Photophysics and kinetics of naphthopyran derivatives, Part 2: Analysis of diarylnaphthopyran kinetics. Degeneracy of the kinetic solution

Article abstract of DOI:10.1002/kin.20130

The photocoloring and fading kinetics of the photochromic 3,3-diphenyl-3H-naphtho|2,1-b|pyran (NPY) have been analyzed using a kinetic treatment for three related species recently developed by us (Int J Chem Kinet 2005, 37(3), 162). The concepts of identifiability and distinguishability are developed with respect to the elucidation of the NPY kinetics. We find that neither set of kinetic data on its own, nor the two data sets combined, allow the true mechanism to be distinguished. In addition, even for a chosen mechanism, there exist degenerate sets of parameters i.e. the solutions are also unidentifiable. A discussion of the limits and uniqueness of the results' interpretation is presented.

Full text of DOI:10.1002/kin.20130

Photophysics and Kinetics
of Naphthopyran
Derivatives, Part 2: Analysis
of Diarylnaphthopyran
Kinetics. Degeneracy of the
Kinetic Solution
M. MAAFI, R. G. BROWN
School of Applied Sciences, University of Glamorgan, Pontypridd, Rhondda Cynon Taff, Cardiff, CF37 1DL, UK
Received 27 June 2003; accepted 21 June 2005
DOI 10.1002/kin.20130
Published online in Wiley InterScience (www.interscience.wiley.com).
ABSTRACT: The photocoloring and fading kinetics of the photochromic 3,3-diphenyl-3H-
naphtho[2,1-b]pyran (NPY) have been analyzed using a kinetic treatment for three related
species recently developed by us (Int J Chem Kinet 2005, 37(3), 162). The concepts of identifia-
bility and distinguishability are developed with respect to the elucidation of the NPY kinetics.
We find that neither set of kinetic data on its own, nor the two data sets combined, allow the
true mechanism to be distinguished. In addition, even for a chosen mechanism, there exist de-
generate sets of parameters i.e. the solutions are also unidentifiable. A discussion of the limits
and uniqueness of the results’ interpretation is presented. ꢀ 2005 Wiley Periodicals, Inc. Int J
C
Chem Kinet 37: 717–727, 2005
steps driving the interconversion of the parent ther-
mally stable, ring-closed structure and its ring-opened
isomers. The structures of these species are shown in
Scheme 1 for 3,3-diphenyl-3H-naphtho[2,1-b]pyran
(NPY). The parent compound is colorless, absorb-
ing only in the UV, whereas the ring-opened isomers
are colored, absorbing in the blue end of the visible
spectrum.
A vast array of naphtho[2,1-b]pyrans have been re-
ported in the Patent literature [7–10], but data on their
kinetic properties are relatively sparse; mainly involv-
ing measurement of the “half-life” (t1/2) of the colored
species [11–19], i.e. the time for the absorbance of the
colored species to reduce to one half of its initial value
INTRODUCTION
An understanding of the fundamental kinetics govern-
ing photochromism can inform the development of new
materials endowed with specific properties required for
particular technological applications [1–6]. In the case
of photochromic systems such as the naphthopyrans,
this represents an interesting theoretical issue since
their kinetic behavior has not yet been fully elucidated.
The reaction dynamics of this family of compounds in-
volves a combination of photochemical and thermal
Correspondence to: Mounir Maafi; e-mail: maafim@yahoo.com.
2005 Wiley Periodicals, Inc.
c
ꢀ

718
MAAFI AND BROWN
Scheme 1 Structures of NPY and the four possible ring-opened species following photoexcitation.
upon removal of the irradiating light. Given that the
available data on the fading kinetics of these pyrans
are largely found not to be monoexponential, the in-
terpretation of these t1/2 values has to be performed
cautiously. The decay of the colored species can be
generally described as being bicomponent; a fast (sec-
onds) component which comprises the majority of the
decay and a slower component occupying hundreds of
seconds.
Most of the studies of the thermally driven recy-
clization reaction have used fitting procedures [20–27]
and have established that the kinetic profiles of the
fading reaction are well fitted by bi-exponential-type
equations. However, there has been little discussion
devoted to the actual mechanism taking place during
the bleaching. The general assumption is that because
the kinetics is fitted by bi-exponential models, there
are two (visibly absorbing) species which contribute
to the kinetics. The exact nature of these species (see
Fig. 1), their UV-visible spectra, and the kinetic re-
lationship between them are a matter of conjecture.
As a consequence, the parameters derived from the bi-
exponential fitting procedures cannot be mapped onto
rate constants etc. As a general observation, the two
“rate constants” derived from the bi-exponential fits
(often interpreted as the rate constants of the slow and
fast processes) have been found to vary between 1 and
3 orders of magnitude, according to the naphthopyran
structure and reaction conditions [20–26].
