1. Soft Matter and Statistical Physics 3SMS 16 January, 2007 Lecture 1: Introduction to Soft Matter

2. 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 carbon dioxide Image: http://wps.prenhall.com/wps/media/objects/602/616516/Chapter_10.html

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

4. Condensed Matter and Origin of Surface Tension From 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. Meniscus Increasing density Liquids and gases are separated by a meniscus; they differ only in density but not structure (i.e. arrangement of molecules in space).

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

6. Soft Condensed Matter Refers to condensed matter that exhibits characteristics of both solids and liquids The 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 + elastic Examples: rubbers, gels, pastes, creams, paints, soaps, liquid crystals, proteins, cells

7. Types of Soft Matter: Colloids A colloid is a sub-  m 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 NameExamples L/SG aerosol fog, hair spray; smoke GL/S foam beer froth; shaving foam; poly(urethane) foam LL (S) emulsion mayonnaise; salad dressing SL sol latex paint; tooth paste SS solid suspension pearl; mineral rocks

8. Interfacial Area of Colloids r For 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. Consider a 1 cm 3 phase dispersed in a continuous medium: No. particlesParticle volume(m 3 ) Edge length (m) Total surface area(m 2 ) 110 -6 10 -2 0.0006 10 3 10 -9 10 -3 0.006 10 6 10 -12 10 -4 0.06 10 9 10 -15 10 -5 0.6 10 12 10 -18 10 -6 6.0 10 15 10 -21 10 -7 60 10 18 10 -24 10 -8 600

9. Types of Soft Matter: Polymers A 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. Physicist’s view of a polymer:

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

11. 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 Crystals This 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

12. Interfacial tension,  Typical  values for interfaces with water - carbon tetrachloride: 45 mN/m; benzene: 35 mN/m; octanol: 8.5 mN/m Work (W) is required to increase the interfacial area (A): “oil” water Types of Soft Matter: Surfactants Surfactants reduce . Are used to make emulsions and to achieve “self assembly” (i.e. spontaneous organisation) A surfactant (surface active agent) molecule has two ends: a “hydrophilic” one (attraction to water) and a “hydrophobic” (not attracted to water) one. emulsion

13. Hydrophobicity and Hydrophilicity water solid  Hydrophilic water solid  Hydrophobic  is small  is large solid water Fully wetting

14. Contact Angle: Balance of Forces Three interfaces: solid/water (sw); water/air (wa); solid/air (sa) Each interface has a surface tension:  sw ;  wa ;  sa  sa  wa  sw  Contact angles thus provide information on surface tensions and the effect of surfactants. At equilibrium, tensions must balance:

15. Acrylic Latex Paint Monodisperse Particle Size Vertical scale = 200nm (1) Length scales between atomic and macroscopic Top view 3  m x 3  m scan Characteristics of Soft Matter (4 in total) Example of colloidal particles

16. Typical Length Scales Atomic spacing: ~ 0.1 nm “Pitch” of a DNA molecule: 3.4 nm Diameter of a surfactant micelle: ~6-7 nm Radius of a polymer molecule: ~10 nm Diam. of a colloidal particle (e.g. in paint): ~200 nm Bacteria cell: ~2  m Diameter of a human hair: ~80  m

17. 15  m x 15  m Poly(ethylene) crystalCrystals of poly(ethylene oxide) 5  m x 5  m

18. Intermediate Length Scales Mathematical 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.

19. (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.

20. Thermal fluctuations Soft 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. VxVx VyVy VzVz V The kinetic energy for a particle of mass, m, is 1/2 m v 2 = 3/2 kT. When m is small, v becomes significant.

