MIT 3.071 Amorphous Materials

7: Viscoelasticity and Relaxation

Juejun (JJ) Hu

1 After-class reading list

 Fundamentals of Inorganic Glasses

 Ch. 8, Ch. 13, Appendix A

 Introduction to Glass Science and Technology

 Ch. 9 (does not cover relaxation)

Mathematics is the language with which God wrote the Universe. Galileo Galilei

2 Where and why does liquid end and glass begin?

“What don’t we know?” Science 309, 83 (2005).

3 Is glass a solid or a viscous liquid?

V Supercooled Solid Liquid liquid Elasticity Viscosity instantaneous, time-dependent, transient permanent  E Glass xx  xy xy   xy  G xy t

E : Young’s  : viscosity modulus (unit: Pa·s or Poise) G : shear modulus Tm T 4 Viscoelasticity: complex shear modulus

 Consider a sinusoidally varying shear strain

xy  0 expit  

 Elastic response: xyG  xy  G 0 exp i  t 

 Viscous response:   xy i   exp i  t   i    xy t 0 xy Loss modulus  In a general viscoelastic solid:

* * xy G  i    xy  G   xy G G  i G '  iG" G* : complex shear modulus Shear/storage modulus

5 Phenomenological models of viscoelastic materials

 Elasticity: Hookean spring

xy  G xy

 Viscosity: Newtonian dashpot    xy xy t

 Models assume linear material response or infinitesimal stress  Each dashpot element corresponds to a relaxation mechanism

6 The Maxwell element

  G  V EE V t  Serial connection of a Hookean spring and a Newtonian dashpot

 Total stress:   E  V

 Total strain:   E  V

7 The Maxwell element

  G  V EE V t

 Constant stress (creep):   E V constant t  0      E V  t G 

 Constant strain (stress relaxation):   E V constant t  0  t t      Relaxation  E  expV  1  exp    G time

8 The Maxwell element

  G  V EE V t  Oscillatory strain:

 E V 0 expit 22 G '  G 122  G"  G 122

9 The Voigt-Kelvin element

 Parallel connection of a Hookean EE G spring and a Newtonian dashpot

 Total stress:   E V

 Total strain:   E V

 Constant strain:   G

 Constant stress:   V V t  t  1  exp  G 

 Oscillatory strain:  0 expit   G '  G G"  G

10 Generalized Maxwell model

 For each Maxwell component:   G i 11    i t Gtii    1  i i 1 G1   i  tG ii Gi    1 ... i  ...  2 G2  Total stress: … G1 ... i ...

11 Generalized Maxwell model

 Stress relaxation:  constant t  0 G   Prony series:  G  ... ... 1 G1  1 i t GGexp      i  i0  i 2 G2   Gexp  tG  R …

In real solids, a multitude of microscopic relaxation processes give rise to dispersion of relaxation time (stretched exponential)

12 Elastic, viscoelastic, and viscous responses

Stress Stress Stress

t t t

Elastic strain Viscoelastic strain Viscous strain

t t t

13 Viscoelastic materials

Mozzarella cheese Human skin Turbine blades

Volcanic lava Memory foams Naval ship propellers Image of Naval ship propellers is in the public domain. Various images © unknown. Lava image © Lavapix on YouTube. All rights reserved. This content is excludedfrom our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. 14 Boltzmann superposition principle

 In the linear viscoelastic regime, the stress (strain) responses to successive strain (stress) stimuli are additive

Strain Strain Strain  2

 1   1 t 2 t t

t1 t2

Stress Stress Stress

t t t

t1 t2

1  G1  t  t1  2  G 2  t  t2  12  

15 Boltzmann superposition principle

 In the linear viscoelastic regime, the stress (strain) responses to successive strain (stress) stimuli are additive

Strain  Viscoelastic response   i i is history-dependent  Relaxation function  t d   i dt ' dictates time-domain t 0 dt ' response

Stress   i G i  t  ti  i i

ttd d t  i dt  G  t  t '  dt ' 00dt dt '

16 V Structural relaxation in glass

Supercooled liquid Glass transition

Glassy state T

Elastic regime Viscoelastic regime Viscous regime

 re  tobs  re ~ tobs  re  tobs All mechanical and Glass structure and Structural changes thermal effects only properties are history- are instantaneous: affect atomic vibrations dependent equilibrium state can be quickly reached Ergodicity breakdown 17 V Structural relaxation in glass

Supercooled liquid Glass transition

Glassy state T

Elastic regime Viscoelastic regime Viscous regime DN 1 DN ~1 DN 1

Debroah number (DN): DN re t obs

“… the mountains flowed before the Lord…” Prophetess Deborah (Judges 5:5) 18 Comparing stress relaxation and structural relaxation

V

T

 “Equilibrium” state  “Equilibrium” state  Zero stress state  Supercooled liquid state  Driving force  Driving force

 Residual stress  Free volume Vfe VV   Relaxation kinetics  Relaxation kinetics  Exponential decay with  Exponential decay with a single relaxation time a single relaxation time  Relaxation rate scales  Relaxation rate scales with driving force with driving force 19 Free volume model of relaxation (first order kinetic model)