Much less attention has been focused on the light-
initiated reaction [28–38]. There have been some recent
reports that have attempted to solve the ring-opening
photochemicalkinetics. Theseinvestigationsweregen-
erally based on a unique, preconceived (and rather sim-
plified) mechanism in addition to several assumptions,
and the fitting procedures were conducted on the col-
oring and fading kinetics independently.
In summary, in most of the latter studies [11–38],
segregation of the fading and coloring reactions has
been accepted as a working procedure. When analyzed
from a conceptual viewpoint, it may be argued that
the color-generating reaction and the thermal back-
isomerization sequence are two separate processes; the
reactions obey distinguishable basic processes, are re-
alizedundersubstantiallydifferentexperimentalcondi-
tions, and can be observed individually. This statement
is acceptable even though both reactions concern the
same set of reactive compounds, held in the same re-
action medium, and the thermal ring-closure steps are
implicitly included in the kinetics of the ring-opening
reaction. However, the important question of whether
such a discriminatory strategy is the best method of
elucidating the NPY kinetics has not been addressed
properly.
In this paper, we investigate separately both NPY re-
actions in order to ascertain whether the former strategy
is viable or not. In the course of our contribution to the
study of photochromic kinetics, we have developed an
analytical formalism obtained in a closed-form integra-
tion [39] that allows kinetic laws for mechanisms of the
form ABC(nk,mφ) to be determined. NPY dynamics is
well described with this formalism, and it will be used
later on in this paper as a tool to model the studied
kinetic behavior.
Figure 1 Absorption spectra of NPY (2.4 × 10−5 M in
acetonitrile) as a function of irradiation time. The visi-
ble absorption increases with irradiation time. [Color figure
can be viewed in the online issue, which is available at
www.interscience.wiley.com.]

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
719
sorption vs. time profiles attained a plateau indicating
that the photostationary states were fully achieved. In
a final stage of 500 s, the irradiation light was switched
off, and the data corresponding 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 Lovenberg-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 be-
tween experimental and calculated results. Evaluation
of the parameter values was achieved using Mathcad
premium software version 2001i to solve the system
equations.
EXPERIMENTAL
The naphthopyran derivative (NPY) used in this
study (3,3-diphenyl-3H-naphtho[2,1-b]pyran) was a
gift from James Robinson Ltd. and was used as re-
ceived. The spectrophotometric grade acetonitrile sol-
vent was supplied by Aldrich.
The absorption spectra and kinetic profiles were
recorded on an Agilent 8453 diode array spectropho-
tometer equipped with a Chemstation for data acqui-
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. 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 monochromator, which
is a special f /2.5 monochromator with a 1200 groove/
300 nm blaze grating to allow irradiation 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 sam-
ple cuvette. Hence, the directions of the irradiation and
analysis light beams were perpendicular so as to avoid
interference. A shutter situated between the monochro-
mator and the optical fiber allowed the excitation light
to be manually switched on and off. An excitation
wavelength of 357 nm corresponding to the wavelength
of the red shifted isosbestic point was chosen.
All measurements were carried out in a quartz cell.
The sample was maintained at a constant tempera-
ture (of 24◦C) and was stirred continuously during the
experiment. Typically, an experiment was carried out
on a 2-mL volume of 2.4 × 10−5 M solution, mon-
itored at the wavelength of the open form isomer(s)
(λobs = 440 nm). Measurements were repeated at least
three times and the standard deviations calculated.
The experimental kinetic profiles, corresponding to
the fading reaction, were corrected for the absorbance
generated by irradiation with the spectrophotometer-
analyzing beam. Indeed, because the reaction medium
is continuously (at a sampling rate of 0.5 s) analyzed
by a wide wavelength range (190–600 nm), a certain
amount of photochemical reaction is always present.
