4550
J. Phys. Chem. 1986, 90, 4550-4554
LASER CHEMISTRY, MOLECULAR DYNAMICS, AND ENERGY TRANSFER Relaxation in Molecular Infrared Absorption. 1. A Rate Law for I R Absorption'aib Ernest Grunwald* and Shu-Huei Lidc Department of Chemistry, Brandeis University, Waltham. Massachusetts 02254 (Received: August 21, 1985; In Final Form: March 26, 1986)
The absorption'of infrared laser radiation by molecules in the gas phase is known to depend strongly on rotational and vibrational relaxation. In order to make a kinetic analysis of such relaxation, one needs a rate law for infrared absorption in which the relaxation time is practically the only unknown. Such a rate law is developed here for conditions that are readily realized experimentally. The result is tried and tested with a precision of 5% by using data for CHClF2 at a pressure of 2.5 Torr and at IR pulse-average intensities ranging from 28 to 480 kW/cm2. Over a significant range of intensities the rate of infrared absorption is nearly proportional to Z1j2.
It is well-known from theory and model calculations that rotational and vibrational relaxation play key roles in molecular absorption from infrared lasers.*-' It follows, therefore, that infrared absorption can provide a method for measuring such relaxation. A method based on infrared absorption might be of interest not only because the required equipment is relatively simple and available, but also because the kinetic assumptions made in the model calculations might gain direct support. This study attempts a kinetic analysis of relaxation during IR absorption in the gas phase. The substrates are CHCIFz and CHC1,F at a series of pressures above 1.5 Torr. Conditions are relatively mild-infrared fluences range from 0.001 to 0.8 J/cm2. Under these conditions, molecular energies remain in the domain of conventional vibrational and rotational spectroscopy and are well below levels at which high-energy phenomena such as mode-mixing, vibrational quasi-continua, quantum ergodicity, and chaos, are dominants8 The present part 1 develops a suitable rate law for IR absorption. Although there are several parameters, the only real and significant unknown is the desired relaxation rate, which can thus be obtained with useful accuracy. Part 2 reports results for IR absorption and relaxation of CHCIFz at a single wavenumber. The results are consistent with a relaxation .'~ mechanism consisting both of direct rotational r e l a x a t i ~ n , ~and (1) (a) Work supported in part by grants from the National Science Foundation and the Edith C. Blum Foundation. (b) Abstracted from the Ph.D. Thesis of Shu-Huei Liu, Brandeis University, 1983; Diss. Abstr. Int., B 1983, 44, 1138. (c) Gillette Fellow, 1979-82. (2) Ambartzumian, R. V.; Letokhov, V. S. Acc. Chem. Res. 1977,10, 61. ( 3 ) (a) Black, J. G.; Yablonovitch, E.; Bloembergen, N.; Mukamel, S. Phys. Rev. Lett. 1977, 38, 1131. (b) Black, J. G.; Kolodner, P.; Shultz, M. J.; Yablonovitch, E.; Bloembergen, N. Phys. Reu. A 1979, 19, 704. (4) (a) Thiele, E.; Goodman, M. F.; Stone, J. Opt. Eng. 1980, 19, 10. (b) McDonald, J. D. Annu. Rev. Phys. Chem. 1979, 30, 29. (c) Kay, K. G. J . Chem. Phys. 1981, 7 5 , 1690. (5) (a) Starov, V.; Steel, C . ; Harrison, R. G. J . Chem. Phys. 1979, 71, 3304. (b) Starov, V.; Selamoglu, N.; Steel, C. J . Am. Chem. Soc. 1981, 103, 7276. (6) (a) Judd, 0. P. J . Chem. Phys. 1979, 71, 4515. (b) Golden, D. M.; Rossi, M. J.; Baldwin, A. C.; Barker, J. R. Acc. Chem. Res. 1981, 14, 56. (7) Duignan, M. T.; Garcia, D.; Grunwald, E. J . A m . Chem. SOC.1981, 103, 7281. (8) (a) Dai, H. L.; Korpa, C. L.; Kinsey, J. L.; Field, R. W. J . Chem. Phys. 1985,82, 1688. (b) Hudgens, J. W.; McDonald, J. D. Ibid. 1981, 74, 1510; 1982, 76, 173. (c) Dai, H. L.; Field, R. W.; Kinsey, J. L. Ibid. 1985, 82,2161. (d) Sundberg, R. L.; Abramson, E.; Kinsey, J. L.; Field, R. W. Ibid. 1985, 83, 466. (9) (a) McGurk, J. C . ;Schmalz, T. G.; Flygare, W. H. Ado. Chem. Phys. 1974, 25, I . (b) Schwendeman, R. H. Annu. Rev. Phys. Chem. 1978, 29, 537.
of resonant V-V whose rate is nearly proportional to the instantaneous vibrational energy. Part 3 reports results for CHC12F a t several wavenumbers and validates the present methods within statistical precision limits.
