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Introduction to Soft MatterProf.Dr.Ir. J.G.E.M.(Hans) Fraaije
Secretary Mrs. Ferry SoesmanTel 4523
f.soesman@chem.leidenuniv.nlhttp://www.chem.leidenuniv.nl/scm
Course material/downloads!
2
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 genomicsPatterned surface films
Origin of Life
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Course materials1. This presentation (downloadable)2. “Introduction to Soft Matter”, Ian Hamley
3. Handouts Supramolecular Chemistry4. Handout Statistical Mechanics (Hill)5. 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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01 Metaphysik der Natur: oder sie beschäftigt sich mit einer besonderen Natur 02 dieser oder jener Art Dinge, von denen ein empirischer Begriff gegeben 03 ist, doch so, daß außer dem, was in diesem Begriffe liegt, kein anderes 04 empirisches Princip zur Erkenntniß derselben gebraucht wird (z. B. sie 05 legt den empirischen Begriff einer Materie, oder eines denkenden Wesens 06 zum Grunde und sucht den Umfang der Erkenntniß, deren die Vernunft 07 über diese Gegenstände a priori fähig ist), und da muß eine solche Wissenschaft 08 noch immer eine Metaphysik der Natur, nämlich der körperlichen 09 oder denkenden Natur, heißen, aber es ist alsdann keine allgemeine, sondern 10 besondere metaphysische Naturwissenschaft (Physik und Psychologie), 11 in der jene transscendentale Principien auf die zwei Gattungen der Gegenstände 12 unserer Sinne angewandt werden.
13 Ich behaupte aber, daß in jeder besonderen Naturlehre nur so viel 14 eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik 15 anzutreffen ist. Denn nach dem Vorhergehenden erfordert eigentliche 16 Wissenschaft, vornehmlich der Natur, einen reinen Theil, der dem17 empirischen zum Grunde liegt, und der auf Erkenntniß der Naturdinge 18 a priori beruht. Nun heißt etwas a priori erkennen, es aus seiner bloßen 19 Möglichkeit erkennen. Die Möglichkeit bestimmter Naturdinge kann aber 20 nicht aus ihren bloßen Begriffen erkannt werden; denn aus diesen kann 21 zwar die Möglichkeit des Gedankens (daß er sich selbst nicht widerspreche), 22 aber nicht des Objects als Naturdinges erkannt werden, welches außer 23 dem Gedanken (als existirend) gegeben werden kann. Also wird, um die 24 Möglichkeit bestimmter Naturdinge, mithin um diese a priori zu erkennen, 25 noch erfordert, daß die dem Begriffe correspondirende Anschauung26 a priori gegeben werde, d. i. daß der Begriff construirt werde. Nun ist die 27 Vernunfterkenntniß durch Construction der Begriffe mathematisch. Also
http://linux-s.ikp.uni-bonn.de/cgi-bin/Kant/lade.pl?/default.htm
Compared to this, Introduction to Soft
Matter is easy!
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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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• Polymers (chapter 2)• Colloids (chapter 3)• Amphiphiles (chapter 4)• Liquid Crystals (chapter 5)• Supramolecules (hand out)
Constituents of Soft Materials
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
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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:
(A and B same degeneracy)
When the energy difference is 1 kT,The probability ratio is:
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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 energyF 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
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Free energy, entropy and energy are related through derivatives
• Relations:
So, find explicit expressionfor F and your are done!
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Entropy Scales• Ω= (some number) (degrees of freedom)
• S= k ln (some number) (degrees of freedom)
• S= degrees of freedom*k*ln(some number)• Remember:
• “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”
S/k= “degrees of freedom”
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• Exact, ‘ab initio’, rigorous, lots of equations
• Intuitive, small ‘scaling’ relations, for example
• Approximative
Theoretical approaches
A goes like B2, or A scales like B2
(we do not care about prefactors)
With hand waving
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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. bond can rotate in three positions, molecule containes 10 such bonds, what is the molecular entropy? (intramolecular effect)
2. single molecule moves around in container with volume V (ideal gas).What is the entropy of the molecule? (effect of freedom of position)
3. n molecules in ideal gas. What is the entropy? (effect of the interchange of particles)
4. Mix two different molecules (mixing entropy)5. 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
3 orientations per bond(of same energy)
Assume chain is ideal
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How many ways to label?Start with empty balls
The Number “8” can be put into 8 different placesThe number “7” then in any of the remaining 7The 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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How many ways to distribute n labeled molecules?
Non-ideality due to reduction of available spaceVan der Waals
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Two level model
A B
A molecule can be in two different levels
What is the formula for the entropy?
A
B
Degeneracy = 4
Degeneracy = 2
Ener
gy e
V
example
0.05
0
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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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Back to Hamley’s book:Intermolecular interactions 1.2
Typical molecular interaction curve
r0
repulsion
attraction
Pote
ntia
l ene
rgy
distance
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Different curves…(check Atkins, Physical Chemistry)
Long range repulsionBetween molecules of same charge,
or neutral flexible molecules
Hard core repulsion(neutral colloids)
Orientation dependent interactionBetween molecular dipoles
V V
V V
r r
r r
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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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Attraction between atoms and molecules
Between permanent dipoles of
opposite orientation
Between fluctuating dipolesDispersion 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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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 energyTypical molecular interaction curve
r0
repulsion
attraction
Pote
ntia
l ene
rgy
distance
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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 interactionsThe molecules are separated by a distance d,
And feel the interaction
d
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1
11
1The number of molecules “2” around central “1”:
Each contact adds an interaction
“z” is geometrical factor(coordination number)
2
Exercise: estimate z
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Mean field interactionThe total interaction between “1” and “2”
Exercise: why the factor ½?
Repeat for 1-1: And 2-2
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Mixing energyCase 2: polymers
Assume all polymers are random coilsMonomers exposed to solventSolvent 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
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1
11
1 If we thread a polymer throughthe interaction shell, it remains the same2
Calculate the interactions on monomer basis
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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 diagramsWhen 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?
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Phase diagramsThe meaning of the bump is: the system is unstable
The system phase separatesInto two different phases
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Phase diagramsIn equilibrium,
the chemical potentials in the two phases are identicalfor each component
If this were not true, one could find a set of concentrationswith total lower mixing free energy
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Advanced topicSwap 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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The chemical potentials are the samein the two phases
Two non-linear equations, two unknowns(remember volume fractions add up to 1)
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Phase diagram regular solution
Usually plotted with Temperature on vertical scale
χ=(cst/T)
1/χχ
unstable unstable
: critical point
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Phase diagram polymer solutionFollow the recipe, try a dimer
The phase diagramis asymmetric,The more so
for longer polymers
Tangent line
2
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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 potentialsDimer N=2, chi=2
solvent
dimer
In coexistence,Chemical potentials
Of solvent and polymerMust be the samein the two phases(indicated by box)
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Example of calculation withMathematica
Define functions
Set parametersN=2, chi=2
Plot chemical potentialsIn interval (0,1)
Find solution
X = concentration polymer in AY= concentration polymer in B
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Typical polymer solution phase diagrams
Advanced Exercise: derive explicit expression for critical point
N=1
10100
1000
N=1000
Hill, page 409
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Structural Organization 1.3Soft matter is usually ordered
on a mesoscopic scale 1nm-1000nmThe ordering is NOT perfectly crystalline
But contains lots of defects
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Advanced topic:Microphase diagrams
In the classical phase diagram theoriesThe 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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As a rule, in soft materials molecules are relatively disordered, but the molecular
aggregates can be (weakly) ordered
Con
cent
ratio
n pr
ofile
φ(r
)