Tagged-particle dynamics in confined colloidal liquids

Gerhard Jung,1, ∗ Lukas Schrack,1 and Thomas Franosch1

1Institut fur Theoretische Physik, Universitat Innsbruck, Technikerstraße 21A, A-6020 Innsbruck, Austria

We present numerical results for the tagged-particle dynamics by solving the mode-coupling theoryin confined geometry for colloidal liquids (cMCT). We show that neither the microscopic dynamicsnor the type of intermediate scattering function qualitatively changes the asymptotic dynamics invicinity of the glass transition. In particular, we find similar characteristics of confinement in thelow-frequency susceptibility spectrum which we interpret as footprints of parallel relaxation. Wederive predictions for the localization length and the scaling of the diffusion coefficient in the su-percooled regime and discover a pronounced non-monotonic dependence on the confinement length.For dilute liquids in the hydrodynamic limit we calculate an analytical expression for the inter-mediate scattering functions, which is in perfect agreement with event-driven Brownian dynamicssimulations. From this, we derive an expression for persistent anti-correlations in the velocity auto-correlation function (VACF) for confined motion. Using numerical results of the cMCT equationsfor the VACF we also identify a cross-over between different scalings corresponding to a transitionfrom unconfined to confined behaviour.

I. INTRODUCTION

Nanoscopic confinement has a strong impact on thestructural and dynamical properties of liquids. Mostpronounced are inhomogeneities in the density pro-file (known as “layering”) emerging from the confin-ing boundaries [1–3]. This leads amongst others toanisotropic structure factors [4] and position-dependentdiffusion coefficients [3, 5]. Moreover, the interplay oflayering and confinement qualitatively changes processeslike transport [6], crystallization [7–9] and glass forma-tion [5].

Colloidal liquids under spatial confinement exhibit asimilar rich phenomenology [10]. Different from molec-ular liquids, colloids additionally display diffusive short-time behavior and long-range hydrodynamic interactionsbetween different colloids and of colloids with the confin-ing walls [11]. A huge advantage of colloidal compared tomolecular liquids is the increase in length scale, enablingthe experimental observation of the single-particle mo-tion [12]. The above discussed phenomena can thus beinvestigated in detail using theory, simulations and ex-periments. Additionally, the dynamics of confined meso-scopic particles is in itself an important problem in bio-physics, including drug delivery in the human body [13],the flow of red blood cells through capillaries [14] or themotion of organelles in the cytoplasm of the cells [15].

In this manuscript, we study the arguably simplestmodel for confined colloidal liquids, namely hard sphereswithout hydrodynamic interactions, confined betweenparallel and planar hard walls. This system was alreadyextensively studied in experiments [12, 16–25] and sim-ulations [1, 3, 4, 26–41], giving valuable insight in theeffect of confinement on inhomogeneous density profilesand diffusivities [34], anisotropic structure factors [10, 20]and glass-formation [5, 12].

∗ jung.gerhard@umontpellier.fr; gerhard.jung.physics@gmail.com

Most noteworthy for the context of this manuscript, isthe emergence of an oscillatory behaviour of the paral-lel diffusivity and the critical packing fraction with wallseparation, leading to a multiple-reentrant glass transi-tion [5]. This behaviour is qualitatively explained withthe concepts of “commensurate” and “incommensurate”packing. Commensurate packing denotes wall separa-tions, H ≈ nσ, with integer n and particle diameter σ.In these systems, the particles create n layers, consis-tent with the inhomogeneous density profiles from theindividual walls. In the case of incommensurate pack-ing for wall separations H ≈ (n + 0.5)σ particles haveto be squeezed in between the pronounced layers whichsignificantly reduces diffusion and also lowers the criticalpacking fraction. This effect highlights the complicatedinterplay of structural relaxation on the one hand andconfinement on the other hand. Investigating this con-nection in detail, however, is a promising approach for abetter understanding of structural relaxation in general[12, 17, 27].

To systematically investigate colloidal liquids, sev-eral first-principle theories have been developed, includ-ing mode-coupling theory (MCT) [42–44], self-consistentgeneralized Langevin dynamics [45] and dynamical mean-field theory [46, 47]. For supercooled bulk liquids thesetheories successfully predict several important featureslike the slowing down of transport, stretching of the inter-mediate scattering function, as well as a two-step power-law relaxation behavior [48–50]. Mode-coupling theorywas recently extended to describe hard spheres in slitgeometry (cMCT) [51–54]. The microscopic picture ofMCT is the trapping of particles in transient “cages”formed by their respective neighbors, which has beeninvestigated systematically in terms of the mean-squaredisplacement close to the glass transition [55]. It is ex-pected that cMCT can similarly describe the influence ofconfinement on the formation of these cages. Indeed, thetheory successfully describes the effect of commensurateand incommensurate packing and thus, amongst others,the multiple-reentrant glass transition [5, 52]. Further-

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more it has been shown that all quantities connected tothe glass transition such as the power law exponents forthe critical and structural relaxation display a similaroscillatory dependence [56]. Interestingly, the reentrantscenario also applies to confinement without layeringwhich was shown by studying cylindrical, quasi-confinedsystems with both simulations and mode-coupling the-ory [57, 58]. From these studies one can conclude thatalthough layering is the most important aspect for con-fined liquids, also the confinement itself plays an impor-tant role.

In this manuscript, we apply and extend the theoryfor the tagged-particle dynamics [53] in colloidal liquids.We numerically evaluate cMCT predictions for the self-intermediate scattering function, the mean-square dis-placement and the velocity autocorrelation function closeto the glass transition. Since these quantities are usu-ally easier accessible (and also less noisy) than collec-tive variables, this work creates a basis for more system-atic comparisons of (mode-coupling) theory with simu-lations and experiments. In addition to this analysis ofstructural relaxation in dense liquids, we also study thelong-wavelength dynamics in dilute systems. This servestwo purposes: First, we can directly compare theoreticalresults obtained from the Zwanzig-Mori projection for-malism (which is the foundation of mode-coupling the-ory) with Brownian event-driven computer simulations[59] (without hydrodynamic interactions) and thus con-firm the validity of the underlying theoretical formalismfor confined liquids. Second, we can derive and also nu-merically evaluate predictions for long-time anomalies inconfined systems, showing a cross-over from an expo-nent connected to three-dimensional motion [60] to two-dimensional diffusion.

The manuscript is organized as follows: In Sec. II, werecapitulate the mode-coupling theory for colloidal liq-uids in slit geometry for the collective and tagged-particlemotion. We then investigate the equations of motion inthe hydrodynamic limit and derive expressions for the in-termediate scattering functions as well as the long-timeanomalies in the velocity autocorrelation function. Af-terwards, we study in Sec. III the structural relaxationin dense liquids and present the numerical solution ofthe cMCT equations for the self intermediate scatteringfunction, the mean-square displacement and the veloc-ity autocorrelation function. The effect of confinementon the above presented results is extensively discussed inSec. IV. In Sec. V, we then numerically evaluate and an-alyze the long-time dynamics in the hydrodynamic limit.We summarize and conclude in Sec. VI.

II. MODE-COUPLING THEORY FORCOLLOIDAL LIQUIDS IN SLIT GEOMETRY

To introduce the underlying equations of motion forthe dynamics of colloidal liquids confined between twoparallel, flat and hard walls, we will mainly follow the

lines of Refs. [52],[53] and [54].

A. Summary of the cMCT equations

The density modes of N particles confined in a channelof accessible width L can be introduced as,

ρ(c)µ (q, t) =

N∑n=1

exp [iQµzn(t)] eiq·rn(t), (1)

with particle positions xn = (rn, zn), confined to the po-sitions −L/2 ≤ zn ≤ L/2, wave vectors q = (qx, qy) andwavenumbers Qµ = 2πµ/L. In the following, we refer tothe indices µ ∈ Z as mode indices. The packing fraction isdefined as ϕ = Nπσ3/6V , with particle diameter σ (unitof length), volume V = HA, physical wall distance H =L+σ, and wall surface area A. The density modes can beinterpreted as Fourier transformation of the fluctuating

density, ρ(c)(r, z, t) =∑Nn=1 δ [r − rn(t)] δ [z − zn(t)],

ρ(c)µ (q, t) =

∫ L/2

−L/2dz

∫A

dr exp(iQµz)eiq·rρ(c)(r, z, t).

