Soft Matter Physics
11 February, 2010
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
a 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
Phase diagram of water
Image: http://wps.prenhall.com/wps/media/objects/602/616516/Chapter_10.html
Condensed Matter and the Origin
of Surface Tension
Meniscus
Increasing
density
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)
G cosq  F
F
d
d
Gq
Interface with air = “surface”
For mercury, G = 0.486 N/m
Mercury has a very high surface energy!
For water, G = 0.072 N/m
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
Hydrophobicity and Hydrophilicity
Fully wetting
water
solid
water
q
Hydrophilic
solid
water
solid
q is <90
q
http://scottosmith.com/2007/10/03/water-beads/
Hydrophobic
q is >90
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
At equilibrium, lateral tensions must balance:
Gsa - Gsw
Gsa  Gsw  Gwa cos q ⇒
 cos q
Gwa
Contact angle measurements thus provide information on surface
tensions.
Gsa
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, tissue, humans?
Types of Soft Matter: Colloids
• A colloid consists of sub-mm particles (but not single molecules) of one
phase dispersed in a continuous phase.
• The size scale of the dispersed phase is between 1 nm and 1 mm.
• The dispersed phase and the continuous phases 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
sol
S
S
solid suspension
emulsion
mayonnaise; salad dressing
latex paint; tooth paste
pearl; mineral rocks
There is no “gas-in-gas” colloid, because there is no interfacial tension
between gases!
Interfacial Area of Colloids
For a spherical particle, the ratio of surface area (A) to
volume (V) is:
2
A 4r
1
=
≈
3
4
V
r

r
3
r
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.
Physicist’s view of a
polymer:
•
•
•
Each “pearl” on the string represents a repeat unit of atoms, linked together by
strong covalent bonds. For instance, in a protein molecule the repeat units are
amino acids. Starch consists of repeat units of sugar.
The composition of the “pearls” is not important (for a physicist!).
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 (three-dimensional array).
Image: http://wps.prenhall.com/wps/media/objects/602/616516/Chapter_10.html
This form of soft matter is interesting and useful because of its
anisotropic optical and mechanical properties.
Characteristics of Soft Matter (4 in total)
(1) Length scales between atomic and macroscopic
Top view
3 mm x 3 mm scan
Vertical scale = 200nm
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
Typical Length Scales
Poly(ethylene) crystal
15 mm x 15 mm
Crystals of poly(ethylene oxide)
5 mm x 5 mm
Polymer crystals can grow up to millimeters in size!
Spider Silk: An Example of a Hierarchical Structure
Amino acid units
P. Ball, Nanotechnology (2002) 13, R15-R28
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
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, the 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).
Vz
V
• kT is a useful “gauge” of bond energy.
Vy
The kinetic energy for a particle of mass, m, is 1/2
mv2 = 3/2 kT. 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 nano-scale. 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, kS, of ~ 10 N/m.
• The energy required for deflection of the beam by a distance X is Ed
= ½ kSX 2.
• At a temperature of 300 K, the thermal energy, E, is on the order of
kT = 4 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.
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
2mm x 2mm
Poly(styrene) and poly(methyl
methacrylate) copolymer
Layers or “lamellae” form spontaneously in diblock copolymers.
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
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
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.
Surfactants at Interfaces
emulsion
“oil”
water
Interfacial tension, G
Work (W) is required to increase the
Typical G values for interfaces with water carbon tetrachloride: 45 mN/m; benzene: 35
mN/m; octanol: 8.5 mN/m
interfacial area (A):
∫
W = GdA
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)
Examples of Self-Assembly
(a)
(b)
Spherical end is hydrophilic.
(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
Surfactant micelles
are packed together
SiO2 (silica) is grown
around the micelles
P. Ball, Nanotechnology (2002) 13, R15-R28
Micelles are removed to
leave ~ 10 nm spherical
holes. Structure has low
refractive index. Can be
used as a membrane.
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 can dominate.
If the 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
• Work associated with changing an interfacial area:
dW = GdA
• Doing work on a system will raise its internal energy (U)
and hence its free energy (F).
• 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.
Coalescence in Emulsions
Liquid droplet volume before and after coalescence:
r
R
Surface area of N particles:
4Nr2
Surface area of droplet made from
coalesced droplets: 4R2
Change in area, DA = - 4r2(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.
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 ~ kT.
• 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. Is not easy 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:
va
Re 

(Compares the effects of inertia (momentum) to viscous drag)
a
 = density of the
continuous medium
V = velocity
 = viscosity of the
continuous medium
When Re is low, the 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.
http://physicsworld.com/cws/article/print/19961
What are the 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:
dw
F 
dr
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 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, attractive 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
w (r ) = C
r
n
s = molecular
spacing
Applies
to pairs
r
•
s
 = #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
Volume of thin shell:
-n
v = 4r 2dr
2
N
(
r
)
=

(
4

r
dr )
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
E = w (r )N (r ) = 4C r
r L
4C 1
E
(n 3) r n 3 r s
But L >> s!
s
E=
n +2
dr
4C
(n
[
n 3 1
3)s
r -n+2=r -(n-2)
(s L )n
When can we neglect the term?
3
]
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=
4C
(n 3)s n
s )n
1
(
3
L
[
3
]
4C
≈
(n 3)s n
3
The Case of n = 3
w (r )  Cr
3
N (r )  4r dr
2
r L
4Cdr
E  
 4Cln L  ln s 
r
r s
s will be very small (typically 10-9 m), but lns is not
negligible. L cannot be neglected in most cases.
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
m2
m1
r
Gm1m2
w (r ) =
r
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
Q2
Q1
r
Q1Q2
w (r ) =
4 or
Sign of w depends on whether charges are
alike or opposite.
• 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).
• When molecules are in close contact, w(r) is typically
~ 10-18 J, corresponding to about 200 to 300 kT at room temp
• The interaction potential is additive in crystals.
van der Waals Interactions
(London dispersion energy): n = 6
u2
a2
u1
a1
r
w (r ) =
C
( 4 o )r
6
• 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 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
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 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.
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.