Photochromism and Thermochromism of Phenanthrospirooxazine in Poly(AlkylMethacrylates)

Marcia Levitus and Pedro F. Aramendıa*INQUIMAE, Departamento de Quı´mica Inorganica, Analıtica y Quımica Fısica, Facultad de Ciencias Exactasy Naturales, UniVersidad de Buenos Aires, Pabello´n 2, Ciudad UniVersitaria, 1428 Buenos Aires, Argentina

ReceiVed: NoVember 5, 1998; In Final Form: January 13, 1999

The decay of the photomerocyanine (MC) formed after flash irradiation of a phenanthrospirooxazine (SO)was studied in polymer films of poly(methyl-), poly(ethyl-), and poly(isobutyl methacrylate) and comparedwith similar data of a benzospiropyran (SP). The decays in the dark depart from monoexponential behavior.The data were analyzed by a Gaussian distribution model and by a matrix relaxation model. The rate constantat the center of the Gaussian distribution is equal in all polymers and equal to the unimolecular rate constantreported in nonpolar liquids. This may be related to the inability of the polymer to solvate the transition state.The polymer matrix affects the width of the activation energy distribution; this distribution is broader inmore rigid media. The relaxation model is able to provide a mean value for the relaxation time of the matrix,which is relevant for the decay. The Arrhenius activation energy of this relaxation time is similar in eachpolymer for the MC forms of SO and of SP and is similar to the activation energy of theâ-relaxation of thepolymers. This indicates that local movements are responsible for the relaxation, which also explains the factthat all kinetic parameters show an Arrhenius-type behavior in temperature ranges including the glass transitiontemperature. The thermochromism of SOa MC transformation was studied in poly(methyl-) and in poly-(ethyl methacrylate). The equilibrium constants depart from the normal behavior in fluid media. This departureis interpreted by assuming a Gaussian distribution of energy difference between the two forms of SO. Thewidths of the distributions obtained from kinetic and equilibrium data are very similar.

Introduction

Spirooxazines (SO) are photochromic compounds formed bytwo heterocyclic parts linked together by a common tetrahedralsp3 carbon atom. The absorption of UV radiation leads, bycleavage of a carbon-oxygen bond, to the formation of an openmerocyanine structure (MC) which absorbs in the visible region(see Figure 1)1. MC reverts thermally to the thermodinamicallystable closed form, with a decay kinetics strongly dependenton the environment.2-5

In some media the proportion of MC in thermal equilibriumwith the closed form is high enough to be detectable byabsorption spectroscopy.1-3,6

In comparison to other photochromic compounds, somespirooxazines are excellent in light fatigue resistance. Thisallows these compounds to be used in commercial applications,such as the manufacturing of ophthalmic lenses, sunglasses, andski goggles.1

Most studies of the photochromism and thermochromism ofspirooxazines and related compounds have been performed inliquid solution.1-9 To extend these studies to polymer media isimportant because practical applications of photochromismrequire a polymer matrix.10-17

Kinetics in solid media is normally quite complex due to therigidity of the environment. For this reason, great departure froman exponential behavior is observed.13,16-22 If the matrix doesnot change around the probe during the decay, different probemolecules may be placed in statistically different environments,which may result in different local probability of decay. The

microheterogeneity is inherent of the solid phase and results incomplex kinetics. In the most general formulation, the prob-ability of probe survival at timet, P(t)/P0, would be19,20

whereg(k) is the distribution function of decay probabilities(or first-order rate constants) and the integral is extended overall possible values ofk. Equation 1 states that the decay kineticsis the Laplace transform of the rate constant’s distribution. Thisview of the system is a limit that considers the medium ascompletely frozen during the decay.

On the other limit, single-exponential decay of unimolecularreactions is often observed in polymers above the glass transitiontemperature,Tg. In this limit, the polymer is flexible enough inorder that every probe molecule experiences the same averageenvironment during the decay. In the picture of eq 1,g(k)approaches the limit of a Diracδ function.

Taking into account the matrix relaxation, the system shouldchange gradually, as temperature is increased, from a situation* To whom correspondence should be addressed.

Figure 1. Structures of spirooxazine (SO) and merocyanine (MC).

