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The solution of the diffusion equation with a sink term using the radiation boundary condition
Journal of Luminescence 27 (1982) 441—448 North-Holland Publishing Company
441
THE SOLUTION OF THE DIFFUSION EQUATION WITH A SINK TERM USING THE RADIATION BOUNDARY CONDITION J.M.G. MARTINHO and J.C. CONTE Centro de Quimica Fisica Molecular, Complexo I, Instituto Superior T~cnico, Av. Rovisco Pals, 1000 Lisbon, Portugal
Original manuscript received 23 April 1982 Revised manuscript received 16 June 1982
The diffusion equation with an added sink term describing the energy transfer by a dipole—dipole mechanism was solved under steady state conditions using the radiation boundary condition. An equation for the non-radiative energy transfer rate constant between organic fluorescent molecules was obtained. This general equation contains the particular cases of a total reflecting boundary condition and the so-called Smoluchowski boundary condition already obtained by other authors.
1. Introduction The non-radiative electronic energy transfer between an energy donor (D) and an acceptor (A) takes place essentially by a Coulombic [1,2] and/or an exchange [3] interaction mechanism. The exchange interaction is a short-range process and is particularly important for triplet—triplet transfer. The Coulombic interaction is a long-range process and in general dominates when the transitions of the donor and the acceptor are spin allowed. The Coulombic interaction integral can be developed as a series of multipolar terms [2].When the transitions in the donor and acceptor are allowed the dipole—dipole term is dominant. In this case the transfer probability per second, to(r) is given by w(r)=—1
6 R ~-j 0\
a
(1)
=~
where i~ is the decay time of the donor, r is the separation of the donor and acceptor molecules and R 0 is the so-called critical distance 2q~ In 10f FD(i’)fA(i’) di’, (2) 128Tr5n4N R~= 90O0~
being the quantum efficiency of the energy donor emission, n the refractive index, x an orientation factor for the donor and acceptor dipole moments, N
qD
0022-2313/82/0000—0000/$02.75
©
1982 North-Holland
442
J. M. G. Martinho, J. C. Conte
/
Solution of the diffusion equation
the Avogadro number, FD(i’) the normalized molecular fluorescence emission of the donor and EA(i’) the decadic molar extinction coefficient of the acceptor. In solution, when the energy donor molecules are excited with a 6-pulse of e.m. radiation there is the possibility of energy transfer to any of the energy acceptors surrounding the excited energy donors. To obtain the rate constant k(t) for the energy transfer process one must use a statistical analysis. Gösele et al. [4,5] used a method, developed by Waite [6], based on pair probabilities and obtained, for a spheric symmetrical process, k(t)
=
4irDR~
au. r
r~R,
—~-~
~1
2
r—R, ~(r)L~,(r)4~r
Assuming that the pairs (D,
—
dr.
(3)
A
1) have no correlated movements [7,8] for any value of r, the diffusion coefficient D is the sum of the diffusion coefficients for the donor and the acceptor. Hence the normalized distribution function satisfies the differential equation
(4) The rate constant k(t) is then the sum of two terms. The first term gives the transfer at the encounter distance Re. The second term gives the transfer due to Coulombic interactions or exchange interactions for distances greater than Re. Eqs. (3) and (4) are quite general and as shown by Gosele et al. [9] they can be used to study the particular cases of bimolecular diffusion-controlled reactions [10] when ~,(r) 0, and Forster’s transfer [11] if there is no material diffusion (D 0). However we are interested in studying the more general case of energy transfer in solution. It this case it is necessary to solve eq. (4) using adequate boundary conditions. We will consider only the case of a uniform initial distribution of either donors or acceptors. Then all are identical and =
=
U(r, t=0)= 1.
Eq. (4) has no analytical solution even for these conditions and so we will seek a solution for long times so the steady state approximation, viz. aU/at
=
0
can be considered to be valid. Under these conditions eq. (4) is soluble for adequate boundary conditions. For dipole—dipole interactions two different cases were presented [5.12], namely (i) U(r,t=0)=l
forr>R~
U(r=R~,t)=0, (ii) U(r,
i
=
0)
I
=
(aU/ar)~R
=
(5)
for r
>
Re
(6)
0.
