Analyses of kinetic glass transition in short-range attractive colloids based on time-convolutionless mode-coupling theory
aa r X i v : . [ c ond - m a t . d i s - nn ] D ec Analyses of kinetic glass transition in short-range attractive colloidsbased on time-convolutionless mode-coupling theory
Takayuki Narumi ∗ and Michio Tokuyama Faculty of Engineering, Kyushu Sangyo University, Fukuoka 813-8503, Japan Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan (Dated: October 15, 2018)The kinetic glass transition in short-range attractive colloids is theoretically studied by time-convolutionless mode-coupling theory (TMCT). By numerical calculations, TMCT is shown to re-cover all the remarkable features predicted by the mode-coupling theory for attractive colloids,namely the glass-liquid-glass reentrant, the glass-glass transition, and the higher-order singularities.It is also demonstrated through the comparisons with the results of molecular dynamics for the bi-nary attractive colloids that TMCT improves the critical values of the volume fraction. In addition,a schematic model of three control parameters is investigated analytically. It is thus confirmed thatTMCT can describe the glass-glass transition and higher-order singularities even in such a schematicmodel.
PACS numbers: 64.70.qd, 05.20.Jj 64.70.kj, 82.70.Dd,Keywords: kinetic phase diagram, square-well system, liquid-glass-liquid reentrant, glass-glass transition,higher-order singularity, schematic model
I. INTRODUCTION
Short-range attractive colloids are prominent in studiesof the glass transition. In colloidal systems of a high vol-ume fraction, since each particle is stuck in the “cage”made of the neighboring particles, the structural rear-rangement rarely occurs. The glass driven by the exclu-sive volume effect is classified as repulsive glass. On theother hand, there is a different glass-forming mechanismin systems of attractive interaction. At a low tempera-ture, each particle is trapped in potential well and stickstogether to form clusters. The glass originated from thecluster formation is called attractive glass. The attrac-tion length in atomic or molecular systems is comparableto the particle size; nevertheless, in colloidal systems, theshort-range attraction can be materialized [1–11]. Forsystems of the attraction range smaller than about onetenth of the particle diameter, the mode-coupling theory(MCT) predicted melting of a glass by cooling and thedirect transition between repulsive and attractive glasses[12–15]. Eckert et al. and Pham et al. then observed theglass-liquid-glass reentrant [4, 5], and Chen et al. con-firmed the glass-glass transition in experiments [7]. Nu-merical simulations for attractive colloids have also sup-ported such rich phenomena [16–22]. We here study theglass transition of short-range attractive colloids to val-idate a theory recently proposed by Tokuyama, time-convolutionless mode-coupling theory (TMCT) [23, 24].A glassy state is ideally characterized by the presenceof an arrested part in correlation functions [25], and MCTdescribes the kinetic glass transition as a nonlinear bifur-cation, so-called nonergodic transition [26–29]. However,while some extensions and modifications have been done ∗ [email protected] [30–33], MCT has a few shortcomings that remain to besolved. A fundamental problem is the case that the tran-sition point predicted by MCT is far from the calorimet-ric glass transition points observed by experiments andalso by simulations. In order to overcome such a diffi-culty, TMCT has been proposed as an alternative theoryof MCT [23].The way of extracting macroscopic (i.e., slow) dynam-ics differs between MCT and TMCT. The starting equa-tion of both MCT and TMCT is the Heisenberg equationof motion, ˙ A ( t ) = i L A ( t ), where A ( t ) denotes a vectorof macroscopic variables and i L is the Liouville operator.To derive a coarse-grained equation of the density fluctu-ation, MCT employs the Mori projection operator [34].This formalism derives an equation that contains a mem-ory function as a form of the time-convolution integral.On the other hand, TMCT employs the Tokuyama–Moriprojection operator [35, 36], where the derived equationcontains the memory function as a form of the time-convolutionless integral. The hypothesis concerning thememory function of TMCT is the same as that of MCT,and consequently the memory functions of MCT andTMCT have the same form. TMCT thus can be studiedby the theoretical framework of MCT [23, 24, 37].TMCT predicts some different features from MCT. Forexample, the initial value of the non-gaussian parame-ter is a non-zero value in MCT, but 0 in TMCT [23].In addition, TMCT improves the quantitative features.For the monodisperse hard-sphere system, Kimura andTokuyama have solved the TMCT equation by using thestatic structure factor under the Percus–Yevick approxi-mation (PYA) [38]. The solution has predicted the criti-cal volume fraction φ c = 0 . φ c = 0 .
