Photophysics and kinetics of naphthopyran derivatives, part 6: A fundamental system of equations to describe ABC(3k,6o) dynamics. A contribution to the elucidation of [2H]-naphthopyran kinetics

Photophysics and Kinetics

of Naphthopyran

Derivatives, Part 6:

A Fundamental System of

Equations to Describe

ABC(3k,6φ) Dynamics.

A Contribution to the

Elucidation of [2H]-

Naphthopyran Kinetics

MOUNIR MAAFI,¹ROBERT G. BROWN²

¹Leicester School of Pharmacy, De Montfort University, The Gateway, Leicester, LE1 9BH, UK

²Faculty of Health, Sport and Science, University of Glamorgan, Pontypridd, Cardiff CF37 1DL, UK

Received 6 September 2005; revised 10 July 2007; accepted 2 August 2007

DOI 10.1002/kin.20306

Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: A new spectrokinetic method free of any assumption is developed in this paper

to investigate photochemical systems obeying ABC(3k,6φ) kinetic mechanisms that can be

characterized by up to 14 unknown parameters for a nonpreconceived mechanism. The math-

ematical framework of this method is derived as an application from an original formalism

previously established to deal with the kinetic law descriptions relating to ABC reactive sys-

tems. The necessary number of linearly independent (but nonlinear) equations has been fully

deﬁned together with seven other supplementary equations useful as constraints during the

iterative calculations. As a practical example, this spectrokinetic method has been used to

Correspondence to: Mounir Maaﬁ; e-mail: mmaaﬁ@dmu.ac.uk.

2008 Wiley Periodicals, Inc.

c

ꢀ

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES

269

investigate the dynamics of a new photochromic derivative belonging to the naph-

thopyran family (2,2-di(4-methoxyphenyl)-5-carbomethoxy-6-(4-methoxyphenyl)-9-methoxy-

2H-naphtho[1,2-b]-pyran). The system of equations has been solved, and the problems of

identiﬁability and distinguishability have been raised with relation to the degeneracy of the ki-

netic solution. To get more insight into the photochemical and thermal reaction steps involved,

a progressive overall discussion leads to several quantitative conclusions whos_ꢀe background

C

sets the dynamics for the present derivative to at least ABC(1k,4φ) kinetics.

2008 Wiley

Periodicals, Inc. Int J Chem Kinet 40: 268–281, 2008

INTRODUCTION

studied individually. Second, their electronic absorp-

tion spectra have similar features (often overlapping

entirely) and mix with the spectrum of the initial, ring-

closed, species over the whole ultraviolet region. As a

consequence, femtosecond–nanosecond time-resolved

studies are restricted to limited investigation options. In

the case of spectrokinetic approaches, in addition to the

above, another challenge has to be considered as well.

It is understood that an NPY system involving three

species is deﬁned by up to 17 unknowns (potentially

including six photochemical yields, six thermal rate

constants, two extinction coefﬁcients, and three con-

centrations of the species at the photostationary state

(PSS)). The total number of unknowns reduces to 14 if

we assume that not more than three thermal processes

are active in the dynamics, as is the case for nonther-

mochromic 2H- and 3H-NPY systems. The challenge

(so far not met) is to derive as many equations that

describe the dynamics as the number of the spectro-

scopic and kinetic unknown parameters that deﬁne the

reactive system so that these unknowns can (in theory)

be determined.

One way to simplify this problem is to propose a

mechanism involving a reduced number of photochem-

ical reaction steps. However, it is important to bear in

mind that as the true mechanism is not known, several

equally plausible sequences can usually be postulated

and it is difﬁcult to argue rationally in favor of a partic-

ular one as the deﬁnitive true mechanism. In addition,

as we will see in this paper, it is not even possible to

be perfectly certain that the parameter values found for

a particular mechanism are unique, even though they

yield a good ﬁt to the experimental data.

Naphthopyrans (whether 2H- or 3H-derivatives

(NPY); see Scheme 1) are frequent components of the

photochromic element employed in plastic ophthalmic

lenses [1,2]. Judging by the ongoing intensive research

effort, their impact can be foreseen in future techno-

logical applications in areas such as photoswitches,

recording media, domestic glass dying, and liquid

crystals [3,4].

Although these expectations have triggered a huge

interest in this family of photochromes, most of the

literature is involved with the preparation of new struc-

tures (see, e.g., [5]). There are only a few investigations

focusing on understanding and elucidating the mech-

anistic processes [4,6–12]. In this ﬁeld, femto-, pico-,

and nanosecond studies represent the core of the in-

formation available that contributes to shed new light

on the mechanism of the 2H-NPY dynamics. Spec-

trokinetic and photokinetic studies of these molecules

are scarcer, and deﬁnitive procedures are yet to be pro-

posed and implemented to allow systematic investi-

gations. From this point of view, understanding and

unraveling the 2H-NPY mechanism offers a valuable

means of assisting the design of structures possessing

optimal features for speciﬁc applications.

The primary event of the photoreaction of a

2H-NPY is the ring opening of the pyran unit through

the cleavage of its intracyclic ether “C O” bond. This

process yields a number of geometric and rotation iso-

mers of the “open” form of the photochrome—the ex-

act number depending on the substitution pattern in the

molecule under study. Among these, only a few con-

formations are long lived enough to be observed over

time scales of seconds or greater (i.e., interesting from

both application and spectrokinetic aspects).

It is still unclear, however, as to the involvement

of these longer living species in the overall dynamics

when the reaction medium is submitted to continu-

ous irradiation. These difﬁculties arise from the con-

junction of two unfavorable factors. First, as has been

alluded to above, the observable species, even though

the most stable, cannot be physically isolated and

In our attempts to contribute to the study of ABC

systems in general and to those involving 2H- and 3H-

NPY in particular, we have developed a mathematical

formalism [13] that provides the kinetic laws for re-

active trimolecular systems of the ABC(nk,mφ) type

when exposed continuously to a monochromatic light

beam at a wavelength corresponding to an isosbestic

point. This formalism has been used as a theoretical

basis to develop several spectroscopic methods and

tools aimed at solving complex ABC kinetics [14–17].

International Journal of Chemical Kinetics DOI 10.1002/kin

270

MAAFI AND BROWN

Scheme 1 The three basic structural variations of the naphthopyran family.

Most of these studies have been applied to gain more

insight in the reactivity of 3H-NPY derivatives. In

this respect, one of our studies [16] has shown that

the reaction of 3,3-diphenyl-3H-naphtho[2,1-b]pyran

is governed by an ABC(1k,6φ) mechanism. The math-

ematical treatment employed for this study was based

on monoexponential kinetic traces and a system of 13

equations [16].

In the present paper, we propose a new and gen-

eral spectrokinetic method based on the mathematical

framework allowed by the formalism, capable of inves-

tigating an ABC(3k,6φ) system for the ﬁrst time. It is

applied here to study the kinetics of a 2H-substituted

naphthopyran derivative 1 (see Scheme 2).

sition and treatment. The sample holder (designed for

a 1-cm cell) is equipped with a temperature control

Peltier system model Agilent 8453. The latter equip-

ment has a precision of 0.1^◦C.

The irradiation equipment was manufactured by

Photon Technology International Corporation (Ford,

UK). The light source was an Ushio 1000 W xenon

arc-lamp located in a housing shell model A6000 and

powered by a model LPS-1200 power supply. The lamp

housing was connected to a model 101 monochroma-

tor, which is a special f /2.5 monochromator with a

1200-groove/300-nm blaze grating to allow irradia-

tion wavelength selection. The lamp was continuously

cooled by a ﬂow of fresh water delivered by a Watson

peristaltic pump model 505S. The excitation beam was

guided through an optical ﬁber to impinge upon the top

of the sample cuvette. Hence, the directions of the ir-

radiation and analysis light beams were perpendicular

so as to avoid interference. A shutter situated between

the monochromator and the optical ﬁber allowed the

excitation light to be manually switched on and off.