Advantage of a Kinetic Approach In our experiments, IR absporption takes place approximately under photophysical steady-state conditions. ("Slow passage" in the language of NMR.13 See Test section below.) Transient optical phenomena,14 which depend on excitation with coherent radiation, can be neglected. Under such conditions the optical Bloch equations14 become equivalent to the rate equations of ordinary chemical kinetics, provided that the relaxation mechanism is a simple transfer of energy between the absorbing system and an undifferentiated "heat bath", as implied by Bloch's T , and T2. If the relaxation mechanism is more complicated, the kinetic equations (but not the Bloch equations) can be simply expanded. The coherence timeI5 of the radiation used in our experiments is not well-defined, but its magnitude is comparable to that of the desired relaxation times. This is not a problem, however, because in the photophysical steady state the rate of absorption depends on the IR intensity, which can be measured without reference to coherence. (10) (a) Mader, H.; Lalowski, W.; Schwarz, R. 2.Naturforsch. A 1979, 34, 1181. (b) Orr, B. J.; Haub, J. G.; Nutt, G . F.; Steward, J. L.; Vozzo, 0. Chem. Phys. Lett. 1981, 78, 621. (c) Rordorf, B. F.; Knight, A. E. W.; Parmenter, C. S. Chem. Phys. 1978,27, 11. (d) Schrepp, W.; Bestmann, G.; Dreizler, H. Z . Naturforsch. A 1979, 34, 1467. (e) Bomsdorf, H.; Dreizler, H. Z . Naturforsch. A 1981, 36,473. (f') Bischel, W. K.; Kelly, P. J.; Rhodes, C. K. Phys. Rev. A 1976, 13, 1829. (9) Bridges, T. J.; Haus, H. A,; Hoff, P. W. Appl. Phys. Lett. 1968, 13, 316. (h) Casleton, K. H.; Chien, K.-R.; Foreman, P. B.; Kukolich, S.G. Chem. Phys. Lett. 1975, 36, 308. (i) Williams, J. R.; Kukolich, S. G. Chem. Phys. 1979, 36, 201. 6 ) Tanaka, K.; Hirota, E. J . Mol. Spectrosc. 1976,59,286. (k) Roodhart, L. P.; Wegdam, G. H. Chem. Phys. Lett. 1979, 61, 449. (1) Vaccaro, P. H.; Redington, R. L.; Schmidt, J.; Kinsey, J. L.; Field, R. W. J . Chem. Phys. 1985, 82, 5755. (1 1) Mkrtchyan, M. M.; Platonenko, V. T. Sou. J . Quantum Electron (Engl. Trans/.) 1978, 8, 1187. (12) Schwartz, R. N.; Slavsky, 2. I.; Herzfeld, K. F. J. Chem. Phys. 1952, 20, 1591; especially eq 19, p 1595. (13) Pople, J. A,; Schneider, W. G.; Bernstein, H. J. High Resolution Nuclear Magnetic Resonance; McGraw-Hill: New York, 1959; pp 30-39. (14) (a) Allen, L.; Eberly, J. H. Optical Resonance and Two-Leuel Atoms; Wiley-Interscience: New York, 1975. (b) Pantell, R. H.; Puthoff, H. E.; Fundamentals of Quantum Electronics; Wiley: New York, 1969. (15) Loudon, R. The Quantum Theory ofLight; Clarendon: Oxford, U. K., 1973.
0022-3654/86/2090-4550$0 1.5010 0 1986 American Chemical Society
Relaxation in Molecular Infrared Absorption
The Journal of Physical Chemistry, Vol. 90, No. 19, I986 4551
’AS’
A+S‘
’S’A I
I1
I v +=+T/RI Figure 1. Energy flow pattern. Direct exchange between (S’,S’’) and V
(dashed line) is relatively negligible. It is reasonable to infer that, in our experiments, IR absorption takes place essentially under steady-state conditions. If we neglect the mode beats in the pulse profiles, the correlation times for changes of intensity during the pulses are at least an order of magnitude longer than the desired relaxation times. The mode beats, on the other hand, are fast, and model calculations showib that their perturbation of the smooth IR pulse profiles has tolerably little effect on the overall absorption.
