2014年11月4-7日　北大物理 集中講義

「ガラス系の物理」　

吉野　元 (大阪大学 サイバーメディアセンター)

1. 序論：2. クローン液体論の基本的なアイディア3. スピングラス平均場理論(1)

4. スピングラス平均場理論(2)

5. 液体論の基礎

6. クローン液体論

7. レオロジー：粘性、弾性、塑性

8. ジャミング

講義の内容

Lecture 1 : 序論

過冷却液体・ガラスと熱力学過冷却液体のダイナミックススピングラスからのヒント

過冷却液体・ガラスと熱力学

F. H. Stillinger and T. A. Weber, J. Chem. Phys. 83, 4767 (1985); P. G. Debenedetti and F. H. Stillinger, Nature (London) 410, 259 (2001).

Kauzmann, W. (1948). The Nature of the Glassy State and the Behavior of Liquids at Low Temperatures. Chemical Reviews, 43(2), 219-256.

温度

エントロピー

結晶

過冷却液体

”Tg”

液体

Tm融解

ガラス

Tc“疑似”動的転移(モード結合理論)

T �“Entropy crisis”

熱力学的なガラス転移？Kauzmann温度 TK

S(T ) = S(Tref)�� Tref

T

dT

TC(T )

温度を下げながら比熱を測定

エントロピーにみられる過冷却液体・ガラスの特徴

　エネルギーランドスケープ描像

全体の自由エネルギー

��F = ln Z = ln�

�

e�N�f� = ln�

dfe�N(�f��(f,T ))

“deeply” supercooled liquid= 多数の準安定状態の間を熱活性によって遷移している

1つの準安定状態: エネルギー的に局所安定な粒子配位 “inherent structure(IS)”

　　+ そのまわりの振動モード　(この中では固体的なdynamics)

ex. F. H. Stillinger and T. A. Weber (1995)

f� = e� � Ts�

1粒子あたりの自由エネルギー

inherent sturcture のエネルギー

振動のエントロピー

Complexity

�

エネルギー

�(f, T ) � 1N

ln�

�

�(f � f�(T ))

H =N�

i=1

p2i

2m+

�

i<j

v(rij)Hamiltonian (単純液体の場合)

ZN =1

N !

� N�

i=1

ddxiddpi

hdexp

�

���N�

i=1

|pi|2

2m� �

�

i<j

v(rij)

�

�

=1

N !

� N�

i=1

ddxi

�dth

exp

�

����

i<j

v(rij)

�

�

統計力学 :　始めの一歩

��FN (T ) = lnZN (T )自由エネルギー

�th = h/�

2�mkBTthermal de Brogile wave length

分子（粒子）間の相互作用

rij = |xi � xj |

理想気体 � = N/V数密度

Z id =1

N !(V/�d

th)N � F id/N = kBT [ln(��dth)� 1]

xi = xi;� + ui;�

� N�

i=1

ddxi

�dth

exp

�

����

i<j

v(rij)

�

� ��

�

�

basin of �

N�

i=1

ddxi

�dth

exp

�

����

i<j

v(rij)

�

�

rij;� � |xi;� � xj;�|

e� =1N

�

i<j

v(rij;�)

s� =kB

Nln

� N�

i=1

ddui

�dth

exp

�

���

2

�

i<j

�2v(rij;�)(ui;� � uj;�)2�

�

e�N�f� � 1N !

�

basin of �

N�

i=1

ddxi

�dth

exp

�

����

i<j

v(rij)

�

�

� exp

�

����

i<j

v(rij ; �)

�

�� N�

i=1

ddui

�dth

exp

�

���

2

�

i<j

�2v(rij;�)(ui;� � uj;�)2�

�

f� = e� � Ts�

　エネルギーランドスケープ描像を仮定すると

inherent structure での配位

v(rij) = v(rij;�) +12�2v(rij;�)|ui;� � uj;�|2 + . . .

　Kauzmann転移

F/N = �kBT

Nln

�dfe�N(�f��(f,T )) = f� � T�(f�, T ))

��(f, T ))�f

����f=f�(T )

=1T

Kauzmann転移(entropy crisis)

limT�T+

K

�(f�(T ), T ) = 0

0

傾き 1/T

f

Σ(f)/kB

f∗(T )

S/N = s� + kB�(f�, T )準安定状態のエントロピー

s� � ��f�

�T構造エントロピー sc(T )

過冷却液体のダイナミックス

Bouchaud, J. P., & Biroli, G. (2004). On the Adam-Gibbs-Kirkpatrick-Thirumalai-Wolynes scenario for the viscosity increase in glasses. The Journal of chemical physics, 121(15), 7347-7354.

F. H. Stillinger and T. A. Weber, J. Chem. Phys. 83, 4767 (1985); P. G. Debenedetti and F. H. Stillinger, Nature (London) 410, 259 (2001).

Kirkpatrick, T. R., D. Thirumalai, and Peter G. Wolynes. "Scaling concepts for the dynamics of viscous liquids near an ideal glassy state." Physical Review A 40.2 (1989): 1045.

