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In the previous lecture:. Characteristics of Soft Matter. Length scales between atomic and macroscopic (sometimes called mesoscopic ) (2) The importance of thermal fluctuations and Brownian motion

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characteristics of soft matter

In the previous lecture:

Characteristics of Soft Matter
  • 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.

slide2

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

slide3

What are the forces that operate over short distances and hold soft matter together?

http://www.cchem.berkeley.edu/rmgrp/about_gecko.jpg

interaction potentials
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
slide5

+

w(r)

-

Repulsive potential

r

s

wrep(r) = (s/r)

“Hard-Sphere” Interaction Potential

+

w(r)

-

Attractive potential

r

watt(r) = -C/rn

slide6

Hard-Sphere Interaction Potentials

+

w(r)

-

Total potential:

s

r

w(r) = watt + wrep

Minimum of potential = equilibrium spacing in a solid = s

The force, F, acting on particles with this interaction energy is:

interaction potentials1
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?
slide8

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!

slide9

Q2

Q1

r

Coulombic Interactions: n = 1

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 crystals.

slide10

van der Waals Interactions (London dispersion energy): 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.

slide11

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.!

slide12

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.
  • 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.
significance of interaction potentials
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, F is compressive (+); If r > re, F is tensile (-).
  • When dF/dr = d2w/dr2 =0, attractive Fis at its maximum.

re = equilibrium spacing

slide14

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?

slide15

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:

r -n+2=r-(n-2)

But L >> s! When can we neglect the term?

conclusions about e

E=

Conclusions about E
  • There are three cases:
  • When n<3, then the exponent is negative. As L>>s, 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 in a solar system.)
  • 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!
slide17

The Third Case: n = 3

swill be very small (typically 10-9 m), but lnsis not negligible. L cannot be neglected in most cases.

What values of n apply to molecular interaction potentials? Is it >, < or = 3?

polarity of molecules
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.

_

+

+

_

slide19

when charges of+qand - qare 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.

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

+

-

slide20

H

H

C

109º

C

CH4

H

H

H

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

slide21

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.

slide22

+ -

C=Ou = 0.11 D

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.

slide23

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

charge dipole interactions
Charge-Dipole Interactions

+

q

Q

-

  • 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 as qdecreases to 0.

w(r) = -Cr-2

r

dipole dipole interactions
Dipole-Dipole Interactions

+

+

q1

q2

-

-

  • 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 q1and q2.
fixed dipole interactions
Fixed-dipole Interactions

+

f

+

q1

q2

-

-

  • 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

slide27

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

From Israelachvili, Intermol.& Surf. Forces, p. 59

End-to-end

q1 = q2 = 0

W(r) = -Cr-3

-2 x10-19

0.4

r (nm)

freely rotating dipoles
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

polarisability

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:
polarisability of nonpolar molecules
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

slide31

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:

slide32

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 the molecule!

units of electronic polarisability

Units of volume

Units of Electronic Polarisability

Polarisability is often reported as:

electronic polarisabilities

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

slide35

Example: Polarisation Induced by an Ion

Ca2+ dispersed in CCl4 (non-polar).

-+

What is the induced dipole moment in CCl4 at a distance of 2 nm?

By how much is the electron cloud of the CCl4 shifted?

From Israelachvili, Intermol.& Surf. Forces, p. 72

slide36

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 close contact when 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).

slide37

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

origin of the london or dispersive energy
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

slide39

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:

u1

u2

We can now calculate the interaction energy between the two dipoles (using the equation for permanent dipoles - slide 27):

slide40

This result:

compares favourably with the London result (1937) that was derived from a quantum-mechanical approach:

-

+

-

+

r

hn is the ionisation energy, i.e. the energy to remove an electron from the molecule

Origin of the London or Dispersive Energy

slide41

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.

slide42

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

van der waals interactions
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.
slide44

Comparison of the Dependence of Interaction Potentials on r

n = 1

Coulombic

n = 2

Charge-fixed dipole

n = 6

n = 3

van der Waals

Dipole-dipole

Not a comparison of the magnitudes of the energies!

slide45

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:

+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).

measuring polarisability
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.

slide47

Frequency dependence of polarisability

From Israelachvili, Intermol. Surf. Forces, p. 99

slide48

PV diagram for CO2

Van der Waals Gas Equation:

Non-polar gasses condense into liquids because of the dispersive (London) attractive energy.

it.wikipedia.org/wiki/Legge_di_Van_der_Waals

measuring polarisability1
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:

wheresis the molecular diameter (= closest molecular spacing). We can thus use the C-M, L-L and v.d.W. equations to find values for aoanda.

measuring polarisability2
Measuring Polarisability

From Israelachvili, Intermol.& Surf. Forces

slide51

ze

q

r

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", An, are given below for each of the three cubic lattices.

SC BCC FCC

A6 8.40 12.25 14.45

A12 6.20 9.11 12.13

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 q with relation to r, as shown below.

(ii) Evaluate your expression for a Mg2+ 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.

hydrogen bonding
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.
  • 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.
hydrophobic interactions
Hydrophobic Interactions

A water “cage” around another molecule

  • “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.
hydrophobic interactions1
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: DG = DH - TDS = -4.3 +28.7 = +24.5 kJ mol-1. Unfavourable (+ve DG) because of the decrease in entropy!
  • This value of DG is consistent with a “surface area” of n-butane of 1 nm2 and g40 mJ m-2 for the water/butane interface; an increase in DG = gDA 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.
slide58

Micellisation

Immiscibility

Association of molecules

Protein folding

Adhesion in water

De-wetting

“Froth flotation”

Coagulation