Soft Matter Physics 3 SM 1618 January 2008

Soft Matter Physics 3 SM 16/18 January, 2008 Lecture 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 ~ k. T. Phase diagram of carbon dioxide Image: http: //wps. prenhall. com/wps/media/objects/602/616516/Chapter_10. html

Condensed Matter and Origin of Surface Tension Increasing density Meniscus From I. W. Hamley, Introduction to Soft Matter Liquids and gases are separated by a meniscus; they differ only in density but not structure (i. e. arrangement of molecules in space). • 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.

Interfacial Energy An interfacial energy G is associated with the interface between two phases (units of Jm-2) (also called an interfacial tension: Nm-1) F d Gq Interface with air = “surface” For mercury, G = 0. 486 N/m For water, G = 0. 072 N/m Mercury has a very high surface energy! For ethanol, G = 0. 022 N/m What characteristics result from a high surface energy? Image: http: //wps. prenhall. com/wps/media/objects/602/616516/Chapter_10. html

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

Types of Soft Matter: Colloids • A 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 Name Examples L/S G aerosol fog, hair spray; smoke G L/S foam beer froth; shaving foam; poly(urethane) foam L L (S) S L S S emulsion solid suspension mayonnaise; salad dressing latex paint; tooth paste pearl; mineral rocks

Interfacial Area of Colloids For a spherical particle, the ratio of surface area (A) to volume (V) is: r 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. particles Particle volume(m 3) Edge length (m) Total surface area(m 2) 1 10 -6 10 -2 0. 0006 103 10 -9 10 -3 0. 006 10 -12 10 -4 0. 06 109 10 -15 10 -5 0. 6 1012 10 -18 10 -6 6. 0 1015 10 -21 10 -7 60 1018 10 -24 10 -8 600

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

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. Physicist’s view of a polymer: • 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.

Types of Soft Matter: Liquid Crystals • 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 (threedimensional array). Image: http: //wps. prenhall. com/wps/media/objects/602/616516/Chapter_10. html This form of soft matter is interesting because of its anisotropic optical and mechanical properties.

Types of Soft Matter: Surfactants emulsion “oil” water Interfacial tension, G Work (W) is required to increase Typical G values for interfaces with water - carbon tetrachloride: 45 m. N/m; benzene: 35 m. N/m; octanol: 8. 5 m. N/m the interfacial area (A): A surfactant (surface active agent) molecule has two ends: a “hydrophilic” one (attraction to water) and a “hydrophobic” (not attracted to water) one. Surfactants reduce G. Are used to make emulsions and to achieve “self assembly” (i. e. spontaneous organisation)

Hydrophobicity and Hydrophilicity water Fully wetting solid q water Hydrophilic q is small solid water solid q Hydrophobic q is large

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

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

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 mm • Diameter of a human hair: ~80 mm

Poly(ethylene) crystal 15 mm x 15 mm Crystals of poly(ethylene oxide) 5 mm x 5 mm

Spider Silk: An Example of a Hierarchical Structure Amino acid units

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.

Characteristics of Soft Matter (2) The importance of thermal fluctuations and Brownian motion can be though of as resulting from a slight imbalance of momentum being transferred between liquid molecules and a colloidal particle.

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 1/2 k. T of thermal energy. • For a colloidal particle able to undergo translation in the x, y and z directions, thermal energy is 3/2 k. T. • k = 1. 38 x 10 -23 JK-1, so k. T = 4 x 10 -21 J per molecule at room temperature (300 K). Vz V • k. T is a useful “gauge” of bond energy. Vy The kinetic energy for a particle of mass, m, is 1/2 mv 2 = 3/2 k. T. When m is small, v becomes significant. Vx

Thermal motion of a nano-sized beam • In atomic force microscopy, an ultra-sharp tip on the end of a silicon cantilever beam is used to probe a surface at the nanoscale. By how much is the beam deflected by thermal motion? 100 mm x 30 mm x 2 mm X • For AFM applications, the cantilever beam typically has a spring constant, k. S, of ~ 10 N/m. • The energy required for deflection of the beam by a distance X is E = ½ k. SX 2. • At a temperature of 300 K, thermal energy is on the order of k. T = 6 x 10 -21 J. • This energy will cause an average deflection of the beam by X = (2 E/k. S)0. 5 1 x 10 -7 m or 100 nm.

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

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

Designed Nanostructures from DNA Strands of DNA only bind to those that are complementary. DNA can be designed so that it spontaneously creates desired structures. N C Seeman 2003 Biochemistry 42 7259 -7269

DNA Base Pairs Adenine (A) complements thymine (T) with its two H bonds at a certain spacing. Guanine (G) complements cytosine (C) with its three H bonds at different spacings. Example of DNA sequence: ATCGAT TAGCTA

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

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

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

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

Examples of Self-Assembly The “plumber’s nightmare” From RAL Jones, Soft Condensed Matter • 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.

