Dynamic glass transition in two dimensions

Dynamic glass transition in two dimensions

M. Bayer,¹J. M. Brader,¹ F. Ebert,¹M. Fuchs,¹ E. Lange,¹G. Maret,¹R. Schilling,^2,*_{M. Sperl,}³and J. P. Wittmer⁴

1Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany

2Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 7, D-55099 Mainz, Germany

3Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft-und Raumfahrt, 51170 Köln, Germany

4Institut Charles Sadron, 6 rue Boussingault, 67083 Strasbourg, France 共Received 7 March 2007; published 20 July 2007兲

The question of the existence of a structural glass transition in two dimensions is studied using mode coupling theory共MCT兲. We determine the explicitddependence of the memory functional of mode coupling for one-component systems. Applied to two dimensions we solve the MCT equations numerically for mono- disperse hard disks. A dynamic glass transition is found at a critical packing fraction ␸cd=2⬵0.697 which is above␸c

d=3⬵0.516 by about 35%.␸c

dscales approximately with␸rcp

d , the value for random close packing, at least ford= 2, 3. Quantities characterizing the local, cooperative “cage motion” do not differ much ford= 2 and d= 3, and we, e.g., find the Lindemann criterion for the localization length at the glass transition. The final relaxation obeys the superposition principle, collapsing remarkably well onto a Kohlrausch law. Thed= 2 MCT results are in qualitative agreement with existing results from Monte Carlo and molecular dynamics simula- tions. The mean-squared displacements measured experimentally for a quasi-two-dimensional binary system of dipolar hard spheres can be described satisfactorily by MCT for monodisperse hard disks over four decades in time provided the experimental control parameter⌫共which measures the strength of dipolar interactions兲and the packing fraction␸are properly related to each other.

DOI:10.1103/PhysRevE.76.011508 PACS number共s兲: 64.70.Pf, 61.20.Lc, 61.43.Fs

I. INTRODUCTION

The static and dynamic behavior of macroscopic systems depends sensitively on the spatial dimensiond. For example, one-dimensional systems with short-range interactions do not exhibit an equilibrium phase transition. In two dimen- sions there is no long-range order if the ground state exhibits a spontaneously broken continuous symmetry and Anderson localization occurs for almost all eigenstates of a disordered system ford= 1 andd= 2, but not in d= 3, if the disorder is small. Critical exponents at continuous phase transitions de- pend on dimensionality. Concerning dynamical features it is known, for instance, that the velocity autocorrelation func- tion of a liquid exhibits a long-time tail proportional tot^−d/2. Consequently, the diffusion constant is infinite for d艋2.

These few examples demonstrate the high sensitivity of vari- ous physical properties of the dimensiond.

Let us consider a liquid ind= 3. If crystallization can be bypassed, a liquid undergoes a structural glass transition. Al- though not all features of this transition are completely un- derstood, recently significant progress has been made con- cerning its microscopic understanding. Following many decades of several phenomenological descriptions with less predictive power, the mode coupling approach introduced in 1984 by Bengtzelius, Götze, and Sjölander关1兴has led to a microscopic theoryof the structural glass transition.

This theory, called mode coupling theory 共MCT兲, has been discussed theoretically in great detail by Götze and co- workers共see Ref.关2兴for a review兲. Its numerous predictions were largely successfully checked by experiments and simu- lations关3,4兴. The main prediction of MCT is the existence of

adynamicalglass transition at which the dynamics changes from ergodic to nonergodic behavior. Thermodynamic共equi- librium兲quantities—e.g., the isothermal compressibility and structural ones like the static structure factor S共q兲—do not become singular at the glass transition singularity of MCT.

Hence, the MCT glass transition is of pure dynamical nature.

It can be smeared-out by additional relaxation channels and then marks a crossover关2,3兴.

Amicroscopictheory predicting a structural glass transi- tion with pure thermodynamic origin was derived by Mézard and Parisi 关5兴. Their replica theory is a first-principles ap- proach which yields a so-called Kauzmann temperatureT_Kat which the configurational entropy vanishes.TK is belowTc, the MCT glass transition temperature. In low dimensions,TK

may mark a crossover关6兴. For a review of both microscopic theories as well as phenomenological approaches to the structural glass transition the reader may consult Ref.关7兴.

An important question is now, what is the dependence of the structural glass transition on the spatial dimensionality?

This question has already been asked by several researchers some time ago. Before we come to a short review of this work, let us consider monodisperse hard spheres and hard disks ind= 3 and d= 2, respectively.

The most dense packing of four hard spheres corresponds to a regular tetrahedron. However, three-dimensional space cannot be covered completely by regular tetrahedra, without overlapping. This kind of geometrical frustration is absent in two dimensions. The densely packed configuration of three hard disks corresponds to an equilateral triangle. Since the two-dimensional plane can be tiled completely without over- lap by equilateral triangles, there is no frustration. Therefore one may be tempted to conclude that there is no structural glass transition in two dimensions. However, the link be- tween frustration and glass transition has proven subtle. Ex-

*Electronic address: rschill@uni-mainz.de

1539-3755/2007/76共1兲/011508共12兲 Konstanzer Online-Publikations-System (KOPS) 011508-1 ©2007 The American Physical Society URL: http://www.ub.uni-konstanz.de/kops/volltexte/2007/3852/

periments 关8兴 and simulations 关9,10兴 of monodisperse hard spheres in three dimensions have shown crystallization.

Hence, the existence of geometric frustration is not sufficient for glass formation. What one needs is bidispersity or poly- dispersity.

