Intermolecular and

Surface Forces
Young-Ki Kim
Department of Chemical Engineering
2025 Fall Semester 00
 Lecture Outline
• Prof. Young-Ki Kim
E-Mail : ykkim@postech.ac.kr Tel. : 054-279-2267
Office : RIST 3, Room 3329
• What We Learn
Fundamental and experimental insights into various
intermolecular and surface interaction
• Goal : Understand
1) Interaction between Atoms and Molecules
2) Interaction between Particles and Surface
3) Self-Assembly

Intermolecular and Surface Forces 01

 Summary
➢ 강의 전체 내용 Summary
1) 수기 (그림은 copy 가능)
2) 분량 자유
3) 자유 형식 (서술형, bullet point 등 자신만의 방식으로)
4) 한글로 작성
5) Scientific Writing
- 자신을 방문판매원이라 생각하고 작성
- Reader/Audience 를 비전문가로 생각하고 작성
6) 단순히 Lecture note 를 옮겨 적는게 아니라10 년 후 자신이 필요할때 꺼내보았을때
도움이 될수 있게 Summary
- e.g.) 자세한 내용을 lecture note 에서 찾아볼수 있게 page 또는 chapter 표기
7) 수식 및 Terminology 의 의미가 중요!

Intermolecular and Surface Forces

1. Historical Perspective

Intermolecular and Surface Forces 02

 Four Forces of Nature
1) Strong and 2) Weak Interactions
✓ Between neutrons, protons, electrons, and other elementary
particles
✓ Short range interactions (< 10-5 nm)
✓ Domain of nuclear and high-energy physics

• Strong nuclear interaction

: Protons and neutrons together in atomic nuclei

• Weak Interactions
: electron emission

Intermolecular and Surface Forces 03

 Four Forces of Nature
3) Electromagnetic and 4) Gravitational Interactions
✓ Between atoms and molecules & Between elementary particles
✓ Long range interaction (subatomic scale to ∞)
➢ Electromagnetic Forces
✓ Source of all intermolecular interactions
✓ Determine properties of solids, liquid crystals,
liquid, and gas
✓ Determine Particle behavior in solution,
chemical reactions, and their organization
✓ e.g.) Electrostatic (intermolecular) forces :
Determine the cohesive forces that hold atoms
and molecules together
Intermolecular and Surface Forces 04
 Four Forces of Nature
3) Electromagnetic and 4) Gravitational Interactions
✓ Between atoms and molecules & Between elementary particles
✓ Long range interaction (subatomic scale to ∞)

➢ Gravitational Forces
✓ Account for all motions and phenomena
( e.g., tide, falling objects, satellite, floatation)
✓ Gravitational + Intermolecular Forces determine
the maximum possible size of building, trees,
mountains, and animals.

Intermolecular and Surface Forces 05

 Brief History

➢ Scientist who made major

contributions to our
understanding of Intermolecular
and Surface forces

Intermolecular and Surface Forces 06

 Brief History : Greek and Medieval
➢ Greek & Medieval notions of intermolecular forces
✓ First to consider forces in a nonreligious way
✓ Only two fundamental forces for all natural phenomena
➔ Love (Attraction) & Hate (Repulsion)
✓ Empedocles (493~430 B.C.), Aristotle (384~322 B.C.), Archimedes (287~212 B.C.)

• Archimedes Principle • Archimedes Screw • Archimedes Death Ray

Intermolecular and Surface Forces 07

 Brief History : 17th Century
➢ First Scientific Period
✓ Conducting experiments to find out how nature works
➢ Galileo Galilei (1564~1642)
✓ Classic experiments on gravity, motion of bodies, optics, astronomy, vacuum
➢ Robert Boyle (1627~1691)
✓ Gas Law : PV = Constant
➢ Isaac Newton (1642~1727)
✓ Law of Gravity
✓ Molecules of a gas must ultimately attract each other

Contradictions ➔ Scientific Advances

Intermolecular and Surface Forces 08
 Brief History : 18th Century
➢ Confusion, Contradictions, and Controversy Period
1) Inverse distance law for the behavior of gas
✓ Becomes unphysical to account for many molecules or large distance
2) Newton’s law of gravity
✓ Instantaneous forces to act across a vast vacuum
3) Force between particles over distances
✓ Attractive (gravity force) ➔ Repulsive (gases) ➔ Attractive (solids, liquids)
➔ Repulsive (matters never disappear)
4) Capillary rise of liquids
✓ Height of rise does not depend on the capillary wall thickness, but thin!

✓ Tube with macroscopic inner radii ➔ Substantial height

Intermolecular and Surface Forces 09
 Brief History : 19th Century
➢ Continuum vs Molecules
✓ Controversy regarding Matter = Continuum vs Assembly of Molecules
➢ First half of 19th Century ➔ Continuum theory Win
Maxwell : “All those who believe in molecules are now dead”
➢ Capillary rise of liquids does not depend on the wall thickness, but thin!
(Short-range intermolecular force)
✓ Clairaut : It works if the attraction between the liquid and glass molecules ≠
attraction between liquid molecules
➢ Latter half of 19th Century ➔ Return to molecules
✓ Kinetic theory and Van der Waals Eq require attracting molecules
✓ Unifying continuum and molecular theories ➔ Origin still in question
Intermolecular and Surface Forces 10
 Long- and Short- Range Forces (early 19th century)
➢ Simple universal Force law or Potential energy function can eventually
explain all intermolecular attractions
✓ A number of interaction potentials were proposed
𝑑𝑤 𝑟 𝑚1 𝑚2 n = 4, 5 ➔ usual interactions
𝐹 𝑟 =− = −𝑛𝐶 𝑛+1 n =1 ➔ Gravitational interaction
𝑑𝑟 𝑟
✓ Short-range interaction (Newton) ➔ n > 3 & Long-range interaction ➔ n ≤ 3
𝐿
𝐶
𝜀 = න 𝑤 𝑟 𝜌 4π 𝑟2 d𝑟 ← 𝑤 𝑟 = − 𝑛
𝜎 𝑟

Intermolecular and Surface Forces 11

 Long- and Short- Range Forces (early 19th century)
𝑑𝑤 𝑟 𝑚1 𝑚2 𝐶
𝐹 𝑟 =− = −𝑛𝐶 𝑛+1 , 𝑤 𝑟 ∝− 𝑛
𝑑𝑟 𝑟 𝑟

➢ Intermolecular force ➔ n > 3 (Short-Range Forces)

✓ Intrinsic bulk properties of solids, liquids, liquid crystals, and gases
Independent on the volume of materials or on the size of the container
Only Dependent on the forces between molecules in close proximity

➢ Gravity (n = 1), Magnetic or Electrical Dipoles (n = 3) ➔ Long-Range Force

✓ Magnetic dipoles align along the same direction and Magnetic forces can be felt
over very large distances (e.g., earth magnetic field)
✓ Electrical dipoles provide long-range “ordering” of the molecules (e.g., LCs)

Intermolecular and Surface Forces 12

 Long- and Short- Range Forces (Later 19th century)
➢ Simple universal Force law or Potential energy function can explain all
intermolecular attractions

➢ Return to molecular theories

✓ Concept of surface tension
(capillary rise)
✓ Short-range forces can lead to
Long-range effects
✓ Long-range effects is not always
associated with Long-range forces
✓ Strength of the interaction is matter !
➔ Typically SR forces > LR forces

Intermolecular and Surface Forces 13

 Phenomenological Theories (Later 19th – 20th century)
➢ Boyle’s Gas Law : PV = nRT = NkT ➔ Molecules repel each other
✓ Real gas does not obey the law!!
➢ van der Waals Equation
✓ Consider the effects of intermolecular attraction (van der Waals forces)
b : the finite size of molecules
𝑃 + 𝑎/𝑉 2 𝑉 − 𝑏 = 𝑅𝑇 a/V2 : for the attractive intermolecular forces

➢ Intermolecular force is not a simple nature

✓ New terms start to be added ➔ Mie Potential (1903)
𝐴 𝐵
𝑤 𝑟 =− 𝑛 + 𝑚
𝑟 𝑟
✓ First principle to account for a wide range of phenomena
Intermolecular and Surface Forces 14
 Recent Trends in Intermolecular & Surface Science
➢ Recent Main Areas of Activities in Intermolecular and Surface Sciences
1) Interactions between simple atoms and molecules in gases
✓ Quantum mechanics & Statistical Mechanics
2) Chemical bonding of ion, atoms, and molecules in solids
3) Long-range interaction between surfaces and collides Bacteria
✓ Colloidal Science
4) Broadening to interactions in Liquid structure, Surface
phenomena, Complex fluids, Soft matter, Self-assembly,
Quantum dots, Smart material, biological molecules
5) Beyond equilibrium Smart Liquid Crystal 50 µm

Intermolecular and Surface Forces 15

2. Thermodynamic and
Statistical Aspects of
Intermolecular Forces

Intermolecular and Surface Forces 16

 Intermolecular Interactions in a Medium
➢ Pair potential (= free energy): Interaction potential between two molecules or particles
𝐶 𝑑𝑤(𝑟)
𝑤 𝑟 =− 𝑛, 𝐹=−
𝑟 𝑑𝑟
➢ Pair potential in a Medium (liquid, liquid crystal, solid..)
✓ Many body interaction: solute-solute + ???
➢ Solvent effects (Medium effect)
1) Force inversion
✓ Solute molecule can approach another only by displacing solvent molecules

• Attracting molecules in free space

can repel each other in a medium

Intermolecular and Surface Forces 17

 Intermolecular Interactions in a Medium
2) Solvation (reordering) Force
✓ Solute molecule can perturb the local ordering of
solvent molecules (e.g., complex fluids, liquid
crystals, polymer) ➔ Solvation energy
3) Change in the properties of solute
✓ Solute-solvent interaction can change the properties of solute molecules such as
dipole moment and charge.
(e.g., particle in a liquid crystal acts like a dipole or quadrupole)

4) Cavity Energy
✓ To introduce solute molecules into a medium, it
require to overcome the solvent-solvent interaction
to make a cavity for the solute molecules

Intermolecular and Surface Forces 18

 Cohesive Energy and Pair Potential
➢ Pair Potential: Interaction between two molecules w(r) = ‒ C / rn for r ≥ σ (n > 3)
➢ Cohesive Energy (𝜇𝑖 ): Interactions of a molecule with all surrounding molecules
∞
4𝜋𝐶𝜌
𝜇𝑖 = න 𝑤 𝑟 𝜌 4π 𝑟2 d𝑟 = −
𝜎 (𝑛 − 3)𝜎 𝑛−3

Intermolecular and Surface Forces 19

 Cohesive Energy and Pair Potential
➢ Pair Potential: Interaction between two molecules w(r) = ‒ C / rn for r ≥ σ (n > 3)
➢ Cohesive Energy (𝜇𝑖 ): Interactions of a molecule with all surrounding molecules
∞
4𝜋𝐶𝜌
𝜇𝑖 = න 𝑤 𝑟 𝜌 4π 𝑟2 d𝑟 = −
𝜎 (𝑛 − 3)𝜎 𝑛−3
➢ When a molecules in vapor is introduced into its condensed phase,
𝜇𝑖 is required to include the cavity energy
✓ Net energy change ➔
✓ Cohesive energy for an introduced molecule

➢ Molar cohesive energy (for 1 mole of molecules) ➔ 𝑈 = −𝑁𝑜 𝜇Intro

𝑖

Intermolecular and Surface Forces 19

 Cohesive Energy and Pair Potential
𝑖 1 ∞ 12
➢ Cohesive Energy 𝜇Intro = ‫ 𝜌 𝑟 𝑤 𝜎׬‬4π 𝑟 2 d𝑟 ≈ 𝑤(𝜎)
2 𝑛−3

➢ van der Waals Interaction (n = 6) 𝜇𝑖 ≈ 4𝑤 𝜎 , 𝑈 ≈ −4𝑁𝑜 𝑤 𝜎

➢ Cohesive ε in a pure liquid or solid = 4𝑤 𝜎 ~ 6𝑤(𝜎)

➢ Accurate calculation of 𝜇𝑖 is extremely difficult

✓ # of surrounding molecules is not known
✓ ρ is not uniform locally ➔ Density distribution function ρ(r)

