1. Characteristics of Soft Matter (1)Length scales between atomic and macroscopic (sometimes called mesoscopic) (2) The importance of thermal fluctuations and Brownian motion (3) Tendency to self-assemble into hierarchical structures (i.e. ordered on multiple size scales beyond the molecular) (4) Short-range forces and interfaces are important. In the previous lecture:

2. Lecture 2: Polarisability and van der Waals’ Interactions: Why are neutral molecules attractive to each other? Soft Matter Physics 18 February, 2010 See Israelachvili’s Intermolecular and Surface Forces, Ch. 4, 5 & 6

3. What are the forces that operate over short distances and hold soft matter together? http://www.cchem.berkeley.edu/rmgrp/about_gecko.jpg

4. Interaction Potentials For two atoms/molecules/segments separated by a distance of r, the interaction energy can be described by an attractive potential energy: w att (r) = - Cr -n = -C/r n, where C and n are constants. There is also repulsion because of the Pauli exclusion principle: electrons cannot occupy the same energy levels. Treat atoms/molecules like hard spheres with a diameter, . Use a simple repulsive potential: w rep (r) = +(  /r)  The interaction potential w(r) = w att + w rep r 

5. “Hard-Sphere” Interaction Potential +w(r)-+w(r)- Attractive potential r w att (r) = -C/r n +w(r)-+w(r)- Repulsive potential r  w rep (r) = (  /r) 

6. Hard-Sphere Interaction Potentials +w(r)-+w(r)- Total potential: r w(r) = w att + w rep  Minimum of potential = equilibrium spacing in a solid =  The force, F, acting on particles with this interaction energy is:

7. Interaction Potentials Gravity: all atoms/molecules have a mass! 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! How can we describe their potentials?

8. Gravity: n = 1 r mm m2m2 G = 6.67 x 10 -11 Nm 2 kg -1 When molecules are in contact, w(r) is typically ~ 10 -52 J Negligible interaction energy!

9. Coulombic Interactions: n = 1 r QQ Q2Q2 With Q 1 = z 1 e, where e is the charge on the electron, and z 1 is an integer value.  o is the permittivity of free space and e is the relative permittivity of the medium between ions (can be vacuum with  = 1 or can be a gas or liquid with  > 1). The interaction potential is additive in crystals. When molecules are in close contact, w(r) is typically ~ 10 -18 J, corresponding to about 200 to 300 kT at room temp. Sign of w depends on whether charges are alike or opposite.

10. van der Waals Interactions (London dispersion energy): n = 6 r  22 u2u2 u1u1 Interaction energy (and the constant, C) depends on the dipole moment (u) of the molecules and their polarisability (  ). 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.

11. 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.!

12. Hydrogen bonding 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. H O H H H O ++ ++ ++ ++ -- --

13. When w(r) is a minimum, dw/dr = 0. Solve for r to find equilibrium spacing for a solid, where r = r e. (Confirm minimum by checking curvature from 2 nd derivative.) The force between two molecules is F = -dw/dr Thus, F = 0 when r = r e. If r r e, F is tensile (-). When dF/dr = d 2 w/dr 2 =0, attractive F is at its maximum. Significance of Interaction Potentials r e = equilibrium spacing

14. r How much energy is required to remove a molecule from the condensed phase? Q: Does a central molecule interact with ALL the others? Applies to pairs L  = molecular spacing when molecules are in contact  = density = number of molec./volume Individual molecules 

15. Total Interaction Energy, E Interaction energy for a pair: w(r) = -Cr -n Volume of thin shell: Number of molecules at a distance, r : Total interaction energy between a central molecule and all others in the system (from  to L), E: But L >>  ! When can we neglect the term? r -n+2 =r -(n-2)

16. Conclusions about E There are three cases: When n > , then (  / L) n-3 >>1 and is thus significant. In this case, E varies with the size of the system, L! (This result applies to gravitational potential in a solar system.) But when n>3, (  /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=

17. The Third Case: n = 3  will be very small (typically 10 -9 m), but ln  is not negligible. L cannot be neglected in most cases. What values of n apply to molecular interaction potentials? Is it >, < or = 3?