An excitation wavelength of 381 nm corresponding to

a wavelength, where the range of absorption variation,

measured at 15^◦C before irradiation and at the PSS,

causes less than 6% difference in the photokinetic fac-

tor values (F as given by Eq. (8); vide infra), i.e., F

can be approximated to a constant.

EXPERIMENTAL

The naphthopyran derivative (1) used in this study (2,2-

di(4-methoxyphenyl)-5-carbomethoxy-6-(4-methoxy-

phenyl)-9-methoxy-2H-naphtho[1,2-b]-pyran)

is

a commercially available photochromic dye that

has been generously donated by PPG Industries

(Pittsburgh, PA, USA) (synthesis procedure as

previously described [18]).

The spectrophotometric-grade toluene solvent was

supplied by Aldrich (Gillingham, UK) and was used

without further puriﬁcation.

The absorption spectra and kinetic proﬁles were

recorded on an Agilent 8453 diode array spectropho-

tometer equipped with a Chemstation for data acqui-

All measurements were carried out in a quartz cell.

The sample was maintained at a constant temperature

(of 15^◦C) and was stirred continuously during the ex-

periment. Typically, an experiment was carried out

on a 2-mL volume of 7.52 × 10⁻⁵M solution, mon-

itored at the wavelength of the open form isomer(s)

(λ_obs= 530 nm).

The experimental kinetic proﬁles were corrected for

the absorbance generated by irradiation with the spec-

trophotometer analyzing beam. Indeed, because the

reaction medium is continuously (at a sampling rate

of 5 s) analyzed by a wide wavelength range (190–

600 nm), a certain amount of photochemical reaction

is always present. Accordingly, the data acquisition be-

gan with a ﬁrst (dark) period of 300 s in which only

the spectrometer analysis beam was allowed to reach

the sample. Then, the excitation beam from the lamp

Scheme 2 Structure of the 2,2-di(4-methoxyphenyl)-5-

carbomethoxy-6-(4-methoxyphenyl)-9-methoxy-2H-naph-

tho[1,2-b]-pyran derivative (1) used in this study.

International Journal of Chemical Kinetics DOI 10.1002/kin

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES

271

was allowed to impinge upon the sample for a sec-

ond period of 400-s duration. At the end of each of

these time periods, the absorption versus time proﬁles

attained a plateau indicating that the photostationary

states were fully achieved. In a ﬁnal stage of 400 s, the

irradiation light was switched off, and the data corre-

sponding to the thermal fading reaction accumulated.

These data have been corrected for the interference

(using the trend observed during the ﬁrst period) due

to the diode array beam prior to analysis. (A typical

trace is shown as Fig. 2 later in this paper.)

Typically experimental errors were less than 5%.

A Levenberg–Marquardt iterative program and a

least-squares analysis tool within the Origin 6.0 soft-

ware package were used to run both nonlinear ﬁtting

on the kinetic experimental data and linear correlation

between experimental and calculated results.

The set of equations describing the kinetics of the

system under consideration was input into Mathcad

(premium software version 2001i) and solved by the

ﬁnd function command, which allows the solution of a

system of nonlinear equations. A set of parameter val-

ues, corresponding to the possible solutions of the non-

linear system of equations, were accordingly generated

(by varying the values given to initial parameters).

T, respectively. We also consider here the realistic sit-

uation of NPYs in which only one species (e.g., A)

is thermally stable, and the reaction starts out from an

initial concentration of that species (C₀= C_A⁰).

In this situation, the kinetic traces are readily de-

scribed by the generic equation given by the following

expression:

ꢀ

ꢁ

m^λ₀− M₀^λ− θ₄^λ· ϑ

ϑ − ϑ

−

₊·t

M^λ(t) = −

· e^ϑ

−

+

ꢀ

ꢁ

m^λ₀− M₀^λ− θ₄^λ· ϑ

₋·t

+

· e^ϑ+ θ₄^λ

(1)

ϑ − ϑ

−

+

In this formula, M^λ(t) represents the observed absorp-

tion intensity due to species B and C at different time

delays and at a given observation wavelength (λ). Be-

cause the same set of equations can be applied to the

photocoloring trace as well as the one corresponding

to the fading, all the coefﬁcients are normally related

to the speciﬁc trace by indexing them with the letters

P or T (e.g., M_P^λ(t) and M_T^λ(t)). The remaining factors

of Eq. (1) are deﬁned in [13] by the series of equations

given below.

The initial velocities recorded at the earliest stages

of each trace are expressed by

ꢀ

ꢁ

ꢀ

ꢁ

THE MATHEMATICAL FRAMEWORK

m^λ₀= ε_B^λa₄+ ε_C^λa₇· C_A⁰+ ε_B^λa₅+ ε_C^λa₈· C_B⁰

ꢀ

ꢁ

+ ε_B^λa₆+ ε_C^λa₉· C_C⁰

(2)

As has been stated above, the dynamics of NPY sys-

tems can be characterized by at least 14 unknown

parameters if it is set to conform to an ABC(3k,6φ)

mechanism. So to tackle the kinetics of these pho-

tochromes, at least 14 nonlinearly dependent equa-

tions are required. To achieve this objective, we make

use of the ﬂexibility of the general formalism to de-

velop a coherent spectrokinetic method for a general

ABC(3k,6φ) system. The full potential of the method

is attained when the reaction reaches a photostation-

ary state (PSS) within a reasonable period, the species

spectra intersect at an isosbestic point, and the pho-

tochemical buildup and the thermal relaxation traces

are collected at different wavelengths to that of the

isosbestic point (λ_Irr).^∗

The cumulative absorption intensities at the beginning

of the trace (where z = 0 or PSS) are

M_z^λ= ε_B^λ· C_B^z+ ε_C^λ· C_C^z

(3)

The coefﬁcient θ₄^λrepresents the amount of absorption

recorded after a long-reaction period for a mechanism

that reaches equilibrium. Depending on the type of

the trace, it may correspond to the absorption intensity

at the photostationary state (i.e., w = PSS) or to the

absorbance reached at a long period of the fading reac-

tion (where w = ∞) if the latter generates species in

equilibrium.

The labeling system used for the general formalism

[13] will be adopted here including the labeling of the

various coefﬁcients, characterizing the photocoloring

and the thermal fading processes by the letters P and

θ₄^λ_,w= ε_B^λ· C_B^w+ ε_C^λ· C_C^w

(4)

The four exponential coefﬁcients (ϑ and ϑ _T) cor-

respond to the overall kinetic rate constants that are

given by the following general analytical expressions:

P

∗

The differential equations describing the reactive system can-

not be integrated in closed form when the irradiation of the reactive

medium is carried out at wavelength where two or all species absorb

differently (nonisosbestic irradiation) because the photokinetic fac-

tor is then time dependent [13]. The only known kinetic model for

such conditions is that of the simplest AB(1φ) photoreaction [19].

ꢃ

ꢂ

ꢄ

ϑ

= 0.5 · a₁₉

a₁²₉+ 4 · a₂₀

(5)

International Journal of Chemical Kinetics DOI 10.1002/kin

272

MAAFI AND BROWN

The a₁–a₉coefﬁcients used in the preceding equa-

tions represent the speciﬁc rate constants of the various

species’ reactions. They are deﬁned and labeled ac-

cording to the most general ABC(6k,6φ) mechanism,

as it has been given in [13].

In the subsequent analysis, we will work with the

nine a_icoefﬁcients as the system variables instead of

the absolute k_iand φ_iquantities. This eases the mathe-

matical manipulation of the equations and establishes a

general method that can handle various kinetic mecha-

nisms (as different sets of k_iand φ_ican be equal to zero

for different speciﬁc sequences). The set of Eq. (6) cor-

responding to the mechanism depicted in Scheme 3, for

example, is obtained by replacing all the missing k_iby

zero. For the ABC(3k,6φ) systems, this formal strategy

does not alter the basic information of the system (and

hence the number of unknowns).