For cases (i) and (ii) the boundary conditions for the encounter distance are
J. M. C. Martinho, J. C. Conte
/
Solution of the diffusion equation
443
different. Case (i) corresponds to a perfect absorbing barrier and is sometimes referred to as Smoluchowski’s condition [13]. It implies that there is an instantaneous reaction at the encounter. Case (ii) corresponds to a perfect reflecting barrier and implies that there is no reaction at the encounter. The solutions for the transfer rate constant are for (i) (7)
k~=41TD[RF+f(ZO)ReI,
for (ii) 1
3/4(zO)
~
k~=41TDRF I
—
8
,
3/4t Z0
where R
F(1/4) ~D)1/4 2F(3/4)(a\
F
4~/~
9
‘
KI/ 4(z0)
2
[r(1/4)]
/
1/2
z0
(10)
,
Il/4l~zO)
F(z) being the gamma function, K,,(z) and I~(z)the modified Bessel functions of second and first kind respectively for order 1
a
i’
and argument z, and
1/2
(11) When z0>> 1 (i.e., small values of D), both theories predict the same value since all the transfer processes occur at distances greater than Re, and then (12) A similar result was obtained by Yokota and Tanimoto [14], and more recently by Allinger and Blumen [15] in the case of low concentrations of acceptor, using a more complex statistical approach. For z0 << 1 and case (i) one finds the value predicted by the theories of the diffusion-controlled reactions k~,Z=41rDRF.
(13)
k~=41TDRe
while for case (ii) (14)
k~=~f—~.
In this case the rate constant does not depend on the diffusion coefficient D. This is so because in the limit of large values of D(z0 ~ 1) there is perfect mixing of donor and acceptor molecules [16] (i.e. U(r) 1) and then from (3) =
=J
k~
47Tr2~~
dr.
(15)
444
f.M. G. Mart,nho, f. C. Conte
/
Solution of the diffusion equation
This is also the result obtained by Allinger and Blumen [15], who implicitly use a total reflecting barrier, in the case of rapidly moving molecules.
2. The general case In the vapour phase the energy transfer occurs at the encounter distance “during the collision time”. The values obtained for the rate constants [17,181 are of the order of l0b0_~l00 !mol~ s’. depending, among other factors, on the values of the overlap integral for the transfer for a dipolar or exchange mechanism. In the liquid phase one must also have a finite value for the rate constant at the encounter and then one must use a more general boundary condition. If we assume that the flux which tends to form the pairs at the encounter equals the rate of reaction at the. encounter, we obtain the so-called radiation boundary condition [10] (16)
4~DR~(~)=k~U(R..t).
where k~(in cm3 s 1) represents the rate constant for the disappearance of the pairs in a thin spherical shell. If we make the substitutions =
~
(17)
U=vr”~2 in (4) we obtain the modified Bessel equation
(18)
2v dv 2 2d —~+z~——(~+z )v=O
(19)
whose solution, under conditions (16) and U(r
I ~ F(l/4) +
~
K
—~ ~.
t)=
1. is
1/4
K_ 3/4(z~)
F( 1/4) I~.l/4(zQ)
—
4$U(z (a/D)’ 0) 3~2z7/4I— 2 3/4(Z~)
(~)k
I ~
(20)
J
where /3=k~/4irDR~
(21)
is a parameter that takes into account the possibility that the encounter pair has a probability of reaction (y) different from one.
/
f.M. G. Martinho, J. C. Conte
Then, for z
445
Solution of the diffusion equation
we obtain
=
3’~4z”~’4
U(z )= 2 ° ~ F(l/4)
~2z ‘
(2zO)3~2L
1
\3/2 °‘
3/4(zO)+$(a/D)’~
(22)
—
4I,/ 4(zO)
Z0
The value of k~ is now easily obtained from eq. (3). Applying Gauss’ theorem to this equation, GOsele [4] obtained for steady state conditions / ~dU\
k~ = lim 4irD1 r ~ r-~ u~j
(23)
Hence 13/4(zO)
= 41rDRF~
I_3/4
Z0
4 2/3(a/D)~
+ ~~[(2zo)3/2J
3/4(zo)+$(a/D)~4JI/4(zo)]
Kl/4(zO)L3/4(zO)+K.3/4(zO)I1/4(zO)
(24)
.
