Theory of activated dynamics and glass transition of hard colloids in two dimensions Bo-kai Zhang, Hui-shu Li, Wen-de Tian, Kang Chen, and Yu-qiang Ma Citation: The Journal of Chemical Physics 140, 094506 (2014); doi: 10.1063/1.4866903 View online: http://dx.doi.org/10.1063/1.4866903 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Colloidal rotation near the colloidal glass transition J. Chem. Phys. 135, 054905 (2011); 10.1063/1.3623489 Theory of dynamic barriers, activated hopping, and the glass transition in polymer melts J. Chem. Phys. 121, 1984 (2004); 10.1063/1.1756854 Slow dynamics of supercooled colloidal fluids: Spatial heterogeneities and nonequilibrium density fluctuations AIP Conf. Proc. 519, 21 (2000); 10.1063/1.1291518 The glass transition in binary mixtures of hard colloidal spheres AIP Conf. Proc. 519, 3 (2000); 10.1063/1.1291516 The glass transition of charged and hard sphere silica colloids J. Chem. Phys. 111, 8209 (1999); 10.1063/1.480154
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THE JOURNAL OF CHEMICAL PHYSICS 140, 094506 (2014)
Theory of activated dynamics and glass transition of hard colloids in two dimensions Bo-kai Zhang,1,2 Hui-shu Li,2 Wen-de Tian,2 Kang Chen,2,a) and Yu-qiang Ma1,2,a) 1
National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjng 210093, People’s Republic of China 2 Center for Soft Condensed Matter Physics and Interdisciplinary Research, Soochow University, Suzhou 215006, People’s Republic of China
(Received 15 December 2013; accepted 12 February 2014; published online 4 March 2014) The microscopic nonlinear Langevin equation theory is applied to study the localization and activated hopping of two-dimensional hard disks in the deeply supercooled and glass states. Quantitative comparisons of dynamic characteristic length scales, barrier, and their dependence on the reduced packing fraction are presented between hard-disk and hard-sphere suspensions. The dynamic barrier of hard disks emerges at higher absolute and reduced packing fractions and correspondingly, the crossover size of the dynamic cage which correlates to the Lindemann length for melting is smaller. The localization lengths of both hard disks and spheres decrease exponentially with packing fraction. Larger localization length of hard disks than that of hard spheres is found at the same reduced packing fraction. The relaxation time of hard disks rises dramatically above the reduced packing fraction of 0.88, which leads to lower reduced packing fraction at the kinetic glass transition than that of hard spheres. The present work provides a foundation for the subsequent study of the glass transition of binary or polydisperse mixtures of hard disks, normally adopted in experiments and simulations to avoid crystallization, and further, the rheology and mechanical response of the two-dimensional glassy colloidal systems. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4866903] I. INTRODUCTION
Slow dynamics and the glass transition of glass-forming liquids have long been the subjects of intensive study in condensed matter physics.1–4 Colloids are excellent model systems for these studies due to advantages in several aspects: (i) the elementary unit is unambiguous; (ii) the interaction can be modelled as simple as hard-sphere potential or be tuned easily;5–8 (iii) as micron-sized “atoms,” the colloidal particles are large enough that the packing structures and motions can be observed by optical microscopies.9 Lots of theoretical, experimental, and simulation work has been done to investigate the phase behaviors of various colloidal systems.10–12 Compared with three-dimensional (3D) systems, the motions of colloids in (quasi) two dimensions (2D) are easier to be traced and analysed. However, the phase transitions in 2D are significantly different from those in 3D. For example, there is no rigorous crystal phase in 2D particle system; the melting transition consists of two steps: (i) solid to hexatic phase transition with the loss of quasi-long-range positional order, (ii) hexatic to isotropic liquid phase transition with the loss of quasilong-range orientational (sixfold) order.13 Therefore, it is of fundamental interest to study the influences of dimensionality on the slow dynamics and glass transition in colloidal suspensions. Interesting questions include, for example, how the localization occurs and depends on area fraction; whether universal glassy phenomena observed in 3D still hold in 2D; a) Authors to whom correspondence should be addressed. Electronic
addresses: [email protected] and [email protected]
0021-9606/2014/140(9)/094506/8/$30.00
qualitatively and quantitatively, what are the differences between 2D and 3D in the features of dynamics, elasticity, and the structural relaxation near the glass transition; and so on. In two dimensions, monodisperse colloids are very easy to crystallize even at very fast quench rate. Hence, most experiments and simulations chose polydisperse or binary mixtures of hard disks as the model system. König and co-workers14 realized the first experimental quasi-2D glass-forming system by confining binary mixtures of superparamagnetic colloidal particles at water-air interface. The interactions between colloidal particles are dominated by magnetic dipole moments which can be controlled by external magnetic field B.14, 15 The kinetic glass transition is practically defined as the point at which the viscosity reaches 1013 P or typically the alpha relaxation time is around 100–10 000 s that is beyond experimental patience. But, no consensus has been achieved on the existence of an ideal thermodynamic glass transition.1, 16 Molecular dynamics (MD) simulations of 2D Lennard-Jones liquid found that the pair distribution function parameter, nonGaussian parameter, and the self-part of the van Hove density correlation function all display a marked transition at a dimensionless density ∼0.83 and the dimensionality has little effect on the Lindemann melting criterion.17 Monte Carlo (MC) and MD simulations,18, 19 which incorporate the idea of inherent structure theory,20 showed that there is no ideal glass transition in a binary hard-disk mixture. In past decades, quite a bit of work has been reported in studying the glass transition of binary hard- or soft-disk mixtures from the aspects of thermodynamics, kinetics, and structure.21–25 Recently, Zheng et al.26 studied quasi-2D suspensions of monodisperse