Santen and Krauth have performed a Monte Carlo共MC兲 simulation of a two-dimensional system of polydispersehard disks. The polydispersity has been quantified by a parameter

␧^pol. Their results clearly demonstrate 共i兲 the absence of a thermodynamic glass transition and 共ii兲 the existence of a dynamic glass transition at a critical packing fraction

␸c共␧^pol兲. The kinetic glass transition shifted outside the re- gion of crystallization for␧^pol艌␧_min^pol⬇10% 关11,12兴. Further- more, the diffusivity was found to be consistent with the MCT result关2,3兴

D⬃ 共␸c−␸兲^␥, ␸艋␸c,

where␥⬇2.4 and␸csim⬇0.80. This critical value agrees with what has been found for a related system by Doliwa and Heuer共see Fig. 2 in Ref.关13兴兲. The absence of a thermody- namic glass transition has been strengthened recently by Donevet al. 关14兴for a binary hard-disk mixture.

There are a few investigations of glass formation in two- dimensional systems with soft potentials. Lançon and Chaudhari 关15兴 studied a binary system with modified Johnson potential. They found that the structural relaxation time seems to diverge when approaching a critical tempera- ture. Similar behavior was observed by Ranganathan关16兴for a monodisperse Lennard-Jones systems and by Perera and Harrowell关17兴for abinarymixture of soft disks with a 1 /r¹² potential. It is surprising that the intermediate self-scattering functionS^共s兲共q,t兲 of the monodisperse system 关16兴 exhibits strong stretching, one of the characteristics of glassy dynam- ics. However,S^共s兲共q,t兲does not produce a well-pronounced plateau 关16兴 under an increase of the density; i.e., the cage effect does not become strong enough. This is quite different to the binary system关17兴. S^共s兲共q,t兲 develops atwo-step re- laxationprocess upon supercooling with a well-pronounced plateau over four to five decades in time, at lower tempera- tures. This behavior is qualitatively identical to that found for, e.g., the Lennard-Jones mixture investigated and ana- lyzed in the framework of MCT by Kob and Anderson关18兴.

The authors of Refs.关11–17兴conclude that there is a struc- tural glass transition in two dimensions. Their conclusion is supported by recent experiments on colloidal particles with repulsive dipolar interactions in two dimensions关19兴. Since these simulational and experimental findings strongly re- semble the MCT predictions obtained for d= 3, it is impor- tant to apply MCT to two-dimensional liquids. MCT has been applied to the two-dimensional Lorentz model of over- lapping hard disks 关20兴 and a charged Bose gas with quenched disorder and logarithmic interactions at zero tem- perature 关21兴, but to our best knowledge not to a two- dimensional liquid-glass problem with self-generated disor- der. To accomplish this is the main motivation of the present contribution.

The outline of our paper is as follows. The MCT equa- tions forarbitrary dimensions and the major predictions of MCT will be presented in Sec. II. The theory requires as only

input the static structure factor, which is computed in Sec.

III. In Sec. IV we apply MCT to a two-dimensional system of monodisperse hard disks and will demonstrate that there is a dynamic glass transition. The dynamic behavior close to that transition is qualitatively identical to that of monodis- perse hard spheres in three dimensions. It will also be shown that the MCT result for the time-dependent mean-squared displacement describes the experimental result of Ref. 关19兴 for both species rather satisfactorily over four decades in time. The final section V contains a short summary and some conclusions.

II. MODE COUPLING EQUATIONS

In this section we will shortly review the MCT equations for the collective and tagged particle correlator of density fluctuations of a one-component liquid and will present the properties of their solution close to the glass transition sin- gularity. The only dependence on dimension d comes through the integrations element 共2␲兲^−dd^dk⬃共2␲兲^−dk^d−1dk which appears in the memory kernels.

MCT provides equations of motion for the normalized intermediate scattering function␾q共t兲and the tagged particle correlator␾_q^共s兲共t兲. The mathematicalstructureof these equa- tions doesnotdepend ond. For Brownian dynamics which is appropriate for colloidal systems they read ford arbitrary

␥q␾^˙q共t兲+␾q共t兲+

冕

0 t

dt⬘^mq共t−t⬘^兲␾^˙q共t⬘^{兲= 0, 共1兲} with the memory kernelmq共t兲containing fluctuating stresses and playing the role of a generalized friction coefficient. It arises because the density fluctuations captured in ␾q共t兲 are affected by all other modes in the system. In MCT, one as- sumes that the dominating contributions at long times are given by density pair fluctuations and approximates共in the thermodynamic limit兲

mq共t兲 ⬅Fq关␾k共t兲兴=

冕

共2^d␲^dk兲^dV共q,k,p兲␾k共t兲␾p共t兲. 共2兲 The vertices express the overlap of fluctuating stresses with the pair density modes and are uniquely determined by the equilibrium structure

V共q,k,p兲=n 2

S_qS_kS_p

q⁴ 关q·kc_k+q·pc_p兴²␦共q−k−p兲, 共3兲 where n is the number of particles per d-dimensional vol- ume,S_q the static structure factor, andc_q the direct correla- tion function related toS_qby the Ornstein-Zernike equation.

␥q is a characteristic microscopic time scale. The reader should note that the vertices, Eq. 共3兲, have been approxi- mated by neglecting static three-point correlations.

The corresponding equations for the tagged-particle cor- relator andd arbitrary are of the same form

␥q共s兲␾^˙q共s兲共t兲+␾q共s兲共t兲+

冕

0 t

dt⬘^mq共s兲共t−t⬘兲␾^˙q共s兲共t⬘兲= 0, 共4兲 with

m_q^共s兲共t兲 ⬅Fq共s兲关␾k共t兲,␾k共s兲共t兲兴=

冕

共2^d␲^dk兲^dV^共s兲共q,k,p兲␾k共t兲␾p共s兲共t兲 共5兲 and

V^共s兲共q,k,p兲=nS_kq⁻⁴共q·k兲²共c_k^共s兲兲²␦共q−k−p兲. 共6兲 c_k^共s兲=具␳q

共s兲*␳q典/共nS_q兲 is the tagged-particle direct correlation function. If the tagged particle is one of the liquid’s particles, it isc_k^共s兲=ck.