G. Adam and J. H. Gibbs, J. Chem. Phys. 43, 139 (1965).

Angell, C. Austin, Kia L. Ngai, Greg B. McKenna, Paul F. McMillan, and Steve W. Martin. "Relaxation in glassforming liquids and amorphous solids." Journal of Applied Physics 88, no. 6 (2000): 3113-3157.

C.7. What is the relation between secondaryrelaxation in glasses due to fast-translatingspecies, on the one hand, and nontranslatingentities „e.g., side chains… on the other?

C.8. Other questionsC.8.1. Can comparisons of dielectric relaxation of

small dipolar molecules in molecular orpolymer glasses, and of ions in ionic glasses,throw light on the origin of nonexponentialrelaxation in ionic glasses; e.g., does CO ino-terphenyl !OTP" give a nonexponential re-laxation like Na! in Na2O•B2O3?

C.8.2. Can we compare mechanical relaxation dueto small atoms in met glasses, small mol-ecules in polymers, and small ions in ionicglasses?

C.8.3. What is the relation of the high frequencyconstant loss in conducting glasses to thehigh frequency loss in dielectric relaxation,given that the first is a secondary relaxationand the second one a primary relaxation?

Domain D: Short time dynamics

D.1. Boson peaks in raman and neutron scatteringD.1.1. How good are the data?

D.1.2. Is there a correlation of Boson peaks withstrong/fragile behavior in the longer-timeproperties?

D.1.3. Is there a correlation with the structural fea-tures responsible for the first sharp diffrac-tion peak !FSDP"?

D.1.4. Do Boson peaks measure cluster dynamics orcage rattling dynamics?

D.1.5. Are there relaxation dynamics at time earlierthan the Boson peak?

D.2. Extrapolated breaks in the Debye–Waller factorversus temperature plot—what do they signify?Are they related to FSDPs?

D.3. What is vibrational localization? What specialdynamical characteristics are associated withlocalized modes? And how might it berelated to relaxation? Are there fractal dynamicsin disordered but Euclidean structures?How helpful are photon echo studies?

D.4. What is the decay time for the FSDP inrelation to the wave vector Q0 , at the peak of thestructure?

D.5. What are the newest, and most important stepsforward?

A. RELAXING GLASSFORMERS IN INTERNALEQUILIBRIUMA.1. The temperature dependence of transportconstants and relaxation times for various types ofperturbations and probes

A.1.1 Relevant measurements!What are the most appropriate measurements for char-

acterizing the temperature dependence?"Liquids of all classes, now including even liquid metals,

have been extensively characterized with respect to the tem-perature dependence of their transport properties and theirrelaxation times down to, and often below, their Tg . Thetransport properties studied have been mainly viscosity # toa lesser extent diffusivity D, and also conductivity $, whereappropriate.1

The relaxation times studied have been dielectric %D ,2longitudinal !1, and sometimes shear mechanical %srelaxations,3 and !with modeling" nuclear spin lattice corre-lation %c ,n , and electron spin correlation %c ,e .4 Recently, en-

FIG. 2. Tg scaled Arrhenius plots for viscosities of different glassformingliquids showing spread of data between strong and fragile extremes. Predic-tions of mode coupling theory power law are shown by dashed line for twocases. Inset shows the pattern of behavior obtained by varying the D param-eter in the modified Vogel–Fulcher equation, Eq. !6".

3118 J. Appl. Phys., Vol. 88, No. 6, 15 September 2000 Applied Physics Reviews: Angell et al.

Downloaded 28 Oct 2010 to 133.1.157.167. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

log �

Tg/T

Angell, C. Austin, Kia L. Ngai, Greg B. McKenna, Paul F. McMillan, and Steve W. Martin. "Relaxation in glassforming liquids and amorphous solids." Journal of Applied Physics 88, no. 6 (2000): 3113-3157.

Vogel-Fulcher則

m =� log10 �

�(1/T )

����Tg

“Fragility”

� = �0 exp�

DT0

T � T0

�

粘性 � � �緩和時間

粘性（緩和時間）とエントロピーの相関

� = �0 exp�

�/c

kBT

TK

T � TK

�Vogel-Fulcher則!

Volume 102, Number 2, March–April 1997Journal of Research of the National Institute of Standards and Technology

Fig. 4. Test of the Adam-Gibbs equation for the relaxation time of butyronitrile vapor-deposited samples (from Ref. [34] by permission).

Fig. 5. Test of the Adam-Gibbs equation for viscosity of tri-napthyl benzene at temperaturesabove Tg. Sc has been assessed in two different ways leading to two different plots (see text) , eachof which is seen to be linear over a wide range of the variable (TSc)–1 (from McGill, Ref. [35] bypermission). (DSc in the paper of Magill is the Sc of this paper.)