Accordingly, the data acquisition began with a first
(dark) period of 200 s in which only the spectrometer
analysis beam was allowed to reach the sample. Then,
the excitation beam from the lamp was allowed to im-
pinge upon the sample for a second period of 300-s du-
ration. At the end of each of these time periods, the ab-
RESULTS AND DISCUSSION
Spectral Characteristics of Open
and Closed Form Isomers
The absorption spectrum of the colorless starting iso-
mer (NPY) is characterized by three sets of transition
bands located mainly in the UV region (Fig. 1). In ace-
tonitrile, the shortest wavelength bands situated at 202
and 244 nm have almost the same, relatively high mo-
lar absorption coefficients, i.e., 64,600 and 64,400 M−1
cm−1, respectively. These shortest wavelength bands
can be interpreted as belonging to ꢀ, ꢀ* 1 B transi-
tions [40]. The subsequent lower energy transitions,
located at 302, 316, 346, and 358 nm, have much
lower extinction coefficients ca. 6600, 7600, 4500, and
4300 M−1 cm−1, respectively. They may be attributed
1
1
1
1
to A − La and A − Lb electronic transitions due
to the aromatic ꢀ,ꢀ* transitions [40]. These features
are bathochromically shifted relative to the naphtha-
lene spectrum because of the more extensive rigid ꢀ
system and the participation of oxygen lone-electron
pairs in the electronic resonance. Accordingly, it might
be postulated that the lowest energy bands should over-
lap the n,ꢀ∗ transition due to the oxygen nonbonding
electrons. Similartrendsofspectralshiftshavebeenob-
served in the absorption spectra of several naphthalene
derivatives relative to the spectrum of the unsubstituted
naphthalene structure [40].
The appearance and increase in intensity of a new
lower energy band (380–500 nm) in the absorption
spectra (Fig. 1) reveals the progress of the photochem-
ical reaction of NPY. The transformation consists of a
ring opening through photolytic breakage of the sp3
C O bond. A total of four principal conformations
(Scheme 1), rotamers and cis–trans isomers, are then
expected to result.

720
MAAFI AND BROWN
Practically, only the two most stable and longer liv-
ing isomers (CT and TT) are believed to be detected
during the spectrophotometric experiment [23,25,31].
Their structures are characterized by a long-range con-
jugation chain that includes the benzene rings as well
as the sp2 oxygen atom. It might be also expected that
the rigidity and the planarity of the open-ring struc-
tures will contribute to lowering the energy of their
ꢀ,ꢀ∗ and n,ꢀ∗ transitions. As a consequence, the new
broadband absorption, belonging jointly to the CT and
the TT, is located at longer wavelength compared to
NPY (65 nm red shift), with a maximum at 425 nm.
Because the emerging band does not show any new
features (e.g., shoulders) and/or changes in spectral
position during the forward and reverse reactions, it
may be postulated that either the cis–trans conforma-
tional differences between the CT and TT structures
have an insignificant impact on their relative spectral
shapesortherelativeamountsofthetwospeciesremain
constant.
Scheme 2 The overall fading reaction scheme for NPY.
the maximum number of chemical steps (n) should not
exceed 4. Scheme 2 gives the framework of these in-
terrelated elementary steps within the naphthopyran’s
fading processes.
Our analysis will avoid any preconceived reaction
options because the true bleaching reaction mecha-
nism is not known with certainty. Hence, on the basis
of the kinetic scheme depicted in Scheme 2, we can
review and extract all the possible mechanisms that
might lead to the fading of the colored species. Eight
different but theoretically plausible reaction routes
(1–8), including consecutive, divergent, cyclic, and op-
posing sequences, can be identified (Table I). How-
ever, it should be emphasized that the observation con-
ditions prevent some of them from being practically
differentiated. The main reason for that is related to
the aforementioned similarity between the spectral fea-
tures of both photoisomers. Therefore, the sequences
1, 4, and 6 are formally equivalent to their homologues
2, 5, and 7, respectively. This reduces the total number
of genuinely different options to five (i.e., 1, 3, 4, 6,
and 8), and hence, only these will be examined in this
study.
Mechanistic Options
Evidence from several steady-state and flash studies
[20,29,41,42], which have dealt with naphthopyran
derivatives, suggests that the photochemical reaction of
these materials leads to the production of two observ-
able colored isomers. However, for particular experi-
ments (such as ours), it is not clear how many species
are responsible for the observed absorption changes.
The bi-exponentiality of the kinetic traces does not
necessarily mean that two colored species are actually
observed, but only that the overall mechanism should
encompass at least three different species (NPY plus
two others). This conclusion derives directly from the
bi-exponential formulae for the concentrations of the
individual species [39]. Quantitatively the best option
is to treat the observed visible spectra as comprising
contributions from both isomers, otherwise we would
be concluding that one of the merocyanine-like isomers
(i.e. CT or TT) does not absorb at all in the visible re-
gion, and has exactly the same spectral shape as the
NPY closed form.