The static correlation functionsSk,Sp,Sq,ck, andcp and the correlators ␾k共t兲, ␾_k^共s兲共t兲, ␾p共t兲, and ␾_p^共s兲共t兲 depend on q=兩q兩, k=兩k兩, and p=兩p兩 only, due to the isotropy of the liquid and glass phase. Therefore thed-dimensional integrals in Eqs.共2兲and共5兲can be reduced to a twofold integral over k and p. This will make explicit the d dependence of the vertices. As a result one obtains, similarly to the case in d= 3关1兴,

Fq关␾k共t兲兴=n⍀d−1

共4␲兲^d S_q q^d+2

冕

0

⬁

dk

冕

_兩q−k兩 q+k

dp

⫻ kpS_kS_p

关4q²k²−共q²+k²−p²兲²兴^{共3−d兲/2}关共q²+k²−p²兲c_k +共q²−k²+p²兲cp兴²␾k共t兲␾p共t兲 共7兲 and

Fq共s兲关␾k共t兲,␾k共s兲共t兲兴= 2n⍀d−1

共4␲兲^d 1 q^d+2

冕

0

⬁

dk

冕

_兩q−k兩 q+k

dp

⫻ kpSk

关4q²k²−共q²+k²−p²兲²兴^{共3−d兲/2}

⫻关共q²+k²−p²兲ck共s兲兴²␾k共t兲␾共ps兲共t兲, 共8兲 with

⍀d= 2␲^d/2

⌫

冉

^d2

冊

^{, 共9兲}

the well-known result for the surface of ad-dimensional unit sphere.⌫共x兲is the gamma function. Note that Eq.共9兲yields 关with⌫共₂¹兲=␲^1/2兴

⍀1= 2, 共10兲

which is consistent with the fact that the “one-dimensional unit sphere” is an interval of length two with a “surface”

consisting of two points.

The behavior for q→0 of both functionals can be ob- tained by a Taylor expansion of 兰_k−q^k+qdp共¯兲. Although straightforward it is rather tedious. Alternatively, one can start directly from Eqs.共2兲and共5兲, in order to obtain forq

→0

Fq关␾k共t兲兴→n 2

⍀d

2␲^dS0

冕

0

⬁

dkk^d−1S_k²

冋

^{ck2+d2kckck⬘}

+ 3

d共d+ 2兲k²c_k⬘²

册

^{关␾k共t兲兴2+O共q兲 共11兲}

and

Fq共s兲关␾k共t兲,␾k共s兲共t兲兴→n ⍀d

d共2␲兲^d 1 q²

冕

0

⬁

dkk^d+1Sk共ck共s兲兲²

⫻␾k共t兲␾k共s兲共t兲+O共1/q兲. 共12兲 Note that F₀关␾k共t兲兴 exists, whereas F_q^共s兲关␾k共t兲,␾k

共s兲共t兲兴 di- verges likeq⁻². This divergence is related to the absence of momentum conservation for the tagged particle.

Takingd= 3 in Eqs.共7兲,共8兲,共11兲, and共12兲, one arrives at the well-known representations of the memory kernel for fi- nite q andq→0 关1,22,23兴. Note that Refs. 关22,23兴 already present the integrals in Eqs.共7兲and共8兲in discretized form.

With knowledge of the number density, of the static cor- relators S_q, c_q, and c_q^共s兲 as functions of the thermodynamic variables, and of␥q and␥q

共s兲 one can solve Eqs. 共1兲 and共4兲 for initial conditions ␾q共0兲= 1 and ␾_q^共s兲共0兲= 1. There exist several quantities which characterize the solutions. These quantities can be found in Refs.关2,22,23兴. In order to keep our presentation self-contained as much as reasonable we discuss those for which results will be reported in the next section. We start with the glass transition singularity. At the glass transition the nonergodicity parameters共NEPs兲

f_q= lim

t→⬁␾q共t兲 共13a兲

change discontinuously from zero to a positive nonzero value, smaller or equal to 1. The corresponding quantity

f_q^共s兲= lim

t→⬁␾q共s兲共t兲 共13b兲 can change discontinuously at the same point or in a continu- ous fashion at higher densities or lower temperatures. Both NEPs fulfill the nonlinear algebraic equations共2兲

fq

1 −fq

=Fq关fk兴, f_q^共s兲

1 −f_q^共s兲=Fq共s兲关fk,f_k^共s兲兴. 共14兲 f_q^c and f_q^共s兲care the NEP at the critical point—e.g., atn=nc. Since we will apply MCT to d-dimensional hard spheres with diameter 2R, we use in the following the packing frac- tion ␸⁼n⍀d−1共R兲^d/d. Above, but close to ␸⁼␸c—i.e., for 0⬍␧⬅共␸⁻␸c兲/␸cⰆ1—it is

f_q=f_q^c+h_q关

冑

_␴/共1 −␭兲+␴共K¯

q+␬兲/

冑

1 −␭兴, 共15兲 with the critical amplitude

hq=共1 −f_q^c兲²e_q^c, 共16兲 the separation parameter

␴共␧兲=C␧+O共␧²兲, 共17兲 and the so-called exponent parameter ␭, which obeys 0⬍␭⬍1. The second term on the right-hand side of Eq.共15兲

is the leading asymptotic result for fq, andK¯

q+␬ ^{yields the} next-to-leading order correction.C,␭,e_q^c, andK¯

q+␬ ^follow fromFq关fk兴and its derivatives with respect tofkat␸c. Par- ticularly,e_q^cis the right eigenvectoreqbelonging to the larg- est eigenvalue E_max共␸兲 of the stability matrix 共⳵Fq关fk兴/⳵^fk兲 at the critical point.E_max共␸兲 is not degenerate since the sta- bility matrix is non-negative and irreducible. At the critical point the maximum eigenvalue becomes 1; i.e., ␸c can be determined from the conditionE_max共␸c兲= 1.