177

“構造エントロピ”ーの温度依存性

sc(T )/kB � c(T/TK � 1)

H. Fujimori and M. Oguni, (1994)

Adam-Gibbs Di-Mazzio (1958)の仮説 � = �0 exp

��

Tsc(T )

�

Adam-Gibbs Di-Mazzioの予想: この背後には過冷却液体におけるある種の相関長　　 の発達があるのではないか？“collective rearranging region” (CRR)

動的不均一性を観測・測定する試みが続けられた

�(T )

is the frequency with maximum dielectric loss for the corre-sponding fitting function being used in Eq. !3". We use f minstead of the relaxation time, such as #HN and #CC , becausef m does not depend on the choice of fitting function. There isa relation f m!1/2$%#&. The relaxation frequency of the 'process, f m

' , for all the materials follows the VFT mannerdescribed by

log10 f m!A"B

T"T0, !4"

where A!log10(1/2$#VFT), B!BVFT/2.303, and T0 is theVFT temperature. A represents the attempt frequency at T

→(. On the other hand, the relaxation frequency of the )process, f m

) , for all the materials follows the Arrhenius man-ner described by

log10 f m!C"EA

2.303RT , !5"

where C!log10(1/2$#A) and EA is the activation energy. Crepresents the attempt frequency at T→( .The temperature dependence of f m

' in various glass form-ers cannot be described precisely by using only one VFTfunction. It has been reported that two VFT functions arerequired to describe the temperature dependence of f m

' forsome systems *33–35+. We have reported that the ' processin sorbitol at temperatures near the glass transition tempera-ture Tg* , which is the temperature with 0.01 Hz of f m' , can-not be described using only one VFT function determinedusing data at higher temperatures *21+. Especially, in the casethat the relaxation strength of the ) process, ,-), is rela-tively large, the behavior of the ' and ) processes near TMcannot be distinguished clearly due to the strong correlationbetween the two processes. To discuss the behavior of theloci for the ' process in the Arrhenius diagram is not the aimof this paper. Therefore, in this work, VFT curves are deter-mined using data below TM just to compare the behavior ofthe ' processes among the materials.As shown in Fig. 4!a", the curves of the VFT functions for

all the materials are different corresponding to the differenceof Tg* . Tg* of sorbitol and xylitol are 268 K *21+ and 248 K,respectively. Tg of sorbitol and xylitol obtained by DSCmeasurement are 267 K and 249 K, respectively *30+. Tg* ofthe mixture is 258 K, which is intermediate value betweenthose of sorbitol and xylitol. The fragility index m was givento be 101 for sorbitol, 86 for xylitol, and 93 for the mixture,respectively.On the other hand, the ) processes in sorbitol, xylitol, and

the mixture show the Arrhenius-type temperature depen-

FIG. 1. Dielectric losses of sorbitol !a", sorbitol-xylitol mixture!b", and xylitol !c" at various temperatures. Solid curves are thefitting function described by Eq. !3".

FIG. 2. Dielectric losses of sorbitol at 282 K !solid diamond",xylitol at 263 K !solid circle", and mixture at 272 K !open triangle".

DIFFERENCE AND SIMILARITY OF DIELECTRIC . . . PHYSICAL REVIEW E 68, 031501 !2003"

031501-3

log f [HZ]

Difference and similarity of dielectric relaxation processes among polyols

Ayumi Minoguchi, Kei Kitai, and Ryusuke NozakiDivision of Physics, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan

!Received 27 December 2002; revised manuscript received 25 March 2003; published 5 September 2003"

Complex permittivity measurements were performed on sorbitol, xylitol, and sorbitol-xylitol mixture in thesupercooled liquid state in an extremely wide frequency range from 10 #Hz to 500 MHz at temperatures nearand above the glass transition temperature. We determined detailed behavior of the relaxation parameters suchas relaxation frequency and broadening against temperature not only for the $ process but also for the %process above the glass transition temperature, to the best of our knowledge, for the first time. Since super-cooled liquids are in the quasi-equilibrium state, the behavior of all the relaxation parameters for the % processcan be compared among the polyols as well as those for the $ process. The relaxation frequencies of the $processes follow the Vogel-Fulcher-Tammann manner and the loci in the Arrhenius diagram are differentcorresponding to the difference of the glass transition temperatures. On the other hand, the relaxation frequen-cies of the % processes, which are often called as the Johari-Goldstein processes, follow the Arrhenius-typetemperature dependence. The relaxation parameters for the % process are quite similar among the polyols attemperatures below the $% merging temperature, TM . However, they show anomalous behavior near TM ,which depends on the molecular size of materials. These results suggest that the origin of the % process isessentially the same among the polyols.

DOI: 10.1103/PhysRevE.68.031501 PACS number!s": 64.70.Pf, 61.25.Em, 77.22.Gm

I. INTRODUCTION

There are two dielectric relaxation processes in many su-percooled glass formers, i.e., the primary $ relaxation pro-cess and the secondary % relaxation process. The $ processshows non-Arrhenius-type temperature dependence de-scribed by Vogel-Fulcher-Tammann !VFT" function &1–3',

()*!)VFTexp! BVFT

T"T0" , !1"

where ()* is the mean relaxation time, )VFT represents therelaxation time at T→+ , T0 is the VFT temperature wherethe mean relaxation time diverges and BVFT is connected tothe strength parameter, which is related to the fragility &39'.It is considered that the $ process is the structural relaxationprocess and closely connected to glass transition phenomena.Recently, one has been trying to understand the $ processusing a concept of ‘‘cooperative molecular motion’’ andthere are several theories or models &4–10'.On the other hand, secondary % process !so-called the

Johari-Goldstein process &12'" is observed in many glassformers at higher-frequency side of the $ process. The %process obeys the Arrhenius-type temperature dependencegiven by

