Magnetic field and magnetic isotope effects in micelles. The kinetics of photochemical reactions

Chemical Physics 147 ( 1990) 369-375 North-Holland

Magnetic field and magnetic isotope effects in micelles. The kinetics of photochemical reactions I.A. Shkrob and V.F. Tarasov Institute of Chemical Physics, USSR Academy of Sciences, Moscow, USSR Received 14 May 1990

Radical pair dynamic models are developed for analyzing the kinetics of spin-selective geminate reactions in micellar media. It is found that by using our approach one can reach a quantitative agreement between the experimental data on magnetic isotope effects and laser flash photolysis for photodissociation of ketones.

1. Introduction Magnetic spin effects in photoreduction [l-6] and photodissociation [ 7-101 reactions in aqueous micellar solutions have been intensively studied by means of laser flash photolysis (LFP). The interest in micelles is due to the fact that in these systems the lifetime of radical pairs (RP) is rather long which makes it possible to observe the radical decay kinetics within nanosecond or microsecond scales. Moreover, in these systems geminate processes are prevalent and magnetic effects, especially magnetic isotope effects (MIE), are quite appreciable (e.g., ref. [ 111). Let us consider in more detail the currently accepted interpretation of LFP experiments. Commonly, LFP experiments yield the Z(t) curve, reflecting a dependence of the decrease of optical absorption of the solution, caused by decay of the radicals, on time t after the exciting pulse. As a rule, this curve has two regions: a region of fast decrease and a region of slow decrease. The former region corresponds to reaction decay of geminate RPs (t < 1 ps) and the latter one describes slow decay of the radicals in miscellaneous secondary reactions ( t m 1 ms). In the following we shall deal exclusively with the fast processes occurring in the microsecond time scale. At sufficiently large times t, when the curve Z(t) is plateau-like (when changes occur within 1 ms only) the radical concentration Z(t) is related to the cage effect P by Z(t) /I( 0) = 1 -P. Apparently, P is the probabil0301-0104/90/$03.50

ity of geminate spin-selective radical reactions. Note that there is a number of non-spin-selective reactions occurring in micelles (e.g., fast decay of radicals due to capture by micelles, chemical transformations of detected radicals). We shall always include these processes by introducing the total escape rate constant. This constant reflects all causes of RP dissociation: both the chemical and the mechanical ones. It is known that for a great number of systems the dependence Z( t ) is approximately mono-exponential in any magnetic field [ l-3 1. On the basis of this observation Scaiano et al. [ 1 ] proposed the following procedure for treatment of kinetic curves. Z(t ) is described by two constants: kr, the rate constant of RP reaction decay in micelles and k,,, the escape rate constant. In this approach the component index /cobs which corresponds to the Z(t) curve and the RP reaction decay probability P is given by P= kflkoss , kt,r=kf+k,.

(1)

Using eq. ( 1) one can easily obtain kf and 5, from experimental data. Eqs. ( 1) are very heuristic and undoubtedly useful, since the introduced parameters unambiguously describe the curve shape. However, the physical meaning of the constants is unclear as well as their relationship with the parameters of RP spin and molecular dynamics. Indeed, a simple iden-

0 1990 - Elsevier Science Publishers B.V. (North-Holland)