At the critical point␾q共t兲 decays to the plateau value f_q^c, 0⬍f_q^c⬍1. Its time dependence is given by

␾q

c共t兲=f_q^c+hq共t/t0兲^−a兵1 +关Kq+␬共a兲兴共t/t0兲^−a其, tⰇt₀. 共18兲 K_q 共not to be confused with K¯

q兲 and ␬共x兲 are again deter- mined by Fq关f_k兴 and its derivatives at ␸⁼␸c. They are a measure of the next-to-leading order contribution with re- spect to the leading asymptotic result

␾q

c共t兲=f_q^c+h_q共t/t₀兲^−a, tⰇt₀, 共19兲 the critical law. This critical decay occurs on a time scalet much larger than a typical microscopic timet₀. The exponent ais determined by ␭, only.

For␧⬍0 two␴-dependent, divergent time scales exist:

t_␴=t₀兩␧兩^−1/2a, ␧ ⬎ ⬍0, 共20a兲 and

t_␴⬘^=t0⬘兩␧兩^−␥, ␧ ⬍0, 共20b兲 with␥⁼_2a¹⁺_2b^{1 and t}0⬘^=t0/B^1/b, where B is a constant. The so-called von Schweidler exponent b follows from ␭ only.

␾q共t兲 exhibits a two-step relaxation. The relaxation for t/t_␴Ⰶ1 to the critical plateau valuef_q^cfollows from Eq.共18兲 by replacing 共t/t₀兲 through 共t/t_␴兲, and the decay from that plateau to zero is initiated by the von Schweidler law for t_␴ⰆtⰆt_␴⬘^:

␾q共t兲=f_q^c−h_q共t/t_␴⬘兲^b兵1 −关K_q+␬共−b兲兴共t/t_␴⬘兲^b其. 共21兲 K_q+␬共−b兲determines again the next-to-leading-order contri- bution.

For␧⬎0 there is a single relaxation process only. ␾q共t兲 relaxes for t/t_␴Ⰶ1 like for ␧⬍0, and finally the plateau value fq is reached by an exponentially long time decay.

For␧⬍0, the final or ␣-relaxation process describes the decay of the correlators from the plateau f_q^c down to zero.

Asymptotically close to the transition, the functional form of the ␣ process is given by a master function ␾^˜q共t˜兲 of the rescaled timet˜=t/t_␴⬘ ^via

␾q共t兲=␾^˜q共t˜兲+␧␾^˜q共2兲共t˜兲+O共␧²兲for␧→0 − . 共22兲 The master function␾^˜qobeys an equation similar to Eq.共1兲 with vertices evaluated right at the critical point,␧= 0. Thus it does not depend on separation␧ and control parameters, and Eq. 共22兲 expresses the often observed “共time- temperature兲 superposition principle” 关2,3兴. The von Sch- weidler series, Eq.共21兲, gives the short-time behavior of␾^˜q

for t˜→0, and the corresponding result for the correction is

␾

˜q

共2兲共t˜→0兲→hqB₁Ct˜^−b, with C from Eq. 共17兲 and B₁ a known constant.

Similar leading-order and next-to-leading-order contribu- tions can be derived for the tagged-particle correlator

␾q

共s兲共t兲 and, e.g., the mean-squared displacement

␦^r2共t兲=具关r共t兲−r共0兲兴²典 关23兴. Since

␦^r2共t兲= lim

q→0

2d

q²关1 −␾q共s兲共t兲兴, 共23兲 the long-wave limit of Eq. 共4兲yields after integration with respect tot,

␦^r2共t兲+D₀^共s兲

冕

0 t

dt⬘^m˜0共s兲共t−t⬘兲␦^r2共t⬘兲= 2dD₀^共s兲t, 共24兲 where the memory kernel m˜₀^共s兲共t兲 follows from Eqs.共5兲and 共12兲:

m˜₀^共s兲共t兲 ⬅lim

q→0q²Fq共s兲关␾k共t兲,␾k共s兲共t兲兴

=n ⍀d−1

d共2␲兲^d

冕

0

⬁

dkk^d+1Sk共ck共s兲兲²␾k共t兲␾k共s兲共t兲. 共25兲

Furthermore, we have used

␥q共s兲= 1/共D₀^共s兲q²兲, 共26兲 withD₀^共s兲the short-time diffusion constant of the tagged par- ticle. Equations.共13b兲and共23兲imply that the long-time limit of␦^r2共t兲,

lim

t→⬁␦^r2共t兲= 2dr_s², 共27兲 is related to the tagged particle’s localization lengthr_sgiven by

r_s²= lim

q→0

1 −f_q^共s兲

q² . 共28兲

In the liquid phase where f_q^共s兲= 0, Eq.共27兲givesr_s=⬁—i.e., the particle is delocalized—while rs becomes finite at the glass transition.

Besides the correlators ␾q共t兲 and ␾_q^共s兲共t兲 one can also study the corresponding susceptibilities ␹q共␻兲 and ␹q

共s兲共␻兲, respectively. Similar asymptotic laws and next-to-leading- order corrections exist for them 关22,23兴. Independent of whether the correlators or their susceptibilities are consid- ered, the dependence on d of the leading- and next-to- leading-order terms enters only via the d dependence ofFq

关Eq.共7兲兴andF_q^共s兲 关Eq.共8兲兴.