➢ Solute molecule dissolved in a different medium

𝑖
𝜇Intro ≈ −6𝑤mm 𝜎 + 12𝑤sm 𝜎

Intermolecular and Surface Forces 20

 Boltzmann Distribution Function
➢ Boltzmann Distribution
✓ For the molecule having different values of μi in two regions (coexisting phase)

𝑋1 = 𝑋2 exp[−(𝜇1i − 𝜇2i )/𝑘𝑇] 𝜇1i + 𝑘𝑇 log𝑋1 = 𝜇2i + 𝑘𝑇 log𝑋2

➢ If there are many different regions

𝜇𝑛i + 𝑘𝑇 log𝑋𝑛 = 𝜇 = ? ? ? ? 𝑛 = 1,2,3, …
✓ Material flow occurs until Boltzmann distribution is satisfied
✓ The requirement provides a general rule for formulating equilibrium condition within a
molecular framework
✓ Can be used to find the spatial or density distribution of molecules in different region
Intermolecular and Surface Forces 21
 Molecular Distribution at Equilibrium
𝑋1 = 𝑋2 exp[−(𝜇1i − 𝜇2i )/𝑘𝑇] 𝜇1i + 𝑘𝑇 log𝑋1 = 𝜇2i + 𝑘𝑇 log𝑋2
➢ Earth’s Atmosphere
𝜌𝑧 = 𝜌0 exp[−(𝜇𝑧i − 𝜇0i )/𝑘𝑇] = 𝜌0 exp[−𝑚𝑔𝑧/𝑘𝑇]
➢ Charged Molecules / Ions
𝜌2 = 𝜌1 exp[−(𝜇2i − 𝜇1i )/𝑘𝑇] = 𝜌0 exp[−𝑒(𝜓2 − 𝜓1 )/𝑘𝑇] ➔ Nernst Eq.
➢ Solute in a Solvent (e.g., salt that is not fully dissolved in water)
𝑋2 = 𝑋1 exp[−(𝜇2i − 𝜇1i )/𝑘𝑇] ➔ X2 = Solubility of solute in a solvent
➢ For independent interactions
𝑋2 = 𝑋1 exp[−(Δ𝜇𝑖 + 𝑚𝑔Δ𝑧 + 𝑒Δ𝜓)/𝑘𝑇]
✓ Many interactions are NOT independent (e.g., van der Waals, structural forces)

Intermolecular and Surface Forces 22

 van der Waals Equation
➢ Chemical potential of Gas: 𝜇 = 𝜇gas
i + 𝑘𝑇 log𝑋gas =
∞
4𝜋𝐶𝜌
𝑖
𝜇gas = න 𝑤 𝑟 𝜌 4π 𝑟 2 d𝑟 = −
𝑛 − 3 𝜎 𝑛−3
= −𝐴𝜌 • A = 4πC/(n-3)σn-3 = Constant
𝜎

𝑣0 𝑣0 𝜌 • vo : Actual volume of molecule

𝑋gas = = • v : Gaseous volume occupied by per molecule (System volume)
(𝑣 − 𝐵) (1 − 𝐵𝜌)
• B = 4πσ3/3 (Excluded volume)
➢ Thermodynamic relation
𝜌
𝜕𝑃 𝜕𝜇 𝜕𝜇
=𝜌 ⟹𝑃=න 𝜌 d𝜌 =
𝜕𝜌 𝑇
𝜕𝜌 𝑇 0 𝜕𝜌 𝑇

➢ van der Waals Eq.

1 2𝜋𝐶 1 2 3
𝑎= 𝐴= 𝑏 = 𝐵 = 𝜋𝜎
𝑃 + 𝑎/𝑉 2 𝑉 − 𝑏 = 𝑘𝑇 2 𝑛 − 3 𝜎 𝑛−3 2 3
➔ force ➔ force

Intermolecular and Surface Forces 23

 van der Waals Equation
1 2𝜋𝐶 1 2 3
𝑃 + 𝑎/𝑉 2 𝑉 − 𝑏 = 𝑘𝑇 𝒂= 𝐴=
2 𝑛 − 3 𝜎 𝑛−3
𝒃 = 𝐵 = 𝜋𝜎
2 3
➢ Relationship between the range of the forces and the size of molecules
𝐶 2𝜋𝜎 3
𝒂= 𝑛 ∝ 𝑤0 ⋅ 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑎𝑟 𝑣𝑜𝑙𝑢𝑚𝑒 Real World 𝒂 ∝ 𝑤0 ⋅ 𝒊𝒏𝒕𝒆𝒓𝒂𝒄𝒕𝒊𝒐𝒏 𝑣𝑜𝑙𝑢𝑚𝑒
𝜎 𝑛−3

Intermolecular and Surface Forces 24

 van der Waals Equation
∞
𝑖 2
4𝜋𝐶𝜌 𝐶 1 2𝜋𝐶
𝜇gas = න 𝑤 𝑟 𝜌 4π 𝑟 d𝑟 = − 𝑛−3 = −𝐴𝜌, 𝑤 𝑟 =− 𝑛, 𝒂= 𝐴=
𝜎 𝑛 − 3 𝜎 𝑟 2 𝑛 − 3 𝜎 𝑛−3

➢ Simple example for Real World Case (square-well potential)

𝑤 𝑟 =∞ for r < σ
𝑤 𝑟 = −𝑤0 for σ ≤ r ≤ σ + Δ
𝑤 𝑟 =0 for r > σ + Δ

Intermolecular and Surface Forces 25

 How Strong is the Intermolecular Interaction?
➢ Intermolecular Interaction in a Condensed Phase at T = 273 K & P = 1 atm
✓ Relationship between Cohesive (interaction) energy and Boiling point
𝑋L 𝑁0 𝜇L𝑖 𝑈𝐵 • VG (per mole) = 22,400 cm3
𝜇G𝑖 − 𝜇L𝑖 = 𝑘𝑇 log 𝑖
⟹ −𝜇L ≈ 𝟕𝒌𝑻𝐁 or − = ≈ 𝟕𝑹
𝑋G 𝑇B 𝑇B • VL (per mole) = 20 cm3

➔ 𝑇B ∝ Energy for taking a molecule from liquid into gas

➢ Latent Heat of Vaporization (Lvap)

✓ Energy in hidden form which is supplied or extracted to change the state of matter
𝐿vap = 𝑈vap + 𝑃𝑉 ≈ 𝑈vap + 𝑅𝑇B ⟹ 𝐿vap /𝑇B ≈ 8𝑅 ≈ 70 J ∙ K −1 ∙ mol−1

➔ Trouton’s Rule: 𝐿vap /𝑇B ≈ 80 J ∙ K −1 ∙ mol−1

Intermolecular and Surface Forces 26

 How Strong is the Intermolecular Interaction?
• Cohesive ε vs TB • Latent Heat vs TB
−𝝁𝒊𝑳 ≈ 7𝑘𝑻𝐁 𝑳𝒗𝒂𝒑 /𝑻𝑩 ≈ 80 J ∙ K −1 ∙ mol−1

✓ Boiling point: Indication of the strength

of the cohesive forces/energies holding
molecules together in condensed phases
✓ Latent heat of vaporization corresponds
to 𝜇L𝑖 ≈ 9𝑘𝑇 (per molecule)
or 𝜇L𝑖 ≈ (3/2)𝑘𝑇 (per bond)
✓ For solid,
Lvap = Heat of sublimation
TB = Sublimation temperature

Intermolecular and Surface Forces 27

 How Strong is the Intermolecular Interaction?
➢ Density Distribution of Liquid and Gas
−𝜇L𝑖 = 𝑘𝑇 log 𝑋L /𝑋G ⟹ 𝜌G exp(−𝜇L𝑖 /𝑘𝑇) ≈ 𝜌L

➢ Boltzmann Distribution for Orientational Distribution

𝜇2i (𝑟, 𝜃2 ) − 𝜇1i (𝑟, 𝜃1 )
𝑋(𝜃2 ) = 𝑋(𝜃1 )exp −
𝑘𝑇

Intermolecular and Surface Forces 28

Common Types of Interactions and Pair-Potential

• Q : Electrical charge [C]

• u : Electric dipole moment [C·m]
• α : Electric polarizability [C2·m2 ·J-1]
• k : Boltzmann constant [1.381 ·10-23 J·K-1]
• h : Plank’s constant [6.626 ·10-34 J·s]
• v : Electronic absorption frequency [s-1]
• ε0 : Dielectric permittivity [8.854 ·10-12 C2·J-1·m-1]

Intermolecular and Surface Forces 29

 Molecular Systems
➢ Pair potentials w(r) in multicomponent multimolecular systems
✓ The medium cannot be treated as a continuum defined solely in terms of its bulk
properties (e.g., bulk density ρ, bulk dielectric constant ε)
➔ Simultaneous interaction of many molecules (e.g., solvent and solute) are involved
➔ It becomes severe in case of short-range interactions
➢ Two basic approaches
✓ Continuum Theory (Bulk → Molecular level)
• Assume some of bulk properties hold right down to molecular level
(Analytical solution)
✓ Molecular or Atomic Theory (Molecular level → Bulk)
• Build up the properties of the system by integrating interactions at atomic or molecular
levels (Numerical solution)
Intermolecular and Surface Forces 30
 Molecular Theory via Simulations
➢ Monte Carlo Simulation
1) Randomly chosen molecule
moved to a randomly chosen
position
2) Determine the acceptation or
rejection of this motion through
the changes in total ε
3) Repeat the simulation until
there is no change in ε of the
system (thermodynamic
equilibrium)
✓ Different results <Diffusion Limited Aggregation>
✓ Less calculation burden
Intermolecular and Surface Forces 31
 Molecular Theory via Simulations
➢ Molecular Dynamics Simulation
1) Calculate the force on each molecules
arising from all others
2) Determine how the molecules moves in
response to the force by solving
Newton’s eq of motion
3) Repeat the simulation until the system
reaches the equilibrium state
✓ SAME final result ➔ More revealing
(time-dependent phenomena, non-
equilibrium effects, fluid flow, transport
phenomena)
✓ Larger calculation burden <Water-Oil Phase Separation>
Intermolecular and Surface Forces 32
3. Strong Intermolecular
Forces : Covalent and
Coulomb Interactions

Intermolecular and Surface Forces 33

 Covalent and Physical Forces
➢ Covalent Force (= Chemical Bonding Force)
✓ Force to tightly bind the atoms together within a molecule
✓ The interatomic bonds are called “Covalent” or “Chemical bonds”
➔ Characterized by the sharing of electrons between atoms
✓ Short-range force (order of interatomic separations 0.1-0.2nm)

➢ Physical Force : Forces between unbonded discrete atoms and molecules

➔ Physical bonds

Intermolecular and Surface Forces 34

 Coulomb Forces
➢ Coulomb Force (= Ionic Force)
✓ Strongest physical force that acts between two charged atoms or ions
(even stronger than most chemical bonding force)
𝑄1 • ε0 : Dielectric permittivity at vacuum [8.854 ·10-12 C2·J-1·m-1]
𝐹 𝑟 = 𝑄2 𝐸1 = 𝑄2
4𝜋𝜀0 𝜀𝑟 𝑟 2 • ε : Relative permittivity of the medium

➔ Very strong & Long-range Force

➢ Free energy for the Coulomb Interaction (= Electrical potential energy)

𝑟
𝑄1 𝑄2 1 𝑧1 𝑧2 𝑒 2 1 • z : Ionic valency
𝑤 𝑟 = න −𝐹 𝑟 𝑑𝑟 = =
∞ 4𝜋𝜀0 𝜀𝑟 𝑟 4𝜋𝜀0 𝜀𝑟 𝑟 • e : Ionic electron charge [1.602 ·10-19 C]

Intermolecular and Surface Forces 35

 Coulomb Forces
➢ e.g.) Binding Energy / Force of NaCl
𝑧1 𝑧2 𝑒 2 1
𝑤 𝑟 =
4𝜋𝜀0 𝜀𝑟 𝑟