P(t)P0

) ∫0

∞g(k) exp(-kt) dk (1)

1864 J. Phys. Chem. B1999,103,1864-1870

10.1021/jp9843151 CCC: $18.00 © 1999 American Chemical SocietyPublished on Web 03/02/1999

in which the medium responds very slowly to probe decay (i.e.,it is frozen) to an intermediate regime in which probe andmedium change in comparable time scales and should finallyreach a limit of high temperature in which the medium isstatistically equal in time during the probe decay. In a previouspaper,17 we developed a kinetic model for the intermediateregime, which explicitly fits a value for the relevant relaxationtime of the matrix,τm, from the kinetic data.

In this paper we investigate the decoloration kinetics andthermochromism of a spiro[indoline phenanthrooxazine] incor-porated in poly(alkyl methacrylates). Kinetic results are analyzedon the basis of a distribution model and a matrix relaxationmodel. The behavior of the equilibrium constants responsiblefor the thermochromism is also analyzed taking into accountthe inhomogeneity of the matrix. Furthermore, we compare thedistribution information obtained from kinetic and equilibriumdata. Finally, the information obtained from the kinetic relax-ation analysis is compared with that obtained with a spiropyran(SP) in a previous work.17

Experimental Section

Chemicals.Spirooxazine (spiro[indoline-2,2′-[2H]phenanthr-[9,10-b][1,4]oxazine]), spiropyrane (1,3′,3′-trimethyl-1-6-nitro-spiro-[2H-1-benzopyran-2,2′-indoline]), and poly(alkyl meth-acrylates) (polymethyl (PMMA), polyethyl (PEMA), andpolyisobutyl (PIBMA) methacrylate, secondary standards) werepurchased from Aldrich and used as received. Molecular weight(Mw) and polydispersity (Mw/Mn) were, respectively, 102 600and 2.1, for PMMA 340 000 and 2.7, for PEMA, and 300 000and 2.1, for PIBMA. Dichloromethane was HPLC grade.

Thermal Analysis. Tg values of the different polymer sampleswere determined by DSC in a DuPont calorimeter model v2.2A990. The values were 103°C for PMMA, 61 °C for PEMA,and 51°C for PIBMA.

Preparation of Polymer Films. Weighted amounts ofpolymer and SO or SP were dissolved in dichloromethane. Thesolution was placed in a silanized Petri dish (see below) andthe solvent was allowed to evaporate. The sample was furtherdried overnight in a vacuum oven at room temperature, andafterward the film (ca. 100µm thick) was removed from theplate. The film thickness was measured with a micrometer withan accuracy of 10%. The SO or SP concentrations in the polymerfilm were always less than 0.5% w/w.

To avoid the film adhering to the glass plate, Petri disheswere previously treated with chlorotrimethylsilane (Merck,CGL). This treatment confers to the glass surface a nonpolarcharacter, which avoids the adherence of polymers with polargroups. After the treatment, Petri dishes were exhaustivelywashed and dried.

Kinetic Measurements.Kinetic experiments were performedfollowing the absorbance in the visible band of MC, in a UV-vis spectrophotometer or in a flash photolysis arrangementdepending on the time scale of the experiment. In both cases,the film was placed in a holder consisting in a brass paral-lelepiped of the same size as a standard 1 cm path cuvette holder.This piece has a longitudinal cut across one of the diagonals ofits basis so that the film can be sandwiched between the twohalves. The whole piece has two perpendicular holes to allowirradiation and analysis. The diameter of the irradiation hole is8 mm, while that of the analysis hole is 3 mm. Measurementswere performed between 5 and 75°C. Temperature wascontrolled by water circulation and measured directly on thefilm surface by means of a flat square Pt100 calibrated resistancewith an accuracy of 1°C. This setup allows a more efficient

thermostatization and a more accurate temperature determinationthan the one used in the study of the photochromism of SP.17

For this reason, and to improve the quality of the previousresults, we repeated the measurements made on SP with thisnew setup.

For decays requiring measuring times longer than 50 s,absorbance was measured as a function of time in a Shimadzu3101-PC UV-vis spectrophotometer. In this case, samples wereirradiated by means of a photography flash, whose plastic coverwas removed to avoid filtering the UV. The flash output wasfiltered with a UG1 Schott optical filter.

A flash photolysis arrangement was used for shorter measure-ments times. In this case, photocoloration was achieved bymeans of a pulsed Nd:YAG laser (Spectron Lasers, England,354 nm, 8 ns fwhm, 80 mJ per pulse). A UG1 Schott opticalfilter was placed before the sample to filter the small amountof the 532 nm component of the laser output and neutral densityfilters attenuated the excitation energy to adequate values.Spectrophotometric analysis was performed by means of a 100W tungsten lamp previously filtered with a GG435 Schott opticalfilter. The detector (Hamamatsu R928) was placed after amonochromator tuned at the maximum of the visible band ofMC. The sample was placed in the holder described above insuch a way that the specular reflection of the laser beam didnot enter the detector.

