Previous Lecture• Lennard-Jones potential energy is used for van der Waals

energy between pairs of atoms and for pairs within molecular crystals

• Young’s (elastic) modulus for molecular crystals can be obtained from the Force-r relation derived from L-J potentials.

• Response of soft matter to shear stress can be both Hookean (elastic) solids and Newtonian (viscous) liquids

• Viscoelasticity can be described with a transition from elastic to viscous response with a characteristic relaxation time,

• An important relationship between the elastic and viscous components: = GM

PH3-SM (PHY3032)

Soft Matter Physics

Lecture 4

Time Scales, the Glass Transition and Glasses

25 October, 2010

See Jones’ Soft Condensed Matter, Chapt. 2 and 7

Response of Soft Matter to a Step Strain

time

Constant shear strain applied

Stress relaxes over time as molecules re-arrange:

eqGγσ viscoelastic solid

viscoelastic liquid

time

Hookean

Newtonian

τt

M eγGtσ-

)(

Response of Soft Matter to a Constant Shear Stress

t

σ

tγtJ s )()(

Elastic response

Viscous response

(provides initial and recoverable strain)

(strain increases linearly over time)

Slope:

ησ

γ

σ

γ

dt

d 1)(

Jeq

Steady-state compliance = Jeq= 1/Geq

In the Maxwell model, Geq = /

So, = Jeq and Jeq = /

Relaxation and a Simple Model of Viscosity

» When a “simple” liquid is subjected to a shear stress, immediately the molecules’ positions are shifted but the same “neighbours” are kept.

• Thereafter, the constituent molecules re-arrange to relax the stress, and the liquid begins to flow.

• A simple model of liquids imagines that relaxation takes place by a hopping mechanism, in which molecules escape the “cage” formed by its neighbours.

• Molecules in a liquid vibrate with a frequency, , comparable to the phonon frequency in a solid of the same substance.

• Thus can be considered a frequency of attempts to escape a cage.• But what is the probability that the molecule will escape the cage?

F

0

Molecular configuration

Potential

Energy

(per molecule)

Intermediate state: some molecular spacings are greater: thus higher w(r)!

)_exp(1

kTf

The frequency, f, at which the molecules overcome the barrier and relax is an exponential function of temperature, T.

Molecular Relaxation in Simple Viscous Flow

Molecular Relaxation Times• is the energy of the higher state and can be considered an energy barrier

per molecule.• Typically, 0.4 Lv/NA, where Lv is the heat of vapourisation per mole and NA

is the Avogadro number (= number of molecules per mole).• A statistical physics argument tells us that the probability P of being in the

high energy state is given by the Boltzmann distribution:

P exp(-/kT)• T is the temperature of the reservoir. As T 0, then P 0, whereas when

T, then P 1 (= 100% likelihood).• Eyring proposed that the frequency of successful escapes, f, is then the

product of the frequency of attempts () and the probability of “success” (P):

)_exp(= kTf

The time required for a molecule to escape its cage defines a molecular configurational relaxation time, config, which is comparable in magnitude to

the macroscopic stress relaxation time, . And so, config = 1/f.

Arrhenius Behaviour of Viscosity• In liquids, the relaxation time, , is very short, varying between 10-12 and 10-10

s. Hence, as commonly observed, stresses in liquids are relaxed nearly instantaneously.

• In melted polymers, can be on the order of several ms or s. Whereas in solids, is very large, such that flow is not observed on realistic time scales.

• We can approximate that Geq, where Geq is the equilibrium shear modulus of the corresponding solid. Hence an expression for can be found from the Eyring relationship:

)exp(kT

ε

ν

GτGη eq

eq

)exp(RT

E

ν

GτGη eq

eq

Alternatively, an expression based on the molar activation energy E can be written:

This is referred to as an Arrhenius relationship.

Non-Arrhenius Temperature Dependence

• Liquids with a viscosity that shows an Arrhenius dependence on temperature are called “strong liquids”. An example is melted silica.

• “Fragile liquids” show a non-Arrhenius behaviour that requires a different description. An example is a melted polymer.

• The viscosity of a melted polymer is described by the Vogel-Fulcher (VF) relationship:

• We see that diverges to , as the liquid is cooled towards To. It solidifies as temperature is decreased. In the high-temperature limit, approaches o - a lower limit.

where B and To are empirical constants. (By convention, the units of temperature here are usually °C!)

)exp(=o

o TTB

Temperature-Dependence of Viscosity

P = Poise = 0.1 Pa-s

Arrhenius

Vogel-Fulcher

Viscosity Relaxation Time

Configurational Re-Arrangements

• As a liquid is cooled, stress relaxation takes longer, and it takes longer for the molecules to change their configuration, as described by the configurational relaxation time, config.

• From the Vogel-Fulcher equation, we see that:

)exp(~oo

oconfig TT

BG

We see that the relaxations become exceedingly slow ( becomes v. large) as T decreases towards To.

An Example of the Glass Transition

Video: Shattering polymer

Experimental Time Scales• To distinguish a liquid from a solid, flow (or other liquid-like behaviour) must

be observed on an experimental time scale, exp. A substance can appear to be a solid on short time scales but a liquid on long time scales!