(2)We also introduce the density Fourier amplitudes,

nµ =

∫ L/2

−L/2n(z) exp [iQµz] dz, (3)

and the inverse Fourier amplitudes vµ as the Fouriertransformation of the inhomogeneous density profilen(z) =

⟨ρ(c)(r, z, t)

⟩and the inverse density profile,

v(z) = 1/n(z), respectively. In this paper, we investigatethe time evolution of a tagged particle, s, with densitymodes,

ρ(s)µ (q, t) = exp [iQµzs(t)] e

iq·rs(t). (4)

The tracer has the same diameter σ as the bath particles.Therefore, it also has the same accessible width anddensity profile n(z). The superscripts (c), for collective,and (s), for self, will be used throughout this manuscript.

To characterize the time-evolution of the density fluc-tuations we introduce the coherent,

S(c)µν (q, t) =

1

N

⟨δρ(c)µ (q, t)∗δρ(c)

ν (q, 0)⟩, (5)

and incoherent,

S(s)µν (q, t) =

⟨δρ(s)µ (q, t)∗δρ(s)

ν (q, 0)⟩, (6)

intermediate scattering functions (ISF), as the Fouriertransform of the usual Van Hove correlation function[52, 61]. Due to translational symmetry in the directionsparallel to the walls and rotational invariance around awall normal, the ISF only depends on the wavenumber

3

defined as the magnitude of the wave vector q = |q|. Thefluctuating density modes are defined as,

δρ(c,s)µ (q, t) = ρ(c,s)

µ (q, t)−⟨ρ(c,s)µ (q, t)

⟩, (7)⟨

ρ(c)µ (q, t)

⟩= δq,0Nn

(c)µ /n0, (8)⟨

ρ(s)µ (q, t)

⟩= δq,0n

(s)µ /n0. (9)

The initial value S(c)µν (q) = S

(c)µν (q, t = 0) is the

anisotropic structure factor, generalized to the slit geom-

etry [52]. We also find S(s)µν = S

(s)µν (q, t = 0) = n

(s)∗µ−ν/n0

[53]. The following derivation is practically identical forthe coherent and the incoherent dynamics. We will thusonly use the explicit notation with the superscripts (c)and (s) whenever the distinction has to be made.

We focus on overdamped, colloidal particles withouthydrodynamic interactions. Therefore, the underlyingmicroscopic equations of motion are given by the Smolu-chowski equation [54, 62]. To derive the equations ofmotion for the coherent and incoherent ISF, we chose the

density modes{ρ

(c,s)µ (q, t)

}as set of distinguished vari-

ables and apply the Zwanzig-Mori formalism [44, 61, 63].We find,

S(t) + DS−1S(t) +

∫ t

0

δK(t− t′)S−1S(t′)dt′ = 0, (10)

with the diffusion matrix [D]µν = D0n∗µ−ν/n0(q2 +

QµQν). The short-time diffusion coefficient D0 defines

the unit of time as τ = σ2D−10 . Here and in the follow-

ing, we will occasionally suppress the explicit dependenceon the wavenumber q in the notation.

The contracted force kernels δK(c)(t) and δK(s)(t) de-scribe the non-Markovian dynamics of the ISF and areformally defined as the time correlation function of the’fluctuating force’ albeit with projected dynamics. Dueto the decomposition of the density modes in the direc-tions parallel and orthogonal to the walls, the memorykernels split naturally into multiple relaxation channels[54]. We therefore introduce the contraction,

Aµν(q, t) = C{Aαβµν (q, t)} (11)

:=∑

α,β=‖,⊥bα(q,Qµ)Aαβµν (q, t)bβ(q,Qν),

with the selector bα(x, z) = xδα,‖+zδα,⊥. In this way, we

define the force kernels δK as δKµν(q, t) = C{δKαβµν (q, t)}.We refer to the indices α, β as channel indices. The ma-trix notation with calligraphic symbols δK has to be readwith respect to the super-index (α, µ).

The (exact) equations of motion for the force kernelscan now be formally written in terms of an irreduciblememory kernel (for details see Ref. [54]),

δK(t) = −DM(t)D −∫ t

0

DM(t− t′)δK(t′)dt′, (12)

with the channel diffusion matrices [D]αβµν =

D0δαβn∗µ−ν/n0. For the irreducible memory kernel

M(t) we now derive an approximate mode-couplingfunctional [54],

Mαβ,(c)µν (q, t) = Fαβ,(c)µν

[S(c)(t); q

], (13)

with,

Fαβ,(c)µν [S(t); q] =1

2N

∑q1,

q2=q−q1

∑µ1,µ2ν1,ν2

Yα,(c)µµ1µ2(q, q1, q2)

× S(c)µ1ν1(q1, t)S

(c)µ2ν2(q2, t)Yβ,(c)νν1ν2(q, q1, q2)∗, (14)

where the vertices Yα,(c)µµ1µ2(q, q1, q2) are smooth functionsof the control parameters,

Yα,(c)µµ1µ2(q, q1, q2) =

n20

L4(15)

×∑κ

v∗µ−κ [bα(q1 · q/q,Qκ−µ2)cµ1,κ−µ2(q1) + (1↔ 2)] .

Here, we have introduced the direct correlation func-tion cµν(q) which is defined via the generalized Ornstein-Zernike equation [51, 52, 61, 64],[

(S(c))−1]µν

=n0

L2(vµ−ν − cµν) . (16)

Similarly, we find for the memory kernel M(s)(t) relatedto the self dynamics,

Mαβ,(s)µν (q, t) = Fαβ,(s)µν

[S(c)(t);S(s)(t); q

], (17)

with the mode-coupling functional,

Fαβ,(s)µν [S(t); q] =1

N

∑q1,

q2=q−q1

∑µ1,µ2ν1,ν2

Yα,(s)µµ1µ2(q, q1, q2)

× S(c)µ1ν1(q1, t)S

(s)µ2ν2(q2, t)Yβ,(s)νν1ν2(q, q1, q2)∗, (18)

and the vertices,

Yα,(s)µµ1µ2(q, q1, q2) =

n0

LLs(19)

×∑σ

[(S(s))−1

]µσbα(q1 · q/q,Qσ−µ2

)cµ1,σ−µ2(q1).

Most of the following analytical work will be performedin the Laplace domain using the convention,

A(q, z) = i

∫ ∞0

eiztA(q, t)dt, z ∈ C, Im[z] ≥ 0. (20)

For convenience, we introduce a modified memory ker-nel K(q, z) = δK(q, z) + iD, which includes explicitly

the high-frequency limit K(q, z) → iD as z → ∞. Thisimplies a δ-function at the time origin. Using the con-traction we find similarly, K(q, z) = δK(q, z) + iD(q).

4

To summarize, we have derived the equations of motionfor the coherent and incoherent intermediate scatteringfunctions, which are given in the Laplace domain by,

S(q, z) = −[zS(q)−1 + S(q)−1K(q, z)S(q)−1

]−1

, (21)

K(q, z) = −[iD(q)−1 + M(q, z)

]−1

. (22)

Here, we omitted the explicit notation of the superscript(c,s). The equations are closed by the contraction definedin Eq. (11) and the approximate mode-coupling function-als.

B. Velocity autocorrelation function andmean-square displacement

In addition to the intermediate scattering functions,we also study the tagged-particle dynamics in the long-wavelength limit. In this limit, we can then derive ex-plicit equations of motion for the velocity autocorrelationfunction (VACF),

Z‖(t) =1

2〈v(t)v(0)〉 , t > 0, (23)

parallel to the wall, with the in-plane “velocity” v(t) =r(t), and the mean-square displacement,

δr2‖(t) =

⟨|r(t)− r(0)|2

⟩, (24)

of the particle. Although this information is, in princi-ple, included in the self intermediate scattering function,in practical implementations it is very hard to extractit a posteriori from the numerical solution of the MCTequations.