Materials with controlled structure obtained through self-assembly Micelles are packed together Si. O 2 (silica) is grown around the micelles Micelles are removed to leave ~ 10 nm spherical holes. Structure has interesting optical properties, e. g. photonic band gap.

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 attractive and repulsive interactions between molecules that dominate. If free energy decreases (DF < 0), then the process is spontaneous. DF = DU - TDS Internal Energy (U) decrease is favourable Entropy (S) increase is favourable

Importance of Interfaces • Free energy change: d. F = Gd. A • 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.

Particle Coalescence Same particle volume before and after coalescence: r Surface area of N particles: 4 Npr 2 R Surface area of particle made from coalesced particles: 4 p. R 2 Change in area, DA = - 4 pr 2(N-N 2/3) 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 m 2 With G = 3 x 10 -2 J m-2, DF =GDA = - 390 J.

Characteristics of Soft Matter (4) Short-range forces and interfaces are important. From Materials World (2003) 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.

Chemical Bonds in Soft Matter • 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 ~ k. T. • Hence, bonds are easily broken and re-formed. • The strength of molecular interactions (e. g. charge attractions) decays with distance, r. • At nm distances, they become significant. r

Nanotechnology Science Fact or fiction? A vision of “nanorobots” travelling through the a blood vessel to make repairs (cutting and hoovering!). http: //physicsworld. com/cws/article/print/19961 An engine created by downscaling a normal engine to the atomic level K Eric Drexler/Institute for Molecular Manufacturing, www. imm. org.

Key Limitations for Nanorobots and Nanodevices (1) Low Reynolds number, Re : viscosity is dominant over inertia. (2) Brownian and thermal motion: there are no straight paths for travel and nothing is static! (Think of the AFM cantilever beam. ) (3) Attractive surface forces: everything is “sticky” at the nano-scale. Lubrication is needed to slide one surface over another. Why not make use of the length scales and self assembly of soft matter?

Viscous Limitation for “Nanorobot Travel” Reynolds’ Number: (Compares the effects of inertia (momentum) to viscous drag) a r = density of the continuous medium V = velocity h = viscosity of the continuous medium When Re is low, viscosity dominates over inertia. There is no “coasting”!

Alternative Vision of a Nano-Device Closed state: K+ cannot pass through Open state: K+ can pass through A channel that allows potassium ions to pass through a cell membrane but excludes other ions. The nanomachine can be activated by a membrane voltage or a signalling molecule. Flexible molecular structure is not disrupted by thermal motion.

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

Interaction Potentials s r • Interaction between two atoms/molecules/ segments can be described by an attractive potential: watt(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, s. A simple repulsive potential: wrep(r) = (s/r) • The interaction potential w(r) = watt + wrep

Simple Interaction Potentials + Attractive potential w(r) r - watt(r) = -C/rn + w(r) - Repulsive potential s r wrep(r) = (s/r)

Simple Interaction Potentials + w(r) - s r Total potential: w(r) = watt + wrep Minimum of potential = equilibrium spacing in a solid = s The force acting on particles with this interaction energy is:

Potentials and Intermolecular Force + re = equilibrium spacing

Interaction Potentials • When 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 2 nd derivative. • The force between two molecules is F = -dw/dr • Thus, F = 0 when r = re. • If r < re, F is compressive (+). • If r > re, F is tensile (-). • When d. F/dr = d 2 w/dr 2 =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? Individual molecules s = molecular spacing Applies to pairs r • s r = #molec. /vol. L Q: Does a central molecule interact with ALL the others?

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 s to L), E: Entire system L dr r-n+2=r-(n-2) s E= But L >> s! When can we neglect the term?

Conclusions about E • There are two cases: • When n<3, then the exponent is negative. As L>>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<<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=

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?

Gravity: n = 1 m 2 m 1 r 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!

Coulombic Interactions: n = 1 Q 2 Q 1 r Sign of w depends on whether charges are alike or opposite. • With Q 1 = z 1 e, where e is the charge on the electron and z 1 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). • When molecules are in close contact, w(r) is typically ~ 10 -18 J, corresponding to about 200 to 300 k. T at room temp • The interaction potential is additive in crystals.

van der Waals Interactions (London dispersion energy): n = 6 u 2 a 2 u 1 a 1 r • 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 k. T 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 Energies From Israelachvili, Intermolecular and Surface Forces 1 k. J mol-1 = 0. 4 k. T per molecule at 300 K Homework: Show why this is true. Therefore, a C=C bond has a strength of 240 k. T at this temp. !

Hydrogen bonding d- O H d+ d- Hd+ H O d+ Hd+ • 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 k. T. • 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.

Hydrophobic Interactions A water “cage” around another molecule • “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.

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