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:  = G M 

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: viscoelastic solid viscoelastic liquid   time Hookean Newtonian

Response of Soft Matter to a Constant Shear Stress t Elastic response Viscous response (provides initial and recoverable strain) (strain increases linearly over time) Slope: J eq Steady-state compliance = J eq = 1/G eq In the Maxwell model, G eq =  /  So,  =  J eq and J eq =  / 

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)! 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 L v /N A, where L v is the heat of vapourisation per mole and N A 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): 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 and 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   G eq , where G eq is the equilibrium shear modulus of the corresponding solid. Hence an expression for  can be found from the Eyring relationship: 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 T o. It solidifies as temperature is decreased. In the high-temperature limit,  approaches  o - a lower limit. where B and T o are empirical constants. (By convention, the units of temperature here are usually °C!)

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

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 Strain Apply a shear strain at an angular frequency of  = 2  /  exp. This defines an experimental timescale. t  2/2/ Dynamic mechanical analysis

Hookean versus Newtonian Responses tt tttt  Hookean: Stress is in phase tt tt tttt 22 Newtonian: Stress is  /2 out of phase tt 22 

Response of a Viscoelastic Material Stress oscillates at the same frequency as the strain, but it leads the strain by a phase angle,  : 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:  = J eq  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, T g, then  config ~  exp. Below T g, 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 = T g,  is typically Pa-s. Compare this to  = Pa-s for water at room temperature.

Competing Time Scales Reciprocal Temperature (K -1 ) Log(1/  ) =1/  vib f = 1/  config 1/  exp 1/T g  config <  exp  config >  exp Melt (liquid) Glass (solid) Molecular conformation does not change when passing through the glass transition.

Effect of Cooling Rate on T g T g 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 T g. Various experimental techniques have different associated  exp values. Hence, a value of T g 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? b a The “b” phase is stable below the critical temperature, T c. TcTc

Temperature, T Free energy, G Free Energy of the Melting/Freezing Transition Crystalline state Liquid (melt) state TmTm Below the melting temperature, T m, the crystalline state is stable. The thermodynamic driving force for crystallisation,  G, increases when cooling below T m. During a transition from solid to liquid, we see that (dG/dT) P will be discontinuous. GG Undercooling,  T, is defined as T m – 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: V T liquid crystalline solid TmTm Viscosity is also discontinuous at T m. There is a heat of melting, and thus H is also discontinuous at T m. (Or S)

Thermodynamics of Glass Transitions V T Crystalline solid TmTm Liquid Glass TgTg

Determining the Glass Transition Temperature in Polymer Thin Films Poly(styrene) h o ~ 100 nm TgTg Melt Glass Keddie et al., Europhys. Lett. 27 (1994) ~ Thickness

Is the Glass Transition Second-Order? Thus in a second-order transition, C P will be discontinuous. Recall that volume expansivity, , is defined as: 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. Note that S is found from -(  G /  T) P. Then we see that the heat capacity, C p, can be given as:

Experimental Results for Poly(Vinyl Acetate) “Classic” data from Kovacs   Expansivity is not strictly discontinuous – there is a broad step. Note that T g depends on the time scale of cooling!

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 Crystals 12 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 Argon Experimentally, vary a wave vector: Scattering occurs when: (where d is the spacing between scatterers). Can vary either  or in experiments

R.D.F. for Liquid Sodium Compared to the BCC Crystal: Correlation at Short Distances 4r2(r)4r2(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 C P /T versus T (since C p = 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 T m. 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, T K, we expect that S glass = S crystal. 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 S glass cannot be less than S crystal because glasses are more disordered! Yet by extrapolation, we can predict that at sufficiently slow cooling rate, S glass will be less than S crystal. This prediction is a paradox! Paradox is resolved by saying that T K defines a lower limit to T g as assumed in the V-F equation. Experimentally, it is usually found that T K  T o (V-F constant). Viscosity diverge towards  when T is reduced towards T K. Typically, T g - T o = 50 K. This is consistent with the prediction that at T=T o,  config will go to . T g equals T K (and T o ) when  exp is approaching , which would be obtained via an exceedingly slow cooling rate.

Problem Set 2 1. Calculate the energy required to separate two atoms from their equilibrium spacing r o 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/r 6 + B/r 12, where A = Jm 6 and B = Jm 12. 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 10 9 Pa. Assume that this modulus is comparable to the instantaneous shear modulus of water G o and estimate the characteristic frequency of vibration for water,. Temp (  C)  (10 -4 Pa s) Temp (  C)  (10 -4 Pa s) In poly(styrene) the relaxation time for configurational rearrangements  follows a Vogel-Fulcher law given as  =  o exp(B/T-T o ), where B = 710  C and T o = 50  C. In an experiment with an effective timescale of  exp = 1000 s, the glass transition temperature T g of poly(styrene) is found to be  C. If you carry out a second experiment with  exp = 10 5 s, what value of T g would be obtained?