Experiments at different analyzing lamp power were per-formed in order to verify that the analysis lamp output did notaccelerate the decay of MC by photolysis. It was also verifiedthat the decay shape was independent of the laser pulse energy.

Spectra at different times after irradiation were taken at roomtemperature on a HP-8452 diode array spectrophotometer. Inthis case, the photography flash was also used to produce theMC.

Equilibrium Measurements. The equilibrium constant be-tween SO and MC (Keq ) [MC]/[SO]) was measured from thedark absorption spectra at different temperatures. The equilib-rium MC concentration was determined from the absorbancein the maximum of the visible band, where the closed formdoes not absorb. The molar absorption coefficient was consid-ered equal to the one measured in toluene,2 εMC (585 nm))68 300 dm3 mol-1 cm-1. The equilibrium SO concentration wasobtained from the absorbance at 340 nm. At this wavelength,MC and SO have similarε values, but since the SO concentra-tion is around 100 times greater than the MC concentration (Keq

< 0.016 in all cases), the contribution of MC to the totalabsorbance at 340 nm can be considered as negligible. Underthis assumption,Keq can be calculated as

The molar absorption coefficient of SO was taken from literatureas the one measured in toluene,2 εSO (340 nm)) 6000 dm3

mol-1 cm-1.To verify the previous assumptions, the equilibrium concen-

tration of SO was alternatively calculated from a mass balancebetween the total SO concentration and the equilibrium MCconcentration. The total SO concentration was measured dis-solving a known mass of film in a measured volume of toluene,whereKeq is small enough to neglect thermal equilibrium. Theconcentration of SO in this solution was calculated from theabsorbance at 340 nm. The total SO concentration in the filmwas calculated from this value taking into account the thickness

Keq )A585/ε585

MC

A340/ε340SO

(2)

Photochromism and Thermochromism J. Phys. Chem. B, Vol. 103, No. 11, 19991865

of the film and the density of the polymer. Results obtainedwith the two methods were the same within experimental error.

To record the spectra after every temperature change, enoughtime elapsed to allow for chemical equilibration of the system.

Results

Photochromism.Spectra measured after UV irradiation areat any time proportional to each other in the visible region,indicating that only one species is formed (see Figure 2). Thisallowed the absorption to be monitored at a fixed wavelengthas a function of time to obtain the kinetic data. The decays ofMC were nonmonoexponential in the whole temperature range,in all polymers. Figure 3 shows data at two extreme temperaturesin PIBMA. The samples show measurable thermochromism inPMMA and PEMA, but since the equilibrium constant is small([MC]/[SO] < 0.016 in all cases, see below), it was notnecessary to take into account this equilibrium in the kineticanalysis.

Two kinetic models were used to fit the decay data. On oneside, a distribution-type model was used. Within this picture,the most widespread model postulates that the probe is locatedin different environments having a distribution of activationenergies that is Gaussian with a dispersion that decreases withincreasing temperature.16,18

HereP0 represents the initial probe concentration,E is the energyof the probe in its ground state,Em is the maximum of the energybarrier,ν0 is an energy-independent frequency factor, andσ andE0 represent the width and the value of the energy at the centerof the distribution,g(E), respectively.

The origins for the nonmonoexponential decay kinetics arenot only due to the heterogeneity but also to the fact that thepolymer matrix can relax from a nonequilibrium situation in acomparable time scale as the decay kinetics. In a previous paper,we quantitatively described the thermal ring closure of a

spiropiran in poly(alkyl methacrylates) introducing a kineticmodel which takes into account the simultaneous relaxation ofthe matrix and the probe.17 This model (subsequently referredto as the relaxation model) assumes that the matrix relaxationmodifies the average first-order rate constant of the probe decayfrom an initial value,k0, immediately after excitation, to arelaxed valuek∞. The rate of change of the rate constant fromk0 to k∞ is governed by a matrix mean relaxation timeτm. Ifthis change is assumed to be monoexponential

The integrated form of eq 5 gives the following expression forthe time dependence of probe concentration:

The three parameters,k0, k∞, and τm, must be regarded asaverage values. For this reason, this model fails to account forthe nonmonoexponential decay at low temperatures, when therelaxation is much slower than probe decay. In this case, thereal distribution of decay times of the probe and relaxation timesof the matrix cannot be adequately replaced by mean values.

