J. Phys. Chem. 1988, 92, 3001-3005
3001
1. In polar solvents, such as water, the ground state of PCA is involved in a two-step equilibrium: PCOO-
+ H+
PCOOH
pK = 3.8
The first singlet excited state exhibits similar behavior, but the px”s differ significantly from those corresponding to the ground state: *PCOO330
39 0
*PCOOH
4 50
+ H+ +
H+
*PCOOH
e PC4+
it is not possible to place a meaningful pH to the whole silica surface as local interactions of the PCA with the OH site control the photophysics. Geminal OH Groups. The Matheson Coleman Bell Co. silica, which contains geminal OH groups, exhibits photophysical properties of PCA which indicate that the acid is entirely in the neutral form, Le., an extremely acidic surface. The emission spectrum of PCA on this surface indicates that only *PCOOH2+is present. Transient experiments show that the proton uptake by *PCOOH is very rapid and well short of S.
The emission of *PCOOH2+ at 415 nm decays with a rate constant of 1.9 X lo8 s-l, indicating that quenching of *PCOOH2+ by H+ takes place on the surface. The data indicate that the “microenvironment” of PCA on the surface corresponds to a pH of 1.6 in an aqueous solution. The same caution used in pH studies of vicinal OH also apply here.
Conclusions The photophysics of PCA is strongly affected by medium polarity:
pX=6.1
%OH
WAVELENGTH ( n m )
Figure 8. Emission and excitation spectra of 1 X lo-’ mol/g of PCA adsorbed on silica surfaces. The solid line corresponds to the MCB silica gel/cyclohexane matrix, while the dashed line corresponds to the Fisher silica/cyclohexane matrix.
pK = 5.2
OH
2. In moderately polar media, such as methanol, due to hydrogen bonding the ground-state equilibria are altered compared with those in water: PCOO-
+ H+ + PCOOH
2PCOOH
dimer
Consequently, the excited-state equilibria are
+ H+ t *PCOOH dimer + hv excimer
*PCOO-
-
The excimer formation is controlled by the H+ concentration as the first reaction step is a hydrogen bond cleavage involving
H+. 3. In media of low-polarity PCA exists as a dimer in its ground state. However, no excimer was found on excitation which can be attributed to strong hydrogen bonding. 4. The photophysical properties of PCA respond in an acknowledged way to the environment of a cationic micelle, in particular pH. 5. PCA may be used to distinguish between geminal and vicinal OH groups on silica surfaces. Registry No. PCA, 19694-02-1; PCOO-, 93638-12-1; PCOOH2, 113748-27-9;CTAC, 112-02-7;HCI, 7647-01-0; glycerol, 56-81-5;water, 7732-18-5;methanol, 67-56-1; cyclohexane, 110-82-7.
Dielectric Relaxation with Kinetic Irreversible Transition Stefan Smigasiewiczt Z a k k d Polimerdw Polskiej Akademii Nauk, 41 -800 Zabrze, Poland (Received: March 31, 1987; In Final Form: September 29, 1987)
A model of dielectric relaxation accompanied by an irreversible transformation of relaxing moieties is presented. Calculations show that for a heating with constant rate there are some maxima on the e’ vs T plots. The heights of the peaks decrease when the frequency of electrical field increases. That result is qualitatively consistent with some experimental data published elsewhere. In addition, the model predicts an influence of the heating rate on the heights and positions of the e’( T ) peaks. Unfortunately, that conclusion has not been confirmed yet experimentally.
Introduction Recently, two papers have appeared (ref 1 and 2 ) in which the pronounced maxima of dielectric permittivity 6’ as a function of temperature T have been observed. In the first reference some cured polyimides have been investigated, and in the second reference the polyepichlorohydrin-nitrobenzene system was treated. Present address: Institute of Chemical Engineering, Polish Academy of Sciences, Baltycka 2, 48-100 Gliwice, Poland.
It was shown there that at nearly constant rate heating the heights of the maxima decrease when frequency of the electric field increases. The observed behavior, i.e., appearing of the maxima, have been suggested to be either a kinetic transformation leading to a new conformation of the epichlorohydrin polymer chains2 or a reordering of molecular movements connected with an irre(1) Destruel, P.; Ypon. D. Y . Muter. Sci. 1984, 10, 5 5 . (2) Trzebicka, B.; Smigasiewicz, S.;Turska, E. Polymer 1986, 27,
0022-365418812092-3001$01 .SO10 0 1988 American Chemical Society
1067.