370

LA. Shkrob, V.F. Tarasov /Magnetic

tification of bwith the RP intersystem crossing (ISC) rate constant and k,, with the radical escape rate constant results in k, being dependent on the strength of the external magnetic field [4,5] and the isotope configuration of the RP precursor [ 3,9]. In some cases k,, as calculated for kinetics in weak fields was found to exceed kobsin high fields [ 9 1. The theory of ref. [ 1 ] has been extended in ref. [ 2 ] on the basis of describing the RP spin dynamics in terms of a formal kinetics equations approach. A similar approach has been offered in ref. [ 61 for description of photoreduction via micelles. There are no detectable MIE for these reactions, though cage effects and magnetic effects are appreciable [2-61. The shape of external magnetic field dependences of P and kobs (so-called MARY spectra) indicates that the relaxation mechanism of ISC is prevalent in these systems. However, when applying the model it is impossible to reconcile the low values of the RP decay constants (approximately 10’ s- ’ ) with the high MIE at i3C/‘*C isotope separation (e.g., the corresponding isotope selection efficiency coefficient (r for photodissociation of aromatic ketones is 1.2- 1.5 [ 111). According to ref. [ 21, the system in the absence of magnetic field is described by the following three constants: K, the rate constant of transition between S and degenerated T sublevels (see fig. 1A); k,, the constant of the singlet RP reaction decay, and k_ the escape rate constant. At k,=x K: kb =k, + kJ4, at k,z+ K: kobr= k, + K. In the former case MIE is utterly impossible, in the latter case /+x K and MIE is possible, but the observed RP decay constant kf must weakly depend on the micelle type, being determined exclusively by the magnetic interactions of radical spins. But experimental data show that kf is strictly dependent on the micelle shape, size, viscosity etc. [ 121. On the other hand the K value estimations in quasi-classical models of RP theory (cf. below) and in perturbation theory [ 21 yield Kx (S-20) x 10’s_‘, which exceeds the observed kobs by an order of magnitude. The reason for the unsuccessful application of the scheme [2] lies in the fact that the approach completely ignores the RP molecular dynamics, in particular the separation of RP between spatial (or distant ) and contact states. Only contact RP may react, the distant one cannot do it even in singlet state. But within the given model any singlet RP is reactive. It

field effects in mice//es

F

F

ke I

K

I ke

G-S-T

A

ke + -F R

kt? G-

S

ke

b--

k,

B Fig. I. The kinetic scheme in the static model, (A) in a zero, (B) in a strong magnetic field. G and F denote the products of geminate reactions and reactions of free radicals.

is the major shortcoming of the approach. The aim of this paper is to propose models, which describe the RP decay kinetics in micelles more adequately. Such models allow us to interpret various experimental data on photodissociation in micelles.

2. Dynamic models of RP The static model describing the RP decay kinetics has been developed in ref. [ 2 1. The meaning of the kinetic constants, used in the model, is clear from fig. 1. In zero magnetic field, due to triplet sublevel degeneration, the ISC is described by a single effective constant K, whilst in a high field the ST+ and TOT, transitions are suppressed by Zeeman splitting of the triplet sublevels and there are two types of transition rates in RP: fast rates for ST0 transitions (constant Qx K) and slow rates for ST+, ToT+ transitions (constant R c Q) . Introduce ps for the part of singlet

LA. Shkrob, V.F. Tarasov / Magneticjeld eflects in micelles

states at the moment of RP generation (for instance, ps=O for triplet-born RP and ps= l/4 for random RP) and assume that triplet states have equal initial populations, i.e. neglect the triplet mechanism polarization for RP involved. Then one obtains ,_s(O)-f(l-4Ps)[g(4K)+2g(2Q+2K)l 4+g(O)+g(4R)+2g(2Q+2R)

G t

I

%

’

Z

(2) g(w)=kl(k+w)

%

(3)

7

F-S

where k, is the reaction decay rate constant for singlet RP, and k, is the escape rate constant. For triplet-born RP in zero magnetic field eq. (2 ) may be simplified, P=

g(0) -g(4K) 4+g(O)+3g(4K)

371

K

Tc

-

‘i

2-l

d

fc-’

z v

-Ta-F

ke

K Fig. 2. The kinetic scheme in the dynamic RP model (zero field).

’

k0t,&,+f[k,+4K-(k:+4k,K+16K2)“2].