516 [26]. In this paper, we thus show notonly how TMCT qualitatively recovers the MCT predic-tions for short-range attractive colloids but also how thecritical values are quantitatively improved. θ φ FIG. 1. Comparison be-tween the MCT (open cir-cle) and TMCT (filled cir-cle) results of the transi-tion lines of SWS based onPYA: ε = 0 .
03 (red), 0 . .
05 (green), 0 . .
09 (black),from right to left. TheMCT results are identicalto those in Ref. [14].
The present paper is organized as follows. Section IIexplains the model we study and the numerical schemes.Section III presents and discusses the kinetic phase di-agram obtained numerically. To support validity of theresults, in Sec. IV, a schematic model is investigated an-alytically. Section V summarizes this paper. The detailsof the analysis for the schematic model are mentioned inthe Appendix.
II. METHOD
The square-well system (SWS) has been studied asa simple model of short-range attractive colloids [12–15, 39–44]. The pairwise potential of SWS is described as U ( r ) = ∞ (0 < r < d ) , − u ( d < r < d + ∆) , d + ∆ 6, and thedimensionless temperature θ = k B T /u , where ρ denotesthe number density. The molecular dynamics (MD) sim-ulations of SWS have been done for the one-componentsystem [17] and binary systems [18, 46].Similarly to the MCT equation for the correlation func-tion of the mode ρ q ( t ) of the density fluctuation, theTMCT equation is solved numerically by using the staticstructure factor S q = D | ρ q ( t ) | E as the initial condition,where the brackets denote an average over an equilib-rium ensemble. The nonergodic transition is intuitivelyquantified by the Debye–Waller factor f q , which is thelong-time limit of the intermediate scattering function F q ( t ) = (cid:10) ρ q ( t ) ρ ∗ q (0) (cid:11) , i.e., f q = lim t →∞ F q ( t ) /S q . Forboth MCT and TMCT, the memory function F q at thelong-time limit is described as F q = 132 π ρ Z ∞ dk Z ′ dp kpq S q S k S p v ( q, k, p ) f k f p , (1)where the prime at the p -integral means that the inte-gration range is restricted to | q − k | ≤ p ≤ q + k , and v ( q, k, p ) = ( q + k − p ) ρc k + ( q − k + p ) ρc p withthe direct correlation function c q = ( S q − / ( ρS q ). Thefunctional F q of f q is called the mode-coupling polyno-mial which is a central concept of the MCT framework.The Debye–Waller factor obeys the fixed-point equa-tion f q = T ( f q ) with T ( f q ) = 11 + 1 / F q [MCT] , exp (cid:18) − F q (cid:19) [TMCT] . (2)An ordinary scheme was employed to obtain f q numeri-cally [47]. The static structure factor of SWS was numer-ically obtained under PYA [14]. The wavenumber inte-grals were discretized to M = 500 points spaced equally,and the cutoff wavenumber was set as q cut = 200 /d .The cutoff was equalized to the previous study for MCT[14, 29]. Note that we carried out the numerical calcu-lations with q cut = 400 /d to guarantee the independenceof the transition points from q cut . III. RESULTS AND DISCUSSIONA. glass-liquid-glass reentrant The numerical solution of TMCT describes the liquid-glass-liquid reentrant at small ε . Figure 1 shows the linesconnecting the transition points of each ε . Each transi-tion point was characterized by the maximum eigenvalue E , where the bifurcation occurs at which E = 1 [48]. Theliquid-glass transition of TMCT appears at higher vol-ume fractions compared to the MCT results. The volumefraction of the high temperature limit slightly exceeds thevalue φ c = 0 . 582 for the monodisperse hard spheres [38]because of the attractive interaction [14]. The shapes ofline are qualitatively similar to those of MCT; they areswollen rightward around θ ≃ ε . This in-dicates the glass-liquid-glass reentry with a decrease oftemperature. φ θ φ θ (a) (b) (c) FIG. 2. Numerical results at ε = 0 . 