So far, we have identiﬁed a number of equations rel-

evant to the analysis of the kinetics. Considering the set

of Eqs. (1)–(12) for the photocoloring reaction (where

all the quantities are indexed P), we ﬁnd seven linearly

independent equations that describe the reactive sys-

tem including Eqs. (2), (9), and (10). Note that Eqs. (4),

(10b), and (10c) are linearly dependent (vide infra), so

that only two of them can be considered for the fun-

damental system of equations to be used to solve the

kinetic system. Hence, at this stage seven more equa-

tions are required to reach the necessary quorum.

The ﬁrst two extra equations derive from the photo-

stationary state condition. This condition assumes that

the variation of the species concentrations is negligible.

However, because these are linearly dependent,

a₁= −k_AB− φ_AB· γ − k_AC− φ_AC· γ

a₂= +k_BA+ φ_BA· γ

a₃= +k_CA+ φ_CA· γ

a₄= +k_AB+ φ_AB· γ

a₅= −k_BA− φ_BA· γ − k_BC− φ_BC· γ

a₆= +k_CB+ φ_CB· γ

a₇= +k_AC+ φ_AC· γ

a₈= +k_BC+ φ_BC· γ

a₉= −k_CB− φ_CB· γ − k_CA− φ_CA· γ

(6a)

(6b)

(6c)

(6d)

(6e)

(6f)

(6g)

(6h)

(6i)

where k_iare the thermal rate constants, φ_ithe quantum

yields of the elementary photochemical reactions, and

γ a constant factor

γ = ε_isos· l · I₀· F

(7)

dependent on the extinction coefﬁcient of the species at

the isosbestic point (ε_isos), the optical path length (l),

the incident light intensity (I₀), and the photokinetic

factor (F), which is considered constant throughout

the reaction, as discussed in the experimental section

(and where A_totis the total absorbance of the medium

at the irradiation wavelength).

dC_A^PSS

dt

dC_B^PSS

dt

dC_C^PSS

dt

=

+

(11)

only two may be selected. From the basic system of

differential equations describing the trimolecular sys-

tem [13], we write those corresponding to species B

and C as equal to zero at the PSS:

1 − 10^−A

tot

F =

(8)

A_tot

The a_icoefﬁcients (Eq. (6)) are related by three linear

equations, as

a_4P· C_A^PSS+ a_5P· C_B^PSS+ a_6P· C_C^PSS= 0 (12a)

a_7P· C_A^PSS+ a_8P· C_B^PSS+ a_9P· C_C^PSS= 0 (12b)

a₁= −a₄− a₇

a₅= −a₂− a₈

a₉= −a₃− a₆

(9a)

(9b)

(9c)

The concentrations of the species at the PSS have

been previously deﬁned as exclusive functions of the

a₁–a₉coefﬁcients and the initial concentration

(Eq. (13)). They represent three additional basic equa-

tions of the dynamics.

The a₁₉–a₂₁coefﬁcients deﬁning the overall kinetic

rate constants (Eq. (5)) are analytically expressed by

the following formulae:

a_3P· a_8P− a_2P· a_9P

C_A^PSS

C_B^PSS

C_C^PSS

=

· C₀

(13a)

(13b)

(13c)

a₁₉= a₁+ a₅+ a₉

a₂₀= a₇· (a₃− a₂) + a₈· (a₁− a₃)

(10a)

(10b)

ϑ_+P· ϑ

−P

a_3P· a_4P− a_1P· a_6P

+ a₉· (a₂− a₁)

ϑ_+P· ϑ

−P

ꢀ

a₂₁= (a₃· a₄− a₁· a₆) · ε_B^λ

a_2P· a_7P− a_1P· a_8P

_+P· ϑ

ꢁ

+ (a₂· a₇− a₁· a₈) · ε_C^λ· C₀(10c)

ϑ

−P

International Journal of Chemical Kinetics DOI 10.1002/kin

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES

273

Scheme 3 Photochemical and thermal reactions of 1 (A) and its postulated open forms B and C. The mechanism encompasses

all nine possible reaction steps, which are labeled according to the notation used in [13].

ꢅ

ꢆ

ε_B^λ

ε_C^λ

ε_B^λ

ε_C^λ

Two more equations are supplied by expressions found

in the intermediary stage of the general formalism

development. These relate to the formula for the ab-

sorbance of species B at the photostationary state:

a₁^λ₁= a₆

+ a₉− a₅+ a₈

− a₆

− a₉

1

·

(16b)

ꢀε^λ· ε_C^λ

ꢅ

ꢆ

C₀

ꢀε^λ

C_A^PSS

ꢀε^λ

M_P^λ_SS

ε_C^λ

ε_B^λ

ε_C^λ

1

ꢀε^λ

A^P_B^SS,λ

=

−

(14)

a₁^λ₂

=

a₅+ a₈

− a₆

− a₉

·

· C₀

ε_C^λ· ꢀε^λ

(16c)

with

The ﬁnal series of equations is obtained from Eqs. (1)–

(12) for the thermal reaction (i.e., the fading process).

For this process, the coefﬁcients are indexed by the let-

ter “T” and all the quantum yields in a_1T–a_9T(Eq. (6))

are equal to zero. In total, we can identify ﬁve equa-

tions to characterize the fading, i.e., Eqs. (2), (3), and

(10). However, if the fading does not lead to a ther-

mal equilibrium (this cannot be postulated for NPYs)

the coefﬁcient a_21T(Eq. (10c)) is equal to zero. Fur-

thermore, we cannot consider the initial absorbance

of the thermal reaction (Eq. (4)) because it is redun-

dant and equal to the absorbance recorded at the PSS

(M_P^λ_SS= M₀^λ_,T). Thus, only three equations are effec-

tively useful (Eqs. (2), (10a), and (10b)).

ε_C^λ− ε_B^λ

ε_C^λ· ε_B^λ

ꢀε^λ=

(15)

and the derivative of the cumulative absorbance at the

PSS (which may be equated to zero):

a₁^λ₀· C_A^PSS+ a₁^λ₁· M_P^λ_SS+ a₁^λ₂= 0

(16)

Here the a₁₀–a₁₂coefﬁcients are deﬁned by the expres-

sions below [13], but are not considered as independent

variables themselves.

a₁^λ₀= ε_B^λ· a₄+ ε_C^λ· a₇

Accordingly, the fundamental system of lin-

early independent equations that describes fully any

ABC(3k,6φ) dynamics is given by the 14 equations

deﬁned above and derived from the photocoloring (P)

ꢅ

ꢆ

ε_C^λ

ε_B^λ

ε_C^λ

1

ꢀε^λ

− a₅+ a₈− a₆− a₉

(16a)

ε_B^λ

International Journal of Chemical Kinetics DOI 10.1002/kin

274

MAAFI AND BROWN

and the fading (T) reactions. These are Eq. (2) (T,P),

Eq. (10a) (T,P), Eq. (10b) (T,P), Eq. (10c) (P), Eq. (12)

(2 × P), Eq. (13) (3 × P), Eq. (14) (P), and Eq. (16) (P).

It is important to note that if the method has been

successful in describing the kinetic system by achiev-

ing the required number of linearly independent equa-

tions to match that of the unknown parameters in-

volved, the latter system of equations still holds a

challenge to a complete kinetic elucidation due to the

nonlinear character of the equations.

equations) that have to be fulﬁlled by the solutions.

In this group of seven equations, we ﬁnd Eqs. (4) and

(9), the mass balance equation (17), and two equations

endowed with the same meaning but expressed with

different sets of coefﬁcients. They deﬁne the concen-

tration of species A at the PSS, one (Eq. (18)) is equiv-

alent to the forms of Eqs. (12), and the other (Eq. (19))

is found in the body of the bulk development of the

formalism [13] and will not be demonstrated further

here:

It is well known that a set of n linear and linearly

independent equations can readily be solved to give a

unique solution, represented by a set of values for the

n unknowns. For a kinetic system, this unique solu-

tion represents the true kinetic solution. However, the

situation is different for a set of n nonlinear and lin-

early independent equations. Indeed, more than one set

of values (for the n unknowns) can be found to mathe-

matically represent an exact solution to the set of equa-

tions. Furthermore, depending on the size and/or the

complexity of the latter set of equations, the total num-

ber of solutions can exceed the number of equations or

even be inﬁnite. Since each set of values found for the

n unknowns is a solution to the system of equations,

the solution is said to be degenerate. In this respect, in

kinetics it is most important not to confuse any “plau-

sible” unique solution amongst the degenerate solution

set, which can be worked out based on sound assump-

tions, with the true solution because degeneracy means

that a similar probability of occurrence is allocated to

all options represented by the degenerate solutions.