Taking into account the NPY structure, the overall
dynamic system may be described as ABC(nk, mφ);
which comprises three distinct structures kinetically
linked by n thermal and m photochemical elementary
reactions. In this specific case, the spontaneous ther-
malbleachingreactionofthenaphthopyranunderstudy
may be described as a NPY–CT–TT (nk) system. Fur-
thermore, as no color changes were observed in a so-
lution of NPY when kept in the dark, this suggests that
no thermochromism (through ring-opening reactions)
has to be considered in the mechanism. As a result,
Table I All Possible Mechanistic Options for the
Thermal Bleaching of NPY
Rate Constants
Label of the ABC(nk) System
Involveda
1
2
3
4
5
6
7
8
(2k) consecutive
(2k) consecutive
(2k) convergent
k2,k1
k3,k4
k1,k4
(3k) consecutive, opposing
(3k) consecutive, opposing
(3k) cyclic
k2,k3,k1
k2,k3,k4
k1,k4,k2
k1,k4,k3
k1,k2,k3,k4
(3k) cyclic
(4k) cyclic, opposing
a First-order rate constants (in s−1) are labeled according to
Scheme 2.

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
721
Although no literature reports have experimentally
demonstrated whether or not the geometric cis/trans
isothermal isomerization (processes k2 and k3) between
CT and TT takes place, it is conceptually possible and
we consider it here for the sake of completeness and to
avoidconsideringonlytheconvergentsequence. Onthe
other hand, in sequences 3, 6, and 8, the direct trans-
formation of TT into NPY which would presumably
necessitate isomerization of the first exocyclic double
bond is also envisaged. Given its structure, the recy-
clization of TT should then probably proceed through
a short-living intermediate species, so it is necessary to
stipulate that the latter is not observed at λobs and/or k4
is the limiting step.
As a consequence of this, for the five mechanistic
schemes under study, k1 values may be expected to
have a higher magnitude than k4. However, if we stip-
ulate that it is not reasonable to envisage TT returning
directly to NPY due to the trans–cis energy barrier to
be overcome, only two mechanisms (1 and 4) remain
for consideration. As we will see later, this lessens the
number of studied cases but will not lead us to a defini-
tive answer as to which mechanism is the true one.
Figure 2 Experimental ( ), and calculated (solid line) pro-
◦
file of absorbance vs. time data for the CT and TT fading
reaction ([NPY] = 2.4 × 10−5 M in acetonitrile at 24◦C).
[Color figure can be viewed in the online issue, which is
available at www.interscience.wiley.com.]
absorption spectrum of NPY is fully recovered at the
end of the fading reaction. This, together with the
four isosbestic points observed at 230, 257, 338, and
357 nm, indicates that the reaction proceeds without
side reactions and that Eq. (1) truly represents the
dynamics.
Kinetic-Law Models
According to case I from the previous formalism [39],
it can be seen that the absorbance vs. time trends,
for all the mechanisms under study, should obey bi-
exponential model equations of the following general
form:
440
440
TT
M(t) = A + A
(2)
CT
Under these reaction conditions, the initial value M0
(at t = 0 s) of M(t) is obviously identical for all the
mechanisms listed below (Eq. (3)).
m0 − M0 · ϑ
−
+·t
M(t) = −
· eϑ
ϑ − ϑ
−
+
440
M0 = ε4C4T0 · CC0T + ε · CT0T
(3)
TT
m0 − M0 · ϑ
ϑ − ϑ
+
−·t
+
· eϑ
(1a)
(1b)
−
+·t
+
The remaining constants of Eq. (1) are defined [39],
with the following expressions (Eqs. (4)–(18)):
M(t) = θ1 · eϑ + θ2 · eϑ
−·t
– Mechanism 1: ABC(2k) consecutive
where M(t) represents the cumulative absorbance of
both photoisomers at λobs at any time (Eq. (2)). The
remaining symbols m0, M0, ϑ , and ϑ denote con-
k2
k1
TT −→ CT −→ NPY
+
−
ꢀ
ꢁ
stants which can be worked out in explicit forms [39].
They may be expressed as functions of the rate con-
stants (k1 − k4); the molar absorption coefficients εCT
and εTT of compounds CT and TT, respectively; and the
initial concentrations CC0T, CT0T of the photoisomers at
the photostationary state (where the initial time, t = 0,
corresponds to the moment where the irradiation light
is cut off—this is also the time at which the open forms
have the maximum absorption intensity under the irra-
diation conditions).
