Week Chapter Week Chapter

Mar. 04, 06 General Overview & Perspectives May 01 Adsorptions, Micelles, Membranes

Mar. 11, 13 Chap. 1: Various Interactions May 06, 08 Chap. 2: Polymers; Conformation

Mar. 18, 20 Phase Transitions, Scaling Laws May 13, 15 Plastics, Rubbers, Copolymers

Mar. 25, 27 Chap. 3: Colloids; Stabilization May 20, 22 Take-Home Exam (Midterm II) Chap. 6: Biological Matters; Lipids

Apr. 01, 03 Emulsions, Dispersions May 27, 29 Proteins, Assemblies

Apr. 08, 10 Chap. 5: Liquid Crystals; Phases Jun. 03, 05 Special Topics on Soft Matters

Apr. 15, 17 Midterm Exam I Elastic Properties, Applications

Jun. 10, 12 Term Paper (Final Exam)

Apr. 22, 24 Chap. 4: Amphipiles: Types, Layers

2013. Spring: Introduction to Electro-Physics (Prof. Sin-Doo Lee, Rm. 301-1109); http://mipd.snu.ac.kr Introduction to Soft Matter (Revised Edition) Ian W. Hamely (John Wiley, Hoboken, 2007)

Chap. 1. Introduction

1.1. Introduction

- 19th century: the age of iron and steel (hard materials) -> 20th century: new types of engineering materials (soft materials), (ex) polymers in the form of plastics

- What are soft materials? polymers, colloids, amphiphiles (surfactants), liquid crystals, biomaterials

- Inspired by nature, complex structures formed by biopolymers self-organization of polymers

- Soft materials: an interdisciplinary subject: physics, chemistry, material science, biochemistry, chemical and mechanical engineering

=> understanding of thermodynamics and dynamics of soft materials

(thermodynamic and statistical principles)

1.2. Intermolecular Interactions

- "soft": macroscopic mechanical properties, flow under certain conditions, weak ordering due to lack of 3-dim. atomic long-range order, but a certain degree of local order at least.

- What is responsible for the ordering of soft materials? a simple description of a balance between repulsive interactions at short distances and attractive interactions over larger length-scales.

- Repulsive interactions: steric or excluded volume interactions (ex1) Hard sphere model; ∞ ≤

[Homework] Calculate the excluded volume of two hard rods which accounts for the orientational ordering of liquid crystals (Onsager model)

(ex2) A self-avoiding random walk for the conformation of a polymer chain with segments by excluded volume interactions (Flory lattice model)

- Inclusion of attractive interactions: typically varies with. for repulsive term varying with (often n=12).

=> Lennard-Jones (12, 6) potential

with the depth of the potential energy minimum the intermolecular separation.

For exponential decay of atomic orbitals at large distances, form used.

- Attractive interactions in uncharged molecules: van der Waals forces (dipolar origin) 1 debye (D) = 3.336 x Cm for small molecules

- Dipolar molecules (like H20) can also induce dipole moments in other molecules producing dipole-induced dipole forces.

1.3. Structural Organization

- Common and distinct features in the ordering of different types of "soft" materials: *Common: 1) intermediate bet a crystalline solid and a liquid. partial translational and/or orientational order of molecules

(ex) mesophase by LC, amphiphiles in water, polymer melts and solutions

-> high viscosity and/or viscoelasticity different from conventional liquids

-> numerous defects such as lattice dislocations or disclinations

2) structural periodicity in the range of 1-103 nm: nanoscale ordering

-> mesoscopic ordering bet microscopic (atomic) and macroscopic scale

- Nematic, lamellar, hexagonal, cubic phases: (orientational. 1-D, 2-D, 3-D translational order)

*Distinct:direct ordering of molecules and "indirect" ordering of molecules via supermolecular aggregates

-> The molecules within the aggregate do not have the same orientational and translational order as the mesoscopic structure itself; relatively disordered.

1.4. Dynamics

- Macromolecules, colloidal particles, and micelles undergo Brownian motion: random forces from thermal motions of the surrounding molecules.