The velocity autocorrelation function can be ex-tracted immediately from the long-wavelength limit of

δK‖‖,(s)00 (q, t) as has been shown for Newtonian dynamics[53]. The same argument, based on the single particleconservation law, applies in the present case of Browniandynamics. We thus obtain,

limq→0

δK‖‖,(s)00 (q, t) = Z‖(t), t > 0. (25)

In the long-wavelength limit, we also find that the de-

cay channels decouple, limq→0Kαβ,(c,s)µν (q, t) = Kα,(c,s)µν (t)δαβ

and limq→0Mαβ,(c,s)

µν (q, t) = Mα,(c,s)µν (t)δαβ . The equation

of motion for the force kernel in the long-wavelength limitcan be directly extracted from Eq. (12) specialized to thecase q = 0 [53],

δKα,(c,s)µν (t) =−Dα,(c,s)µα Mα,(c,s)αβ (t)Dα,(c,s)βν (26)

−∫ t

0

Dα,(c,s)µα Mα,(c,s)αβ (t− t′)δKα,(c,s)βν (t′)dt′,

with the memory kernels,

Mα,(c)µν (t) = n0

∫ ∞0

dkk (27)

×∑µ1,µ2ν1,ν2

Yα,(c)µµ1µ2(k)S(c)

µ1ν1(k, t)S(c)µ2ν2(k, t)Yα,(c)νν1ν2(k)∗,

Mα,(s)µν (t) = n0

∫ ∞0

dkk (28)

×∑µ1,µ2ν1,ν2

Yα,(s)µµ1µ2(k)S(c)

µ1ν1(k, t)S(s)µ2ν2(k, t)Yα,(s)νν1ν2(k)∗,

and the vertices,

Yα,(c)µµ1µ2(k) =

n0√2πL4

∑κ

v∗µ−κ (29)

×[bα(k/

√2, Q

(c)κ−µ2

)c(c)µ1,κ−µ2

(k) + (1↔ 2)],

Yα,(s)νν1ν2(k) =1√

2πLLs

∑κ

[(S(s))−1

]µκ

(30)

×bα(k/√

2, Q(s)κ−µ2

)c(s)κ−µ2,µ1

(k).

From the memory kernel M‖,(s)µν (t) we can also calculatethe long-time diffusion coefficient D‖ = D0 +

∫∞0Z‖(t)dt

and the localization length 4l2‖ = limt→∞

δr2‖(t), as,

D‖ =

([∫ ∞0

M‖,(s)(t)dt

]−1)

00

, (31)

and

l‖ =

√([limt→∞

M‖,(s)(t)]−1)

00

. (32)

These equations can be evaluated using the numericalsolution of the cMCT equations for the coherent and in-coherent intermediate scattering function. Results willbe presented in Sec. III.

C. Long-time anomalies of the cMCT equations

Before solving the equations numerically we analyzetheir behavior in the hydrodynamic limit. Since thederivations are rather technical we will show the detailsin App. B and in this section only present the importantresults. We define the hydrodynamic limit as the simul-taneous limit q → 0 and z → 0 while z/q2 = const. Inthis limit, we find for the in-plane coherent dynamics,

S(c)00 (q, z) ' −S(c)

00 (q)

z + iq2D0

[S

(c)00 (q)

]−1 . (33)

This result highlights that the long-time diffusion coef-ficient of the collective dynamics is indeed equivalent to

5

the short-time diffusion constant D0. For the incoherentdynamics we obtain,

S(s)00 (q, z) ' −1

z + iq2D‖. (34)

Here, we have introduced the long-time diffusion coeffi-

cient D‖ = limz→0

(−i)δK‖,(s)00 (z)+D(s)0 . Using these results,

we finally find in the hydrodynamic limit,

S(c)00 (q, t) ' S(c)

00 exp(−D(c)

0 q2t/S(c)00

), (35)

S(s)00 (q, t) ' exp

(−D‖q2t

), (36)

with S(c)00 = S

(c)00 (q → 0). The derivation of the higher

modes involves careful application of the Zwanzig-Moriprojection formalism. Details are presented in Ap-pendix B 2. As a final result we find,

S(c,s)µν (q, t) '

S(c,s)µ0 S

(c,s)ν0

S(c,s)00 S

(c,s)00

S(c,s)00 (q, t) = S(c,s)

µν S(c,s)00 (q, t).

(37)This result for the long-wavelengths dynamics of the

coherent and incoherent scattering functions is also phys-ically intuitive. At short times t . L2/D0, only the dif-fusion in the direction perpendicular to the wall can leadto a significant change in the scattering functions. Thisleads to a fast initial decay in the modes µ 6= 0 and ν 6= 0,since only these modes depend on the instantaneous posi-tions in z-direction. For long times L2/D0 � t . 1/q2D0

the system has lost all information on the position inperpendicular direction and only diffusion in lateral di-rection is relevant, which is given by the hydrodynamicpole in the 00-mode. Therefore, on this time scale, allother modes only decay via their coupling to the modeS00(t), given by the coupling strength Sµν . Interestingly,this coupling depends inherently on the modes for whichµ = 0 or ν = 0 and not on the diagonal modes, whichindeed questions the quality of the “diagonal approxima-tion” in the hydrodynamic limit (see Sec. III A).

Using the behaviour of S(c,s)(q, t) in the hydrodynamiclimit, Eqs. (35) and (36), we find for the memory kernelin Eq. (28),

M‖,(s)µν (t) ' n0

∑µ1,µ2ν1,ν2

S(s)µ1µ2

S(c)ν1ν2

∫ ∞0

dkkY‖,(s)µµ1µ2(k) (38)

× exp(−D(c)

0 k2t/S(c)00 (0)

)exp

(−D‖k2t

)Y‖,(s)νν1ν2(k)∗,

where we assumed that the integrals are dominated bythe intermediate scattering functions in the hydrody-namic limit. These Gaussian integrals can then approxi-mated (for details we refer to App. B 3),

M‖,(s)µν (t) ' B‖,(s)µν t−2. (39)

From two-times partial integration and application ofTauber’s theorem we can finally conclude that,

δK‖‖µν(t) ' −[D‖,(s)L

]µαB‖,(s)αβ

[D‖,(s)L

]βνt−2. (40)

For the velocity autocorrelation function Z‖(t) we thus

find a tail −t−2. This tail is similar to the persistentanti-correlations, Zbulk

‖ (t) = −Bt−2.5, recently discussed

in Ref. [60] for Brownian particles in bulk. Here, dueto the confined motion, we find for long times the two-dimensional analogue.

III. ASYMPTOTIC DYNAMICS CLOSE TOTHE GLASS TRANSITION

In this section, we first discuss the introduction of theeffective memory kernel and the diagonal approximation,which are necessary to obtain a stable numerical solutionfor the full time-dependence of the intermediate scatter-ing functions. Afterwards, we present numerical resultsof the cMCT equations of motion for the tagged-particledynamics in colloidal liquids asymptotically close to the(ideal) glass transition. The results are compared to thecoherent dynamics in Newtonian liquids as discussed inRef. [56] and to bulk liquids [55].