440
m0 = k2 · ε4C4T0 − ε
· CT0T − k1 · ε4C4T0 · CC0T
TT
(4)
ϑ
= −k2;
ϑ
= −k1
(5,6)
+
−
– Mechanism 3: ABC(2k) convergent
k4
k1
TT −→ NPY ←− CT
m0 = k4 · εT44T0 · CT0T − k1 · εC44T0 · CC0T
(7)
The evolution of M(t) during the fading reaction
is shown in Fig. 2. It is important to stress that the
ϑ
= −k4;
ϑ
= −k1
(8,9)
+
−

722
MAAFI AND BROWN
– Mechanism 4: ABC(3k) consecutive
they represent rate constant based expressions. This
means that interpretation of the values of ϑ and ϑ is
fraught with difficulty, unless reliable and strong evi-
dence can be established beforehand about the mech-
anism involved. For example, it may not be reliable to
calculate activation energies on the basis of the vari-
+
−
k1
k2
TT ꢀ CT −→ NPY
ꢁ
k3
ꢂ
ꢃ
m0 = −(k1 + k3) · εC44T0 + k3 · εT44T0 · CC0T
ꢀ
ꢁ
440
TT
+ k2 · ε4C4T0 − ε
· CT0T
(10)
ation of ϑ and ϑ values with temperature and the
+
−
Arrhenius equation, as has been reported in several pa-
pers [34,38]. Application of the Arrhenius formula as-
ꢄ
−(k1 + k2 + k3)
(k1 + k2 + k3)2 − 4 · k1 · k2
ϑ
=
sumes a single-step reaction, and therefore, ϑ and ϑ
+
−
2
should correspond each to a single rate constant as in
mechanisms 1 and 3. However, if the true mechanism
is one of the remaining three (4, 6, or 8), application of
the Arrhenius equation is problematic.
(11,12)
– Mechanism 6: ABC(3k) cyclic
Fitting Analysis
Figure 2 shows the time evolution of the absorbance
at 440 nm after the excitation light has been removed.
Usually, this kind of data have been fit with the aim
of extracting the two pre-exponential and exponential
m0 = (k2 + k4) · ε4T4T0 · CT0T
+ (k2 · CT0T − k1 · CC0T) · ε
440
CT
(13)
terms (θ1, θ2 and ϑ , ϑ , respectively) of Eq. (1b)
+
−
[17,20,21,23,24]. Although, this procedure will always
give values for these parameters, in our treatment the
fitting has been carried out using the model Eq. (1a); the
fact that the two pre-exponential coefficients in Eq. (1a)
share parameters imposes a constraint to the fitting cal-
culation which undoubtedly increases the reliability of
quantification of all the factors. It also generates di-
rectly the four individual parameters that may subse-
quently constrain the estimation and refinement of the
kinetic results for the reaction.
ϑ
= −k1;
ϑ
= −(k2 + k4)
(14,15)
+
−
– Mechanism 8: ABC(4k) cyclic–opposing
ꢂ
ꢃ
m0 = −(k1 + k3) · εC44T0 + k3 · εT44T0 · CC0T
ꢂ
ꢃ
440
+ k2 · ε4C4T0 − (k2 + k4) · ε ) · CT0T (16)
TT
The fitting outputs (Eqs. (19)–(22)) indicate that er-
rors on the estimated parameter values are very small
(<5 × 10−4), and the fit to the experimental profile
is good (χ2 = 10−7, rE2xp. vs. calc. > 0.99). As expected,
M0 has the value of the initial absorbance, and the ini-
tial velocity, m0, is negative. It may be noted that two
ꢀ
1
2
ϑ
=
· − (k1 + k2 + k3 + k4)
ꢅ
ꢄ
(k1 + k2 + k3 + k4)2 − 4 · (k1 · k2 + k1 · k4 + k3 · k4)
(17,18)
orders of magnitude separate the values of ϑ and ϑ ,
+
−
in agreement with values reported for the same com-
pound in several solvents (toluene [20], ethanol [20],
or methylcyclohexane [34]).
It can be seen that ϑ and ϑ are always negative
+
−
entities. The quantity m0, representing the value of the
first derivative of M(t) at the initial time (or initial ve-
locity), is also negative because of the negative slope
of the cumulative absorbance vs. time data when the
UV-light is switched off (Fig. 2). The above formalism
(Eqs. (3)–(18)) shows clearly that M(t), as written in
Eq. (1b), is dependent on at least six constants (for the
simplest mechanisms 1 and 3) that have to be deter-
mined.
M0 = 0.09244 6 × 10−5
(19)
(20)
(21)
(22)
m0 = −0.01253 1.7 × 10−4
ϑ
= −0.15776 5.2 × 10−4
= −0.00229 1 × 10−5
+
ϑ
−
Most importantly, the exponential factors ϑ and
ϑ , correspond respectively to specific and unique rate
constants only for the first two mechanisms (1 and 3).
Otherwise, for the remaining sequences (4, 6, and 8)
+
Distinguishability and Identifiability
−
Resolution of the kinetics and determination of the
relevant rate constants and spectroscopic parameters

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
723
involved in the dynamics of NPY photochromism
present two challenges. First, it is necessary to be able
to decide upon the actual mechanism involved in the
fading reaction. Once this has been achieved, the ques-
tion of whether we can determine a unique set of kinetic
and spectroscopic parameters describing the dynamics
must be confronted.