03 obtained in the TMCT analysis. (a) The contour map of the Debye–Waller factor f q of q = 7 . /d . The white region corresponds to ergodic state, f q = 0. The black bold lines indicate the transition line, and thegray ones are the contour line per 0 . f q at φ = 0 . q = 7 . /d , the square (green) for q = 10 . /d , and thetriangle (blue) for q = 3 . /d . (c) The φ dependence of the maximum eigenvalue E : θ = 1 . 110 (blue circle), 1 . 125 (red triangle),1 . 130 (black plus mark), and 1 . 150 (green cross mark). In (b) and (c), the results of the attractive glass is represented by filledsymbols and those of the repulsive glass is done by open symbols. B. glass-glass transition At ε = 0 . 03 and 0 . 04 in Fig. 1, the TMCT lines corre-sponding to the attractive glass transition penetrate intothe glassy state. To clarify whether the bifurcation inthe glassy state is the glass-glass transition or not, wenext focus on the peak value of f q . Figure 2 (a) illus-trates the contour map of the peak value at ε = 0 . f q appears around q = 7 . /d , which corre-sponds to the wavenumber where S q has a peak. Thedirections of the contour lines are distinguished with re-spect to each area of the repulsive and attractive glasses.Although the peak value continuously changes almost ev-erywhere, it discontinuously changes on the bifurcationline in the glassy state. Figure 2 (b) shows the value of f q for three wavenumbers at ε = 0 . 03. The volume fractionwas selected at φ = 0 . θ c = 1 . 05. The behavior of f q near the glass-glass tran-sition point is the same as that of MCT [14]. In θ < θ c , | f q − f c ,q | asymptotically holds the square root variationof | θ − θ c | , where f c ,q = lim θ ր θ c f q . This means that theattractive glass appears/disappears as a fold bifurcation. C. higher-order singularities In this subsection, we confirm that the glass-glass tran-sition line of TMCT ends as well as that of MCT. Figure2 (c) shows the φ -dependence of the maximum eigenvalue E for several θ at ε = 0 . 03. It clearly shows that thereis a marginal temperature θ ∗ such that the value of E reaches the unity in θ < θ ∗ and it does not in θ > θ ∗ with controlling φ . At θ = θ ∗ , the eigenvalues of boththe repulsive and attractive glasses reach the unity. Itis thus concluded that the glass-glass transition line of ε = 0 . 03 terminates at ( φ ∗ , θ ∗ ) ≃ (0 . , . A singularity (equiva-lently, cusp bifurcation), which is a higher-order singular-ity [12–15, 42–44]. In this context, the nonergodic tran-sition is classified as the A singularity. Chen et al. haveexperimentally proved the existence of the A singular-ity [7]. Note that the A singularity of ε = 0 . 04 is at( φ ∗ , θ ∗ ) ≃ (0 . , . ε , theglass-glass transition line disappears at a certain point.This parameter set is called A singularity (equivalently,swallow-tail bifurcation) point [14, 42–44]. The TMCTvalue of ε at the A singularity point is around 0 . 05. Asthe MCT value is around 0 . 04 [14], TMCT extends the ε range within which the glass-glass transition occurs. D. quantitative comparison of transition points We finally compare TMCT with MCT in a quanti-tative manner. Figure 3 shows the kinetic phase dia-gram at ε = 0 . 03, in which the TMCT critical valuesfor the one-component SWS are compared with thoseof MCT and also the MD results for the binary SWS(A : B = 50 : 50) [18]. The transition line of the MDsimulation was determined by the contour of the normal-ized diffusivity ˜ D = 5 × − of the A particle, where˜ D = D/D , D = d A p k B T /m , D denotes the long-timeself-diffusion coefficient, and d A the diameter of the Aparticle. The long-time self-diffusion coefficient is an ap-propriate physical value for a unified comparison betweendifferent systems [49]. The value 5 × − was chosen forthe iso-diffusivity line in the high T limit to approach φ ≃ . 