18. Polarity of Molecules All interaction potentials (and forces) between molecules are electrostatic in origin. A neutral molecule is polar when its electronic charge distribution is not symmetric about its nuclear (+ve charged) centre. In a non-polar molecule the centre of electronic (-ve) charge does not coincide with the centre of nuclear (+ve) charge. + _ _ +

19. Dipole Moments A “convenient” (and conventional) unit for polarity is called a Debye (D): 1 D = 3.336 x 10 -30 Cm The polarity of a molecule is described by its dipole moment, u, given as: when charges of +q and - q are separated by a distance. Typically, q is the charge on the electron: 1.602 x10 -19 C and the magnitude of is on the order of 1Å= 10 -10 m, giving u = 1.602 x 10 -29 Cm. + -

20. Examples of Nonpolar Molecules: u = 0 CO 2 O-C-O CH 4 C H H H H C H H H H 109 º CCl 4 Cl C 109 º methane Have rotational and mirror symmetry 120  Top view

21. Examples of Polar Molecules CH 3 Cl CHCl 3 Cl C H C H H H Have lost some rotational and mirror symmetry! Unequal sharing of electrons between two unlike atoms leads to polarity in the bond.

22. Dipole moments C=O u = 0.11 D + - N H H H u = 1.47 D - + H H O - + u = 1.85 D S O u = 1.62 D +-+- Bond moments N-H 1.31 D O-H 1.51 D F-H 1.94 D What is the S-O bond moment? Find from vector addition knowing O- S-O bond angle. V. High! Vector addition of bond moments is used to find u for molecules.

23. HH Given that the H-O-H bond angle is 104.5° and that the bond moment of OH is 1.51 D, what is the dipole moment of water? /2/2 O 1.51 D u H 2 O = 2 cos(  /2)u OH = 2 cos (52.25 °) x 1.51 D = 1.85 D Vector Addition of Bond Moments

24. Charge-Dipole Interactions There is an electrostatic (i.e. Coulombic) interaction between a charged molecule (an ion) and a static polar molecule. The interaction potential can be compared to the Coulomb potential for two point charges (Q 1 and Q 2 ): Ions can induce ordering and alignment of polar molecules. Why? Equilibrium state when w(r) is minimum. w(r) decreases as  decreases to 0. Q  r + - w(r) = -Cr -2

25. Dipole-Dipole Interactions There are Coulombic interactions between the +ve and -ve charges associated with each dipole. In liquids, thermal energy causes continuous motion, i.e. tumbling, of dipoles in relation to each other. In solids, dipoles are usually fixed on a lattice with a certain orientation, described by  1 and  2. 11 22 + + - -

26. Fixed-dipole Interactions The interaction energy, w(r), depends on the relative orientation of the dipoles : Molecular size influences the minimum possible r. For a given spacing r, the end-to-end alignment has a lower w, but usually this alignment requires a larger r compared to side-by-side (parallel) alignment. 11 22  r Note: W(r) = -Cr -3 - + - +

27. w(r) (J) r (nm) At a typical spacing of 0.4 nm, w(r) is about 1 kT. Hence, thermal energy is able to disrupt the alignment. -10 -19 -2 x10 -19 0 0.4 kT at 300 K End-to-end Side-by-side W(r) = -Cr -3  1 =  2 = 0  1 =  2 = 90° From Israelachvili, Intermol.& Surf. Forces, p. 59

28. Freely-Rotating Dipoles In some cases, dipoles do not have a fixed position and orientation on a lattice but constantly move about. This occurs when thermal energy is greater than the fixed dipole interaction energy: Interaction energy depends inversely on T, and because of constant motion, there is no angular dependence: Note: W(r) = -Cr -6

29. Polarisability All molecules can have a dipole induced by an external electromagnetic field, The strength of the induced dipole moment, |u ind |, is determined by the polarisability, , of the molecule: Units of polarisability:

30. Polarisability of Nonpolar Molecules An electric field will shift the electron cloud of a molecule. The extent of polarisation is determined by its electronic polarisability,  o. + _ E + _ Initial state In an electric field