Intermolecular and Surface Forces 36

 Ionic Crystal
➢ Lattice Energy
✓ Coulomb energy of an ion with all the other ions in the lattice
• Total interaction energy of Na+, Cl‒ ions in the lattice
𝑧 𝑧 𝑒 2 1 𝑒 2
1 2 r = 0.276 nm
𝜇𝑖 = ෍ 𝑤(𝑟) = ෍ = −1.748 = −1.46 ∙ 10−18 J
4𝜋𝜀0 𝜀𝑟 𝑟i 4𝜋𝜀0 𝜀𝑟 𝑟
𝑖

✓ Madelung constant (1.638 ~ 1.763) for crystal

• Molar Lattice Energy (Cohesive energy) of NaCl
𝑈 = −𝑁0 𝜇𝑖 = 880 kJ/mol

✓ 15 % is higher than the measured value

➔ We neglect the repulsive steric forces at constant Na+ Cl‒
(short-range force)
Intermolecular and Surface Forces 37
 Range of Coulomb Force
➢ Coulomb Force ➔ Long-Ranged & Strong Force r = 0.276 nm

𝑄1 𝑄2 1 𝐹C
𝐹 𝑟 = ~ 1033
4𝜋𝜀0 𝜀𝑟 𝑟 2 𝐹G

➢ In a real world (e.g., crystal lattice)

✓ Ions do not play alone ➔ Form dipole or quadrupole
1 1 1
𝐹m ∝ 𝐹d ∝ 𝐹q ∝
𝑟2 𝑟3 𝑟4
✓ Outside the ionic lattice, the field is seen to be
short-ranged even though it is made up of many Na+ Cl‒
long-ranged contribution

Intermolecular and Surface Forces 38

 Born Energy of an Ion
➢ Born Energy ( = Solvation Energy)
✓ Electrostatic work done in forming the ion in a medium
✓ It determine how much the ions dissolve or partition into a different solvent
• Charging an atom from 0 to full charge Q in the sphere of radius a
𝑞 𝑑𝑞 1
𝑑𝑤 𝑟 =
4𝜋𝜀0 𝜀𝑟 𝑎

• Total free energy of charging the ion (= Born Energy)

𝑄
𝑞 𝑑𝑞 𝑄2 (𝑧𝑒)2
𝜇𝑖 = න 𝑑𝑤 = න = =
0 4𝜋𝜀0 𝜀𝑟 𝑎 8𝜋𝜀0 𝜀𝑟 𝑎 8𝜋𝜀0 𝜀𝑟 𝑎

Intermolecular and Surface Forces 39

 Partitioning of Ions between Two Solvents

➢ Transfer an ion from a medium with ε1 to one with ε2 (ε1 < ε2)

(𝑧𝑒)2 1 1
𝑖 𝑖 𝑖
∆𝜇 = 𝜇2 − 𝜇1 = − −
8𝜋𝜀0 𝑎 𝜀1 𝜀2

➢ Transfer 1 mole of monovalent cations or anions from gas (ε1= 1) to water (ε2= 78)
2
𝑖
𝑧𝑒 1 1
∆𝐺 = 𝑁0 ∆𝜇 = −𝑁0 −
8𝜋𝜀0 𝑎 𝜀1 𝜀2
≈ −490 kJ/mol  a = 0.14 nm

✓ Basis for calculating the partitioning of ions between different solvents

Intermolecular and Surface Forces 40

 Solubility of Ions in Different Solvent
➢ Solubility of Ions
✓ Coulomb and Born energies provide insights into understanding why ionic
crystals dissociate in solvents with high ε in spite of very high lattice energies
1) Decrease in Coulomb Interaction in a solvent with high ε
𝑧1 𝑧2 𝑒 2 1
𝑤 𝑟 =
4𝜋𝜀0 𝜀𝑟 𝑟

2) Decrease in ∆μi from the associated to dissociated states (e.g., NaCl ➔ Na+ + Cl‒)
(𝑧 𝑒)2 (𝑧 𝑒)2 𝑧1 𝑧2 𝑒 2
1 2
∆𝜇𝑖 = 𝜇𝑑𝑖 − 𝜇𝑎𝑖 = + −
8𝜋𝜀0 𝜀𝑎1 8𝜋𝜀0 𝜀𝑎2 𝑑
4𝜋𝜀0 𝜀(𝑎1 + 𝑎2 ) 𝑎

Intermolecular and Surface Forces 41

 Solubility of Ions in Different Solvent
➢ Solubility Xs of ions in a solvent containing a solid
(e.g., salt that is not fully dissolved in water)

𝑋𝑑 = 𝑋𝑎 exp[−(𝜇𝑑i − 𝜇𝑎i )/𝑘𝑇]  Xa = 1 (pure solid)

𝑒2 NaCl
≈ exp −
4𝜋𝜀0 𝜀 𝑎1 + 𝑎2 𝑘𝑇

➔ Higher solubility of ions for larger ions and higher ε

• For NaCl in water (𝑎1 + 𝑎2 = 0.276 𝑛𝑚, 𝜀 = 78, 𝑇 = 298 K)

𝑋𝑑 ≈ 𝑒 −2.6 ≈ 0.075 (mole fraction) ➔ Measured Xd ≈ 0.11

Intermolecular and Surface Forces 42

 Solubility of Ions in Different Solvent
• Solubilities of NaCl and Glycine in solvents
• Static ε of common liquid and solids at 25oC
with different ε

Intermolecular and Surface Forces 43

 Ion-Solvent Effect : Continuum Approach
➢ Role of the Solvent in Coulomb Interactions
✓ Must investigate how a solvent affects the electric fields around dissolved ions
➔ need to consider Dielectric permittivity ε
∞ 2 2
Born Energy 1 𝑄 (𝑧𝑒)
𝜇𝑖 = 𝜀0 𝜀𝑟 න 𝐸 2 𝑑𝑉 = න 4𝜋𝑟 2 𝑑𝑟 =
of an Ion 2 𝑎 4𝜋𝜀0 𝜀𝑟 𝑟 2 2 8𝜋𝜀0 𝜀𝑟 𝑎

➢ Important Insights from the Born Energy

1) Bulk value of ε of a medium = Local value of ε surrounding an ion
𝑅
𝑄2 𝑄 2
1 1 For a = 0.1 nm
න 2
4𝜋𝑟 𝑑𝑟 = − ➔ 50 % energy within R = 0.2 nm
2a 𝑎 4𝜋𝜀0 𝜀𝑟 𝑟 2 2 8𝜋𝜀0 𝜀𝑟 𝑎 𝑅
➔ 90 % energy within R = 1 nm

a • All region in the medium has ε including the surrounding area of ions.
However, 0.1 nm is even larger than the smallest solvent molecule
➔ Can be explained upon continuum theory
Intermolecular and Surface Forces 44
 Ion-Solvent Effect : Continuum Approach
➢ Important Insights from the Born Energy
2) Coulomb interactions in a medium is not determined by ε in the region between
two charges but ε in the region surrounding the charges
∞ 2 2
1 𝑄 (𝑧𝑒)
Born Energy 𝜇𝑖 = 𝜀0 𝜀𝑟 න 𝐸 2 𝑑𝑉 = න 4𝜋𝑟 2 𝑑𝑟 =
of an Ion 2 𝑎 4𝜋𝜀0 𝜀𝑟 𝑟 2 2 8𝜋𝜀0 𝜀𝑟 𝑎

✓ Coulomb interaction in a medium can also be derived from the change in E energy,
integrated over the whole of space, when two charges are brought together.
(= Change in the born energies of two charges as they approach)
✓ Therefore, strength of the Coulomb interaction of two oppositely charged ions will
be reduced in a solvent even if the ions still remain in contact even before there
any solvent molecules between them!! (NaCl ➔ Na+ + Cl‒)
➔ Shows why the strong ionic bond is readily disrupted in a medium of high ε
Intermolecular and Surface Forces 45
 Ion-Solvent Effect : Molecular Approach
➢ Ion-Solvent Effect via Continuum Approach
𝑅
𝑄2 𝑄 2
1 1 For a = 0.1 nm
2a න 2
4𝜋𝑟 𝑑𝑟 = − ➔ 50 % energy within R = 0.2 nm
4𝜋𝜀0 𝜀𝑟 𝑟 2 2 8𝜋𝜀0 𝜀𝑟 𝑎 𝑅
𝑎 ➔ 90 % energy within R = 1 nm
a
• 0.1 nm is even larger than the smallest solvent molecule
➔ Can be explained upon continuum theory

➢ Ion-Solvent Effect via Molecular Approach

• Solvation Effect : Local ordering of solvent molecules

around ions
➔ Decrease in the effective size of solvent molecules

Intermolecular and Surface Forces 46

4. Interaction Involving
Polar Molecules

Intermolecular and Surface Forces 47

 Polar Molecule
➢ Polar Molecules (= Dipolar Molecules) : Molecules having a permanent dipole
2δ‒ 𝜇Ԧ
δ+ δ‒
δ+ δ+

No net charge
➢ Dipolar Ions
✓ Dipoles of some molecules depend on their environment
✓ Dipoles can change substantially when they are transferred into a different medium
(Especially, when molecules are ionized in a solvent)

Water
Glycine H H H

Low pH High pH

Intermolecular and Surface Forces 48

 Polar Molecule
➢ Dipole Moment
𝜇 = 𝑞𝑙
l : Distance between + q and ‒ q

✓ Debye : unit of dipole moment

(1 D = 3.336 ·10-30 C m)
✓ Small molecules have μ of
the order of 1D
✓ Permanent dipole moment
occur only in asymmetric
molecules

Intermolecular and Surface Forces 49

 Dipole Self-Energy
➢ Dipole Self-Energy
✓ Born energy for dipole
✓ Born energies of two charges ±q at infinity (> 0)
+ Coulomb energy of bringing the two charges together to from the dipole (< 0)
2 2 2 2
(+𝑞) (−𝑞) 𝑞 1 𝑞
𝜇𝑖 = + − = (for l = 2a)
8𝜋𝜀0 𝜀𝑟 𝑎 8𝜋𝜀0 𝜀𝑟 𝑎 4𝜋𝜀0 𝜀𝑟 𝑙 8𝜋𝜀0 𝜀𝑟 𝑎
➔ Solubility of polar molecules increases as εr increases
➢ Dipole Moment and Self-Energy for Dipolar Molecules
1) Unable to estimate the charge distribution of a dipolar molecule from μ
2) μ is dependent on solvent
3) Size of dipolar molecule is much bigger than ions
➔ Additional large energies from nonelectrostatic solute-solvent interactions
Intermolecular and Surface Forces 50
 Ion-Dipole Interactions
➢ Ion-Dipole Interactions
✓ Electrostatic pair interaction between a charged atom and a polar molecule
✓ Sum of Coulomb energies of Q with – q and Q with + q
−𝑄𝑞 1 𝑄𝑞 1 𝑄𝜇 cos 𝜃
𝑤 𝑟, 𝜃 = + ≈− = −𝜇𝐸(𝑟) cos 𝜃
4𝜋𝜀0 𝜀𝑟 𝐴𝐵 4𝜋𝜀0 𝜀𝑟 𝐴𝐶 4𝜋𝜀0 𝜀𝑟 𝑟 2
Finite-sized dipole Point dipole

