriers. Thermal fluctuations areincluded via a random force which destroys the �naive�IMCT glass transition and restores ergodicity via barrier hop-ping. We do not claim any rigorous derivation, but the theoryis predictive and contains no adjustable parameters or postu-lated dynamic or thermodynamic divergences. The basicspirit of our approach might also be viewed as a microscopicquantification of ‘‘trap models’’ 13,19 which describe the cag-ing process in a highly coarse-grained fashion via a staticrandom potential which is modeled in various phenomeno-logical manners.

Good agreement of our theory with the experimental ki-netic glass transition volume fraction of hard-sphere suspen-sions is found. Within the uncertainties of the variousexperiments,24,27–31 reasonable agreement is found for themagnitude and volume fraction dependences of the single-particle relaxation time, apparent localization length, andlength scale of maximum non-Gaussian behavior. The basicphysical picture differs from IMCT since the slow dynamicsand transient localization arise from the emergence of en-tropic barriers and activated transport. Experiments whichmeasure the relaxation time and other transport properties athigher volume fractions, and possible dependences on non-universal system parameters such as hard-sphere diameterand solvent viscosity, will provide further tests of the theory.Further experiments are also required to clearly distinguishthe predictions of our theory from fits of IMCT to hard-sphere suspension experiments. Multiple connections arealso predicted between thermodynamics, short-time dynam-ics in the temporarily localized state, and long-time barriercrossing. Many of these connections seem qualitatively simi-lar to puzzling observations made for thermal glass-formingpolymer and molecular liquids.1

It is worth noting that the present approach does notinvoke concepts such as an underlying thermodynamic phasetransition, entropy catastrophe, kinetic singularities, free vol-ume, percolation, frustration, dynamic heterogeneity, coop-eratively rearranging regions, or other special collectivemotions.1 Of course, we are not claiming that these conceptsare irrelevant since the single-particle nature and restrictedfocus of our theory precludes rigorously drawing such a con-clusion. But the importance of such ‘‘highly many-body’’aspects on the observables we have studied is not obvious.


We are also not suggesting our naive IMCT based approachis generically ‘‘better’’ than the full idealized mode-couplingtheory. Indeed it is not, and IMCT appears to be a powerfultool for understanding precursor dynamical phenomenawhich our approach is not meant to accurately address.

Quantitative comparisons of our theoretical results forthe relaxation time with vastly different fit formulas havebeen carried out. We find that the � dependence of � can befit rather well over limited ranges by many theoretically mo-tivated functions which all have ‘‘idealized divergences’’ andthree adjustable parameters. For hard-sphere colloids, wheredynamic light scattering and shear viscosity measurementsare limited to rather modest volume fractions of �0.57 orless, there is not much to choose among the various func-tions. Based on fit quality, the empirical critical power lawwith a hypothetical divergence at �or near� random closepacking and a large critical exponent appears to provide thebest global description of our theoretical results. However,the derivative Stickel analysis shows that all the consideredforms are not fundamental over a wide range in �, an un-avoidable conclusion since our results do not contain anysingular features. On the other hand, the ability of the fitfunctions to reproduce the theoretical trends is strongly con-tingent on building in an apparent singular state. As empha-sized by Stillinger and co-workers2 and others,56,58 this raisesquestions about the existence and/or practical relevance oftrue divergences �equilibrium or dynamic� in the glass tran-sition problem.

By design the theory has invoked multiple simplifica-tions. For colloidal suspensions many-body hydrodynamicinteractions remain poorly understood and have not been ex-plicitly taken into account. However, it is widely believedthat at high volume fractions the direct interparticle forcesare dominant.59,60 Non-Markovian effects on rate processescan be important37 and deserve further study. However, thesegenerally result in corrections only to the prefactor in Eq.�16� and, hence, are likely of minor significance. In a one-component fluid the physical origin of a non-Markovian as-pect is the lack of a clear time-scale separation between col-lective cage and single-particle dynamics. For such acorrection to be large would seem to require the confiningdegrees of freedom to relax much more rapidly than thetagged particle. This is generally not the case since collectivedensity fluctuations and stress relaxation are tightly coupledto single-particle motion,3–6,15,17,24 although dynamic hetero-geneity61 close to the glass transition might introduce com-plications.

Our focus on single-particle dynamics on relatively shortlength scales implies additional limitations. In contrast to thefull IMCT, collective wave-vector-dependent quantities suchas S(q ,t) cannot be rigorously addressed. However, theprime importance at high densities of single-particlemotion15–18,24 suggests that progress may be possible usingthe present theory combined with general MCT and statisti-cal dynamical expressions. This aspect is addressed for theshear viscosity and long-time self-diffusion constant in thefollowing companion paper.62 There is also a need for gen-eralization to treat the smooth dynamical crossover from anactivated hopping regime, to the precursor mode-coupling



regime, to the lower-density normal-fluid regime. This is adifficult and largely unsolved task within IMCT and all othertheoretical approaches.

The present work provides a foundation to treat the el-ementary trapping and activated hopping aspects of morecomplex charged or sticky colloidal and nanoparticle suspen-sions. Building on recent progress of IMCT,63,64 the influ-ence of thermally activated hopping on gelation should beamenable to study. Most excitingly, molecular and polymericthermal glass-forming liquids can be treated where activatedprocesses can control ten or more orders of magnitude varia-tion of the � relaxation time. Work is in progress in all thesedirections and will be reported in future publications.

Finally, after this paper was submitted we became awareof two very recent simulation studies65,66 of activated dy-namics and potential energy landscapes for a model binarymixture of Lennard-Jones particles �BMLJ�. Both studiesconclude that the long-time dynamical behavior is controlledby activated barrier hopping processes even well above theempirically deduced critical temperature Tc . IMCT was ar-gued to still be relevant for the short-time �in cage� dynam-ics, but long-time motion and transport is controlled by rareactivated escape from deep traps.65 The analysis of Ref. 66suggests that activated processes determine the diffusionconstant up to temperatures far above the apparent criticaltemperature, perhaps up to �2Tc which curiously is close tothe a priori calculated Tc of mode-coupling theory for theBMLJ system. Also, there is no indication of a change indiffusion mechanism around the MCT temperature.66 Hence,these authors question the commonly cited idea that barriersand activated hopping become important only as T→Tc .These simulation results seem consistent with our theory inthe sense that when we empirically analyze our relaxationtime predictions based on the MCT scenario �Fig. 7� an ap-parent critical volume fraction is identified which is in agree-ment with the experimental kinetic value of �g�0.57. How-ever, at this volume fraction the entropic barrier is large(�7kBT from Fig. 5�, implying relaxation is controlled byactivated hopping over barriers of significant height.

ACKNOWLEDGMENTS

Helpful conversations and correspondence with M. D.Ediger are gratefully acknowledged. We thank V. Ganesanfor making us aware of the work of Bouchard and collabo-rators, D. R. Reichman for informing us of Ref. 65, and S.Granick for encouragement and valuable comments on themanuscript. This work was supported by the Nanoscale Sci-ence and Engineering Initiative of the National ScienceFoundation under NSF Grant No. DMR-0117792.

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