• For example, if a sample is being cooled at a rate of 1 K per min., then exp is ~1 min. at each temperature increment.

At higher temperatures, exp > config, and flow is observed on the time scale of the measurement.

Debonding an Adhesive

Flow can be observed on long time scales, exp

Oscillatory StrainApply a shear strain at an angular frequency of = 2/exp. This defines an experimental timescale.

t

2/

)sin( tωγγ o

Dynamic mechanical analysis

Hookean versus Newtonian Responses

t

tt

Hookean: Stress is in phase

t

)sin()( tωγGtσ o

t

tt

2

Newtonian: Stress is /2 out of phase

t

)2sin()( πtωωηγtσ o

2

Response of a Viscoelastic Material

Stress oscillates at the same frequency as the strain, but it leads the strain by a phase angle, :

)sin()( δtωσtσ o

The relative values of the viscous and the elastic components depend on the time-scale of the observation (exp = 2/) in relation to the relaxation time: = Jeq

If exp > : the material appears more liquid-like

If exp > : the material appears more solid-like

The Glass Transition• At higher temperatures, exp > config, and so flow is observed on

the time scale of the measurement.

• As T is lowered, config increases.

• When T is decreased to a certain value, known as the glass transition temperature, Tg, then config ~ exp.

• Below Tg, molecules do not change their configuration sufficiently fast to be observed during exp. That is, exp < config. The substance appears to be solid-like, with no observable flow.

• At T = Tg, is typically 1013 Pa-s. Compare this to = 10-3 Pa-s for water at room temperature.

Competing Time Scales

Reciprocal Temperature (K-1)

Log(1/)

=1/vib

f = 1/config

1/exp

1/Tg

config < exp

config > exp

Melt (liquid)

Glass (solid)

Molecular conformation does not change when passing through the glass transition.

Effect of Cooling Rate on Tg

• Tg is not a constant for a substance.

• When the cooling rate is slower, exp is longer.• For instance, reducing the rate from 1 K min-1 to 0.1 K min-1,

increases exp from 1 min. to 10 min. at each increment in K.• With a slower cooling rate, a lower T can be reached before

config exp.

• The result is a lower observed Tg.• Various experimental techniques have different associated

exp values. Hence, a value of Tg depends on the technique used to measure it and the frequency of the sampling.

Are Stained-Glass Windows Liquid?

Window in the Duomo of Siena

Some medieval church windows are thicker at their bottom.

Is there flow over a time scale of exp 100 years?

Thermodynamics of Phase Transitions• At equilibrium, the phase with the lowest Gibbs’ free energy will

be the stable phase.• How can we describe this transition?

ba

The “b” phase is stable below the critical temperature, Tc.

Tc

Temperature, T

Free

ene

rgy,

G

Free Energy of the Melting/Freezing Transition

Crystalline state

Liquid (melt) state

Tm

• Below the melting temperature, Tm, the crystalline state is stable.

• The thermodynamic driving force for crystallisation, G, increases when cooling below Tm .

• During a transition from solid to liquid, we see that (dG/dT)P will be discontinuous.

G

Undercooling, T, is defined as Tm – T.

Classification of Phase Transitions

• A phase transition is classified as “first-order” if the first derivative of the Gibbs’ Free Energy, G, with respect to any state variable (P,V, or T) is discontinuous.

• An example - from the previous page - is the melting transition.

• In the same way, in a “second-order” phase transition, the second derivative of the Gibbs’ Free Energy G is discontinuous.

• Examples include order-disorder phase transitions in metals and superconducting/non-SC transitions.

Thermodynamics of First-Order Transitions

• Gibbs’ Free Energy, G: G = H-ST so that dG = dH - TdS - SdT

• Enthalpy, H = U + PV so that dH = dU + PdV + VdP

• Substituting in for dH, we see: dG = dU + PdV + VdP - TdS - SdT

• The central equation of thermodynamics tells us: dU = TdS - PdV

• Substituting for dU, we find: dG = TdS - PdV + PdV + VdP - TdS - SdT

Finally, dG = VdP-SdT

H = enthalpy

S = entropy

U = internal energy

Thermodynamics of First-Order Transitions• dG = VdP - SdT• In a first order transition, we see that V and S must be discontinuous:

TPG

V

S

TG

P

V

T

liquid

crystalline solid

Tm

Viscosity is also discontinuous at Tm.

There is a heat of melting, and thus H is also discontinuous at Tm.

(Or S)

Thermodynamics of Glass Transitions

V

T

Crystalline solid

Tm

Liquid

Glass

Tg

Determining the Glass Transition Temperature in Polymer Thin Films

Poly(styrene)

ho ~ 100 nm

Tg

Melt

Glass

Keddie et al., Europhys. Lett. 27 (1994) 59-64

~ Thickness

Is the Glass Transition Second-Order?

• Thus in a second-order transition, CP will be discontinuous.• Recall that volume expansivity, , is defined as:

Po T

V

Vβ

1

And V = (G/P)T. So,

Expansivity is related to a second differential of G, and hence it is likewise discontinuous in a second-order phase transition.

