Preferential Adsorption and Co-nonsolvency of ... - au one net

Macromolecules ARTICLE
(bl,m Kl,m/(n þ nAl þ nBm)). This function is called binding
polynomial in the literature. It is equivalent to the grand partition
function of a H-bonding chain in the mixed solvent.
The degree of binding for each component is defined by the
average number of bound molecules per binding site on the
polymer chain and derived from the binding polynomial by
differentiation
θRðyA, yBÞ
1 ∂ ln p
n ∂ ln yR
ðR ¼ A, BÞ ð2.20Þ
In summary, we have found
βΔμ0, 0 ¼ 1 þ ln x - nv S þ ng0, 0 ð2.21aÞ
βΔμ fA ¼ 1 þ ln yA - nAv S þ nAgfA
βΔμ fB ¼ 1 þ ln yB - nBv S þ nBgfB
ð2.21bÞ
ð2.21cÞ
with
v S ¼ φ=n þ yA=nA þ yB=nB ð2.22Þ
Substituting the equilibrium distribution function (2.16) back
into the original free energy, we find
βΔF=Ω ¼ F FH þ F AS ð2.23Þ
where
F FH ¼ φ
n ln φ þ φA ln φA þ
nA
φB ln φB þ gðfφgÞ ð2.24Þ
nB
is the conventional Flory-Huggins free energy, and
F AS ¼ φ x
ln
n φ þ φA ln
nA
yA
þ
φA φB ln
nB
yB
þðθA þ θBÞφ
φB ð2.25Þ
is the free energy due to the molecular association.
3. PREFERENTIAL ADSORPTION BY POLYMERS IN
MIXED SOLVENTS
The total degree of binding is
θ θA þ θB ð3.1Þ
If there is no interaction between the bound molecules, the total
is given by the simple mixing law
ð3.2Þ
where θR° is the degree of binding in each pure solvent, and
xR NR=ðNA þ NBÞ ðR¼A, BÞ ð3.3Þ
are the mole fractions of the solvent. The excess binding is then
defined by
Δθ E
θ° θ °
A xA þ θ °
B xB
θ - θ° ¼ θA - θ °
A xA þ θB - θ °
B xB
ð3.4Þ
For cosolvency, the excess is positive, while for co-nonsolvency
it is negative. There is a possibility that the excess binding
changes its sign at a certain composition of the mixed solvent.
The mole fraction of the bound molecules at such a particular
concentration takes exactly the same value as the mole fraction of
the solvent in the bulk outside the coil region. Such a particular
point is called azeotropic binding. 26
In what follows we focus on co-nonsolvency, and derive Gibbs
matrix for the study of phase separation. As for the composition
of the solvent mixture, we use its volume fraction
vR nRNR=ðnANA þ nBNBÞ ðR¼A, BÞ ð3.5Þ
as well as the mole fraction. We then have
φA ¼ð1-φÞvA, φB ¼ð1-φÞvB ð3.6Þ
We regard the volume fraction vB of the second solvent as the
controlling parameter, and write it as vB ξ. We then have vA =1
- ξ.
3.1. Transition Matrix. Let us first relate the variation of the
volume fraction yA and yB of the free solvents by the variation of
the independent variable φA and φB (solvent composition) which
are controlled variables in the experiments. We first define a
matrix JR,β by the equation
d ln yR ¼ X
JR, βdφβ ð3.7Þ
Or, in the matrix form
2 3
dlnyA
4 5 ¼
dlnyB
JA,
2 3
A
4
JB, A
JA, B
5
JB, B
dφ 2 3
A
4 5
dφB ð3.8Þ
By taking the derivatives of the material conservation laws
(2.18a) to (2.18c), we have
p dx þ φðθA dlnyA þ θB dlnyBÞ ¼ - ðdφA þ dφBÞ=n ðyA þ nAφθAKA, AÞ dlnyA þ nAφ AθAKA, B dlnyB
2981 dx.doi.org/10.1021/ma102695n |Macromolecules 2011, 44, 2978–2989
β
¼ð1 þ nAθAÞ dφ A þ nAθA dφ B
nBφθBKB, A dlnyB þðyB þ nBφθBKB, BÞ dlnyB
¼ nBθB dφA þð1þ nBθBÞ dφB where the matrix KR,β is defined by
∂ ln θR
KR, β
∂ ln yβ
Solving these equations for d lnyR, we find
JA, A ¼fð1-φB- yAÞ½yB þðφB - yBÞKB, BŠ
- ðφA - yAÞðφB - yBÞKA, Bg=φΦ
ð3.9Þ
ð3.10Þ
JA, B ¼ðφ A - yAÞ½yB - φKA, B þðφ B - yBÞðKB, B - KA, BÞŠ=φΦ
JB, A ¼ðφ B - yBÞ½yA - φKB, A þðφ A - yAÞðKA, A - KB, AÞŠ=φΦ
JB, B ¼fð1-φA- yBÞ½yA þðφA - yAÞKA, AŠ
- ðφB - yBÞðφA - yAÞKB, Ag=φΦ ð3.11Þ
where
ΦðyA, yBÞ yAyB þðφA - yAÞyBKA, A þðφB - yBÞyAKB, B
þðφA - yAÞðφB - yBÞK ð3.12Þ
and
K KA, AKB, B - KA, BKB, A ð3.13Þ
is the determinant of the matrix ^K.