APPENDIX VI

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123

APPENDIX VI

Derivation of the relaxation times for a system of coupled reactions using the Castellan’s method

Consider the reaction scheme (charges omitted)

Fe ⇄ FeOH + H (VI.1)

2 FeOH ⇄ D (VI.2)

D + FeOH ⇄ T + H (VI.3)

Following an external perturbation of the system equilibrium, reaction (VI.1) reaches the equilibrium much faster than reactions (VI.2) and (VI.3), so the

€ FeOH

[

]

Fe

[ ]

= K_H H

[ ]

ratio

remains constant for each [H] value (in excess) considered. This fact has enabled us to reduce the reaction system to the simpler form

cD 2M ⇄ D (VI.4) c-D cT D + M ⇄ T (VI.5) c-T

where [M] = [Fe] + [FeOH].

Let’s denote the exchange rates of reactions (VI.4) and (VI.5) as rD and rT, respectively.

rD = cD [M] 2_{= c}

-D [D] (VI.6)

rT = cT [D][M] = c-T [T] (VI.7)

According to Castellan (Castellan, 1963), the coupling coefficients of the reaction system (VI.4)-(VI.5) are defined as:

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Appendix VI 124 € g₁₁ = 4 M

[ ]

+ 1 D

[ ]

(VI.8) € g₁₂ = g21 = 2 M

[ ]

− 1 D

[ ]

(VI.9) € g22 = 1 M

[ ]

+ 1 D

[ ]

+ 1 T

[ ]

(VI.10)

The two relaxation times, 1/τk, of the system of coupled reaction (VI.4)-(VI.5) depend on

the equilibrium concentrations according to the determinant equation (VI.11)

€ rDg11− 1

τ

k rDg12 rTg21 rTg22− 1

τ

k = 0 (VI.11)

which provides the expressions for 1/τk in the form (VI.12)

€

1

τk

= TR ± TR2 − 4DET

2 (VI.12)

where TR is the trace and DET is the determinant of equation (VI.11). The k = 1 value corresponds to 1/τf (fast) and k = 2 corresponds to 1/τs (slow). Instead of considering

separately the two expressions (VI.12), we have applied the properties

(VI.13)

(VI.14)

where TR = rDg11+rTg22 and DET = rDrT(g11g22−g12g21).

Introduction of equations (VI.6)-(VI.10) into equations (VI.13) and (VI.14) yields equations (VI.15) and (VI.16)

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Appendix VI 125 € 1 τf + 1 τs = 4

(

χD+χT

)

[ ]

M +

(

χ−D+χ−T

)

(VI.15) € 1 τf × 1 τs = 6χDχT

[ ]

M 2 + 4χDχ−T

[ ]

M +χ−Dχ−T (VI.16)

Since the concentration of the total monomer, [M] over the explored range of concentrations is much higher than that of the aggregated species, one can replace [M] by CM.

Equations (VI.15) and (VI.16) are reported in the text (equation 4.7 and 4.8 of chapter 4) with this approximation.

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