Characteristics of Soft Matter

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:

Interaction Potentials: w = -Cr -n

• If n<3, molecules interact with all others in the system of size, L. If n<3, molecules interact only with the nearer neighbours.

• Gravity: negligible at the molecular level. W(r) = -Cr -1

• Coulombic: relevant for salts, ionic liquids and charged molecules. W(r) = -Cr -1

• van der Waals’ Interaction: usually quite weak; causes attraction between ANY two molecules. W(r) = -Cr -6

• Covalent bonds: usually the strongest type of bond; directional forces - not described by a simple potential.

• Hydrogen bonding: stronger than van der Waals bonds; charge attracting resulting from unshielded proton in H.

In the previous lecture:

Lecture 2:

Polarisability and van der Waals’ Interactions:

Why are neutral molecules attractive to each other?

3SM Soft Matter Physics

24 January, 2008

See Israelachvili’s Intermolecular and Surface Forces, Ch. 4, 5 & 6

Polarity of Molecules• The intermolecular force is found from F = - dw/dr

• 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.

+_

_ +

Dipole Moments

A “convenient” (and conventional) unit for polarity is called a Debye (D):

1 D = 3.336 x 10-30 Cm

qu =

The polarity of a molecule is described by its dipole moment, u, given as:

where 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.

+ -

Examples of Nonpolar Molecules: u = 0

CO2 O-C-O

CH4C

H

H

H

H C

H

HH

H109º

CCl4

ClC

Cl

ClCl

109º

methane

Have rotational and mirror symmetry

120

Top view

Examples of Polar Molecules

CH3Cl CHCl3

Cmxu 3010246 _.= Cmxu 3010543 _.=

ClC

H

ClCl

C

Cl

HH

H

Have lost some rotational and mirror symmetry!

Unequal sharing of electrons between two unlike atoms leads to polarity in the bond.

Dipole moments

C=O u = 0.11 D

+ -

N

H HH u = 1.47 D

-

+

H

HO

- +

u = 1.85 D

SO Ou = 1.62 D

+

-

Bond moments

N-H 1.31 D

O-H 1.51 D

F-H 1.94 D

What is 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.

H H

Given that 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

O

1.51 D

uH2O = 2 cos(/2)uOH = 2 cos (52.25 °) x 1.51 D = 1.85 D

Vector Addition of Bond Moments

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 (Q1 and Q2):

• Ions can induce ordering and alignment of polar molecules.• Why? Equilibrium state when W(r) is minimum. W(r) decreases as

decreases to 0.

24 r

Qurw

ocos

_=)(

rQQ

rwo4

21=)(

Qu

r

+

-W(r) = -Cr -2

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 21u

2u

+ +

--

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 21u 2u

]cossinsin_coscos[_=)( 21213

21 24 r

uurw

o

r

Note: W(r) = -Cr -3

-

+

-

+

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.

nm10.=

nm10.=

-10-19

-2 x10-19

0

0.4

kT at 300 K

Dqu 1=||=||

End-to-end

Side-by-side

W(r) = -Cr -3

1 = 2 = 0

1 = 2 = 90°

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

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:

321

4 r

uukT

o>

62

22

21

43 kTr

uurw

o )(_=)(

Note: W(r) = -Cr -6

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, , of the molecule:

E

uind

=

Units of polarisability: J

mCNmC

CNCm

CmJCm 222

===

E

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

Simple Illustration of e- Polarisability

eEu oind

==

Force on the electron due to the field: EeFext

=

Attractive Coulombic force on the electron from nucleus:

32

2

2

2

int 4=

4=sin

4=

)(=

R

ueRR

e

R

edR

RdwF

o

ind

oo

At equilibrium, the forces balance:int= FFext

Without a field:With a field:

Fext

Fint

Simple Illustration of e- Polarisability

int= FFext

eEu oind

==

34 R

ueEe

o

ind

=Substituting expressions

for the forces:

Solving for the induced dipole moment: ERu oind

34=

So we obtain an expression for the polarisability: 34 Roo =

From this crude argument, we predict that electronic polarisability is proportional to the size of the molecule!

Units of Electronic Polarisability

3112

122

mmJC

JmC=__

_

Units of volume

o is often reported as:o

o

4

Electronic Polarisabilities

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

Largest

Smallest

Units:

(4o)10-30 m3

=1.11 x 10-40 C2m2J-1

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

Example: Polarisation Induced by an Ion

Ca2+ dispersed in CCl4 (non-polar). Eu oind

=

Affected by the permittivity of CCl4: = 2.2

2

24 r

eu

o

oind

=

330105104

mxo

o _.=From the literature,

we find for CCl4:

24

2

r

eE

o=

Field from the Ca2+ ion:

We find when r = 2 nm: u = 3.82 x 10-31 Cm

Thus, an electron with charge e is shifted by:

02010382 12 .=.== _ mxeu

Å

Polarisability of Polar MoleculesIn 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 <cos2> = 1/3

The induced dipole moment is: 22

cos=kT

Euuind

An external electric field can partially align dipoles:

E +

-

The molecule still has electronic polarisability, so the total

polarisability, , is given as:

kTu

o 3

2

+=Debye-Langevin equation

kTu

orient 3

2

=As u = E, we can define an orientational polarisability.

