Last Lecture: Lennard-Jones potential energy for pairs of atoms and for molecular crystals Evaluation of the Young’s and bulk moduli from the L-J potentials Response of soft matter to shear stress: Hookean (elastic) solids versus Newtonian (viscous) liquids Description of the viscoelastic response with a transition at the characteristic relaxation time, 

Time Scales, the Glass Transition and Glasses, and Liquid Crystals 3SCMP 10 February, 2004 Lecture 4 See Jones’ Soft Condensed Matter, Chapt. 2 and 7

Response of Soft Matter to a Constant Shear Stress: Viscoelasticity t Slope: We see that 1/G o  ( 1 /  )    is the relaxation time An alternative expression for viscosity is thus   G o 

Relaxation and a Simple Model of Viscosity When a liquid is subjected to a shear stress, immediately the molecules’ positions are shifted but the same “neighbours” are kept. Thereafter, the stress falls to zero, as the constituent molecules re-arrange to relax the stress. 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.

Molecular Relaxation Time The probability P of successful escape is given by a Boltzmann distribution: P ~ exp(-  / kT) We see that  must be a vibrational energy barrier per molecule. As T  0, then P  0, whereas when T , then P  1 (100% success) Typically,   0.4 L v /N A, where L v is the heat of vapourisation per mole. Eyring proposed that the frequency of successful escapes, f, is then the product of the frequency of attempts and the probability of success: The time required for a molecule to escape its cage defines a relaxation time, , which is comparable in magnitude to the macroscopic relaxation time. And so, f = 1/ .

Arrhenius Behaviour of Viscosity In liquids,  is very short, varying between and s. Hence, as commonly observed, stresses in liquids are relaxed nearly instantaneously. In melted polymers,  is on the order of several ms. From our discussion of viscoelasticity, we know that   G o . 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”. “Fragile liquids” show non-Arrhenius behaviour that requires a different description. An example of a fragile liquid is a melted polymer, which is described by the Vogel-Fulcher relationship: We see that  diverges to , as the liquid is cooled. In the high-temperature limit,  approaches  o. where B and T o are empirical constants.

Temperature-Dependence of Viscosity P = Poise Arrhenius

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 as T decreases towards T o.

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. For example, if a sample is being cooled at a rate of 1 K per min., then  exp is 1 min. at each temperature. At higher temperatures,  exp >  config, and flow is observed on the time scale of the measurement. Debonding of an Adhesive Flow is observed on long time scales,  exp

Are Stained-Glass Windows Liquid? Window in the Duomo of Siena Medieval church windows are thicker at their bottom. Is there flow over a time scale of  exp  100 years?

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. The substance appears to be solid-like, with no observable flow. At T = T g,  is typically Pas, compared to  = Pas 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

An Example of 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.

Thermodynamics of Phase Transitions How can we classify the glass transition? During a transition from one phase to another, we see that will be discontinuous:

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

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

Thermodynamics of First-Order Transitions dG = VdP - TdS 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 discontinuous at T m. (Or H)

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

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

Experimental Results for Poly(Vinyl Acetate) Data from Kovacs

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

Entropy of Glasses Since the glass transition is not first-order, entropy, S, is not discontinuous. The disorder in a glass is similar to that in the melt. Compare to crystallisation in which S jumps down at T m. S can be determined experimentally from integrating plots of C P versus T. S for a glass depends on the cooling rate. As the cooling rate becomes slower, S becomes lower. At a temperature called the Kauzmann temperature, S glass = S crystal. 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 Liquid

Kauzmann Paradox S glass cannot be less than S crystal. But 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 given by the V-F equation. Experimentally, it is usually found that T K  T o (V-F constant). 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 when  exp is approaching , which would be obtained via an exceedingly slow cooling rate.

Why does the Glass Transition Occur? Adam and Gibbs (1965) proposed that as the temperature of a liquid is lowered, more and more atoms must co-operatively re-arrange. If the number of atoms/molecules required for co- operativity is z*, and the barrier for each molecule to move is , then  will vary with T as: Z* = 9

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 distances.

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

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.

Liquid Structure r

R.D.F. for Liquid Argon Experimentally, vary a wave vector: Scattering occurs when: (where d is the spacing). Can very either  or in experiments

R.D.F. for Liquid Sodium Compared to the BCC Crystal 4  r 2  (r) r (Å) 3 BCC cells Each Na has 8 nearest neighbours.

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

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

LC Characteristics LCs display 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. When there is a shear stress along the director, a nematic LC flows. In a “splay” deformation, order is disrupted, and there is an elastic response.

Polarised Light Microscopy of LC Phases Nematic LC Why do LCs show birefringence? (That is, their refractive index varies with direction in the substance.)

Birefringence of LCs The bonding and atomic distribution along the longitudinal axis of a calamitic LC molecule is different than along the transverse axis. Hence, the electronic polarisability (  o ) differs in the two directions. Polarisability in the bulk nematic and crystalline phases will mirror the molecular. The Clausius-Mossotti equation relates the molecular characteristic  o to the bulk property (  or n 2 ): In the isotropic phase:

LC Orientation Director   Distribution function Higher order Lower order NN n

Diffraction from LC Phases L a

Order Parameter for a Nematic- Isotropic LC Transition Discontinuity at T c : First-order transition Isotropic Nematic S The molecular ordering in a LC can be described by a so-called order parameter, S: 1 0

Problem Set 2 1. 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?