The measured decays of MC were analyzed by the two kineticmodels described above (eqs 3 and 6). The relaxation model ofeq 6 fits adequately the data above 45-50 °C. The threeparameters of this model have an Arrhenius-type temperaturedependence, as shown in Figure 4. Arrhenius parameters arelisted in Table 1. Results show thatτm decreases and thatk0

and k∞ tend to the same value as the temperature increases,reaching the monoexponential limit.

Figure 2. Spectra of a PIBMA film containing SO after flash irradiationin the UV at room temperature. Continuous lines correspond to spectrataken at different times after the photolysis. Times in seconds are, fromtop to bottom: 0, 2, 4, 6, 8, 12, 18, 28, 36, 48, 70. The dotted line isthe spectrum before the flash, which coincides with the spectrum atlong times after photolisis.

Figure 3. First-order plots for the decay of MC in PIBMA at (a) 6.7°C and (b) 75°C. The dotted line in plot b is the slope of the graph att > 0.4 s.

k(t) ) k∞ + (k0 - k∞) exp(-t/τm) (5)

P(t) ) P0 exp{-k∞t - [k0 - k∞]τm[1 - exp(-t/τm)]} (6)

P(t) ) P01

x2πσ∫-∞

∞exp[-(E0 - E)2

2σ2- (ν0e

-(Em-E)/kT)t] dE

(3)

g(E) ) 1

x2πσexp[-(E0 - E)2

2σ2 ] (4)

1866 J. Phys. Chem. B, Vol. 103, No. 11, 1999 Levitus and Aramendı´a

The Gaussian distribution of activation energies of eq 3 fitsadequately the data in the whole temperature range. Thetemperature dependence of the two parameters of this model isshown in Figure 5.

Arrhenius parameters ofk(E0), the rate constant at the centerof the distribution, are also listed in Table 1. At temperaturesfar below theTg, theσ value is independent of the polymer. Attemperatures near and above theTg, σ decreases more rapidlyfor PIBMA, the polymer with the lowestTg. On the other hand,σ remains almost constant for PMMA, whoseTg is about 30°C higher than the highest studied temperature. A remarkablefeature of these results is the fact that the value ofk(E0) at agiven temperature is independent of the polymer and verysimilar to the first-order decay constant of the same spirooxazinemeasured in nonpolar solvents.2 According to the descriptiongiven by the Gaussian distribution model, the influence of thematrix would be given exclusively through the existence of adistribution of activation energies. At low temperatures, where

the value ofσ is independent of the polymer, all decays arevery similar in the three polymers. As the temperature increases,the distribution for PIBMA becomes narrower more rapidly andthe decay becomes accordingly faster than in the other polymers.

As outlined in the Experimental Section, we repeated withthis new experimental setup previous measurements made onSP and informed in reference.17 Results show that there was atemperature shift of around 4-6 °C in the published results dueto the fact that temperature was not measured directly on thesurface of the film. This temperature shift does not invalidatethe discussion and conclusions of the paper, but in order toadequately compare results obtained with SP and SO, we presenthere the corrected results in Table 2. We also present in thispaper the kinetic analysis made with the model of eq 3 on thekinetic data of the decoloration of the MC form of SP, whichwas not included in the previous work.

The temperature dependence of the two parameters of theGaussian distribution model for SP is shown in Figure 6. Thebehavior ofσ with temperature is very similar to that obtainedwith SO, except that values are always smaller under the sameconditions. On the other hand, a different behavior was foundfor k(E0), since these values depend on the nature of the polymerfor SP.

Thermochromism. Samples of SO in PEMA and PMMAshow thermochromism; that is, there is a measurable concentra-tion of MC in thermal equilibrium with SO. Figure 7 showsthe dark spectra of the SO-MC system in PMMA at differenttemperatures. The equilibrium constant was calculated at each

Figure 4. Arrhenius plots for the kinetic parameters of eq 6 for thedecay of MC in (a) PMMA, (b) PEMA, and (c) PIBMA: (b) k0; (O)k∞; ([) 1/τm.

