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

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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.
Macromolecules ARTICLE 3.2. Preferential Adsorption Coefficients. The preferential adsorption coefficient ΓR,β is defined by the change d ln θR of the adsorbed R-component by an infinitesimal increment dφβ of the β-component solvent. By multiplication of K matrix to the transition J matrix, we find 2 3 dlnθA 4 5 ¼ dlnθB KA, 2 3 A KA, B 4 5 KB, A KB, B JA, 2 3 A JA, B 4 5 JB, A JB, B dφ 2 3 A 4 5 dφB ¼ ΓA, " # " # A ΓA, B dφA ð3.14Þ ΓB, A ΓB, B dφB where ΓA, A ¼fð1-φB- yAÞ½yBKA, A þðφB - yBÞKŠ þðφB - yBÞyAKA, Bg=φΦ ΓA, B ¼fð1 - φ A - yBÞyAKA, B þðφ A - yAÞyBKA, A þðφ A - yAÞðφ B - yBÞKg=φΦ ΓB, A ¼fð1 - φ B - yAÞyBKB, A þðφ B - yBÞyAKB, B þðφ A - yAÞðφ B - yBÞKg=φΦ ΓB, B ¼fð1-φA- yBÞ½yAKB, B þðφA - yAÞKŠ þðφA - yAÞyBKB, Ag=φΦ ð3.15Þ If the volume fraction φ of the polymer is fixed, dφ A and dφ B are not independent, but related by dφ A ¼ - ð1 - φÞdξ ð3.16aÞ dφB ¼ð1-φÞdξ ð3.16bÞ and hence the preferential adsorption coefficient is ∂θA Γ ∂ξ ð1 - φÞθA ¼ φΦðyA, yBÞ ½ð - yBKA, A þ yAKA, BÞφ - ðφB - yBÞφKŠ Here yA and yB are the solutions of the coupled equations ð3.17Þ yA þ nAφθAðyA, yBÞ ¼ð1 - φÞð1 - ξÞ ð3.18aÞ yB þ nBφθBðyA, yBÞ ¼ð1 - φÞξ ð3.18bÞ 4. PHASE SEPARATION IN MIXED SOLVENTS Although we are ready to find the binodals by using the chemical potentials, we are involved in many technical difficulties for numerical calculations of three-component systems, in particular in the presence of the cooperativity in the H-binding. Therefore, we attempt to find spinodal lines only by the calculation of Gibbs matrix. 4.1. Construction of the Gibbs Matrix. In tertiary systems, two of the three chemical potentials are independent due to the Gibbs-D€uhem relation. Here we use the chemical potentials of the free solvent molecules. The Gibbs matrix is then defined by GR, β ∂Δμ f R ∂φ β Because we have dv S vA dφA þ vB dφB for the variation of ν S , where and also vA vB - 1 n - 1 n þ yA nA þ yA nA JA, A þ yB JA, B þ yB JB, A nB JB, B nB " # dgfA ¼ dgfB gA, " # " # A gA, B dφA gB, A gB, B dφB ð4.1Þ ð4.2Þ ð4.3aÞ ð4.3bÞ ð4.4Þ for the variation of the interaction, we find " # dΔμfA ¼ dΔμfB " # " # JA, A - nAvA þ nAgA, A JA, B - nAvB þ nAgA, B dφA JB, A - nBvA þ nBgB, A JB, B - nBvB þ nBgB, B dφB ð4.5Þ for the variation of the chemical potentials, where gA, A - 2χAφ - ðχA - χB þ χABÞφB ð4.6aÞ gA, B - χ B ðφ - φ BÞ - ðχ A - χ AB Þð1 - φ AÞ ð4.6bÞ gB, A - χ A ðφ - φ AÞ - ðχ B - χ AB Þð1 - φ BÞ ð4.6cÞ gB, B - 2χ B φ þðχ A - χ B - χ AB Þφ A ð4.6dÞ The Gibbs matrix is ^G ¼ JA, " A - nAvA þ nAgA, A JB, A - nBvA þ nBgB, A # JA, B - nAvB þ nAgA, B JB, B - nBvB þ nBgB, B ð4.7Þ 4.2. Spinodal Condition. The spinodal condition is given by |^G| = 0. After lengthy calculation, we finally find - nnAnB~χφΦ - 2nAnBχABðΦ þ nΨÞ - 2nðχAΦA þ χBΦBÞ þ ΨA þ ΨB þ nð1 þ nAθA þ nBθBÞ 2 φ ¼ 0 ð4.8Þ where Φ is defined by eq 3.12. The rests are Ψ yBθAðφA - yAÞþyAθBðφB - yBÞ þ 1 2 φ2ðθAKA, B þ θBKB, AÞþðφA - yAÞðφB - yBÞ½θAðKB, B - KA, BÞ and þ θBðKA, A - KB, AÞŠ ð4.9Þ ΨA nAyA þðφ A - yAÞðnAKA, A þ nBKA, BÞ ð4.10aÞ ΨB nByB þðφ B - yBÞðnBKB, B þ nAKB, AÞ ð4.10bÞ 2982 dx.doi.org/10.1021/ma102695n |Macromolecules 2011, 44, 2978–2989
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