()*!)Aexp! EA

RT " , !2"

where ()* is the mean relaxation time, )A represents therelaxation time at T→+ , and EA is the activation energy. Ithas been considered that the % process is connected to localmolecular motions, in contrast with the $ process. It is alsoconsidered that the % process is concerned with the essenti-ality of amorphous matters, such as the supercooled liquidsand the glassy solids, because the % process is observed not

only in the supercooled liquid state but also in the glassystate. Although several understandings have been examined&9–11,13,14', the mechanism of the % processes is stillunclear.While these two processes are observed separately at

lower temperatures above the glass transition temperature,they approach each other with increasing temperature andmerge together at a higher temperature TM . There would bea kind of strong correlation between the $ and the % pro-cesses at near the merging temperature TM , which some-times cannot be determined precisely. It must be very impor-tant to study the relation between the behavior of the $process and the appearance of the % process to understandthe glass transition phenomena in detail.The merging of the $ and % processes is classified into

two scenarios by Donth et al. &15,16'. Scenario I shows acontinuity of the relaxation frequency of the % process belowTM and the relaxation frequency of the process above TMwith a bend in the Arrhenius diagram. The $ process appearsat around TM . The values of the relaxation strength of the $and % processes have the same order of magnitude. ScenarioII shows a quasicontinuity of the relaxation frequency of the$ process below TM and the relaxation frequency of theprocess above TM in the Arrhenius diagram. The value of therelaxation strength of the % process is smaller than that of the$ process.However, some glass formers do not show the %

process clearly, but an excess wing, which is the high-frequency wing of the $ process, is observed. It is proposedthat glass formers are divided into two classes: type A withan excess wing and type B with a % process &17'. Manyglass formers are classified into type B. Glycerol&CH2OH(CHOH)CH2OH' and propylene carbonate arewell-known glass formers classified into type A &17,19,20'.Sorbitol &CH2OH(CHOH)4CH2OH' and xylitol

&CH2OH(CHOH)3CH2OH' are typical glass-forming liq-

PHYSICAL REVIEW E 68, 031501 !2003"

1063-651X/2003/68!3"/031501!7"/$20.00 ©2003 The American Physical Society68 031501-1

dence and the behavior is quite similar to each other. Figure3 shows the behavior of the ! processes in sorbitol, xylitol,and the mixture. The temperature dependences of f m

! for allthe materials are described well by the solid lines obtainedfrom Eq. "5#. The values of parameter C for all the materialsare roughly 17. The values of the activation energy are EA!80 "kJ/mol# for sorbitol, EA!78 "kJ/mol# for xylitol, andEA!79 "kJ/mol# for the mixture.The values of the activation energy of the ! processes in

sorbitol and xylitol are somewhat larger than those reportedin a literature $29%. However, it should be noted that thevalues of the activation energy in this literature were ob-tained from the data below Tg . It is not very strange thatthere is difference in the activation energies below and aboveTg . Actually, such a difference in sorbitol has been reportedin Ref. $13%.As mentioned earlier, only one dielectric relaxation pro-

cess for the & process is observed in the mixture. f m& , '(&,

and the Tg* of the mixture are intermediate value betweenthose of sorbitol and xylitol. Namely, the mixture behaves

FIG. 3. Arrhenius diagrams of sorbitol "a#, mixture "b#, andxylitol "c#. Dotted lines represent TM of sorbitol "320 K#, mixture"300 K#, and xylitol "288 K#, respectively. "a# Solid and open dia-monds represent f m

& and f m! for sorbitol, "b# solid and open triangles

represent f m& and f m

! for mixture, and "c# solid and open circlesrepresent f m

& and f m! for xylitol. Solid curves were obtained from

Eq. "4# "sorbitol: A!12.7, B!572 K, and T0!229 K, xylitol: A!12.85, B!640 K, and T0!205 K, mixture: A!12.77, B!606 K, and T0!217 K). Solid lines were obtained from Eq. "5#$sorbitol: C!17.2 and EA!80 "kJ/mol#, xylitol: C!17.12 and EA!78 "kJ/mol#, sorbitol-xylitol mixture: C!17.16 and EA!79 "kJ/mol#%. The fitting uncertainties near TM are larger than those forbelow and above TM . The base of the logarithm is 10.

FIG. 4. Dotted lines represent TM of sorbitol "320 K# and xylitol"288 K#, respectively. "a#Arrhenius diagram of sorbitol, xylitol, andmixture. Symbols are the same as those used in Fig. 3. "b# Tem-perature dependences of '(! and )! for sorbitol, xylitol, and themixture. Open diamonds, open circles, and open triangles represent'(! for sorbitol, xylitol, and the mixture, respectively. Solid dia-monds, solid circles, and solid triangles represent )! for sorbitol,xylitol, and the mixture, respectively. The base of the logarithm is10.

MINOGUCHI, KITAI, AND NOZAKI PHYSICAL REVIEW E 68, 031501 "2003#

031501-4

Adam-Gibbs vs “Mosaic picture”Adam-Gibbs(1965)’s original argument

sc(T ) = kB(a/�(T ))3 ln � � � 2

A kind of Mattis model?

