Kinetic, thermodynamic, and Adam-Gibbs Fragilities Srikanth Sastry - - PowerPoint PPT Presentation

Srikanth Sastry

TIFR Centre for Interdisciplinary Sciences, Hyderabad, India Jawaharlal Nehru Centre for Advanced Scientific Research Bengaluru, India

Kinetic, thermodynamic, and Adam-Gibbs Fragilities

Symposium on Fragility, JNCASR, Bengaluru, India January 8, 2014

In collaboration with: Anshul Deep Singh Parmer (TCIS) Shiladitya Sengupta (JNCASR/TCIS) Frederic Affouard (Lille) Filipe Vasconcelos (Lille)

Sengupta et al, J Chem Phys, 135, 194503 (2011) + ongoing (unpublished) work

Outline

 How do we understand the rapid rise of structural relaxation times in glass forming liquids?  How does fragility – the rapidity of rise of relaxation times - depend on the nature of the interparticle interactions ?  Can we understand the fragility in terms of the variation of the configurational entropy?  What role is played by the high temperature activation energy? Approach: Study relaxation times and configurational entropy in model liquids using computer simulations to address these questions.

• Tune the “softness” of the interaction potential
• Configurational entropy and relaxation times for different

softnesses

• Tune the barrier heights and study its role in determining fragility.

• Viscosities of supercooled liquids rise rapidly at low temperatures, well described

(but not uniquely) by Vogel-Fulcher-Tammann relation

• The glass transition – When viscosity reaches 1013 poise, they stop flowing on

experimental time scales (relaxation times ~ 100 s)

• “Laboratory glass transition” seen for wide range of substances, and is a kinetic effect.
• How do we understand the rapid rise of viscosities, relaxation times?

The glass transition

Fragility (kinetic definition)

• Data collapse when T scaled

with Tg? – No, Instead, a range of behavior.

• How rapidly the viscosity

changes is called the fragility

• f the glass former.
• Well described by the VFT

(Vogel Fulcher Tammann) form.

KVFT – kinetic fragility index

Fig: C. A. Angell, Jl. of Non-Crystalline Solids, 102, 205 (1988).

฀  h  h0 exp 1 KVFT T T0 1                

The Adam-Gibbs Relation

• Builds on Gibbs-DiMarzio (GD) theory describing the glass transition as

an entropy vanishing transition.

• Connection to dynamics.
• Views a liquid as divided into cooperatively rearranging regions of size z.
• Probability of rearrangement:
• Entropy of each CRR roughly constant regardless of size.
• Total entropy of the system
• Results in expression for relaxation times:
• Tested and found to be a good description of experimental and

simulation data.

• Configurational entropy in simulations estimated via the multiplicity of

local energy minima, or inherent structures.

Fragility (thermodynamic definition)

                  1 exp

K T c c

T T TS TS A  

• Thermodynamic definition of fragility (KT)

from configuration entropy

• Kinetic and thermodynamic definitions

connected if AG relation holds

• Relation with energy landscape of the liquid:

Broader distribution of potential energy minima  more fragile

Fig.: S. Sastry, Nature, 409, 164 (2001)

But, can we say something about how fragility depends on the nature of interatomic interactions? Potential energy minima = Inherent Structures (IS)

Ref.Talk by Shiladitya Sengupta

Fragility and the nature of interactions I softness of interactions

Bordat et al found that as the interaction potential became softer, the fragility increased. Computer simulation study using (p,q) potentials [Bordat et al PRL 2004]

The slope is proportional to fragility.

Fragility from VFT fits to relaxation times

Fragility and the nature of interactions

Weitz and co-workers studied colloidal fluids of microgel particles, whose softness can be varied.

Mattson et al Nature 2009

They find: Hmm…

Fragility and the nature of interactions: What to expect?

Intuitively, a system with softer interactions can have better and more robust packing of particles, a narrower distribution of energies, and therefore should be stronger. Various findings are consistent with this intuitive expectation:

• "..more deformable molecules fill space better than hard

molecules, leading to stronger fluids that are less sensitive to the structural changes induced by temperature variation.” (Douglas & coworkers JCP 2007).

