Appendix I
111
APPENDIX I
Derivation of equations for the analysis of dye self-aggregation
Assuming that the self-aggregation process could be represented by the reaction (I.1)
KD
mD ' Dm (I.1)
its equilibrium constant can be written as eq. (I.2)
1 m m D [D] [D]
]
K [D −
= × (I.2)
Thus
m)log[D]
(1 [D] logK
]
log[Dm = D − − (I.3)
Eq. (I.3) could be written in the form
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛ ×
− +
=
D D D
m
C 1 [D]
m)log C (1
[D] logK ]
log[D (I.4)
and mass conservation in the form of eq. (I.5)
[D]
] [D 1 m
[D]
CD ⋅ m
+
= (I.5)
Introduction of eq. (I.5) into eq. (I.4) yields eq. (I.6)
⎟ ⎟
⎟ ⎟
⎠
⎞
⎜ ⎜
⎜ ⎜
⎝
⎛ + ×
− +
=
D m D
m
C [D]
] m [D 1 m)log (1
[D] logK ]
log [D
(I.6)Appendix I
112
For λ = 425 nm and λ = 525 nm it turns out that A425 = εD425 [D] and A525 = εDm525 [Dm]. Since for BO εD425 ≈ εDm525, then [Dm]/[D] = A525/A425 and eq.
(I.6) is reduced to eq. (5.2) of the text.
From the kinetic point of view, consider the reaction
(I.7)
with rate law (I.8)
] [D k [D]
dt k ] d[D
2 d 2 f
2 = − (I.8)
and mass conservation expressed by eq. (I.9)
CD = [D] + 2[D2] (I.9)
Denoting as δ[D] and δ[D2] the change of reactant concentrations following a jump of temperature, then:
[D]
] D [
δ[D] = −
(I.10)] [D ] D [ ]
δ[D2 = 2 − 2 (I.11)
where
[ D ]
and [D2] are the equilibrium concentrations.Introduction of eqs. (IV.10) and (IV.11) into eq. (IV.8) yields:
]) δ[D ] D ([
k δ[D]) ]
D ([
dt k ]) δ[D ] D d([
2 2
d 2 f
2
2 − = − − −
(I.12)
Eq. (I.9) can be written in differential form as
δD = -2 δD2 (I.13)
2D k
fD
2k
dAppendix I
113 Moreover,
δD 2[D]
δ[D]
2<< ⋅
(I.14)and
k
f[ D ]
2= k
d[ D
2]
(I.15)Introducing eq. (I.13) on the relationship (I.12), one obtains
] δ[D ) k ] D [ k dt (
] [D dδ
2 d f
2 = + ⋅
− 4 (I.16)
or
dt ) k ] D [ ] k
δ[D [D dδ
d f
2
2 = + ⋅
− ] (4
(I.17)
Integration of eq. (I.17) yields
t/τ 1 0 2](t) A A e
[D = + − (I.18)
where
d f[D] k 1 = k +
τ 4 (I.19)
If the aggregation process is sufficiently limited, then one can state ][D ≅ CD.
Appendix I
114