A. Effective memory kernel and diagonalapproximation

Based on the above equations of motion, we can definean effective memory kernel M(z) implicitly via,

K(z) = −[iD−1 + M(z)

]−1

. (41)

Using the effective memory kernel the equations of mo-tion reduce in the time domain to the standard MCTequation for Brownian dynamics,

D−1S(t) + S−1S(t) +

∫ t

0

M(t− t′)S(t′)dt′ = 0. (42)

To solve this equation numerically and determine the fulltime dependence of the relaxation process we apply the“diagonal approximation” discussed in the previous liter-ature (see Ref. [56], section II. B). The diagonal approxi-mation is a technical approximation which becomes exactin the planar and bulk limits [65]. It has been applied todetermine the critical packing fraction and the nonergod-icity parameters of confined glasses and for these quanti-ties successfully compared to computer simulations [4, 5]and the full model [51] with good agreement. A similarapproximation has been successfully applied to molecularliquids [66]. Within the diagonal approximation, we as-sume that the matrix-valued quantities discussed beforeare diagonal, and the coupling between different modes isthus purely based on the mode-coupling functional. Forbetter readability, we introduce the notation, Aµ = Aµµ.In light of the diagonal approximation, we can derive an

6

0

0.2

0.4

0.6

0.8

1

10−4 10−2 100 102 104 106 108 1010

S(s)

0(q

m,t)

t/(σ2D−10 )

c

n=6

n=9

n=6 n=9

ε < 0

ε > 0L/σ = 2.0,qmσ = 6.52

FIG. 1. Normalized incoherent intermediate scattering func-

tion S(s)0 (qm, t)/S

(s)0 (qm, 0) for accessible width L = 2.0σ,

for wavenumber qmσ = 6.52 corresponding to the first sharpdiffraction peak in the structure factor. The control parame-ter ε = (ϕ− ϕc)/ϕc = ±10−n/3, n ∈ N, increases from left toright for ε < 0 and from bottom to top for ε > 0. The criticalcorrelator for ϕ = ϕc (or ε = 0) is displayed as thick line andlabeled as “c”.

explicit expression for the effective memory kernel,

Mµ(q, t) +D20

∫αµ(q, t− t′)Mµ(q, t′)dt′ = (43)

D20βµ(q, t) +D2

0Dµ(q)−1

∫ t

0

M‖µ(q, t− t′)M⊥µ (q, t′)dt′,

with,

αµ(q, t) = Dµ(q)−1(Q2µM‖µ(t) + q2M⊥µ (t)), (44)

βµ(q, t) = Dµ(q)−2(q2M‖µ(t) +Q2µM⊥µ (t)). (45)

With the above equations the incoherent and coherentscattering functions can be determined using the staticinput functions, n(z) (density profiles), S(c)(q) (staticstructure factor) and c(q) (direct correlation functions).These (anisotropic) static input functions are connectedvia the Ornstein-Zernike equation [51–53]. For details onthe numerical evaluation of the static input functions andthe numerical solution of the cMCT equations we referthe reader to Appendix A.

B. Incoherent scattering function and dynamicsusceptibility

For packing fractions smaller than the critical pack-ing fraction, the incoherent scattering function shows theusual two-step relaxation scenario known for supercooled

0

0.2

0.4

0.6

0.8

1

1.2

10−2 100 102 104 106 108 1010 1012

S0(q

m,t)/S0(q

m,0)

t/(σ2D−10 )

Brownian, S(s)

Brownian, S(c)

Newtonian, S(c)

L/σ = 2.0, qmσ = 6.52

FIG. 2. Comparison of Brownian and Newtonian dynamicsin the normalized incoherent intermediate scattering function

S(s)0 (qm, t)/S

(s)0 (qm, 0) for accessible width L = 2.0σ, control

parameter ε = −10−5 and wavenumber qmσ = 6.52. Theresult for Newtonian dynamics corresponds to ε = −10−5

shown in Fig. 1 in Ref. [56].

colloidal liquids, i.e. an initial exponential and then al-gebraic decay to an extended plateau, followed by theprimary α-relaxation [55] (see Fig. 1). In MCT the re-laxation times diverge as the glass transition ϕc is ap-proached [44], such that above the critical packing frac-tion no α-relaxation occurs and the system becomes non-ergodic.

It has been elaborated via MCT and computer sim-ulations that the long-time relaxation close to the glasstransition is not affected by the microscopic dynamics[67, 68]. Therefore, we expect that the only visible differ-ence between the ISF for Brownian dynamics and New-tonian dynamics is the early exponential decay for shorttimes t . 10−2σ2D−1

0 . Similarly, since the incoherentand coherent ISF describe the same relaxation scenario,we expect that both functions display similar critical be-haviour. The above statements can be confirmed by com-paring the asymptotic colloidal tagged-particle dynamicspresented here with the coherent dynamics of Newtonianliquids discussed in Ref. [56] (see Fig. 2). Indeed, exceptfor the short-time behaviour the dynamics are in perfectagreement, which holds in particular for the power lawexponents a (critical exponent) and b (von-Schweidler ex-ponent). The numerical values are summarized in Tab. I.

Analogous to the calculation of the dynamic suscep-tibility, we can define the susceptibility of the self-

dynamics as χ′′(s)µ (q, ω) = ωS

(s)µ (q, ω), where, S

(s)µ (ω) =∫∞

0cos(ωt)S

(s)µ (q, t)dt is the Fourier cosine transform.

This allows for a more direct observation of the differ-ent relaxation processes in the system. For moderateconfinement (L = 2.0σ) the incoherent dynamic sus-

7

L/σ ϕc λ a b γ l‖/σ

1.0 0.4497 0.795 0.282 0.484 2.81 0.0537

1.25 0.4029 0.629 0.354 0.761 2.07 0.0784

1.5 0.3817 0.629 0.354 0.761 2.07 0.0963

1.75 0.4352 0.672 0.338 0.688 2.21 0.0723

2.0 0.4495 0.668 0.340 0.694 2.19 0.0688

TABLE I. Critical packing fractions ϕc and asymptotic coef-ficients for the confinements lengths considered in this work.The critical exponents are related via Gotze’s exponent re-lation Γ(1 + b)2/Γ(1 + 2b) = λ = Γ(1− a)2/Γ(1− 2a) andγ = 1/(2a) + 1/(2b). The localization length l‖ was extractedusing Eq. (32). See Ref. [56] Appendix B for the determi-nation of the critical packing fraction ϕc and the asymptoticanalysis.

10−3

10−2

10−1

100

10−10 10−8 10−6 10−4 10−2 100 102

χ′′(

s)

0(q

m,ω

)

ω/(D0σ−2)

c n=6n=9

n=6n=9ε < 0

ε > 0

L/σ = 2.0, qmσ = 6.52

FIG. 3. Frequency-dependent susceptibility of the self-

dynamics χ′′(s)0 (qm, ω) for the same parameters as in Fig. 1.

The dashed, black line shows a Debye peak, χ′′D(ω) =2χmaxωτD/

[1 + (ωτD)2

](χmax = 0.37, τD = 1.56 · 108σ2D−1

0 )for comparison.

ceptibility is visualized in Fig. 3. The high-frequencypeak (ω ≈ 102D0σ

−2) stems from the short-time relax-ation and is thus qualitatively different from the high-frequency spectrum in Newtonian liquids (see Fig. 2 inRef. [56]). For frequencies much smaller than the micro-scopic ones one then observes a power-law in the dynamicsusceptibility, χ′′0 ∝ ωa, which corresponds to the criticaldecay displayed in the intermediate scattering function.In the case of supercooled liquids a second peak emergesfor very small frequencies describing the structural relax-ation. The right flank of this peak is the von-Schweidlerlaw, χ′′0 ∝ ω−b, while the left-flank corresponds to a De-bye peak resulting from the exponential decay of the ISFat long times.

