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Previous Lecture • Lennard-Jones potential energy is applied to find the 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-distance relation derived from the L-J potentials. • Response of soft matter to shear stress can be like 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,t • An important relationship between the elastic and viscous components:h = GMt (or 1/Je = t/h)

PH3-SM (PHY3032) Soft Matter Physics Lecture 4 Time Scales, the Glass Transition and Glasses 25 October, 2011 See Jones’ Soft Condensed Matter, Chapt. 2 and 7

Response of Soft Matter to a Step Strain g Constant shear strain applied time Stress relaxes over time as molecules re-arrange: s Hookean viscoelastic solid viscoelastic liquid Newtonian t time

Viscous response Elastic response Steady-state compliance, Jeq= 1/Geq In the Maxwell model, Geq = h/t So, t = hJeq and Jeq = t/h Response of Soft Matter to a Constant Shear Stress Slope: (strain increases linearly over time) Jeq (provides initial and recoverable strain) t What is the physical reason for t and h?

Simulation of 32 Hard Smooth Particles in a Box Liquid: particles are “caged” but can occasionally swap places. Solid: particles vibrate on a lattice Alder and Wainwright, J. Chem. Physics (1959) 31, 459.

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 their neighbours. Molecules in a liquid vibrate with a frequency,n, comparable to the phonon frequency in a solid of the same substance (kT/h, where h is Planck’s constant). Thusncan be considered a frequency of attempts to escape a cage. But what is the probability that the molecule will escape the cage? F

Molecular Relaxation in Simple Viscous Flow The frequency, f, at which the molecules overcome the barrier and relax is an exponential function of temperature, T. Intermediate state: some molecular spacings are greater: thus higher w(r)! e Potential Energy (per molecule) 0 Molecular configuration

Molecular Relaxation Times • eis the energy of the higher state and can be considered an energy barrier per molecule. It is related to the energy required to “push apart” neighbouring molecules and hence is proportional to the cohesive energy. • Typically,e 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(-e /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 (n) and the probability of “success” (P): The time required for a molecule to escape its cage defines a molecular configurational relaxation time, tconfig, which is comparable in magnitude to the macroscopic stress relaxation time, t. And so, tconfig= 1/f.

Arrhenius Behaviour of Viscosity In liquids, the relaxation time,t, is very short, varying between10-12 and 10-10 s. Hence, as commonly observed, stresses in liquids are relaxed nearly instantaneously. In melted polymers,tcan be on the order of several ms or s. Whereas in solids, t is very large, such that flow is not observed on realistic time scales. We can approximate thath Geqt, where Geq is the equilibrium shear modulus of the corresponding solid. Hence an expression forhcan 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: where B and To are empirical constants. (By convention, the units of temperature here are usually °C!) • We see that hdiverges to , as the liquid is cooled towards To. It solidifies as temperature is decreased. In the high-temperature limit, happroaches ho- a lower limit.

Temperature-Dependence of Viscosity Relaxation Time Viscosity Arrhenius Vogel-Fulcher P = Poise = 0.1 Pa-s

Temperature-Dependence of Polymer Viscosity Viscosity above Tg: 1 Pa-s = 10 Poise T0Tg – 50 C Tg G. Strobl, The Physics of Polymers, 2nd Ed. (1997) Springer, p. 229

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,tconfig. From the Vogel-Fulcher equation, we see that: We see that the relaxations become exceedingly slow (t becomes v. large) as T decreases towards To.

Experimental Time Scales To distinguish a liquid from a solid, flow must be observed on an experimental time scale,texp. 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., thentexpis ~1 min. at each temperature increment. Debonding an Adhesive Flow can be observed on the time scale of the experiment, texp, becausetexp > tconfig.

Oscillatory Strain Dynamic mechanical analysis Apply a shear strain at an angular frequency of  = 2p/texp. This defines an experimental timescale. g t 2p/

Hookean versus Newtonian Responses g(t) s(t) Hookean: Stress is in phase Stress set by the shear modulus, G. wt 2p p g(t) s(t) Newtonian: Stress is p/2 out of phase wt Stress set by the viscosity, h. p 2p

Response of a Viscoelastic Material Stress oscillates at the same frequency as the strain, but it leads the strain by a phase angle,d: The relative values of the viscous and the elastic components depend on the time-scale of the observation (texp = 2p/w) in relation to the relaxation time:t = Jeqh. If texp > t : the material appears more liquid-like If texp < t : the material appears more solid-like

The Glass Transition At higher temperatures,texp > tconfig, and so flow is observed on the time scale of the measurement. As T is lowered,tconfigincreases. When T is decreased to a certain value, known as the glass transition temperature, Tg, thentconfig ~ texp. Below Tg, molecules do not change their configuration sufficiently fast to be observed duringtexp. That is, texp < tconfig. The substance appears to be solid-like, with no observable flow. At T = Tg,his typically 1013 Pa-s. Compare this to h = 10-3 Pa-s for water at room temperature.