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colloidal ellipsoids by video microscopy and found two-step glass transitions corresponding to freezing of rotational and translational degrees of freedom, respectively. Microscopic Mode-coupling theory (MCT) has also been used to investigate the glass transition and dynamics in 2D colloidal suspensions. By making use of the modifiedhypernetted-chain (MHNC) theory, which like all the other integral equation theories predicts only fluidlike structure, to calculate the structural correlation functions, Bayer and coworkers were able to study the dynamic glass transition of monodisperse hard disks in the framework of MCT.27 They found the critical packing fraction φc2D is about 35% above φc3D but the ratios of φ c /φ RCP are very close in 2D and 3D, where φ RCP is the area/volume fraction of random close packing (RCP). This suggests that φ RCP might be the appropriate scale for φ to do comparisons between 2D and 3D. The nonergodicity parameters at the “glass transition” were found to change more abruptly in 2D. Whereas, the localization length differs only by less than 3% and, thus, the original workers conclude that the Lindemann criterion for melting is almost dimension-independent. The work was then extended to study the mixing effects and the influence of composition changes on the glass transition for binary hard disk mixtures.28 Recently, Hajnal and co-workers investigated binary mixtures of dipolar particles in 2D based on the point dipole model and within the framework of MCT to compare with the experimental system of Ref. 14. They demonstrated that mixing always stabilizes the liquid state (plasticization effect) and deemed that partial clustering of the smaller particles is the physical origin for such an effect. It is well known that MCT describes well the precursor dynamics in moderately supercooled liquid and predicts the ergodicity-to-nonergodicity transition before the experimental glass transition.29 A common interpretation of the MCT transition is that it signals the emergence of strongly activated dynamical barrier.30–32 Some developments of the ideal MCT to include coupling to currents or new closure can restore the ergodicity beyond the ordinary MCT transition.33–35 In 2003, Schweizer and Saltzmann developed the nonlinear Langevin equation (NLE) theory for glassy colloidal suspensions which contains the elements of MCT and can address the single-particle transient localization and activated hopping dynamics.36 In the current paper, we extend and apply the NLE theory for 3D colloidal suspensions to monodisperse hard disks. The localization and activated hopping in the deeply supercooled regime and that beyond the kinetic glass transition are investigated. The focus is on the packingfraction dependence of the characteristic length and time in 2D. As aforementioned, polydisperse or binary, instead of monodisperse, hard disks are normally investigated in experiments and simulations to avoid crystallization. This restricts the comparisons of our theoretical predictions with experiments and simulations. While we still try to make comparisons with MCT and some simulation predictions, the accent is on the qualitative and quantitative differences and similarities between 2D and 3D systems. This work provides a foundation for the future study of binary or polydisperse hard disks, where detailed comparisons between theory and experiment/simulation will be possible.
J. Chem. Phys. 140, 094506 (2014)
The paper is structured as follows. Section II briefly reviews the idea and key elements of the nonlinear Langevin equation theory for glassy particle system. A simple extension of formula to two dimensions is given. As the primary inputs to the NLE theory, static structures and direct correlations in 2D are discussed in Sec. III. Section IV presents our model studies of 2D hard colloids on the crossover behaviour, the characteristic length scales of localization, and time scales of activated barrier hopping, etc. Quantitative comparisons with the results in 3D hard spheres are also presented. A short summary is then given in Sec. V. II. NONLINEAR LANGEVIN EQUATION THEORY
NLE theory focuses on the motion of a “tagged” particle localized in a cage formed by neighbour particles at short time and escaping through activated hopping process. The central quantity is the time-dependent scalar displacement r(t) of a “tagged” particle from its initial position. Another way to think of this quantity is that r(t) is the time-dependent deviation of the Gaussian distribution of a vibrational tagged particle, i.e., r(t) measures the size of the motional space of a particle, which depends on time due to thermal fluctuations. r(t), which serves as the order parameter, is described by a stochastic nonlinear Langevin equation (in the overdamped limit): ζs
∂Feff [r(t)] ∂r(t) =− + δf (t), ∂t ∂r(t)
(1)
where ζ s = kB T/Ds is the short time friction constant and the white noise random force satisfies δf(0)δf(t) = 2kB Tζ s δ(t). These two quantities account for the local fast motions. The “dynamic free energy” Feff is the key quantity which dictates the caging constraints caused by the effective force exerted by surrounding particles: d q ρC 2 (q)S(q) βFeff (r) = −3 ln(r) − (2π )3 2 2 q r −1 −1 −1 [1 + S (q)] , ×[1 + S (q)] exp − 6 (2) where β ≡ (kB T)−1 . C(q) and S(q) are the Fourier-transformed site-site direct correlation function (DCF)37 and the static structure factor, respectively. ρ is the number density of particles. The first entropy-like term stands for the Fickian diffusion that favors delocalized liquid state. The second intermolecular interaction term leads to the caging effect. Initially, the dynamic free energy was constructed requiring that Eq. (1) restores the Naïve MCT localization equation38 in the absence of thermal noise. Later, a detailed derivation of NLE theory was accomplished, using the nonequilibrium statistical mechanics.39 In the derivation a local equilibrium approximation, Einstein solid perspective, and elements from dynamic density-functional theory (DDFT) were invoked. The basic feature of NLE theory is that a minimum and a barrier emerge in the profile of the dynamic free energy when the temperature (volume fraction) is below (above) the crossover temperature Tc (volume fraction φ c ) and, consequently,
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activated hopping process dominates the long-time relaxation. For in-depth discussion of the theoretical basis of the approach and its achievements in segmental relaxation, aging, rejuvenation, and nonlinear mechanical responses of polymer glasses, the readers are referred to earlier papers40–44 and two recent reviews.45, 46 Following the derivation,39 we can easily achieve NLE for 2D monodisperse colloidal suspensions. The form of the Langevin equation and dynamic free energy does not change except the dimension-related coefficients:
(4) f (q) = (2/q) × J1 (q), 1 16 h(q) = 1 − x 2 (f (q) − a 2 x 2 f (aqx))dx, π 1/a where J1 (q) is the first order Bessel function and the scaling functions are C(x = 0; φ) = −
2 + 0.2733φ 2 − 0.2836φ a= . 1 + φ + 0.2733φ 2 − 0.2836φ
d q ρC 2 (q)S(q) (2π )2 2 2 q r −1 −1 −1 × [1 + S (q)] exp − [1 + S (q)] . 4 (3)
βFeff (r) = −2 ln(r) −
However, the static structure factor and the direct correlation function which enter the force-force correlation or memory function vertices differ greatly between 2D and 3D.