31. Simple Bohr Model of e - Polarisability Force on the electron due to the field: Attractive Coulombic force on the electron from nucleus: At equilibrium, the forces balance: Without a field: With a field: F ext F int

32. Substituting expressions for the forces : Solving for the induced dipole moment : So we obtain an expression for the polarisability: From this crude argument, we predict that electronic polarisability is proportional to the size of the molecule! Simple Bohr Model of e - Polarisability

33. Units of Electronic Polarisability Units of volume Polarisability is often reported as:

34. Electronic Polarisabilities He0.20 H 2 O1.45 O 2 1.60 CO1.95 NH 3 2.3 CO 2 2.6 Xe4.0 CHCl 3 8.2 CCl 4 10.5 Largest Smallest Units  o /(4  o ): 10 -30 m 3 Numerically equivalent to  o in units of 1.11 x 10 -40 C 2 m 2 J -1  o /(4  o ) ( 10 -30 m 3 )

35. Example: Polarisation Induced by an Ion Ca 2+ dispersed in CCl 4 (non-polar). What is the induced dipole moment in CCl 4 at a distance of 2 nm? - + By how much is the electron cloud of the CCl 4 shifted? From Israelachvili, Intermol.& Surf. Forces, p. 72

36. Example: Polarisation Induced by an Ion Ca 2+ dispersed in CCl 4 (non-polar). Affected by the permittivity of CCl 4 :  = 2.2 From the literature, we find for CCl 4 : Field from the Ca 2+ ion: We find at close contact when r = 2 nm: u ind = 3.82 x 10 -31 Cm Thus, an electron with charge e is shifted by: Å

37. Polarisability of Polar Molecules In a liquid, molecules are continuously rotating and turning, so the time- averaged dipole moment for a polar molecule in the liquid state is 0. Let  represent the angle between the dipole moment of a molecule and an external E-field direction. The spatially-averaged value of = 1/3 The induced dipole moment is: An external electric field can partially align dipoles: + -  The molecule still has electronic polarisability, so the total polarisability, , is given as : Debye-Langevin equation As u =  E, we can define an orientational polarisability.

38. Origin of the London or Dispersive Energy The dispersive energy is quantum-mechanical in origin, but we can treat it with electrostatics. Applies to all molecules, but is insignificant in charged or polar molecules. An instantaneous dipole, resulting from fluctuations in the electronic distribution, creates an electric field that can polarise a neighbouring molecule. The two dipoles then interact. 11 22 22 - + + + -- r

39. Origin of the London or Dispersive Energy ++-- r u1u1 u2u2  The field produced by the instantaneous dipole is: So the induced dipole moment in the neighbour is: We can now calculate the interaction energy between the two dipoles (using the equation for permanent dipoles - slide 27): Instantaneous dipole Induced dipole

40. Origin of the London or Dispersive Energy ++-- r This result: compares favourably with the London result (1937) that was derived from a quantum-mechanical approach: h  is the ionisation energy, i.e. the energy to remove an electron from the molecule

41. London or Dispersive Energy The London result is of the form: In simple liquids and solids consisting of non-polar molecules, such as N 2 or O 2, the dispersive energy is solely responsible for the cohesion of the condensed phase. where C is called the London constant: Must consider the pair interaction energies of all “near” neighbours.

42. Summary Type of Interaction Interaction Energy, w(r) Charge-charge Coulombic Nonpolar-nonpolar Dispersive Charge-nonpolar Dipole-charge Dipole-dipole Keesom Dipole-nonpolar Debye In vacuum:  =1

43. van der Waals’ Interactions Refers to all interactions between polar or nonpolar molecules, varying as r -6. Includes Keesom, Debye and dispersive interactions. Values of interaction energy are usually only a few kT.