Intermolecular and Surface Forces 51

 Ion-Dipole Interactions
✓ Exact solution (solid line)
≈ Point-dipole approximation (dashed line)
✓ For l < 0.1 nm, the Point-dipole approximation
is valid at all physical realistic interactions
(Typical interatomic separation = 0.2‒0.4 nm)
✓ KT (at RT) << Ion-Dipole Interaction
✓ w(r, θ) for Na+ vs Water molecule
• Na+ → 39 KT
• Li+ → 50 KT
• Mg2+ → 100 KT
• Be2+ → 150 KT
• Charge (+q) vs Dipole (1D) in Vac
Intermolecular and Surface Forces 52
 Solvated Ions
➢ Solvated Ion (= Hydrated Ion for water solvent) :
✓ Ions that orientationally bound solvent molecules around them by the strong
orientation dependent Ion-Dipole interaction (cation → θ = 0o, anion → θ = 180o)
➢ Hydration number : # of water molecules bound to the ion (typically 4 ~ 6)
✓ The bound water molecules are NOT completely immobilized (but less fluctuating)
➔ exchange with bulk water
➢ Effective (hydrated) Radius
✓ Smaller Ion ➔ More intense E ➔ More
hydrated ➔ Higher effective radius
✓ Very small ions (e.g., BE2+) have lower
hydration number ➔ Too small for more
than 4 water molecules to pack around
them
Intermolecular and Surface Forces 53
 Solvated Ions
➢ Rotational correction time (= Reorientation time = Life time = Exchange rate)
✓ Average time that solvent molecules remain bound to ions (~ 10-11 s)
✓ Negative hydration : Ions with t ≤ 10-11 s (hydrogen bonding life time)
✓ Cation vs Anion : cations are more solvated (tcation > tanion) as they are smaller
➢ t3+ (s to hrs) > t2+ (10-8 to 10-6) > t1+ (10-9 s)
✓ Trivalent cations have strong binding ➔
Ion-Water complexes form (quasi-stable)
✓ Small divalent / Trivalent cations have
weak but well-defined 2nd hydration shell
➢ Hydronium (= oxonium) Ion
✓ Proton H+ + One water molecule (H3O+)
✓ 3 H2O are solvated ➔ H3O+(H2O)3
Intermolecular and Surface Forces 54
 Solvated Ions
➢ Rotational correction time (= Reorientation time = Life time = Exchange rate)
✓ Average time that solvent molecules remain bound to ions (~ 10-11 s)
✓ Negative hydration : Ions with t ≤ 10-11 s (hydrogen bonding life time)
✓ Cation vs Anion : cations are more solvated (tcation > tanion) as they are smaller
➢ t3+ (s to hrs) > t2+ (10-8 to 10-6) > t1+ (10-9 s)
✓ Trivalent cations have strong binding ➔
Ion-Water complexes form (quasi-stable)
✓ Small divalent / Trivalent cations have
weak but well-defined 2nd hydration shell
➢ Hydronium (= oxonium) Ion
✓ Proton H+ + One water molecule (H3O+)
✓ 3 H2O are solvated ➔ H3O+(H2O)3
Intermolecular and Surface Forces 54
 Solvation, Structural, Hydration Effects
➢ First (= Primary) Hydration shell
✓ First shell of water molecules around a strongly solvated ion
✓ Solvent molecules in 1st shell are Structured (Positionally and Orientationally ordered)
✓ Short range ordering : Ordering propagates beyond the 1st shell, but more weakly
➔ H2O in 1st shell directly interact with the ion (Solute-Solvent interaction)
➔ H2O beyond 1st shell indirectly interact with the ion (Solute-Solvent interaction)
and directly interact with other H2O (Solvent-Solvent interaction)
➢ Solvation Zone
✓ Region where the solvent properties (e.g., density, ordering, mobility) are significantly
different from the bulk value due to modified molecular ordering
1) ε decreases in the zone due to restricted mobility of solvent molecules
2) Modification of short-range interactions (e.g., Coulomb interaction)
➔ Solvation or Structural Forces (= Hydration forces for Water)
Intermolecular and Surface Forces 55
 Dipole ‒ Dipole Interactions
μ1 μ2
θ1 ϕ θ2 𝜇1 𝜇2
r 𝑤 𝑟, 𝜃1 , 𝜃2 , ∅ = − 2 cos 𝜃1 cos 𝜃2 − sin 𝜃1 sin 𝜃2 cos ∅
4𝜋𝜀0 𝜀𝑟 𝑟 3

✓ Max Attraction ➔
✓ Max Repulsion ➔
✓ Half Attraction ➔
✓ Half Repulsion ➔
✓ For two dipoles of 1D in a vacuum
• Dipoles in line : 2A = KT (for r = 0.36 nm)
• Parallel (or antiparallel): A = KT (for r = 0.29 nm)
➔ Only very polar molecules can bind each other
via dipole-dipole interaction at room T

Intermolecular and Surface Forces 56

 Hydrogen Bond
➢ Hydrogen bond
✓ Particularly strong type of dipole-dipole interaction associated an electron-depleted H (H+)

✓ H+ is small and forms highly polar group (A‒‒ H+)

➔ Electro negative atoms (e.g., N‒, O‒, F‒) can get quite close to the highly polar group,
thus experiencing an unusually strong field
➔ The resulting bond (‒A‒ ··· H+‒) is known as Hydrogen Bond
✓ Strongly attractive and Moderately directional
➔ can even orient neighboring molecules (e.g., H2O, NH3, HF) in solid to gas phases
➔ Such liquids are called Associated Liquid

Intermolecular and Surface Forces 57

 Rotating Dipoles and Angle-Averaged Potentials
➢ When Dipole Interaction w(r) < kT (at large separation distance or in a medium of high ε)
✓ Dipoles can rotate freely
➢ Angle-Averaged Free Energy w(r) (Potential distribution theorem)
1 2𝜋 𝜋
𝑒 −𝑤(𝑟)/𝑘𝑇 = න 𝑒 −𝑤(𝑟,Ω)/𝑘𝑇 𝑑Ω / න 𝑑Ω = 𝑒 −𝑤(𝑟,Ω)/𝑘𝑇 = න 𝑑∅ න 𝑒 −𝑤(𝑟,𝜃,∅)/𝑘𝑇 sin 𝜃 𝑑𝜃
4𝜋 0 0

𝑤 2 (𝑟,Ω)
➔ 𝑤(𝑟) = 𝑤 𝑟, Ω −
2𝐾𝑇
➢ w(r) for Ion-Dipole Interaction
𝑄 2 𝜇2 𝑄𝜇 𝑄𝜇
𝑤 𝑟 ≈− 𝑓𝑜𝑟 𝑘𝑇 > or 𝑟 >
6 4𝜋𝜀0 𝜀𝑟 2 𝑘𝑇𝑟 4 4𝜋𝜀0 𝜀𝑟 𝑟 2 4𝜋𝜀0 𝜀𝑟 𝑘𝑇

➢ w(r) for Dipole-Dipole Interaction (Orientation or Keesom Interaction)

1/3
𝜇12 𝜇22 𝜇1 𝜇2 𝜇1 𝜇2
𝑤 𝑟 ≈− 𝑓𝑜𝑟 𝑘𝑇 > or 𝑟 >
3 4𝜋𝜀0 𝜀𝑟 2 𝑘𝑇𝑟 6 4𝜋𝜀0 𝜀𝑟 𝑟 3 4𝜋𝜀0 𝜀𝑟 𝑘𝑇

Intermolecular and Surface Forces 58

5. Interactions Involving the
Polarization of Molecules

Intermolecular and Surface Forces 59

 Polarizability of Atoms and Molecules
➢ Polarizability (α): μind = α E
✓ Tendency to induce the dipole moments in molecules by an external E
➢ Electronic Polarizability (α0 for nonpolar molecules)
• Induced μ by E • External Force on e by E
𝜇ind = 𝐹ext =

• Internal Force on e along E direction

𝑒2
𝐹int =− sin 𝜃 ≈
4𝜋𝜀0 𝜀𝑟 𝑅2

• At equilibrium, Fext = Fint

𝜇ind =

Intermolecular and Surface Forces 60

 Polarizability of Atoms and Molecules
Electronic polarizabilities α0 of Atoms, Molecules, Bonds, and Molecular Groups
in the unit of 4𝜋𝜀0 Å3 = 1.11 ∙ 10−40 C2 m2 J−1

Intermolecular and Surface Forces 61

 Dipolar Polarizability
➢ Dipolar (or Orientational) Polarizability (αdip for polar molecules)
✓ Polarizability of freely rotating dipolar molecules (time averaged μ = 0)
✓ In the presence of E, the orientation of dipolar molecules are weighted along E
𝜇2 μ
𝜇ind = 𝜇 cos 𝜃 𝑒 𝜇𝐸cos𝜃/𝑘𝑇 = 𝐸 = 𝛼dip 𝐸 (for 𝜇𝐸 ≪ 𝑘𝑇)
3𝑘𝑇
θ E
• For a polar molecule of μ = 1D at 300K
𝜇2 3.336 ∙ 10−30 2
𝛼dip = = −23
= 9 ∙ 10−40 C2 m2 J−1 ≈ 8 ∙ 4𝜋𝜀0 Å3
3𝑘𝑇 3 1.38 ∙ 10 300
✓ When μE >> kT (very high E or Low T), αdip is no longer valid (α0 is still valid)
➔ Dipole is fully aligned along E
➢ Total Polarizability of a Polar Molecule (Debye-Langevin Eq.)
𝛼 = 𝛼0 + 𝛼dip = 4𝜋𝜀0 𝜀𝑟 𝑅3 + 𝜇2 /3𝑘𝑇

Intermolecular and Surface Forces 62

 Effect of Polarization on Electrostatic Interactions
➢ External E ➔ Induced Dipole μind ➔ “Reaction” dipole field Er
✓ Er enhances or opposes E, depending on the location
➢ In a condensed phase consisting of many polarizable molecules
✓ Sum of Er from all the induced dipoles = “Polarization” field Ep (always opposes E)
➢ Net force on a charge Q in a medium
𝐸 𝜀𝑟 − 1
𝐹 = 𝐸 + 𝐸p 𝑄 = 𝐸eff 𝑄 ⟹ 𝐸eff = , 𝐸p = − 𝐸
𝜀𝑟 𝜀𝑟

• For 𝜀𝑟 = 1, Eeff = E, Ep=0 (no polarization)

• For 𝜀𝑟 >> 1, Eeff → 0, Ep → ‒E
✓ Dielectric constant 𝜀𝑟 is related to α
(Higher α ➔ Higher 𝜀𝑟 )

Intermolecular and Surface Forces 63

 Interactions between Ions and Nonpolar Molecules
➢ Interaction between an Ion and Nonpolar Molecule
Nonpolar ✓ Induced dipole on the uncharged molecule by the Ion
Ion Molecule
r 𝑧𝑒
𝜇ind = 𝛼𝐸 = 𝛼
α 4𝜋𝜀0 𝜀𝑟 𝑟 2
ze

Ion +q ✓ Reaction field Er from μind

AC
r μ
ϕ
θ
ze 𝐸𝑟 =
AB -q
➢ Attractive Force and Energy between Ion and Nonpolar Molecule
𝑟
𝐹 𝑟 = 𝑄𝐸𝑟 = 𝑤 𝑟 = − න 𝐹 𝑟 𝑑𝑟 =
∞

Intermolecular and Surface Forces 64

 Interactions between Ions and Nonpolar Molecules
• Ion vs Induced Dipole • Ion vs Permanent Dipole
1
𝑤ind 𝑟 = − 𝛼𝐸 2 𝑤pe 𝑟 = −𝜇𝐸 = −𝛼𝐸 2
2
➢ When a dipole is induced, some energy is spent for polarizing the molecule
✓ Energy is used internally in displacing the charges in the molecule
from 0 to l
𝑙
1 2 𝑒2𝑙
𝑤int 𝑟 = − න 𝐹int 𝑑𝑙 = 𝛼𝐸 ⟵ 𝐹int =−
0 2 4𝜋𝜀0 𝜀𝑟 𝑅3

⟹ 𝑤pe + 𝑤int = 𝑤ind

➢ Total Interaction Energy of an Ion in a Solvent

∞ ∞
𝑖 2
𝛼 𝑧𝑒 2 𝜌 4π 𝑟 2 𝜌𝛼 𝑧𝑒 2
𝜇 = න 𝑤 𝑟 𝜌 4π 𝑟 d𝑟 = න − d𝑟 = −
a a
2
2 4𝜋𝜀0 𝜀𝑟 𝑟 4 8𝜋𝜀02 𝜀r2 a

