Soft Matter Review 10 January 2012

Characteristics of Soft Matter (1)Length scales between atomic and macroscopic (sometimes called mesoscopic) (2) The importance of thermal fluctuations and Brownian motion (3) Tendency to self-assemble into hierarchical structures (i.e. ordered on multiple size scales beyond the molecular) (4) Short-range forces and interfaces are important. Lecture 1

Lecture 2: Discussed polar molecules and dipole moments (Debye units) and described charge-dipole and dipole-dipole interactions. Discussed polarisability of molecules (electronic and orientational) and described charge-nonpolar and dispersive (London) interactions. Summarised ways to measure polarisability. Related the interaction energy to cohesive energy and boiling temperatures

Summary Type of Interaction Interaction Energy, w(r) Charge-charge Coulombic Nonpolar-nonpolar Dispersive Charge-nonpolar Dipole-charge Dipole-dipole Keesom Dipole-nonpolar Debye In vacuum:  =1

Measuring Polarisability From Israelachvili, Intermol.& Surf. Forces Polarisability determined from van der Waals gas (a) and u measurements. Polarisability determined from dielectric/index measurements. <<<<<< High f Low f

Lecture 3 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 

Interaction Potentials: w = -Cr -n If n 3, molecules interact only with the nearer neighbours. Gravity: negligible at the molecular level. W(r) = -Cr -1 Coulombic: relevant for salts, ionic liquids and charged molecules. W(r) = -Cr -1 van der Waals’ Interaction: three types; usually quite weak; causes attraction between ANY two molecules. W(r) = -Cr -6 Covalent bonds: usually the strongest type of bond; directional forces - not described by a simple potential. Hydrogen bonding: stronger than van der Waals bonds; charge attracting resulting from unshielded proton in H. In the previous lecture:

Comparison of Theory and Experiment Evaluated at close contact where r = . Note that  o and C increase with . Non-polar London equation (Per mole, n = 1)

Lecture 4 Viscosity and relaxation times increase strongly with decreasing temperature: Arrhenius and Vogel-Fulcher equations First and second-order phase transitions are defined by derivatives of Gibbs’ free energy. The glass transition occurs at a temperature where  config   exp and is dependent on thermal history. In a glass,  config >  exp. Glass structure is described by a radial distribution function. The Kauzmann temperature could represent the temperature at which there is a first-order phase transition underlying the glass transition – possibly at a temperature of T 0.

Lecture 5 For mixing to occur, the free energy (F) of the system must decrease;  F mix < 0. The change in free energy upon mixing is determined by changes in internal energy (U) and entropy (S):  F mix =  U - T  S. The  interaction parameter is a unitless parameter to compare the interaction energy between dissimilar molecules and their self-interaction energy. The change of  F mix with  (and T) leads to stable, metastable, and unstable regions of the phase diagram. For simple liquids, with molecules of the same size, assuming non-compressibility, the critical point occurs when  = 2. At the critical point, interfacial energy,  = 0.

Constructing a Phase Diagram T1T1 T2T2 T3T3 T4T4 T5T5 T 1 <T 2 <T 3 …. Co-existence where: Spinodal where : GG  =2  >2

Phase Diagram for Two Liquids Described by the Regular Solution Model GG Immiscible Miscible Low T High T Spinodal and co-existence lines meet at the critical point.

Lecture 6 The thermodynamics of polymer phase separation is similar to that of simple liquids, with consideration given to the number of repeat units, N. For polymers, the critical point occurs at  N=2, with the result that most polymers are immiscible. As  N decreases toward 2, the interfacial width of polymers becomes broader. The Stokes’ drag force on a colloidal particle is F s =6  av. Colloids undergo Brownian motion, which can be described by random walk statistics: 1/2 = n 1/2, where is the step-size and n is the number of steps. The Stokes-Einstein diffusion coefficient of a colloidal particle is given by D = kT (6  a) -1.

Lecture 7 The viscosity of colloidal dispersions depends on the volume fraction of the particles (Einstein equation): The Peclet number, Pe, describes the competition between particle disordering because of Brownian diffusion and particle ordering under a shear stress. At high Pe (high shear strain rate), the particles are more ordered; shear thinning behaviour occurs and  decreases. van der Waals’ energy acting between a colloidal particle and a semi-  slab (or another particle) can be calculated by summing up the intermolecular energy between the constituent molecules. Macroscopic interactions can be related to the molecular level. The Hamaker constant, A, contains information about molecular density (  ) and the strength of intermolecular interactions (via the London constant, C): A =  2  2 C

Israelachvili, p. 177 If R 1 > R 2 : Colloidal particles Summary of Molecular and Macroscopic Interaction Energies

Polymer crystals have a hierarchical structure: aligned chains, lamella, spherulites. Melting point is inversely related to the crystal’s lamellar thickness. Lamellar thickness is inversely related to the amount of undercooling. The maximum crystal growth rate usually occurs at temperatures between the melting temperature and the glass transition temperature. Tacticity and chain branching prevents or interrupts polymer crystal growth. Lecture 8

Lecture 9 The root-mean-squared end-to-end distance, 1/2, of a freely- jointed polymer molecule is N 1/2 a, when there are N repeat units, each of length a. Polymer coiling is favoured by entropy. The elastic free energy of a polymer coil is given as Copolymers can be random, statistical, alternating or diblock. Thinner lamellar layers in a diblock copolymer will increase the interfacial energy and are not favourable. Thicker layers require chain stretch and likewise are not favourable! A compromise in the lamellar thickness, d, is reached as:

Lecture 9 Elastic (entropic) effects cause a polymer molecule to coil up. Excluded volume effects cause polymer molecules to swell (in a self-avoiding walk). Polymer-solvent interactions, described by the  - parameter, can favour tight polymer coiling into a globule (large  ) or swelling (low  ). Thus there is a competition between three effects! The radius-of-gyration of a polymer, R g, is 1/6 of its root-mean- square end-to-end distance 1/2.

Lecture 9 When  = 1/2, excluded volume effects are exactly balanced by polymer/solvent interactions. Elastic effects (from an entropic spring) lead to a random coil: 1/2 ~ a N 1/2 When  1/2 ~ a N 3/5 When  > 1/2, polymer/solvent interactions are dominant over excluded volume effects. They lead to polymer coiling: a globule results.

Lecture 10: Applies for higher N: N>N C when chains are entangled. G.Strobl, The Physics of Polymers, p. 221 Data shifted for clarity! Viscosity is shear-strain rate dependent. Usually measure in the limit of a low shear rate:  o 3.4 Reptation occurs when polymer chains are entangled (in melts or in concentrated solutions where chains overlap).

Testing of Scaling Relation: D ~N -2 M=Nm o -2 Experimentally, D ~ N -2.3 Data for poly(butadiene) Jones, Soft Condensed Matter, p. 92

Relaxation Modulus for Polymer Melts Viscous flow TT Gedde, Polymer Physics, p. 103