Energy Flow Pattern In order to account for relaxation and harvesting of absorbed energy in our experiments with CHCIF, and C H Q F , one needs to divide the “heat bath” into at least three subsystems, each with its own time-dependent temperature. These subsystems are (1) the absorbing vibrational mode, with temperature TM; (2) the nonabsorbing vibrational modes, with temperature Tv; (3) rotation and translation, with temperature TT/R. This three-part division is approximate but seems to be adequate. By comparing the time scales for relaxation reported in parts 2 and 3 with known results for V-V’, T/R relaxation under comparable conditionsi6 one arrives at the approximate energy flow pattern shown in Figure 1. The dominant relaxation process occurs within the absorbing mode, by direct rotational relaxation, and by kinetically parallel resonant V-V relaxation, the latter in any mode, including the absorbing mode. Resonant V-V relaxation can transfer a molecule from any state S’ or S” to a state in the “A” reservoir (Figure 1) if accompanied by a change of rotational state and/or an anharmonic frequency shift. The rate of absorption is sensitive to the temperature TM, not only because it defines the concentrations of vibrationally excited A molecules, but also (as will be shown) because it affects the ratios of [S”]/[S’] in the photophysical steady state. T T p determines the rotational distribution and hence the band shape of the overall absorption band. Tv will be shown to determine much of the rate of resonant V-V relaxation.
1
(B)
Figure 2. (a) Kinetic mechanism for IR absorption and relaxation within the absorbing mode. (b) Equivalent mechanism after transformation of variables. See text.
experiments. Thus, when a laser-active subspecies relaxes to a different rotational state, it will be assumed that the new subspecies is in the A reservoir. The rate law for I R absorption and relaxation within the absorbing mode will be obtained in the following steps: (1) I R absorption by a single laser-active two-state system; (2) mathematical transformation to separate IR absorption from relaxation; (3) line-shape function; (4) total absorption, by summing over all subspecies; (5) replacement of unknown parameters. (1) The formal kinetic scheme for two-state absorption/emission of S’ and S” is shown in Figure 2a. The j’s denote intrinsic rates (pseudo-first-order rate constants, reciprocals of lifetimes) for the respective steps. We use the symbol j (rather than the more standard symbol k or 1/7) to stress that the j’s are formal quantities and still need to be related to measurable physical S‘ SI’ variables. The rates for forward cycling (u+: A A) and for reverse cycling (u-: A S” S’ A) in the steady state are obtained as functions of the j’s by Christiansen’s method,18 as described in textbooks.I9 (2) While the steps A + S’ and A + S” in Figure 2a are entirely relaxation, the step S‘ * S” is a sum of parallel processes of absorption/emission (intrinsic rate j,r) and spontaneous or T, relaxation (l/Tup, 1/TdOwn), according to eq 1. The net rate w
- - -- - -
~S,V =
1/Tup+ j,l;
~ S J , S !=
1 / Tdown
+ jJ
(1 )
of I R absorption therefore is not simply proportional to u+ - ubut needs to be corrected, as in eq 2. w / [A] denotes the net rate
w / [AI = NOhvL (u+ - u- - [st]/Tup+
[s”] / Tdown)/ [AI
(2)
of energy absorption per mole of absorbing substrate at the infrared laser frequency vL. To express the absorption band shapes of the rotational subspecies (see below) and analyze the relaxation mechanism, it is desirable to gather up all relaxation from S’ and S” in single (pseudo-first-order) rate constants k’and k ” according to eq 3.
Kinetics of Relaxation within the Absorbing Mode It is convenient to examine the steps in the mechanism of Figure 1 individually. In this section we shall consider the rate law for IR absorption and relaxation within the absorbing mode, ignoring for the time being the V and T / R modes. The absorption then is formally equivalent to that of a set of two-state systems relaxing into a nonabsorbing heat bath. However, the equivalence is formal rather than physical. Lorentzian absorption bands” have wide wings, and in principle each molecule absorbs at least a tiny amount. The only subdivision within the absorbing mode which makes physical sense is based on whether the absorbing two-state system is optically saturated. If the ratio of [S’’]/[S’] agrees with a Boltzmann factor based on TM, the pair of subspecies is assigned to the A reservoir. If there is significant saturation, Le., the ratio exceeds that for T M by a specified amount, then the pair is called “laser-active”. A reasonable criterion for laser activity will be given in eq 20. By this criterion, the laser-active mole fraction is small, -0.007 in a typical experiment, and 4 are quite negligible. In practice, terms with m > 4 may therefore be omitted from (15). Thus anharmonicity need be introduced only in first order of approximation. Accordingly, the m-dependent parameters will be evaluated as follows. vm,m+l
rm,m+l
= v0.1 + m6 = (m
+ 1)r
[I‘ by integration at room temperature according to (13).] aA/m,m+l(vL)
In addition, T,,,+~ will be replaced by a common average value T . The final result is eq 17.