TABLE 1: Kinetic Parameters for the Decay of the MCForm of SO in the Different Polymers

k (E0)a k0b k∞

b 1/τmb

PIBMAEa (kJ/mol) 64( 4 51( 7 98( 10 53( 7log Ac 10.5( 0.6 8.6( 0.8 15( 1 9 ( 1

PEMAEa (kJ/mol) 65( 3 62( 6 93( 9 77( 5log A 10.6( 0.5 10.2( 0.7 15( 1 12( 1

PMMAEa (kJ/mol) 66( 4 73( 8 90( 9 73( 6log A 10.8( 0.7 12( 1 14( 2 12( 1

toluened

Ea (kJ/mol) 66.5log A 10.8

ethanold

Ea (kJ/mol) 65.6logA 12.0

a Model of eq 3.b Model of eq 6.c A in s-1. d Decays are monoex-ponential. Values taken from ref 2.

Figure 5. Top: Arrhenius plot fork(E0) of eq 3 for the decay of theMC form of SO in (O) PMMA; (b) PEMA; (]) PIBMA. Full anddotted lines correspond to the Arrhenius parameters of the unimolecularrate constant in toluene and ethanol, respectively (data taken from ref2). Bottom: Dispersion parameter,σ/k (k is Boltzmann’s constant),from eq 3 for the decay of the MC form of SO in (O) PMMA; (b)PEMA; (]) PIBMA.

Photochromism and Thermochromism J. Phys. Chem. B, Vol. 103, No. 11, 19991867

temperature from the dark spectra as outlined in the Experi-mental Section. Results are shown in Figure 8. The logarithmicplots ofKeq vs 1/T are highly curved, whereas in solution theyare linear.2

The curvature of the plots in Figure 8 could be interpretedas a temperature dependence of the energy difference betweenMC and SO. Under this assumption, the appointed curvature isconsistent with a difference in the heat capacities of SO andMC (∆Cp) of about 14R, which is very unlikely for thisreaction.

For both polymers, the slope of the plot of Figure 8 at thehighest measured temperatures is similar to the slope of thecorresponding plots in solution. These facts suggest that thematrix is responsible for the deviations in the temperaturedependence ofKeq, as it will be discussed in the next section.

Discussion

Photochromism. Regardless of the kinetic model used tointerpret the results, it is observed that under the same conditionsdeviations from first-order kinetics are more pronounced forSO than for SP. This may be related to the fact that decays arefaster for SO, causing the process to experience a lessequilibrated environment.

The relaxation model was adequate to describe the decay nearand above 45-50 °C, when the real distribution of relaxation

TABLE 2: Kinetic Parameters for the Decay of the MCForm of SP in the Different Polymers

k (E0)a k0b k∞

b 1/τmb

PIBMAEa (kJ/mol) 69( 4 66( 8 120( 11 41( 6log Ac 9.7( 0.7 9( 1 17( 2 5 ( 1

PEMAEa (kJ/mol) 81( 3 69( 4 128( 9 103( 8log A 11.4( 0.4 9.6( 0.6 18( 1 11( 1

PMMAEa (kJ/mol) 80( 2 79( 3 104( 5 74( 15log A 11.2( 0.4 11.2( 0.5 14.4( 0.7 10( 2

benzened

Ea (kJ/mol) 70.6logA 11.2

acetoned

Ea (kJ/mol) 101logA 15.6

a Model of eq 3.b Model of eq 6.c A in s-1. d Decays are mono-ecponential. Values taken from ref 25.

Figure 6. Same caption as Figure 5 for the decay of the MC form ofSP. In the top graph, full and dotted lines correspond to the Arrheniusparameters of the unimolecular rate constant in benzene and acetone,respectively (data taken from ref 25).

Figure 7. Spectra of films of SOh MC in PMMA. Temperatures are6.8, 13.2, 25.7, 36.6, 44.4, 51.7, 55.3, 61.3, and 75.3°C. The arrowsindicate the direction of absorbance change with increasing temperature.

Figure 8. Logarithmic plot of the equilibrium constant of SOh MCin (a) PMMA and (c) PEMA: ([) experimental values; (O) fitted valuesaccording to eq 9. Also shown are the temperature dependence of theGaussian energy dispersion for (b) PMMA and (d) PEMA: (O)thermodynamic values from eq 9; (b) kinetic data from eq 3.

1868 J. Phys. Chem. B, Vol. 103, No. 11, 1999 Levitus and Aramendı´a

times can be satisfactorily replaced by mean values. At lowtemperatures, the probe decays in a frozen environment and themodel of eq 3 gives a more reasonable picture through thedistribution of sites.