Accordingly, the prospect of ﬁnding several sets of

parameter values as solutions of our system of equa-

tions is signiﬁcantly increased. Despite the fact that all

of them are exact solutions, they are genuinely differ-

ent and most dramatically they are indiscernible, since

there are no effective means to allow us to eliminate

particular solutions and to single out the unique and

true kinetic solution. In general terms, this outcome

relates to the issues of identiﬁability and distinguisha-

bility [20–23], which we have discussed previously in

relation to NPY kinetics [14,16,17].^†

C₀= C_A^PSS+ C_B^PSS+ C_C^PSS

(17)

a_1P· C_A^PSS+ a_2P· C_B^PSS+ a_3P· C_C^PSS= 0 (18)

a₁^λ₃· C_A^PSS+ a₁^λ₄· M_P^λ_SS+ a₁^λ₅= 0

(19)

where the a₁₃–a₁₅coefﬁcients are deﬁned by the fol-

lowing expressions:

ꢅ

ꢆ

a_2P

ε_B^λ

a_3P

ε_C^λ

1

ꢀε^λ

a₁^λ₃= a_1P

−

(19a)

(19b)

(19c)

ꢅ

ꢆ

a_3P

ε_C^λ

a_2P

ε_B^λ

a_3P

1

a₁^λ₄

a₁^λ₅

=

−

ꢆ

ε_C^λꢀε^λ· ε_C^λ

ꢅ

a_2P

ε_B^λ

a_3PC₀

ε_C^λ

−

ꢀε^λ

APPLICATION TO 2H-NPY KINETICS

Absorption Spectra

The absorption spectrum of the closed form of 1 is

characterized in toluene by a strong absorption band at

280 nm and a second band with vibrational structure

between 300 and 350 nm. The bands are attributed to

π, π^∗transitions [7], and the last band envelop ends

at the lower limit of the visible (400 nm; Fig. 1). A

broad absorption band spanning the spectral region

between 400 and 650 nm appears after the solution

containing the closed form has been exposed to UV

irradiation. Absorption maxima are situated at 530 and

437 nm with a shoulder at 420 nm. These new spectral

features are due to the open forms that emerge from the

photochemical cleavage of the intracyclic C O bond

(Scheme 3).

The strong donor effects of the multiple p-methoxy-

phenyl groups cause a signiﬁcant bathochromic shift

of the maximum of the absorption band of open

forms (λ_max= 530 nm) of compound 1 relative to the

unsubstituted 2,2-diphenyl-2H-naphtho[1,2-b]-pyran

(λ_max= 467 nm in acetonitrile [24] and 480 nm in

ethanol [7]). Also, the absorption maximum of com-

pound 1 is red-shifted by about 40–80 nm compared

In view of this complication, it is advisable to use as

much information and/or equations as possible to char-

acterize the dynamic system, to reduce the number of

the resulting solutions. With this aim, the following ex-

tra equations were used in the solving process for the

system, mainly as constraints (and not as fundamental

†

Note that in the much simpler cases where the three species’

individual C_i(t) experimental traces are available, we have found

(Maaﬁ, M., 2006, unpublished data) that it is possible to analytically

solve up to ABC(6k,6φ) kinetics by using our formalism [13] (i.e.,

distinghuishability and identiﬁability problems are readily solved,

and the true kinetic solution is obtained).

International Journal of Chemical Kinetics DOI 10.1002/kin

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES

275

Figure 2 Photocoloring and fading kinetic traces of 1,

7.52 × 10⁻⁵M solution in toluene at 15^◦C (λ_irr= 381 nm

and λ = 530 nm). Experimental data ( ) and ﬁtting curves

◦

obs

using Eq. 1 (—). [Color ﬁgure can be viewed in the online

issue, which is available at www.interscience.wiley.com.]

Figure 1 Absorption spectra of 1 before and after

irradiation. [Color ﬁgure can be viewed in the online issue,

which is available at www.interscience.wiley.com.]

utes. These features represent a reliable indication of a

trimolecular mechanism [14].

The results corresponding to the fading trace

(Table I) strongly suggest that species C is virtually

thermally stable since one of its overall rate constants

(ϑ_−T) can be approximated to zero. This indicates that

the trans–cis isomerization (C → B) is very slow, i.e.,

we can set k_CB= 0 in the mechanism in Scheme 3.

Since the initial condition for the thermal reaction

implies that all three species are present (the concen-

trations of the closed forms, B and C, are likely to

be nonzero at the PSS), the apparent monoexponential

trend of the fading trace may account for two mech-

anistic options. One assumes that B undergoes a di-

vergent reaction (k_BAand k_BCboth nonzero, k_CB= 0),

and the other only takes into account the recyclization

of B (k_BC= k_CB= 0). From a theoretical viewpoint, the

difference between these two possibilities can be trans-

lated from the formalism [13] into Eq. (20) where the

absorbance recorded after a long-fading period (A^∞_C) is

either equal to or greater than the absorbance of species

C at the PSS (A_C^PSS). If k_BAand k_BCare both nonzero,

then A_C^∞> A^P_C^SS, whereas A_C^∞= A_C^PSSif k_BC= 0.

to the absorption maxima of several substituted 2H-

naphthopyrans recorded in various solvents, includ-

ing halogen-substituted structures with one, two, or

three identical or different F, Cl, Br atoms [24],

derivatives incorporating methyl-groups (Lapachenol)

(λ_max= 452 nm in acetonitrile/water) [25,26], methyl-

and/or carbomethoxy-groups (λ_max= 450–470 nm in

ethanol) [7], and that modiﬁed by the presence of an

adamantylidene group [27].

However, the spectrum of the open form(s) of 1

is blue-shifted compared to those corresponding to

rigid structures containing a benzene linked to the 2H-

structure through a cyclopentene ring (λ_max∼ 560 nm

in ethanol) [7]. These results indicate that the deep pink

color of the open forms of 1 may contribute to expand-

ing the application capabilities of these photochromes

as dyes.

Analysis of the Traces

The experimental data corresponding to the kinetic

traces recorded at 530 nm are given in Fig. 2. The

model equation (Eq. (1)) has been used to ﬁt the pho-

tochemical (θ₄= 0) and the thermal (θ₄= 0) evolution

of the cumulative absorbance of the open forms with

reaction time (respectively, for the sample exposed to

UV light and in the dark). The ﬁtting results reported

in Table I are characterized by a good accuracy and by

correlation coefﬁcients over 0.99 for the experimental

versus calculated data of both traces.

k_BC

A^∞_C= A_C^PSS+ C_B^PSS· ε_C⁵³⁰

·

(20)

k_BA+ k_BC

The ¹⁹F-NMR experimental evidence reported

by Arnall-Culliford et al. [25] for a 2,2-bis(4-

ﬂuorophenyl)-2H-benzo[h]-chromene proved that the

thermal cis–trans isomerization (B → C) does not oc-

cur for this compound. In general, all literature reports

dealing with such photochromes [4,7,12, 26–28] have

ignored, de facto, this reaction step in the mechanism.

In the present study, we assume that the results ob-

tained for the 2,2-bis(4-ﬂuorophenyl)-2H-benzo[h]-

chromene derivative are plausible and can be adopted

As is usually observed for such compounds, the pho-

tocoloring trace obeys a biexponential model (Eq. (1))

with fast and slow kinetic regimes (ϑ_+P/ϑ = 6.35),

−P

and a photostationary state reached within a few min-

International Journal of Chemical Kinetics DOI 10.1002/kin

276

MAAFI AND BROWN

Table I Fitting Results and the Deﬁning Coefﬁcient Values for the Traces of 1 in Toluene (15^◦C)

Photocoloring trace^a

Coefﬁcients^b

Value^c

M₀^λ_,P

0

m^λ_0,P

ϑ

M_P^λ_SS

0.1838

+P

−P

0.00483

−0.0277

−0.00436

Fading trace^a

d

Coefﬁcients^b

Values^c

M₀^λ_,T

0.1838

M₀^λ_,T

−0.00317

ϑ

A^∞_C,T

+T

−T

−0.01858

0.00001

0.0143

^aExperimental traces are ﬁt by model Eq. (1), with correlation coefﬁcients exceeding 0.99.

