Flashcards in Chapter 4 - The real chain Deck (17)
Which types of solvents do we have?
Describe each of them with regards to the excluded volume.
Athermal, good, theta, poor, non-solvent
What is the pervaded volume?
It is the volume spanned by the polymer as if it was a cube with length R. The pervaded volume is much bigger than the actual volume of the polymer.
What is the overlap volume fraction?
It is the fraction between the actual volume of the polymer and the pervaded volume:
phi = Nb^3/R^3 = b^3N/(bN^1/2)^3 = N^-1/2
(That is, the overlap fraction is very small, as N >> 1, N^-1/2 << 1)
What is the number of monomer-monomer contacts in a mean-field approximation of the real chain?
It is proportional to the overlap volume fraction:
#number of monomer-monomer contacts = N * phi = N* N^-1/2 = N^1/2
(That is, even though the overlap fraction is very small, since there are so many monomers, the number of monomer-monomer interactions is still very high).
Draw the energy cost curve of the following three situations:
Bringing two monomers from ∞ -> r when..
i) ... the monomer likes the monomer better than the solvent.
ii) ... the monomer has no preference between monomer and solvent.
iii) ... the monomer likes the solvent better than the monomer.
i) The curve will have an attractive part when the monomers are a short distance from each other (that is, negative energy cost), before the hard-core repulsion hits in.
ii) The curve will have no energy cost until the hard-core repulsion hits in.
iii) The curve will have an additional repulsive region before the hard-core repulsion hits in.
What is the Mayer f-function?
The Mayer f-function is defined as
f(r) = exp[-U(r)/kT] - 1,
that is the relative probabilty of finding two monomers seperated by r at T minus unity. This means that attractive parts will have a positive value, repulsive parts will have a negative value.
What is the interpretation of the integral of the Mayer f-function?
The integral of the Mayer f-function is the negative of the excluded volume.
What is the excluded volume?
The excluded volume refers to the effect that one monomer can't occupy the same space as another. This can be negative, in the case where there is extra attractive forces between the monomers. Defined as
-∫f(r) d^3r = -∫(1-exp[-U(r)/kT) d^3r
What does it mean when the excluded volume is negative? Positive?
When the excluded volume is negative, there is net attraction. When the excluded volume is positive there is net repulsion.
What is an athermal solvent? What is the excluded volume?
An athermal solvent is a solvent in which there is only hard-core repulsion, as is the case when a polymer is solvated in a solvent of same nature (e.g. polystyrene in ethyl benzene).
The excluded volume is v = b^3
What is a good solvent? What is the excluded volume?
A good solvent is a solvent in which the monomer-solvent interaction is slightly stronger than the monomer-monomer. This leads to a small net repulsion.
The excluded volume is 0 < v < b^3
What is a theta solvent? What is the excluded volume?
This is the case where the contribution of the attractive well cancels the contribution from the hard-core repulsion, giving the polymer an ideal chain-like behavior. v = 0, at the theta-temperature.
The excluded volume is v = 0.
What is a poor solvent? What is the excluded volume?
This is the case when the contribution from the attractive well is higher than the contribution from the hard-core repulsion, causing a breakdown in the chain.
The excluded volume is -b^3 < v < 0.
What is a non-solvent? What is the excluded volume?
A non-solvent is the limiting case of a poor solvent, when the monomers have a very strong preference for each other, and that nearly all solvent is excluded.
The excluded volume is v ≈ -b^3.
What is the conformation of a real chain in an athermal or good solvent?
Swelling of chain due to repulsion energy between two monomers. This causes entropy loss, because the end-to-end vector is decreased.
What is the free energy contribution of interaction in a good solvent?
F_int = kT * v * N * N/R^3
([b^3 * N * N/R^3] is the number of monomer-monomer contacts, so v/b^3 is a measure of the interaction between monomer and solvent)