(4) ks is the decay rate constant for the contact singlet

(5)

In the following we shall consider only the zero field limit. It is easy to see that at k, x 5 X 1O6s - ‘, /cobs= k,=107s-1,Kx5~107s-’ (andK*=2x108s-‘for typical RP which contains a “C magnetic nucleus) the isotope selection efficiency coefficient a for recombination product, 1-xP (y=I-xp*, where P* is the corresponding value for 13CRP and x is the contribution of the recombination reaction to the singlet RP total decay (note that according to ref. [ lo] this factor is independent of RP spin dynamics parameters), is too small - (~2 1.05 - compared to experimental values ( z 1.2- 1.5 ) . There are two possible ways of improving the scheme of fig. 1: (i ) adding new states into the kinetic scheme that describes the contact and spatialseparated or distant pairs; and (ii) describing the RP dynamics in terms of diffusion-controlled reactions theory. Let us consider both variants. 2.1. The kinetic model The scheme which corresponds to selecting RP into contact and distant pairs is presented in fig. 2 (S,, T, and S,, Td are the populations of the sublevels of contact and distant RP respectively, T is the total popu-

lation of triplet states, z is the frequency of radical encounter, 7 is the lifetime of the contact state, and

RP. Eqs. (2 ) and (4) can be shown to be applicable for this model too, provided that an alternating g function is used, g(w)=

ks7 l+(k,+w)7-z/(k,+z+w)’

(6)

Eq. (6) enables us to understand how the scheme in fig. 2 can result in the considerable MIE. Note that the following conditions are the most real: z7-=x 1, k,7Q: 1, since zx lo’-10’s_‘, k;x (2-50) x lo5 s-i, 7s 0.1-5 ns. The recombination rate of the contact pairs in the singlet state is very high ( 1O’O-10” s- ’ ) and the reactions are normally diffusion controlled: &rz+ 1. At w=O g(O)=k:/k, and k+k7z>z. If 4K7e 1, the P values coincide with those calculated via eqs. (2 ) and (4) at k, = z. In this case as well as with the scheme in fig. 1 one fails to explain large MIE. However, at 4K7 3 1 g( 4K) a ks/4K> k:/4K. Hence, the increase ofg( w) with w, calculated via eq. (6)’ is sufficiently slower than via eq. (3). Therefore, the dependence of P on K is stronger and MIE is possible even at K=B z. At 4K7 w lkobs%z+ k, and one can see why &b and z are close values. If 4K7 w z/K eq. (4) is simplified, p=

(a+l)(k+z)-k z(a+

l)+/&(a+4)

’

0)

where (I!= 4Kr for the slowest decay constant kob,

I&‘&+=.

(7b)

LA. Shkrob. V.F. Tarasov / Magneticjield effects in micelles

312

Transforming (7a) and (7b) one obtains (Y -= 4

‘=

k, k&,(1_P)-1

I

’

kobs -k l- $k,,,,( 1 -P)/k,

(7c)

*

On the basis of eq. (7~) one can find the dynamic parameters from experimental data. The equation was used in ref. [ 13 ] for analysis of data on a-methyldeoxybenzoin photodissociation in sodium dodecyl sulphate micellar aqueous solution. The reaction decay for RP of set-phenethyl and benzoyl radicals is known to occur via two channels: recombination (83%) and disproportionation ( 17%) [ lo], the total probability P of the RP reaction decay being 0.62 + 0.02, which is supported by both the LFP and photoracemization experiments [ 7, lo]. For the RP in question kb, is about 1.2~ 10’ s-’ [ 71, so KTX 0.16, z x 2 x 10’ s - I. Let us use the quasi-classical evaluation for K [ 141, K=

f CA,z,(z,+l) ( c

>

1

0.10 time,

mcs

I

-0

112 )