3002 The Journal of Physical Chemistry, Vol. 92, No. 10, 1988 versible transition during the curing of the polyimides. This paper presents a simple model which, besides the above-mentioned experimental behavior, predicts a new dependence of the height and position of the peaks on the rate of heating.
Model Suppose that a system undergoes a kinetic, irreversible transformation from one form characterized by complex polarizability a*]to another form with polarizability a*2. During the transformation a whole polarization, P*, (Le., electric moment per unit volume) is a sum of that for the first and the second forms: P* = P*l + P*2 = Nia*iF*I + N2”*2F*2 (1) where N , and N2 mean the number of polarized units of the two forms per unit volume and F*l and F*2 are local electric fields. Now, the following simplifying assumptions are made: (1) each polarized unit of the first form transforms to one of the second form; ( 2 ) the local electric fields are of Lorentz type for the two forms;3 (3) dipolar orientational polarizabilities are that of the Debye theory of dielectric r e l a ~ a t i o n . ~ From the first assumption it follows that before and after the 03, respectively transformation, Le., for time t = 0 and for t
-
N,(t=O) = N2(t--)
=N
(2)
Smigasiewicz
where eo is the permittivity of free space and e , and e, mean highand low-frequency limiting values of the dielectric permittivity, respectively. After the introduction of eq 5a-6b into eq 4 and rearranging, the e * is given by t*(t,o)
=
1 + 2X,[a, + 61/(1 1 - xl[al + 61/(1
+ jW7,)] + 2 X 2 [ U 2 + 62/(1 + joT2)] + l’o7dI - x2[a2 + W ( 1 + jo72)l
(7)
where
This is the general solution of the kinetic transformation, defined by the function x , ( t ) ,accompanied by the dielectric relaxation. As it is seen, the dielectric permittivity is dependent on both frequency and time. It will be valuable, at first, to consider the simplest case, Le., when the second form does not relax (62 = 0). The introduction of that condition into eq 7 leads to
and for any instant the following holds:
N,(t)
+ N2(t) = N
(3a) where
or
xl(r) + x2(t) = 1 (3b) where x 1 = N I / N and x2=N21N. Now, it is assumed that a relative change of dielectric permittivity (due to the change in the composition) during one period of the electric field oscillations is negligible. When, moreover, the measurements start up a sufficiently long time after the electric field is switched on (Le., when the transient solution for polarization practically vanishes), then steady-state conditions will be fulfilled. For that case, taking assumptions ( 2 ) and (3) and eq 3a,b into account the polarization is P*(t,o) = N
[ (
)+
xI(t) aDI+ lrI2 3kT 1 j o r l
+
The above solution reminds us of the Debye equations but with e,, e,, and 7 (now, the macroscopic relaxation time) all dependent on time, the Debye expressions obtained are for time tending either to zero or to infinity. When one wants to know the exact solution for e* as the function of time or temperature, T, then the kinetic reaction scheme and a heating program must be specified. One of the simple cases of the kinetic transformation is vanishing of x 1 according to the first-order reaction scheme dx1/dt = -K( T ) x ,
where aD,p, and T mean the dispersion polarizability, dipole moment, and microscopic relaxation time, respectively. E*(t) is the time-dependent external electric field which changes with angular frequency o, and e* is the complex dielectric permittivity. Regarding definitions of the polarization (P*= to(€*- l)E*), eq 4 can be, for time t = 0, (before transformation), written in the forms
where K ( T ) is the temperature-dependent reaction rate constant. If K( T ) is assumed to fulfill the Arrhenius law, then for constant rate heating procedure it follows from eq 9 that
and
where Tois the starting temperature, A is the preexponential factor in the Arrhenius formula, Q is the rate of heating, and AH is the enthalpy of activation for the reaction. The solution of eq 10 can be satisfactorily approximated by the Hastings
eo(es,l - 1)
=N
(
aDI
+2 , +- 3
f o r o = 0 (Sb)
Similarly, for time tending to infinity (Le., after transformation)
and (3) Chelkowski, A. Fizyka Dielektrykbw; Panstw Wydawn Nauk: Warsaw, 1913; p 26. (4) Hill, N. E.; Vaughan, W. E.; Price, A. H.; Davies, M. Dielectric Properties and Molecular Behauiour; Van Nostrand: London, 1969: p 60.
