Introduction to Soft Matter
Prof.Dr.Ir. J.G.E.M.(Hans) Fraaije
Secretary Mrs. Ferry Soesman
Tel 4523
f.soesman@chem.leidenuniv.nl
http://www.chem.leidenuniv.nl/scm
Course material/downloads!
1
versions
• 1.0 Handout 020903
• 1.1 Embarrassing mistakes removed (thanks
to Jan van Male), clarification ‘level’ and
state’, and extension phase diagrams
180903
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We study: the design, synthesis and
analysis of (bio)macromolecular
assemblies
Applications:
Smart polymeric drug delivery systems
Microgels for genomics
Patterned surface films
Origin of Life
3
What we need
Thermodynamic theory and computer
simulations
Synthesis
Characterization
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Course materials
1.
2.
3.
4.
5.
This presentation (downloadable)
“Introduction to Soft Matter”, Ian Hamley
Handouts Supramolecular Chemistry
Handout Statistical Mechanics (Hill)
Handout Home Soft Lab
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Summary course
• September: statistical thermodynamics, phase
diagrams, dynamics and simulations (8 hrs)
• October: properties colloids, polymers and
amphiphiles (8 hrs)
• November: supramolecules and molecular
building blocks (4 hrs)
• November: demonstration and exercises Home
Soft Lab (4 hrs)
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Motto
“Ich behaupte nur dass in jeder
besonderen Naturlehre nur so viel
eigentliche Wissenshaft angetroffen
könne als darin Mathematic
anzutreffen ist” (Kant)*
Citation from preface “On Growth
and Form”
D’Arcy Wenthworth Thompson
* See next slide
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Metaphysik der Natur: oder sie beschäftigt sich mit einer besonderen Natur
dieser oder jener Art Dinge, von denen ein empirischer Begriff gegeben
ist, doch so, daß außer dem, was in diesem Begriffe liegt, kein anderes
empirisches Princip zur Erkenntniß derselben gebraucht wird (z. B. sie
legt den empirischen Begriff einer Materie, oder eines denkenden Wesens
zum Grunde und sucht den Umfang der Erkenntniß, deren die Vernunft
über diese Gegenstände a priori fähig ist), und da muß eine solche Wissenschaft
noch immer eine Metaphysik der Natur, nämlich der körperlichen
oder denkenden Natur, heißen, aber es ist alsdann keine allgemeine, sondern
besondere metaphysische Naturwissenschaft (Physik und Psychologie),
in der jene transscendentale Principien auf die zwei Gattungen der Gegenstände
unserer Sinne angewandt werden.
Compared to this,
Introduction to Soft
Matter is easy!
Ich behaupte aber, daß in jeder besonderen Naturlehre nur so viel
eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik
anzutreffen ist. Denn nach dem Vorhergehenden erfordert eigentliche
Wissenschaft, vornehmlich der Natur, einen reinen Theil, der dem
empirischen zum Grunde liegt, und der auf Erkenntniß der Naturdinge
a priori beruht. Nun heißt etwas a priori erkennen, es aus seiner bloßen
Möglichkeit erkennen. Die Möglichkeit bestimmter Naturdinge kann aber
nicht aus ihren bloßen Begriffen erkannt werden; denn aus diesen kann
zwar die Möglichkeit des Gedankens (daß er sich selbst nicht widerspreche),
aber nicht des Objects als Naturdinges erkannt werden, welches außer
dem Gedanken (als existirend) gegeben werden kann. Also wird, um die
Möglichkeit bestimmter Naturdinge, mithin um diese a priori zu erkennen,
noch erfordert, daß die dem Begriffe correspondirende Anschauung
a priori gegeben werde, d. i. daß der Begriff construirt werde. Nun ist die
Vernunfterkenntniß durch Construction der Begriffe mathematisch. Also
http://linux-s.ikp.uni-bonn.de/cgibin/Kant/lade.pl?/default.htm
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Polymers, Colloids, Amphiphiles
and Liquid Crystals
• Hard matter versus Soft Matter: scales of
time
• Hard: rocks, metals,…
• Soft: soil, gels, living tissue
• Soft matter is microstructured (1-1000 nm)
• Interdisciplinary: physics, chemistry,
mathematics and biology
Introduction 1.1
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Applications of soft materials
•
•
•
•
•
•
•
•
•
‘everyday’ world
Detergents
Paints
Plastics
Soils
Food
Drug delivery
Cosmetics
All living systems
Introduction 1.1
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Constituents of Soft Materials
•
•
•
•
•
Polymers (chapter 2)
Colloids (chapter 3)
Amphiphiles (chapter 4)
Liquid Crystals (chapter 5)
Supramolecules (hand out)
Introduction 1.1
1-100 nm
10-1000 nm
1-10 nm
1-10 nm
1-10 nm
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Intermolecular Interactions
• Soft materials can often be induced to flow
• Weak ordering due to absence of long range
crystalline order
Introduction 1.2
13
INTERMEZZO
Recapitulation statistical
thermodynamics
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Energy scales
• For different phenomena, we use different
scales
• High-energy physics: (sub-)atomic particle
energy measured in MeV-TeV (mega-terra
electronvolt);
• Atomic quantum states: eV
• Unhuman atomic bomb: tons of TNT (or
‘Hiroshima’ equivalents)
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Energy scales
• Macroscopic human world: energy in Joule
• 1 Joule = 1 Nm = energy required to lift 0.1 kg 1
meter, or lift 1 kg 0.1 meter
• Exercise: Lift Hamley’s book above your head.