where A,, is the hfi constant in frequency units, and Zp is the spin of the Z&hnucleus. In the case of secphenethyl and benzoyl RP Kx4.9~10’ s-’ and r’x 3.2 ns, while the same RP with 13Cin its carbonyl group has K* = 1.8 x 1O* s--I. Since the recombination contribution xc 0.83, one obtains cr x 1.39. The experimental value for this RP is 1.45 [ lo]. Application of exact solution (4), (6) at r= 4.7 ns and k,r= 10 leads to (Y= 1.455. Thus, there is good agreement with experiment. Note that the constant k& for “C RP in this case is about 1.6~ 10’s_‘. It is impossible to find the dependence I( t ) in the analytical form for the general situation. Fig. 3A shows the calculated dependences in semilogatithmic coordinates for some typical RP with ksr= 10, z=1.7x107 s-‘, k,=5x106 s-i, K=5.6x107 s-‘, K* = 1.72 x 1O*s - ‘. Table 1 lists the corresponding constants, cage effects, selection coefftcients cy and /I= k&Jkobs. Calculations indicate that k,,, obtained from ( 1), drastically falls provided K grows, i.e. ISC accelerates, and k, may be notably different from k,. Suggestion was made in ref. [ 9 ] to consider k, in high magnetic field (at Q =BR ) as k,, but even in this case

0.00

0

0 10

time,

mcs

Fig. 3. RP decay kinetics in the kinetic (a) and diffusion (b) models for “C RP ( I ) and for ‘% RP (2) under conditions given in table 1.

the calculations indicate that there may be lo-20% difference between k, and k+ In spite of the fact that the kinetic model provides a close description of the experiment and has predictive power, its application cannot be considered as absolutely correct even within the formal kinetic approach. In reality, at time scale of x K-’ the motion of the radicals has a diffusive character and cannot be described by the restricted number of states. 2.2. Diflbsion model To describe the RP dynamics one employs the model of a microreactor (or supercage) proposed in refs. [ 15,161. In this model the micelle is viewed as

I.A. Shkrob, V.F. Tarasov /Magnetic $eld eflrcts in micelles Table I Calculated parameters for decay kinetics of geminate RPs in dynamic models. In the kinetic model: r=2 ns, .r= 1.7X IO’s_‘; in the diffusion model: D=~x IO-’ cmr/s, r.=6 A, L= 15 A; in hothmodelsk,=5~106s-‘,~7=10,K=5.6~10’~-’for’~C RPand~=1.72x10ss-‘for”CRP Diffusion model

4xr2 as =4xrfAk,SI ’ ar ,+

P k$s,;’ ) bar(us-‘)

‘%I RP

0.64

0.713

13.7 8.8 4.9

16.0 11.4 4.6

(Y B

1.253 1.17

‘*C RP

9.67 5.0 4.68

0.644 11.35 7.31 4.04

$=

59

-3KS+KT-kJ,

dT -= at

Da’(rT) -r ar2

+3KS-KT-k,T,

T(t=O)=

&r-r,) c

>

&r-r,) (1 -PSI ~9

dtSl,,,

c

.

0

By integrating the system of equations one can obtain the equations (2) and (4) with dtexp(--wt)p(t)

1.36 1.174

a sphere of radius L with absolutely elastic walls (as it was mentioned above, the exit rate for radicals is much less than total decay in non-spin-selective reactions as a rule, so it is neglected). The radical motion inside a micelle is stochastic with diffusion coefficient D. Reactions occur when the distance between radicals becomes approximately equal to r,, the reaction encounter radius. The reaction zone is considered as a spherical layer of the radius r, and thickness A. Within the zone singlet RP reacts with decay rate constants ks. Without any loss to generality it can be accepted that one of the radicals always is located in the micellar center [ 15 1. In a more realistic way both radicals move, but (i) analytical solution is not possible in this case, (ii ) the computations demonstrate that results, obtained within the given framework, for usual reactions are quite close to exact solutions provided mere doubling of diffusion coefficient [ 17 1. So for the present system

S(t=O)=PsF

I

13CRP

0.516

I=lc,

00

Kinetic model

P=hrfbA 12CRP

373

.

(8)

In this manner g(w) coincides with Laplace transformed reencounter function p (t ) in the kinetic limit with an accuracy up to some factor. Eq. (8) is valid at any molecular dynamics of RP. The concrete form of the function for the microreactor model is

g(w)=

ksr l+x(l+Y)I(l-Y)

’

x= [ (I&+w)rD]“‘) 1 + Lx/r, Y= 1 _Lx,r 7, = r,Z/D ,

c

expVx(

1 -L/d1

7= Arc/D ,

,

7, = L2/D .

