characteristics of soft matter
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/1.jpg)
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)
In the previous lecture:
![Page 2: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/2.jpg)
PH3-SM (PHY3032)
Soft Matter PhysicsLecture 2:
Polarisability and van der Waals’ Interactions:
Why are neutral molecules attractive to each other?
11 October, 2010
See Israelachvili’s Intermolecular and Surface Forces, Ch. 4, 5 & 6
![Page 3: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/3.jpg)
What are the forces that operate over short distances to hold condensed matter together?
Nitrogen condensed in the liquid phase.
![Page 4: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/4.jpg)
What are the forces that operate over short distances to cause adhesion?
http://www.cchem.berkeley.edu/rmgrp/about_gecko.jpg
![Page 5: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/5.jpg)
Interaction Potentials
• For two atoms/molecules/segments separated by a distance of r, the interaction energy can be described by an attractive potential energy: watt(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, s. Use a simple repulsive potential:
wrep(r) = +(s/r)
• The interaction potential w(r) = watt + wrep
r
s
![Page 6: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/6.jpg)
“Hard-Sphere” Interaction Potential
+
w(r)
-
Attractive potential
r
watt(r) = -C/rn
+
w(r)
-
Repulsive potential
rswrep(r) = (s/r)
r
s
![Page 7: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/7.jpg)
Hard-Sphere Interaction Potentials
+
w(r)
-Total potential:
rw(r) = watt + wrep
s
Minimum of potential = equilibrium spacing in a solid = s
The force, F, acting between atoms (molecules) with this interaction energy is:
drdwF
where a negative force is attractive. As r , F 0.
![Page 8: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/8.jpg)
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?
![Page 9: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/9.jpg)
Gravity: n = 1
r
m1m2
rmGmrmGmrw 211
21)(
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)!
![Page 10: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/10.jpg)
Coulombic Interactions: n = 1
r
Q1Q2 rQQ
rwo4
21=)(
• With Q1 = z1e, where e is the charge on the electron, and z1 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 ionic crystals.
• When molecules are in close contact, w(r) is typically ~ 10-18 J, corresponding to about 200 to 300 kT at room temp.
The sign of w depends on whether charges are alike or opposite.
![Page 11: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/11.jpg)
van der Waals Interactions (London dispersion energy): n = 6
ra1
a26)(rCrw
u2 u1
• 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.
![Page 12: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/12.jpg)
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.!
![Page 13: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/13.jpg)
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. 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.• 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.
HO
HH
HO
d+
d+
d+d+
d-d-
![Page 14: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/14.jpg)
• 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, attractive F is at its maximum.
Significance of Interaction Potentials
re = equilibrium spacing
![Page 15: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/15.jpg)
r
How much energy is required to remove a molecule from the condensed phase?
Q: Does a central molecule interact with ALL the others?
nrCrw =)(
Applies to pairs
L
s = molecular spacing when molecules are in contact
r = density = number of molec./volume
Individual molecules
•
s
![Page 16: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/16.jpg)
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:
drrv 24)4()( 2drrrN r
Lr
rnrn
CE
s
r3-
1)3-(
4- 3-3 )(1
)3(4 n
n LnCE ssr
But L >> s ! When can we neglect the term?
r -n+2=r -(n-2) System L
nrCrNrwEs
r 24)()(
![Page 17: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/17.jpg)
Conclusions about E
• There are three cases:• (1) 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.) • (2) But when 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!
[ ] 33
3 )3(4
≈)(1)3(
4n
nn n
CLn
Csrs
sr
E=
33 )(1
)3(4
nn Ln
CE ssr
![Page 18: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/18.jpg)
The Third Case: n = 33)( Crrw
drrrN 24)( r
srr
s
lnln44
LCrCdrE
Lr
r
s will be very small (typically 10-9 m), but lns is not negligible. L cannot be neglected in most cases.
Which values of n apply to various molecular interaction potentials? Are they >, < or = 3?
![Page 19: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/19.jpg)
http://www.chem1.com/acad/webtext/atoms/atpt-4.html
Electron Probability Distributions
1s orbitals in the H atom H orbitals with n = 3
• Symmetric distribution electrons in atomic orbitals• Position of the electrons cannot be known with certainty.