• Fragility decreases with increasing polydispersity (Bagchi & co-

workers PRL 2008). So: Are the simulation results incorrect, or they are telling us something new?

Tuning softness to tune fragility

Softness is a measure of how steeply the repulsive interaction rises

• Simulation * of model glass

formers with tunable LJ like interactions  kinetic fragility increases with increasing softness  soft is fragile ?

• P. Bordat, F. Affouard and M. Descamps,
• Phys. Rev. Lett., 93, 105502 (2004).

** J. Mattsson et. al., Nature, 462, 83 (2009)

• C. A. Angell and K. Ueno, Nature, 462, 45 (2009) (News and views)

                      

p q

r r q r r p q p r V

min min

) ( 

• Experiments ** on deformable

colloid particles  kinetic fragility decreases with increasing softness  soft is strong ?

• What about thermodynamic

fragility ? Not known. Study the model systems of Bordat et al, and do thermodynamic analysis, by calculating the configurational entropy.

Simulation Details

80:20 binary mixture, N = 1500. Density = 1.2 Model I: q = 12, p =11 Model II: q = 12, p = 6 Model III: q = 8, p = 5 Temperatures above and below onset temperature. 3 – 5 sample runs for each temperature, each run > 100 a



Diffusion coefficients and relaxation times (from overlap function q(t)) to obtain kinetic fragilities. Configurational entropy to obtain thermodynamic fragility.



Kinetic Fragility vs. softness

Fragility plot

Activation energy E(T) scaled by high T value E0 vs. kBT / E0 Increasing softness Increasing softness Increasing softness Increasing softness

Diffusivity Relaxation time Diffusivity Relaxation time Kinetic fragility increases with increasing softness  supporting Bordat et. al. (2004)

Thermodynamic fragility vs. softness

Increasing softness

Thermodynamic fragility decreases with increasing softness

           1

K T c

T T TS

KT = slope

• How to reconcile the opposite trends?
• Possibility 1: Kinetic (KVFT) and thermodynamic (KT) fragility are
• unrelated. Incorrect if Adam Gibbs relation holds for all softness.
• Possiblity 2: Coefficient A in AG relation depends on softness/system.

Reconciling the trends : AG fragility

• Adam Gibbs relation holds for all softness.
• Estimate kinetic fragility using full AG

relation:

• Both KT and „A‟ decreases with softness.

kinetic fragility increases.

Diffusivity Relaxation time

                                          1 exp 1 1 exp

K T c c VFT

T T TS TS A T T K    

) (assuming

K T AG

T T A K K  

Increasing softness Increasing softness

Understanding softness dependence of „A‟

Increasing softness

Diffusivity

• Extrapolate AG relation to high T
• Estimate „A‟ and AG fragility from high T

activation energy

est T AG B c est B c

A K K k S E A T k E T TS A T                      ) ( exp ) ( exp ) (   

Diffusivity Relaxation time Model A Aest A Aest I 2.27 2.52 2.88 3.91 II 1.35 1.20 1.79 1.87 III 0.71 0.65 1.02 1.00 Diffusivity Relaxation time KVFT

I

KVFT

II

KAG KVFT

I

KVFT

II

KAG 0.34 0.24 0.22 0.20 0.19 0.14 0.38 0.26 0.27 0.21 0.19 0.17 0.40 0.30 0.32 0.26 0.21 0.21

KVFT

I = kinetic fragility from VFT fit. KVFT II = kinetic fragility from VFT fit using T0 = TK

Kinetic fragility estimate using full AG relation (KAG) and high T behaviour agrees quantitatively with direct values (KVFT

II).

Partial success predicting A behavior on density scaling (not shown).

Fragility and the nature of interactions II Dependence on high T activation energy

est T AG B c est B c

A K K k S E A T k E T TS A T                      ) ( exp ) ( exp ) (   

Recap : Contribution from thermodynamics Contribution from high temperature Activation energy Can we tune the interatomic interaction such that static structure and thermodynamics remains same but the kinetics are different ?