44. Comparison of the Dependence of Interaction Potentials on r Not a comparison of the magnitudes of the energies! n = 1 n = 2 n = 3 n = 6 Coulombic van der Waals Dipole-dipole Charge-fixed dipole

45. Interaction energy between ions and polar molecules Interactions involving charged molecules (e.g. ions) tend to be stronger than polar-polar interactions. For freely-rotating dipoles with a moment of u interacting with molecules with a charge of Q we saw: One result of this interaction energy is the condensation of water (u = 1.85 D) caused by the presence of ions in the atmosphere. During a thunderstorm, ions are created that nucleate rain drops in thunderclouds (ionic nucleation). +Q+Q

46. Measuring Polarisability Polarisability is dependent on the frequency of the E-field. The Clausius-Mossotti equation relates the dielectric constant (permittivity)  of a molecule having a volume v to  : At the frequency of visible light, however, only the electronic polarisability,  o, is active. At these frequencies, the Lorenz-Lorentz equation relates the refractive index, n (n 2 =  ) to  o : So we see that measurements of the refractive index can be used to find the electronic polarisability.

47. Frequency dependence of polarisability From Israelachvili, Intermol. Surf. Forces, p. 99

48. it.wikipedia.org/wiki/Legge_di_Van_der_Waals PV diagram for CO 2 Non-polar gasses condense into liquids because of the dispersive (London) attractive energy. Van der Waals Gas Equation:

49. Measuring Polarisability The van der Waals’ gas law can be written (with V = molar volume) as: The constant, a, is directly related to the London constant, C: where  is the molecular diameter (= closest molecular spacing). We can thus use the C-M, L-L and v.d.W. equations to find values for  o and .

50. Measuring Polarisability From Israelachvili, Intermol.& Surf. Forces

51. Problem Set 1 1. Noble gases (e.g. Ar and Xe) condense to form crystals at very low temperatures. As the atoms do not undergo any chemical bonding, the crystals are held together by the London dispersion energy only. All noble gases form crystals having the face-centred cubic (FCC) structure and not the body-centred cubic (BCC) or simple cubic (SC) structures. Explain why the FCC structure is the most favourable in terms of energy, realising that the internal energy will be a minimum at the equilibrium spacing for a particular structure. Assume that the pairs have an interaction energy, u(r), described as where r is the centre-to-centre spacing between atoms. The so-called "lattice sums", A n, are given below for each of the three cubic lattices. SCBCCFCC A 6 8.4012.2514.45 A 12 6.209.1112.13 Then derive an expression for the maximum force required to move a pair of Ar atoms from their point of contact to an infinite separation. 2. (i) Starting with an expression for the Coulomb energy, derive an expression for the interaction energy between an ion of charge ze and a polar molecule at a distance of r from the ion. The dipole moment is tilted by an angle  with relation to r, as shown below. (ii) Evaluate your expression for a Mg 2+ ion (radius of 0.065 nm) dissolved in water (radius of 0.14 nm) when the water dipole is oriented normal to the ion and when the water and ion are at the point of contact. Express your answer in units of kT. Is it a significant value? (The dipole moment of water is 1.85 Debye.) 3. Show that 1 kJ mole -1 = 0.4 kT per molecule at 300 K. r  ze

55. Hydrogen bonding 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. H O H H H O ++ ++ ++ ++ -- --

56. Hydrophobic Interactions “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. A water “cage” around another molecule

57. Hydrophobic Interactions The decrease in entropy (associated with the ordering of molecules) makes it unfavourable to mix water with “hydrophobic molecules”. For example, when mixing n-butane with water:  G =  H - T  S = -4.3 +28.7 = +24.5 kJ mol -1. Unfavourable (+ve  G) because of the decrease in entropy! This value of  G is consistent with a “surface area” of n-butane of  1 nm 2 and   40 mJ m -2 for the water/butane interface; an increase in  G =  A is needed to create a new interface! Although hydrophobic means “water-fearing”, there is an attractive van der Waals’ force (as discussed later in this lecture) between water and other molecules - there is not a repulsion! Water is more strongly attracted to itself, because of H bonding, however, in comparison to hydrophobic molecules.

58. De-wetting “Froth flotation” Protein folding Adhesion in water Immiscibility Micellisation Association of molecules Coagulation