The mean relaxation time obtained with the relaxation modelis an average in the observation time window of the MC decay,and it is consequently lower for SO than for SP under the sameconditions. Activation energies forτm are very similar for thetwo compounds, suggesting that the decays are regulated bythe same type of matrix relaxation in both cases. These activationenergies are also very similar to the activation energy for theârelaxation (70-90 kJ/mol),23 which involves the rotation of theester group, although this is not enough evidence to state thatthe â relaxation is the one relevant in this case. Nevertheless,the fact that there are no discontinuities around theTg for anyof the kinetic parameters, and the fact that energy differencesfor the glass transition (R relaxation) are much larger (180 kJ/mol for PIBMA, 385 kJ/mol for PEMA, and 418 kJ/mol forPMMA)23 than the activation energies ofτm, suggest that thisparticular reaction is not influenced by the large cooperativemovements occurring during the glass transition, but probablyby relaxations involving local movements. Another piece ofevidence supporting this hypothesis is the fact that the temper-ature dependence ofτm aboveTg is not well represented by theWLF equation.24 Neither does this equation describe thebehavior of any other kinetic parameter of the systems studiedin this work. This has been previously reported when spirocompounds were noncovalently incorporated into the matrix.12

In the analysis of the SO results with the model of distributionof activation energies (eq 3) is very remarkable the fact thatk(E0) is independent of the nature of the polymer and also verysimilar to the value measured in toluene, where decays aremonoexponential. Studies performed in liquid solution show thatthe influence of solvent polarity is purely entropic.2 As can beseen from the data of Table 1, the activation energy is the samein ethanol and toluene. Thus, the difference in the MC decayrate constant arises in the preexponential factor. Activationenergies, and consequently activation enthalpies (∆Hq), arealmost independent of solvent polarity, predicting Arrheniusplots for solvents of different polarity to be parallel to each other.The polarity dependence of the entropic factor is related to thereorganization of the solvent around the transition state.Although the polymer molecules have polar groups, the verylow chain mobility does not allow the matrix to solvate a probeas if it were a fluid solvent, giving the same∆Sq (and then thesame intercept in Arrhenius plots) as a nonpolar solvent.

On the other hand, the polarity dependence for SP is bothentropic and enthalpic,25 as can be seen in Table 2. In this case,both ∆Hq and ∆Sq increase with increasing solvent polarity.The activation energy in PIBMA is very similar to that measuredin benzene and most nonpolar solvents, while the activationenergies in PEMA and PMMA are intermediate between valuesmeasured in benzene and acetone. The intercepts of theArrhenius plots, on the other hand, are all similar to the valuesmeasured in nonpolar solvents, as found for SO.

Thermochromism. As outlined above, the deviations in thetemperature dependence ofKeq cannot be explained in terms ofa reasonable change in the enthalpy of reaction with temperature.To qualitatively and quantitatively explain this behavior, wepropose a model in which the matrix is responsible for thedeviations by providing levels of lower energy for the isomer-ization.

Similarly to the model of distribution of activation energies,in a microheterogeneous environment the energy levels of SO

and MC can be represented by a distribution. This distributionoriginates a distribution in the internal energy difference of thereaction (∆E), which would be responsible for the deviationsfrom the expectedKeq temperature dependence, in the same wayas the distribution inEa is responsible for the deviations fromthe first-order kinetics. This causesKeq to diminish more slowlywith T than what would be predicted extrapolating from thebehavior at high temperatures.

When the energies of two species in chemical equilibrium,A and B, are measured from the same origin, the equilibriumconstant can be computed from the molecular partition functionsas

Each sum extends over all levels of energy,εj, of each species;gj is the degeneracy of each level, andâ ) (kT)-1. If the totalenergy for each species is assumed to have a normal Gaussiandistribution, the degeneracy can be expressed as

whereE0 represents the center of the distribution andσ is relatedto its width.

Since equilibrium measures only energy differences, therelevant distribution is the difference between the two distribu-tions, which is also normal26 with E0 ) Ea

0 - Eb0 and σ2 )

σA2 + σB

2.Taking this into account,Keq can be expressed as

whereA is related to the entropic factor.Experimental results ofKeq(T) were fitted to eq 9. For the

transformation SOh MC, the difference between∆E and∆Hof the reaction is negligible. The width of the distribution,determined byσ, depends on temperature as it was previouslydiscussed for photochromism. To perform the fits and accordingto the photochromism results, theσ temperature dependencewas assumed to be linear.