An Example of the Glass Transition Video: Shattering polymer

Viscosity, h 1013 Pa-s Tg Temperature

Competing Time Scales 1/texp 1/Tg  =1/tvib Log(1/t) f = 1/tconfig tconfig < texp Melt (liquid) Molecular conformation does not change when passing through the glass transition. tconfig > texp Glass (solid) Reciprocal Temp., 1/T (K-1)

Effect of Cooling Rate on Tg Tg is not a constant for a substance. When the cooling rate is slower,texpis longer. For instance, reducing the rate from 1 K min-1 to 0.1 K min-1, increasestexpfrom 1 min. to 10 min. at each increment in K. With a slower cooling rate, a lower T can be reached beforetconfig texp. The result is a lower observed Tg. Various experimental techniques have different associatedtexpvalues. Hence, a value of Tgdepends on the technique used to measure it and the frequency of the sampling.

Viscosity relates a velocity gradient to a shear stress. Are Stained-Glass Windows Liquid? Some medieval church windows are thicker at their bottom. Is there flow over a time scale oftexp100 years? Window in the Duomo of Siena

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

Free Energy of the Melting/Freezing Transition • Below the melting temperature, Tm, the crystalline state is stable. • The thermodynamic driving force for crystallisation, DG, increases when cooling below Tm . • During a transition from solid to liquid, we see that (dG/dT)P will be discontinuous. DG Free energy, G Crystalline state Liquid (melt) state Tm Undercooling, DT, is defined as Tm – T. Temperature, 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-STso 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 H = enthalpy S = entropy U = internal energy Finally, dG = VdP-SdT

V There is a heat of melting, and thus H is also discontinuous at Tm. liquid (Or S) crystalline solid Tm T Thermodynamics of First-Order Transitions • dG = VdP - SdT • In a first order transition, we see that V and S must be discontinuous: Viscosity is also discontinuous at Tm.

Glass Tg Tm Thermodynamics of Glass Transitions V Liquid Crystalline solid T

Thermodynamics of Glass Transitions Faster-cooled glass Glass Tg Tm Tfcg Tg is higher when there is a faster cooling rate. We see that the density of a glass is a function of its “thermal history”. V Liquid Crystalline solid T

Melt Glass Tg Determining the Glass Transition Temperature in Polymer Thin Films Poly(styrene) ho ~ 100 nm ~ Thickness Keddie et al., Europhys. Lett. 27 (1994) 59-64

Note that S is found from -(dG/dT)P. Then we see that the heat capacity, Cp, can be given as: And V = (dG/dP)T. So, Expansivity is related to a second differential of G, and hence it is likewise discontinuous in a second-order phase transition. Is the Glass Transition Second-Order? • Thus in a second-order transition, CP will be discontinuous. • Recall that volume expansivity,b, is defined as:

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

Sample is heated at a constant rate. Calorimeter measures how much heat is required. Heat capacity is not strictly discontinuous – the step occurs over about 10 C. Glass Transition of Poly(vinyl chloride) Heat flow  heat capacity T Data from H. Utschick, TA Instruments

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 Local order is identical in both structures Crystalline Glassy (amorphous) Going from glassy to crystalline, there is a discontinuous decrease in volume.

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 r atoms per volume. • For the entire substance, let there be ro atoms per unit volume. Then g(r) =r(r)/ro. • At shortr, there is some predictability of position because short-range forces are operative. • At long r,r(r) approachesroand 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 eitherqorlin experiments

R.D.F. for Liquid Sodium Compared to the BCC Crystal: Correlation at Short Distances 4pr2r(r) 3 BCC cells Each Na has 8 nearest neighbours. r (Å)

Entropy of Glasses • Entropy, S, can be determined experimentally from integrating plots of CP/TversusT (since Cp =T(dS/dT)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 Melt (Liquid) Glass Crystal

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,tconfigwill go to. • Tg equals TK (and To) whentexpis 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 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 waterhis 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, n. Temp (C)0 10 20 30 40 50 h(10-4 Pa s) 17.93 13.07 10.02 7.98 6.53 5.47 Temp (C)60 70 80 90 100 h(10-4 Pa s) 4.67 4.04 3.54 3.15 2.82 3. In poly(styrene) the relaxation time for configurational rearrangementstfollows a Vogel-Fulcher law given as t = toexp(B/T-To), where B = 710 C and To = 50 C. In an experiment with an effective timescale oftexp= 1000 s, the glass transition temperature Tg of poly(styrene) is found to be 101.4 C. If you carry out a second experiment withtexp = 105 s, what value of Tg would be obtained?

Liquid Crystals Rod-like (= calamitic) molecules Molecules can also be plate-like (= discotic)

Density Temp. LC Phases N = director Isotropic The phases ofthermotropicLCs depend on the temperature. Nematic Attractive van der Waals’ forces are balanced by forces from thermal motion. Smectic

Density Order in LC Phases N = director OrientationalPositional None weak 1-D 1-D None High High High Isotropic Nematic Smectic

In a “splay” deformation, order is disrupted, and there is an elastic response with an elastic constant, K When there is a shear stress along the director, a nematic LC will flow. LC Characteristics • LCs exhibit more molecular ordering than liquids, although not as much as in conventional crystals. • LCs flow like liquids in directions that do not upset the long-ranged order. From RAL Jones, Soft Condensed Matter

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