These two challenges are respectively known in the
literature as the distinguishability and identifiability
problems [43–54]. They raise the fundamental ques-
tions of the uniqueness of the specific attributes of a
kinetic problem and whether there may be ambiguity
in the reaction model even with a good fit between the
experimental data and the chosen model. In the case
where more than one reaction scheme reproduces the
experimental data with equal goodness of fit, the sys-
tem is said to be nondistinguishable. The identifiability
problem has meaning in the context of unknown model
parameters. The usual question is whether or not it is
possible to find a unique solution of the unknown pa-
rameter set of the model, from data collected in a real-
sample experiment. These issues have been discussed
only from a conceptual point of view in the literature.
In this work we give an experimental and conceptual
example as well as raising them for the first time in
relation to the naphthopyran kinetics.
Figure 3 Linear correlation of experimental versus cal-
culated data for mechanism 8a (995 points). [Color figure
can be viewed in the online issue, which is available at
www.interscience.wiley.com.]
As expected, all the mechanisms accurately repro-
ducetheexperimentaldata(ofFig. 2)withexemplarsof
the parameterization given in Table II. In all cases, ex-
cellent agreement (slopes and correlation coefficients
for plots such as those of Fig. 3 are almost unity, to-
gether with nil intercepts for data sets of the order of
1000 points) is obtained between the observed and
calculated absorbance values (M(t)) at all times, as
shown by Fig. 3. Therefore, the five possible reaction
sequences are all equally good candidates to be the
true reaction mechanism. Under these conditions and
for the experimental data available, the mechanism of
the NPY-fading kinetics is totally nondistinguishable.
By corollary, it becomes evident within the scope of
this problem that a good fit for a set of results cannot
be used as a supporting argument for the occurrence of
a particular mechanism.
In our case, energetic and steric considerations can
be put forward to exclude the retro-cyclization reaction
TT → NPY (i.e. k4 = 0). However, it is still not possi-
ble to distinguish between the remaining sequences 1
and 4, Table II. This points out the special caution that
should prevail when arguing in favor of a particular
mechanism for these photochromic systems.
Distinguishing the True Reaction Mechanism. The
distinguishability definition flows directly from the
uniqueness of the M(t) model (Eq. (1)) that describes
the kinetics of NPY. Indeed, Eq. (1) defines formally
the output data trace of all the mechanisms (1–8) with-
out regard to the specific arrangement of the reaction
steps within the sequences. Therefore, discrimination
between any two mechanisms cannot be established on
the basis of distinctive mathematical kinetic models.
Table II Examples of Parameter Sets Satisfying Each Reaction Mechanism
Reaction Mechanism
Kinetic Parameters
Concentrationsa
1
3
4
6
8
CC0T
CT0T
k1
k2
k3
k4
εCT
εTT
3.21 × 10−6
1.09 × 10−6
0.15776
0.00229
–
6.89 × 10−6
2.09 × 10−6
0.15776
–
5.14 × 10−6
3.28 × 10−8
0.11301
0.00319
0.04385
–
4.72 × 10−6
1.74 × 10−7
0.1386
0.0196
–
2.93 × 10−6
8.70 ×10−8
0.12830
0.00138
0.02926
0.00144
30892
Rate constantsb
–
–
0.00229
11486
6311
0.0015
19189
10544
Molar absorption
coefficientsc
24817
11706
17927
8536
20451
a Concentrations in mol L−1
b Rate constants in s−1
c Molar absorption coefficients in M−1 cm−1
.
.
.

724
MAAFI AND BROWN
Identifying the True Parameter Set. The second
part of the kinetic elucidation must focus on the de-
termination of the parameter set values. In order to
achieve this we rely on the linearly independent equa-
tions of the kinetic systems as given by the formalism
(Eqs. (1)–(18)). In each case we have four equations
situation is to consider the application of mechanism 1
to the actual data for NPY bleaching (Table III). In our
hands, the previous formalism [39] sets the minimum
condition for the solvability of models 6 and 8 to be
knowledge of the k values [58]; however, model 1 is
not solvable even with a known ratio of εCT and εTT
at a particular wavelength (demonstration of which is
beyond the scope of this paper).
for ϑ , ϑ , M0, and m0. Leaving aside the rate con-
+
−
stants (which are combined in the relationships for ϑ
+
and ϑ ), the initial absorbance and velocity expres-
Similar conclusions on identifiability and distin-
guishability have been reached by Vajda and Rabitz
[43] for many first-order reaction systems. Their math-
ematical treatment, based on the application of either
Laplace transforms or the state-space similarity trans-
formation, has set the limits for what can be achieved
in terms of elucidating trimolecular-system kinetics.