Intermolecular and Surface Forces 65

 Interactions between Polar and Nonpolar Molecules
➢ Interaction between a Polar molecule and a Nonpolar molecule
✓ E from the dipole acting on the nonpolar molecule
Polar Nonpolar
Molecule Molecule 𝐸 = 𝜇 (1 + 3cos 2 𝜃) / 4𝜋𝜀0 𝜀𝑟 𝑟 3
+q AC
μ r ✓ Interaction energy of the dipole with the nonpolar molecule
θ 1 2 𝛼𝜇2 (1 + 3cos 2 𝜃)
-q AB 𝑤 𝑟, 𝜃 = − 𝛼𝐸 = −
2 2 4𝜋𝜀0 𝜀𝑟 2 𝑟 6
✓ For typical μ and α0, w(r, θ) is NOT sufficient to mutually orient the molecules
𝛼𝜇2
𝑤 𝑟 = 𝑤 𝑟, 𝜃 =−
4𝜋𝜀0 𝜀𝑟 2 𝑟 6
➔ Angle-Averaged Energy

Intermolecular and Surface Forces 66

 Interactions between Polar and Polar Molecules
➢ For Two Polar Molecules ( μ1, α1 and μ2, α2), Net Dipole – Induced Dipole Energy
𝛼1 𝜇1 + 𝛼2 𝜇2
𝑤 𝑟 =−
4𝜋𝜀0 𝜀𝑟 2 𝑟 6 ➔ Debye (or Induction) Interaction
➢ General Equation for Angle-Averaged Interactions
𝑄12 3𝑘𝑇𝛼1 𝛼2 𝛼 = 𝛼0 + 𝛼dip
Molecule 1 Molecule 2
+q
𝑤 𝑟 =−
2𝑟 4
+
𝑟6 4𝜋𝜀0 𝜀𝑟 2  = 4𝜋𝜀0 𝜀𝑟 𝑅3 + 𝜇2 /3𝑘𝑇
μ1 +q μ2
r
θ1 θ2 𝑄12 𝜇12 3𝐾𝑇𝛼01 1 𝜇22
=− + + + 𝛼02
-q -q 2𝑟 4 𝑟 6 𝑟6 4𝜋𝜀0 𝜀𝑟 2 3𝑘𝑇

✓ For typical values of 𝑄1 = 𝑒, 𝜇1 = 1D, 𝛼01 = 3 4𝜋𝜀0 Å3 , 𝑟 = 0.5 nm, 𝑇 = 300 K

• Permanent dipole associated
𝑄12 𝜇12 3𝑘𝑇𝛼01 𝜇12 3𝑘𝑇𝛼01 interactions always dominate
∶ ∶ = 800 ∶ 3 ∶ 1 Water ∶ = 20 ∶ 1
2𝑟 4 𝑟6 𝑟6 𝑟6 𝑟6 over electronic polarization
effect
Intermolecular and Surface Forces 67
 Solvent Effects : Excess Polarizability
➢ Solvent Effects
1) Intrinsic μ and α of molecules change upon dissolution in a solvent
➔ Depend on their interactions with surrounding molecules and only be found by experiment
2) Cavity Effect ➔ α of dissolved molecules must represent the “Excess Polarizability”
3) For the molecules with same properties as the solvent, they are “invisible” in the solvent and
consequently do not experience a force.
➢ Excess (or Effective) Polarizability (αi)
𝜀𝑖 − 𝜀𝑟
𝜇ind = 4𝜋𝜀0 𝜀𝑟 𝑎i3 𝐸 = 𝛼𝑖 𝐸
𝜀𝑖 + 2𝜀𝑟 r

𝜀𝑖 − 𝜀𝑟 3 𝜀𝑖 − 𝜀𝑟
⟹ 𝛼𝑖 = 4𝜋𝜀0 𝜀𝑟 𝑎 = 3𝜀0 𝜀𝑟 𝑣
𝜀𝑖 + 2𝜀𝑟 i 𝜀𝑖 + 2𝜀𝑟 𝑖

Intermolecular and Surface Forces 68

 Solvent Effects : Excess Polarizability
➢ General Eq. for Angle-Averaged Interactions with αi
r
𝑄12 3𝑘𝑇𝛼1 𝛼2 𝜀𝑖 − 𝜀
𝑤 𝑟 =− + ⟸ 𝛼𝑖 = 4𝜋𝜀0 𝜀 𝑎i3
2𝑟 4 𝑟6 4𝜋𝜀0 𝜀𝑟 2 𝜀𝑖 + 2𝜀

𝑄12 3𝑘𝑇 𝜀1 − 𝜀𝑟 3
𝜀2 − 𝜀𝑟
=− + 𝑎1 𝑎23
8𝜋𝜀0 𝜀𝑟 𝑟 4 𝑟 6 𝜀1 + 2𝜀𝑟 𝜀2 + 2𝜀𝑟

1) The net force between dissolved molecules or particles in a medium can be

Zero, Attractive, or Repulsive, depending on ε1, ε2, and εr
2) Ions will be attracted to dissolved molecules of high ε2 (highly polar molecule, ε2 > εr), but
repelled from the molecules of low ε2 (nonpolar molecule, ε2 < εr)
3) Any two identical uncharged molecules (ε1 = ε2) are always attractive regardless of the
nature of the suspending medium.

Intermolecular and Surface Forces 69

 Solvent Effects : Excess Polarizability
➢ Clausius-Mossotti Equation
✓ For isolated molecule in the gas phase (ε1 = 1), Total polarizability Gas
Phase
𝛼𝑖 𝜀𝑖 − 1 3𝑣𝑖 𝜀𝑖 − 𝜀𝑟
= ⟸ 𝛼𝑖 = 3𝜀0 𝜀𝑟 𝑣
(4𝜋𝜀0 ) 𝜀𝑖 + 2 4𝜋 𝜀𝑖 + 2𝜀𝑟 𝑖

➢ Lorenz-Lorentz Equation
✓ For the Electronic polarizability

𝛼0 𝑛2 − 1 3𝑣
= 2
(4𝜋𝜀0 ) 𝑛 + 2 4𝜋

⟸ 𝜀𝑖 = 𝑛2
⟸ 𝑣 = 𝑀/𝜌𝑁0

Intermolecular and Surface Forces 70

6. Van der Waals Forces

Intermolecular and Surface Forces 71

 London Dispersion Force
➢ Physical forces arising from Electrostatic interactions
✓ Charge-Charge, Charge-Dipole, Dipole-Dipole interactions
➢ Another type of physical forces Always acting between all atoms and molecules
(including neutral ones)
✓ Gravitational force, London Dispersion Force (LDF, dispersion force, London force, charge-
fluctuation force, induced dipole-dipole force, Van der Waals force)
➢ General Characteristics of London Dispersion Force
1) Long range force that can be effective from interatomic spacings (~0.2 nm) to > 10 nm
2) Repulsive or Attractive (in general, do not follow a simple power law)
3) Not only bring molecules together but also align/orient them (weaker than dipolar interaction)
4) Nonadditive force
5) Play a role in various phenomena such as adhesion, surface tension, adsorption, wetting,
properties of phases, aggregation of particles in liquids, structures of macromolecules
Intermolecular and Surface Forces 72
 London Dispersion Force
➢ London Dispersion Force
✓ For a nonpolar atom (e.g., helium, carbon dioxide, hydrocarbons), the time average of its
dipole moment is zero, but a finite μ1 exists at any instant.
✓ The instantaneous dipole generates an electric field ➔ Polarize nearby neutral atom and
Induce μ2 in it ➔ Result in instantaneous dipole-dipole attraction (= LDF)

➢ Bohr Radius (a0)

✓ Smallest distance between the electron and proton in the Bohr atom
𝑒2 h : Plank constant (6.626·10-34 m2Kg/s)
𝑎0 = = 0.053 nm
8𝜋𝜀0 ℎ𝑣 v : Orbiting frequency of an electron (3.3·1015 s-1)

➔ I = hv : Energy needed to ionize the atom (First ionization potential) a0

Intermolecular and Surface Forces 73

 London Dispersion Force
➢ Interaction between two Bohr atoms (London Dispersion Force)
Bohr
✓ No Permanent dipole but Instantaneous dipole exists (μ = a0e) Atom
μ
➔ Polarize a nearby neutral atom ➔ Introduce attractive interaction r
α0
𝑤 𝑟 = 𝑤 𝑟, 𝜃 = a0

➢ Interaction between two Bohr atoms (by London using Quantum mechanical theory)
• For two identical atoms • For two dissimilar atoms
2
3 𝛼0 ℎ𝑣 3 6 ℎ𝑣 3 𝛼01 𝛼02 ℎ𝑣1 𝑣2
𝑤 𝑟 =− = − 𝑎0 6 𝑤 𝑟 =−
4 4𝜋𝜀0 𝑟 6 4 𝑟 2 4𝜋𝜀0 2 𝑟 6 𝑣1 + 𝑣2

✓ 1/r6 dependence interactions : 1) μ vs Induced μ 2) μ vs Permanent μ (Keesom)

3) Net μ vs Induced μ (Debye) ➔ Together contribute to the net van der Waals Force

Intermolecular and Surface Forces 74

 Strength of Dispersion Interactions
➢ Dispersion Interaction between Two Bohr Atoms
3 6 ℎ𝑣 • a0 (H) = 5.3·10-11 m
𝑤 𝑟 = − 𝑎0 6 = 5.02 ∙ 10−23 J < 𝑘𝑇(= 4.14 ∙ 10−21 J)
4 𝑟 • r = 3·10-10 m
Simple London Eq.
Small nonpolar atoms/molecules ➔ Gas phase at TR
✓ 𝑤(𝑟) ∝ 𝑎06
Larger nonpolar atoms/molecules ➔ Liquid or Solid
➢ van der Waals Solids
✓ Solids in which atoms are held solely by the Dispersion Interaction
✓ Weak undirected bonds ➔ Low melting point and Low latent heat of melting
✓ For small spherical molecules ➔ Closely packed structure (12 nearest neighbors per atom)
Lattice Energy (per molecule) ≈ 6𝑤 𝑟 ⟹ 7.22𝑤 𝑟
2
3 𝛼0 ℎ𝑣
Molar Lattice or Cohesive Energy ∶ 𝑈 ≈ 7.22𝑁0 ≈ 𝐿m + 𝐿v (~10 KJ/mol)
4 4𝜋𝜀0 𝑟6

Intermolecular and Surface Forces 75

Strength of Dispersion Interactions

Intermolecular and Surface Forces 76

Dispersion Interaction for Large / Asymmetric Molecules
➢ Large Spherical Molecules (σ > 0.5 nm)
✓ Simple London Eq. is no longer valid
✓ Dispersion interaction acts between the centers of electronic polarization but not of molecules
➔ r is smaller than the intermolecular separation (e.g., CCl4)
➢ Asymmetric Molecules (e.g., alkane, polymer)
✓ Simple London Eq. is no longer valid
✓ 1) Molecular packing and 2) Different contributions from different parts of the molecules
must be known for computing U

Intermolecular and Surface Forces 77

 Van der Waals Equation of State
➢ Boyle’s Gas law ➔ Van der Waals Eq.
2𝜋𝐶 2 3
𝑎= 𝑏 = 𝜋𝜎
𝑛 − 3 𝜎 𝑛−3 3
𝑃𝑉 = 𝑛𝑅𝑇 = 𝑁𝑘𝑇 𝑃 + 𝑎/𝑉 2 𝑉 − 𝑏 = 𝑘𝑇 per molecule ➔ Attraction ➔ Repulsion

➢ London Dispersion Force

2 2
3 𝛼0 ℎ𝑣 𝐶 3ℎ𝑣 𝛼0
𝑤 𝑟 =− =− 6 for 𝑟 ≥ 𝜎 ⟹ 𝒏 = 𝟔, 𝒅𝑪 = −
4 4𝜋𝜀0 𝑟6 𝑟 4 4𝜋𝜀0

𝑤 𝑟 = ∞ (for 𝑟 < 𝜎)

➢ Van der Waals Eq. (per mole)

2𝜋𝑁02 𝐶 2
𝑃+ 𝑎′/𝑉′2 𝑉′ − 𝑏′ = 𝑁0 𝑘𝑇 per mole 𝑉′ = 𝑁0 𝑉 𝑎′ = 𝑏′ = 𝜋𝑁0 𝜎 3
3𝜎 3 3
1/3
3𝜎 3 𝑎′ 9𝑎′𝑏′ −72 𝑎 ′ 𝑏 ′ mol3 3𝑏′
𝐶= = = 1.04 ∙ 10 ⟸𝜎=
2𝜋𝑁02 4𝜋 2 𝑁03 2𝜋𝑁0