+a 2 φ(a 2 f 2 (aq/2) + h(q; a))},
1 − dφ 2 , (1 − 2φ + dφ 2 )2
d = 0.98946,
III. STATIC STRUCTURES AND CORRELATIONS IN TWO DIMENSIONS
C(q, φ) = φC(x = 0; φ){4(1 − a 2 φ) × f (q)
(5)
Later, Guo and Riebel48 modified the approximate analytic DCF expression of Baus and Colot by employing a simpler compressibility factor proposed by Santos50 which has √ a singularity at φ0 = 3π/6, i.e., the area fraction of a crystalline close packing in two dimensions, instead of physically unreachable density φ = 1. The compressibility is written as Z(φ) = [1 − 2φ + φ 2 (2φ0 − 1)/φ02 ]−1 . The scaling functions, therefore, become C(x = 0; φ) = −
Integral equation theory is a powerful approach to calculate the pair correlation functions and static structure factors. It is based on the Ornstein-Zernike (OZ) equation which defines the direct correlation function.37 A supplemental closure relation is required for completeness. Most integral equation theory builds on liquid state and only predicts the liquid structure, i.e., the crystallization is obviated even at high density. This enables the theories such as MCT and NLE which utilize the correlation functions predicted by the integral equation theory as input to study the glass transition of monodisperse hard disks, not having the problem of crystallization. In pure hard-core colloidal system, Percus-Yevick (PY) approximate relation is widely used as the closure. In 3D, the resulting integral equation has analytical solutions for the DCF, but a full numerical calculation is required in 2D. Some efforts have been made to derive analytical approximate formula for DCF of hard disks.47–49 Baus and Colot47 started from the equation of state and proposed the virial expansion of the compressibility factor n c φ ]/(1 − φ)D which diverges Z(φ) = βp/ρ = [1 + ∞ n n=1 at φ = 1 inspired by the scaled particle theory. p is the pressure; D is the dimension; and cn ’s are expansion coefficients. To calculate the direct correlation function, C(r, φ), they assumed the expression of the rescaled DCF at low density still valid at high density but with density-dependent diameter for rescaling. The two scaling functions C(0, φ) and a(φ) are then determined by combining the OZ equation and the compressibility equation of state which relates the structural functions to thermodynamics. The final expressions for 2D hard disks are
0.384φ 2 + 1 − 0.128φ 3 + φ , (1 − φ)3
(6)
a = 0.3699φ 4 − 1.2511φ 3 + 2.0199φ 2 −2.2373φ + 2.1. Rosenfeld49 derived the DCF of hard disks from a “fundamental measures” free-energy model. Generally, this approach first postulates an excess free-energy functional, βFex [{ρ(r )}] = d r [{nα (r )}], where the free-energy density is assumed as a function of the fundamental geomet ric measures of the particles, nα (r ) = d r ρ(r )ω(α) (r − r ). ω(α) are the “weight functions” for the particles’ geometry. DCF is then obtained by functional derivative of excess free energy to density. In 2D, pair exclusion terms analogous to the results in 1D and 3D are introduced by hand, and the final approximate structure factor is given by 1/S(q) = 4φ 4a0 J12 (q/2)/q 2 +2b0 J0 (q/2) × J1 (q/2)/q + 2g0 J1 (q)/q + 1, a0 =
1+(2φ −1)(1+φ)/(1−φ)3 +2φ/(1−φ) , φ
(1 + φ)/(1 − φ)2 − 1 − 3φ/(1 − φ) , b0 = φ
(7)
g0 = 1/(1 − φ). In this paper, for simplicity, we use the above three approximate analytic theories, noted as Baus-Colot, GuoRiebel, and Rosenfeld in the following, respectively, to calculate correlation functions as input to the dynamic free energy (Eq. (3)) in the NLE theory for 2D hard disks. Lacking the simulation and/or experimental data of the “unrealistic” amorphous structures of monodisperse hard disks above freezing point, the accuracy or validity of the extrapolation of these structural theories to high-density glass state is unclear. PY theory is used in the calculations of 3D hard spheres.
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FIG. 1. Structure factors of monodisperse hard disks at packing fractions φ = 0.612 (dashed lines) and 0.68 (solid lines), calculated by the approximate analytic theories of Baus-Colot, Guo-Riebel, and Rosenfeld. The inset zooms in the data of φ = 0.612 around the first peak.
On the one hand, it is of interest to see the differences in the NLE predictions based on these three 2D structural theories. On the other hand, the NLE predictions may also provide judgment on the accuracy of these approximate theories, especially at high density.