With this assumption, the adjustable parameters of eq 9 are theentropic factor,A; the center of the energy distribution,E0; ands1 ands2, which determine the temperature-dependent distribu-tion width. Fitting results are shown in Table 3 and Figure 8.

Figure 8 shows that theσ values obtained from these fits arevery similar to the ones obtained from kinetic measurements

TABLE 3. Equilibrium Parameters for the ReactionSOhMC

A E0 (kJ/mol) s1 (kJ/mol) s2 (J/K mol)

PMMA 54.04 24.5 12.69 29.0PEMA 6.87 21.1 9.80 18.5

Keq )NB

NA

)qB

qA

)

∑j

gB,je-âεB,j

∑j

gA,je-âεA,j

(7)

g(Ei) ) 1

x2πσj

exp[-(Ei - Ei0)2

2σ2 ] i ) A, B (8)

K ) A∫g(EB)e-EB/RT dEB

∫g(EA)e-EA/RT dEA

)

A∫-∞

∞ 1

x2πσexp[-(E - E0)2

2σ2 ] e-E/RT dE (9)

σ(T) ) s1 - s2T (10)

Photochromism and Thermochromism J. Phys. Chem. B, Vol. 103, No. 11, 19991869

using eq 2. We think that the differences between kinetic andthermodynamic results are not relevant. First, because in thethermodynamic fitsσ was forced to have a linear dependencewith T. Second, because of the fact that there are four adjustableparameters in eq 9, and 10-15 experimental points, it makes itpossible to change the value ofσ in a quantity of the order ofthese differences, without losing quality in the fits.

The E0 values are similar to the∆H values measured insolution2 (10.9 kJ/mol in ethanol and 9.6 kJ/mol in toluene).Differences are not discussed for the same reasons outlinedabove.

Conclusions

The two kinetic models applied in this work to explain theresults imply different insights into the kinetic scenario. TheGaussian distribution model offers a rather static picture of eachpolymer site because it assumes that it does not change duringthe decay of the probe. Nevertheless, it provides a good fit ofthe present decay data in the whole temperature range, as italso does in other cases.27 The relaxation model takes intoaccount the possibility of change of the environment during thedecay and provides a value for the relevant relaxation time ofthe matrix. The coincidence of the Arrhenius activation energiesof τm for both probes is a success of the model. On the otherhand, it does not consider the microheterogeneity of solidsamples, and in this respect, it is predicted to fail in the kineticdescription in highly rigid media. An attempt to incorporate arelaxation parameter in a model of distribution renderedcomplicated equations with additional parameters, which didnot improve the quality of the fits (see Appendix of ref 17).Nevertheless, the relaxation model can correctly account forthe kinetics in the intermediate range between a highly rigidpolymer and a very labile, liquidlike environment. The relaxationapproach offers a new insight into the kinetics with simpleequations. Furthermore, a cage relaxation is likely to occur foran isomerization reaction such as the ring opening of SO, whichinvolves a great molecular form and dipole moment change. Ina study of second harmonic generation of SO-doped polymerfilms, the authors postulate a polymer reorganization aroundthe MC to explain their results.28

The thermochromism of spirooxazines can be accuratelydescribed taking into account a distribution of the energy ofeach probe. The solid environment is responsible for thisdistribution. The standard deviation of the energy distributionderived from the kinetic and from the equilibrium data is similar.The application of a distribution model for the equilibrium anda relaxation view of the kinetics is not incompatible because,probably, the actual situation for the kinetics is that of a time-dependent distribution of reaction sites, but this information isquite difficult to extract from kinetic data only. The two modelsapplied in this paper provide complementary information.

Acknowledgment. M.L. is a Research Fellow of Universityof Buenos Aires. P.F.A. is a member of Carrera del Investigador

Cientıfico from Consejo Nacional de Investigaciones Cientı´ficasy Tecnicas (Research Staff from National Research Council,CONICET, from Argentina). Research was supported by grantsfrom UBA (EX-025), CONICET (PIA 6633), and VolkswagenFoundation (Germany).

References and Notes

(1) Chu, N. Y. C. InPhotochromism. Molecules and Systems; Durr,H., Bouas-Laurent, H., Eds.; Elsevier: Amsterdam, 1990; Chapter 10.

(2) Favaro, G.; Masetti, F.; Mazzucato, U.; Ottavi, G.; Allegrini, P.;Malatesta, V.J. Chem. Soc., Faraday Trans.1994, 90, 333-338.