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- +1u

+ +- - 2u

1u

r


+ +- - 2u

1u

r

2123

1 314

/)cos+(= r

uE

o

u1

u2

The field produced by the instantaneous dipole is:

)(===

fr

uEuu

o

ooind 3

12 4

So the induced dipole moment in the neighbour is:

62

21

3

31

1

21321

44

4

4 r

u

r

r

uu

fr

uurw

o

o

o

o

o

o )(=

)(

=),,(=)(

We can now calculate the interaction energy between the two dipoles (using the equation for permanent dipoles - slide 12):

Instantaneous dipole

Induced dipole


+ +- - 2u

1u

r

62

21

4 r

urw

o

o

)(=)(

This result:

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

62

2

443

r

hrw

o

o

)(=)(

his the ionisation energy,

i.e. the energy to remove an electron from the molecule

London or Dispersive Energy

62

2

443

r

hrw

o

o

)(=)(

The London result is of the form: 6r

Crw =)(

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.

where C is called the London constant:

2

2

443

)(=

o

o hC

Must consider the pair interaction energies of all “near” neighbours.

Measuring Polarisability

• Polarisability is dependent on the frequency of the E-field.

• The Clausius-Mossotti equation relates the dielectric constant of a molecule with a volume v to :

43

21

4v

o

)(

43

21

4 2

2 vnn

o

o )(

•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 (n2 = ) to o:

Frequency dependence of polarisability

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

it.wikipedia.org/wiki/Legge_di_Van_der_Waals

PV diagram for CO2

RTbVV

aP ))(( 2


• The van der Waals’ gas law can be written (with V = molar volume) as:

RTbVV

aP ))(( 2

33

2

C

a =

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 .


SummaryType of Interaction Interaction Energy, w(r)

Charge-charge rQQ

o421 Coulombic

Nonpolar-nonpolar 62

2

443

r

hrw

o

o

)(_=)(

Dispersive

Charge-nonpolar 42

2

42 rQ

o )(_

Dipole-charge24 r

Qu

ocos_

42

22

46 kTruQ

o )(_

Dipole-dipole

62

22

21

43 kTruu

o )(_

Keesom

321

22

21

4 rfuu

o ),,(_

Dipole-nonpolar

62

2

4 ru

o )(_

Debye

62

22

4231

ru

o )()cos+(_

In vacuum: =1

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.

Comparison of the Dependence of Interaction Potentials on r

Not a comparison of the magnitudes of the energies!

n = 1

n = 2

n = 3n = 6

Coulombic

van der Waals

Dipole-dipole

Interaction 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:

42

22

46 kTruQ

o )(_

• 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).

Cohesive Energy• Def’n.: Energy required to separate all molecules in

the condensed phase or energy holding molecules in the condensed phase.

• In Lecture 1, we found that for a single molecule, and with n>3:

• with = number of molecules per unit volume -3, where is the molecular diameter. So, with n = 6:

• For one mole, Esubstance = (1/2)NAE

• Esubstance = sum of heats of melting + vaporisation.

• Predictions agree well with experiment!

334

nn

CE

)(

63 34

34

CC

E 1/2 to avoid double counting!

Boiling Point• At the boiling point, TB, for a liquid, the thermal energy of a monoatomic molecule, 3/2 kTB, will exactly equal the energy of attraction between molecules.

• Of course, the strongest attraction will be between the “nearest neighbours”, rather than pairs of molecules that are farther away.

• The interaction energy for van der Waals’ interactions is of the form, w(r) = -Cr -6. If molecules have a diameter of , then the shortest centre-to-centre distance will likewise be .

• Thus the boiling point is approximately:

k

wTB

23

)(=

Comparison of Theory and Experiment

63

42

CNE A

mole ~

Evaluated at close contact where r = .

k

rwTB

23

)(=Note that o and C increase with .

C can be found experimentally from deviations from the ideal gas law:

RTbVV

aP =)_)(+( 2

Cohesive energy = energy required to separate molecules to a large distance (solid gas)

Additivity of Interactions

Molecule Mol. Wt. u (D) TB(°C)

Ethane: CH3CH3 30 0 -89

Formaldehyde: HCHO 30 2.3 -21

Methanol: CH3OH 32 1.7 64

C=OH

H

C-O-HH

HH

C-CH

HH

H

HH Dispersive only

Keesom + dispersive

H-bonding + Keesom + dispersive

Problem Set 11. 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 14.45 12.25A12 6.20 12.13 9.11

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 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.)

,=)(6

6

12

122r

Ar

Aru

r

ze

Interaction betw een ions andpolar molecules

• Interactions involving charged molecules (e.g. ions)tend to be stronger than polar-polar interactions.

• For freely-rotating dipoles with a moment of uinteracting with molecules with a charge of Q we saw:

42

22

46 kTruQ

o )(_

• One result of this interaction energy is the condensation ofwater (u = 1.85 D) caused by the presence of ions in theatmosphere.

• During a thunderstorm, ions are created that nucleate raindrops in thunderclouds (ionic nucleation).

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.

HO

HH

HO

+

+

++

--

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

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 - TS = -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 nm2 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.

De-wetting

“Froth flotation”

Protein folding

Adhesion in water

ImmiscibilityMicellisation

Association of molecules

Coagulation

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Characteristics of Soft Matter