Aggregates Flashcards
Chemical Potential of Monomers
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
Chemical Potential of Dimers
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
Chemical Potential of Aggregates of Size N
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
Concentration of Molecules in Aggregates of Size N
Xn = N * [X1*exp((µ10-µn0)/kT)]^N
Total Concentration of Molecules
C = X1 + X2 + … = Σ Xi
Chemical Potential
Phase Separation
- 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
Chemical Potential
Micelle/Vesicle Formation
- 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
Chemical Potential
Functional Form
- the functional form of the variation of µn0 with N determines the physical properties of the aggregates
- e.g. size, size distribution etc.
Variation of µn0 with N
1D Aggregates
µn0 = µ∞0 + αkT/N
Variation of µn0 with N
2D Aggregates
µn0 = µ∞0 + αkT/(N^(1/2))
Variation of µn0 with N
3D Aggregates
µn0 = µ∞0 + αkT/(N^(1/3))
Critical Micelle Concentration
Formula
Xn ≈ N * [X1 e^α]^N
Total Concentration At Low Monomer Concentration
-for low monomer concentration X1, such that X1e^α«1 which means that X1
Critical Micelle Concentration
Description
- 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)
Critical Micelle Concentration
Definition
- 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
What is a typical critical micelle concentration for most single chain surfactants?
~ 10^(-2) - 10^(-5) M
What is a typical critical micelle concentration for most double chain surfactants?
~ 10^(-10) M
Alkanes vs Amphiphilic Molecules
- 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
Size of Spherical Micelle
- 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
Conditions Required for Aggregation
- 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
Major Driving Forces for Amphiphilic Self-Assembly
- 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
Optimal Headgroup Area
Attractive Interaction
- 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
Optimal Headgroup Area
Repulsive Interaction
- 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
Optimal Headgroup Area
μ_No
μ_No = γa + K/a
Optimal Headgroup Area
Optimal Surface Area Per Molecule
-differentiate μ_No with respect to a to find the value of a that minimises μ_No
=>
ao = √[K/γ]
Optimal Headgroup Area
μ_No|min
μ_No|min = 2γa