PTo

TP

G

Vβ

/1

• Note that S is found from -(G/T)P. Then we see that the heat capacity, Cp, can be given as:

PPP T

GT

T

STC

2

2

Experimental Results for Poly(Vinyl Acetate)

“Classic” data from Kovacs

Po T

V

V

1

Expansivity is not strictly discontinuous – there is a broad step.

Note that Tg depends on the time scale of cooling!

75.06°C(H)0.3804J/g/°CH

eat F

low

(W

/g)

20 40 60 80 100

Temperature (°C)

Data from H. Utschick, TA Instruments

Glass Transition of Poly(vinyl chloride)

Heat flow heat capacity

Sample is heated at a constant rate. Calorimeter measures how much heat is required.

T

Heat capacity is not strictly discontinuous – the step is over about 10 C.

Structure of Glasses

• There is no discontinuity in volume at the glass transition and nor is there a discontinuity in the structure.

• In a crystal, there is long-range order of atoms. They are found at predictable distances.

• But at T>0, the atoms vibrate about an average position, and so the position is described by a distribution of probable interatomic distances, n(r).

Atomic Distribution in Crystals12 nearest neighbours

And 4th nearest!

FCC unit cell (which is repeated in all three directions)

Comparison of Glassy and Crystalline Structures

2-D Structures

Going from glassy to crystalline, there is a discontinuous decrease in volume.

Local order is identical in

both structures

Glassy (amorphous) Crystalline

Simple Liquid Structure

r r = radial distance

Structure of Glasses and Liquids• The structure of glasses and liquids can be described by a

radial distribution function: g(r), where r is the distance from the centre of a reference atom/molecule.

• The density in a shell of radius r will have atoms per volume.

• For the entire substance, let there be o atoms per unit volume. Then g(r) = (r)/o.

• At short r, there is some predictability of position because short-range forces are operative.

• At long r, (r) approaches o and g(r) 1.

R.D.F. for Liquid ArgonExperimentally, vary a wave vector:

sin=||

4q

Scattering occurs when:

(where d is the spacing between scatterers).d

q2

=||

Can vary either or in experiments

R.D.F. for Liquid Sodium Compared to the BCC Crystal: Correlation at Short Distances

4r2(r)

r (Å)

3 BCC cells

Each Na has 8 nearest neighbours.

Entropy of Glasses• Entropy, S, can be determined experimentally from integrating

plots of CP/T versus T (since Cp = T(S/T)P)

• The disorder (and S) in a glass is similar to that in the melt. Contrast this case to crystallisation in which S “jumps” down at Tm.

• Since the glass transition is not first-order, S is not discontinuous through the transition.

• S for a glass depends on the cooling rate.• As the cooling rate becomes slower, S of the glass becomes lower.• At a temperature called the Kauzmann temperature, TK, we expect

that Sglass = Scrystal.• Remember that the structure of a glass is similar to the liquid’s,

but there is greater disorder in the glass compared to the crystal of the same substance.

Kauzmann Paradox

Crystal

Glass

Melt (Liquid)

Kauzmann Paradox• Sglass cannot be less than Scrystal because glasses are more disordered!

• Yet by extrapolation, we can predict that at sufficiently slow cooling rate, Sglass will be less than Scrystal. This prediction is a paradox!

• Paradox is resolved by saying that TK defines a lower limit to Tg as assumed in the V-F equation.

• Experimentally, it is usually found that TK To (V-F constant). Viscosity diverge towards when T is reduced towards TK. Typically, Tg - To = 50 K.

• This is consistent with the prediction that at T=To, config will go to .

• Tg equals TK (and To) when exp is approaching , which would be obtained via an exceedingly slow cooling rate.

Problem Set 21. Calculate the energy required to separate two atoms from their equilibrium spacing ro to a very large distance apart. Then calculate the maximum force required to separate the atoms. The pair potential is given as w(r) = - A/r6 + B/r12, where A = 10-77 Jm6 and B = 10-134 Jm12. Finally, provide a rough estimate of the modulus of a solid composed of these atoms.

2. The latent heat of vaporisation of water is given as 40.7 kJ mole -1. The temperature dependence of the

viscosity of water is given in the table below. (i) Does the viscosity follow the expectations of an Arrhenius relationship with a reasonable activation energy?(ii) The shear modulus G of ice at 0 C is 2.5 x 109 Pa. Assume that this modulus is comparable to the instantaneous shear modulus of water Go and estimate the characteristic frequency of vibration for water, .

Temp (C) 0 10 20 30 40 50(10-4 Pa s) 17.93 13.07 10.02 7.98 6.53 5.47

Temp (C) 60 70 80 90 100(10-4 Pa s) 4.67 4.04 3.54 3.15 2.82

3. In poly(styrene) the relaxation time for configurational rearrangements follows a Vogel-Fulcher law given as

= o exp(B/T-To),

where B = 710 C and To = 50 C. In an experiment with an effective timescale of exp = 1000 s, the glass transition temperature Tg of poly(styrene) is found to be 101.4 C. If you carry out a second experiment with exp = 105 s, what value of Tg would be obtained?