Intermolecular and Surface Forces 78

 Van der Waals Equation of State
➢ Van der Waals Coefficient
✓ For Methane (CH4), a’ = 0.2283 J m3 mol-2, b’ = 0.0428·10-3 m3 mol-1
2
9𝑎′𝑏′ 3ℎ𝑣 𝛼0
𝐶 CH4 = 2 3 = 102 ∙ 10−79 J m6 𝒅𝐶 CH4 =− = 102 ∙ 10−79 J m6
4𝜋 𝑁0 4 4𝜋𝜀0
measured theoretical

➢ Van der Waals Eq for 2-Dimensional (per mole)

2𝜋𝑁02 𝐶 2
• 3D 𝑃+ 𝑎′/𝑉′2 𝑉′ − 𝑏′ = 𝑁0 𝑘𝑇 𝑎′ = 𝑏′ = 𝜋𝑁0 𝜎 3
3𝜎 3 3

𝜋𝑁02 𝐶 1
• 2D 𝑃2D + 𝑎2D /𝐴 2
𝐴 − 𝑏2D = 𝑁0 𝑘𝑇 𝑎2D = 𝑏2D = 𝜋𝑁0 𝜎 2
4𝜎 4 2

Intermolecular and Surface Forces 79

 Van der Waals Force between Polar Molecules
➢ Total Long-Range Interaction between Polar Molecules (Van der Waals Force)
✓ 1) Induction, 2) Orientation, 3) Dispersion Forces
𝐶 𝐶ind +𝐶orient +𝐶disp 𝜇12 𝜇22 3𝛼01 𝛼02 ℎ𝑣1 𝑣2
𝑤VDW 𝑟 = − VDW =− =− 𝜇12 𝛼02 + 𝜇22 𝛼01 + + / (4𝜋𝜀0 )2 𝑟 6
𝑟6 𝑟6 3𝑘𝑇 2 𝑣1 +𝑣2

Contribution of
Cdisp to
Theoretical (%)
(NP)
(NP)

(P)
(P)
(NP)

(P)
(P)
(P)

(NP‒NP)
(P‒NP)
(P‒NP)
(P‒NP)

Intermolecular and Surface Forces 80

 Van der Waals Force between Polar Molecules
➢ Important Properties of Van der Waals Interactions
✓ Dominance of dispersion forces
• Dispersion forces generally are dominant except for small highly polar molecules (e.g., H2O)
• In the hydrogen halides, the dipolar forces are clearly unimportant
✓ Comparisons with experimental data
• The theoretical and measured values for CVDW are in good agreement
✓ Interaction of dissimilar molecules
• VDW interaction energy between two dissimilar molecules A and B ≈
0.5 (VDW of A-A + VDW of B-B)
• 𝐶VDW (𝐴 − 𝐵) ≈ 𝐶(𝐴 − 𝐴) ∙ 𝐶(𝐵 − 𝐵)
➔ Not valid for the interactions with highly polar molecules (e.g., H2O-CH4)

𝐶VDW H2O−H2O > 𝐶VDW CH4−CH4 > 𝐶VDW H2O−CH4

Intermolecular and Surface Forces 81

 General Theory of Van der Waals Force
➢ Shortcoming of Classic Van der Waals Eq.
✓ Cannot handle the interactions of molecules in a solvent (one absorption frequency v)
➢ General Eq. of VDW Interaction (McLachlan’s Expression)
∞
𝐶VDW 6𝑘𝑇 ′ 𝛼1 (𝑖𝑣𝑛 )𝛼2 (𝑖𝑣𝑛 )
𝑤 𝑟 =− 6 =− ෍ 2
𝑟 (4𝜋𝜀0 )2 𝑟 6 𝜀3 (𝑖𝑣𝑛 )
𝑛=0,1,2,..

2𝜋𝑘𝑇
⟹ 𝑣𝑛 = 𝑛 ≈ 4 ∙ 1013 𝑛 s−1 at 300 K
ℎ

𝛼0 𝜇2
⟹ 𝛼 𝑖𝑣𝑛 = 2 + 3𝑘𝑇(1 + 𝑣 /𝑣
1 − 𝜅 𝑣𝑛 /𝑣I + 𝑣𝑛 /𝑣I 𝑛 rot )

⟹ 𝛼(0) = for one v approximation

• κ : Damping coefficient (<< 1)

• vI : Ionization frequency (~ 1015 s-1)
• vrot : Rotational relaxation frequency (~ 1011 s-1)

Intermolecular and Surface Forces 82

 General Theory of Van der Waals Force
∞
6𝑘𝑇 ′ 𝛼1 (𝑖𝑣𝑛 )𝛼2 (𝑖𝑣𝑛 ) 𝛼0 𝜇2
𝑤 𝑟 =− ෍ 2 ⟹ 𝛼 𝑖𝑣𝑛 = 2 + 3𝑘𝑇(1 + 𝑣 /𝑣
(4𝜋𝜀0 )2 𝑟 6 𝜀3 (𝑖𝑣𝑛 ) 1 − 𝜅 𝑣𝑛 /𝑣I + 𝑣𝑛 /𝑣I 𝑛 rot )
𝑛=0,1,2,..

➢ 1st Term of McLachlan’s Expression (n = 0, i.e., vn = 0)

3𝑘𝑇 3𝑘𝑇 𝜇12 𝜇22
𝑤 𝑟 𝑣𝑛 =0 =− 𝛼 0 𝛼2 0 = − 𝛼 + 𝛼02 +
4𝜋𝜀0 2 𝑟 6 1 4𝜋𝜀0 2 𝑟 6 01 3𝑘𝑇 3𝑘𝑇

➢ Non-zero Frequency Terms of McLachlan’s Expression (n = 1, 2.., vn = 4·1013 n s-1 at 300 K)

✓ α(ivn) is effectively determined solely by the electronic polarizability contribution
∞
ℎ 𝑣𝑛 =∞
✓ vn << vI ➔ Reasonable to convent 𝑘𝑇 ෍ to න
2𝜋 𝑣𝑛 =𝑣1
𝑑𝑣
𝑛=1,2..

∞ London Eq.
3ℎ 3𝛼01 𝛼02 ℎ𝑣I1 𝑣I2
𝑤 𝑟 𝑣𝑛 >0 =− න 𝛼 (𝑖𝑣 )𝛼
(4𝜋𝜀0 )2 𝜋𝑟 6 0 1 𝑛 2 𝑛
(𝑖𝑣 )𝑑𝑣𝑛 ≈ − ∙
2(4𝜋𝜀0 )2 𝑟 6 (𝑣I1 + 𝑣I2 ) (Complete
McLachlan Eq.)
Intermolecular and Surface Forces 83
 Van der Waals Forces in a Medium
➢ McLachlan’s Expression in a Medium
∞
6𝑘𝑇 ′ 𝛼1 (𝑖𝑣𝑛 )𝛼2 (𝑖𝑣𝑛 ) 𝜀1 𝑖𝑣𝑛 − 𝜀3 𝑖𝑣𝑛
𝑤 𝑟 =− ෍ 2 ⟹ 𝛼1 𝑖𝑣𝑛 = 4𝜋𝜀0 𝜀3 𝑖𝑣𝑛 𝑎13
(4𝜋𝜀0 )2 𝑟 6 𝜀3 (𝑖𝑣𝑛 ) 𝜀1 𝑖𝑣𝑛 + 2𝜀3 𝑖𝑣𝑛
𝑛=0,1,2,..

➢ Zero Frequency Contribution (n = 0, i.e., vn = 0)

3𝑘𝑇𝑎13 𝑎23 𝜀1 0 − 𝜀3 0 𝜀2 0 − 𝜀3 0
𝑤 𝑟 𝑣𝑛 =0 =−
𝑟6 𝜀1 0 + 2𝜀3 0 𝜀2 0 + 2𝜀3 0

➢ Non-zero Frequency Contribution (n = 1, 2..)

3ℎ𝑎13 𝑎23 ∞ 𝜀1 𝑖𝑣𝑛 − 𝜀3 𝑖𝑣𝑛 𝜀2 𝑖𝑣𝑛 − 𝜀3 𝑖𝑣𝑛 (𝑛12 − 1)
𝑤 𝑟 𝑣𝑛 >0 =− න 𝑑𝑣𝑛 ⟸ 𝜀1 𝑖𝑣𝑛 = 1+
𝜋𝑟 6 0 𝜀1 𝑖𝑣𝑛 + 2𝜀3 𝑖𝑣𝑛 𝜀2 𝑖𝑣𝑛 + 2𝜀3 𝑖𝑣𝑛 1 + (𝑣𝑛 /𝑣𝑒 )2

3ℎ𝑣𝑒 𝑎13 𝑎23 (𝑛12 − 𝑛32 )(𝑛22 − 𝑛32 )

≈−
2𝑟 6 (𝑛12 + 2𝑛32 )1/2 (𝑛22 + 2𝑛32 )1/2 (𝑛12 + 2𝑛32 )1/2 +(𝑛22 + 2𝑛32 )1/2

Intermolecular and Surface Forces 84

 Van der Waals Forces in a Medium
➢ McLachlan’s Expression in a Medium (𝑤 𝑟 = 𝑤 𝑟 𝑣𝑛 =0 +𝑤 𝑟 𝑣𝑛 >0 )
✓ For molecule 1 & molecule 2 in medium 3
𝑎13 𝑎23 𝜀1 0 − 𝜀3 0 𝜀2 0 − 𝜀3 0 3ℎ𝑣𝑒 (𝑛12 − 𝑛32 )(𝑛22 − 𝑛32 )
𝑤 𝑟 = − 6 3𝑘𝑇 +
𝑟 𝜀1 0 + 2𝜀3 0 𝜀2 0 + 2𝜀3 0 2 (𝑛12 + 2𝑛32 )1/2 (𝑛22 + 2𝑛32 )1/2 (𝑛12 + 2𝑛32 )1/2 +(𝑛22 + 2𝑛32 )1/2

✓ For two identical molecules 1 in medium 3

2
𝑎16 𝜀1 0 − 𝜀3 0 3ℎ𝑣𝑒 (𝑛12 − 𝑛32 )2
𝑤 𝑟 =𝑤 𝑟 𝑣𝑛 =0 +𝑤 𝑟 𝑣𝑛 >0 = − 6 3𝑘𝑇 +
𝑟 𝜀1 0 + 2𝜀3 0 4 (𝑛12 + 2𝑛32 )3/2

Intermolecular and Surface Forces 85

 Van der Waals Forces in a Medium
2
𝑎16 𝜀1 0 − 𝜀3 0 3ℎ𝑣𝑒 𝑛12 − 𝑛32 2
𝑎16 3𝑘𝑇(𝑛12 − 𝑛32 )2 3ℎ𝑣𝑒 (𝑛12 − 𝑛32 )2
𝑤 𝑟 = − 6 3𝑘𝑇 + =− 6 + for visible λ
𝑟 𝜀1 0 + 2𝜀3 0 4 𝑛12 + 2𝑛32 3/2 𝑟 (𝑛12 + 2𝑛32 )2 4(𝑛12 + 2𝑛32 )3/2

➢ Important Aspects of VDW Forces in a medium

1) Typically, 𝑤 𝑟 𝑣𝑛>0 > 𝑤 𝑟 𝑣𝑛=0 as hve >> kT
• For 𝜀1 = 𝑛12 ≈ 2 & 𝜀3 = 𝑛32 ≈ 1 & 𝑣𝑒 = 3 ∙ 1015 s−1 ,
𝑤 𝑟 𝑣𝑛 >0 /𝑤 𝑟 𝑣𝑛 =0 ≈ ℎ𝑣𝑒 /2 3𝑘𝑇 ≈ 139

2) VDW interaction is much reduced in a solvent medium

• Two identical molecules of n = 1.5 transferred from a free space (n3=1) to a solvent (𝑛3′ =1.4)
2
(𝑛12 − 𝑛32 )2 (𝑛12 + 2𝑛3′ )3/2
𝑤 𝑟, 𝑛3 /𝑤 𝑟, 𝑛3′ ≈ ≈ 32
(𝑛12 + 2𝑛32 )3/2 (𝑛12 − 𝑛3′ 2 )2

3) Simple London Eq. vs Dispersion contribution in McLachlan’s Eq.