(3) Favaro, G.; Ortica, F.; Malatesta, V.J. Chem. Soc., Faraday Trans.1995, 91, 4099-4103.

(4) Favaro, G.; Masetti, F.; Mazzucato, U.; Ottavi, G.; Romani, A.J.Photochem. Photobiol. A: Chem.1995, 87, 235-241.

(5) Guglielmetti, R. InPhotochromism. Molecules and Systems; Durr,H., Bouas-Laurent, H., Eds.; Elsevier: Amsterdam, 1990; Chapter 8.

(6) Chu, N. Y. C.Can. J. Chem.1983, 61, 300-305.(7) Ernsting, N. P.; Dick, B.; Arthen-Engeland, Th.Pure Appl. Chem.

1990, 62, 1483-1488. Ernsting, N. P.; Arthen-Engeland, Th.J. Phys. Chem.1991, 95, 5502-5509.

(8) Gorner, H.; Atabekyan, L. S.; Chibisov, A. K.Chem. Phys. Lett.1996, 260, 59-64. Chibisov, A. K.; Go¨rner, H.J. Photochem. Photobiol.A: Chem.1997, 105, 261-267. Gorner, H.Chem. Phys.1997, 222, 315-329. Chibisov, A. K.; Go¨rner, H. J. Phys. Chem. A.1997, 101, 4305-4312.

(9) Wilkinson, F.; Worrall, D. R.; Hobley, J.; Janseu, L.; Williams, S.L.; Langley, A. J.; Matousek, P.J. Chem. Soc., Faraday Trans.1996, 92,1331-1336. Wilkinson, F.; Hobley, J.J. Chem. Soc., Faraday Trans.1996,92, 1323-1330.

(10) Pimienta, V.; Lavabre, D.; Levy, G.; Samat, A.; Guglielmetti, R.;Micheau, J. C.J. Phys. Chem.1996, 100, 4485-4490.

(11) Krongauz, V. A. InPhotochromism. Molecules and Systems; Durr,H., Bouas-Laurent, H., Eds.; Elsevier: Amsterdam, 1990; Chapter 21.

(12) Smets, G.AdV. Polym. Sci.1983, 50, 17-44.(13) Kryszewski, M.; Nadolski, B.; North, A. M.; Pethrick, R. A.J.

Chem. Soc., Faraday Trans. 21980, 76, 351-368.(14) Bitecu, J.; Chaput, F.; Boilot, J.-P.J. Phys. Chem.1996, 100, 9024-

9031(15) Richert, R.; Ba¨ssler, M.Chem. Phys. Lett.1985, 116, 302-306.(16) Richert, R.Chem. Phys. Lett.1985, 118, 534-538.(17) Levitus, M.; Talhavini, M.; Negri, R. M.; Atvars, T. D. Z.;

Aramendı´a, P. F.J. Phys. Chem. B1997, 101, 7680-7686.(18) Albery, W. J.; Bartlett, P. N.; Wilde, C. P.; Darwent, J. R.J. Am.

Chem. Soc.1985, 107, 1854-1858.(19) Ware, W. R. InPhotochemistry in Organized and Costrained

Media; Ramamurthy, V., Ed.; VCH Publishers: New York, 1991; Chapter13.

(20) James, D. R.; Ware, W. R.Chem. Phys. Lett.1986, 126, 7-11.(21) Richert, R.J. Phys. Chem. B1997, 101, 6323-6326.(22) Richert, R.; Heuer, A.Macromolecules1997, 30, 4038-4041.(23) Mc. Crum, N. G.; Read, B. E.; Williams, G.Anelastic and Dielectric

Effects in Polymeric Solids; Dover Publications Inc.: New York, 1967;Chapter 8.

(24) Sperling, L. H.Introduction to Physical Polymer Science; JohnWiley & Sons: New York, 1986; p 257.

(25) Sueish, Y.; Ohcho, M.; Nishimura, N.Bull.Chem. Soc. Jpn.1985,58, 2608-2613.

(26) Hald, A. Statistical Theory with Engineering Applications; JohnWiley & sons: New York, 1952; p 214.

(27) Richert, R.Chem. Phys.1988, 122, 455-462.(28) Atassi, Y.; Delaire, J. A.; Nakatani, K.J. Phys. Chem.1995, 99,

16320-16326.

1870 J. Phys. Chem. B, Vol. 103, No. 11, 1999 Levitus and Aramendı´a