58 [18]. The iso-diffusive line is much closer to thekinetic glass transition line of TMCT without any scal-ing. Although the critical temperatures of TMCT overes- θ φ FIG. 3. The transition lines at ε = 0 . 03. The line with filledcircles (red) indicates the TMCT result and the line with opencircles (red) the MCT result. The broken line with squaresindicates the MD result of the iso-diffusivity ˜ D = 5 × − for A particle of the binary SWS [18]. timate the MD results, we do not judge whether TMCTfails to predict the critical temperature or not. Approx-imation methods (e.g., PYA) for S q affect the tempera-ture dependence. A characteristic T of SWS based on themean-spherical approximation (MSA) is about five timessmaller than that based on PYA, while characteristic φ and ε are comparable between PYA and MSA [14]. Infact, the transition line of the TMCT analysis for one-component SWS based on MSA underestimates the MDresult. In addition, the difference might originate fromthe fact that the simulation was done for binary SWS,while TMCT was applied for one-component SWS. IV. SCHEMATIC MODEL Our numerical results in SWS have shown that TMCTleads to the glass-liquid-glass reentrant, the glass-glasstransition, and the higher-order singularities. However,in a schematic model where MCT predicts both theliquid-glass and glass-glass transition with the A sin-gularity [50], G¨otze and Schilling have shown that, al-though the liquid-glass transition occurs in TMCT, theglass-glass transition does not [37]. In this section, weanalyze a modified version of the schematic model to sup-port validity of our numerical results.The model analyzed by G¨otze and Schilling assumes amode-coupling polynomial of a single wavenumber (i.e., M = 1) as F = v f + v f , where f denotes the Debye–Waller factor of M = 1 and positive coefficients v and v correspond to control parameters. We here considerthe modified schematic model in which a mode-couplingpolynomial is defined by F = v f + v n f w , (3) v n v FIG. 4. The bifurcation diagram on the v – v n space ofthe modified schematic model with w = 1 / 6. The blue linescorrespond to MCT and the red ones do to TMCT. The solidlines represent the discontinuous bifurcation and the open cir-cles mark the A singularities. The dashed line represents thecontinuous bifurcation, in which the Debye–Waller factor con-tinuously changes across the bifurcation line [50]. where, in addition to positive coefficients v and v n , apositive coefficient w in the power is another control pa-rameter.In the modified schematic model (3), TMCT predictsthe liquid-glass transition, the glass-glass transition, andthe higher-order singularities, where the details of theanalysis is summarized in the Appendix. The kineticphase diagram of the model with w = 1 / A singularity asshown in Fig. 4. The range of w where the A singularityemerges is limited to w < w ∗ with w ∗ = ( √ − / ≃ . A singularity exists at w = w ∗ .In the MCT analysis to the modified schematic model,the discontinuous bifurcation line of w < w > A singularity occurs w ∗ = 1 in the MCT analysis. Thisdifference of w ∗ between MCT and TMCT is a probablereason why higher-order singularities do not appear inthe TMCT analysis by G¨otze and Schilling.The modified schematic model with MCT does not cor-respond to the short-range attractive colloids because oneof the bifurcation line predicted by MCT indicates thecontinuous bifurcation. On the other hand, TMCT doesnot describe any continuous bifurcations [37]. Thus, iden-tifying ( v , v n , w ) with ( φ, /T, ε ), except for the glass-liquid-glass reentrant, the modified schematic model withTMCT