As anticipated, apart from the high-frequency spec-trum, the dynamic susceptibility does not depend on the

10−3

10−2

10−1

100

10−12 10−10 10−8 10−6 10−4 10−2 100 102

χ′′(

s)

0(q

m,ω

)

ω/(D0σ−2)

c

n=6n=9

n=6n=9ε < 0

ε > 0

L/σ = 1.0, qmσ = 6.52

FIG. 4. Frequency-dependent susceptibility of the self-

dynamics χ′′(s)0 (qm, ω) for channel width L = 1.0σ. The pa-

rameters for the Debye peak (dashed black line) are χmax =0.41, τD = 1.45 · 1011σ2D−1

0 .

microscopic dynamics (Newtonian or Brownian) and thetype of ISF (coherent or incoherent) [56]. This holdsin particular for the “kink” visible in the low-frequencysusceptibility spectrum for strong confinement L = 1.0σ(see Fig. 4) which emerges because the right flank of thelow-frequency peak is much better described by a Debyerelaxation than the usual Kohlrausch function. We ratio-nalize this with the existence of multiple and very differ-ent parallel-relaxation channels as is the case for strongconfinement with accessible width L ≤ 1.0σ. In suchsmall channels, the relaxation parallel and perpendicu-lar to the walls differ significantly, which is described incMCT by distinct couplings to the different modes. Sincethese modes have distinct relaxation times it can occurthat the peaks connected to the different relaxation chan-nels do not coincide. In extreme cases this can lead tomultiple low-frequency peaks [69], while for realistic sce-narios a kink in the low-frequency spectrum emerges (seeRefs. [56] for detailed discussions). For larger wall sepa-rations as shown in Fig. 3, the effect disappears becausethe difference between the relaxation channels becomesnegligible.

C. Mean-square displacement (MSD)

As described in Sec. II B, we can calculate the tagged-particle dynamics in the long-wavelength limit using thenumerical results for the incoherent and coherent scat-tering functions as input for the long-wavelength limit

of the force kernels M‖,(s)00 (t). The mean-square dis-placement can be obtained by solving numerically thee.o.m. presented in App. A 3. The results for the

8

10−4

10−2

100

102

104

106

10−4 10−2 100 102 104 106 108 1010

δr2 ‖(t)/σ

2

t/(σ2D−10 )

c

n=6n=9

n=6 n=9

ε < 0

ε > 0

L/σ = 2.0

FIG. 5. Mean-square displacement δr2‖(t) for accessible widthL = 2.0σ asymptotically close to the glass transition. Thecolor code is the same as used in Fig. 1.

mean-square displacement close to the glass transitionare presented in Fig. 5. The two-step relaxation sce-nario manifests itself in a short-time diffusive regime(δr2‖(t) = 4D0t) followed by an algebraic approach to

a plateau (δr2‖(t) = 4l2‖ − hMSDt

−a + O(t−2a)). At long

times, below the (ideal) glass transition (ε < 0), the dy-namics become eventually diffusive again. The power lawexponent a is the same critical exponent as observed inthe coherent and incoherent scattering functions. Thelocalization length l‖ corresponds to the size of the cagesin which the particles are trapped. When approachingthe glass transition from above, the localization lengthincreases to reach a finite value at the critical point (seeFig. 6). The dependence of l‖ on the control parameterabove the critical packing fraction, ε > 0, can be welldescribed by a square-root function. This corresponds tothe usual Whitney fold bifurcation scenario known fromthe asymptotic behavior of the nonergodicity parameters[69, 70].

The long-time diffusion coefficient D‖ as defined inEq. (31) below the glass transition shows a clear powerlaw dependence on the separation parameter ε = (ϕ −ϕc)/ϕc with exponent γ (see Fig. 7). This exponent isconsistent with γ = 1/(2a) + 1/(2b) describing the in-crease of the α-relaxation time τα ∝ |ε|−γ (see Tab. Ifor numerical values for γ) [55]. One can thus concludethat D‖τα = const., consistent with the Stokes-Einsteinrelation.

Importantly, both the localization length at the criti-cal packing fraction as well as the asymptotic behaviourof the long-time diffusion coefficient depend strongly onconfinement. We will discuss this in more detail inSec. IV.

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

10−5 10−4 10−3 10−2 10−1

l ‖/σ

ε

L/σ = 1.0L/σ = 2.0

bulk

FIG. 6. Localization length l‖ close to the glass transitionwith ε = (ϕ−ϕc)/ϕc. The data was extracted using Eq. (32).The lines correspond to fits l‖(ε) = lc‖ + hl

√ε.

100

102

104

106

108

1010

1012

1014

10−5 10−4 10−3 10−2 10−1

D−1‖D

0

−ε

L/σ = 1.0L/σ = 2.0

bulk

FIG. 7. Inverse diffusion constant close to the glass transition.The fitted (and shifted) power laws plotted in black are γ(L =1.0σ) = 2.81 (full), γ(L = 2.0σ) = 2.19 (long-dashed) andγ(bulk) = 2.52 (short-dashed). The data was extracted usingEq. (31).

D. Velocity autocorrelation function (VACF)

The velocity autocorrelation function (determined hereas the second time-derivative of the mean-square dis-placement, see App. A 3) shows several subtleties that arehidden in the MSD due to the dominating linear growthof diffusion (see Fig. 8).

Also in the VACF a two-step relaxation can be ob-served. For times much larger than the microscopic

9

FIG. 8. Velocity autocorrelation function close to the glasstransition for ε < 0 (a) and ε > 0 (b), and accessible widthL = 2.0σ. The critical correlator is fitted by a power lawwith exponent a + 2 = 2.34, shown as a dashed, black line.The dotted, black line displays as reference a power law withexponent 2− b = 1.31. The color code is the same as used inFig. 1.

times, the decay follows a power law, −Z‖(t) ∝ t−(a+2)

with exponent a + 2 (= 2.34 for L = 2.0σ). If ε > 0,a finite plateau will be reached in the MSD and the de-cay of the VACF eventually becomes exponential. Inthe supercooled regime, the decay of the VACF flattensat the α-relaxation time to follow another power law,−Z‖(t) ∝ t−(2−b), with exponent 2.0 − b (= 1.31 forL = 2.0σ). This corresponds to the decay from theplateau. At long times, the VACF then eventually de-cays exponentially.

It should be noted, that the above analysis of theVACF is mostly of theoretical interest, since the very

0.05

0.06

0.07

0.08

0.09

0.1

1 1.5 2 2.5 3 3.5 4 4.52

2.2

2.4

2.6

2.8

3

l ‖/σ γ

L/σ

l‖/σγ

FIG. 9. Dependence of the critical localization length l‖ andscaling exponent γ on the accessible width L. The exponentswere determined from the asymptotic analysis as described inRef. [56]. The arrows indicate the value of the exponents inthe bulk limit for hard spheres (arrows pointing to the right[70]) and hard discs (arrows pointing to the left [71]).

slow creeping motion of the particles visible here in theVACF is hardly detectable in simulations and laboratoryexperiments. Nevertheless, we find it fascinating to seeall the features of glassy dynamics in the VACF - albeitthey are characterised by very small amplitudes.

IV. THE EFFECT OF CONFINEMENT ONGLASSY DYNAMICS

We have already discussed several effects of confine-ment: First, the low-frequency susceptibility spectrumfeaturing a kink for very strong confinement, second thedifferent scaling of the diffusion coefficient close to theglass transition and third, the significantly decreased lo-calization length for very strong confinement. Here, wediscuss these features more quantitatively.

In Fig. 9 the dependence on the accessible width L fortwo critical parameters are shown. The power law ex-ponent γ = 1/(2a) + 1/(2b) is directly derived from thecorresponding critical exponent a and von-Schweidler ex-ponent b (see Tab. I). It describes the divergence of thestructural relaxation time τα and, as discussed above,also the divergence of the (inverse) long-time diffusioncoefficient D−1

‖ . As discussed in Ref. [56], both a and b

display a clear non-monotonic dependence on L which istherefore similarly true for γ. Interestingly, γ is very dif-ferent between L = 1.0σ and L = 2.0σ, although the re-spective critical packing fractions ϕc are basically equiv-alent. When analyzing larger wall separations it standsout that the convergence to the bulk limit is very slow,

10

0.001

0.01

0.1

1

10

100

1000

2 4 6 8 10 12

δr2 ‖(t∗ )/σ2

L/σ

ϕ = 0.2ϕ = 0.3ϕ = 0.4ϕ = 0.43

t∗ = 100σ2D−10

FIG. 10. Mean-square displacement δr2‖(t) at time t∗ =

100σ2D−10 for different accessible widths L and packing frac-

tions ϕ.

as has already been discussed in Ref. [56].