III. STATIC STRUCTURE

In this section we consider the calculation of accurate equilibrium structural correlation functions for the hard disk system. Within MCT, all information regarding the interpar- ticle interactions is contained in the static structure factor which enters the memory function vertices, Eqs.共3兲and共6兲;

the interaction potential does not enter explicitly in the MCT equations. Experience with MCT calculations in three- dimensional systems has shown that the location of the glass transition somewhat depends on the details of the input struc- ture factor, particularly the height of the main peak. Different approximate theories for the static structure lead to varying values for, e.g., the critical packing fraction␾c

共d=3兲 关24兴. We have therefore considered a number of approximation schemes for the two- dimensional static structure factor in order to obtain the best possible values for the description of the ideal glass transition. The quality of the various approxi- mation schemes is assessed by comparison with computer simulation data. 共Monte Carlo simulations of 6⫻10⁴ par- ticles, as well as event-driven molecular dynamics simula- tions of 1089 particles, were performed, both in the NVT ensemble关26兴.兲

Integral equation theories based on the Ornstein-Zernike 共OZ兲 equation provide a powerful method to calculate the pair correlation functions for a given interaction potential 关25兴. The OZ equation is given by

h共r兲=c共r兲+n

冕

^{ddr⬘c共r⬘兲h共r−r⬘兲, 共29兲}

whereh共r兲⬅g共r兲− 1. This expression must be supplemented by an additional共generally approximate兲closure relation be- tweenc共r兲 and h共r兲. For hard spheres in d dimensions the most widely used closure is the Percus-Yevick共PY兲relation g共r⬍1兲= 0,c共r⬎1兲= 0. In odd dimensions the resulting in- tegral equation can be solved analytically for the direct cor- relation function c共r兲. In even dimensions there exists no analytic solution and a full numerical solution is required 关27兴. Efforts have been made to approximate the numerical PY data by analytic forms关28,29兴but in all cases these fail to reproduce accurately the detailed structure of the numeri- cal solution at high densities. The formally exact closure to the OZ equation for systems with pairwise interactions is given by

h共r兲= − 1 + exp关−␤u共r兲+h共r兲−c共r兲+B共r兲兴, 共30兲 whereu共r兲is the pair potential and B共r兲is the bridge func- tion, an intractable function representing the sum of the most highly connected diagrams in the virial expansion. Setting B共r兲= 0 recovers the familiar hypernetted-chain approxima- tion.

The modified-hypernetted-chain共MHNC兲 approximation is to take B共r兲 from the PY theory solved at some effective densityn^*, different from the true system density n, and to treat this as a variational parameter to ensure thermodynamic consistency between the virial and compressibility routes to the pressure. A detailed description of the MHNC equation can be found in关30兴. The steps taken in solving the MHNC equation are the following: 共i兲 numerically solve the PY equation at density n^*, 共ii兲 use Eq. 共30兲 to find B共r兲

⬅BPY共r;n^*兲,共iii兲solve Eqs. 共29兲 and共30兲 with this bridge function,共iv兲calculate the pressure from the virial and com- pressibility equations, and 共v兲 adjust n^* until the two pres- sures are equal. In three dimensions it is generally recog- nized that the MHNC approximation provides a highly

accurate description of the pair correlations for the hard- sphere fluid, significantly improving upon the PY theory. We find that the same is true in the case of two-dimensional hard disks. At low densities the MHNCSq lies very close to the PY result. As the density increases discrepancies begin to arise, particularly in the region of the main peak, with the MHNC in closer agreement with simulation. Both the MHNC and PY theories are significantly more accurate than the analytical Baus-Colot expression关28兴. Figure1 shows a comparison between the MHNC Sq and the simulation re- sults. The level of agreement is very satisfactory, and the MHNC shows clear improvement over the other theories in- vestigated. The only notable deviation from the simulation results is the height and width of the second 共third兲 peak, which is overestimated 共underestimated兲 by the MHNC theory. It is known that upon approaching the crystallization phase boundary共located at␾F= 0.69 for hard disks兲a shoul- der develops on the second peak of the structure factor, a feature which has been interpreted as an indicator of ap- proaching crystallization关31兴. The development of the shoul- der also suppresses the height of the second peak to some extent and leads to a small shift in the location of the third peak. The MHNC theory, like the PY and all other standard integral equation theories, does not contain information about crystallization and thus predicts fluidlike structure at, and beyond, the freezing transition. While this property leads to some discrepancy with simulation results at high density, it makes such theories ideal for calculating the fluidlike structure factors required as input to the MCT, where we assume crystallization to have been suppressed. We can thus proceed with confidence using MHNC structure factors as input to the MCT.

IV. RESULTS

In this section we will apply MCT to atwo-dimensional 共2D兲 system of monodisperse hard disks with diameter 2R.

0 10

0.0

q

1.0 2.0 3.0 4.0

S( q)

_{30 32}

1 1.1

6 6.5

2 3

MHNC BC PY

(a) (b)

FIG. 1. Comparison between theoretical MHNC structure fac- tors and simulation for hard disks at packing fractions␾= 0.5共+兲, 0.628共䊊兲, and 0.68共⌬兲;qis given in units of 1 /共2R兲, the inverse diameter. Inset共a兲concentrates on the vicinity of the main peak and gives additional comparison with Baus-Colot and PY theories for packing fraction␾= 0.628. Inset共b兲demonstrates how the MHNC theory correctly captures the asymptotic behavior for largeqvalues for packing fraction␾= 0.628.

We will solve the MCT equations共1兲,共4兲, and共24兲and will present results for those quantities discussed in the last sec- tion. As input we will use the static structure factorsSq ob- tained from the MHNC approach. This result is presented in Fig.2for three different packing fractions close to the glass transition and compared with the corresponding Percus- Yevick result for hard spheres in d= 3. For instance, for

␸⁼␸cd=2 and ␸⁼␸cd=3—i.e., for ␧= 0—the peaks are more pronounced ind= 2. In particular the main peak is more nar- row and higher ford= 2 than ford= 3. The direct correlation functioncqfollows from the Ornstein-Zernike equation. Be- cause we choose the tagged particle as one of the liquid particles, we havec_q^共s兲=cq.