(9)
xi(T) = xi(To) ex~(-Af(z)/QR)
(1 1)
where z = AH/RT f(z) =
4 - c1 + (d2 - C d / Z z2
+ d l z + d2
exP(-z)
(5) Hastings, E. Approximarion for Digiral Computers; Princeton University Press: Princeton, NJ, 1955; p 188. (6) van Turnhout, J. Thermally Stimulated Discharge of Polymer Electrets; Elsevier: Amsterdam, 1975; p 34.
Dielectric Relaxation with Irreversible Transition ts 6
I
-----A-
I
I
w)
T,r
Figure 1. Frequency dependence of e'-T relations at constant rate heating, Q = 0.5 Kmin-', for different frequencies: (1) 5, (2) 10, and (3) 100 Hz.
Figure 3. Dependence of tan 6-T relations on frequency, at Q = 0.5 K-min-I: (1) 5, (2) 10, and (3) 100 Hz.
45
40
91
30
1
I
I
#a ba Figure 2. Heating rate dependence of e'-T relations at constant frequency,f= 5 Hz, for Q = (1) 0.2, (2) 0.5, and (3) 1 K-mi&. PI0
200
= 2.334733, ~2 = 0.250621, dl = 3.330657, d2 = 1.681 534, and x,(To)is the starting value of xl. When the relaxation time is assumed to follow the Arrhenius formula ~1
dT) =
70
exp(-W/RT)
(12)
where T~ and AH, are the preexponential factor and enthalpy change for the relaxation process, respectively, then the introduction of that expression together with eq 11 into eq 8 gives the temperature dependence of e* for the constant rate heating experiment. In the numerical calculations of the dielectric permittivity (eq 8) the condition x l ( To) = 1 was used. The values of the remaining parameters used in the calculations were the following: AH,,, = A H = 15 kcal/mol, A = 3 X 1Olo s-l, T~ = 1.1 X s, = 6, and em,' = = 2.8. In Figure 1 the influence of frequency on the real part of the permittivity E'( T ) is shown (the rate of heating is equal to 0.5 Kamin-I). It is seen that the peak height decreases when the frequency increases and only a small shift along the T axis is observed. The similar dependences have been found in cured polyimides (ref 1; Figure 3) and polyepichlorohydrin-nitrobenzene system (ref 2; Figure
Figure 4. Heating rate dependence of tan 6-T relations atf = 5 Hz; (1) 0.2, (2) 0.5, and (3) 1 Kernin-'.
4). Figure 2 visualizes the influence of the heating rate on the 6'-T dependences. The displacement of the peaks to higher
temperatures and the increase of their heights with increase of the Q values is clearly seen. It seems to the author that experimental data on the last influences have not been reported yet. It is worthwhile to note here that the results visualized in Figure 2 are similar to the Q dependences of current-temperature characteristics in the theory of thermally stimulated currents.6 To complete the case of b2 = 0, the results of calculations of the dielectric loss tangent tan 6 ( T ) for different frequencies (Figure 3) and for different heating rates (Figure 4) are presented. Finally, the case b2 # 0 was examined. It was assumed that T2 >> 7 1 and AH,,2 = AH,,' = AH = 15 kcal/mol, T ~= , 1 ~X s, 6s,2. = 5, and other parameters were left unchanged, i.e., as before. Figure 5 presents the e ' ( T ) curves obtained from eq 7 for three frequencies. The nonperturbed second relaxation in the high-temperature region is seen. The picture reminds us, again, of the results of dielectric measurements included in ref 1 and 2. Figure 6 and 7 show that the above examples satisfactorily fulfill the steady-state condition requirements. As it is seen from Figure 6, the relative change of the dielectric permittivity during one
3004 The Journal of Physical Chemistry, Vol. 92, No. IO, I988
Smigasiewicz
E'
45
-2
-l-Q4 5 2-0- it' d n
40
I
t -
-1
-8 30
Figure 5. Dependence of ~ ' ( 7 ' on ) temperature according to eq 7, at Q = 0.5 Kemin-', for different frequencies: (1) 5, (2) 10, and (3) 100 Hz.
'2
--
Figure 7. Change of composition during one period of electric field oscillations (the same data as for Figure 6).
is
25
1
m
Figure 6. Relative change of dielectric permittivity during one period of electric field oscillations, based on data of Figure 1 (curve 2).
period of the electric field oscillations is for the frequencyf= 10 H z and for the heating rate Q = 0.5 K - m i d less than 3 X and at the same time the change of composition in the whole range of temperatures is less than 4 X (for Q = 0.5 Kqmin-I). In the example below those quantities are lower, and for frequency lo5 H z they are less than 6 X and 2 X for t' and xl, respectively.