�

a : 分子のスケール

Kirkpatrick Thirumalai and Wolyness (1989)Bouchaud Biroli (2004)

R

� ミスマッチのエネルギー

Z(R, T ; �) =�

�

e�N(R)�f���(1���,�)�(R/a)�

N(R) � �R3

�mosaic = a(�/�a3Tsc(T ))1

3��

EB = �(�(T )/a)3

ln �(T )/�0 = EB/kBT = (� ln �/Tsc(T ))

活性化エネルギー

「ケージ構造、2段階緩和」

共焦点顕微鏡で捉えられた過冷却液体状態にあるコロイド分散系での粒子の運動 (E. Weeks and D. Weitz (2002))

ケージの中での熱振動β緩和

動的不均一性(Stokes-Einstein関係の破れ）

ケージ構造の組み替えα緩和

スピングラスからのヒント

Edwards, Samuel Frederick, and Phil W. Anderson. "Theory of spin glasses." Journal of Physics F: Metal Physics 5.5 (1975): 965.

Miyako, Yoshihito, et al. "Nonlinear magnetization at the spin glass phase transition temperature." Journal of the Physical Society of Japan 46 (1979): 1951.

Berthier, L., Biroli, G., Bouchaud, J. P., Cipelletti, L., & van Saarloos, W. (Eds.). (2011). Dynamical heterogeneities in glasses, colloids, and granular media. Oxford University Press.

W. Go tze, Complex Dynamics of Glass-Forming Liquids: A Mode-Coupling Theory, International Series of Monographs on Physics Vol. 143 (Oxford University Press, New York, 2009).

スピングラス：強磁性と反強磁性がランダムに存在する磁性体 (例：希薄磁性合金)

Edwards-Anderson模型(1975)

[Jij ] = 0 [J2ij ] = J2正負ランダムなスピン間相互作用

Ag1�xMnx

H = ��

<i,j>

Jijsisj

T > TSG

T < TSG

常磁性相

スピングラス相

qEA = 0

qEA > 0

Edwards-Andeson 秩序パラメータ（非エルゴードパラメータ)

qEA = limt��

1N

N�

i=1

�si(t)si(0)� =1N

N�

i=1

�si�2

さらに、レプリカ対称性の破れ

(多数の乱れた準安定状態の出現)

（ただし、現実の有限次元系では未解決）の臨界ゆらぎ：非線形帯磁率の発散qEA

スピングラス転移「相関長」の発散を伴う、2次相転移

？ フラストレーションによるエネルギー縮重

基底状態: 乱れたスピン配位

スピングラス

�2 = ���SG

�qEA = �SGh2 + . . .

�SG =1N

�

ij

�SiSj�2

非線形帯磁率

スピングラス帯磁率

非線形帯磁率の発散

m = �0h + �2h3 + . . .

qEA = limt��

1N

N�

i=1

�si(t)si(0)� =1N

N�

i=1

�si�2

qi(t) qi

�SG =1N

�

ij

�SiSj�2 = limt��

1N

�

ij

�Si(t)Sj(t)Si(0)Sj(0)�

= limt��

1N

�

ij

�qi(t)qj(t)�

�4(t) = limt��

1N

�

ij

[�qi(t)qj(t)� � �qi(t)��qj(t)�]

dynamicsの空間相関

スピングラス帯磁率

Edwards-Anderson秩序パラメータ

ガラス転移の研究でも盛んに使われるようになった:　動的不均一性をとらえる道具

レビュー Berthier, Ludovic, et al., eds. Dynamical heterogeneities in glasses, colloids, and granular media. Oxford University Press, 2011. glycerol での非線形誘電応答の実験: Saclay group (2010-)

分子動力学(MD)シミュレーションで観測された自己中間散乱関数 (W. Kob and H. C. Andersen (1995))

19

Figure 10: The static structure factor - The static struc-ture factor S(q) in a Lennard-Jones liquid at three differenttemperatures. The relaxation time τR increases by almost 4orders of magnitude, and yet the structure factor shows noparticular change. (Reprinted with permission from [8]).

on than what is shown in Fig.10. We conclude that it isimpossible to use the structure factor, or any other stan-dard structural quantity, to understand whether or notthe sample is close to the glass transition.

The weak modification of the structure factor withtemperature also implies that any lengthscale that can bereasonably extracted from S(q) or g(r) shows a depress-ingly weak dependence on T near the glass transition[45, 46, 47, 48]. This is surprising. The common wisdomis that the presence of a sharply increasing (or diverging)relaxation time should be associated to a sharply increas-ing (or diverging) correlation length. The wisdom comesmainly form the theory of critical phenomena [50], andthe argument is basically that a large relaxation time de-rives from the need to rearrange larger and larger corre-lated regions. Even though common indeed, such wisdomdeserve some carefulness anyway. First, we cannot stresstoo much the (obvious) fact that in a system with a finitenumber of degrees of freedom, the relaxation time mustbe equally finite. A real divergence can only occur inthe thermodynamic limit and only in presence of a phasetransition. Still, the time can be very large, typically be-cause of the presence of large energy or free energy bar-riers to relaxation. In general, a barrier could arise froman external potential, and in this case there is no need toinvoke the existence a large length scale. However, in allinteresting systems barriers almost invariably arise fromthe internal interaction among the degrees of freedom. Inthis case, increasing barriers (and thus increasing relax-ation times), are indeed due to the increasing number ofdegrees of freedom that must be rearranged together torelax the system. Hence: large time, large length.