Similarly, also the localization length features the samenon-monotonic dependence, with a maximum for incom-mensurate packing. The difference between L = 1.0σand L = 1.5σ is actually severe, showing that at theglass transition, the localization (in parallel direction)varies up to a factor of 2. This effect can be partiallyexplained by the fact that for incommensurate packing(L = 1.5σ), the critical packing fraction is much smallerthan for commensurate packing, mostly because of geo-metric constraints in the direction perpendicular to thewalls. Remarkably, the convergence to the bulk limit fea-tures a completely different behavior than shown for thepower law exponent γ. Indeed, for L & 3.5σ no sys-tematic difference to the bulk solution can be observed.One can thus conclude that cMCT predicts a significantimpact of confinement on the cage formation and thatin particular in strongly confined systems localizationalone is not a sufficient criterion for characterizing theglass transition. This highlights the complicated inter-play between geometry, structure and dynamics in con-fined, glassy liquids.

In Fig. 10 the effect of confinement on the structuralrelaxation in colloidal liquids is shown for a broaderrange of channel widths [72]. At small packing fraction,ϕ / 0.30, only a minimal effect of confinement can be ob-served and only for very small wall distances diffusion isslightly affected by the confined geometry. Due to the lowpacking fraction and the reduction of layering the differ-ence between commensurate and incommensurate pack-ing also becomes negligible. Therefore, the dependenceof δr2

‖(t) on the accessible width L is monotonic. This

changes drastically when increasing the packing fraction.Now, already at moderate confinement (L ≈ 6σ) a signif-

icant reduction of diffusion is observed. Furthermore, anon-monotonic dependence of the mean-square displace-ment on channel width is identified for strong confine-ment, which mirrors the non-monotonic dependence ofthe critical packing fraction ϕc on L. In fact, the twosmallest values for L = 1.5σ and ϕ ≥ 0.4 already corre-spond to arrested states. These theoretical results sup-port the experimental results in Ref. [12] by showing thatthe effect of confinement is indeed stronger pronouncedwhen increasing the packing fraction. This effect couldbe accounted to a growing length scale at the glass tran-sition. This conclusion, however, has to be taken witha grain of salt since at least one confinement-specific ef-fect plays an important role: layering. The inhomoge-neous density profiles close to the walls strongly affectthe glass transition which can be highlighted by the de-scribed non-monotonic dependence of basically all glass-related dynamical properties on the channel width. Theobservation that the effects of confinement are enhancedfor larger packing fraction could thus be connected to lay-ering and not (only) to an intrinsic property (like growinglength scales) of glass-forming liquids close to the glasstransition.

V. LONG-TIME ANOMALIES IN DILUTELIQUIDS

In Sec. II C, we have analysed the cMCT equationsin the hydrodynamic limit. We found that the 00-modeof the intermediate scattering functions displays a hy-drodynamic pole and that higher modes couple to thispole with a coupling strength that solely depends on thestructure factor S(c,s). Here, we validate this predictionusing event-driven Brownian dynamics (EDBD) simula-tions [59] and show cMCT results for the velocity auto-correlation function.

The Brownian event-driven simulations are performedfor monodisperse hard-spheres at packing fraction ϕ =0.1. The wall separation is H = 2.0σ (and thus L =1.0σ). We determine the mean-square displacement aswell as the incoherent and coherent intermediate scat-tering functions. The simulations are compared to thetheoretical results presented in Eqs. (35), (36) and (37)using the simulation results for the structure factor S(c,s)

and the long-time diffusion coefficient D‖.The results for the intermediate scattering functions

are presented in Figs. 11 and 12. One finds perfect agree-ment between simulations and theory showing that thelong-time decay is indeed always defined by the 00-mode,as predicted by the theory. All (normalized) modes witheither µ = 0 and/or ν = 0 do not exhibit a short-time de-cay but only the long-time exponential decay connectedto the hydrodynamic pole in the 00-mode. All othermodes exhibit a pronounced short-time decay from dif-fusion in the confined direction and then decay from awell-defined plateau with the same rate as the 00-mode.

Having discussed the hydrodynamic limit for the in-

11

10−3

10−2

10−1

100

10−1 100 101

S(c)

µν(qs,t)/S(c)

µν(qs,0)

t/(σ2D−10 )

µ = 0, ν = 0µ = 1, ν = 0µ = 1, ν = 1

theory, µ, ν = 0theory, µ, ν

L/σ = 1.0, qsσ = 0.271828

FIG. 11. Coherent intermediate scattering function for smallqsσ = 0.272 at packing fraction ϕ = 0.1 and L = 1.0σ. Re-sults are shown for event-driven Brownian simulations and forthe theoretical prediction Eqs. (35) and (37).

10−3

10−2

10−1

100

10−1 100 101

S(s)

µν(qs′ ,t)/S

(s)

µν(qs′ ,0)

t/(σ2D−10 )

µ = 0, ν = 0µ = 1, ν = 0µ = 1, ν = 1

theory, µ, ν = 0theory, µ, ν

L/σ = 1.0, qs′σ = 0.388325

FIG. 12. Incoherent intermediate scattering function for smallqs′σ = 0.388 at packing fraction ϕ = 0.1 and L = 1.0. Resultsare shown for event-driven Brownian simulations and for thetheoretical prediction Eqs. (36) and (37). The value for D‖was determined from the mean-square displacement of thecolloids in the EDBD simulations.

termediate scattering functions we can now focus on theanalysis of the long-time behaviour of the velocity au-tocorrelation function (VACF) which (in mode-couplingapproximation) follows immediately from the scatter-ing functions (see App. B 3). For the numerical solu-tion of the cMCT equations in the hydrodynamic limitwe have used N = 250 equally spaced grid points be-

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

10−1 100 101

−Z‖(t)σ2/D

2 0

t/(σ2D−10 )

L/σ = 1.0L/σ = 10.0

bulk

FIG. 13. Velocity autocorrelation function determined withcMCT for ϕ = 0.005. The figure shows results for confinedand bulk colloidal liquids. Black lines correspond to powerlaws with exponents 2.0 (dashed) and 2.5 (dotted).

tween q0σ = 0.0365 and qmaxσ = 30.0365, M = 10and Nt = 256 (see Ref. [56], App. A for a definition ofthe parameters). Different from the data shown for theVACF in vicinity of the glass transition, for this analysisin the hydrodynamic limit we used the direct integra-tion of Eq. (26) using an adapted version of Alg. (A2).It should be noted that due to the diagonal approxima-tion which is applied here to solve the cMCT equations,the numerical results are expected to deviate from thesolution of the full model, because the amplitudes Sµν(see Eq. 37) explicitly depend on the off-diagonal terms.These differences will, however, be quantitative and notaffect the qualitative behaviour in the long-time dynam-ics. (In fact, one could easily derive the prefactor for thelong-time tail in the diagonal approximation).

The numerical solutions of the cMCT and MCT equa-tions for the VACF in dilute colloidal liquids are shown inFig. 13. The figure perfectly highlights the different per-sistent anti-correlations in bulk, Z‖ ∝ −t−5/2 [60], and

slit geometry, Z‖ ∼ −t−2. Furthermore, one observesa cross-over between these two long-time anomalies inconfined systems with large wall separation. In strongconfinement (L = 1.0σ) the particles “recognize” on veryshort time scales that they are confined to two dimen-sional motion. For larger wall separation (L = 10σ) theparticle’s motion is indistinguishable from bulk motionfor short times, but for longer times it deviates from theunconfined dynamics and exhibits the expected powerlaw, Z‖ ∝ −t−2.

12

VI. CONCLUSION

In this manuscript, we have investigated the tagged-particle dynamics in confined colloidal liquids based onmode-coupling theory. In the first part, we have studiedthe dynamics in the vicinity of the glass transition. Wehave found that the incoherent and coherent scatteringfunctions show the same qualitative behaviour, in par-ticular both display for strong confinement a kink in thelow-frequency susceptibility spectrum.