Comparison with Experiment In the cited paper by Destruel and Yoon,' the real part of the dielectric permittivity of cured polyimides was studied. The polymers were obtained by the thermal cyclization reaction of an appropriate polyamic acid. To explain the appearing of the maximum at about 305 O C on the E'-T curve during the constant rate heating experiment, we propose the following mechanism. The polyimide chains contain some amount of free dipoles -COOH and =NH. When the temperature increases, especially above Tg,the glass-rubber transition temperature, the amplitude of segmental motion increases and t' strongly rises. At the same time the rate of the cyclization or even network-forming reaction (reaction between -COOH and =NH dipoles of the neighboring chains) rapidly develops. Due to that reaction the dipole moment of the moving segments decreases. The new imide bonds are formed, the mobility of which is lower, and then they do not contribute to the dielectric permittivity. When the reaction is completed, the dielectric permittivity drops to the value determined only by atomic and electronic polarizations. The polyimide so formed has lower mobility (higher relaxation time) and another
I 259
T,%
Figure 8. Comparison between experimental data of ref 1 (solid lines) and model calculation results (broken lines): 1 and l', lo5 Hz;2 and 2', lo6 Hz (based on parameters used; see text).
orientational (dipolar) polarization. The thermal cyclization reaction of the polyamic acid was studied first by Kreuz et al.' It was found to be a two-step reaction (assumming that each step is a first-order reaction) with enthalpy of activation equal to 26 f 3 kcal/mol and 23 f 7 kcal/mol for the first step and the second step, respectively. To describe the temperature dependence of the relaxation time, we have chosen, due to lack of data on a relaxation of the polyamic acid, the Williams-Landel-Ferry (WLF) equation? ,with so-called 'universal" constants. Hence, in the simulation procedure we have varied only two parameters, namely, the preexponential factor A in the reaction rate constant expression (see eq 9 and 10) and the glass-rubber transition temperature Tgin the W L F equation for 7,: r1=
(1 X 10-l6)exp(4300/1.98(T- T,)
(13)
where T, = Tg ("C)- 52 ("C).As it is seen from Figure 3 of the paper by Destruel and Yoon,' it is reasonable to assume that e,,' = = 2.8. The value of e , ' = 7 was chosen arbitraly because there are no data on the amount and the configurational arrangement of the -COOH and =NH dipoles in the cured po(7) Kreuz, J. H.; Endrey, A. L.; Gay, F. p.; Sroog, C . E. J . Polym. Sci., Part A-1 1966, 4, 2607. ( 8 ) McCrum, N. G.; Read, B. E.; Williams, G. Anelastic and Dielectric Effects in Polymeric Solids; Wiley: London, 1967; p 169.
J. Phys. Chem. 1988, 92, 3005-3007 lyamic acid. With parameters AH = 26 000 cal/mol, A = 5 X lo6 s-l, and Tg = 524 K, the theoretical calculations of the e'-T dependence were carried out. The results are plotted in Figure 8 (broken lines) together with the experimental plots of Destruel and Yoon, for two frequencies: lo5 and lo6 Hz. The agreement is rather qualitative, but the difference between the results is easy to explain. The tail part in the low-temperature region of the experimental plots is due to the contribution of another process, visible on the d-T plot for lo4 H z (Figure 3, ref l), and due to the fact that the a relaxation in the polymer is most likely
3005
characterized by a very wide relaxation time distribution function.
Conclusions The model presented above, in spite of its rather simplifying assumptions, seems to explain qualitatively the origin of e'( T) maxima observed in recent experimental data. In addition, it describes the time dependence of e* and foresees the heating rate dependence of both positions and heights of the peaks. There are some possibilities to modify the model by changing the kinetic scheme and/or the temperature dependence of the relaxation time.