This argument makes sense, but of course in order to

detect a sharply growing correlation length, we need toidentify a suitable correlation function. This is the cru-cial point: in very viscous liquids and glasses we are com-pletely lost when it comes to identify the correct orderparameter. Structural correlation functions are not up tothe job and fail to provide any exciting characterizationof the viscous and glassy phase. This is unfortunatelytrue for all other standard static correlation functions.

The reason for this can be either because we are usingthe wrong correlation function, or simply because thereis no sharply growing lengthscale (the common wisdomcould be wrong!). In fact, we will see at the end of thesenotes that the first hypothesis is the correct one. For now,we simply note that the standard structural observablesare not good enough to say something about the deeplysupercooled phase. Structure remains qualitatively thesame going through the glass transition. Supercooledliquids and glasses are structurally unexciting.

E. Equilibrium fingerprint of glassiness: two stepsrelaxation

The failure of a standard static approach to find a sig-nature of the glass transition is, in fact, not surprising.The very definition of the glass transition is purely dy-namic in nature, and therefore if something new happensclose to Tg, the dynamics, rather than the static struc-ture, should detect it. In particular, the viscosity, whichmarks the onset of glassiness, is the integral over timeof a dynamic correlation functions, namely the shear re-laxation function, see equation (11). The same is truefor the diffusion coefficient, related to the time integralof the velocity-velocity correlation function (18). An in-tegral wraps up an entire function into a single number,thus loosing a lot of information. Hence, it seems a goodidea to check the dynamic correlation functions, ratherthan their integrals, to see whether they show some qual-itative signature of Tg.

Let us consider, in full generality, the dynamic corre-lation function,

C(t1, t2) =1

N

N!

k=1

⟨ϕk(t1)ϕk(t2)⟩ , (54)

where ϕk(t) is a generic quantity relative to particle k,observed at time t. If the system is at equilibrium, thentime translation invariance (TTI) holds, and the corre-lation function only depends on the difference of times,t = t2 − t1, so that C(t1, t2) = C(t), and we can write,

C(t) =1

N

N!

k=1

⟨ϕk(t)ϕk(0)⟩ . (55)

In liquids, a typical choice for ϕk(t) is the Fourier trans-form of the density fluctuations of a tagged particle k,δρk(q, t) = exp[−iq · rk(t)], at fixed momentum q. In

静的構造因子 温度を低下させても構造に目立った変化なし(緩和時間は急激に増大）

g(r) =1N

N�

i=1

N�

i=1

�(r � |ri � rj |)

密度場

動径分布関数

�(r) =N�

i=1

�3(r� ri)

液体の密度場の様子

MD simulation (2成分Lennard-Jones) Kob-Andersen (1995)

q

冷却

S(q) � 1V

�d3rd3r�eiq·(r�r�)�(r)�(r�)

= �0

�d3reiq·r 1

N

N�

i,j=1

�3(r� (ri � rj))

= 4�2�0

� �

0dreiqrr2g(r)

図 6: 2段階緩和:　 a) 過冷却液体での自己中間散乱関数 Fs(q, t)の時間 t依存性 (波数は静的構造因子 S(q)の第 1ピークの波数 q = 7.25) (2成分Lennard-Jonnes粒子系の分子動力学(MD)シミュレーションW. Kob andH. C. Andersen (1995) [8]より転載)　 b) 中間散乱関数 F (k, t) についてのMCT方程式 (コロイド分散系)の数値解 (宮崎州正 (2007) [9] より転載) 　 MCT 臨界点に対する距離� = 10�n (n = 1, 2, 3, . . .)での緩和関数をプロットしている。

図 7: 線形粘弾性測定に現れる 2段階緩和:　 a)過冷却液体 (フタル酸ジブチル DBP) の貯蔵弾性率の周波数依存性。(Maggi et. al. (2008) [10]より転載)　 Tm = 238 K、MCTによる（疑似)転移温度は Tc � 226 K [11]。b)コロイド分散系 (エマルジョン)の貯蔵弾性率 (Eq. (16) を参照)の周波数依存性 (Mason et. al (1997) [12]より転載) 　縦軸は対数であることに注意。MCTによる（疑似)転移密度は ⇤c � 0.58。

図 8: 熱力学量の振る舞い: 急冷して作成したガラス (トルエン)を昇温しながら測定した比熱 (白丸) (Ya-mamuro et. al. (1998) [13] より転載) 比較のために結晶化させた場合のデータも示されている。(黒丸) b)p=3 球形平均場スピングラス模型での熱平衡状態での比熱 (A. Crisantiand H. -J. Sommers (1992) [14]より転載)

との予想がある。すなわち、より高温側では、MCTの記述する臨界的なダイナミックス、低温側では、MCTでは記述できない熱活性型のダイナミックスが緩和を主導するというものである [17,18]。そのようなクロスオーバー温度の存在自体はMCT以前から推測されていた [19]。