IV. RESULTS AND DISCUSSION
Figure 1 shows the structure factors of monodisperse hard disks at the area fraction of φ = 0.612 and 0.68, calculated based on the above three approximate analytic theories. For φ = 0.612, the three curves are very close to each other, indirectly reflecting the accuracy of all the three approaches at low disk density. But zooming in the picture near the first peak, we still see weak differences (see inset of Fig. 1). At high disk density, marked differences are found especially in the height of the first peak, e.g., the value of the first peak of Baus-Colot is about 20% smaller than that of Rosenfeld. Slow activated hopping dynamics and the glass transition happen at high density. Hence, the difference in the prediction of structure may be reflected in the obtained glassy behaviors. Combining Eqs. (4)–(7) with Eq. (3), we obtain the dynamic free energy Feff ’s of hard disks shown in Fig. 2. At relatively lower packing fraction φ = 0.53, Feff decays monotonically with particle displacement, implying that the particles move in the way analogous to nonlocalized free diffusion. Whereas, at higher disk density φ = 0.7, a minimum and a barrier emerge indicative of the localized motion of hard disks. And consequently, the long-time structural relaxation is realized through the activated hopping events. The dynamicfree-energy curves of the three approaches are close to each other at low density but differ greatly at high density. rLOC , rB , and R∗ (see inset of Fig. 2) are three important length scales characterizing the dynamic “cage” at high density. rLOC is the localization length denoting the location of the minimum; rB is the position of the barrier beyond which the particle escapes; and R∗ is the displacement where the particle receives the maximum restoring force. At the crossover area fraction from the monotonic to non-monotonic dynamic-free-energy
J. Chem. Phys. 140, 094506 (2014)
FIG. 2. Dynamic free energies as functions of particle displacement at area fractions of φ = 0.53, 0.612, and 0.7. The inset shows the schematic profile of the dynamic free energy at high density, marking the height of the barrier and the locations of the local minimum, barrier, and the maximum restoring force.
profile, which corresponds to the ergodic-to-nonergodic transition point predicted by Naïve MCT, the three characteristic length scales converge. Calculations based on the three approximate analytic structural theories give the same crossover area fraction of φc2D = 0.612, lower than the ideal MCT result of φc2D,I MCT = 0.697.27 But, compared with the crossover volume fraction in 3D of φc3D = 0.432,36 φc2D in 2D is larger by 42%. To properly compare the predictions between 2D and 3D systems, we need a scaling for the packing fraction. Here, we choose the random close packing as the appropriate quantity to rescale the packing fractions, because (i) full MCT calculation shows the ratios of the critical packing fraction to the RCP, φ c /φ RCP are close in 2D and 3D;27 (ii) RCP is the upper bound of the amorphous structure (even though it is not a well-defined quantity), and intuitively, the ratio of φ/φ RCP represents the degree of crowding of the amorphous system. Hence, we find the reduced crossover packing 3D 2D = 0.729 > φc3D /φRCP = 0.675, where we fraction φc2D /φRCP 3D 2D = 0.84 and φRCP = 0.64 in the current paper.51 adopt φRCP Hence, the crossover transition in 2D happens at higher absolute and reduced packing fraction than in 3D. Although, the three approaches lead to the same crossover area fraction φc2D , different localization lengths at the crossover are predicted: c,2D = 0.154σ (Baus-Colot), 0.170σ (Guo-Riebel), 0.172σ rLOC (Rosenfeld). The result of Baus-Colot is markedly smaller than the other two. The localization length at the crossover is closely related to the Lindemann length for melting. The crossover localization length in 2D is smaller than its counc,3D = 0.19σ .36 Hence, we can conclude that terpart in 3D, rLOC the crossover transition in 2D happens at higher “density,” i.e., smaller dynamic cage size than in 3D, consistent with the predictions on the crossover packing fractions. The above quantities at the crossover are summarized in Table I. Figure 3 shows the three characteristic length scales in 2D (lines+open symbols) and 3D (scattered solid symbols) as functions of reduced packing fraction. We find the curves of localization length in 2D are above the data in 3D, manifesting that at the same reduced packing fraction, the 2D
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TABLE I. Lists of packing fractions at the crossover and the kinetic glass transition and localization lengths (in unit of particle diameter σ ) at the 2D = 0.84 and crossover in hard-disk and -sphere suspensions. We adopt φRCP 3D φRCP = 0.64. φc2D
φc2D,I MCT
0.612 2D φc2D /φRCP
0.729 φg2D 0.747 c,2D rLOC
φc3D
Baus-Colot Guo-Riebel Rosenfeld 3D
r˜LOC
R˜ ∗
K˜ 0
21.3e−7.09φr 44.7e−8.15φr 39.5e−7.74φr 30e−7.81φr
7.19e−5.10φr 9.07e−5.41φr 6.59e−4.87φr 3.3e−4.22φr
3.09 × 10−4 e17.9φr 3.77 × 10−5 e20.7φr 4.30 × 10−5 e20.1φr 1.02 × 10−3 e16.2φr
0.697
0.432
0.515
2D φc2D,I MCT /φRCP
3D φc3D /φRCP
3D φc3D,I MCT /φRCP
0.830
0.675
0.805
2D φg2D /φRCP
φg3D
3D φg3D /φRCP
0.579
0.905
(Rosenfeld)
c,3D rLOC
local equilibrium time scale. In the Einstein solid picture,
0.172
0.19
the mean square displacement R 2 (t) = (R(t) − R(0))2
0.889 (Baus)
φc3D,I MCT
TABLE II. Lists of scaling relations of dimensionless localization length r˜LOC , displacement of maximum restoring force R˜ ∗ , and well curvature K˜ 0 with the reduced packing fraction φ r = φ / φ RCP .