- Einstein: the statistics of a random walk.

The root-mean-square displacement at time t: where D = diffusion coeff.

For not too high velocity, a frictional (or drag) force in a medium where f is the frictional coeff. and v is the velocity of the a particle.

D and f are related to the kinetic energy via where is an estimate of the translational kinetic energy per particle.

For a spherical particle, f is given by Stokes' law: where R = the hydrodynamic radius (the effective radius presented by the particle to the liquid flowing locally around it).

-> Stokes-Einstein eqn for the diffusion of spherical particle:

* Typical values of D: for molecules in liquids ≈

for polymers (larger R), ≈

for micelles diffusing in water at room temp, ≈× with R = 10 nm

- In the presence of a non-equilibrium concentration gradient (constant in time),

Fick's first law:

with j = the flux, c = the number of moles of the substance, dc/dx = the concentration gradient along x.

Under the real situation that the concentration gradient itself changes with diffusion,

Fick's second law:

1.5. Phase Transitions

- Phase transition: 1st order or 2nd order classified by Ehrenfest.

Changes in various thermodynamic properties, in particular, an order parameter:

-> The changes (first derivatives of chemical potential: enthalpy, entropy, volume) are discontinuous at a first order phase transition.

-> A second order transition is characterized by a continuous first derivative of chemical potential but a discontinuous second derivative.

- Modelling of phase transition in terms of the average order parameter in a mean-field theory: the choice of an appropriate order parameter?

*The order parameter for a system is a function of the thermodynamic state of the system and is uniform throughout the system, and at equilibrium, is time-independent.

*van der Waals' equation of state for the liquid-gas transition: a mean-field theory.

- Landau mean field theory: a general model that has broad applicability to phase transitions in soft matters - it is based on an expansion of the free energy in power series of an order parameter

It describes the ordering at the mesoscopic (not molecular) level. (ex) the orientation of an arbitrary molecule surrounded by all others.

- Free energy density in terms of the order parameter near the transition:

⋅⋅⋅

where = the free energy density in the high-temperature phase

- The symmetry of the phase transition under consideration imposes constraints on the number of non-zero terms as well as odd- and even-orders of the order parameter in the free energy expansion.

⋅⋅⋅ with

*Above transition temperature T*, the free energy has a single minimum at . *D(T) is assumed to be weakly temperature dependent, and then to be a constant.

- The equilibrium state:

(minimization)

-> or ±

∝

where the exponent 1/2 is the characteristic of a mean field behavior.

Then, = the equilibrium order parameter in the low-temperature phase

[Homework] Inclusion of into the free energy: the first order phase transition

- Beyond the mean-field theory, thermal fluctuations included and treated as a weak perturbation with respect to the average order; but many soft materials are strongly fluctuating, especially, near the phase transition -> MFT is broken down.

1.6. Order Parameters

- Phase transitions in condensed phases are characterized by symmetry changes, ie., by transformation in orientational and translational ordering in the system.

-> said to be symmetry-breaking transitions.

- Order parameter at a point r in the system:

where = the appropriate distribution function for an orientational or translational variable

(ex) orientational order in LC; = the 2nd order Legendre polynomial.

1.7. Scaling Laws

- Soft matter is characterized by complexity, both in structure and dynamics.

-> Scaling laws: how one variable depends on another, holding other quantities constant

∼ ; x scales with y to the power z.

1.8. Polydispersity

- Unlike atoms and natural biomolecules, synthetic soft materials (colloidal particles, polymer chains, and micelles) have distributions of sizes; not monodisperse but polydisperse.

*Polydispersity of polymer chain length leads to phase transition in solutions or blends into phases rich in the small and large polymer species.

*Crystallization of colloidal latex particles with polydispersity.

1.9. Experimental Techniques

1.9.1. Microscopy

- Polarizing optical miscroscopy: useful for identifying birefringent structures, textures rather than microstructure itself.

*the microaggregates formed by polymers, colloidal particles.