21. (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”

22. Poly(styrene) and poly(methyl methacrylate) copolymer 2  m x 2  m Layers or “lamellae” form spontaneously in diblock copolymers. Diblock copolymer Polymer Self-Assembly

23. Colloidosomes: Self-assembled colloidal particles A.D. Dinsmore et al., “Colloidosomes: Selectively Permeable Capsules Composed of Colloidal Particles,” Science, 298 (2002) p. 1006. Liquid B Liquid A Colloidal particles (<1  m)

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

25. MRS Bulletin, Feb 2004, p. 86 Colloidal Crystals Colloidal 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.

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

27. Examples of Self-Assembly Surfactants 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 Matter The “plumber’s nightmare”

28. Competitions in Self-Assembly Molecules 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.

29. Importance of Interfaces Free energy change: dF =  dA An increase in area raises the system’s free energy, which is not thermodynamically favourable. So, sometimes interfacial tension opposes and destroys self-assembly. An example is coalescence in emulsions.

30. Particle Coalescence Surface area of N particles: 4N  r 2 Surface area of particle made from coalesced particles: 4  R 2 Same particle volume before and after coalescence: R r Change in area,  A = - 4  r 2 (N-N 2/3 ) In 1 L of emulsion (50% dispersed phase), with a droplet diameter of 200 nm, N is ~ 10 17 particles. Then  A = -1.3 x 10 4 m 2 With  = 3 x 10 -2 J m -2,  F =  A = - 390 J.

31. (4) Short-range forces and interfaces are important. Characteristics of Soft Matter (4 in total) The structure of a gecko’s 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)

32. 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

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

34. Interaction Potentials Interaction between two atoms/molecules/ segments can be described by an attractive potential: w att (r) = -C/r n where C and n are constants There 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, . A simple repulsive potential: w rep (r) = (  /r)  The interaction potential w(r) = w att + w rep r 

35. Simple Interaction Potentials + w(r) - Attractive potential r w att (r) = -C/r n + w(r) - Repulsive potential r  w rep (r) = (  /r) 

36. Simple Interaction Potentials + w(r) - Total potential: r w(r) = w att + w rep  Minimum of potential = equilibrium spacing in a solid = 

37. Potentials and Intermolecular Force + r e = equilibrium spacing

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

39. r How much energy is required to remove a molecule from the condensed phase? Q: Does a central molecule interact with ALL the others? Applies to pairs L  = molecular spacing  = #molec./vol. Individual molecules 

40. Total Interaction Energy, E Interaction energy for a pair: w(r) = -Cr -n Volume of thin shell: Number of molecules at a distance, r: Total interaction energy between a central molecule and all others in the system (from  to L), E: E= But L >>  When can we neglect the term?  L Entire system r -n+2 =r -(n-2)

41. Conclusions about E There are two cases: When n > , then (  /L) n-3 >>1 and is thus significant. In this case, E varies with the size of the system, L! But when n>3, (  /L) n-3 <<1 and can be neglected. Then E is independent of system size, L. When n>3, a central molecule is not attracted strongly by ALL others - just its closer neighbours! E=

42. Interaction Potentials Gravity : acts on molecules but negligible Coulomb : applies to ions and charged molecules; same equations as in electrostatics van 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 <3?

43. Gravity: n = 1 r mm m2m2 G = 6.67 x 10 -11 Nm 2 kg -1 When molecules are in contact, w(r) is typically ~ 10 -52 J Negligible interaction energy!

44. Coulombic Interactions: n = 1 r QQ Q2Q2 With Q 1 = z 1 e, where e is the charge on the electron and z 1 is an integer value.  o is the permittivity of free space and  is the relative permittivity of the medium between ions (can be vacuum with  = 1 or can be a gas or liquid with  > 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

45. van der Waals Interactions (London dispersion energy): n = 6 r  22 u2u2 u1u1 Interaction energy (and the constant, C) depends on the dipole moment (u) of the molecules and their polarisability (  ). 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.

46. Covalent Bond Energies From Israelachvili, Intermolecular and Surface Forces 1 kJ mol -1 = 0.4 kT per molecule at 300 K Homework: Show why this is true. Therefore, a C=C bond has a strength of 240 kT at this temp.!

47. Hydrogen bonding In 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. H O H H H O ++ ++ ++ ++ -- --

48. Hydrophobic Interactions “Foreign” 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