In the long-wavelength limit, we have also studied themean-square displacement (MSD) and the velocity au-tocorrelation function (VACF). From the MSD we ex-tracted the scaling of the long-time diffusion coefficient inthe supercooled regime and the localization length in theideal glass. Similar to all other dynamical variables thatcharacterize the glass transition, both quantities displaya pronounced non-monotonic dependence on the wall sep-aration.

We have also studied the hydrodynamic limit in dilutecolloidal liquids. We have observed a perfect agreementbetween theory and simulations for the intermediate scat-tering functions which confirms the validity of the generalformalism in the hydrodynamic limit for colloidal liquids.We have also found an intriguing cross-over from initiallybulk-like behaviour to confined motion in the VACF forlarge wall separation.

This manuscript represents a throughout theoreticalanalysis of confined colloidal liquids - both in the hydro-dynamic and the supercooled regime. It predicts impor-tant features of the MSD, the VACF and the incoherentscattering function and their dependence on the confine-ment length. We expect, that these features can also beobserved in future investigations using computer simu-lations and laboratory experiments. The results of thispaper on the tagged-particle dynamics will be particu-larly important for these studies, since the incoherentdynamics can be determined with much greater accuracythan the (collective) coherent dynamics.

An interesting open question is the relative influenceof layering and confinement on the discussed quantities,most importantly with regard to possible general insightsinto the glass transition that could be drawn from inves-tigations of glasses in confined geometry. This can bestudied using quasi-confinement [57, 58] or using inho-mogeneous wall profiles [28].

ACKNOWLEDGMENTS

This work has been supported by the Austrian ScienceFund (FWF): I 2887.

Appendix A: Discretization and numericalevaluation of the cMCT equations

In this appendix, we introduce the numerical meth-ods used to obtain the solutions of the mode-couplingequations in slit geometry. Since the methods are mostlyadapted form previous work, the main goal of this ap-pendix is to refer the reader to the respective referenceswhere the techniques were introduced.

1. Static input functions

To determine the static input functions, nµ, cµν and

S(c)µν we first apply fundamental measure theory (FMT),

which gives the density profile n(z). In principle, onecould also determine the direct correlation function fromFMT, but it seems to be more stable to use for thisstep an iterative procedure based on the Ornstein-Zernikeequation with a Percus-Yerwick closure [73]. For detailson the algorithm see Refs. [57, 74], for parameters, seeRef. [56] (Table II).

2. Discretization and numerical evaluation of theintermediate scattering functions

The introduction of the thermodynamic limit and thediscretization of the mode-coupling equations is per-formed in the same way as described in Ref. [56] (Ap-pendix A). Since the equations of motion for the coher-ent and incoherent intermediate scattering functions areformally identical, we apply the same discretization andintegration techniques. The underlying dynamics are,however, slightly different to Ref. [56] (Newtonian vs.Brownian dynamics). For Brownian dynamics, we arriveat the following, discretized equations of motion, (dis-carding the explicit dependence on q and µ),

ASSi = (4Si−1 − Si−2)/3

+2∆tD

3( dM1Si−1 +MiS0 − 0.5Mi−iSi − 0.5MiSi−i)

− 2∆tD

3

i∑j=1

dSj(Mi−j+1 −Mi−j)

− 2∆tD

3

i∑j=2

dMj(Si−j+1 − Si−j)

− ∆tD

3

{dSi−i(Mi+1 −Mi) if i 6= i− i0 otherwise

− ∆tD

3

{dMi−i(Si+1 − Si) if i 6= i− i0 otherwise

,

AS = 1 +2∆tD

3

(S−1

0 + dM1

), (A1)

13

with Si = S(i∆t), dSi = ∆t−1∫ i∆t

(i−1)∆tdt′S(t′) and sim-

ilarly Mi, dMi. The half time i is defined as i = bi/2c.The brackets bjc denote the largest integer less or equalj. Similar to the discussion in Ref. [56] for Newtoniandynamics, we do not integrate the equation for the ef-fective memory kernel (43) directly, but we integrate itsfirst derivative.

AMMi = (4Mi−1 −Mi−2)/3

+∆tD2

0

3(2α[dM1]Mi−1 − α[Mi−i]Mi − α[Mi]Mi−i)

+D20 (β[Mi]− 4/3β[Mi−1] + 1/3β[Mi−2])

− 2∆tD20

3

i∑j=1

dMj(α[Mi−j+1]− α[Mi−j ])

− 2∆tD20

3

i∑j=2

α[dMj ](Mi−j+1 −Mi−j)

− ∆tD20

3

{dMi−i(α[Mi+1]− α[Mi]) if i 6= i− i0 otherwise

− ∆tD20

3

{α[dMi−i](Mi+1 −Mi) if i 6= i− i0 otherwise

+2∆tD2

0D−1

3

(M‖

i−iM⊥i +M⊥i−iM

‖i

)+

2∆tD20D−1

3

i∑j=1

dM‖j (M⊥i−j+1 −M⊥i−j)

+2∆tD2

0D−1

3

i∑j=1

dM⊥j (M‖i−j+1 −M‖i−j)

+∆tD2

0D−1

3

{dM‖

i−i(M⊥i+1−M⊥

i) if i 6= i− i

0 otherwise

+∆tD2

0D−1

3

{dM⊥

i−i(M‖i+1−M‖

i) if i 6= i− i

0 otherwise,

AM = 1 +2∆tD2

0

3α[dM1]. (A2)

where we defined α[B] = Dµ(q)−1(Q2µB‖µ(t) + q2B⊥µ (t))

and β[B] = Dµ(q)−2(q2B‖µ(t)+Q2µB⊥µ (t)). The remaining

decimation and iteration procedure is identical to the onedescribed in Refs. [56, 75, 76].

3. Mean-square displacement and velocityautocorrelation function

The equation of motion for the velocity autocorrelationfunction, Z‖(t), is given by the µ = 0, ν = 0 element ofEq. (26). By integration twice over time utilizing therelation,

δr2‖(t) = 4Z‖(t), (A3)

we find a similar equation for the mean-square displace-ment δr2

‖(t) (now in diagonal approximation) [55],

δr2‖(t) +D0

∫ t

0

M(s),‖00 (t− t′)δr2

‖(t′)dt′ = 4D0t. (A4)

The memory kernel in the long-wavelength limit

M(s),‖00 (t) is defined in Eq. (29). Eq. (A4) has exactly

the same form as the one for the effective memory ker-nel, Eq. (43), enabling us to use the algorithm presentedin Eq. (A2). The velocity autocorrelation function wasintegrated both with algorithm Eq. (A2) and with therespective integration scheme without taking the firstderivative in time. The former methods displayed in-correct plateau values and the latter led to instabilitiesin the numerical integration. We therefore determinedthe VACF by taking the second time-derivative of themean-square displacement which exhibits both long-timestability and proper convergence to zero.

Appendix B: The cMCT equations in thehydrodynamic limit

In Sec. II C, we presented results for the dynamics ofthe coherent and incoherent scattering function in thehydrodynamic limit, as well as long-time anomalies inthe velocity autocorrelation function. Here, we performthe analytical derivations in detail.

1. The hydrodynamic pole in the intermediatescattering function

In Ref. [53] is was shown that the in-plane density fluc-tuations can be written as,

S(c,s)00 (q, z) =

−S(c,s)00 (q)

z + q2D(c,s)(q, z)[S

(c,s)00 (q)

]−1 , (B1)

with,

D(c,s)(q, z) = K(c,s)00 (q, z)/q2 (B2)

+∑ν

K(c,s)0ν

([z1 +QK(c,s)(q, z)

]−1

× QK(c,s)(q, z))ν0/q2.