- Scanning electron microscopy: electron beam is scanned across an object, knocking the secondary electrons out of its surface atoms -> exterior images of an object

- Transmission electron microscopy: intensity of transmitted electrons is inversely proportional to the electron density of a section of a material, requiring sectioning of a bulk sample into nano-meter thick slices.

*TEM provides a higher resolution of 1 nm than SEM of about 5 nm.

1.9.2. Scattering Methods

1) Light scattering:

- Static light scattering: the intensity of elastically scattered light is measured as a function of the scattering angle.

*Particle size vs the wavelength

; Rayleigh scattering->fluctuations of dielectric constant ; Rayleigh-Debye-Gans regime (complex analysis needed)

∼ ; Mie scattering->refactive index difference bet particles and dispersion medium

- Dynamic (Photo correlation or Quasi-elastic) light scattering: temporal fluctuations of scattered light intensity -> decay rates of the relaxation modes -> diffusion coeff.

2) X-ray scattering: to probe much smaller features than with visible light due to smaller wavelengths of 0.1 nm.

*X-ray diffraction comes from the electron density distribution in a material. *Two types: small angle XS (∼) and wide angle XS.

*For elastic scattering, , and then .

*WAXS provides information on the structures of crystalline polymer via Bragg's law with d = the lattice spacing.

3) Small angle neutron scattering (SANS): neutrons are scattered by atomic nuclei and neutron diffraction depends on the nuclear scattering length density.

- Form factor and structure factor from the intensity distribution of scattered radiation:

*In crystallograpgy, total intensity from interfering waves scattered by single particles and from interparticle interferences for a collection of spherical particles

where G = a constant related to geometrical effects and scattering volume, = the scattering contrast between the polymer and solvent (the difference in electron densities in the case of X-ray scatt.)

= the single particle scattering term (the square of the form factor) = the interparticle scattering term (the structure factor)

-> determine particle size and shape from using approximate models

1.9.3. Rheology

- The deformation and flow of materials: information on the mechanical response to a dynamic stress or strain.

Stress = the force per unit area (Nm-2 or Pa) Strain = the relative change in length of the sample (dimensionless)

- Simple shear deformation:

- The viscosity : kgm-1s-1 or Pa s, 1 poise = 0.1 Pa s. For water, 10-3 Pa s.

- Shear rate: (in units of s-1) *Newtonian fluid: (linear relationship and is indep. of )

-The viscoelasticity of soft materials: In stress relaxation measurements, the strain is held constant and the decay of the stress is monitored as a function of time.

*Under a oscillating strain of the form as , for a viscoelastic system, one component of the stress will be in-phase with the strain (elastic part) and the other out-of-phase with it (viscous part)

′ ″ ≡′″ where in-phase shear modulus ′

′ (storage modulus) and

out-of-phase shear modulus ″ ″ (loss modulus)

-> the loss tangent ′″ ; a measure of the energy loss per cycle

1.9.4. Spectroscopic Methods

1) NMR

2) IR, Raman, dielectric spectroscopies: frequency (or wavelength) range

1.9.5. Others: Calorimetry, AFM

1.10. Computer Simulation

1.10.1. Monte Carlo (MC) Method

- Simulation of the thermodynamic properties a box of particles, interacting through a potential specified by the programmer: at each step, the total potential energy change of the system is calculated.

- A new configuration is accepted or not: for , accepted but for , the probability of acceptance according to , Boltzmann factor

1.10.2. Molecular Dynamics (MD) Method

- Molecules obey Newton's law of motion: a reasonable model for molecular dynamics over short-time scales.

- Compute the position of each particle after a short time step (about 10-15 s)

* A major problem: small errors involved in numerical integration of MD equations can build up many time steps and leads to changes in the kinetic energy, and hence to the average temperature of the system. -> corrected by occasional rescaling of the particle velocity.

1.10.3. Brownian Dynamcis Method

- Random collisions with the surrounding molecules is simulated by a random stochastic force: the medium is effectively treated as a viscous continuum.