Soft Matter and Statistical Physics3SMS

16 January, 2007Lecture 1: Introduction to Soft Matter

What is Condensed Matter?Condensed matter refers to matter that is not in the gas phase but is condensed as liquid or solid. (condensed denser!)Matter condenses when attractive intermolecular bond energies are comparable to or greater than thermal (i.e. kinetic) energy ~ kT.

Phase diagram of waterImage: http://wps.prenhall.com/wps/media/objects/602/616516/Chapter_10.html

Condensed Matter and Origin of Surface TensionFrom I.W. Hamley, Introduction to Soft Matter Molecules at an interface have asymmetric forces around them. In reducing the interfacial area, more molecules are forced below the surface, where they are completely surrounded by neighbours. Force associated with separating neighbouring molecules = surface tension.Liquids and gases are separated by a meniscus; they differ only in density but not structure (i.e. arrangement of molecules in space).

Mercury has a very high surface energy!Image: http://wps.prenhall.com/wps/media/objects/602/616516/Chapter_10.htmlWhat characteristics result from a high surface energy?An interfacial energy G is associated with the interface between two phases (units of Jm-2) (also called an interfacial tension: Nm-1)Interface with air = surfaceFor mercury, G = 0.486 N/mFor water, G = 0.072 N/mFor ethanol, G = 0.022 N/m

Soft Condensed MatterRefers to condensed matter that exhibits characteristics of both solids and liquidsThe phrase soft matter was used by Pierre de Gennes as the title of his 1991 Nobel Prize acceptance speech.Soft matter can flow like liquids (measurable viscosity)Soft matter can bear stress (elastic deformation)Viscoelastic behaviour = viscous + elasticExamples: rubbers, gels, pastes, creams, paints, soaps, liquid crystals, proteins, cells

Types of Soft Matter: ColloidsA colloid is a sub-mm particle (but not a single molecule) of one phase dispersed in a continuous phase.The dispersed phase and the continuous phase can consist of either a solid (S), liquid (L), or gas (G):Dispersed Phase Continuous NameExamplesL/SG aerosol fog, hair spray; smokeGL/S foam beer froth; shaving foam; poly(urethane) foamLL (S) emulsion mayonnaise; salad dressingSL sol latex paint; tooth pasteSS solid suspension pearl; mineral rocks

Interfacial Area of ColloidsrFor a spherical particle, the ratio of surface area (A) to volume (V) is:Thus, with smaller particles, the interface becomes more significant. A greater fraction of molecules is near the surface.

Types of Soft Matter: PolymersA polymer is a large molecule, typically with 50 or more repeat units. (A unit is a chemical group.)Examples include everyday plastics (polystyrene, polyethylene); rubbers; biomolecules, such as proteins and starch.

Each pearl on the string represents a repeat unit of atoms, linked together by strong covalent bonds.The composition of the pearls is not important.Physics can predict the size and shape of the molecule; the important parameter is the number of repeat units, N.

Shear thickening behaviour of a polymer colloid (200 nm particles of polymers dispersed in water):At a low shear rate: flows like a liquidAt a high shear rate: solid-like behaviour

A liquid crystal is made up of molecules that exhibit a level of ordering that is intermediate between liquids (randomly arranged and oriented) and crystals (three-dimensional array).Types of Soft Matter: Liquid CrystalsThis form of soft matter is interesting because of its anisotropic optical and mechanical properties.Image: http://wps.prenhall.com/wps/media/objects/602/616516/Chapter_10.html

Interfacial tension, GTypical G values for interfaces with water - carbon tetrachloride: 45 mN/m; benzene: 35 mN/m; octanol: 8.5 mN/moilwaterTypes of Soft Matter: Surfactantsemulsion

Hydrophobicity and HydrophilicitywaterqHydrophobicq is smallq is largewaterFully wetting

Contact Angle: Balance of ForcesThree interfaces: solid/water (sw); water/air (wa); solid/air (sa)Each interface has a surface tension: Gsw; Gwa; GsaContact angles thus provide information on surface tensions and the effect of surfactants.