2.4 線形粘弾性測定：プラトーモジュラス結晶ではそれぞれの分子はまわりの分子に囲まれて作られた平衡位置を中心として熱振動している。ケージ構造が有限の時間とはいえ、存在しているということは、過冷却液体は液体でありながら、構造緩和時間 ⇥�よりも短い時間スケールでは「固体的」に振る舞うことを示唆する。実際、図 7にあるような線形粘弾性 (線形レオロジー)の測定を行うと、過冷却液体は力学的にまさに固体と液体の両方の性格、すなわち弾性と粘性を併せ持っていることが明らかになる。線形粘弾性に詳しくはまた後で (3.3節)議論するのでここでは大づかみの特徴を紹介しよう。測定方法は、系に弱いシア歪みを周波数 ⌅の交流で掛け、応答として発生するシア応力を測定するというものである。図 7には貯蔵弾性率 (storagemodulus)と呼ばれる、摂動に対して位相の遅れのない応答成分を示している。(Eq. (16)参照) 明らかな特徴として、中間的な振動数領域に平坦部 (plateau)がある。これはこの時間スケールで過冷却液体が有限の剛性率 (shear modulus)を持つ弾性体として振る舞うことを意味する。この平坦部の高さで決まる「実効的な剛性率」をプラトーモジュラス (plateau modulus) と呼び、以下これを µと記すことにす

の平均場理論になっている確証がまだない。(スピングラスの平均場理論はそうなっていると強く信じられている。)　高次元極限で厳密に成立する過冷却液体・ガラスの理論の厳密な手続きによる導出は、理論としては重要で、興味深い課題である。

れなくなったのだ。そこで冷却速度 T をより遅くすると、ガラス化はもう少し低温で起こる。冷却速度T は実験室の時間スケール、つまり人間の時間スケールで決まっているわけで、Tgは絶対的な意味を持つ温度ではない。したがって、Tgから TKまでは、どうなっているのかまだ本当のところは何もわかっていない区間である。この過冷却液体のライン上で知られている重要な「道しるべ」としてモード結合理論 (MCT: mode coupling theory)の動的 (疑似)臨界温度 Tcがある。その意味は以下で説明しよう。

2.2 ケージ構造と 2段階緩和　過冷却液体状態では、粒子 (分子) がまわりの粒子に取り囲まれて自由な運動を阻害されている。この囲い込みの構造は、高温の液体にはないもので、「ケージ構造」と呼ばれる。図 5にあるように、コロイドや粉体で実空間での粒子の運動を追跡すると実際に見ることができる。

図 5: 共焦点顕微鏡で捉えられた過冷却液体状態にあるコロイド分散系での粒子の運動 (E. Weeks and D.Weitz (2002) [5]より転載)　体積分率 ⇧ = 0.52 (コロイド結晶への相転移密度 ⇧m = 0.42よりも高密度側、ガラス転移密度 ⇧g ⇥ 0.58よりも低密度側にいる。)、コロイド粒子の粒径は a = 1.18µ m。10分間 (この密度での構造緩和時間 ⌅� の程度)での変位が相対的に大きい粒子を明るい色で表示している。矢印は 0.2µm以上変位した粒子について、その変位の方向を表わす単位ベクトルを 2次元面に射影したもの。差し込み図は、ある粒子の軌跡 (120分間)。ケージ内の運動に加え、ケージの組み替えによる大変位が 1回見られる。交流のシアで揺すった粉体でも同様な運動の階層構造が観測されている [6]。

ケージ構造の存在は、マクロな動的観測量に特徴的な「2段階緩和」として表れる。散乱実験、数値シミュレーションでしばしば調べられる物理量として中間散乱関数F (q, t)や自己中間散乱関数 Fs(q, t)がある。これらは 2つの時刻での粒子の配位の相関を見る時間相関関数で、

F (q, t) =1N

N�

i,j=1

exp(iq · (ri(t)� rj(0)))

Fs(q, t) =1N

N�

j=1

exp(iq · (rj(t)� rj(0))) (1)

のように定義される。ここでri(s) = (xi(s), yi(s), zi(s))は時刻 sでの粒子の位置座標である。図 6を見ると、平坦部 (plateau)が現れることが注目される。これは粒子の配置が時間的に凍結することを表わし、平坦部の高さはスピングラスで提案された Edwards-Anderson(EA)秩序パラメータ qEA [7]に相当する。この平坦部に入って行く緩和をベータ (⇥)緩和、平坦部を抜けて 0まで落ちて行く過程をアルファ(�)緩和と呼ぶ。⇥ 緩和では、粒子はケージ内で運動している。その運動によって時間相関はある程度減衰するが、ケージが崩れない限り、0までは減衰しない。しかしケージは有限の寿命で崩壊し、新たなケージが作り直される。この過程が �緩和で、その緩和時間 ⌅�は構造緩和時間とも呼ばれる。

2.3 モード結合理論: (疑似)臨界温度 Tc

モード結合理論 (MCT)は過冷却液体におけるこの 2段階緩和を捉えることに成功した初めての微視的な理論である [9,15](図 6 b))。MCTでは、上記の自己中間散乱関数 Fs(q, t)や中間散乱関数 F (q, t) のような時間相関関数の時間発展を自己無撞着に定めるMCT方程式と呼ばれる微積分方程式をある種の近似のもとで導き、これを解析する。MCTの重要な帰結はある温度 Tcに向かって緩和時間 ⌅�が臨界温度までの”距離”⇤ = |T/Tc � 1|のベキ乗で発散し、エルゴード性が破れる動的転移がある、ということである。後述するように、これはMCTが本質的に平均場理論であることによっていて、実際には、真の転移はそこにはない。3しかしこの Tcが、緩和の機構が切り替わるクロスオーバ-温度となっている