(9)

At wr,~ 1 g(w)xkJ(k,+w), which coincides with the formula of the static model with k, = 1&7z, where z= 3r,D/ (L’- r:) is the encounter frequency. At wr,z+l g(w)xbr/(l+x) corresponds to the expression for free spatial unrestricted diffusion. Since at Kzs 10’ s-i 4K7L 3 1, then kb and P exceed the corresponding parameters in the kinetic model: g( w) falls at large w more slowly than (6 ). At equal dynamic parameters 7 and z, the diffusion model yields greater kbs and P and the kinetic model yields greater (1!and j9. This is supported by fig. 3b, table 1 and fig. 4. Under similar conditions kob and P are about 20% greater than their corresponding values in the kinetic model, with roughly the same B and markedly lower (Y. Calculation of I(t) even for the simplified formal kinetic schemes is a complicated problem. I( t ) is related to the probability of the RP reaction decay at time t through

LA. Shkrob, V.F. Tarasov /Magnetic field

374

16 ‘; U-J CD 0 12 X

2

*

I I

4

0

1

100 effective

1

200

HFI,

xl O6

s -’

Fig. 4. The dependence of RP decay constant in zero field on the ISC rate in the diffusion ( 1) and the kinetic (2) models under conditions given in table I.

al(t) =-P(l)

-

at

)

i.e. for corresponding expressions SZ(S)- 1=-P(S)

Laplace

.

transformed (10)

One can easily calculate P(S) via eqs. (2)-( 9) by replacing k. by the “shifted” value k: = k, + s. The inverse Laplace transform can be carried out practically by using the FFT algorithm (we used a 1024point one). If the Z(t ) kinetics is close to exponential at large t, then the slowest component index may be found from kobs= lim (ZVP(N-‘)/P(N)) N-a,

,

(11)

where PcN) is the Nth order derivative of P(s) at s=O. Even the third or fourth iteration ( 11) allows us to obtain k,,b to better than 2% accuracy. Figs. 3a and 3b show the RP decay kinetics in both models. At close t and 7 the kinetic model demonstrates a sharp decrease of Z(t) (note that its magnitude grows with increasing K), at t B 7 the kinetics is mono-exponential. The pulse duration being as a rule 1O-30 ns, the experimental error connected with de-

effects in micelles

termination of initial concentration I( 0) can be large. On the other hand, when the diffusion model is used no sharp decreases are observed for kinetic curves of triplet-born RPs: the curves smoothly reach a linear section at t z 7D (fig. 3b). So the diffusion model, on the contrary, predicts that Z(0) in nanosecond LFP experiments is determined with good accuracy. Considering the fact that the various data on cage effects, obtained by means of different techniques, are in good agreement [ 7- 10 1,the diffusion model appears to be more suitable for describing the initial stage of the kinetics. At 4K7,. B 1 eqs. (7) are valid and one should assume that (Y= (4K7p) “*, i.e. as for the kinetic model the parameters can be determined directly from experiment. Fig. 4 demonstrates dependences k,,(K) calculated in both models at the conditions given in table 1. Fig. 4 indicates that at Kx z the linear growth of kobsturns into a plateau. This means that the largest MIE in RP decay kinetics can be expected at KG z< P [ 15 ] (in this case j? may be 2-5 ), whilst relatively small MIE are to be expected for RP with z d KG P. The last limit is likely to realize for H/D substitution in ref. [ 9 1, the /I observed were not large ( X.1.13 ) . In general, both the kinetic and diffusion models provide a good description of the experiment, yielding comparable results. Since the kinetic model expressions are more heuristic it appears to be more handy for description of the experiment, except at the initial stage of RP life. Another merit of this model is the simplicity of the expressions for g functions. There is a quite sufficient method for bulky calculations within the frame of more complicated RP theories based upon the solution of Liouville’s matrix equations. The calculations require computation of g for each transition between a few hundreds of spin sublevels. Formula (6) is the simplest one, reliably reflecting the features of the radical motion via micelles. The applicability of the formal kinetics scheme to spin dynamics of RP in substantially high field is restricted. Only the limit of strong Zeeman splitting can be studied quantitatively. At QB R the kinetics becomes close to bi-exponential. This is confirmed by the calculations shown in fig. 5 (Q=5x107s-*, R=lxl@s-‘,otherparameters are presented in table 1): the dependence is