An ab Initio Quantum Chemical Investigation on the Effect of the Magnitude of the T-0-T Angle on the Bransted Acid Characteristics of Zeolites PPdraig J. O'Malley* and J. Dwyer* Department of Chemistry, The University of Manchester Institute of Science and Technology, Manchester M60 1 QD, United Kingdom (Received: June 15, 1987; In Final Form: December 2, 1987)
Ab initio molecular orbital calculations are used to monitor the effect of increasing SiOAl angle (LSiOAl) on the acidic properties of bridged hydroxyl groups in zeolites. A 3-21G basis set is employed in the calculations, and the bridged hydroxyl group is modeled via an aluminosiloxaneunit. Increasing LsiOAl is shown to give rise to a decrease in the stretching frequency of the OH group (pOH) with a corresponding decrease in the ionicity of the OH bond. The decrease in ionicity of the OH bond indicates that increasing CSiOAl gives rise to a decrease in acidity. The decreased strength of the OH bond as evidenced by the decreased poH value suggests that the OH bond is unstable when situated at a high LSiOAl link with the decreased polarity of the bond tending to favor homolytic dissociation over heterolytic dissociation. It is also shown that &OH exhibits decreased flexibility with increasing LSiOAl, suggesting that the SiOH bending frequency should be greater for zeolites with higher TOT angles.
Introduction Many of the catalytic properties of zeolites can be directly related to Br~rnstedacidity. Experimentally, it has been demonstrated that the acidity of zeolites varies with structure and composition.' Recently, quantum chemical calculations (both semiempirical and a b initio) have been shown to give a good account of the variations in acidity brought about by compositional properties such as the Si/Al ratio? increased electronegativity,' or isomorphous s~bstitution.~The investigation of the effect of structural characteristics, such as variation of the TOT angle on the acidic properties, has not received the same attention, h ~ w e v e r . ~ This is surprising as a striking difference between low-silica zeolites such as faujasite and the novel high-silica forms such as ZSM-5 is the larger T O T angle of the high-silica forms.6 In this report the effect of increasing TOT angles on the acidic properties of the bridging OH group is studied. Increasing the TOT angle is shown to give rise to a decreasing strength of the bridging OH bond with a concomitant decrease in the ionicity of the bond. The potential energy curve for the SiOH bond angle indicates that increasing the TOT angle also leads to an increase in the SiOH bending force constant.
Models and Methods Units used to model the zeolite framework are generally determined by balancing computer cost with the complexity of the (1) Dwyer, J. Chem. Znd. 1984, 258. (2) Kazansky, V. E. In Structure and Reactivity of Modified Zeolites;
Jacob, P. A., et al., Eds.; Elsevier: Amsterdam, 1984; p 61. (3) Datka, J.; Geerling, P.; Mortier, W.; Jacobs, P. J . Phys. Chem. 1985, 89, 3483. (4) O'Malley, P. J.; Dwyer, J. J . Chem. Soc., Chem. Commun. 1987, 72. ( 5 ) (a) Senchenya, I. N.; Kazansky, V. B.; Beran, S . J . Phys. Chem. 1986, 90,4857. (b) Zhidomirov, G. M.; Kazansky, V. B. Adv. Catal. 1986,34, 131. (6) Olson, D. H.;Kolcotailo, G. T.; Lawton, S.L.; Meier, W. N. J. Phys. Chem. 1981,85, 2238.
TABLE I: Effect of Variation of the SiOAl Angle on the Si0 and AI0 Bond Length and the SiOH Angle rAo/A LSiOH, deg LSiOA1, deg rsio/b, 127 1.73 1.93 120.4 1.73 1.93 115.2 140 150 1.73 1.95 111.7 160 1.74 1.97 107.9 170 1.74 2.00 104.2 TABLE II: Effect of Variation in the SiOAl Angle on the Acidic Characteristics of the Bridging OH Group
LSiOA1, deg
rc,OH/A
vOH/cm-'
127 140 150 160 170
0.966 0.970 0.974 0.978 0.981
3931 3870 3828 3773 3717
qH 0.472 0.465 0.456 0.445 0.432
k H
401
0.442 0.431 0.418 0.404 0.386
calculation available. Many studies of zeolite properties have been performed using semiempirical methods such as CNDO, INDO, etc. Up to the present time most ab initio calculations have been performed with the STO-3G basis set. While the STO-3G basis set has been successful in describing the bonding characteristics of many molecules, it has been found' that splitting the valence shell region into a single and double Gaussian contraction (giving rise to a 3-21G basis set) improves the flexibility of the valence shell region and hence allows more adaptability to various bonding situations in molecules.' The 3-21G basis set was chosen therefore for the present study. The SiOAl fragment was modeled via the aluminosiloxane unit of Figure 1. This unit allows extensive geometry optimization to be performed at a reasonable computer cost, and the unit has been previously shown to give a good account ~
~
(7) Gordon, M. S.;Binkley, J. S.;Pople, J. A,; Pietro, W. J.; Hehre, W. J. J . Am. Chem. Soc. 1982, 104, 2797.
0 1988 American Chemical Society