^bThe coefﬁcient meaning and expressions are deﬁned in the text, with indices P and T indicating, respectively, the data from the photocoloring

and the thermal fading.

^cValues are obtained at 15^◦C and at a monitoring wavelength λ_obs= 530 nm.

^dThis value is provided by the kinetic trace from the fading reaction (Fig. 2).

the minimum values of the extinction coefﬁcients ε_B^λ

and ε_C^λ(2000 dm³mol⁻¹cm⁻¹), and the accuracy index

on the predicted set of results (convergence tolerance

<10⁻⁸), in carrying out the calculations. The results

obtained from the simultaneous resolution of the sys-

tem of equations (Appendix) are reported in Table II.

for the thermal reaction of our compound 1. Ac-

cordingly, the overall mechanism of 1 reduces to an

ABC(1k,6φ), where the fading is represented by a sin-

gle reaction step (k_BAin Scheme 3). This reduces the

total number of the unknown parameters of the system

to 12.

Parameterization

DISCUSSION

By setting A^P_C^SS= A_C^∞as has been discussed earlier, we

ﬁnd that the coefﬁcient a_19T(Eq. (10a)) is equivalent

Time-resolved and NMR studies have shown that for

the 2H-benzopyrans and 2H-naphthopyrans the ring-

opening reaction is the prelude to the formation of sev-

eral intermediate species as the open forms can adopt

a number of different conformations dependent on the

orientation of the three exocyclic (two double and one

single) bonds. However, it has been strongly suggested

that many of these are short living species (less than a

nanosecond) and in general only two of these structures

have lifetimes longer than a millisecond. We attribute

to the cis–trans isomer the highest thermal rate constant

because it requires a single bond rotation to achieve re-

cyclization (species B in Scheme 2). The trans–trans

isomer (C) is now considered thermally inert during the

course of the photoreaction (500 s; Fig. 2), as is shown

in Scheme 3 (with k_BC= k_CB= 0). Since our experi-

mental setup allows us to collect data in time intervals

not less than 0.5 s, we can only observe the kinetics

due to these three species (A, B, C). And hence, the dy-

namics of 1 can be considered as a trimolecular system

that obeys an overall assumed ABC(1k,6φ) kinetics.

The fading reaction is characterized by rate constant

values comparable to that of other similar 2H-NPY

ranging between 1 × 10⁻³and 2 × 10⁻²s⁻¹depend-

ing on structure, solvent, and temperature. In general

terms, these values are of the same order of magnitude

as those of the unsubstituted 2,2-diphenyl-2H-NPY. It

seems that the variation of the rate constant values is

more sensitive to geometry and steric hindrance than

to the negative value of the thermal rate constant k_BA

,

a_20T(Eq. (10b)) is equal to zero and the numerical value

of the absorbance of species B at the PSS (Eq. (14))

is deﬁned experimentally by using the expression of

the cumulative absorbance (Eq. (4)) and A^P_C^SS. This

also implies that one parameter of the system has been

identiﬁed, meaning that only 11 equations are now

required to fully describe the reactive system under

study.

Taking into account the discussion above, it is pos-

sible to deﬁne the set of equations that will be used for

the parameterization. We give in the appendix the series

of fundamental as well as the supplementary equations

employed in the present study to describe and to solve

the dynamics of 1. As can be seen, many equations

have been attributed a numerical value by using those

supplied by the ﬁtting results as given in Table I. For

the a₁₉–a₂₁coefﬁcients, the values are determined on

the basis of the following expressions that we have set

previously [15]:

a₁₉= ϑ + ϑ

(21)

(22)

(23)

+

−

a₂₀= −ϑ · ϑ

+

−

a_21P= −a_20P· M_P^λ_SS

Beside the supplementary equations (Appendix), we

use constraints on the sign of the a₁–a₉coefﬁcients,

International Journal of Chemical Kinetics DOI 10.1002/kin

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES

277

Table II Six Predicted Sets of Kinetic and Spectroscopic Parameter Values for 1 Photochemical Reaction in Toluene

at 15^◦C^a

Result Sets

S

a

S

f

b

c

d

e

Kinetic Data^b

ABC(1k,6φ)

ABC(1k,5φ)

ABC(1k,4φ)

φ

γ

0.00505

0.00240

0.00171

0.00176

0.00238

0.00708

0.00203

0.00432

0.00225

0.00268

0.00118

0.00306

0.00785

0.00089

0.00038

0.00580

0.00139

0.00205

0

0.00424

0

4.63 × 10⁴

8.04 × 10³

6.98 × 10⁻⁵

3.66 × 10⁻⁶

1.78 × 10⁻⁶

0.00832

0.00079

0

BA

AB

BC

CB

CA

AC

0

0.00157

0.00250

0.00029

3.11 × 10⁴

3.03 × 10³

6.50 × 10⁻⁵

5.45 × 10⁻⁶

4.71 × 10⁻⁶

0

0.00428

0.00006

0.00429

0.00007

7.99 × 10⁴

1.28 × 10⁴

7.20 × 10⁻⁵

2.12 × 10⁻⁶

1.12 × 10⁻⁶

0.00017

0

ε_B⁵³⁰

2.65 × 10⁴

2.67 × 10³

6.34 × 10⁻⁵

6.40 × 10⁻⁶

5.35 × 10⁻⁶

2.86 × 10⁴

3.82 × 10³

6.55 × 10⁻⁵

5.93 × 10⁻⁶

3.75 × 10⁻⁶

7.04 × 10⁴

1.14 × 10⁴

7.16 × 10⁻⁵

2.40 × 10⁻⁶

1.25 × 10⁻⁶

ε_C⁵³⁰

C_A^PSS

C_B^PSS

C_C^PSS

^aThe value of the rate constant (k_BA) of the single thermal reaction step is given by −ϑ from Table I.

+T

^bThe product φ_i· γ in s⁻¹; the extinction coefﬁcients of species B and C at the observation wavelength (530 nm) in dm³mol⁻¹cm⁻¹; and

the species concentrations at the PSS in mol dm⁻³

.

to any electronic effects introduced by the substitutent

groups. As a matter of fact, it has been difﬁcult to cor-

relate such variation to either the electronic properties

of the substitution groups or the other spectroscopic or

kinetic features of series of these compounds.

ruled out. Consequently, to give an account of these

observations, the two metastable open forms of 3,3-

diphenyl-3H-naphtho[2,1-b]pyran that are produced

either have a common excited state intermediate pre-

cursor or thermally interconvert between one another.

Conversely, the photoreactivity of the open forms

has been much more difﬁcult to study. Nonetheless,

Go¨rner and Chibisov [10] have suggested on the ba-

sis of microsecond transient spectra of 3,3-diphenyl-

3H-naphtho[2,1-b]pyran that the closed form and the

trans–trans species are both photochemically reactive

to produce the other two structures, but that species B

(cis–trans isomer) reacts exclusively toward the closed

form (by both thermal and photochemical routes). It

is obvious from our experimental results on 1 that the

thermally stable isomer (C) is necessarily photoactive,

otherwise it can be accumulated over a long time and

the reaction will end by complete transformation of the

initial species into C. Accordingly, the very fact of ob-

serving a photostationary state for 1 (k_BC= k_CB= 0), is

a good indication for the deactivation of excited mero-

cyanine C* into at least one of its counterparts (φ_CB= 0

and/or φ_CA= 0, in Scheme 3). On the basis of this dis-

cussion, it emerges that no reaction step among the six

possible photoreactions of Scheme 3 can objectively

be ruled out with certainty.