I.A. Shkrob, V.F. Tarasov / Magneticjeld effmts in micelles

375

marizing all the current data on the kinetics of spinselective geminate reactions in micelles, especially of photodissociation. In terms of the developed approach one can not only interpret the known experimental results but also estimate the molecular dynamics parameters and predict magnetic field and magnetic isotope effects in radical reactions.

References [ 11 J.C. Scaiano, E.B. Abmin and L.C. Stewart, J. Am. Chem.

time,

mcs

Fig. 5. The RP decay kinetics in the diffusion model in high magnetic field: Q=SxlO’s-‘and (1) R=lXl@ s-‘, (2) R=O. Other constants are listed in table I.

approximately b&exponential, the component indexes are (10+1)x106 and 6~10~ s-‘. Thus, dynamic RP models, as the static ones, predict multiexponential behavior for decay kinetics in high fields. B&exponential character of reaction kinetics was reliably observed in photodissociation of some aromatic ketones [ 91. For other systems the component indexes are probably too close to be separate. The evaluation of parameters Q and R via perturbation theory was presented in ref. [2]. However, the perturbation theory application for real RP is doubtful because the transition rates and rates of chemical or diffusion processes are close. Moreover, the ISC cannot be described within the formal kinetics approach for at least some fields, since taking account of quantum effects is necessary [ lo]. 3. Conclusion The dynamic RP models are instrumental in sum-

Sot. 104 (1982) 2673. [2] H. Hayashi and S. Nagakura, Bull. Chem. Sot. Japan 57 (1984) 322. [ 3 ] Y. Sakaguchi, H. Hayashi and S. Nagakura, J. Phys. Chem. 86 (1982) 3177. [4] Y. Sakaguchi and H. Hayashi, J. Phys. Chem. 88 ( 1984) 1437. [ 51 Y. Tanimoto, H. Udagava and M. Itoh, J. Phys. Chem. 87 (1983) 724. [6] P.P. Levin and V.A. Kuzmin, Isv. Akad. Nauk. SSSR Ser. Khim. ( 1988) 298, in Russian. (71 N.J. Turro and J. Mattay, J. Am. Chem. Sot. 103 (1981) 4200. [8]N.J.TurroandG.C. Weed, J.Am.Chem.Soc. 105 (1983) 724. [9] N.J. Turro, M.B. Zimmt and I.R. Gould, J. Phys. Chem. 92 (1988) 433. [ IO] V.F. Tarasov, LA. Shkrob, E.N. Step and A.L. Buchachenko, Chem. Phys. 135 (1989) 391. [ I 1] E.N. Step, V.F. Tarasov and A.L. Buchachenko, Russ. J. Phys. Chem. 64 ( 1990) 193. [ 121 N.J. Turro and J. Mattay, Tetrahedron Letters 21 (1980) 1799. [ 131 V.F. Tarasov and LA. Shkrob, Chem. Phys. Russ. 9 ( 1990) 812. [ 141 K.M. Salichov, Yu.N. Molin, R.Z. Sagdeev and A.L. Buchachenko, in: Spin Polarization and Magnetic Effects in Radical Reactions, ed. Yu.N. Molin (Elsevier, Amsterdam, 1984). [ 15) V.F. Tarasov, A.L. Buchachenko and V.Ph. Ma&v, Russ. J. Phys. Chem. 55 ( 198 1) 1905. [ 161 L. Sterna, D. Ronis, S. Wolfe and A. Pines, J. Phys. Chem. 73 (1980) 5439. [ 171 V. Gbsel, U.K.A. Klein and M. Kauser, Chem. Phys. Letters 68 (1980) 291.