 Relaxation kinetics at V constant temperature

Vfet Vt V

Vtf 0 expt  re VV T ff t  re

 Temperature dependence of relaxation  EEaaV VVff re  0 exp  exp kTBt t  0 kTB

20 Model predicted relaxation kinetics

Cooling rate: 10 °C/s Slope: volume CTE Varying reheating rate of supercooled liquid

10 °C/s 1 °C/s 0.1 °C/s

21 Model predicted relaxation kinetics

Varying cooling rate Reheating rate: 1 °C/s

100 °C/s

10 °C/s Fictive temperature and glass structure are functions of cooling rate 1 °C/s

22 Tool’s Fictive temperature theory

 Fictive temperature T : the V, H Supercooled f temperature on the supercooled liquid liquid curve at which the glass Increasing would find itself in equilibrium cooling rate with the supercooled liquid state if brought suddenly to it 3  With increasing cooling rate:

2  Tf1 < Tf2 < Tf3 1  A glass state is fully described by thermodynamic parameters

Tf1 Tf2 Tf3 T (T, P) and Tf  Glass properties are functions of VVTT , f  HHTT , f  temperature and Tf (structure)

23 Tool’s Fictive temperature theory

 Glass property change in the V, H Supercooled glass transition range consists liquid of two components Increasing  Temperature-dependent cooling rate property evolution without modifying glass structure 3 VT    Volume:  T Vg, 2 f Enthalpy: HTC    T Pg, 1 f  Property change due to

Tf1 Tf2 Tf3 T relaxation (Tf change)

Volume: VT f   V,, e  V g VVTT , f  HHTT , f  T Enthalpy: HTCC    f T P,, e P g

24 Tool’s Fictive temperature theory

 Glass property change in the V, H Supercooled glass transition range consists liquid of two components Increasing

cooling rate dV T, Tf  VVdTf    dT T T dT Tf f T 3 dTf dT V,,, g   V e   V g   2 dt dt 1 dH T, Tf  HHdTf    Tf1 Tf2 Tf3 T dT T T dT Tf f T

dTf dT VVTT , f HHTT , f     CCCP,,, g  P e  P g   dt dt

25 Predicting glass structure evolution due to relaxation: Tool’s equation

Consider a glass sample Pe Pg with Fictive temperature T PPge TTf   f TT P (V, H, S, etc.) Take time derivative: Supercooled dP P dT liquid g Pe g f     dt T T dt    Assume first-order relaxation:    dPg P g P e  Glass   dt  re Pg - Pe

T - T dTff T T f  Tool’s equation dt  re T Tf 26 Tool’s Fictive temperature theory

Line D'M : Tf  T

dTff T T  Tool’s equation dt  re

dH T, Tf  HHdTf    dT T T dT Tf f T

dTf dT CCCP,,, g  P e  P g   dt dt J. Am. Ceram. Soc. 29, 240 (1946)

27 Difficulties with Tool’s Tf theory

 Ritland experiment: two groups of glass samples of identical composition were heat treated to obtain the same refractive index via two different routes  Group A: kept at 530°C for 24 h  Group B: cooled at 16°C/h through the glass transition range

 Both groups were then placed in a furnace standing at 530°C and their refractive indices were measured as a function of heat treatment time

 Glass structure cannot be fully characterized by the

single parameter Tf

J. Am. Ceram. Soc. 39, 403 (1956). 28 Explaining the Ritland experiment n

expt  re,1 t

expt  re,2

Structural relaxation in glass involves multiple structural entities and is characterized by a multitude of relaxation time scales

Tool-Narayanaswamy-Moynihan (TNM) model 29 What is relaxation?

30 Relaxation: return of a perturbed system into equilibrium

 Examples  Stress and strain relaxation in viscoelastic solids

 Free volume relaxation in glasses near Tg

 Glass structural relaxation (Tf change)  Time-dependent, occurs even after stimulus is removed

 Debroah Number: DN re t obs  DN >> 1: negligible relaxation due to sluggish kinetics  DN << 1: system always in equilibrium  DN ~ 1: system behavior dominated by relaxation

31 P PP Driving Modeling relaxation Relaxation   e force rate t  re

 Maxwellian relaxation models t P t   P0  Pee  exp   P  re

 t P t   P0  Pee  exp   P  re

 Boltzmann superposition principle in linear systems d  G tt '   t R IRF t  t'  S  t dt

ttd d t   dt  G tt '  dt ' R IRFt  t '  S t '   dt ' 00dt dt ' 0

32 “The Nature of Glass Remains Anything but Clear”

 Free volume relaxation theory

V Vf E   exp  a t  0 kB T  Tool’s Fictive temperature theory

 Uses a single parameter Tf to label glass structure

 Tool’s equation (of Tf relaxation) dT T T ff dt  re  Structural relaxation in glass involves multiple structural entities and is characterized by a multitude of relaxation time scales  The Ritland experiment

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3.071 Amorphous Materials Fall 2015

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