As far as we known, eq 14 and 17, in which the relaxation time in principle is the only unknown, are new. An experimental test will be given later in this paper. T
Mole Fraction of Laser-Active Subspecies It has been assumed in the above that the total mole fraction of laser-active subspecies ((S’) and (Sf’)in Figure 1) is small so that a relaxation event nearly always transfers an S’ or S” molecule into the A reservoir rather than into another laser-active rotational state. As stated earlier, a reasonable criterion for distinguishing S’, S” from A is based on photophysical saturation. Consider a subspecies whose rate of absorption is expressed by (10). In the terminology of eq 14, the saturation factorf,,, for IR absorption then is given by eq 18. We shall regard saturation
- gT/R)
or evaluated.
w,(tOtal) -- aA/m.m+l(vL) I
X m = p(1- g)
gM/m,m+l = g
= (m + l)aA/O,l(vL - m6)
[Shift of band center without change of band shape.] In this approach aA/O,I is a function of T T and is identified with aA at TT/Rmeasured under, or extrapojated from, conditions where vibrational excitation of the absorbing mode can be neglected. xm+l/xm eXp(-hUo/kTM) = g [harmonic oscillator populations] (23) (a) Crawford, B. L., Jr. J . Chem. Phys. 1958, 29, 1042. (b) Fujiyama, T.; Crawford, B. L., Jr. J . Phys. Chem. 1968, 72, 2174.
as negligible (and assign the subspecies to the A reservoir) if 0.98 > 1. On introducing the familiar expression for the half-width Avllz of a two-state system14 (in our notation, eq 19),
+
A q p = ( 2 ~ 7 ) - ~ ( 14~~(I’/s)l/NOh)’/~
(19)
the condition for saturation to be negligible becomes (20). Accordingly, molecules will be assigned to the A reservoir if IAvl12/(vL - vb)l 2 7. The laser-active subspecies then account for a segment of the absorption band (Figure 3a) centered on vL and extending to vL f 7Avl 2. This segment subtends a band area of 14 +he full area assignable to a single “branch” of all rotational subspecies is r/s. The mole fraction Xadwof laser-active subspecies is therefore given by 21. On introducing representative Xactivc
= 14sAvl/~Q~(v~)/(vor)
(21)
values of s = 9, T = IO-* s, r = 4 x io4 cm2/mol, I = io5 W/cm2, A v l / ? = 0.01 1 cm-I, vo = 1090 cm-I, aA = 2 X lo5 cm2/mol, we = 0.0069, which is small. obtain Xactive
M-V and V-T/R Energy Flow Let EM, Ev, and E T / R respectively denote energy per mole24 of substrate A in the absorbing mode a t TM, the nonabsorbing vibrational modes V at Tv, and the T / R modes at TT R. Let dEM,/dt and dEv,T R/dt respectively denote net rates ofl energy flow per mole of s d s t r a t e A from M to V and from V to T / R . According to the flow pattern of Figure 1, the time dependence of EM, Ev, and ETIRthen is given by eq 22. Phenomenological dEM/dt = w(total)/[A] - dE,,,/dt dEv/dt = dEM,v/dt
- dEV,T/R/dt
dET/R/dt = dEV!T/R/dt
(22)
rate equations for collisional M-V and V-T/R energy exchange are given in eq 23 and 24. To obtain (23), for instance, assume dEM,V/dt = kM,VIAl [EM(TM) dEV,T/R/dt =
kV,T/RIAl
-
EM(TV)I
(23)
EV(TT/R)I
(24)
(24) E M and EVare thermal or disposable vibrational energies, in excess of the zero point energy.