Acrylic Latex Paint Monodisperse Particle Size

Vertical scale = 200nm(1) Length scales between atomic and macroscopicTop view3 mm x 3 mm scan

Characteristics of Soft Matter (4 in total)Example of colloidal particles

Typical Length ScalesAtomic spacing: ~ 0.1 nmPitch of a DNA molecule: 3.4 nm

Diameter of a surfactant micelle: ~6-7 nmRadius of a polymer molecule: ~10 nm

Diam. of a colloidal particle (e.g. in paint): ~200 nmBacteria cell: ~2 mmDiameter of a human hair: ~80 mm

15 mm x 15 mmPoly(ethylene) crystalCrystals of poly(ethylene oxide)5 mm x 5 mm

Intermediate Length ScalesMathematical descriptions of soft matter can ignore the atomic level.Mean field approaches define an average energy or force imposed by the neighbouring molecules.Physicists usually ignore the detailed chemical make-up of molecules; can treat molecules as strings, rods or discs.

(2) The importance of thermal fluctuations and Brownian motion

Characteristics of Soft Matter (4 in total)Brownian motion can be though of as resulting from a slight imbalance of momentum being transferred between liquid molecules and a colloidal particle.

Thermal fluctuationsSoft condensed matter is not static but in constant motion at the level of molecules and particles.The equipartition of energy means that for each degree of freedom of a particle to move, there is 1/2kT of thermal energy. For a colloidal particle able to undergo translation in the x, y and z directions, thermal energy is 3/2 kT.k = 1.38 x 10-23 JK-1, so kT = 4 x 10-21 J per molecule at room temperature (300 K).kT is a useful gauge of bond energy.

(3) Tendency to self-assemble into hierarchical structures (i.e. ordered on large size scales)Characteristics of Soft Matter (4 in total)Diblock copolymer molecules spontaneously form a pattern in a thin film.(If one phase is etched away, the film can be used for lithography.)Image from IBM (taken from BBC website)Two blocks

2mm x 2mmLayers or lamellae form spontaneously in diblock copolymers.Diblock copolymerPolymer Self-Assembly

Colloidosomes: Self-assembled colloidal particlesA.D. Dinsmore et al., Colloidosomes: Selectively Permeable Capsules Composed of Colloidal Particles, Science, 298 (2002) p. 1006.Liquid BLiquid AColloidal particles (

I. Karakurt et al., Langmuir 22 (2006) 2415Hydrophilically-driven self-assembly of particles

MRS Bulletin,Feb 2004, p. 86Colloidal CrystalsColloidal particles can have a +ve or -ve charge.In direct analogy to salt crystals of +ve and -ve ions, charge attractions can lead to close-packing in ordered arrays.

Examples of Self-AssemblySurfactants can assemble into (a) spherical micelles, (b) cylindrical micelles, (c) bi-layers (membranes), or (d) saddle surfaces in bicontinuous structuresFrom I.W. Hamley, Introduction to Soft Matter(a)(b)(c)(d)

Examples of Self-AssemblySurfactants can create a bi-continuous surface to separate an oil phase and a water phase.The hydrophilic end of the molecule orients itself towards the aqueous phase.The oil and water are completely separated but both are CONTINUOUS across the system.From RAL Jones, Soft Condensed MatterThe plumbers nightmare

Competitions in Self-AssemblyMolecules often segregate at an interface to LOWER the interfacial energy - leading to an ordering of the system.This self-assembly is opposed by thermal motion that disrupts the ordering.Self-assembly usually DECREASES the entropy, which is not favoured by thermodynamics.But there are attractive and repulsive interactions between molecules that dominate.

Importance of Interfaces

Free energy change: dF = GdA An increase in area raises the systems free energy, which is not thermodynamically favourable.So, sometimes interfacial tension opposes and destroys self-assembly.An example is coalescence in emulsions.

Particle CoalescenceSurface area of N particles: 4Npr2Surface area of particle made from coalesced particles: 4pR2Same particle volume before and after coalescence: RrChange in area, DA = - 4pr2(N-N2/3)In 1 L of emulsion (50% dispersed phase), with a droplet diameter of 200 nm, N is ~ 1017 particles. Then DA = -1.3 x 104 m2 With G = 3 x 10-2 J m-2, DF =GDA = - 390 J.

(4) Short-range forces and interfaces are important.Characteristics of Soft Matter (4 in total)The structure of a geckos foot has been mimicked to create an adhesive. But the attractive adhesive forces can cause the synthetic hairs to stick together.From Materials World (2003)

In hard condensed matter, such as Si or Cu, strong covalent or metallic bonds give a crystal strength and a high cohesive energy (i.e. the energy to separate atoms). In soft matter, weaker bonds - such as van der Waals - are important. Bond energy is on the same order of magnitude as thermal energy ~ kT. Hence, bonds are easily broken and re-formed. Chemical Bonds in Soft Matter The strength of molecular interactions (e.g. charge attractions) decays with distance, r. At nm distances, they become significant.r

What are these forces that operate over short distances and hold soft matter together?