How much energy do you need? (the book weighs
436 gram)
• Exercise: Take the stairs to the top of the Gorlaeus
building. Calculate the energy you need.
• Exercise: how much energy is stored in one
sandwich? Can you use all of it?
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Energy Scales
•
•
•
•
Microscopic world: 1 kT (katé)
“T” = Temperature (in Kelvin)
“k” = Boltzmann’s constant = R/Nav
Exercise: how much Joule is 1 kT at room
temperature (T = 300 K)
• Fundamental relation:
When the energy difference is 1 kT,
The probability ratio is:
(A and B same degeneracy)
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Energy scales
•
•
•
•
Hard: intermolecular interaction >> kT
Soft: intermolecular interaction ~kT
Hard: assembly of small things (atoms)
Soft: assembly of large things (polymers,
colloids, amphiphiles, liquid crystals,…)
Introduction 1.2
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Entropy Scales
• What we really need is FREE energy
• Free Energy F= U – TS
• S = klnΩ (Boltzmann, when all states same
energy)
• Ω = multiplication of things you can do
(configurations, at constant energy)
• Ω=Ω (1)*Ω (2)*Ω (3)*…
Notice: we use the symbol F for the Helmholtz energy,
and G for the Gibbs energy
F is free energy, G is free enthalpy G=H-TS
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Recall: why do we need Free Energy???
• Optimise total entropy (natural law)
• This is the same as: minimize free energy
when mechanical work on the system is zero.
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What is the advantage of F?
• The total entropy is sum of system and
environment
• F contains system variables only
• From now on, we will abbreviate
21
Free energy, entropy and energy
are related through derivatives
• Relations:
So, find explicit expression
for F and your are done!
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Entropy Scales
•
•
•
•
(degrees of freedom)
Ω= (some number)
(degrees of freedom)
S= k ln (some number)
S= degrees of freedom*k*ln(some number)
Remember: S/k= “degrees of freedom”
• “k” is the natural scale for the entropy
• In applications, we need to figure out: the value of
“some number” and the value of “degrees of
freedom”
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Theoretical approaches
• Exact, ‘ab initio’, rigorous, lots of equations
• Intuitive, small ‘scaling’ relations, for example
A goes like B2, or A scales like B2
(we do not care about prefactors)
• Approximative
With hand waving
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Statistical Thermodynamics
Exercise: what is the dimension of Q?
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When there is only one level…
(or, equivalent, all configurations
have same energy)
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Road Map Modelling with
Statistical Thermodynamics
•
•
•
•
From molecule, or assembly, or…
Work out the states
Find the energy for each state
Find the degeneracy for each level (the toughest
part)
• Calculate the partition function, by summation
over the levels
• Calculate the free energy
• From free energy, calculate entropy, energy, …
the properties you are interested in
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Exercises Statistical
Thermodynamcis
1.
2.
3.
4.