Much less quantitative and/or qualitative results are

available in the literature on the photocoloring process

of 2H-naphthopyrans. Picosecond and submillisecond

investigation of some 2H-benzopyrans (2,4-diphenyl-

2H-benzopyran and 2,2,4-triphenyl-2H-benzopyran)

[28] concluded that the photochemical C O bond

cleavage occurs exclusively via the ﬁrst excited sin-

glet state within 2 ps. It has been suggested for these

species that the photocleavage yields only the open

forms (20 ps–12 µs) that can revert to the closed form

by a single bond rotation. In a subsequent photoexci-

tation of the initially generated open forms, Kodama

et al. [29] have suggested other open forms (lifetime

>1 ms) are formed (which need two double bond ro-

tations for reversion to the closed form). Femtosecond

results dealing with the unsubstituted 3,3-diphenyl-

3H-naphtho[2,1-b]pyran derivative [6,9] in organic

media show that the ring-opening reaction proceeds

mainly from a singlet excited state and is much faster

(450 fs) than that reported for 2H-benzopyrans [29].

Aubard et al. [6] observed that the transient spectra

recorded after a delay of 10 ps do not exhibit any fur-

ther spectral changes (until 100 ps) and resemble those

obtained in continuous irradiation experiments. Given

the excitation pulse width used in these experiments

(≤100 fs) and the similarity of the visible region spec-

tra of species B and C [15], the eventuality of them

being simultaneously produced cannot be deﬁnitely

The approaches applied so far with relation to naph-

thopyrans studied under continuous irradiation have

dealt with three structures including 3,3-diphenyl-

3H-naphtho[2,1-b]pyran [11,12] and two 2H-NPYs

[26,27]. Except for the 3H-NPY derivatives for which

an ABC(1k,1φ) kinetic has been adopted [12], each

of these studies has been carried out on the basis of

International Journal of Chemical Kinetics DOI 10.1002/kin

278

MAAFI AND BROWN

a preconceived ABC(1k,3φ) mechanism that ignores

some of the reaction steps discussed above. In the

case of the 2H-NPYs, it has been suggested that the

closed form species undergoes a divergent photoreac-

tion yielding both open form species, but considered

the cis–trans open form (isomer B) to be photochem-

ically inactive [26,27]. In addition, the methods used

assumptions and worked with a deﬁcit in the number

of equations in comparison with that of the unknown

parameters. In this respect, the mathematical formal-

ism we have developed [13] is an interesting tool to

elaborate new spectrokinetic methods [15,17] without

the need to introduce any simplifying assumptions. In

a previous study [15], we demonstrated that the kinet-

ics of systems such as these naphthopyrans can only be

solved analytically for ABC(1 or 2k,2φ) mechanisms if

the only available experimental data are a photocolor-

ing fading data set. However, the formalism yields, as

described in this paper, the prerequisite number of in-

dependent equations to match the unknown parameters

and can hence fully describe an ABC(3k,6φ) system.

The solution for such cases can only be achieved by

solving this system of linearly independent (but non-

linear) equations.

We have applied the analytical solution of the

ABC(1k,2φ) kinetics to the experimental results of 1,

but no consistent set of values could be found. This is

interpreted as meaning that the mechanism governing

the reaction of 1 cannot be described by the various ki-

netic models allowed by an ABC(1k,2φ) system. This

is in agreement with the results depicted in Table II,

and are analogous to those reported for a 3H-NPY

derivative [15].

In the case of the present kinetics of compound 1, the

number of fundamental equations exceeds by one the

number of parameters (see the appendix). However,

despite this fact, it is not possible to reach a unique

solution (the true solution). This situation arises from

the nonlinearity of the equations, with the consequence

that the kinetic solution is degenerate, i.e., multiple

solutions are found that ﬁt the experimental results.

In this study, solving the set of equations for the

experimental traces of compound 1 (Fig. 2) yields

three different generic mechanisms, i.e., ABC(1k,4φ),

ABC(1k,5φ), and ABC(1k,6φ) (as given in Table II).

These mechanisms are the only ones that represent a

realistic physical meaning (other options, which in-

volved negative values for any of the system parame-

ters or values over unity for the quantum yields, were

discarded).

the true sequence, the kinetics are said to be nondistin-

guishable [14,16,20–23]. Moreover, for each individ-

ual mechanism our results show that several different

parameter sets can be found. This outcome underlines

the nonidentiﬁability of the solution [14,16,20–23].

These two problems of distinguishability and identi-

ﬁability prove to be central for such systems and out-

line the increased difﬁculty of determination of the true

solution (if achievable).

It is important to note that tens of plausible se-

quences can often be postulated for such kinetics. Ex-

amples in the literature show that the few kinetic stud-

ies available use preconceived mechanisms that opt

for simplicity. They involve three or less phototrans-

formations (ABC(1k,1φ) [12], ABC(1k,2φ) [26], and

ABC(1k,3φ) [27]) to circumvent a common problem

in the methods employed, namely a lack of indepen-

dent equations that describe the reaction to match the

number of unknown parameters.

The present treatment shows that it is possible to

avoid postulating a mechanism for the reaction prior

to the analysis of the experimental data and that the

sequences proposed earlier in the literature seem to

be unsuitable for 1 since the simplest option in our

study involves at least four photochemical reactions

ABC(1k,4φ). Such discrepancies underline two gen-

eral but important issues. First, it is possible to reach

a unique solution for a given complex kinetic system

by means of introducing a number of “reasonable”

assumptions. However, as demonstrated here for 1

(and compared to the literature ﬁndings), the achieved

unique solution might not be the true solution. This

then casts doubts on the usefulness of the results ob-

tained and raises the question of their validity. Second,

reproducing the experimental traces by using the pa-

rameter values of a unique solution (through a model

or numerical integration) is a necessary but not a suf-

ﬁcient condition to conclude that the unique solution

at hand is the true solution. This stems from the de-

generacy of the solution where all the options of the

degenerate solution would allow a faithful reconstruc-

tion of the experimental traces. This is the case for a

good correlation between the options in Table II and

the experimental traces of Fig. 2. Hence, no solution

can be deemed acceptable for such systems (as the true

solution) before the degeneracy issue has been properly

addressed.

In the present study, the analysis of the kinetic data

shows that the true mechanism describing the reaction

of compound 1 is to be found among six options in-

volving between four and six photoreactions (Table II).

This represents the best result that can be achieved us-

ing kinetic data alone. A deﬁnitive decision about the

mechanism will need extra independent data. However,

Without imposing any prerequisite conditions or as-

sumptions, our treatment generates only a few options

that represent possible mechanisms for the dynamics

of 1. Since there is not a means to select among them

International Journal of Chemical Kinetics DOI 10.1002/kin

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES

279

we can obtain more details about the reaction of 1 by

looking at the common features of the options laid out

in Table II.

low values of φ_ABdispute the ﬁndings of earlier studies,

where this quantum yield has been found to approach

unity (φ_AB> 0.8) [10–12]. The reason for this latter

ﬁnding can be traced to the treatment used therein,

which was originally developed for AB systems. Those

Inspection of the results presented in Table II re-

veals that the A ↔ B (i.e., φ_BAand φ_AB) and C → A

(φ_CA) photochemical reactions are always predicted to

occur. The occurrence and efﬁciency of the other three

photochemical reactions varies from case to case but

it is very noticeable that φ_ACis always very low or

zero. The A → C photochemical reaction, therefore,

only constitutes a small percentage (0%–12%) of the

photoreaction proﬁle of the closed form, which is dom-

inated by the ring-opening reaction to form species B

(φ_AB). This is unsurprising given the geometric rear-

rangement that needs to take place in each case. In

those parameter sets where φ_AC= 0, it is reassuring to

observe that φ_BCis nonzero and vice versa. In other

words, there is always a (one or two photon) photo-

chemical route predicted for the formation of species

C from the initial closed form. If it is possible to draw

the conclusion that the A → C direct photochemical

reaction is negligible, which corroborates the experi-

mental data corresponding to several unsubstituted and

substituted derivatives of benzopyran [30,31], then pa-

rameter sets S_b, S_d, and S_fmay be eliminated from con-

sideration (as φ_BC≈ 0) leaving three (S_a, S_c, and S_e)

mechanisms to account for the kinetics of compound

1. It is worth noting that two among these sequences

(S_aand S_c) represent a similar generic ABC(1k,5φ)

mechanism (as φ_AC≈ 0) and can be seen as an exam-

ple of the unidentiﬁability of the solution.

methods consider the overall kinetic rate (such as ϑ

+P

in Table I) as if it is due to the exclusive reaction of

species A, resulting in an overevaluation of φ_AB

.

