Glass Crossover of Super-Cooled Liquids

Tommaso Rizzo

IPCF-CNR and Sapienza University, Rome

Capri, September 11, 2014

• The glass problem: the dramatic increase upon cooling of the relaxation timeof super-cooled liquids

• Mode-Coupling-Theory, (’80) captures many important features of the prob-lem, in particular the two-step nature of the relaxation but predicts a glasstransition at too high temperatures

• Structural-Glass/Spin-Glass analogy: (Kirkpatrick-Thirumalay-Wolynes, 87-89): p-spin-glass models with (p > 2) obey the same dynamical equations.

• Structural-Glass/Spin-Glass analogy: (Kirkpatrick-Thirumalay-Wolynes) :at TMCT the phase space is split into an exponential number of states, dis-appearance of the transition through nucleation arguments.

• In Spin-glass models the transition occurring at TMCT has the features of asecond order phase transition (diverging correlation length and fluctuations)and these features can be also obtained directly in the MCT framework

• Recently (2010-) it has been discovered that the critical properties of thetheory are related to those of the Random field Ising model: unexeptectedbut solid connections.

• The problem: Why and how the dynamical singularity of Mode-Coupling-theory (MCT) becomes a crossover?

• The dynamical singularity at TMCT is assumed to be a genuine second orderphase transition and dynamical field theoretical methods are applied to it.

• The procedure leads instead to a rather intuitive dynamical model (StochasticBeta Relaxation) that predicts that the transition is changed into a dynamicalcrossover.

T. Rizzo, arXiv:1307.4303, EPL 2014

Solution of Stochastic-Beta-Relaxation equations in 3D

• Above TMCT : power-law increase of dynamical fluctuations and dynamicalcorrelation length, scale invariance.

• Near TMCT : (i) Dynamical arrest is avoided (ii) relaxation time grows muchfaster, from power-law to eponential.

• The structure of fluctuations also displays a qualitative change below TMCT :rare faster regions dominates the relaxation: Dynamical Heterogenities. Dy-namics slows down because these regions are rare not because they are larger.

• Dynamical correlation length decreases and decorrelates from the relaxationtime. Decoupling between di↵erent observables: Deviations from the Stokes-Einstein relationship.

T. Rizzo & T. Voigtmann, ’14

Critical Slow Dynamics in MCT

�(k, t) = f (k) +G(t) ⇠Rc (k) (� regime)

� = G2(t) (1� �) +Z t

0(G(t� s)�G(t))G(s)ds

� / TMCT � T, � / ⇢� ⇢MCT

limt!1

G(t) = �B(�)tb for � < 0 (liquid)

limt!1

G(t) =r

�/(1� �) for � > 0 (glass)

� =�2(1� a)

�(1� 2a)=

�2(1 + b)

�(1 + 2b)

• scaling limit: |�| ⌧ 1, |G(t)| ⌧ 1, 1 ⌧ t = O(⌧�) ⌧ ⌧↵;

⌧↵ / B(�)�1/b

Stochastic Beta Relaxation

• Stochastic �-relaxation (SBR) equation: Extension of the MCT equation forthe critical correlator with random fluctuations of the separation parameter.

� = G2(t) (1� �) +Z t

0(G(t� s)�G(t))G(s)ds (MCT)

#

� + s(x) = �r2 g(x, t) + (1� �) g2(x, t) +Z t

0(g(x, t� s)� g(x, t))

dg

ds(x, s)ds

[s(x)s(y)] = ��2 �(x� y)

G(t) = [1

V

Zg(x, t)dx]

T. Rizzo, arXiv:1307.4303, EPL 2014

The replicated Field Theory

L =1

2

Zdx

0

@�⌧X

ab�ab +

1

2

X

ab(r�ab)

2 +m2X

abc�ab�ac +m3

X

abcd�ab�cd+

�1

6w1

X

abc�ab�bc�ca �

1

6w2

X

ab�3ab

1

A

• It is possible to resum the loop expansion unveiling a mapping to a quadraticequation in a random field

• This allows to identify non-perturbative e↵ects that show that the glassyphase is not stable: the singularity at Tc is avoided.

• Non-trivial features: (i) typically loop expansions are limited to few orders,here we have control at all orders, similar to the RFIM (ii) but the mappingseems natural in the RFIM model while there is no quenched random field inthis case

S. Franz, G. Parisi, F. Ricci-Tersenghi, and T. Rizzo, Eur. Phys. J. E, 34, 102, (2011).

From Replicas to MCT Dynamics

• One can formulate the dynamics in such a way that the dynamical field theoryis precisely the same of the Replicated theory (essentially di↵erent replicasare equivalent to di↵erent times).

⌧ = (w1 � w2)�2(t) + w1

Z t

0(�(t� y)� �(t))�(y)dy

� =w2

w1.

G. Parisi and T. Rizzo, Phys. Rev. E 87, 012101 (2013). F. Caltagirone, U. Ferrari, L.

Leuzzi, G. Parisi, F. Ricci-Tersenghi, T. Rizzo, Phys. Rev. Lett. 108, 085702 (2012)

The B-Profile

• At any value of �, the solution g(x, t) goes at large times as �B(x)tb, i.e.all regions of space are liquid.

• SBR equations induce a mapping between the realisation of the random tem-peratures s(x) and a positive function B(x), such a mapping is not possiblein the static treatment.

• the B-profile allows a compact description of dynamical quantities like the↵-relaxation time, the di↵usion coe�cient and the correlation length.

-5-4-3-2-101

x

gHx,tLêtb

Figure 1: Top: Pictorial representation of the solution of SBR equations in finite dimension for a givenrealisation of the random �(x). (generated from actual solution of a 1D system). At large times the solutionconverges to the form �B(x)tb, thus inducing a mapping between the realisation of the random �(x) and apositive function B(x), the B-profile. Bottom: Plot of g(x, t)/tb vs. x for increasing times t: increasing thetime the curves converge to the B-profile.