5.
bond can rotate in three positions, molecule containes 10 such
bonds, what is the molecular entropy? (intramolecular effect)
single molecule moves around in container with volume V (ideal
gas).What is the entropy of the molecule? (effect of freedom of
position)
n molecules in ideal gas. What is the entropy? (effect of the
interchange of particles)
Mix two different molecules (mixing entropy)
molecule can be in two different states, A and B, give formulas for
probability it is in A (effect of different energy levels), the entropy
and the energy
6. Phase diagrams
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Bond rotations
Polymer with 10 bonds, 11 monomers
Assume chain is ideal
3 orientations per bond
(of same energy)
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Single molecule in container
Assume intramolecular interactions
are decoupled from position
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Single molecule in container
How many possible positions?
Method 1: lattice model:
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Single molecule in container
Method II: Assume it is a quantum particle
Particle-wave duality
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n molecules
Start with 8
First label the molecules
1
7
2
3
3
8
4
6
8
5
7
6
1
1
2
5
4
6
5
4
7
2
3
8
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How many ways to label?
Start with empty balls
The Number “8” can be put into 8 different places
The number “7” then in any of the remaining 7
The number “6” then in any of the remaining 6
And so on
The total number is 8*7*6*5*4*3*2*1=8!
For n labels this is n!
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The molecules are
undistinguishable
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How many ways to distribute n
labeled molecules?
Ideal molecules: we do not care about overlap
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How many ways to distribute n
labeled molecules?
Non-ideality due to reduction of available space
Van der Waals
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Two level model
A
B
Energy eV
A molecule can be in two different levels
example
B
0.05
Degeneracy = 4
0
A
Degeneracy = 2
What is the formula for the entropy?
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Two level model entropy
S/k
T
What are the limiting values?
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Advanced topic: Multiple states
entropy formula
Levels:
States:
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Ideal Mixing
+
1
Pure “1”
2
Pure “2”
Mixture 12
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Ideal mixing entropy
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Properties ideal mixing entropy
•
•
•
•
Independent of molecular volume!
Always > 0
It is therefore entropically favourable to mix
Maximum when volume fractions are equal
to 0.5
• The mixing entropy is then
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Do the same, with one extra
molecule
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Continued…
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Limiting mixing cases
Case 1: Molecules have the same size
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Case 2: polymers
Component 1 is solvent,
Component 2 is polymer, with N monomers
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Case 3: collloid, emulsions…
(big things)
Solvent molecule
Colloidal particle
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Now, if we want to calculate the
free energy of mixing…
…we need a model for the mixing energy
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Back to Hamley’s book:
Intermolecular interactions 1.2
Potential energy
Typical molecular interaction curve
repulsion
r
0
distance
attraction
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Different curves…
(check Atkins, Physical Chemistry)
V
V
r
Long range repulsion
Between molecules of same charge,
or neutral flexible molecules
r
Hard core repulsion
(neutral colloids)
V
V
r
Orientation dependent interaction
Between molecular dipoles
r
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Mathematical forms
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Repulsion between atoms
Electron clouds (orbitals) do not like to overlap
(unless a bond is formed, as in a reaction)
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Actually, electron cloud repulsion
is better represented by
From quantum theory
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Hard core repulsion
Like a solid wall!
V
d
r
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Attraction between atoms and molecules
Between permanent
dipoles of
opposite orientation
Between fluctuating dipoles
Dispersion interactions
Every atom has a fluctuating dipole
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Lennard-Jones (12,6) potential
Exercise: what is the relation between
the two sets of parameters?
At which position is the minimum?
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Graphical representation LJ
potential
V
r
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Hierarchy of interactions
•
•
•
•
•
Coulombic ~100-300 kJ/mol
Van der Waals ~1 kJ/mol
Exercise: how much kT is this?
Hydrogen bonding (in water): a few kT
Hydrophobic interactions (in water): a few
kT
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Now we can make a simple
model for mixing energy
Potential energy
Typical molecular interaction curve
repulsion
r
0
distance
attraction
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Mean-field approximation
On average, the concentration around a
given molecule is the same
as the average concentration
We shall assume
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Mean field model interactions
2
1
d
1
1
1
The molecules are separated by a distance d,
And feel the interaction
The number of molecules “2” around central “1”:
2
2
“z” is geometrical factor
(coordination number)
Each contact adds an interaction
Exercise: estimate z
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Mean field interaction
The total interaction between “1” and “2”
Exercise: why the factor ½?
