Aggregates

Question 1

Chemical Potential of Monomers

Answer 1

µ1 = µ10 + kTlog(X1)
-where µ1 = chemical potential, total free energy per molecule
µ10 = standard free energy per molecule in a given state
kTlog(X1) = entropy of confining the molecules also known as ideal gas entropy, configurational entropy, entropy of confinement, ideal solution entropy, translational entropy, entropy of dilution, entropy of mixing
-and X1 is the concentration of aggregates of size 1

Question 2

Chemical Potential of Dimers

Answer 2

µ2 = µ20 + kT/2 * log(X2/2)
-where µ2 = chemical potential, total free energy per molecule in an aggregate of size 2
µ20 = interaction free energy per molecule in aggregates of size 2
X2 is the concentration of aggregates of size 2

Question 3

Chemical Potential of Aggregates of Size N

Answer 3

µn = µn0 + kT/2 * log(Xn/N)
-where µn = chemical potential, total free energy per molecule in an aggregate of size N
µn0 = interaction free energy per molecule in aggregates of size N
Xn is the concentration of aggregates of size N

Question 4

Concentration of Molecules in Aggregates of Size N

Answer 4

Xn = N * [X1*exp((µ10-µn0)/kT)]^N

Question 5

Total Concentration of Molecules

Answer 5

C = X1 + X2 + … = Σ Xi

Question 6

Chemical Potential

Phase Separation

Answer 6

• if µn0 decreases as N increases, there will be no limit to aggregate size
• they will continue to grow until all molecules are consumed in a single aggregate => phase separation

Question 7

Chemical Potential

Micelle/Vesicle Formation

Answer 7

• if µn0 has a minimum finite value of N, then aggregates of size N would be the preference
• multiple aggregates of size N would spontaneously arise

Question 8

Chemical Potential

Functional Form

Answer 8

• the functional form of the variation of µn0 with N determines the physical properties of the aggregates
• e.g. size, size distribution etc.

Question 9

Variation of µn0 with N

1D Aggregates

Answer 9

µn0 = µ∞0 + αkT/N

Question 10

Variation of µn0 with N

2D Aggregates

Answer 10

µn0 = µ∞0 + αkT/(N^(1/2))

Question 11

Variation of µn0 with N

3D Aggregates

Answer 11

µn0 = µ∞0 + αkT/(N^(1/3))

Question 12

Critical Micelle Concentration

Formula

Answer 12

Xn ≈ N * [X1 e^α]^N

Question 13

Total Concentration At Low Monomer Concentration

Answer 13

-for low monomer concentration X1, such that X1e^α«1 which means that X1

Question 14

Critical Micelle Concentration

Description

Answer 14

• if X1 approaches e^(-α), then, from the critical micelle concentration formula, X1 will not increase further
• however the higher cluster number aggregates Nn continue to increase
• the concentration X1 at which this occurs is known as the critical micelle concentration (CMC)

Question 15

Critical Micelle Concentration

Definition

Answer 15

• there comes a point when the number of monomers cannot increase and the molecules must become involved in aggregates
• at low concentrations, almost all of the molecules exist as monomers, but beyond a certain concentration, aggregates form
• this concentration is known as the CMC

Question 16

What is a typical critical micelle concentration for most single chain surfactants?

Answer 16

~ 10^(-2) - 10^(-5) M

Question 17

What is a typical critical micelle concentration for most double chain surfactants?

Answer 17

~ 10^(-10) M

Question 18

Alkanes vs Amphiphilic Molecules

Answer 18

• for alkanes (oils), μNo tends to increase with N
• this leads to phase separation, two immiscible phases
• for amphiphilic molecules, those with a hydrophilic head and hydrophobic tail, assemble into structures in which μ_No reaches a minimum or is constant at a finite value of N leading to micelle formation

Question 19

Size of Spherical Micelle

Answer 19

• there is not a random collection of spherical micelles of all sizes
• there is an optimal packing number for phospholipids to form a spherical micelle

Question 20

Conditions Required for Aggregation

Answer 20

• aggregation involves a decrease in entropy so can only occur if there is a significant decrease in overall free energy
• aggregation only occurs at concentrations exceeding the critical micelle concentration

Question 21

Major Driving Forces for Amphiphilic Self-Assembly

Answer 21

• the hydrophobic attraction which occurs at the hydrocarbon-water interface and induces the molecules to associate (entropic effect due to breakage of H-bonds)
• the hydrophilic, ionic or steric repulsion of the headgroups
• > these interaction act in opposite senses to decrease/increase the interfacial region

Question 22

Optimal Headgroup Area

Attractive Interaction

Answer 22

• hydrophobic force (interfacial tension) acts at the hydrocarbon-water interface
• can be represented by a positive interfacial free energy per unit area, γ=50mJ/m^2
• this gives a contribution to the free energy, μ_No, of aγ where a is the area per head group

Question 23

Optimal Headgroup Area

Repulsive Interaction

Answer 23

• hydrophilic, ionic or steric repulsion of the head groups
• not well defined since it is due to a combination of terms
• however, summed together they are inversely proportional to the are occupied per headgroup

Question 24

Optimal Headgroup Area

μ_No

Answer 24

μ_No = γa + K/a

Question 25

Optimal Headgroup Area

Optimal Surface Area Per Molecule

Answer 25

-differentiate μ_No with respect to a to find the value of a that minimises μ_No
=>
ao = √[K/γ]

Question 26

Optimal Headgroup Area

μ_No|min

Answer 26

μ_No|min = 2γa