c,2D rLOC
(Guo)
c,2D rLOC
0.154
0.170
hard disks are dynamically less compact (larger cage size) than the 3D hard spheres. Both the locations of the minimum and the displacements of maximum restoring force decay exponentially with packing fraction in 2D and 3D. The localization lengths can be described by exponential law at 2D /σ ∼ 21.3e−7.09φ/φRCP (Baus-Colot), high densities as rLOC −8.15φ/φRCP (Guo-Riebel), ∼ 39.5e−7.74φ/φRCP (Rosen∼ 44.7e feld). The exponents are close to (
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THE JOURNAL OF CHEMICAL PHYSICS 140, 094506 (2014)
Theory of activated dynamics and glass transition of hard colloids in two dimensions Bo-kai Zhang,1,2 Hui-shu Li,2 Wen-de Tian,2 Kang Chen,2,a) and Yu-qiang Ma1,2,a) 1
National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjng 210093, People’s Republic of China 2 Center for Soft Condensed Matter Physics and Interdisciplinary Research, Soochow University, Suzhou 215006, People’s Republic of China
(Received 15 December 2013; accepted 12 February 2014; published online 4 March 2014) The microscopic nonlinear Langevin equation theory is applied to study the localization and activated hopping of two-dimensional hard disks in the deeply supercooled and glass states. Quantitative comparisons of dynamic characteristic length scales, barrier, and their dependence on the reduced packing fraction are presented between hard-disk and hard-sphere suspensions. The dynamic barrier of hard disks emerges at higher absolute and reduced packing fractions and correspondingly, the crossover size of the dynamic cage which correlates to the Lindemann length for melting is smaller. The localization lengths of both hard disks and spheres decrease exponentially with packing fraction. Larger localization length of hard disks than that of hard spheres is found at the same reduced packing fraction. The relaxation time of hard disks rises dramatically above the reduced packing fraction of 0.88, which leads to lower reduced packing fraction at the kinetic glass transition than that of hard spheres. The present work provides a foundation for the subsequent study of the glass transition of binary or polydisperse mixtures of hard disks, normally adopted in experiments and simulations to avoid crystallization, and further, the rheology and mechanical response of the two-dimensional glassy colloidal systems. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4866903] I. INTRODUCTION
Slow dynamics and the glass transition of glass-forming liquids have long been the subjects of intensive study in condensed matter physics.1–4 Colloids are excellent model systems for these studies due to advantages in several aspects: (i) the elementary unit is unambiguous; (ii) the interaction can be modelled as simple as hard-sphere potential or be tuned easily;5–8 (iii) as micron-sized “atoms,” the colloidal particles are large enough that the packing structures and motions can be observed by optical microscopies.9 Lots of theoretical, experimental, and simulation work has been done to investigate the phase behaviors of various colloidal systems.10–12 Compared with three-dimensional (3D) systems, the motions of colloids in (quasi) two dimensions (2D) are easier to be traced and analysed. However, the phase transitions in 2D are significantly different from those in 3D. For example, there is no rigorous crystal phase in 2D particle system; the melting transition consists of two steps: (i) solid to hexatic phase transition with the loss of quasi-long-range positional order, (ii) hexatic to isotropic liquid phase transition with the loss of quasilong-range orientational (sixfold) order.13 Therefore, it is of fundamental interest to study the influences of dimensionality on the slow dynamics and glass transition in colloidal suspensions. Interesting questions include, for example, how the localization occurs and depends on area fraction; whether universal glassy phenomena observed in 3D still hold in 2D; a) Authors to whom correspondence should be addressed. Electronic
addresses: [email protected] and [email protected]
0021-9606/2014/140(9)/094506/8/$30.00
qualitatively and quantitatively, what are the differences between 2D and 3D in the features of dynamics, elasticity, and the structural relaxation near the glass transition; and so on. In two dimensions, monodisperse colloids are very easy to crystallize even at very fast quench rate. Hence, most experiments and simulations chose polydisperse or binary mixtures of hard disks as the model system. König and co-workers14 realized the first experimental quasi-2D glass-forming system by confining binary mixtures of superparamagnetic colloidal particles at water-air interface. The interactions between colloidal particles are dominated by magnetic dipole moments which can be controlled by external magnetic field B.14, 15 The kinetic glass transition is practically defined as the point at which the viscosity reaches 1013 P or typically the alpha relaxation time is around 100–10 000 s that is beyond experimental patience. But, no consensus has been achieved on the existence of an ideal thermodynamic glass transition.1, 16 Molecular dynamics (MD) simulations of 2D Lennard-Jones liquid found that the pair distribution function parameter, nonGaussian parameter, and the self-part of the van Hove density correlation function all display a marked transition at a dimensionless density ∼0.83 and the dimensionality has little effect on the Lindemann melting criterion.17 Monte Carlo (MC) and MD simulations,18, 19 which incorporate the idea of inherent structure theory,20 showed that there is no ideal glass transition in a binary hard-disk mixture. In past decades, quite a bit of work has been reported in studying the glass transition of binary hard- or soft-disk mixtures from the aspects of thermodynamics, kinetics, and structure.21–25 Recently, Zheng et al.26 studied quasi-2D suspensions of monodisperse
140, 094506-1