Their study also comes to the conclusion that such sys-
tems are nondistinguishable and unidentifiable and has
underlined the dependency between elucidation per-
formance and the number of unknowns and the need
to be able to measure and/or define more quantities
(including for the ABC(2k) consecutive reactions). A
particular application of these methods has been car-
ried out by Sienicki and Durocher [44] to analyze first-
order or pseudo-first-order photophysical (excimer and
exciplex) reactions monitored by time resolved flu-
orescence spectroscopy. The two-state photophysical
models investigated were characterized by eight un-
known parameters, including, lifetimes (τA and τB),
thermal rate constants (k1 and k−1), fluorescence in-
tensity functions (A(ν) and B(ν)), and initial concen-
trations of the species (A0 and B0). The overall decay
profiles obey double exponential equations dependent
on the latter parameters. It should be noted that even
though these systems are quite different from the kinet-
ics of NPY, they involve the same fundamental issues
and criteria so that the conclusions can be compared
in a qualitative approach. The earlier study [44] has
−
sions are dependent on four variables (i.e., the molar
absorption coefficients at the observation wavelength
and the initial concentrations of the photoisomers). As
a matter of fact, the number of parameters to be defined
(a minimum of six) exceeds the number of equations
for all the mechanisms (1–8). Owing to this situation,
the models do not admit an algebraic solution. In other
words, there is not a single thermal model among those
possibly describing the NPY system that is uniquely
identifiable. Furthermore, in the absence of tight re-
strictions over the allowed parameter values, a large
number of solutionscan beobtained, allwithhighaccu-
racy of fit. Consequently, the individual kinetic models
are unidentifiable.
A few examples, such as those shown in Table III,
demonstrate also that restricting the values of εCT and
εTT to be similar as might be anticipated, does not nec-
essarily lead to an improvement in the refinement of
the kinetic analysis.
Thus, the results of the fading experiment are not ob-
jectivelysufficienttoallowthedeterminationofthetrue
parameter set values. For NPY-like kinetic systems, the
situation is one of the worst since the latter statement
is correct even when the reaction model is known with
certainty and all the kinetic rate constants are known,
i.e. where the only purpose of the analysis is to obtain
the spectroscopic parameters and photostationary state
concentrations of the isomers. A typical example of this
Table III Examples of Multiple Solutions for Mechanisms 1 and 8, ABC(2 and 4k)
Mechanism 1
Mechanism 8
1a
1b
1c
8a
8b
8c
Initial concentrations (M)
CC0T
CT0T
1.11 × 10−5
2.03 × 10−6
4.44 × 10−6
8.13 × 10−7
1.76 × 10−6
8.5 × 10−6
1.15 × 10−5
4.37 × 10−9
1.09 × 10−7
7.21 × 10−6
3.23 ×10−7
1.2 × 10−8
Molar absorption coefficients (M−1 cm−1
)
440
ε
7168
6399
17890
45000
10842
7980
6000
15721
12576
CT
440
TT
ε
15973
40178
13222
Rate constants (s−1
)
k1
k2
k3
k4
0.15776
0.00229
0.15776
0.00229
0.15776
0.00229
0.13970
0.00146
0.01788
0.0010
0.12830
0.00389
0.02936
0.00199
0.00119
0.01563
0.00121
0.14199
–
–
–
–
–
–

PHOTOPHYSICS AND KINETICS OF NAPHTHOPYRAN DERIVATIVES
725
proved the unidentifiability and nondistinguishability
of such kinetic models on a similar basis to ours (the
number of equations was found to be less than the num-
ber of unknowns). In addition to this, the analysis has
been amended by special conditions where some of
the unknown parameters could be available, to set the
casesforwhichtheidentifiabilityanddistinguishability
problems can be solved. None of these solvable cases
has more than three unknowns.
Evaluation of NPY-Photocoloring Process. The
procedure which has been applied above to the
NPY-fading process can be equally well applied to
the photocoloring reaction by using the appropriate
series of equations that derive from the general
formalism [39]. The inclusion of the photochemical
reactions adds up to six extra reaction steps into the
overall mechanism and thereby greatly multiplies the
number of possible overall reaction mechanisms. As
with the fading reaction, the formalism developed in
[39] predicts that the intensity (absorbance) of the
colored forms developed as a function of irradiation
time will be bi-exponential. Except for the cases
ABC(2k,2φ) as discussed in [58], we find that all
the bi-exponential kinetic laws (all higher level
kinetic mechanisms ABC(2–4k, 3–6φ)) describing the
photocoloring dynamics are nondistinguishable and
unidentifiable as well. The huge number of different
possible mechanisms, far more than that encountered
for the fading reaction, makes an exhaustive study
of the photocoloring mechanism too long. Instead,
as an illustration we have studied the identifiability
problem for one single kinetic model, that is, the
ABC(2k,3φ) sequence proposed by Favaro et al.