I_3/4(zO)
Since for Bessel functions [19] Kl/4(zO)L3/4(z0)+K3/4(zO)II/4(z0)=
I/z0
(25)
the following relationship is obtained 13/4(zO) kw=4ITDRF
j —
3/4 ~
4
26
2$(a/D)~
+ ~~zOI
3/4(zO)[(2zO)J3/4(zO)+$(a/D)Il/4(zO)]
.
(
)
From the general expression of k~ (26) the two particular cases already mentioned can be obtained: if $ = 0 we obtain case (ii) and if /3 is large (fiRe>> I) the Smoluchowski boundary condition is obtained and then case (i) is recovered. In fig. 1 the variation of the effective distance for the energy transfer process Ret = k~,/4irD with ~ for Re = 7.0 A and different values of /3 is shown. The values of ~ vary between the two particular cases already reported, according to the /3
446
f.M. G. Martinho, J. C. Conte
/
Solution of the diffusion equation
4
/ ~i7—~ 0
.7’
I 1.0
I — ~zo
2.0
Fig. I. Theoretical variation of the effective radius (R (1) /3 (5) /3
= =
011) with ~fzo, from eq. (26) with R~= 7.0 A. lOx lO’~cm~ (2) j3 = lOx 10~cm~ (3) /3 = 2.Ox lO~cm ‘~ (4) /3 = lOx l0~cm~: 5.Ox 106 cm (6) /3 = lOx 106 cm (7) /3 = 0.
value. From fig. 1 we can distinguish three cases according to the values of z0. If z0>> 1.0, Ret1 does not depend on the /3 value and is given by RF (9) in accordance with the results already proposed by different authors [5,14.15]. However for z0 ~ 1 R eff — ~ +$R~ $R~R ~ — — 1 kc/4ITDRe + kc/4~DReR
e
( 27 )
which is the result obtained on the study of partially diffusion-controlled reactions [10]. For values of z0 between these two particular cases (z0 1.0) the values of RCff are obviously given by eq. (26). The value of /3 is related to the transfer processes at the encounter. If it is assumed that $R~is only related to the probability of reaction of the pair at
J. M. C. Martinho, J.C. Conte
/
Solution of the diffusion equation
447
the encounter, then $Re=
(28)
1~.
Under these circumstances the possibility of re-encounters, that are particularly important for values of y ~ 1, is not considered. On the contrary if this possibility is considered, then, assuming the equivalence of the treatment of Noyes [20] based on particle pair approach and the concentration gradient approach of Collins and Kimball [10] it is possible to obtain a “new” relationship [21,22], viz. (29)
$Re=~Y P0 Po
where Po is the total reencounter probability (with or without reaction).
3. Transient effects The treatment developed is only valid under steady state conditions. The steady state is only achieved for values of ~ >> R~~~/irD. Gösele [4,5] proposed for the case of z0>> 1 an approximate time-dependent equation, based on the known solutions for long and short times k(t)=41TDRF+r~/~.
(30)
This eq. (30) seems to give results just as good as the more complicated relationships presented by Yokota and Tanimoto [14] and Allinger and Blumen [15]. In the transfer between cresol violet and azulene this equation was shown [23] to be valid for times greater than 1 ps. However the more general equation k(t) = 4ITDRet1[l + ~
J
(31)
proposed by Gösele [4,5] is also a good approximation for any value of z0 as shown by Butler and Pilling [24] using a numerical integration method. This equation is reduced to the known equation for the partially diffusion-controlled reactions, when z0 << 1. For z0>> I it gives the same results as eq. (30) within 13%. So it is easy to understand that the set of equations presented in this work should be a good approximation for all cases. 4. Conclusions The relationships developed in this paper take into account the fact that the probability of reaction at the encounter (y) may be less than one. This
448
f.M. G. Martinho, f.C. Conte
/
Solution of the diffusion equation
probability is given by ~
where kte is the transfer rate constant at the encounter and kD is the rate constant for the “dissociation” of the pair. They constitute a general set of equations incorporating the known and already verified cases obtained by other authors and should be of help when the dipole—dipole interaction is the dominant process. These equations are relevant for low values of k1~and/or large values of kD. However, for systems with low values of R1~(R0
< 20 A) other mechanisms of transfer (multipole or exchange processes) may become important
[25].
The application of these equations to experimental results will be left to future publications.