3ただし、現在の標準的なMCTは、まだそのままでは高次元極限で厳密になるような真の意味での平均場理論にはなっていない [16]。 後述のクローン液体論についても、「ある種の平均場近似」である以上に、無限大次元で厳密になる真の意味で

α緩和

β緩和自己中間散乱関数

2段階緩和ケージ内のダイナミックス

構造緩和（ケージの崩壊、組み替え）

��「構造緩和時間」は温度の低下とともに急激に増大

冷却

t

0.8 0.60.55 0.5

0.475

1.02.0

3.04.0

モード結合理論(MCT): 第一原理から2段階緩和を導き、非エルゴード転移を予言

(review) 「ガラス転移とモード結合理論」-最近の発展と課題 - 宮崎州正, 物性研究 88-5 (2007-8), p621

「肩の高さ」：ケージの小ささ

Edwards-Anderson秩序パラメータ

qEA = limt��

1N

N�

i=1

�Si(0)Si(t)�

に概念的に非常に近い！

W. Gotze et. al. 1980s

1. 序論：2. クローン液体論の基本的なアイディア3. スピングラス平均場理論(1)

4. スピングラス平均場理論(2)

5. 液体論の基礎

6. クローン液体論

7. レオロジー：粘性、弾性、塑性

8. ジャミング

講義の内容

Lecture 2 : クローン液体論：基本的なアイディア

Random Energy Model (REM)

クローン液体論：基本的なアィデイア

Random Energy Model (REM)

B. Derrida, Phys. Rev. Lett. 45(1980) 79

Kirkpatrick, Thirumalai, Wolynes 1980’s

Kirkpatrick, T. R., D. Thirumalai, and Peter G. Wolynes. "Scaling concepts for the dynamics of viscous liquids near an ideal glassy state." Physical Review A 40.2 (1989): 1045.

Kirkpatrick, T. R., & Wolynes, P. G. (1987). Stable and metastable states in mean-field Potts and structural glasses. Physical Review B, 36(16), 8552.

ある種のスピングラス平均場理論(1段階のRSB)と構造ガラスの物理とのアナロジー

Lecture 3-4でp-spin Ising mean-field spinglass modelを議論する。

クローン液体論：基本的なアィデイア

M. Mezard and G. Parisi, J. of Chem. Phys. 111 (1999) 1076

R. Monasson, Phys. Rev. Lett. 75 2847 (1995).S. Franz and G. Parisi, J. de Phsyqie I 55 (1995), 1401.

同じハミルトニアンに従うm個のレプリカ

�A

ケージサイズ

A =�液体

a = 1, 2, . . . ,mxa

i i = 1, 2, . . . , N

A <�固体

R. Monasson, Phys. Rev. Lett. 75 2847 (1995).

TK

Liquid

Glass

10 m∗

T

Tc

0

a) b)slope m/T

f

Σ(f)

fsp(T,m)m

構造エントロピー(complexity)

Zm =�

�

e�Nm⇥f� =�

dfe�N(m⇥f��(f,T ))クローン系の分配関数

準安定状態についての和

クローン液体論 (基本的なアイディア)

A = lim��0

limN��

�(xai � xb

i )2�

H =m�

a=1

H[{xai }] +

�

4

�

i

�

a,b

�(xai � xb

i )2�

スピングラスの平均場理論（連続的なレプリカ対称性の破れRSB) :

Edwards-Anderson (1975), Sherrington-Kirkpatrick(1975), Parisi (1979), Sompolinski-Zippelius (1982).

Random First Order Transition (RFOT)理論: Kirkpatrick, Thirumalai, Wolynes 1980’s

スピングラス平均場理論(1段階のRSB)と構造ガラスの物理とのアナロジー

✦ エルゴード性の破れ モード結合理論 (MCT)の臨界温度

✦ 静的なスピングラス転移 Kauzmann 転移

レプリカ理論の発展

✦ Franz-Parisi ポテンシャル (1995)

✦ complexity (構造エントロピーの計算) Monasson (1995)

✦ クローン液体論 Mezard-Parisi (1999), Parisi-Zamponi, Rev. Mod. Phys. 82, 789 (2010).

動的不均一性

モザイク描像 - Adam-Gibbsの現象論の基礎付け(Wolynes 1997, Bouchaud-Biroli 2004, Franz-

Montanari 2007), ..

ランダム系の統計力学の発展：スピングラス、構造ガラス

�4

Tc

TK

非平衡ダイナミックス(エイジング効果)の　動的平均場理論　Cugliandolo-Kurchan (1994)

J. Kurchan, G. Parisi, P. Urbani and F. Zamponi, (2013),P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani and F. Zamponi (2013).

1+連続RSB

W. Gotze et. al. 1980s

課題1

ランダムエネルギー模型と構造ガラスの熱力学との類似性をまとめよ。