Repeat for 1-1:
And 2-2
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Mixing energy
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Sign of exchange
parameter
11
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Cohesive energy
Typical values χ in the range 0-3
12
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Mixing energy
Case 2: polymers
Assume all polymers are random coils
Monomers exposed to solvent
Solvent exposed to monomers
The connectivity of the monomers is irrelevant
for the mixing energy
We approximate:
Bonds do not matter!
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Mixing energy polymer and
solvent
2
1
1
1
2
1
2
If we thread a polymer through
the interaction shell,
it remains the same
Calculate the interactions on monomer basis
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Mixing free energy polymers
Mixing energy:
Mixing entropy:
Mixing
Free Energy:
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Fig. 2.15, page 82
Comparison: F or G?
Hamley’s book eq.2.28, page 84
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Comparison:lattice theory
Hill’s equation 21-15, page 406
Fig 21-1
Page 402
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Mixing Free energy Polymers
Case 3: mix two polymers
Case 4: mix three polymers
Etc.
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Why is it that in general polymers do
not mix?
Take long polymers
The mixing entropy is reduced due to
the connectivity of the polymers
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Phase diagrams
When we try to mix two pure fluids
of unlike character,
In general the result is mixture of
two coexisting phases
Question:
what are the two concentrations in the two phases?73
Phase diagrams
Two fluids of equal molecular volume
χ=0
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Phase diagrams
Dissimilar molecules (like two hydrocarbons)
χ=1
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Phase diagrams
Repelling molecules (like oil and water)
χ=2.5
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Phase diagrams
The meaning of the bump is: the system is unstable
The system phase separates
Into two different phases
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Phase diagrams
In equilibrium,
the chemical potentials in the two phases are identical
for each component
If this were not true,
one could find a set of concentrations
with total lower mixing free energy
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Advanced topic:
A
why the chemical potentials should be the
same
B
}
Exercise: check
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Advanced topic
Swap one molecule ‘1’ from B to A: the change in mixing free energy is
Swap one molecule ‘2’ from B to A: the change in mixing free energy is
In the minimum:
a small shift in composition leaves mixing free energy unaffected. Hence:
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Chemical potentials
We already have expressions for
mixing entropy and mixing energy…
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We have all the ingredients
82
Hence, the chemical potentials are
Notice reference potentials
are absent…why is that?
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The chemical potentials are the same
in the two phases
Two non-linear equations, two unknowns
(remember volume fractions add up to 1)
84
Phase diagram regular solution
χ=(cst/T)
Usually plotted with
Temperature on vertical scale
1/χ
χ
unstable
: critical point
unstable
85
Phase diagram polymer solution
Follow the recipe, try a dimer
2
Tangent line
The phase diagram
is asymmetric,
The more so
for longer polymers
86
The chemical potentials are…
(try yourself)
And solve for the two concentrations in the two phases
(not so easy)
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Graphs of the chemical potentials
Dimer N=2, chi=2
solvent
In coexistence,
Chemical potentials
Of solvent and polymer
Must be the same
in the two phases
(indicated by box)
dimer
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Example of calculation with
Mathematica
Define functions
X = concentration polymer in A
Y= concentration polymer in B
Set parameters
N=2, chi=2
Plot chemical potentials
In interval (0,1)
Find solution
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Typical polymer solution phase
diagrams
Hill, page 409
1000
100
10
N=1
N=1000
Advanced Exercise: derive explicit
expression for critical point
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Structural Organization 1.3
Soft matter is usually ordered
on a mesoscopic scale 1nm-1000nm
The ordering is NOT perfectly crystalline
But contains lots of defects
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Structural Organisation 1.3
Block copolymer
In a melt,
the blocks are oriented
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Advanced topic:
Microphase diagrams
In the classical phase diagram theories
The phases are homogeneous
(the variables are the concentrations in the phases)
In Microphase diagram theories,
the phases are heterogeneous
(the variables are, for example:
-positions of the molecules
- concentration profiles
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Advaned topic:Model for Microphase
diagram
Block copolymers
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Concentration profile φ(r)
As a rule, in soft materials molecules
are relatively disordered, but the molecular
aggregates can be (weakly) ordered
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Now, we understand what this is!
End of file ISM01
(do we?)
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