Introduction to Soft Matter

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1 Introduction to Soft Matter Prof.Dr.Ir. J.G.E.M.(Hans) Fraaije Secretary Mrs. Ferry Soesman Tel Course material/downloads! 1

2 versions 1.0 Handout Embarrassing mistakes removed (thanks to Jan van Male), clarification level and state, and extension phase diagrams

3 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

4 What we need Thermodynamic theory and computer simulations Synthesis Characterization 4

5 Course materials 1. This presentation (downloadable) 2. Introduction to Soft Matter, Ian Hamley 3. Handouts Supramolecular Chemistry 4. Handout Statistical Mechanics (Hill) 5. Handout Home Soft Lab 5

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) 6

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 7

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. Compared to this, Introduction to Soft Matter is easy! 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 dem 17 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 Anschauung 26 a priori gegeben werde, d. i. daß der Begriff construirt werde. Nun ist die 27 Vernunfterkenntniß durch Construction der Begriffe mathematisch. Also 8

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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 ( nm) Interdisciplinary: physics, chemistry, mathematics and biology Introduction

11 Applications of soft materials everyday world Detergents Paints Plastics Soils Food Drug delivery Cosmetics All living systems Introduction

12 Constituents of Soft Materials Polymers (chapter 2) Colloids (chapter 3) Amphiphiles (chapter 4) Liquid Crystals (chapter 5) Supramolecules (hand out) nm nm 1-10 nm 1-10 nm 1-10 nm Introduction

13 Intermolecular Interactions Soft materials can often be induced to flow Weak ordering due to absence of long range crystalline order Introduction

14 INTERMEZZO Recapitulation statistical thermodynamics 14

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) 15

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? 16

17 Energy Scales Microscopic world: 1 kt (katé) T = Temperature (in Kelvin) k = Boltzmann s constant = R/N av 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) 17

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

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 energy F is free energy, G is free enthalpy G=H-TS 19

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. 20

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 21

22 Free energy, entropy and energy are related through derivatives Relations: So, find explicit expression for F and your are done! 22

23 Ω= (some number) S= k ln (some number) Entropy Scales (degrees of freedom) (degrees of freedom) 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 23

24 Theoretical approaches Exact, ab initio, rigorous, lots of equations Intuitive, small scaling relations, for example Approximative A goes like B 2, or A scales like B 2 (we do not care about prefactors) With hand waving 24

25 Statistical Thermodynamics Exercise: what is the dimension of Q? 25

26 When there is only one level (or, equivalent, all configurations have same energy) 26

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 27

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 28

29 Bond rotations Assume chain is ideal Polymer with 10 bonds, 11 monomers 3 orientations per bond (of same energy) 29

30 Single molecule in container Assume intramolecular interactions are decoupled from position 30

31 Single molecule in container How many possible positions? Method 1: lattice model: 31

32 Single molecule in container Method II: Assume it is a quantum particle Particle-wave duality 32

33 n molecules Start with 8 First label the molecules

34 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! 34

35 The molecules are undistinguishable 35

36 How many ways to distribute n labeled molecules? Ideal molecules: we do not care about overlap 36

37 How many ways to distribute n labeled molecules? Non-ideality due to reduction of available space Van der Waals 37

38 Two level model A molecule can be in two different levels A B Energy ev 0.05 example B Degeneracy = 4 0 A Degeneracy = 2 What is the formula for the entropy? 38

39 Two level model entropy S/k T What are the limiting values? 39

40 Advanced topic: Multiple states entropy formula Levels: States: 40

41 Ideal Mixing Pure 1 Pure 2 Mixture 12 41

42 Ideal mixing entropy 42

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 43

44 Do the same, with one extra molecule 44

45 Continued 45

46 Limiting mixing cases Case 1: Molecules have the same size 46

47 Case 2: polymers Component 1 is solvent, Component 2 is polymer, with N monomers 47

48 Case 3: collloid, emulsions (big things) Solvent molecule Colloidal particle 48

49 Now, if we want to calculate the free energy of mixing we need a model for the mixing energy 49

50 Back to Hamley s book: Intermolecular interactions 1.2 Typical molecular interaction curve Potential energy 0 repulsion r distance attraction 50

51 Different curves (check Atkins, Physical Chemistry) V V Long range repulsion Between molecules of same charge, or neutral flexible molecules r r V r Hard core repulsion (neutral colloids) V r Orientation dependent interaction Between molecular dipoles 51