𝑣𝑒 = 𝑣I 3/ 𝑛12 + 2
Intermolecular and Surface Forces 86
 Van der Waals Forces in a Medium
2
𝑎16 𝜀1 0 − 𝜀3 0 3ℎ𝑣𝑒 𝑛12 − 𝑛32 2
𝑎16 3𝑘𝑇(𝑛12 − 𝑛32 )2 3ℎ𝑣𝑒 (𝑛12 − 𝑛32 )2
𝑤 𝑟 = − 6 3𝑘𝑇 + =− 6 + for visible λ
𝑟 𝜀1 0 + 2𝜀3 0 4 𝑛12 + 2𝑛32 3/2 𝑟 (𝑛12 + 2𝑛32 )2 4(𝑛12 + 2𝑛32 )3/2

➢ Important Aspects of VDW Forces in a medium

4) For similar molecules, Dispersion force is always attractive, while that between dissimilar
molecules can be attractive or repulsive
➔ Repulsion occurs when n3 is intermediate between n1 and n2
3ℎ𝑣𝑒 𝑎13 𝑎23 (𝑛12 − 𝑛32 )(𝑛22 − 𝑛32 )
𝑤 𝑟 𝑣𝑛 >0 ≈−
2𝑟 6 (𝑛12 + 2𝑛32 )1/2 (𝑛22 + 2𝑛32 )1/2 (𝑛12 + 2𝑛32 )1/2 +(𝑛22 + 2𝑛32 )1/2

5) 𝑤(𝑟) ∝ (𝑛12 − 𝑛32 )2 ➔ If (𝑛12 − 𝑛32 )2 is large enough, molecule 1 is immiscible with a medium 3

6) n1 & n2 are close to n3 ➔ Dipolar contribution (𝑤 𝑟 𝑣𝑛 =0 ) dominates

• For Alkanes in water (𝑛CH4 ≈ 1.30, 𝑛H2O ≈ 1.33, 𝜀CH4 (0) ≈ 2, 𝜀H2O 0 ≈ 80)
𝑘𝑇𝑎16
𝑤 𝑟 ≈𝑤 𝑟 𝑣𝑛 =0 ≈ − 6 ➔
𝑟

Intermolecular and Surface Forces 87

 Dispersion Self-Energy of a Molecule in a Medium
• For two identical atoms • For two dissimilar atoms
Simple 3 𝛼0
2
ℎ𝑣 3 𝛼01 𝛼02 ℎ𝑣1 𝑣2
London Eq. 𝑤 𝑟 =−
4 4𝜋𝜀0 𝑟6
𝑤 𝑟 =−
2 4𝜋𝜀0 2 𝑟 6 𝑣1 + 𝑣2

➢ Transfer of a Molecule 1 (diameter σ) from Free Space into a Medium 2

✓ Breaking 6 solvent-solvent bonds + Forming 12 new solute-solvent bonds
𝑖 3ℎ𝑣I 2
∆𝜇disp = 2 6
6𝛼02 − 12𝛼01 𝛼02 ⟸ 𝑣I = 𝑣I1 = 𝑣I2
4(4𝜋𝜀0 ) 𝜎

✓ If Molecule 1 = Medium 2

𝑖 3ℎ𝑣I 2
∆𝜇disp = 2 6
−6𝛼01
4(4𝜋𝜀0 ) 𝜎

Intermolecular and Surface Forces 88

 Dispersion Self-Energy of a Molecule in a Medium
• For two identical atoms • For two dissimilar atoms
Simple 3 𝛼0
2
ℎ𝑣 3 𝛼01 𝛼02 ℎ𝑣1 𝑣2
London Eq. 𝑤 𝑟 =−
4 4𝜋𝜀0 𝑟6
𝑤 𝑟 =−
2 4𝜋𝜀0 2 𝑟 6 𝑣1 + 𝑣2
➢ Transfer of a Molecule 1 (diameter σ) from Medium 3 into a Medium 2 (𝑣I = 𝑣I1 = 𝑣I2 = 𝑣I3)
✓ Breaking 12 1-3 bonds + Forming 6 new 3-3 bonds
➔ Breaking 6 2-2 bonds + Forming 12 new 1-2 bonds
𝑖 3ℎ𝑣I 2 2
∆𝜇disp = −6𝛼 03 + 12𝛼 01 𝛼03 + 6𝛼 02 − 12𝛼01 𝛼02
4(4𝜋𝜀0 )2 𝜎 6
𝑖
∆𝜇disp ∝ −(𝛼02 − 𝛼03 )(2𝛼01 − 𝛼02 − 𝛼03 ) ∝ −(𝑛22 − 𝑛32 )(2𝑛12 − 𝑛22 − 𝑛32 )

✓ If Molecule 1 = Medium 2
𝑖 3ℎ𝑣I
∆𝜇disp =− 6 𝛼01 − 𝛼02 2 ∝ −( 𝑈1 − 𝑈3 )2 ∝ −(𝑛12 − 𝑛32 )2
4 4𝜋𝜀0 2 𝜎 6

Intermolecular and Surface Forces 89

 7. Repulsive Steric Forces,
Total Intermolecular Pair Potential,
And Liquid Structure

Intermolecular and Surface Forces 90

 Repulsive Steric Forces
• Effective packing radii (nm)
➢ Steric Repulsion (= Exchange-, Hard Core-, Born-Repulsions)
✓ Repulsive forces arising when electron clouds overlap
✓ Determine how close two molecules can approach each other
✓ Very short-range force, Increasing very sharply as two molecules
come together
✓ No general Eq. for the distance dependence but empirical potential
Eq.
(1. Hard sphere-, 2. Power law-, 3. Exponential-potentials)

➢ Hard Sphere Potential

✓ Atoms = Hard Sphere (i.e., incompressible) ➔ Repulsive force = ∞
at a certain interatomic separation
✓ Reflect the observation when atoms pack together in liquids and
solids, they often do behave as hard spheres of fixed radii called as
hard sphere- or van der Waals packing- or bare ion-(for ions) radius

Intermolecular and Surface Forces 91

 Repulsive Steric Forces
➢ Hard Sphere Potential ➢ Power Law Potential ➢ Exponential Potential
𝑤 𝑟 = +(𝜎/𝑟)𝑛 where 𝑛 → ∞ 𝑤 𝑟 = +(𝜎/𝑟)𝑛 𝑤 𝑟 = + 𝑐𝑒 −𝑟/𝜎0
➔𝑤 𝑟 >𝜎 =0 & 𝑤 𝑟 <𝜎 =∞ ➔ n = integer (9 ~ 16) ➔ c and σ0 = constant
✓ Power Law & Exponential Potentials are more realistic as they allow the finite compressibility
or softness of atoms

Intermolecular and Surface Forces 92

 Total Intermolecular Pair Potentials
➢ Total Intermolecular Pair Potential
✓ Steric Repulsive + Inverse Power Attractive Forces
✓ Best description of the pair potential (Lennard-Jones Potential)
𝑤 𝑟 = +𝐴/𝑟12 − 𝐵/𝑟 6 = 4 ∈ (𝜎/𝑟)12 −(𝜎/𝑟)6
𝑤 𝑟 = 𝜎 = 0 & Minimum 𝑤 𝑟 = 21/6 𝜎 =−∈

✓ Pair potential from Hard Sphere Repulsion

• Minimum w(r) at r = σ
• Still in excellent agreement with experiments for atoms
and small molecules where the attraction is stronger than
the prediction by simple London Eq. and the repulsion is
steeper than 1/r12 (e.g., nonpolar solids)

Intermolecular and Surface Forces 93

Total Intermolecular Pair Potentials

Intermolecular and Surface Forces 94

 Role of Repulsive Forces in Noncovalently Bonded Solids
➢ Role of Steric Repulsive Forces
✓ Size & Shape in the condensed state
(lattice structure, density, rigidity, internal energy..)
• Orientation dependent Steric force (Reflection of asymmetric
molecular shape) influence how molecules are packed
✓ Melting Point
• Asymmetric molecular shape ➔ Bad packing ➔ Low TM
(but TB is dominantly influenced by attractive forces)
• Existence of Branch or Double bond dramatically lower TM but
do not influence much on TB
➢ Exception
✓ For some molecules (e.g., H2O) with Covalent bonds or
Strongly directional dipolar or Hydrogen-bonding interactions,
the orientation dependence of their attractive forces plays the
dominant role in determining their condensed phase properties
Intermolecular and Surface Forces 95
 Packing of Molecules and Particles in Solids
Close Packing

➢ Close Packing (CP, 12 nearest neighbors)

✓ Crystalline solids with FCC and HCP lattices Random Close Packing
✓ Long-range crystalline order & No material flow
➢ Random Close-Packing (RCP)
✓ Non-crystalline solid = Amorphous solids
✓ Occur at the packing density (P) ≤ 63.5 % (~ 8 contact points with its neighbors)
Random Loose Packing
✓ No long-range order & No material flow
➢ Random Loose-Packing (RLP)
✓ Occur at P ≤ Critical density (RLP density = 55.5%)
✓ No long-range order & Flow
✓ Can be induced at P > 55.5% by forming ordered domains (Voronoi cells)
Intermolecular and Surface Forces 96
 Role of Repulsive Forces in Liquids
➢ Liquid Structure
✓ Solid melt at TM ➔ Molecular ordering is not completely lost in Liquid
✓ Arise first and foremost from the molecular geometry (reflected into the repulsive force)

Intermolecular and Surface Forces 97

 8. Special Interactions:
Hydrogen-Bonding and
Hydrophobic, Hydrophilic
Interactions

Intermolecular and Surface Forces 98

 Unique Properties of Water
1) Low Molecular Weight but Unexpectedly High TM, TB, Lvap
✓ Extraordinarily strong intermolecular interaction

2) ρice < ρwater (Max ρ at 4 oC)

✓ In the ice lattice, the molecules prefer to be farther apart than in the liquid
✓ Intermolecular bonds in ice are strongly orientation-dependent
• Tetrahedral coordination (4 nearest neighbors per molecule) rather than close packing

3) Very Low Compressibility and Unusual Solubility

4) Increase in ε upon Liquid-to-Solid Transition
✓ Ice lattice affords easy pathways for the charge transfer along
the hydrogen-bonding network (Proton hopping mechanism)
• Same mechanism persist into the liquid state (Grotthus
Mechanism)
Intermolecular and Surface Forces 99
 Hydrogen Bond
➢ Hydrogen bond
1) Unusually strong and orientation dependent type of dipole-dipole interaction associated an
electron-depleted H (H+)

Ice Formamide Alcohols, HF Fatty acids Intermolecular Shared H-Bond

(3D: Tetrahedral) (2D: Layer, Sheet) (1D: Chain, Ring) (Dimer) H-Bond

2) In the ice lattice, DO-H (~ 0.1 nm) < DO···H (~ 0.176 nm) < DVDW (~ 0.26 nm)
➔ Possess some covalent character (Quasi-Covalent bond ??)