Here, [P]µν = δµ0[1/S(c,s)00 ]δν0 is a projection operator

on the 00-mode and Q =[S(c,s)

]−1 − P the respective“pseudo” orthogonal projector [77]. The above equa-tions were derived using Gram-Schmidt orthogonalisa-tion procedure, however, as can be seen the solution isindependent of the procedure, in particular and mostimportantly it does not depend on the transformationmatrix. Although derived for Newtonian dynamics, the

14

above results also hold for the present case of Browniandynamics due to the similarity of the equations of mo-tion for the intermediate scattering function. Here wehave K(q, z) = δK(q, z) + iD(q). In the hydrodynamiclimit with q → 0 and z → 0 for z/q2 = const. we findimmediately for the coherent dynamics,

S(c)00 (q, z) ' −S(c)

00 (q)

z + iq2D(c)0

[S

(c)00 (q)

]−1 , (B3)

due to the decoupling property limq→0Kαβ,(c,s)µν (q, t) =

Kα,(c,s)µν (t)δαβ and since M‖,(c)µσ (t) = O(q2). Conse-quently, all other terms in the denominator scale asO(q4). The incoherent dynamics are given by,

S(s)00 (q, z) ' −1

z + q2(iD(s)0 + δK‖,(s)00 (z))

, (B4)

with the finite force kernel δK‖,(s)00 (z) in the hydrody-namic limit q → 0 and z → 0 with z/q2 = const. Sincethe memory kernel contains the “fast” variables (con-nected to a ‘fluctuating force’ in Brownian dynamics) weassume that in this limit it reduces to the long-time dif-

fusion coefficient D‖ = limz→0

(−i)δK‖,(s)00 (z)+D(s)0 and thus

we find,

S(s)00 (q, z) ' −1

z + iq2D‖. (B5)

2. Derivation of the higher modes of theintermediate scattering function

In this part we want to show that,

Sµν(t) =Sµ0Sν0

S200

S00(t). (B6)

Our derivation is based on Eq. (C7) in Ref. [53],[z1 + K

]P0S +

[z1 + K

]Q0S = −1, (B7)

where A = L†AL, P0 is the projector onto the distin-guished subspace |Π0(q)〉 =

∑µ |ρµ(q)〉Lκ0, Q0 = 1−P0

is the orthogonal projector and [L]µν = Lµν is the Gram-Schmidt triangular transformation matrix. Note that thederivation is completely independent of the coherent andincoherent dynamics, we will therefore not use the ex-plicit notation (c,s). In the following, all quantities withthe symbol ˆ denote the Laplace transformed quantitiesand thus carry an explicit z-dependence.

First we sandwich Eq. (B7) with LQ0(...)P0L†, to find,

QSP = −[z1 +QK

]−1

QKPSP, (B8)

corresponding to Eq. (C9) in Ref. [53]. Here, PSP isthe projection on the 00-mode an thus corresponds to ahydrodynamic pole (see Sec. II C). The projection KP =O(q) is of higher order, because the off-diagonal terms

vanish in the hydrodynamic limit limq→0Kαβ,(c,s)µν (q, t) =

Kα,(c,s)µν (t)δαβ , as discussed in the main manuscript and

thus,[K]µ0

= q2K‖‖µν + O(q). We therefore conclude

that,

QKP = O(q). (B9)

The O is a notation to indicate that the order of ap-proximation has to be read relative to the hydrodynamicpole. From Eq. (B9), using the explicit expressions for

[P]µν = δµ0 [1/S00] δν0 and Q = [S]−1 − P, we immedi-

ately find,

Sµ0(z) =Sµ0

S00S00(z) + O(q), (B10)

which in the hydrodynamic limit means that the Sµ0(t)only decay due to a coupling to the hydrodynamic polein S00(t).

Similarly, we can sandwich Eq. (B7) withLQ0(...)Q0L

†, to find,

QKPSQ+[z1 +QK

]QSQ = −Q, (B11)

which we can reorganize to obtain,

QSQ = −[z1 +QK

]−1

(Q+QKPSQ) = O(q). (B12)

As already indicated we can use the argument from be-fore to show that, QSQ = O(q), and thus conclude,

Sµν(z) =Sµ0Sν0

S200

S00(z) + O(q), (B13)

which is what we wanted to show in the hydrodynamiclimit. It should be noted that the amplitude Tµν =Sµ0Sν0 is positive-definite which can be readily concludedfrom the positive-definiteness of S(t) for all times t.

3. Long-time anomalies in the velocityauto-correlation function

We start with Eq. (38) in the hydrodynamic limit,

M‖,(s)µν (t) ' n0

∑µ1,µ2ν1,ν2

S(s)µ1µ2

S(c)ν1ν2

∫ ∞0

dkkY‖,(s)µµ1µ2(k)

(B14)

× exp(−D(c)

0 k2t/S(c)00 (0)

)exp

(−D‖k2t

)Y‖,(s)νν1ν2(k)∗,

and the knowledge that the vertices Y‖,(s)νν1ν2(k) are smoothfunctions of the control parameters, which themselves

15

converge in the hydrodynamic limit to a finite value.Eq. (B15) can therefore be rewritten as,

M‖,(s)µν (t) ' n0

∑µ1,µ2ν1,ν2

S(s)µ1µ2

S(c)ν1ν2Y‖,(s)µµ1µ2

Y‖,(s)∗νν1ν2 (B15)

×∫ ∞

0

dkk2 exp(−D(c)

0 k2t/S(c)00 (0)

)exp

(−D‖k2t

),

with,

Y‖,(s)µµ1µ2=

1

2√πLLs

∑σ

[(S(s))−1

]µσc(s)σ−µ2,µ1

(0). (B16)

The above expression for the memory kernel correspondsto a Gaussian integral which can be trivially integrated

to find M‖,(s)µν (t) ' B‖,(s)µν t−2, with,

B‖,(s)µν =n0S

(c)00 (0)

2(D

(s)L +D

(c)0 /S

(c)00 (0)

)2 (B17)

×∑µ1,µ2ν1,ν2

S(s)µ1µ2

S(s)ν1ν2Y‖,(s)µµ1µ2

Y‖,(s)∗νν1ν2 .

From the long-time tail in the memory kernel M‖,(s)µν (t)we can now conclude using Taubers theorem that wehave in Laplace space a leading singularity scaling as(−iz) ln(−iz) (see Ref. [78] Eq. (48-51), and Ref. [79],Ch.13.5.)

M‖,(s)µν = iA0,µν + iA1,µν(−iz) + iB‖,(s)µν (−iz) ln(−iz)

+O(z2 ln(z2)). (B18)

From Eq. (22) we find,

A0 =[D‖,(s)L

]−1

−[D‖,(s)

]−1

(B19)

where,[D‖,(s)L

]µν

= limz→0

(−i)δK‖,(s)µν (z) +D‖‖,(s)µν , (B20)[D‖,(s)

]µν

= D‖‖,(s)µν . (B21)

Similar to Eq. (22) we can derive an equation for

δK(s)

(z),

δK(s)

(q, z) =[iD(s)(q)−1 + M

(s)(q, z)

]−1

×(−iD(s)(q)M

(s)(q, z)

), (B22)

which we can expand using (A + B)−1 = A−1 +A−1BA−1 for small B,

δK‖‖µν(z) = ...−[iD‖,(s)

−1+ A0

]−1

µαiB‖,(s)αβ (−iz) ln(−iz)

×[iD‖,(s)

−1+ A0

]−1

βν+ ... (B23)

= ...−[D‖,(s)L

]µα

iB‖,(s)αβ (−iz) ln(−iz)

[D‖,(s)L

]βν

+ ...

(B24)From the leading singularity in Laplace space we there-fore find,

δK‖‖µν(t) ' −[D‖,(s)L

]µαB‖,(s)αβ

[D‖,(s)L

]βνt−2. (B25)

In particular, for the velocity autocorrelation function,

Z‖(t) ' −[D‖,(s)L

]0αB‖,(s)αβ

[D‖,(s)L

]β0t−2, (B26)

we thus find a power law tail, Z‖(t) ∝ −t−2.

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