1

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

3

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

4

What we need

Thermodynamic theory and computer simulationsSynthesis

Characterization

5

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

6

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)

7

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

8

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!

9

10

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

11

Applications of soft materials

• ‘everyday’ world• Detergents• Paints• Plastics• Soils• Food• Drug delivery• Cosmetics• All living systems

Introduction 1.1

12

• 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

13

Intermolecular Interactions

• Soft materials can often be induced to flow• Weak ordering due to absence of long range

crystalline order

Introduction 1.2

14

Recapitulation statistical thermodynamics

INTERMEZZO

15

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)

16

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?

17

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:

18

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

19

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

20

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.

21

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

22

Free energy, entropy and energy are related through derivatives

• Relations:

So, find explicit expressionfor F and your are done!

23

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”

24

• 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

25

Statistical Thermodynamics

Exercise: what is the dimension of Q?

26

When there is only one level…(or, equivalent, all configurations

have same energy)

27

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

28

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

29

Bond rotations

Polymer with 10 bonds, 11 monomers

3 orientations per bond(of same energy)

Assume chain is ideal

30

Single molecule in container

Assume intramolecular interactionsare decoupled from position

31

Single molecule in containerHow many possible positions?

Method 1: lattice model:

32

Single molecule in containerMethod II: Assume it is a quantum particle

Particle-wave duality

33

n molecules

1

524

3

6

7

8

First label the molecules

11

2 2

33

4

4

55

6

67

78

8

Start with 8

34

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!

35

The molecules are undistinguishable

36

How many ways to distribute n labeled molecules?

Ideal molecules: we do not care about overlap

37

How many ways to distribute n labeled molecules?

Non-ideality due to reduction of available spaceVan der Waals

38

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

39

Two level model entropy

T

S/k

What are the limiting values?

40

Advanced topic: Multiple states entropy formula

Levels:

States:

41

Ideal Mixing

+

1 2

Mixture 12

Pure “1” Pure “2”

42

Ideal mixing entropy

43

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

44

Do the same, with one extra molecule

45

Continued…

46

Limiting mixing cases

Case 1: Molecules have the same size

47

Case 2: polymersComponent 1 is solvent,

Component 2 is polymer, with N monomers

48

Case 3: collloid, emulsions…(big things)

Solvent molecule

Colloidal particle

49

Now, if we want to calculate thefree energy of mixing…

…we need a model for the mixing energy

50

Back to Hamley’s book:Intermolecular interactions 1.2

Typical molecular interaction curve

r0

repulsion

attraction

Pote

ntia

l ene

rgy

distance

51

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

52

Mathematical forms

53

Repulsion between atoms

Electron clouds (orbitals) do not like to overlap

(unless a bond is formed, as in a reaction)

54

Actually, electron cloud repulsion is better represented by

From quantum theory

55

Hard core repulsion

Like a solid wall!

d r

V

56

Attraction between atoms and molecules

Between permanent dipoles of

opposite orientation

Between fluctuating dipolesDispersion interactions

Every atom has a fluctuating dipole

57

Lennard-Jones (12,6) potential

Exercise: what is the relation between the two sets of parameters?

At which position is the minimum?

58

Graphical representation LJ potential

V

r

59

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

60

Now we can make a simple model for mixing energyTypical molecular interaction curve

r0

repulsion

attraction

Pote

ntia

l ene

rgy

distance

61

Mean-field approximation

On average, the concentration around a given molecule is the same

as the average concentration

We shall assume

62

Mean field model interactionsThe molecules are separated by a distance d,

And feel the interaction

d

22

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

63

Mean field interactionThe total interaction between “1” and “2”

Exercise: why the factor ½?

Repeat for 1-1: And 2-2

64

Mixing energy

65

Sign of exchange parameter

11 22

12

Typical values χ in the range 0-3

Cohesive energy

66

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!

67

Mixing energy polymer and solvent

22

1

11

1 If we thread a polymer throughthe interaction shell, it remains the same2

Calculate the interactions on monomer basis

68

Mixing free energy polymers

Mixing entropy:

Mixing energy:

MixingFree Energy:

69

Comparison: F or G?Hamley’s book eq.2.28, page 84

Fig. 2.15, page 82

70

Comparison:lattice theoryHill’s equation 21-15, page 406

Fig 21-1Page 402

71

Mixing Free energy PolymersCase 3: mix two polymers

Case 4: mix three polymers

Etc.

72

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

73

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?

74

Phase diagramsTwo fluids of equal molecular volume

χ=0

75

Phase diagramsDissimilar molecules (like two hydrocarbons)

χ=1

76

Phase diagrams

χ=2.5

Repelling molecules (like oil and water)

77

Phase diagramsThe meaning of the bump is: the system is unstable

The system phase separatesInto two different phases

78

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

79

Advanced topic:why the chemical potentials should be the

sameA B

Exercise: check

80

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:

81

Chemical potentials

We already have expressions for mixing entropy and mixing energy…

82

We have all the ingredients

83

Hence, the chemical potentials are

Notice reference potentials are absent…why is that?

84

The chemical potentials are the samein the two phases

Two non-linear equations, two unknowns(remember volume fractions add up to 1)

85

Phase diagram regular solution

Usually plotted with Temperature on vertical scale

χ=(cst/T)

1/χχ

unstable unstable

: critical point

86

Phase diagram polymer solutionFollow the recipe, try a dimer

The phase diagramis asymmetric,The more so

for longer polymers

Tangent line

2

87

The chemical potentials are…(try yourself)

And solve for the two concentrations in the two phases(not so easy)

88

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)

89

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

90

Typical polymer solution phase diagrams

Advanced Exercise: derive explicit expression for critical point

N=1

10100

1000

N=1000

Hill, page 409

91

Structural Organization 1.3Soft matter is usually ordered

on a mesoscopic scale 1nm-1000nmThe ordering is NOT perfectly crystalline

But contains lots of defects

92

Structural Organisation 1.3

Block copolymer

In a melt, the blocks are oriented

93

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

94

Advaned topic:Model for Microphase diagram

Block copolymers

95

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

)

96

Now, we understand what this is!

End of file ISM01

(do we?)