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colloidal ellipsoids by video microscopy and found two-step glass transitions corresponding to freezing of rotational and translational degrees of freedom, respectively. Microscopic Mode-coupling theory (MCT) has also been used to investigate the glass transition and dynamics in 2D colloidal suspensions. By making use of the modifiedhypernetted-chain (MHNC) theory, which like all the other integral equation theories predicts only fluidlike structure, to calculate the structural correlation functions, Bayer and coworkers were able to study the dynamic glass transition of monodisperse hard disks in the framework of MCT.27 They found the critical packing fraction φc2D is about 35% above φc3D but the ratios of φ c /φ RCP are very close in 2D and 3D, where φ RCP is the area/volume fraction of random close packing (RCP). This suggests that φ RCP might be the appropriate scale for φ to do comparisons between 2D and 3D. The nonergodicity parameters at the “glass transition” were found to change more abruptly in 2D. Whereas, the localization length differs only by less than 3% and, thus, the original workers conclude that the Lindemann criterion for melting is almost dimension-independent. The work was then extended to study the mixing effects and the influence of composition changes on the glass transition for binary hard disk mixtures.28 Recently, Hajnal and co-workers investigated binary mixtures of dipolar particles in 2D based on the point dipole model and within the framework of MCT to compare with the experimental system of Ref. 14. They demonstrated that mixing always stabilizes the liquid state (plasticization effect) and deemed that partial clustering of the smaller particles is the physical origin for such an effect. It is well known that MCT describes well the precursor dynamics in moderately supercooled liquid and predicts the ergodicity-to-nonergodicity transition before the experimental glass transition.29 A common interpretation of the MCT transition is that it signals the emergence of strongly activated dynamical barrier.30–32 Some developments of the ideal MCT to include coupling to currents or new closure can restore the ergodicity beyond the ordinary MCT transition.33–35 In 2003, Schweizer and Saltzmann developed the nonlinear Langevin equation (NLE) theory for glassy colloidal suspensions which contains the elements of MCT and can address the single-particle transient localization and activated hopping dynamics.36 In the current paper, we extend and apply the NLE theory for 3D colloidal suspensions to monodisperse hard disks. The localization and activated hopping in the deeply supercooled regime and that beyond the kinetic glass transition are investigated. The focus is on the packingfraction dependence of the characteristic length and time in 2D. As aforementioned, polydisperse or binary, instead of monodisperse, hard disks are normally investigated in experiments and simulations to avoid crystallization. This restricts the comparisons of our theoretical predictions with experiments and simulations. While we still try to make comparisons with MCT and some simulation predictions, the accent is on the qualitative and quantitative differences and similarities between 2D and 3D systems. This work provides a foundation for the future study of binary or polydisperse hard disks, where detailed comparisons between theory and experiment/simulation will be possible.
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The paper is structured as follows. Section II briefly reviews the idea and key elements of the nonlinear Langevin equation theory for glassy particle system. A simple extension of formula to two dimensions is given. As the primary inputs to the NLE theory, static structures and direct correlations in 2D are discussed in Sec. III. Section IV presents our model studies of 2D hard colloids on the crossover behaviour, the characteristic length scales of localization, and time scales of activated barrier hopping, etc. Quantitative comparisons with the results in 3D hard spheres are also presented. A short summary is then given in Sec. V. II. NONLINEAR LANGEVIN EQUATION THEORY
NLE theory focuses on the motion of a “tagged” particle localized in a cage formed by neighbour particles at short time and escaping through activated hopping process. The central quantity is the time-dependent scalar displacement r(t) of a “tagged” particle from its initial position. Another way to think of this quantity is that r(t) is the time-dependent deviation of the Gaussian distribution of a vibrational tagged particle, i.e., r(t) measures the size of the motional space of a particle, which depends on time due to thermal fluctuations. r(t), which serves as the order parameter, is described by a stochastic nonlinear Langevin equation (in the overdamped limit): ζs
∂Feff [r(t)] ∂r(t) =− + δf (t), ∂t ∂r(t)
(1)
where ζ s = kB T/Ds is the short time friction constant and the white noise random force satisfies δf(0)δf(t) = 2kB Tζ s δ(t). These two quantities account for the local fast motions. The “dynamic free energy” Feff is the key quantity which dictates the caging constraints caused by the effective force exerted by surrounding particles: d q ρC 2 (q)S(q) βFeff (r) = −3 ln(r) − (2π )3 2 2 q r −1 −1 −1 [1 + S (q)] , ×[1 + S (q)] exp − 6 (2) where β ≡ (kB T)−1 . C(q) and S(q) are the Fourier-transformed site-site direct correlation function (DCF)37 and the static structure factor, respectively. ρ is the number density of particles. The first entropy-like term stands for the Fickian diffusion that favors delocalized liquid state. The second intermolecular interaction term leads to the caging effect. Initially, the dynamic free energy was constructed requiring that Eq. (1) restores the Naïve MCT localization equation38 in the absence of thermal noise. Later, a detailed derivation of NLE theory was accomplished, using the nonequilibrium statistical mechanics.39 In the derivation a local equilibrium approximation, Einstein solid perspective, and elements from dynamic density-functional theory (DDFT) were invoked. The basic feature of NLE theory is that a minimum and a barrier emerge in the profile of the dynamic free energy when the temperature (volume fraction) is below (above) the crossover temperature Tc (volume fraction φ c ) and, consequently,
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activated hopping process dominates the long-time relaxation. For in-depth discussion of the theoretical basis of the approach and its achievements in segmental relaxation, aging, rejuvenation, and nonlinear mechanical responses of polymer glasses, the readers are referred to earlier papers40–44 and two recent reviews.45, 46 Following the derivation,39 we can easily achieve NLE for 2D monodisperse colloidal suspensions. The form of the Langevin equation and dynamic free energy does not change except the dimension-related coefficients:
(4) f (q) = (2/q) × J1 (q), 1 16 h(q) = 1 − x 2 (f (q) − a 2 x 2 f (aqx))dx, π 1/a where J1 (q) is the first order Bessel function and the scaling functions are C(x = 0; φ) = −
2 + 0.2733φ 2 − 0.2836φ a= . 1 + φ + 0.2733φ 2 − 0.2836φ
d q ρC 2 (q)S(q) (2π )2 2 2 q r −1 −1 −1 × [1 + S (q)] exp − [1 + S (q)] . 4 (3)
βFeff (r) = −2 ln(r) −
However, the static structure factor and the direct correlation function which enter the force-force correlation or memory function vertices differ greatly between 2D and 3D.