52 Mathematical forms 52

53 Repulsion between atoms Electron clouds (orbitals) do not like to overlap (unless a bond is formed, as in a reaction) 53

54 Actually, electron cloud repulsion is better represented by From quantum theory 54

55 Hard core repulsion Like a solid wall! V d r 55

56 Attraction between atoms and molecules Between permanent dipoles of opposite orientation Between fluctuating dipoles Dispersion interactions Every atom has a fluctuating dipole 56

57 Lennard-Jones (12,6) potential Exercise: what is the relation between the two sets of parameters? At which position is the minimum? 57

58 Graphical representation LJ potential V r 58

59 Hierarchy of interactions Coulombic ~ 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 59

60 Now we can make a simple model for mixing energy Typical molecular interaction curve Potential energy 0 repulsion r distance attraction 60

61 Mean-field approximation On average, the concentration around a given molecule is the same as the average concentration We shall assume 61

Mean field model interactions 2 1 1 1 2 1 2 The molecules are separated by a distance d, And feel the interaction The number of

62 Mean field model interactions The molecules are separated by a distance d, And feel the interaction The number of molecules 2 around central 1 : d z is geometrical factor (coordination number) Each contact adds an interaction Exercise: estimate z 62

63 Mean field interaction The total interaction between 1 and 2 Exercise: why the factor ½? Repeat for 1-1: And

64 Mixing energy 64

65 Sign of exchange parameter Cohesive energy 12 Typical values χ in the range

66 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! 66

Mixing energy polymer and solvent 2 1 1 If we thread a polymer through 1 2 1 2 the

67 Mixing energy polymer and solvent If we thread a polymer through the interaction shell, it remains the same Calculate the interactions on monomer basis 67

68 Mixing free energy polymers Mixing energy: Mixing entropy: Mixing Free Energy: 68

69 Fig. 2.15, page 82 Comparison: F or G? Hamley s book eq.2.28, page 84 69

70 Comparison:lattice theory Hill s equation 21-15, page 406 Fig 21-1 Page

71 Mixing Free energy Polymers Case 3: mix two polymers Case 4: mix three polymers Etc. 71

72 Why is it that in general polymers do Take long polymers not mix? The mixing entropy is reduced due to the connectivity of the polymers 72

73 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

74 Phase diagrams Two fluids of equal molecular volume χ=0 74

75 Phase diagrams Dissimilar molecules (like two hydrocarbons) χ=1 75

76 Phase diagrams Repelling molecules (like oil and water) χ=2.5 76

77 Phase diagrams The meaning of the bump is: the system is unstable The system phase separates Into two different phases 77

78 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 78

79 Advanced topic: why the chemical potentials should be the same A B } Exercise: check 79

80 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: 80

81 Chemical potentials We already have expressions for mixing entropy and mixing energy 81

82 We have all the ingredients 82

83 Hence, the chemical potentials are Notice reference potentials are absent why is that? 83

$non-linear equations, two unknowns (remember volume fractions add$

84 The chemical potentials are the same in the two phases Two non-linear equations, two unknowns (remember volume fractions add up to 1) 84

85 Phase diagram regular solution χ=(cst/t) χ 1/χ Usually plotted with Temperature on vertical scale unstable unstable : critical point 85

86 Phase diagram polymer solution Follow the recipe, try a dimer 2 The phase diagram is asymmetric, The more so for longer polymers Tangent line 86

87 The chemical potentials are (try yourself) And solve for the two concentrations in the two phases (not so easy) 87

88 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 88

89 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 89

90 Typical polymer solution phase diagrams Hill, page 409 N=1 N=1000 Advanced Exercise: derive explicit expression for critical point 90

91 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 91

92 Structural Organisation 1.3 Block copolymer In a melt, the blocks are oriented 92

93 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 93

94 Advaned topic:model for Microphase diagram Block copolymers 94

95 Concentration profile φ(r) As a rule, in soft materials molecules are relatively disordered, but the molecular aggregates can be (weakly) ordered 95

96 Now, we understand what this is! End of file ISM01 (do we?) 96

PHASE TRANSITIONS IN SOFT MATTER SYSTEMS

OUTLINE: Topic D. PHASE TRANSITIONS IN SOFT MATTER SYSTEMS Definition of a phase Classification of phase transitions Thermodynamics of mixing (gases, polymers, etc.) Mean-field approaches in the spirit