Interaction PotentialsInteraction between two atoms/molecules/ segments can be described by an attractive potential: watt(r) = -C/rn where C and n are constantsThere is a repulsion because of the Pauli exclusion principle: electrons cannot occupy the same energy levels. Treat atoms/molecules like hard spheres with a diameter, s. A simple repulsive potential: wrep(r) = (s/r)The interaction potential w(r) = watt + wreprs

Simple Interaction Potentials+w(r)-

Attractive potentialrwatt(r) = -C/rn

Simple Interaction Potentials+w(r)-

Total potential:rw(r) = watt + wrepsMinimum of potential = equilibrium spacing in a solid = s

Potentials and Intermolecular Force+re = equilibrium spacing

Interaction PotentialsWhen w(r) is a minimum, dw/dr = 0.Solve for r to find equilibrium spacing for a solid, where r = re.Confirm minimum by checking curvature from 2nd derivative.The force between two molecules is F = -dw/drThus, F = 0 when r = re.If r < re, F is compressive (+).If r > re, F is tensile (-).When dF/dr = d2w/dr2 =0, attr.F is at its maximum.Force acts between all neighbouring molecules!

How much energy is required to remove a molecule from the condensed phase?Q: Does a central molecule interact with ALL the others?

Total Interaction Energy, EInteraction energy for a pair: w(r) = -Cr -nVolume of thin shell: Number of molecules at a distance, r:Total interaction energy between a central molecule and all others in the system (from s to L), E:But L >> s! When can we neglect the term?

Conclusions about EThere are two cases:When n>s, then (s/L)n-3>>1 and is thus significant.In this case, E varies with the size of the system, L! But when n>3, (s/L)n-3

Interaction PotentialsGravity: acts on molecules but negligibleCoulomb: applies to ions and charged molecules; same equations as in electrostaticsvan der Waals: classification of interactions that applies to non-polar and to polar molecules (i.e. without or with a uniform distribution of charge). IMPORTANT in soft matter!We need to consider: Is n>3 or

Gravity: n = 1rm1m2G = 6.67 x 10-11 Nm2kg-1When molecules are in contact, w(r) is typically ~ 10-52 JNegligible interaction energy!

Coulombic Interactions: n = 1 With Q1 = z1e, where e is the charge on the electron and z1 is an integer value. eo is the permittivity of free space and e is the relative permittivity of the medium between ions (can be vacuum with e = 1 or can be a gas or liquid with e > 1). The interaction potential is additive in crystals. When molecules are in close contact, w(r) is typically ~ 10-18 J, corresponding to about 200 to 300 kT at room temp

van der Waals Interactions (London dispersion energy): n = 6ra1a2u2u1 Interaction energy (and the constant, C) depends on the dipole moment (u) of the molecules and their polarisability (a). When molecules are in close contact, w(r) is typically ~ 10-21 to 10-20 J, corresponding to about 0.2 to 2 kT at room temp., i.e. of a comparable magnitude to thermal energy! v.d.W. interaction energy is much weaker than covalent bond strengths.

Covalent Bond EnergiesFrom Israelachvili, Intermolecular and Surface Forces1 kJ mol-1 = 0.4 kT per molecule at 300 KHomework: Show why this is true.Therefore, a C=C bond has a strength of 240 kT at this temp.!

Hydrogen bondingIn a covalent bond, an electron is shared between two atoms.Hydrogen possesses only one electron and so it can covalently bond with only ONE other atom.The proton is unshielded and makes an electropositive end to the bond: ionic character.Bond energies are usually stronger than v.d.W., typically 25-100 kT.The interaction potential is difficult to describe but goes roughly as r-2, and it is somewhat directional. H-bonding can lead to weak structuring in water.HOHHHOd+d+d+d+d-d-

Hydrophobic InteractionsForeign molecules in water can increase the local ordering - which decreases the entropy. Thus their presence is unfavourable.Less ordering of the water is required if two or more of the foreign molecules cluster together: a type of attractive interaction.Hydrophobic interactions can promote self-assembly.A water cage around another molecule