+a 2 φ(a 2 f 2 (aq/2) + h(q; a))},
1 − dφ 2 , (1 − 2φ + dφ 2 )2
d = 0.98946,
III. STATIC STRUCTURES AND CORRELATIONS IN TWO DIMENSIONS
C(q, φ) = φC(x = 0; φ){4(1 − a 2 φ) × f (q)
(5)
Later, Guo and Riebel48 modified the approximate analytic DCF expression of Baus and Colot by employing a simpler compressibility factor proposed by Santos50 which has √ a singularity at φ0 = 3π/6, i.e., the area fraction of a crystalline close packing in two dimensions, instead of physically unreachable density φ = 1. The compressibility is written as Z(φ) = [1 − 2φ + φ 2 (2φ0 − 1)/φ02 ]−1 . The scaling functions, therefore, become C(x = 0; φ) = −
Integral equation theory is a powerful approach to calculate the pair correlation functions and static structure factors. It is based on the Ornstein-Zernike (OZ) equation which defines the direct correlation function.37 A supplemental closure relation is required for completeness. Most integral equation theory builds on liquid state and only predicts the liquid structure, i.e., the crystallization is obviated even at high density. This enables the theories such as MCT and NLE which utilize the correlation functions predicted by the integral equation theory as input to study the glass transition of monodisperse hard disks, not having the problem of crystallization. In pure hard-core colloidal system, Percus-Yevick (PY) approximate relation is widely used as the closure. In 3D, the resulting integral equation has analytical solutions for the DCF, but a full numerical calculation is required in 2D. Some efforts have been made to derive analytical approximate formula for DCF of hard disks.47–49 Baus and Colot47 started from the equation of state and proposed the virial expansion of the compressibility factor n c φ ]/(1 − φ)D which diverges Z(φ) = βp/ρ = [1 + ∞ n n=1 at φ = 1 inspired by the scaled particle theory. p is the pressure; D is the dimension; and cn ’s are expansion coefficients. To calculate the direct correlation function, C(r, φ), they assumed the expression of the rescaled DCF at low density still valid at high density but with density-dependent diameter for rescaling. The two scaling functions C(0, φ) and a(φ) are then determined by combining the OZ equation and the compressibility equation of state which relates the structural functions to thermodynamics. The final expressions for 2D hard disks are
0.384φ 2 + 1 − 0.128φ 3 + φ , (1 − φ)3
(6)
a = 0.3699φ 4 − 1.2511φ 3 + 2.0199φ 2 −2.2373φ + 2.1. Rosenfeld49 derived the DCF of hard disks from a “fundamental measures” free-energy model. Generally, this approach first postulates an excess free-energy functional, βFex [{ρ(r )}] = d r [{nα (r )}], where the free-energy density is assumed as a function of the fundamental geomet ric measures of the particles, nα (r ) = d r ρ(r )ω(α) (r − r ). ω(α) are the “weight functions” for the particles’ geometry. DCF is then obtained by functional derivative of excess free energy to density. In 2D, pair exclusion terms analogous to the results in 1D and 3D are introduced by hand, and the final approximate structure factor is given by 1/S(q) = 4φ 4a0 J12 (q/2)/q 2 +2b0 J0 (q/2) × J1 (q/2)/q + 2g0 J1 (q)/q + 1, a0 =
1+(2φ −1)(1+φ)/(1−φ)3 +2φ/(1−φ) , φ
(1 + φ)/(1 − φ)2 − 1 − 3φ/(1 − φ) , b0 = φ
(7)
g0 = 1/(1 − φ). In this paper, for simplicity, we use the above three approximate analytic theories, noted as Baus-Colot, GuoRiebel, and Rosenfeld in the following, respectively, to calculate correlation functions as input to the dynamic free energy (Eq. (3)) in the NLE theory for 2D hard disks. Lacking the simulation and/or experimental data of the “unrealistic” amorphous structures of monodisperse hard disks above freezing point, the accuracy or validity of the extrapolation of these structural theories to high-density glass state is unclear. PY theory is used in the calculations of 3D hard spheres.
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FIG. 1. Structure factors of monodisperse hard disks at packing fractions φ = 0.612 (dashed lines) and 0.68 (solid lines), calculated by the approximate analytic theories of Baus-Colot, Guo-Riebel, and Rosenfeld. The inset zooms in the data of φ = 0.612 around the first peak.
On the one hand, it is of interest to see the differences in the NLE predictions based on these three 2D structural theories. On the other hand, the NLE predictions may also provide judgment on the accuracy of these approximate theories, especially at high density.