The ﬁnal observation that we can make about the

photoreactions relates to the photochemical paths open

to species C. The parameter sets in Table II make it

clear that the C → A photoreaction is always predicted

to take place (and with good efﬁciency). One could,

therefore, intuitively expect the C → B photoreaction

to also take place. If this argument holds, then the

ABC (1k,4φ) mechanism (S_e) can be eliminated since

it predicts φ_CB= 0.

With regard to the nonphotochemical parameters

provided in Table II, it is interesting to note that ε_B⁵³⁰

is always predicted to be higher than ε_C⁵³⁰by a signif-

icant factor of between 6 and 10. This is somewhat

surprising given the similarity of their structures. The

photostationary state, under our experimental condi-

tions, is determined by the closed form—the maximum

conversion to the open forms being 15.6% for the pa-

rameter set S_a. The relative amount of B and C, which

are present at the PSS, is quite similar for all the cases

presented in Table II. However, the combination of the

extinction coefﬁcients with the stationary state concen-

trations indicates that the experimental absorbance at

the 530-nm observation wavelength is dominated by

the contribution from species B (92%).

It is noticeable that the photochemical recycliza-

tion reactions of both open form isomers are predicted

to take place irrespective of the number of other pro-

cesses that are also occurring in the sequence. These

steps are always signiﬁcantly more efﬁcient than the

corresponding ring-opening reactions and also occur

with a higher yield than their cis–trans isomerizations

(φ_BA> φ_BCand φ_CA> φ_CB). For the A ↔ B reactions,

the relative efﬁciencies of the ring-closure and the ring-

opening reactions lie in the range 2–10, whereas for

A ↔ C the lowest ratio is 9. In the case of the A ↔ B

reactions, this observation may indicate that the inter-

nal conversion of A^∗to the ground state represents

signiﬁcant competition to the ring-opening reaction. A

parallel example can be found in the case of a nitro-

substituted spiropyran [32] for which a femtosecond

vibrational investigation showed that up to 90% of the

excited singlet state population decays to the ground

state. If this proves to be true for NPYs as well, it gives

a ﬁrst clue as to how to improve the performance of the

system (i.e., by targeting the design of structures capa-

ble of higher reactionrates of theinitial species or byre-

ducing the efﬁciency of the internal conversion). These

It is worth noting that similar results for the photore-

actions efﬁciencies have been found for the 3H-NPY

that has been investigated using monoexponential pho-

tochemical kinetics and a speciﬁc mathematical devel-

opment [16]. This indicates that the photochemistry of

both 2H- and 3H-NPY derivatives is achieved through

similar routes despite geometry alteration.

CONCLUSION

The descriptive spectrokinetic method developed in

this paper is fully applicable to the investigation of

complex ABC(3k,6φ) reactive systems, and easily

adapted to the study of photochromic naphthopyran

kinetics. It also proves the versatility of the basic for-

malism and its ability to address a number of reaction

systems.

For the ﬁrst time, this study has shown that it is pos-

sible to derive as many linearly independent equations

as there are unknown parameters that deﬁne the system

under consideration.

International Journal of Chemical Kinetics DOI 10.1002/kin

280

MAAFI AND BROWN

ꢅ

ꢆ

ε_C^λ

ε_B^λ

ε_C^λ

1

ꢀε^λ

The results of the present method have also un-

derlined for the ﬁrst time the importance of the dis-

tinguishability/identiﬁability problems with relation to

the kinetics of 2H-NPY compounds. This conclusion

conﬁrms previous experience with the homologous

3H-NPY systems. It also raises a crucial point of

interest; namely that whether the mechanism is pre-

conceived or not, unavoidable and fundamental issues

of distinguishability, and/or identiﬁability have to be

solved to achieve any consistent progress in under-

standing (and improving) naphthopyran reactions and

similar systems. Effort is being expended in our labo-

ratory to increase our understanding of these issues.

For compound 1, the predictions of our method

point clearly to the reaction steps that are an essen-

tial part of the true mechanism. The phototransforma-

tion of the closed form is dominated by the forward

A → B reaction (A → C is only a negligible event),

and the divergent conversion of species B is likely to

be more efﬁcient in the direction of the recyclization

to the closed form. Similarly, the C → A reaction is

always predicted and hence is considered to be active

within the true mechanism. The occurrence of the ﬁ-

nal possible photochemical step (C → B) is not clearly

conﬁrmed. The spectrokinetic method developed in

this paper identiﬁes an ABC(1k,4φ) mechanism as the

minimum possibility for the kinetics of compound 1.

a₁^λ₂

=

a₅+ a₈− a₆− a₉

·

· C₀

ꢅ

ꢆ

a_2P

ε_B^λ

a_3P

ε_C^λ

1

ꢀε_λ

a₁^λ₃= a_1P

−

ꢅ

ꢆ

a_3P

ε_C^λ

ꢅ

a_2P

ε_B^λ

a_3P

1

a₁^λ₄

a₁^λ₅

=

−

ꢆ

ε_C^λꢀε^λ· ε_C^λ

a_2P

ε_B^λ

a_2PC₀

ε_C^λ

−

ꢀε^λ

Fundamental Equations

Photochemical Dynamics

ꢀ

ꢁ

m^λ_0,P= ε_B^λ· a_4P+ ε_C^λ· a_7P· C₀= 0.00483

a_19P= a_1P+ a_5P+ a_9P= ϑ_+P+ ϑ = −0.03206

−P

a_20P= a_7P· (a_3P− a_2P) + a_8P· (a_1P− a_3P) + a_9P

· (a_2P− a_1P) = −ϑ_+P· ϑ = −1.2 × 10⁻⁴

−P

ꢀ

a_21P= (a_3P· a_4P− a_1P· a_6P) · ε_B^λ

ꢁ

+ (a_2P· a_7P− a_1P· a_8P) · ε_C^λ· C₀

= −a_20P· M_P^λ_SS= 2.22 × 10⁻⁵

a_4P· C_A^PSS+ a_5P· C_B^PSS+ a_6P· C_C^PSS= 0

a_7P· C_A^PSS+ a_8P· C_B^PSS+ a_9P· C_C^PSS= 0

a_3P· a_8P− a_2P· a_9P

APPENDIX

Deﬁnitions

C_P^A_SS

C_P^B_SS

C_P^C_SS

=

· C₀

ϑ

_+P· ϑ

a_3P· a_4P− a⁻_1P^P· a_6P

ϑ

_+P· ϑ

a_2P· a_7P− a⁻_1P^P· a_8P

a₁= −φ_AB· γ − φ_AC· γ a₂= +k_BA+ φ_BA· γ

a₃= +φ_CA· γ a₄= φ_AB· γ

a₅= −k_BA− φ_BA· γ − φ_BC· γ a₆= +φ_CB· γ

a₇= +φ_AC· γ a₈= +φ_BC· γ

ϑ

_+P· ϑ

−P

C₀

C_A^PSS

M_P^λ_SS

ε_C^λ· ꢀε_λ

A^P_B^SS,λ

=

−

ꢀε_λ

a₁^λ₀· C_A^PSS+ a₁^λ₁· M_P^λ_SS+ a₁^λ₂= 0

a₉= −φ_CB· γ − φ_CA.γ a₁₉= ϑ + ϑ

+

−

a₂₀= −ϑ · ϑ

a_21P= −a_20P· M_P^λ_SS

+

−

ε_C^λ− ε_B^λ

Thermal Dynamics

ꢀε^λ=

ε_C^λ· ε_B^λ

m^λ_0,T= −k_BA· ε_B^λ· C_B^PSS= −0.00317

a_19T= −k_BA= −0.01858

a₁^λ₀= ε_B^λ· a₄+ ε_C^λ· a₇

ꢅ

ꢆ

ε_C^λ

ε_B^λ

ε_C^λ

1

ꢀε^λ

−

a₅+ a₈− a₆− a₉

Supplementary Equations

ꢅ

ꢆ

ε_B^λ

ε_C^λ

ε_B^λ

a₁^λ₁= a₆+ a₉− a₅+ a₈− a₆− a₉

ε_C^λ

a_1P+ a_4P+ a_7P= 0

a_5P+ a_2P+ a_8P= 0

1

·

ꢀε^λ· ε_C^λ

International Journal of Chemical Kinetics DOI 10.1002/kin

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES

281

a_9P+ a_3P+ a_6P= 0

Molecular Science; Martin, M. M.; Hynes, J. T. (Eds.);

Elsevier: Amsterdam, 2004; p. 283.