For the friction coefficients in Eqs.共1兲 and 共4兲 we take

␥q=Sq/共D₀^共s兲q²兲 and ␥_q^共s兲= 1 /共D₀^共s兲q²兲 关cf. Eq. 共28兲兴, and choose as our unit of time␶^共BD兲=D₀^共s兲/关10共2R兲²兴; in the fol- lowing all times will be given as rescaled ones,t/␶^{共BD兲. For} the numerical solution of MCT equation one has to discretize q. A compromise between a fine grid and computation time is required, as the computations scale with number of grid points,M³. We choose a grid withM= 250 grid points and a high-q cutoff of 50/共2R兲. A higher cutoff has only a very small effect on the critical packing fraction 共⬍10⁻⁴兲. The effect of the number of grid points is more sensitive due to the form of the Jacobian of the transformation to bipolar coordinates. In this paper the integration is substituted by a rule that can be called a modified trapezoid rule. The value of the function to be integrated is not taken in the middle of the interval关nh,共n+ 1兲h兴 but at共n+ 0.303兲h. By using this rule one gets the best discrete description of the Jacobian and hence it is used here. The difference in␸ ^{between M= 500} andM= 250 is then⬍10⁻³, leading to a system close enough to continuum. Then the solution of Eqs. 共1兲, 共4兲, and 共24兲 will be performed by use of a decimation technique关32兴.

The search for the glass transition singularity can be done either by an iterative solution of the nonlinear equations共14兲 or by calculation ofE_max共␸兲. In this paper a simple bracket- ing algorithm is used starting from two points where pointA

yields a finite NEP forq near the peak position and pointB has fq= 0. The next point C is taken in the middle of the interval. If it yields finitefqthe next point is taken betweenA and,C; if f_q= 0 the other interval关C,B兴is taken. This pro- cedure is continued until the critical packing fraction ␸c is determined to a precision of 10⁻¹⁰. As a result we have found

␸c

d=2⬵0.696810890共317兲, which is above the value ford= 3关22兴:

␸c

d=3⬵0.51591213共1兲.

共The denoted accuracy will be required to reliably compute␧ in the following.兲 Similar to the three-dimensional system, the collective part and self-part of the density fluctuations become nonergodic at the same critical packing fraction

␸c

d=2. The corresponding critical nonergodicity parameters are shown in Fig.3.

While almost no difference between incoherent NEPs for d= 2 and d= 3 can be observed, more pronounced maxima appear in the coherent NEP at higher wave vectors for the lower dimension. Regions of rather abruptqdependences in fq should be observable experimentally. Two length scales appear to be involved in fq. While the average particle dis- tance, connected to the main peak in S_q, somewhat differs fromd= 3 tod= 2, the localization length, which dominates the incoherent NEP, is insensitive to dimensionality. The change of fq when stepping down in dimension thus cannot simply be scaled away.

An important observation in the numerical solution of Eq.

共14兲concerns the convergence of the required integrals. We find here, and for all other integrations performed, that con- vergence at small and large wave vectors holds. No critical anomalies arise connected with the growth of a static corre- lation length 关33兴. We interpret this as indication that the

0 5 10 15 20

q 0

1 2 3 4 5 6

S q

FIG. 2. 共Color online兲Comparison of 2D MHNC structure fac- tors共black/dark兲 to 3D PY共magenta/light兲. The packing fractions correspond to␧= 0共solid lines兲, ␧= −10^−4/3共dotted lines兲, and␧= + 10^−4/3共dashed lines兲. The critical packing fractions are 0.697 for 2D and 0.516 for 3D.

0 10 q 20 30

0 0.2 0.4 0.6 0.8 1

fq,fqs

FIG. 3. Nonergodicity parameter of coherent共2D diamond, 3D dashed line兲and incoherent共2D circles, 3D solid line兲correlators at critical packing fraction␸c

2D= 0.697 and␸c

3D= 0.516. Theqvalues forq⬍1 are not included since they can not be determined accu- rately for numerical reasons. Theq= 0 results are from the analytic expansions, Eqs.共11兲and共12兲, and are included as triangles.

MCT glass transition ind= 2 describes a local phenomenon not affected by long-range correlations, which might sensi- tively depend on dimensionality.

All wave-vector-dependent structure functions describing the glassy structure and its relaxation 共“cage effect”兲 are summarized in Fig. 4. The critical nonergodicity parameter f_q^cand the critical amplitudehqare depicted in Fig.4共b兲and the next-to-leading-order amplitudes Kq and K¯

q 关cf. Eqs.

共15兲,共18兲, and共21兲兴and in Fig. 4共c兲. The q dependence of the shown quantities follows generally the one ofS_q. Onlyh_q exhibits the opposite variation around the main structure fac- tor peak. Figure4共a兲specifies three typical wave numbers for which results will be discussed below. Comparison with the d= 3 result共Fig.2in Ref.关22兴兲reveals a qualitatively similar qdependence of f_q^c,hq,Kq andKq. Like forSq, theq varia- tion of all quantities is more pronounced in d= 2 than in d= 3.

As mentioned in Sec. II, the separation parameter ␴共␧兲 can be calculated fromFq关f_k兴and its derivatives at ␸c. The result is given in the inset of Fig.5. The linear term in Eq.

共17兲describes␴共␧兲for −0.030艋␧艋0.025 with an accuracy better than 10%. This range is similar to that for the corre- sponding result ford= 3关22兴and provides an estimate for the range of validity of the asymptotic expansions. The quality of the leading-order result and the next-to-leading-order con-

tribution forfq关cf. Eq.共15兲兴is demonstrated in Fig.5for the threeqvaluesq₁,q₂, andq₃关Fig.4共a兲兴. It is interesting, that forq₂ the leading asymptote describesfq共␧兲best and for an unexpectedly wide range. This arises because of a cancella- 0.2

0.4 0.6 0.8

f_q^c h_q

0 5 10 15 20

q -2

-1 0

1 ^Kq

K__q 1 2 3 4 5 6 S_q

q₁

q₂

q₃ (a)

(b)

(c)

FIG. 4.共a兲Structure factorS_qas function of wave vectorqfor

␸=␸c⬇0.697 共solid line兲, ␸⬇0.729 共dotted line兲, and ␸⬇0.664 共dashed line兲; the latter correspond to␧= ± 10^−4/3. The arrows mark the wave vectorsq₁= 6.46, q₂= 10.06, and q₃= 18.26. 共b兲 Critical NEP f_q^c 共diamonds兲 and critical amplitude h_q 共squares兲. 共c兲 The amplitudesK_q共triangles兲andK¯

q共circles兲.

