soft matter physics 3sm 16/18 january, 2008 lecture 1: introduction to soft matter

Soft Matter Physics

3SM

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 ~ kT.

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

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.

MeniscusIncreasing density

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!


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

Interfacial Energy

F

dF cos

d

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-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 Name Examples

L/S G aerosol fog, hair spray; smoke

G L/S foam beer froth; shaving foam; poly(urethane) foam

L L (S) emulsion mayonnaise; salad dressing

S L sol latex paint; tooth paste

S S solid suspension pearl; mineral rocks

Interfacial Area of Colloids

r

For a spherical particle, the ratio of surface area (A) to volume (V) is:

rr

rVA 1

≈3

44

=3

2

Thus, with smaller particles, the interface becomes more significant. A greater fraction of molecules is near the surface.

Consider a 1 cm3 phase dispersed in a continuous medium:

No. particles Particle volume(m3) Edge length (m) Total surface area(m2)

1 10-6 10-2 0.0006

103 10-9 10-3 0.006

106 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.

• 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:

• 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.


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):

∫= dAW

“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

Hydrophobicity and Hydrophilicity

water

solid

Hydrophilic

water

solid

Hydrophobic

is small

is large

solid

waterFully wetting

Contact Angle: Balance of ForcesThree 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:

cos=⇒cos+=wa

swsawaswsa

Acrylic Latex Paint Monodisperse Particle Size

Vertical scale = 200nm

(1) Length scales between atomic and macroscopic

Top view3 m x 3 m scan

Characteristics of Soft Matter (4 in total)

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 m• Diameter of a human hair: ~80 m

15 m x 15 m

Poly(ethylene) crystal Crystals of poly(ethylene oxide)

5 m x 5 m

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.

(2) The importance of thermal fluctuations and Brownian motion

Characteristics of Soft Matter

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 1/2kT 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.

Vx

Vy

Vz V

The kinetic energy for a particle of mass, m, is 1/2 mv2 = 3/2 kT. When m is small, v becomes significant.

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 nano-scale. By how much is the beam deflected by thermal motion?

• For AFM applications, the cantilever beam typically has a spring constant, kS, of ~ 10 N/m.

• The energy required for deflection of the beam by a distance X is E = ½ kSX 2.

• At a temperature of 300 K, the thermal energy is on the order of kT = 6 x10-21 J.

• This energy will cause an average deflection of the beam by X = (2E/kS)0.5 1 x 10-7 m or 100 nm.

X100 m x 30m x 2 m

(3) Tendency to self-assemble into hierarchical structures (i.e. ordered on size scales larger than molecular)


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”

Poly(styrene) and poly(methyl methacrylate) copolymer

2m x 2m

Layers or “lamellae” form spontaneously in diblock copolymers.

Diblock copolymer

Polymer Self-Assembly

AFM image

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

ATCGAT TAGCTA

Example of DNA sequence:

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.

DNA Base Pairs

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)

I. Karakurt et al., Langmuir 22 (2006) 2415

Hydrophilically-driven self-assembly of particles

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.

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)

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”

Materials with controlled structure obtained through self-assembly

Micelles are packed together

SiO2 (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 are attractive and repulsive interactions between molecules that dominate.

F = U - TS

If free energy decreases (F < 0), then the process is spontaneous.

Entropy (S) increase is favourable

Internal Energy (U) decrease is favourable

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.

Particle Coalescence

Surface area of N particles:

4Nr2

Surface area of particle made from coalesced particles:

4R2

Same particle volume before and after coalescence:

Rr

Change in area, A = - 4r2(N-N2/3)

In 1 L of emulsion (50% dispersed phase), with a droplet diameter of 200

nm, N is ~ 1017 particles. Then A = -1.3 x 104 m2

With = 3 x 10-2 J m-2, F =A = - 390 J.

(4) Short-range forces and interfaces are important.


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)

• 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

Nanotechnology Science Fact or fiction?

A vision of “nanorobots” travelling through the a blood vessel to make repairs (cutting and hoovering!). An engine created by down-

scaling a normal engine to the atomic level

http://physicsworld.com/cws/article/print/19961K Eric Drexler/Institute for Molecular Manufacturing, www.imm.org.

http://www.imm.org/

(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.

Key Limitations for Nanorobots and Nanodevices

Why not make use of the length scales and self assembly of soft matter?

vaRe

V = velocitya

= viscosity of the continuous medium

= density of the continuous medium

Reynolds’ Number:

When Re is low, viscosity dominates over inertia. There is no “coasting”!

Viscous Limitation for “Nanorobot Travel”

(Compares the effects of inertia (momentum) to viscous drag)

Alternative Vision of a Nano-Device

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.

Closed state: K+ cannot pass through

Open state: K+ can pass through

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

Interaction Potentials

• 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, . A simple repulsive potential: wrep(r) = (/r)

• The interaction potential w(r) = watt + wrep

r

Simple Interaction Potentials

+

w(r)

-

Attractive potential

r

watt(r) = -C/rn

+

w(r)

-

Repulsive potential

rwrep(r) = (/r)

Simple Interaction Potentials

+

w(r)

-

Total potential:r

w(r) = watt + wrep

Minimum of potential = equilibrium spacing in a solid =

The force acting on particles with this interaction energy is:

drdw

F

Potentials and Intermolecular Force

+

re = equilibrium spacing

• 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 2nd 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 dF/dr = d2w/dr2 =0, attr.F is at its maximum.• Force acts between all neighbouring molecules!


r

How much energy is required to remove a molecule from the condensed phase?

Q: Does a central molecule interact with ALL the others?

nrCrw =)(

Applies to pairs

L

= molecular spacing

= #molec./vol.

Individual molecules

•

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:

drrv 24=)(=)( drrrN 24

Lr

rnrn

CE

3

13

4)(

[ ]33 1

3

4 nn Ln

C)(

)(

E=

But L >> When can we neglect the term?

24 +=)()(= nrCrNrwE

LEntire system

r-n+2=r-(n-2) dr

Conclusions about E

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

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!

[ ]3

33 )3(

4≈)(1

)3(

4n

nn n

CLn

C

E=


• 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

r

mm2

rmGm

rw 21=)(

G = 6.67 x 10-11 Nm2kg-1

When molecules are in contact, w(r) is typically ~ 10-52 J

Negligible interaction energy!

Coulombic Interactions: n = 1

r

QQ2 rQQ

rwo4

21=)(

• With Q1 = z1e, where e is the charge on the electron and z1 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

Sign of w depends on whether charges are alike or opposite.

van der Waals Interactions (London dispersion energy): n = 6

r

2

64 r

Crw

o )(=)(

u2 u1

• 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.

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.!

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.

HO

HH

HO

+

+

++

--

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

Phase diagram of water


soft matter physics 3sm 16/18 january, 2008 lecture 1: introduction to soft matter

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