Intermolecular and Surface Forces 100

 Hydrogen Bond
3) Orientation dependent interactions
➔ Shor-range order in liquids can be of significantly longer range
when hydrogen bonds are involved (Associated Liquid)
4) Predominantly electrostatic (Coulombic) interaction

➢ Strength of Hydrogen Bond

wVDW (1 kJ/mol, 1 KT) < wH (10-40 kJ/mol, 5-10 kT) < wC or I (500 kJ/mol, 100 kT)
✓ There is no simple Eq.
✓ Tend to follow a 1/r2 dependence ➔ Expected to apply the charge-dipole interaction Eq
𝑄H+ 𝜇 cos 𝜃
𝑤 𝑟, 𝜃 = −
4𝜋𝜀0 𝜀𝑟 𝑟 2
➔ QH+ is not the full electronic charge +e (not known) and neither is r

Intermolecular and Surface Forces 101

 Hydrophobic Effect
➢ At Hydrophobic–Water Interfaces
✓ Water molecules become more ordered at a hydrophobic-Water interface (e.g., air-water)
✓ One of 2 positive H+ atoms or one of 2 negative O- atoms (higher possibility) point toward
the surface
➔ Water-air interface has a negative potential (~ -15 to -40 mV)
✓ Other 3 arms are interacting with the arms of neighboring tetrahedral structure

Intermolecular and Surface Forces 103

 Hydrophobic Effect
➢ Hydrophobic Solvation (= Hydrophobic Hydration)
✓ H-bonds between water molecules influences their interactions with nonpolar molecules
(e.g., alkanes, hydrocarbons, fluorocarbons, inert atoms, vapor) that cannot form H-bonds
✓ By rearranging the coordination of bulk liquid (cage formation) around the nonpolar
molecule, the water can accommodate the nonpolar molecules with the lowest free energy
✓ The water molecules forming the cages (4 H-bonds/molecule) are more ordered than in the
bulk liquid (3-3.5 H-bonds) ➔ Higher free energy state but surprisingly stable
➔ Enable the trapping of high concentration of gases (e.g., Methane, CO2)

Intermolecular and Surface Forces 102

 Hydrophobic Interaction
➢ Hydrophobic Interaction
✓ Unusually strong attraction between hydrophobic molecules/surfaces in water, which is
closely related to the hydrophobic effect
✓ Often stronger than their interaction in free space
➔ For two methane : wVDW (in free space) = –2.5·10–21 J > wVDW (in water) = –14·10–21
➔ For typical hydrocarbons: σ (in free space) = 15-30 mJ/m2 < σ (in water) = 40-50 mJ/m2
✓ Arise primarily from the rearrangement of H-bond configurations in the overlapping
solvation zones as two hydrophobic species come together
✓ Only few measurements because nonpolar molecules
are so insoluble in water
✓ Difficulty in theoretical modeling as many other
molecules are involved in the interaction of two
molecules

Intermolecular and Surface Forces 104

 Hydrophobic Interaction
➢ Hydrophobic Pair Potential ( wH(r) )
✓ Experiment with two macroscopic curved hydrophobic surfaces in water
➔ In the range of 4-10 nm, the force decayed exponentially
✓ For small solute molecules
𝑤H 𝑟 ≈ −20 𝜎 𝑒 −(𝑟−𝜎)/𝐷H 109 kJ/mol • σ : Diameter of molecule / molecular group
≈ −8 𝜎 𝑒 −(𝑟−𝜎)/𝐷H 109 𝑘B 𝑇 at 298 K • DH : Characteristic hydrophobic decay length [~1 nm]

✓ Free energy of dimerization

∆𝐺dimer = 𝑤H 𝜎 ≈ −20 𝜎 109 kJ/mol ≈ −8 𝜎 109 𝑘B 𝑇 at 298 K
∆𝐺dimer (cyclohexane) = 𝑤H 0.57 nm ≈ −11.4 kJ/mol ≈ −5 𝑘B 𝑇 at 298 K

✓ Hydrophobic force to separate two molecules

𝑑𝑤H 𝑟 𝑟−𝜎
−
𝐹H 𝑟 = ≈3𝜎𝑒 𝐷H 10−2 N at 298 K
𝑑𝑟

Intermolecular and Surface Forces 105

 Hydrophobic Interaction
✓ Hydrophobic force to separate two molecules from contact
𝐹H 𝜎 ≈ 3 𝜎 10−2 N at 298 K 𝐹H cyclohexane ≈ 17 pN at 298 K

✓ Effective range of Hydrophobic interaction

𝑑 ∗ −𝜎
−
𝑤H 𝑑 ∗ ≈ −8 𝜎 𝑒 𝐷H 109 𝑘B 𝑇 > 𝑘B 𝑇 (298 K)

𝑑 ∗ < 𝜎 − 𝐷H ln(109 /8𝜎) [m] 𝑑 ∗ cyclohexane < ~2.1 nm

✓ d* for small molecules (e.g., methane, cyclohexane, benzene) ≈ 1.5 ‒ 2.0 nm

>> Intermolecular separation (0.2 ‒ 0.4 nm)
➔ Hydrophobic Interaction = Long-Range Interaction
✓ Long range has implications for various molecular phenomena in water
(e.g., micelles formation, folding of proteins, hydrophobic aggregation, fusion of membranes)
✓ Addition of electrolyte ions can vary the amplitude of hydrophobic interaction

Intermolecular and Surface Forces 106

 Hydrophilic Interaction
➢ Hydrophilic Interaction
✓ Strong repulsion in water between certain
water-soluble molecules
(prefer to be in contact with water)
✓ Example: Hydrogels, Hydrated ions,
Zwitterions
✓ Uncharged / Nonpolar molecule can be also
hydrophilic if they can form H-bonding
(e.g., O atoms in C=O & -OH, N atoms in
amines, Ethylen-Oxide groups)

Intermolecular and Surface Forces 107

 Hydrophilic Interaction
➢ Hydrophilic Surface
Water Water

Hydrophilic Surface Hydrophobic or Vapour Surfaces

✓ Hydrophilic Surface: Strong and Positionally / Orientationally binding of water

• Additional steric barrier ➔ Increase in the range of steric force and the effective size
• Unidirectional orientation of water molecules ➔ Partially restrict charge transport
✓ Positional & Orientation ordering of H2O affect the interactions between surfaces/molecules
1) Positional Ordering into layers ➔ Affect on Oscillatory forces (P. 97)
2) Orientational Ordering ➔ Affect on Electrostatic / Charge-transfer Interactions
3) Mean Density Variation ➔ Additional steric repulsion between hydrophilic surfaces &
Depletion attraction between hydrophobic surfaces

Intermolecular and Surface Forces 108

 Hydrophilic Interaction
➢ Chaotropic Agent (= Chaotrope)
✓ Molecules, when dissolved in water, having a drastic effect on other solutes by
altering or disrupting the local ordering of water
✓ Example: Urea (NH2)2C=0 vs Proteins in Water
• Addition of Urea unfolds proteins in Water
➢ Hydrophilic & Hydrophobic Interactions
✓ Interdependent but Not Additive as both are determined
by the structure of H-bonds
• ‒OH (Hydrophilic) + ‒(CH2)11CH3 (Hydrophobic)
➔ Complete neutralization
• Attachment of hydrophilic head group onto a
hydrophobic alkyl chain reduce the hydrophobicity of
the chain

Intermolecular and Surface Forces 109

9. Unifying Concept in
Intermolecular and
Interparticle Forces

Intermolecular and Surface Forces 110

 Association of Like Molecules or Particles in a Medium
➢ Any Type of Interaction between Molecules A and B
✓ Proportional to some molecular property of A (A) X some molecular property of B (B)
𝑊AA = −𝑨2 , 𝑊BB = −𝑩2 , 𝑊AB = −𝑨𝑩,

➢ Dispersed State vs Associated State

✓ Liquid consisting of a mixture of A and B (same
concentration)

∆W

• Dispersed State • Associated State

∆𝑊(2D) =

∆𝑊(3D) =
Intermolecular and Surface Forces 111
Association of Like Molecules or Particles in a Medium
✓ For the simplest case of two associating molecules

∆𝑊 =

Dispersed State Associated State

✓ In general,

∆𝑊 = 𝑊ass − 𝑊dis =
• n = # of like “bonds”

∆W < 0 ➔ Association state is energetically preferred in a binary mixture

Intermolecular and Surface Forces 112

 Association of Like Molecules or Particles in a Medium
2
➢ Insights into the Interactions of Like Molecules from ∆𝑊 = −𝑛 𝑨 − 𝑩
1) Eq. can be expressed in numerous different forms
2
∆𝑊 = −𝑛 𝑨 − 𝑩 = −𝑛(𝑨2 + 𝑩2 − 2𝑨𝑩)

2) Coulomb interaction between charged atoms ➔ ∆W > 0

3) Specific, Complementary, Lock-and-Key Interactions

➔ WAB = ‒AB is not valid
• Complementary shapes of molecules
• Inherently specific interatomic bonds (e.g., H-bond)

Intermolecular and Surface Forces 113

 Two Like Surfaces in a Medium
➢ Surface & Interfacial Energy of Flat Surfaces A in a Liquid B
✓ Free energy change between two flat surfaces A in a Liquid B
1 1 2
1 1
∆𝑊 = −2𝛾AB or 𝛾AB = − ∆𝑊 = 𝑛 𝑨 − 𝑩 = − 𝑛𝑊AA − 𝑛𝑊BB + 𝑛𝑊AB
2 2 2 2
✓ Let there be n bonds per area A
• nWAB : Energy change of bringing unit area A into contact with unit area of B in a vacuum
➔ Adhesion Energy or Work of Adhesion per unit area of A-B interface
• nWAA : Energy change of bringing unit area A into contact in a vacuum
➔ Cohesion Energy or Work of Cohesion per unit area of A-B interface

✓ Surface Energy of A (γA)

✓ Interfacial Energy of A-B (γAB, Combining Law)

Intermolecular and Surface Forces 114

 Association of Unlike Molecules / Particles / Surfaces in a Medium
➢ Unlike Molecules A and B in a Media C
✓ Free energy change from “A, B dispersed” to “A, B associated” states
∆𝑊 = 𝑊ass − 𝑊dis =

✓ Equilibrium state is “AA and BB associated”

∆𝑊 𝑎 ⟷ 𝑐 = 𝑊(c) − 𝑊 a =
∆𝑊 𝑏 ⟷ 𝑐 = 𝑊(c) − 𝑊 b =

✓ This procedure can be extended to mixtures with more species

➔ Always an effective attraction between like molecules in a
multicomponent mixture
➔ Unlike molecules can attract or repel each other in a solvent
Intermolecular and Surface Forces 115
 Particle-Surface and Particle-Interface Interactions
➢ Particle C near an Interface by Two Immiscible Liquid A and B
✓ Energy change of particle coming up to the interface
• From A : ∆𝑊 ∝
• From B : ∆𝑊 ∝ Adsorption
Adsorption
✓ Three possible situations (A solid)
1) C intermediate (A > C > B or B > C > A) ➔ ∆𝑊 & ∆𝑊 < 0 Engulfing (A→B)
•
2) B intermediate (A > B > C or C > B > A) ➔ ∆𝑊 < 0 & ∆𝑊 > 0 Engulfing (B→A)
•

3) A intermediate (B > A > C or C > A > B) ➔ ∆𝑊 > 0 & ∆𝑊 < 0 Desorption

•
✓ Repulsion from both sides of an interface cannot occur
Intermolecular and Surface Forces 116
 Engulfing and Ejection
➢ Total Energy of Transfer from A to B for Particle C
Engulfing (A→B)

∆𝑊t ∝ ∆𝑊𝐴→𝐵 = ∆𝑊 − ∆𝑊 ∝ − 𝑪 − 𝑨 𝑩 − 𝑨 + (𝑪 − 𝑩)(𝑨 − 𝑩)

✓ When ∆𝑊t < 0 (i.e., 𝛾BC < 𝛾AC ), engulfing of particle C from medium A to B occurs
✓ Total energy in terms of interfacial energy
∆𝑊t = Surface area ∙ 𝛾BC − 𝛾AC = 4π𝑟 2 ∙ (𝛾BC − 𝛾AC )

✓ Required condition for Engulfing

𝛾BC + 𝛾AC > 𝛾AB
Intermolecular and Surface Forces 117
 Adsorbed Surface Films : Wetting and Nonwetting
➢ At High Concentration of C in Media A or B
✓ Phase separation, Aggregation, or Gelation of C either in A or B,
or at A-B depending on the relative magnitudes of A, B, and C
✓ Formation of adsorbed C films at the A-B interface
(Solid A in contact with a binary mixture of B and C)
1) C intermediate ➔ Adsorbed layer of C (wetting)
2) B intermediate ➔ Engulfing (A→B)
➔ Desorption of C from A (dewetting)
3) A intermediate ➔ Engulfing (B→A)
➔ Both B & C are attracted to the solid surface
➔ No uniform film but droplets of C (partial-wetting)
• Contact angle θ : γAC + γBC cos θ = γAB (Young Eq.)
• Adhesion Energy A-C in B : ΔWABC = γBC (1+cos θ) (Young-Dupre Eq.)
Intermolecular and Surface Forces 118