0 0.02 0.04 0.06 0.08 0.1

0.2 ε

0.4 0.6 0.8 1

f q

-0.1 0 ε0.1

-0.1 0 0.1σ

q₁

q₂ q₃

FIG. 5. Nonergodicity parameterf_qforq₁= 6.46,q₂= 10.06, and q₃= 18.26 共solid lines兲. The leading asymptotes共dashed line兲 de- scribe f_q−f_q^c within 10% up to the values marked by diamonds 共0.005, 0.075, 0.0015兲. The next-to-leading asymptotes共dotted line兲 are within 10% for ␧ up to the values marked by circles 共0.03, 0.033, 0.05兲. The values for the correction amplitudes areC= 2.08,

␬= 1.18, h₁= 0.337, h₂= 0.654, h₃= 0.508,K¯

1= −1.86, K¯

2= −0.456, and K¯

3= 0.844. The inset shows the separation parameter␴ as a function of␧. The dashed line is the linear asymptote␴=C␧. The range where the asymptote deviates less than 10% from␴ is be- tween the diamonds that are at␸= 0.676 and␸= 0.714.

-2 0 2 4 6 8 10

log₁₀(t) 0

0.2 0.4 0.6 0.8

φ

2

(t)

0.2 0.4 0.6 0.8 1

φ

1

(t)

3

6

9 c

3 6 9 c

15 9 12

3 6 n=1

12 15 6 9

3 n=1 ε<0

ε>0 ε<0

ε>0

FIG. 6. Coherent correlators␾1共t兲 共top兲 and␾2共t兲 共bottom兲for different␧= ± 10^−n/3. Curves are labeled byn.

tion of higher-order correction terms. Forq₁andq₃the next- to-leading order has to be taken into account already for rather small␧, as expected from the␴共␧兲curve. The overall behavior also is quite similar tod= 3共cf. Fig.3of Ref.关22兴兲.

Now we turn to dynamical features. Figure6presents the normalized correlators␾i共t兲⬅␾q_i共t兲fori= 1 , 2. The two-step relaxation process for␧⬍0 becomes obvious for both corr- elators. Since f_q

2 c ⬍f_q

1

c 关cf. Figs. 4共a兲 and 4共b兲兴 the plateau heights for␾2共t兲are below those for␾1共t兲. Again, the tand

␻^dependence共the latter is not shown兲is in qualitative agree- ment with the corresponding results ind= 3共cf. Figs.4and6 of Ref.关22兴兲. In the following, we will apply the asymptotic expansions from Sec. II to the correlators in order to charac- terize the long-time dynamics in more detail.

Following Ref. 关22兴 we have calculated typical time scales␶q±共␧⬎ ⬍0兲and ␶q⬘共␧⬍0兲 characterizing the first and second relaxation steps. In the fluid,␶⁻共q兲marks the crossing of the plateau, ␾q共␶⁻共q兲兲=f_q^c, and the ␣-relaxation time is defined by␾q共␶q⬘兲=f_q^c/e. In the glass,␶⁺共q兲captures the ap- proach to the long-time plateau, ␾q共␶q

+兲−f_q^c= 1.001共f_q−f_q^c兲. Results are shown in Fig. 7 for q=q₁. The divergence of these relaxation times at␧= 0 is described by the asymptotic

laws, Eqs.共20a兲and共20b兲. Since␸cd=2has been determined, one can calculate the exponent parameter␭. As a result we find␭^d=2⬵0.7167 which implies a^d=2⬵0.320,b^d=2⬵0.613, and ␥^d=2⬵2.38. These values are close to ␭^d=3⬵0.735, a^d=3⬵0.312, b^d=3⬵0.583, and ␥^d=3⬵2.46 关22兴. Using in Eqs. 共20a兲 and共20b兲 a^d=2 and ␥^d=2 leads to the asymptotes 共solid lines兲in Fig.7.

The microscopic time scalet₀entering the critical power law, Eq.共18兲, can be deduced by plotting关␾q共t兲−f_q^c兴t^aversus logt for ␧⬎0 and␧⬍0 共see Fig.8兲. The value at which a constant plateau is best reached by both curves is 0.114

=hqt₀^a. Withhq₁⬵0.337 anda^d=2we gett₀= 0.034. The qual- ity of the leading-order result 关Eq. 共19兲兴 and its next-to- leading-order correction关second and third terms in the curly bracket of Eq.共18兲兴of the critical law is checked in Fig.9. A similar check for the von Schweidler law共Eq. 共21兲兲is done in Fig.10. Like forf_q^c, the leading order has a large range of validity forq=q₂. Note that both time scales of the structural

-0.0001 -0.01

ε

0 2 4 6 8 10

log10(t)

0.0001 0.01

ε

0 2 4 6 8 10

log 10(t) τ₁^,

τ₁^- τ₁⁺

FIG. 7. Timescales␶^±for the␤process and␶⬘^{for the␣process} in the liquid for wave vector q₁ 共in double-logarithmic presenta- tion兲. The lines are the asymptotic power laws. ␭= 0.7167, a

= 0.320, b= 0.613, ␦⁼_2a^{1= 1.56, c}+= 0.044, c₋= 0.009, ␥=_2a¹+_2b¹

= 2.38, andc⬘^{= 0.0173. 共}For the definitions ofc_± andc⬘^{see Ref.}

关22兴.兲

-1 0 1 2 3 4 5 6

log₁₀(t) 0.06

0.08 0.1 0.12 0.14

(φ1(t)-f1c )*ta

ε>0 ε<0

FIG. 8. Determination oft₀. 关␾1共t兲−f₁^c兴t^ais plotted over logt, for glass and liquid curves close to the transition. The position where the plateau is best reached by both curves is marked by arrows. The value found is 0.114. t₀ is then given by

共

^0.114h1

兲

^1/a

= 0.034, withh₁= 0.337 anda= 0.320.