References [I] Th. Förster, Ann. Phys. 2 (1948) 55. [2] Th. Forster, in Modern Quantum Chemistry, Part III, Action of Light and Organic Crystals. ed. 0. Sinanoglu (Academic Press, New York, 1965). [3] D.L. Dexter, J. Chem. Phys. 21(1953) 836. [4] U.K.A. Klein, R. Frey, M. Hauser and U. Gosele, Chem. Phys. Lett. 41(1976)139. [5] U. Gosele, M. Hauser, U.K.A. Klein and R. Frey, Chem. Phys. Lett. 34(1975) 519. [6] T.R. Waite, Phys. Rev. 107 (1957) 463. [7] PG. Wolynes and J.M. Deutch, J. Chem. Phys. 65 (1976) 450. [8] S.H. Northrup and J.T. Hynes, J. Chem. Phys. 71(1979) 871. [9] U. Gösele, M. Hauser and U.K.A. Klein. Z. Phys. Chem. NF 99 (1976) 81. [10] F.C. Collins and G.E. Kimball, J. Colloid Sci. 4 (1949) 425. [II] Th. Förster, Z. Naturforsch. 4a (1949) 321. [12] N.N. Tunitskii and Kh. S. Bagdarsar’ Yan. Opt. Spectrosc. 15 (1963) 50. [13] M. von Smoluchowski, Z. Physik. Chem. 92 (1917) 192. [14] M. Yokota and 0. Tanimoto, J. Phys. Soc. Japan 22 (1967) 779. [15] K. Allinger and A. Blumen. J. Chem. Phys. 72 (1980) 4608. [16] A. Chalzel, J. Chem. Phys. 67 (1977) 4735. [17] G.M. Breuer and E.K.C. Lee, Chem. Phys. Lett. 14(1972) 407. [18] G.L. Loper and E.K.C. Lee, J. Chem. Phys. 63(1975) 264. [19] M. Abramowitz and IA. Stegun, Handbook of Mathematical Functions (Dover Publications. Inc., New York, 1965). [20] R.M. Noyes, Prog. React. Kinet. 1(1961)129. [21] B. Stevens, J. Phys. Chem. 85 (1981) 3552. [22] K. Razi Naqvi, K.J. Mark and S. Waldenstrøm, J. Phys. Chem. 84 (1980) 1315. [23] D.P. Millar, R.J. Robbins and A.H. Zewail, J. Chem. Phys. 75 (1981) 3649. [24] P.R. Butler and J. Pilling, Chem. Phys. 41(1979) 239. [25] R. Voltz, Radiation Res. Rev. 1(1968) 301.
441
THE SOLUTION OF THE DIFFUSION EQUATION WITH A SINK TERM USING THE RADIATION BOUNDARY CONDITION J.M.G. MARTINHO and J.C. CONTE Centro de Quimica Fisica Molecular, Complexo I, Instituto Superior T~cnico, Av. Rovisco Pals, 1000 Lisbon, Portugal
Original manuscript received 23 April 1982 Revised manuscript received 16 June 1982
The diffusion equation with an added sink term describing the energy transfer by a dipole—dipole mechanism was solved under steady state conditions using the radiation boundary condition. An equation for the non-radiative energy transfer rate constant between organic fluorescent molecules was obtained. This general equation contains the particular cases of a total reflecting boundary condition and the so-called Smoluchowski boundary condition already obtained by other authors.
1. Introduction The non-radiative electronic energy transfer between an energy donor (D) and an acceptor (A) takes place essentially by a Coulombic [1,2] and/or an exchange [3] interaction mechanism. The exchange interaction is a short-range process and is particularly important for triplet—triplet transfer. The Coulombic interaction is a long-range process and in general dominates when the transitions of the donor and the acceptor are spin allowed. The Coulombic interaction integral can be developed as a series of multipolar terms [2].When the transitions in the donor and acceptor are allowed the dipole—dipole term is dominant. In this case the transfer probability per second, to(r) is given by w(r)=—1
6 R ~-j 0\
a
(1)
=~
where i~ is the decay time of the donor, r is the separation of the donor and acceptor molecules and R 0 is the so-called critical distance 2q~ In 10f FD(i’)fA(i’) di’, (2) 128Tr5n4N R~= 90O0~
being the quantum efficiency of the energy donor emission, n the refractive index, x an orientation factor for the donor and acceptor dipole moments, N
qD
0022-2313/82/0000—0000/$02.75
©
1982 North-Holland
442
J. M. G. Martinho, J. C. Conte
/
Solution of the diffusion equation
the Avogadro number, FD(i’) the normalized molecular fluorescence emission of the donor and EA(i’) the decadic molar extinction coefficient of the acceptor. In solution, when the energy donor molecules are excited with a 6-pulse of e.m. radiation there is the possibility of energy transfer to any of the energy acceptors surrounding the excited energy donors. To obtain the rate constant k(t) for the energy transfer process one must use a statistical analysis. Gösele et al. [4,5] used a method, developed by Waite [6], based on pair probabilities and obtained, for a spheric symmetrical process, k(t)
=
4irDR~
au. r
r~R,
—~-~
~1
2
r—R, ~(r)L~,(r)4~r
Assuming that the pairs (D,
—
dr.