qualitatively corresponds to the short-range at-traction colloids well. Nevertheless, it should be notedthat the model is just schematic; there is no knowingwhether a control parameter such as ε can shift the non-linear power. As the modified schematic model is asingle-wavenumber model ( M = 1), it can be interpretedas renormalization from the whole rage of wavenumberalters the power (i.e., w ) of the nonlinear term. On thebasis of this idea, Tokuyama has described simulationresults well by TMCT [51].A similar form of the phase diagram shown in Fig. 4 hasbeen reported by G¨otze and Sperl [42]. They have stud-ied a two-wavenumber model with three control parame-ters: F ( x , x ) = v x + v x and F ( x , x ) = v x x .In their model, there is a marginal value v ∗ such thatthe A singularity exists in v > v ∗ and it does not in v < v ∗ . At v = v ∗ , the discontinuous bifurcation linescollapse. Note that there are no single-wavenumber mod-els in which the A singularity predicted by MCT existsby collapsing the discontinuous bifurcation lines. V. SUMMARY For SWS as a model of the attractive colloids, wehave presented numerical evidence for the existence ofthe glass-liquid-glass reentrant, the glass-glass transition,and the higher-order (i.e., A and A ) singularities in theTMCT analysis. Compared with the results of MD sim-ulation for a binary colloidal system with short-range at-traction, we have clarified quantitative improvement ofthe critical volume fractions. As TMCT has the sameform of the memory function of MCT, our analysis en-hances the utility of the MCT framework for the study ofglass transition. By contrast to the success in the criticalvolume fraction, the difference of the critical tempera-tures between theoretical calculations and MD simula-tions should be addressed. The TMCT analysis by usingthe static structure factor obtained from the MD simu-lation will enable the detailed comparison of the criticaltemperature with that of the simulation result.We have analytically studied the modified schematicmodel defined in Eq. (3) to demonstrate that TMCTpredicts the higher-order singularities within schematicmodels. Except for the glass-liquid-glass reentrant, themodified schematic model qualitatively describes the ki-netic phase diagram of the short-range attractive colloidswell. Recalling that w is introduced in the power of thenonlinear term, TMCT suggests that the A singularityemerges in the nonlinear power more than 3+2 √ ≃ . ACKNOWLEDGMENTS We are deeply grateful to Professor Sow-Hsin Chenfor suggesting us to apply TMCT for short-range attrac-tive colloidal systems. We also wish to thank Dr. YutoKimura for fruitful discussions on numerical calculations.This work was partially supported by JSPS KAKENHIGrant No. JP26400180. APPENDIX The appendix presents details of analysis for the mod-ified schematic model (3).For MCT and TMCT, the fixed-point equation f = T ( f ) leads to v + v n f w = 1 f s , (4)where s is defined as 1 / F : s = f − f [MCT] , /f ) [TMCT] . (5)Further, the stability matrix A on the bifurcation pointsmust be unity, that is, v c + (cid:18) w (cid:19) v cn ( f c ) w = 1( f c ) δ MCT ( s c ) , (6)where the superscript c indicates their critical values and δ MCT = (cid:26) , . (7)Equations (4) and (6) reduce to the parametric represen-tation of v c and v cn : v c = w (cid:20)(cid:18) w (cid:19) G − G (cid:21) , (8) v cn = w ( f c ) w ( G − G ) , (9)with G = 1 f c s c , G = 1( f c ) δ MCT ( s c ) . (10)Figure 4 was drawn based on these equations. The inter-cepts of the discontinuous bifurcation lines are derived asfollows. When v n = 0, the critical values are v c = e inTMCT, and v c = 1 in MCT for w > 1. On the otherhand, when v = 0, Eqs. 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