• Previous work (Saika-voivod et al., PRE, 90, 041401, 2004) suggests

that this is indeed possible by adding a barrier of finite height h but infinitesimal width to the interparticle potential

• Expectation : tunable barrier height “h” changes the bond lifetime

(kinetics) without changing the static structure and thermodynamics

Anticipated by Angell..by pure thought..?! Angell, in NATO ASI E 188:133 (1990)

Square well model with tuning barrier height

Particle sizes : Barrier width Square well width Barrier height Methods and quantities :

• Event driven MD simulation at

constant N,V,T

• Total entropy from thermodynamic

integration

• Relaxation time from self intermediate

scattering function

Comparison of static structure

Pair correlation functions independent of barrier height.

Thermodynamics

Pressures and energies do not depend significantly on barrier heights. Excess entropies calculated from thermodynamic integration of pressures and potential energy.

Excess entropy for different h

Preliminary data for excess entropy suggests that tuning the barrier height does not change thermodynamics of liquid Path of thermodynamic integration

Phase diagram : Saika-voivod et al., PRE, 90, 041401, 2004

A(f0,T40) B(0.55,40) C(0.55,T)

Dynamics

Slow down of dynamics with decreasing temperature, as well as increase of the barrier height h. Relaxation times from fits to Fs(k,t)

Relaxation times for different barrier heights

Relaxation times indicate that tuning the barrier height changes the liquid from fragile to strong.

฀   T

ref

( 104

Kinetic fragility (KVFT) estimated by fitting VFT form. Kinetic fragility decreases as h increases.

But..

Scaling by the barrier height collapses the data at low T…

Which is the correct fragility to consider?

The correct estimation of kinetic fragility requires knowledge both

• f the temperature dependence of the configurational entropy, and

the high temperature activation energy. In the case of the variation of fragilty with density (Shiladitya Sengupta‟s talk), kinetic fragility has weak ~ no density dependence, and KAG = KT/A estimates kinetic fragility correctly. For variation of fragility with softness of interactions, KAG = KT/A again estimates kinetic fragility correctly. For the system with tunable barrier heights, KAG = KT/A should provide the kinetic fragilities (work in progress), BUT kinetic fragilities will depend on barrier heights. HOWEVER, we know that the system has no change in fragility with barrier height if we accept KT as being the correct measure of fragility… Which is the correct fragility to consider?.. Or….

Will the real fragility please stand up?!

Summary

For the model systems we study,

• With increasing softness, kinetic fragility increases and

thermodynamics fragility decreases.

• The opposite trend can be reconciled by taking into

account softness dependence of the coefficient A in Adam Gibbs relation.

• Softness dependence of A can be related to high T activation

energy in Arrhenius law.

• For the system with variable barrier heights, turning the

barrier height changes the kinetic fragility, but the “intrinsic” fragiity may be viewed as not changing.

• Raises the question of which the correct fragility to consider is.

High temperature activation energy

Detour: Density temperature scaling

• In inverse power law (IPL) models :

1.Virial (W) is proportional to potential energy (V). 1.Relaxation time (in appropriate reduced unit) obeys scaling relation.

• Density and temperature (or, length and energy) are not

independent parameters.

• High temperature activation energy must scale as

rgsince  = (rg/T. 

• Recent studies find these properties for many non-IPL

liquids. 

• gn/d for IPL liquids, but different for non-IPL liquids

which show temperature-density scaling a la IPL liquids.

• gmeasured from correlations between potential energy

and virial.    

How to understand the behaviour of high T activation energy, which we need in order to understand fragility vs softness of interactions? Hint: recent analysis of „hidden scaling‟ in many liquids.

                    





T T V d n V r r d W r V

d n i i i n /

) , ( 1 ~ r  r   

Density temperature scaling in KA model

Density – temperature scaling of dynamics holds approximately in KA model over a range of density and temperatures

High T activation energy E0 ~ rg g computed from correlation between potential energy and virial

g g

r r r  r                    ) ( exp ~ ) , ( f E T E T T

Estimating high temperature activation energy from density- temperature scaling

Can we use density-temperature scaling to relate E0 for our models? We expect E0 = E00 rg,gobtained from correlation between potential energy and the virial. Estimated E0 (from fitted E00) has the correct trend but does not fully explain the variation with the softness of interaction.