[34] (Scheme 3) in a previous investigation of NPY
dynamics. Several examples of parameter values,
which fit the experimental data (Fig. 4) with high
accuracy, are reported in Table IV.
Figure 4 Experimental (o) and calculated (solid line) pro-
file of absorbance vs. time data for the photocoloring reaction
of NPY (2.4 × 10−5 M in acetonitrile at 24◦C). Inbox: Fitting
parameters for the bi-exponential model.
In terms of kinetic investigations of photochromic
systems, these findings relative to the issues of identifi-
ability and distinguishability can easily and equally be
raised for many other photochromic systems whether
they exhibit similar photocoloring and fading curves to
NPY (which is typical for the 3H-naphthopyrans) or
different ones as in the case of the 2H-naphthopyrans
[35,38,55–57].
From an application viewpoint, it is generally ac-
cepted that the structures of most interest are those
characterized by a deep and strong visible absorption
in their open forms, so as to confer a specific color
in a light environment to the substrate containing the
photochrome. However, it should not be neglected that
efficiency from this perspective is not exclusively de-
pendent on the absorption properties but also should
include photochemical considerations as well. Indeed,
a higher absorption in the visible could also result in
a higher photochemical reactivity of the colored iso-
mers, particularly through routes that return them to
the starting colorless structure. It is clear that successful
applications of these photodyes have to be monitored
from a perspective that takes into account the balance
between the antagonistic thermal and photochemical
reactions. A complete study cannot be achieved by
looking only at selected reactions of the NPY system.
Breaking through the NPY kinetics will most probably
shape the future developments in this field.
There is considerable variation between the differ-
ent parameter sets and, as for the fading data discussed
above, in the absence of additional information it is
impossible to decide which parameter set most accu-
rately reflects the fundamental properties of the system.
Analytically solving such kinetics requires a particular
method, as will be described in a future investigation
[59].
CONCLUSION
In this work, it has been conceptually and experimen-
tally shown that the elucidation of the true kinetics
Scheme 3 NPY photochromism mechanism as proposed
by Favaro et al. [34]. Labels as per Maafi and Brown [39].

726
MAAFI AND BROWN
Table IV Various Parameter Setsa Satisfying the Conditions of the Proposed Photochemical/Thermal Mechanism as
Given in Scheme 3
1
2
3
4
5
φABγb,c
0.02855
0.01324
0.03908
0.00265
0.00749
0.03016
0.01866
0.02189
0.03087
0.01104
φ
φ
BCγ
CBγ
0.02280
0.02286
0.02693
0.02403
0.02267
CBPSS
CCPSS
ε4B40
3.296 × 10−6
4.513 × 10−6
8.679 × 10−6
2.158 × 10−6
3.565 × 10−6
1.375 × 10−6
2.404 × 104
9.773 × 103
3.828 × 10−7
1.756 × 104
3.511 × 104
3.470 × 10−6
9.129 × 104
3.878 × 103
1.925 × 10−6
3.672 × 104
6.981 × 103
1.214 × 10−6
2.222 × 104
1.108 × 104
ε4C40
a For all cases kBA = 0.15776 s−1 and kCB = 0.00229 s−1
.
b The product φi γ in s−1
.
c γ = εNPY I0Fl is a constant (see [39] for definitions).
related to naphthopyran thermal and photochemical
reactions is inevitably confronted with the degener-
acy of the kinetic solution. The distinguishability–
identifiability problem stems primarily from the fact
that the defining equations of the system are less than
the number of unknown parameters describing the re-
action. It is striking to note that the NPY system pro-
vides a particular combination of unfavorable features
for kinetic elucidation: (i) the reactive species are not
physically isolable and do not exhibit separate absorp-
tion bands; (ii) no major information can be obtained
from the few different experimental operations that can
be carried out on the reactive solutions in the condi-
tions of the generally used UV-Vis experimental set
up (only three variables operate on the experiment:
temperature, the initial concentration of the mother
closed-form compound, and the excitation-light inten-
sity or wavelength); (iii) the number of parameters to
define can be very high; and (iv) an excessive number
of plausible mechanisms. It is impossible on the ex-
clusive basis of the observed kinetic profiles, to reach
any satisfactory interpretation that fully elucidates the
studied dynamics. For this reason, any possible extra
physico-chemical information obtained by alternative
means about the involved reactive compounds and/or
the reaction process, is expected to play a crucial role
in this situation, ultimately leading to a better under-
standing of the fundamental kinetic behavior of these
chromenes. For the time being though, the identifia-
bility and distinguishability concepts and the ways of
uniquely resolving them will remain of central appeal
to NPY kineticists.
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