![Page 20: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/20.jpg)
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.
O nucleus 8+
O nucleus 8+ --
O2 is non-polar CO is polar
O nucleus 8+
C nucleus 6+ --
Centre of +ve charge
Centre of -ve charge
![Page 21: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/21.jpg)
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:
when charges of +q and - q are 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.
+ -
![Page 22: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/22.jpg)
Examples of Nonpolar Molecules: u = 0
CO2 O=C=O
CCl4
ClC
Cl
ClCl
109º
methane
Have rotational and mirror symmetry
120
Top view
C
H
HH
H109º
Tetrahedral co-ordination
CH4C
H
H
HH
Tetrahedron
![Page 23: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/23.jpg)
Examples of Polar MoleculesCH3Cl CHCl3
Cmxu 301024.6
Cmxu 301054.3
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 CH polarity ≠CCl polarity.
![Page 24: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/24.jpg)
Dipole moments
NH H
H 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 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.
![Page 25: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/25.jpg)
H H
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?
q/2
O1.51 D
uH2O = 2 cos(q/2)uOH = 2 cos (52.25 °) x 1.51 D = 1.85 D
Vector Addition of Bond Moments
![Page 26: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/26.jpg)
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 q decreases
to 0.
24cos)(r
Qurwoq
rQQrwo4
)( 21
Q qu
r
+
- w(r) = -Cr -2
![Page 27: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/27.jpg)
Interactions between Fixed Dipoles
• 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.
q1 q21u 2u f
]cossinsincoscos2[4
)( 2121321 fqqqq
ruurwo
r
Note: W(r) = -Cr -3
-
+
-
+
![Page 28: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/28.jpg)
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
q1 = q2 = 0
q1 = q2 = 90°
From Israelachvili, Intermol. & Surf. Forces, p. 59
End-to-end alignment lowers the energy more than side-by-side. But, small values of r cannot be achieved.
![Page 29: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/29.jpg)
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:
321
4 ruu
kTo
>
62
22
21
)4(3)(
kTruurw
o
Note: w(r) = -Cr -6
This interaction energy is called the Keesom energy.
![Page 30: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/30.jpg)
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:
Euind
=a
Units of polarisability: J
mCNmC
CNCm
CmJCm 222
===
E
![Page 31: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/31.jpg)
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, ao.
_E
_
Initial state In an electric field
+ +
![Page 32: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/32.jpg)
Simple Bohr Model of e- Polarisability
eEu oind
==a
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=
)(=
Rue
RRe
Re
dRRdw
Fo
ind
oo q
At equilibrium, the forces balance: int= FFext
Without a field: With a field:
Fext
Fint
![Page 33: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/33.jpg)
int= FFext
eEu oind
==a
34 Rue
Eeo
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 a =
From this crude argument, we predict that electronic polarisability is proportional to the size of a molecule!
Simple Bohr Model of e- Polarisability
![Page 34: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/34.jpg)
Units of Electronic Polarisability
3112
122
mmJCJmC
Units of volume
Polarisability is often reported as:o
oa
4
![Page 35: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/35.jpg)
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 10.5Largest
Smallest
Unitsao/(4o): 10-30 m3
Numerically equivalent to ao in units of 1.11 x 10-40 C2m2J-1
ao/(4o) (10-30 m3)
![Page 36: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/36.jpg)
Example: Polarisation Induced by an Ion’s Field
Consider 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
Ca2+
CCl4
![Page 37: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/37.jpg)
Example: Polarisation Induced by an IonCa2+ dispersed in CCl4 (non-polar). Eu oind
a=
Affected by the permittivity of CCl4: = 2.2
22
4 re
uo
oind
a=
330105.104
mxo
o aFrom the literature, we
find for CCl4:
242
re
Eo
=Field from the
Ca2+ ion:
We find at a “close contact” of r = 2 nm:
uind = 3.82 x 10-31 Cm
Thus, an electron with charge e is shifted by:
02.01038.2 12 mxeu
Å
![Page 38: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/38.jpg)
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.