4554 The Journal of Physical Chemistry, Vol. 90, No. 19, 1986 TABLE I: Proftle(t) at 1088 cm-I (R36)for Experiments with CHCIFZ 104t, ns profile(t) 104t, ns profile(t) i04r, ns profile(?) 370 2.10 140 19.7 1 0 0.00 10 20 30 40 50 60 70 80 90 100 110 120 130
2.27 9.31 23.51 52.65 84.11 99.47 114.69 113.29 93.61 69.61 52.16 38.19 24.54
150 160 170 180 190 200 210 220 230 250 280 310 340
16.61 14.20 1 1.84 10.22 8.20 6.20 4.85 4.23 3.78 3.53 2.76 2.19 2.14
400 430 460 490 520 570 640 710 780 850 9 20 990 1060 1130
1.91 2.14 2.12 1.86 1.86 1.67 1.50 1.34 1 .08 1.14 0.53 0.34 0.18 0.00
Grunwald and Liu TABLE 11: Samole Results for CHCIF,‘ IO~F, lo-41a”: 10-4E,b/F, J/cm2
W/cm2
cm2/mol
4.24 10.05 32.9 72.8
2.81 6.66 21.8 48.2
10.0 7.91 5.73 4.26
T.
ns
1.43 f 0.07 1.42 i 0.07 1.33 f 0.08 1.37 f 0.09
“Conditions: P = 2.47 Torr; vL = 1088 cm-I; yo = 11 16 cm-’; 3.76 X IO4 cm2 mol-’; s = 9; a A = 1.61 X lo5 cm2 mol-’ (298 K). = S I d F / F = J”pd t / S I dt.
r
=
J/cm2. (2) Eab, the energy absorbed per mole of substrate per pulse, as defined in
(3) The pulse profile: profile(t) = Z(t)/F. Measurements of F and Eabs are quantitative and involve calorimetric methods.2* The results are corrected to “thin-cell” that the forward rate ( M V) with mode M at temperature TM* conditions by a convenient, mathematically exact algorism due is given by kM,v[A]EM(TM*). The reverse rate (V M) with to Liu.lb These corrections are mostly less than the 5% experiV at temperature Tv* equals the forward rate when Tv* = TM* mental precision. and thus equals kM,v[A] EM( Tv*). The net rate is the difference Profile(t) is the normalized intensity profile of the laser pulses, (23). as measured with a photon drag detector.28bBc Although in Equation 24 rests on a similar basis and has been confirmed principle Profile(t) may be different in each experiment, under by experiment.,’ Thus for CHC1F2, kV,T/R= 1 X lo9 s-I M-’ the present conditions the variations are small enough so that a (6 X lo4 s-I Torr-I) under similar conditions of IR a b ~ o r p t i o n . ~ ’ ~ single average pulse profile can be applied to an entire series. This value will be used in part 2. For CHC1,F the V-T/R rate Table I lists the pulse profile used in the present test. in ultrasonic relaxatoin is 4 times greater,26and a value of kV,T,R Table I1 lists some test results for CHClF,. The fluence ranges = 4 X lo9 s-I M-’ will be used for this substrate in part 3. from 4 to 73 mJ/cm2, average intensity from 28 to 480 kW/cm2, To estimate kM,Vin (23), let E M = Nohvo(Xl X , + ...), with and Eabsfrom 0.4 to 3.1 kJ/mol. In spite of the moderate inX,,, = [A,]/[A]. The forward rate then is given by eq 25. tensities, the absorption cross section as measured by E,,,/F decreases by more than half, showing that photophysical saturation kM,V[A] E M = NOh%kM,V([Al] + 2[A21 + ...) at TM (25) of the laser-active subspecies is significant. Because IR absorption is slight, temperatures remain moderate, and as a result one expects Accordingly, energy transfer takes place in parallel events and the relaxation time T to be nearly constant. The results listed for the amount of energy transferred per event is proportional to m. T in Table I1 show that this is approximately true. The precision A likely, but not unique, mechanism is that M-V transfer takes of T is about 5%. The decreasing trend with increasing intensity place at each collision and that the m quanta of vibrational energy is consistent with the more detailed investigation reported in part are transferred to other modes all at once. According to propensity 2. a collisional rules suggested particularly by Flynn and co-worker~,~~ The calculation of T involves successive approximations and mechanism of this sort might occur when the density of vibrational numerical integration of eq 14 or 17 (depending on conditions), states per wavenumber is greater than about 1/30. This condition 22, 23, and 24. Instantaneous temperatures required in the inis met for both CHClF, and CHC1,F-see data in parts 2 and tegration are calculated from the instantaneous energies by usual 3. We shall therefore equate kM,vto 2.8 X 10’’ s-l M-’ (1.5 X methods of statistical mechanics.29 An initial value is adopted lo7 s-l Torr-’), the kinetic theory rate constant for bimolecular for T and the corresponding value of Eabis calculated. The initial collisions. value is then changed until the calculated value of Eabsagrees with experiment. In this approach the result obtained for T is the Test effective average during the IR pulse. Moreover, since g, g M r and In our laboratory we used a C 0 2 laser with a square-pulse gT/R in (14) and (1 7) are