-2 0 2 4

log₁₀(t)

0.4 0.6 0.8 1

φq(t)

q₁

q₂

FIG. 9. The critical laws. The leading asymptotes共dashed line兲 describe the solution within 10% up to the points marked by dia- monds共41, 0.52兲 for共q₁,q₂兲. The next-to-leading results共dotted兲 are within 10% up to the circles共0.73, 0.17兲for共q₁,q₂兲. Correction amplitudes are␬共a兲= −0.021,K₁= −1.007, andK₂= −0.120.

-4 -2 0

log₁₀(t~)

0.2 0.4 0.6 0.8 1

φ~ q(t~)

q₁

q₂

FIG. 10. The von Schweidler law. The leading asymptotes 共dashed line兲 describe the solution within 10% up to the points marked by diamonds共0.29, 0.053兲for共q₁,q₂兲. The next-to-leading results共dotted line兲are within 10% up to the circles共0.37, 0.47兲for 共q₁,q₂兲 共b= 0.613 and t_␴⬘^{= 7.45}⫻10⁹兲. Correction amplitudes are

␬共b兲= 0.496,K₁= −1.007, andK₂= −0.120.

relaxation are given by the matching timet₀ determined in Fig. 8, and by the separation parameter ␧. Thus, there remains no adjustable parameter in the test of the von Schweidler law in Fig.10.

The critical law and the von Schweidler law are the short- and long-time expansions of the so-called␤-master function g_±共tˆ=t/t_␴兲 for ␧⭵0, respectively. g_±共tˆ兲 describes the first scaling-law regime. The t and ␧ dependence of ␾^ˆq共t兲

=共␾q共t兲−f_q^c兲/共hqC兲on the time scalet_␴is given byg_±共tˆ兲for 兩␧兩→0, independent of on q 关2兴. For the two-dimensional system this property is demonstrated in Fig.11. We clearly observe that the curves for q₁, q₂, and q₃ collapse onto a master function with increasing n—i.e., for ␧→0. Because of the connection between theq dependences of the correc- tion amplitudes in Eqs.共18兲 and 共21兲, an ordering scheme exists for the functions␾^ˆq共t兲in Fig.11. Their vertical order before and after crossing the plateau needs to coincide; this is obeyed in Fig.11.

The second scaling-law regime共for ␧⬍0兲 is defined by the rescaled time t˜=t/t_␴⬘^{, where t}_␴⬘ often is called the

␣-relaxation time and denoted by␶. Fort˜=O共1兲, thetand␧ dependence of ␾q is given by the ␣-master function ␾^˜q共t˜兲 关2兴. The validity of this second scaling law 关Eq. 共22兲兴 is

presented in Fig. 12. Approaching ␧= 0 from below a col- lapse onto aqdependent master function ␾^˜qoccurs.

For the test of the superposition principle, the

␣-relaxation time was computed using the power law, Eq.

共20b兲, andt₀from Fig.8. Atq₁, the␣-relaxation time obvi- ously deviates early from the asymptotic power law, since there are intersections of the rescaled correlators. Neverthe- less, the range of validity of the␣-scaling law in Fig.12far exceeds the one of the␤-scaling law tested in Fig.11. This originates from the dependence of the leading corrections on the separation parameter ␧. While the corrections to the ␤ process are smaller by a factor

冑

_␧ only, the relative correc- tions to the␣-superposition principle start out in order␧. For example, atq₂, the␣master function␾^˜q₂共t˜兲describes 68% of the decay of the final relaxation better than on a 5% error level at the separation␧= −0.01 共see the circle in Fig. 12兲, while in Fig.11for the test of the␤-scaling law smaller␧are required.

The shape of the ␣-relaxation process—viz, its master functions␾^˜q共t˜兲—often is described by a Kohlrausch law

␾^˜q共t˜兲 ⬇A_qexp

再

⁻

^冉

^˜␶t˜q

^冊

^␤q

冎

^{, 共31兲}

where a possible dependence of the parameters on wave vec- torqis taken into account. The von Schweidler expansion of -0.8

-0.4 0 0.4 0.8

-0.8 -0.4 0 0.4 0.8

φ q(t)^

-2 0 2 4 6 8 10

log₁₀(t)

-0.8 -0.4 0 0.4 0.8

n=8

n=11

n=14

3 2 1

32 1

1 1 2 3

2 3

1 2

3 3

2 1

3 2 1

3 2 1 3

2 1

FIG. 11.共Color online兲Functions␾ˆ

q共t兲=关␾q共t兲−f_q^c兴/共h_q冑C兲for q₁= 6.46,q₂= 10.06, andq₃= 18.26 for␧= ± 10^−n/3for 3 values ofn 共solid lines兲. The scaling asymptotesG共t兲/冑Care shown as dashed lines. The region where the functions for differentqcollapse onto a master function共␤region兲increases with decreasing␧.C= 2.08; the curves for␧⬎0 are shifted down by 1.

0.2 0.4 0.6 0.8 1

φ 1(t~)

-5 -4 -3 -2 -1 0 1

log₁₀(t~) 0.2

0.4 0.6 0.8

φ 2(t~)

10 8 6 4

10

8 6

4

q₁

q₂

FIG. 12. 共Color online兲 Correlators ␾q共t˜兲 for different ␧=

−10^−n/3, n= 4 , 6 , 8 , 10, as function of rescaled time t˜=t/t_␴⬘ 共solid lines兲. The thick solid line is the␣-master function␾˜

q共t˜兲. The dot- ted lines are the short time parts of leading-plus-next-to-leading approximation for the ␣ correlators ␾˜

q共t˜兲+h_qB₁␴t˜^−b according to Eq. 共22兲 with B₁= 0.5/关⌫共1 −b兲⌫共1 +b兲−␭兴= 0.374. Light 共ma- genta兲 curves give the Kohlrausch laws fitted to the ␾˜

q共t˜兲 in the range log₁₀共t˜兲=关−3.86, 2.14兴; the parameters are given in Fig.13.