1. Previous Lecture Lennard-Jones potential energy for pairs of atoms and for pairs within molecular crystals Evaluation of the Young’s (elastic) modulus for molecular crystals starting from the L-J potentials Response of soft matter to shear stress: Hookean (elastic) solids versus Newtonian (viscous) liquids Description of viscoelasticity with a transition from elastic to viscous response at a characteristic relaxation time,  An important relationship between elastic and viscous components:  = G o 

2. 3SM Lecture 4 Time Scales, the Glass Transition and Glasses 4 March, 2010 See Jones’ Soft Condensed Matter, Chapt. 2 and 7

3. 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 

4. 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

5.  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

6. 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. A statistical 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% success) 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 relaxation time. And so,  config = 1/ f.

7. 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. We can approximate that   G o , where G o is the 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.

8. 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!)

9. Temperature-Dependence of Viscosity P = Poise Arrhenius Vogel- Fulcher

10. 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.

11. An Example of the Glass Transition Video: Shattering polymer

12. 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 will 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 of an Adhesive Flow can be observed on long time scales,  exp

13. Oscillatory Stress Apply a shear strain (or stress) at an angular frequency of  = 1 /  exp t ss 2/2/

14. 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 10 13 Pa-s. Compare this to  = 10 -3 Pa-s for water at room temperature.

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

16. 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?

17. 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

18. 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.

19. 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.

20. 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

21. 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)

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

23. 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) 59-64 ~ Thickness

24. 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:

25. Experimental Results for Poly(Vinyl Acetate) Data from Kovacs   Expansivity is not strictly discontinuous – there is a broad step.

26. 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.

27. 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 broad.

28. 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).

29. Atomic Distribution in Crystals 12 nearest neighbours And 4th nearest! FCC unit cell (which is repeated in all three directions)

30. 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

31. Simple Liquid Structure r

32. 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.

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

34. R.D.F. for Liquid Sodium Compared to the BCC Crystal 4r2(r)4r2(r) r (Å) 3 BCC cells Each Na has 8 nearest neighbours.

35. 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 becomes lower. At a temperature called the Kauzmann temperature, T K, we expect that 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.

36. Kauzmann Paradox Crystal Glass Melt (Liquid)

37. Kauzmann Paradox S glass cannot be less than S crystal. 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). 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.

39. 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 = 10 -77 Jm 6 and B = 10 -134 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)01020304050  (10 -4 Pa s)17.9313.0710.027.986.535.47 Temp (  C)60708090100  (10 -4 Pa s) 4.674.043.543.152.82 3. 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 101.4  C. If you carry out a second experiment with  exp = 10 5 s, what value of T g would be obtained?

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

43. 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.

44. Order in LC Phases Isotropic Nematic Smectic Density N = director Orientational Positional None High None weak 1-D

45. 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. When there is a shear stress along the director, a nematic LC will flow. In a “splay” deformation, order is disrupted, and there is an elastic response with an elastic constant, K From RAL Jones, Soft Condensed Matter

46. LC Orientation Distribution function, f(  ) Director  NN n  Higher order Lower order  0

47. Order Parameter for a Nematic- Isotropic LC Transition Discontinuity at T c : Therefore, a first-order transition Isotropic Nematic S The molecular ordering in a LC can be described by a so-called order parameter, S: 1 0 With the greatest ordering,  = 0° and S = 1. The order parameter is determined by the minimum in the free energy, F. Disordering increases S and decreases F, BUT intermolecular energies and F are decreased with ordering.

48. From RAL Jones, Soft Condensed Matter, p. 111 Experimental Example of First-Order Nematic-Isotropic Transition Data obtained from birefringence measurements (circles) and diamagnetic anisotropy (squares) of the LC p-azoxyanisole. TcTc

49. Scattering Experiments d = molecular spacing 

50. Diffraction from LC Phases L a nematic isotropic smectic

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

52. 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 (longitudinal (  l ) versus transverse (  t )). Polarisability in the bulk nematic and crystalline phases will mirror the molecular. The Clausius-Mossotti equation relates the molecular characteristic  to the bulk property (  or n 2 ): In the isotropic phase: With greater LC ordering, there is more birefringence.

53. IsotropicNematicPerfect nematic N N S = 1 S = 0 0< S <1

54. Crossed polarisers Crossed Polarisers Block Light Transmission http://www.kth.se/fakulteter/TFY/kmf/lcd/lcd~1.htm

55. Twisted nematic LC: No applied field and light is transmitted E > E crit : Light is blocked Liquid Crystal Displays The director rotates by 90° going from the top to the bottom of the LC. A strong field aligns the LC director in the same direction - except along the surfaces. d

58. 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

59. Viscous Response of Newtonian Liquids A A y F xx There is a velocity gradient (v/y) normal to the area. The viscosity  relates the shear stress,  s, to the velocity gradient. The viscosity can thus be seen to relate the shear stress to the shear rate: The top plane moves at a constant velocity, v, in response to a shear stress: v  has S.I. units of Pa s. The shear strain increases by a constant amount over a time interval, allowing us to define a strain rate: Units of s -1 

60. 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 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

61.  0 Molecular configuration Potential Energy Need to consider the probability of being in a higher state with an energy of . Intermediate state: some molecular spacings are greater

62. Arrhenius Behaviour of Viscosity In liquids,  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,  is on the order of several ms or s. 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.

63. 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 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 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 temp. here are usually °C!)

64. Temperature-Dependence of Viscosity P = Poise Arrhenius V-F

65. 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.

66. An Example of the Glass Transition: NOT freezing or crystallisation! Video: Shattering polymer

67. 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 will 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 of an Adhesive Flow can be observed on long time scales,  exp

68. Oscillatory Stress Apply a shear strain (or stress) at an angular frequency of  = 1 /  exp t ss 2/2/

69. 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 10 13 Pas. Compare this to  = 10 -3 Pa-s for water at room temperature.

70. 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

71. 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”.

72. 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.

73. Experimental Results for Poly(Vinyl Acetate) Data from Kovacs

74. 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”.

75. 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