a1 a2
a2- +1u
+ +- - 2u1u
r
![Page 39: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/39.jpg)
Origin of the London or Dispersive Energy
+ +- - 2u
1u
r
2123
1 314
/)cos+(= q ru
Eo
u1
u2
q
The field produced by the instantaneous dipole is:
)(=== qa
a fr
uEuu
o
ooind 3
12 4
So the induced dipole moment in the neighbour is:
62
21
3
31
1
21321
444
4 ru
rr
uuf
ruurw
o
o
o
o
o
o )(
)(),,()(
a
a
fqq
We can now calculate the interaction energy between the two dipoles (using the equation for fixed, permanent dipoles - slide 27):
Instantaneous dipole
Induced dipole
![Page 40: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/40.jpg)
Origin of the London or Dispersive Energy
+ +- - 2u
1u
r
62
21
)4()(
rurwo
o
a
This result:
compares favourably with the London result (1937) that was derived from a quantum-mechanical approach:
62
2
)4(43)(
rhrwo
o
a
h is the ionisation energy, i.e. the energy to remove an electron from the molecule
![Page 41: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/41.jpg)
London or Dispersive Energy
62
2
)4(43)(
rhrwo
o
a
The London result is of the form: 6)(rCrw
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
)4(43
o
o hC
a
Must consider the pair interaction energies of all “near” neighbours.
![Page 42: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/42.jpg)
SummaryType of Interaction Interaction Energy, w(r)
Charge-charge rQQ
o421 Coulombic
Nonpolar-nonpolar 62
2
443
rh
rwo
o
)(_=)(
a
Dispersive
Charge-nonpolar 42
2
42 rQ
o )(_
a
Dipole-charge24 r
Quo
qcos_
42
22
46 kTruQ
o )(_
Dipole-dipole
62
22
21
43 kTruu
o )(_
Keesom
321
22
21
4 rfuu
ofqq ),,(_
Dipole-nonpolar
62
2
4 ru
o )(_
a
Debye
62
22
4231
ru
o )()cos+(_
qa
In vacuum: =1
![Page 43: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/43.jpg)
van der Waals’ Interactions
• Refers 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.
![Page 44: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/44.jpg)
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
Fixed dipole-dipole
Charge-fixed dipole
![Page 45: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/45.jpg)
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:
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).
+Q
![Page 46: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/46.jpg)
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 q represent the angle between the dipole moment of a molecule and an external E-field direction.
The spatially-averaged value of <cos2q> = 1/3
The induced dipole moment is: q22
cos=kT
Euuind
An external electric field can partially align dipoles:
E +
-q
The molecule still has electronic polarisability, so the total polarisability, a, is given as:
kTu
o 3
2+=aa Debye-Langevin
equation
kTu
orient 3
2=aAs u = aE, we can define an orientational polarisability.
![Page 47: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/47.jpg)
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 a:
a
43
21
4v
o
)(
a
43
21
4 2
2 vnn
o
o )(
• 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 = ) to ao:
So we see that measurements of the refractive index can be used to find the electronic polarisability.
![Page 48: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/48.jpg)
Frequency dependence of polarisability
From Israelachvili, Intermol. Surf. Forces, p. 99
![Page 49: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/49.jpg)
it.wikipedia.org/wiki/Legge_di_Van_der_Waals
PV diagram for CO2
RTnbVVaP ))(( 2
Non-polar gasses condense into liquids because of the dispersive (London) attractive energy.
Van der Waals Gas Equation:
P
V
![Page 50: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/50.jpg)
Measuring Polarisability• The van der Waals’ gas law can be written (with V = molar
volume) as:
RTnbVVaP ))(( 2
332sCa
The constant, a, is directly related to the London constant, C:
where s 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 ao and a.
![Page 51: Characteristics of Soft Matter](https://reader035.vdocuments.site/reader035/viewer/2022062305/56816357550346895dd40bc2/html5/thumbnails/51.jpg)
Measuring Polarisability
From Israelachvili, Intermol.& Surf. Forces
Polarisability determined from van der Waals gas (a) and u measurements.
Polarisability determined from dielectric/index measurements.
<
<
<
High f
Low f