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Characteristics of Soft Matter

In the previous lecture:. Characteristics of Soft Matter. (1) Length scales between atomic and macroscopic (sometimes called mesoscopic) are relevant. (2) Weak, short-range forces and interfaces are important. (3) The importance of thermal fluctuations and Brownian motion

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Characteristics of Soft Matter

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

  2. PH3-SM (PHY3032) Soft Matter Physics Lecture 2: Polarisability and van der Waals’ Interactions: Why are neutral molecules attractive to each other? 11 October, 2011 See Israelachvili’s Intermolecular and Surface Forces, Ch. 4, 5 & 6

  3. What are the forces that operate over short distances to hold condensed matter together? Nitrogen condensed in the liquid phase.

  4. What are the forces that operate over short distances to cause adhesion? http://www.cchem.berkeley.edu/rmgrp/about_gecko.jpg

  5. Interaction Potentials s r • For two atoms/molecules/segments separated by a distance of r, the interaction energy can be described by an attractivepotential energy: watt(r) = - Cr -n = -C/rn, 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, s. Use a simplerepulsive potential: wrep(r) = +(s/r) • The interaction potential w(r) = watt + wrep

  6. + w(r) - Repulsive potential s r s wrep(r) = (s/r) “Hard-Sphere” Interaction Potential + w(r) - Attractive potential r watt(r) = -C/rn r

  7. The force, F, acting between atoms (molecules) with this interaction energy is: where a negative force is attractive. As r ∞, F  0. Hard-Sphere Interaction Potentials + w(r) - Total potential: s r w(r) = watt + wrep Minimum of potential = equilibrium spacing in a solid = s

  8. 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?

  9. Gravity: n = 1 m2 m1 r G = 6.67 x 10-11 Nm2kg-1 When molecules are in contact, w(r) is typically ~ 10-52 J Negligible interaction energy – much weaker than thermal energy (kT)!

  10. Q2 Q1 r Coulombic Interactions: n = 1 The sign of w depends on whether charges are alike or opposite. • With Q1 = z1e, where e is the charge on the electron, and z1 is an integer value. • eois the permittivity of free space and e is the relative permittivity of the medium between ions (can be vacuum with e = 1 or can be a gas or liquid with e> 1). • When molecules are in close contact, w(r) is typically ~ 10-18 J, corresponding to about 200 to 300 kT at room temp. • The interaction potential is additive in ionic crystals.

  11. van der Waals Interactions: n = 6 u2 u1 a2 a1 r • Interaction energy (and the constant, C) depends on the dipole moment (u) of the molecules and their polarisability (a). • 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.

  12. 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.!

  13. Hydrogen bonding d- H O d+ d- H O H d+ d+ H d+ • 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. It cannot make a covalent network. • The H’s proton is unshielded by electrons and makes an electropositive end (d+) to the bond: ionic character in a covalent bond. • Hydrogen 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.

  14. Significance of Interaction Potentials + • When w(r) is a minimum, dw/dr = 0. • Solve for r to find equilibrium spacing for a solid, where r = re. • (Confirm minimum by checking curvature from 2nd derivative.) • The force between two molecules is F = -dw/dr • Thus, F = 0 when r = re. • If r < re, the external F is compressive (+ve); If r > re, the external F is tensile (-ve). • When dF/dr = d2w/dr2 =0, the attractive F is at its maximum. - re = equilibrium spacing

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

  16. r -n+2=r-(n-2) 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 s to L), E: But L >> s ! When can we neglect the term?

  17. E= Conclusions about E • There are three cases: • (1) When n<3, then the exponent is negative. As s <<L, then (s/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 (n= 1) in a solar system.) • (2) Butwhen n>3, (s/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!

  18. The Third Case: n = 3 swill be very small (typically 10-9 m), but lnsis not negligible. L cannot be neglected in most cases. Which values of n apply to various molecular interaction potentials? Are they >, < or = 3?

  19. Electron Probability Distributions H orbitals with n = 3 s orbitals in the H atom • Symmetric distribution of electrons in atomic orbitals • Position of the electrons cannot be known with certainty. http://www.chem1.com/acad/webtext/atoms/atpt-4.html

  20. Centre of +ve charge Centre of -ve charge - O nucleus 8+ O nucleus 8+ - - O nucleus 8+ C nucleus 6+ - O2 is non-polar CO is polar Polarity of Molecules • All interaction potentials (and forces) between molecules are electrostatic in origin: even when the molecules have no net charge. • In a non-polar molecule the centre of electronic (-ve) charge coincides with the centre of nuclear (+ve) charge. • But, a charge-neutral molecule is polar when its electronic charge distribution is not symmetric about its nuclear (+ve charged) centre.

  21. when charges of+qand - qare separated by a distance . If 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, then we have that u = 1.602 x 10-29 Cm. Dipole Moments The polarity of a molecule is described by its dipole moment, u, given as: A “convenient” (and conventional) unit for polarity is called a Debye (D): 1 D = 3.336 x 10-30 Cm + -

  22. H H C 109º C CH4 H H Tetrahedron H Tetrahedral co-ordination H H H Cl 120 CCl4 109º C Cl Top view Cl Cl Examples of Nonpolar Molecules: u = 0 O=C=O CO2 methane Have rotational and mirror symmetry

  23. Examples of Polar Molecules CHCl3 CH3Cl Cl H C C H Cl H Cl H Cl Have lost some rotational and mirror symmetry! Unequal sharing of electrons between two unlike atoms leads to polarity in the bond  CH polarity ≠CCl polarity.