Viscosity and Di↵usivity

⌘ = ⌧↵ ⇠0

@1

V

ZB(x) dx

1

A�1/b

D ⇠ 1

V

ZB1/b(x) dx

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1010

1012

1014

s

t a

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10-4

10-3

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10-1

sD

Figure 2: Simplified SBR: plot of the ↵ relaxation time ⌧↵ vs. the separation parameter � for �� = .1 and� = .75. The dashed lines correspond to the leading asymptotic behavior. Symbols are dielectric-spectroscopydata for propylene carbonate from Lunkenheimer et al (2000). Inset: plot of the Di↵usion constant vs. theseparation parameter � for �� = .1 and � = .75. The dashed line corresponds to the ideal MCT resultD / 1/��.

Some Comments

• Dynamical arrest is avoided and replaced by a crossover from power-law toexponential

• there is no ad hoc assumption on nucleation or activated processes in thederivation, actually the initial assumption is that TMCT marks a genuinephase transition

• could work in an extended range of temperature, even close to Tg

• used as a fit function but the actual computation of the coupling constantsis feasible for many systems

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0.5

1.0

1.5

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logIC

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logH10L

Figure 3: simplified SBR: plot of the squared thermal susceptibility vs. ⌧↵ for �� = .1 and � = .75. Thedashed lines correspond to the asymptotic behaviors. Symbols are experimental data shifted by arbitraryfactors along the vertical C. Dalle-Ferrier, C. Thibierge, C. Alba-Simionesco, L. Berthier, G. Biroli, J.-P.Bouchaud, F. Ladieu, D. L’Hote, and G. Tarjus, Phys. Rev. E 76, 041510 (2007). (circles: Lennard Jonesmixture; squares: hard spheres; BKS silica: diamonds; triangles: propylene carbonate; inverted triangles:glycerol; open circles: OTP; open squares: salol). The thin solid line is the expression proposed in the samepaper.

Non-Monotonous Correlation Length

�(r) =1

V

ZB(x)B(x + r) dx

�n(r) ⌘�(r)� �(1)

�(0)� �(1)

• �n(r) has a bell-shaped form with rapidly decaying tails both below andabove � = 0

• non-monotonous behavior with �, appears to be related to recent observationsin numerical simulations (Berthier, Kob 2010-)

0 2 4 6 8 10 120.0

0.2

0.4

0.6

0.8

1.0

x

G nHxL

Figure 4: Plot of the normalized autocorrelation of the B profile in 3D for � = �0.006, 0, 0.006, 0.012(dashed, dotted, thick, solid).

104

106

108

1010

1012

1014

t a

-0.04 -0.03 -0.02 -0.01 0.00 0.01 0.020

1

2

3

4

s

x d

-0.03 -0.015 0 0.015

10-9

10-8

10-7

10-6

10-5

10-4

-0.03 -0.015 0 0.015

s

D

Figure 5: Top: ⌧↵ vs. � from SBR in 3D. Inset: D vs. � in 3D. Bottom: The correlation length ⇠d (definedas half-width at half-maximum of �(r)) as a function of �.

s=-0.006 s=0

s=0.006 s=0.012

Figure 6: Normalized B-profile B(x)/B on a plane sliced from a cubic box for di↵erent values of �.

0 10 20 30 40

0.1

0.5

1.0

5.0

10.0

50.0

100.0

x

BHxLêB

Figure 7: Log-plot of the Normalized B-profile B(x)/B on the line x = 20 cut from the planes of figures 6(increasing values of � correspond to increasing line thickness).

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1

h

D-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

1.0

1.5

2.0

2.5

s

Dh

Figure 8: simplified SBR: plot of the Di↵usion constant vs. the viscosity for �� = .1 and � = .75. Thestraight lines correspond to D / ⌘�1 (the Stokes-Einstein relation) and to a fractional SER D / ⌘�.65. The

actual asymptotic behavior is D / ⌘�b(ln ⌘)b��1

2 , the logarithmic prefactor induces a huge correction and as aconsequence a fit with a pure fractional SER gives an exponent 0.65 considerably higher than b = .558. Thepoint where the SER breaks down (intersection of the straight lines) corresponds to a temperature higherthan Tc (dot). Smaller symbols are experimental data for o-terphenyl by Lohfink and Sillescu (1992)(circles:tracer di↵usion at T > Tc, di↵usion of ACR dye, diamonds: di↵usion of TTI dye).

Future Directions and Open Problems

• Quantitative tests with actual coupling constants, (length, shift of TMCT ,Lennard-Jones vs. HARM crossover sharpness)

• di↵erent dimensions and finite-size e↵ects

• better understanding of the ↵ regime

• cannot sustain too large fluctuations and must be abandoned at some point.Conjecture: non-standard activation characterized by elementary events thathave intrinsic time and length scales of an unusual large (but not increasing)size (mesoscopic vs. microscopic).

• Connection with configurational complexity phenomenology

Conclusions

• Dynamical Field Theoretical loop corrections to MCT lead to SBR.

• displays a rich phenomenology common to most super cooled liquids.

• displays simultaneously an increase of several orders of magnitude of therelaxation time and a decrease of the dynamical correlation length.

• below TMCT there are strong dynamical heterogeneities, crossover from scale-invariance to activated-like, SER deviations.

• None of these qualitative and quantitative features was put ad hoc into themodel. In particular one should not confuse the fluctuations of the orderparameter (the standard starting point of the computation) of the originaltheory with the fluctuations of the temperature in SBR (the result of thecomputation).

Thanks!

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Figure 9: Existence of the continuum limit of the B(x) profile.