10. Go¨rner, H.; Chibisov, A. K. J Photochem Photobiol A:

Chem 2002, 149, 83.

11. Ottavi, G.; Favaro, G.; Malatesta, V. J Photochem Pho-

tobiol A: Chem 1998, 115, 123.

12. Frigoli, M.; Pimienta, V.; Mostrou, C.; Samat, A.;

Guglielmetti, R.; Aubard, J.; Maurel, F.; Micheau, J.-C.

Photochem Photobiol Sci 2003, 2, 888.

13. Maaﬁ, M.; Brown, R. G. Int J Chem Kinet 2005, 37(3),

162.

a₁^λ₃· C_A^PSS+ a₁^λ₄· M_P^λ_SS+ a₁^λ₅= 0

a_1P· C_A^PSS+ a_2P· C_B^PSS+ a_3P· C_C^PSS= 0

M_P^λ_SS= ε_B^λ· C_B^PSS+ ε_C^λ· C_C^PSS= 0.1838

A^P_C^SS= C_C^PSS· ε_C⁵³⁰= 0.0143

C₀= C_A^PSS+ C_B^PSS+ C_C^PSS= 7.52 × 10⁻⁵

BIBLIOGRAPHY

14. Maaﬁ, M.; Brown, R. G. Int J Chem Kinet 2005, 37(12),

717.

1. (a) Durr, H.; Bouas-Laurent, H. (Eds.). Studies in Or-

ganic Chemistry, 40: Photochromism Molecules and

Systems; Elsevier Press: Amesterdam, 1990; (b) Van

Gemert, B.; Bergomi, M.; Knowles, D. Mol Cryst Liq

Cryst 1994, 246, 67; (c) Crano, J. C. Frame Mag 1996,

8; (d) Van Gemert, B.; Kish, D. G. PPG Technol J 1999,

5(1), 53.

2. (a) Hughes, F. J., Ippoliti, J. T., Knowles, D. U.S. Patent

6,478,988, 2002; (b) Lin, J. World Patent PCT-WO-

01/51483/A1, 2001; (c) Breyne, O.; Chan, Y.-P.; Jean,

P. World Patent PCT-WO-02/30916/A1, 2002.

3. (a) Zhao, W.; Carreira, E. M. J Am Chem Soc 2002,

124(8), 1582; (b) Ahmed, S. A.; Sallenave, X.; Fages, F.;

Mieden-Gundert, G.; Muller, W. M.; Muller, U.; Vogtle,

F.; Pozzo J.-L. Langmuir 2002, 18, 7096; (c) Brun, P.;

Gugglielmetti, R.; Anguille, S. Appl Organometal Chem

2002, 16, 271.

15. Maaﬁ, M.; Brown, R. G. Int J Chem Kinet 2006, 38(7),

421.

16. Maaﬁ, M.; Brown, R. G. Int J Chem Kinet 2006, 38(7),

431.

17. Maaﬁ, M.; Brown, R. G. Int J Chem Kinet 2007, 39(8),

472.

18. Van Gemert, B.; Kumar, A.; Knowles, D. B. Mol Cryst

Liq Cryst 1997, 297, 131.

19. Maaﬁ, M.; Brown, R. G. J Photochem Photobiol A:

Chem 2007, 187, 319.

20. Vajda, S.; Rabitz, H. J Phys Chem 1988, 92, 701.

21. Sienicki, K.; Durocher, G. J Photochem Photobiol A:

Chem 1992, 63, 279.

22. Berthier, F.; Diard, J.-P.; Pronzato, L.; Walter, E. Auto-

matica 1996, 32(7), 973.

23. Aspey, S. P.; Macchieto, S. Comput Chem Eng 2000,

24, 1261.

4. Crano, J. C.; Guglielmetti, R. J. (Eds.). Topics in Applied

Chemistry, Vols. 1 and 2: Organic Photochromic and

Thermochromic Compounds; Plenum Press: New York,

1999, and references therein.

5. (a) Queiroz, M.-J. R. P.; Plasencia, P. M. S.; Dubest,

R.; Aubard, J.; Guglielmetti, R. Tetrahedron 2003, 59,

2567; (b) Tanaka, K.; Aoki, H.; Hosomi, H.; Ohba, S.

Org Lett 2000, 2(14), 2133; (c) Kaneko, T.; Akutsu, H.;

Yamada, J.-I.; Nakatsuji, S. Org Lett 2003, 5(12), 2127;

(d) Zhao, W.; Carreira, E. M. Org. Lett. 2003, 5(22),

4153.

24. Arnall-Culliford, J. C.; Teral, Y.; Sgarabotto, P.;

Campredon, M.; Giusti, G. J Photochem Photobiol A:

Chem 2003, 159, 7.

25. Kolc, J.; Becker, R. S. J Phys Chem 1967, 71,

4045.

26. Ottavi, G.; Ortica, F.; Favaro, G. Int J Chem Kinet 1999,

31, 303.

27. Favaro, G.; Romani, A.; Becker, R. S. Photochem Pho-

tobiol 2000, 72(5), 632.

28. Brun, P.; Gugglielmetti, R.; Pepe, G.; Anguille, S. J

Photochem Photobiol A: Chem 2003, 156, 77.

29. Kodama, Y.; Nakabayashi, T.; Segawa, K.; Hattori, E.;

Sakuragi, M.; Nishi, N.; Sakuragi, H. J Phys Chem A

2000, 104, 11478.

6. Aubard, J.; Maurel, F.; Buntinx, G.; Poizat, O.; Levi, G.;

Guglielmetti, R.; Samat, A. Mol Cryst Liq Cryst 2000,

345, 215.

7. Jockusch, S.; Turro, N. J.; Blackburn, F. R. J Phys Chem

2002, 106, 9236.

8. Hobley, J.; Malatesta, V.; Hatanaka, K.; Kajimoto, S.;

Williams, S. L.; Fukurama, H. Phys Chem Chem Phys

2002, 4, 180.

30. Kodama, Y.; Nakabayashi, T.; Segawa, K.; Hattori, E.;

Sakuragi, M.; Nishi, N.; Sakuragi, H. J Phys Chem A

2000, 104, 11478.

31. Nakabayashi, T.; Nishi, N.; Sakuragi, H. Sci Prog 2001,

84(2), 137.

9. Brown, R. G.; Maaﬁ, M.; Foggi, P.; Bussotti, L. In

Femtochemistry and Femtobiology—Ultrafast Events in

32. Rini, M.; Holm, A.-K.; Nibbering, E. T. J.; Fidder, H. J

Am Chem Soc 2003, 125, 3028.

International Journal of Chemical Kinetics DOI 10.1002/kin

Article Doi

Photophysics and kinetics of naphthopyran derivatives, part 6: A fundamental system of equations to describe ABC(3k,6o) dynamics. A contribution to the elucidation of [2H]-naphthopyran kinetics

DOI: 10.1002/kin.20306

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Full text of DOI:10.1002/kin.20306

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