IV. RESULTS AND DISCUSSION
Figure 1 shows the structure factors of monodisperse hard disks at the area fraction of φ = 0.612 and 0.68, calculated based on the above three approximate analytic theories. For φ = 0.612, the three curves are very close to each other, indirectly reflecting the accuracy of all the three approaches at low disk density. But zooming in the picture near the first peak, we still see weak differences (see inset of Fig. 1). At high disk density, marked differences are found especially in the height of the first peak, e.g., the value of the first peak of Baus-Colot is about 20% smaller than that of Rosenfeld. Slow activated hopping dynamics and the glass transition happen at high density. Hence, the difference in the prediction of structure may be reflected in the obtained glassy behaviors. Combining Eqs. (4)–(7) with Eq. (3), we obtain the dynamic free energy Feff ’s of hard disks shown in Fig. 2. At relatively lower packing fraction φ = 0.53, Feff decays monotonically with particle displacement, implying that the particles move in the way analogous to nonlocalized free diffusion. Whereas, at higher disk density φ = 0.7, a minimum and a barrier emerge indicative of the localized motion of hard disks. And consequently, the long-time structural relaxation is realized through the activated hopping events. The dynamicfree-energy curves of the three approaches are close to each other at low density but differ greatly at high density. rLOC , rB , and R∗ (see inset of Fig. 2) are three important length scales characterizing the dynamic “cage” at high density. rLOC is the localization length denoting the location of the minimum; rB is the position of the barrier beyond which the particle escapes; and R∗ is the displacement where the particle receives the maximum restoring force. At the crossover area fraction from the monotonic to non-monotonic dynamic-free-energy
J. Chem. Phys. 140, 094506 (2014)
FIG. 2. Dynamic free energies as functions of particle displacement at area fractions of φ = 0.53, 0.612, and 0.7. The inset shows the schematic profile of the dynamic free energy at high density, marking the height of the barrier and the locations of the local minimum, barrier, and the maximum restoring force.
profile, which corresponds to the ergodic-to-nonergodic transition point predicted by Naïve MCT, the three characteristic length scales converge. Calculations based on the three approximate analytic structural theories give the same crossover area fraction of φc2D = 0.612, lower than the ideal MCT result of φc2D,I MCT = 0.697.27 But, compared with the crossover volume fraction in 3D of φc3D = 0.432,36 φc2D in 2D is larger by 42%. To properly compare the predictions between 2D and 3D systems, we need a scaling for the packing fraction. Here, we choose the random close packing as the appropriate quantity to rescale the packing fractions, because (i) full MCT calculation shows the ratios of the critical packing fraction to the RCP, φ c /φ RCP are close in 2D and 3D;27 (ii) RCP is the upper bound of the amorphous structure (even though it is not a well-defined quantity), and intuitively, the ratio of φ/φ RCP represents the degree of crowding of the amorphous system. Hence, we find the reduced crossover packing 3D 2D = 0.729 > φc3D /φRCP = 0.675, where we fraction φc2D /φRCP 3D 2D = 0.84 and φRCP = 0.64 in the current paper.51 adopt φRCP Hence, the crossover transition in 2D happens at higher absolute and reduced packing fraction than in 3D. Although, the three approaches lead to the same crossover area fraction φc2D , different localization lengths at the crossover are predicted: c,2D = 0.154σ (Baus-Colot), 0.170σ (Guo-Riebel), 0.172σ rLOC (Rosenfeld). The result of Baus-Colot is markedly smaller than the other two. The localization length at the crossover is closely related to the Lindemann length for melting. The crossover localization length in 2D is smaller than its counc,3D = 0.19σ .36 Hence, we can conclude that terpart in 3D, rLOC the crossover transition in 2D happens at higher “density,” i.e., smaller dynamic cage size than in 3D, consistent with the predictions on the crossover packing fractions. The above quantities at the crossover are summarized in Table I. Figure 3 shows the three characteristic length scales in 2D (lines+open symbols) and 3D (scattered solid symbols) as functions of reduced packing fraction. We find the curves of localization length in 2D are above the data in 3D, manifesting that at the same reduced packing fraction, the 2D
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TABLE I. Lists of packing fractions at the crossover and the kinetic glass transition and localization lengths (in unit of particle diameter σ ) at the 2D = 0.84 and crossover in hard-disk and -sphere suspensions. We adopt φRCP 3D φRCP = 0.64. φc2D
φc2D,I MCT
0.612 2D φc2D /φRCP
0.729 φg2D 0.747 c,2D rLOC
φc3D
Baus-Colot Guo-Riebel Rosenfeld 3D
r˜LOC
R˜ ∗
K˜ 0
21.3e−7.09φr 44.7e−8.15φr 39.5e−7.74φr 30e−7.81φr
7.19e−5.10φr 9.07e−5.41φr 6.59e−4.87φr 3.3e−4.22φr
3.09 × 10−4 e17.9φr 3.77 × 10−5 e20.7φr 4.30 × 10−5 e20.1φr 1.02 × 10−3 e16.2φr
0.697
0.432
0.515
2D φc2D,I MCT /φRCP
3D φc3D /φRCP
3D φc3D,I MCT /φRCP
0.830
0.675
0.805
2D φg2D /φRCP
φg3D
3D φg3D /φRCP
0.579
0.905
(Rosenfeld)
c,3D rLOC
local equilibrium time scale. In the Einstein solid picture,
0.172
0.19
the mean square displacement R 2 (t) = (R(t) − R(0))2
0.889 (Baus)
φc3D,I MCT
TABLE II. Lists of scaling relations of dimensionless localization length r˜LOC , displacement of maximum restoring force R˜ ∗ , and well curvature K˜ 0 with the reduced packing fraction φ r = φ / φ RCP .
c,2D rLOC
(Guo)
c,2D rLOC
0.154
0.170
hard disks are dynamically less compact (larger cage size) than the 3D hard spheres. Both the locations of the minimum and the displacements of maximum restoring force decay exponentially with packing fraction in 2D and 3D. The localization lengths can be described by exponential law at 2D /σ ∼ 21.3e−7.09φ/φRCP (Baus-Colot), high densities as rLOC −8.15φ/φRCP (Guo-Riebel), ∼ 39.5e−7.74φ/φRCP (Rosen∼ 44.7e feld). The exponents are close to (