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

  25. Vector Addition of Bond Moments 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? O 1.51 D q/2 H H uH2O = 2 cos(q/2)uOH = 2 cos (52.25 °) x 1.51 D = 1.85 D

  26. Charge-Dipole Interactions + q Q - w(r) = -Cr-2 • 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 (Q1 and Q2): • Ions can induce ordering and alignment of polar molecules. • Why? Equilibrium state when w(r) is minimum. w(r) decreases (becomes more negative) as qincreases to 0 degrees. r

  27. Interactions between Fixed Dipoles + f + q1 q2 - - • There are Coulombic interactions between the +ve and -ve charges associated with each dipole. • 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. r Note: W(r) = -Cr-3

  28. 0 w(r) (J) kT at 300 K At a typical spacing of 0.4 nm, w(r) is about 1 kT. Hence, thermal energy is able to disrupt the alignment. Side-by-side q1 = q2 = 90° -10-19 End-to-end alignment lowers the energy more than side-by-side. But, small values of r cannot be achieved. End-to-end q1 = q2 = 0 W(r) = -Cr-3 From Israelachvili, Intermol. & Surf. Forces, p. 59 -2 x10-19 0.4 r (nm)

  29. Freely-Rotating Dipoles • In liquids and gases, dipoles do not have a fixed position and orientation on a lattice, but instead constantly “tumble” about. • Freely-rotating dipoles occur when the thermal energy is greater than the fixed dipole interaction energy: • The interaction energy depends inversely on T, and because of constant motion, there is no angular dependence: This interaction energy is called the Keesom energy. Note: w(r) = -Cr-6

  30. Units of polarisability: Polarisability • All molecules can have a dipole induced by an external electromagnetic field, • The strength of the induced dipole moment, |uind|, is determined by the polarisability, a, of the molecule:

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

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

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

  34. Units of volume Units of Electronic Polarisability Polarisability is often reported as: e0is the permittivity of free space (vacuum)

  35. Smallest Largest Electronic Polarisabilities ao/(4o) (10-30 m3) He 0.20 H2O 1.45 O2 1.60 CO 1.95 NH3 2.3 CO2 2.6 Xe 4.0 CHCl3 8.2 CCl4 10.5 Units ao/(4o): 10-30 m3 Numerically equivalent to ao in units of 1.11 x 10-40 C2m2J-1

  36. Example: Polarisation Induced by an Ion’s Field Consider Ca2+ dispersed in CCl4 (non-polar). Ca2+ CCl4 -+ What is the induced dipole moment in CCl4 at a distance of r = 2 nm? By how much is the electron cloud of the CCl4 shifted? From Israelachvili, Intermol.& Surf. Forces, p. 72

  37. Field from the Ca2+ ion: From the literature, we find for CCl4: Affected by the permittivity of CCl4: e= 2.2 uind = 3.82 x 10-31 Cm We find at a “close contact” of r = 2 nm: Thus, an electron with charge e is shifted by: Å Example: Polarisation Induced by an Ion Ca2+ dispersed in CCl4 (non-polar).

  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 2 - + 2 - + - + r

  39. u1  u2 So the induced dipole moment in the neighbour is: - + - + r Origin of the London or Dispersive Energy Instantaneous dipole Induced dipole The field produced by the instantaneous dipole is: We can now calculate the interaction energy between the two dipoles (using the equation for fixed, permanent dipoles - slide 27):

  40. We note that u can be approximated as eR, where R is the atomic radius. We recall the approximation that - + - + So we see that r Substituting for R2, we find: Origin of the London or Dispersive Energy R

  41. Substituting in the right-hand side of this equation (Z = 1), we see: This resultcompares favourably with London’s result (1937) that was derived from a quantum-mechanical approach using perturbations in the Schrödinger equation: Origin of the London or Dispersive Energy In the Bohr model, then angular momentum of the electron is quantised as mvr =nh/2pand the electron energy is quantised as: when it is orbiting with a frequency of. hn is the ionisation energy, i.e. the energy to remove an electron from the molecule

  42. The London result is of the form: where C is called the London constant: London or Dispersive Energy In simple liquids and solids consisting of non-polar molecules, such as N2 or O2, the dispersive energy is solely responsible for the cohesion of the condensed phase. Must consider the pair interaction energies of all “near” neighbours.

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

  44. van der Waals’ Interactions • Refers collectively to all interactions between polar or nonpolar molecules that vary as r-6. • Includes the Keesom, Debye and dispersive interactions. • Values of interaction energy are usually only a few kT (at RT), and hence are considered weak.

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

  46. 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 on Slide 43: +Q • 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).

  47. An external electric field can partially align dipoles: +  - The induced dipole moment is: Asu = aE, we can define an orientational polarisability. The molecule still has electronic polarisability, so the total polarisability,a, is given as: Debye-Langevin equation 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 qrepresent the angle between the dipole moment of a molecule and an external E-field direction. The spatially-averaged value of <cos2q> = 1/3

  48. Measuring Polarisability • Polarisability is dependent on the frequency of the E-field. • The Clausius-Mossotti equation relates the dielectric constant (permittivity) e of a molecule having a volume v to a: • At the frequency of visible light, however, only the electronic polarisability, ao, is active. • At these frequencies, the Lorenz-Lorentz equation relates the refractive index, n (n2 = e) to ao: So we see that measurements of the refractive index can be used to find the electronic polarisability.

  49. Frequency Dependence of Polarisability From Israelachvili, Intermol. Surf. Forces, p. 99

  50. PV diagram for CO2 P Van der Waals Gas Equation: Non-polar gases condense into liquids because of the dispersive (London) attractive energy. V it.wikipedia.org/wiki/Legge_di_Van_der_Waals

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