(3)
A
1) have no correlated movements [7,8] for any value of r, the diffusion coefficient D is the sum of the diffusion coefficients for the donor and the acceptor. Hence the normalized distribution function satisfies the differential equation
(4) The rate constant k(t) is then the sum of two terms. The first term gives the transfer at the encounter distance Re. The second term gives the transfer due to Coulombic interactions or exchange interactions for distances greater than Re. Eqs. (3) and (4) are quite general and as shown by Gosele et al. [9] they can be used to study the particular cases of bimolecular diffusion-controlled reactions [10] when ~,(r) 0, and Forster’s transfer [11] if there is no material diffusion (D 0). However we are interested in studying the more general case of energy transfer in solution. It this case it is necessary to solve eq. (4) using adequate boundary conditions. We will consider only the case of a uniform initial distribution of either donors or acceptors. Then all are identical and =
=
U(r, t=0)= 1.
Eq. (4) has no analytical solution even for these conditions and so we will seek a solution for long times so the steady state approximation, viz. aU/at
=
0
can be considered to be valid. Under these conditions eq. (4) is soluble for adequate boundary conditions. For dipole—dipole interactions two different cases were presented [5.12], namely (i) U(r,t=0)=l
forr>R~
U(r=R~,t)=0, (ii) U(r,
i
=
0)
I
=
(aU/ar)~R
=
(5)
for r
>
Re
(6)
0.
For cases (i) and (ii) the boundary conditions for the encounter distance are
J. M. C. Martinho, J. C. Conte
/
Solution of the diffusion equation
443
different. Case (i) corresponds to a perfect absorbing barrier and is sometimes referred to as Smoluchowski’s condition [13]. It implies that there is an instantaneous reaction at the encounter. Case (ii) corresponds to a perfect reflecting barrier and implies that there is no reaction at the encounter. The solutions for the transfer rate constant are for (i) (7)
k~=41TD[RF+f(ZO)ReI,
for (ii) 1
3/4(zO)
~
k~=41TDRF I
—
8
,
3/4t Z0
where R
F(1/4) ~D)1/4 2F(3/4)(a\
F
4~/~
9
‘
KI/ 4(z0)
2
[r(1/4)]
/
1/2
z0
(10)
,
Il/4l~zO)
F(z) being the gamma function, K,,(z) and I~(z)the modified Bessel functions of second and first kind respectively for order 1
a
i’
and argument z, and
1/2
(11) When z0>> 1 (i.e., small values of D), both theories predict the same value since all the transfer processes occur at distances greater than Re, and then (12) A similar result was obtained by Yokota and Tanimoto [14], and more recently by Allinger and Blumen [15] in the case of low concentrations of acceptor, using a more complex statistical approach. For z0 << 1 and case (i) one finds the value predicted by the theories of the diffusion-controlled reactions k~,Z=41rDRF.
(13)
k~=41TDRe
while for case (ii) (14)
k~=~f—~.
In this case the rate constant does not depend on the diffusion coefficient D. This is so because in the limit of large values of D(z0 ~ 1) there is perfect mixing of donor and acceptor molecules [16] (i.e. U(r) 1) and then from (3) =
=J
k~
47Tr2~~
dr.
(15)
444
f.M. G. Mart,nho, f. C. Conte
/
Solution of the diffusion equation
This is also the result obtained by Allinger and Blumen [15], who implicitly use a total reflecting barrier, in the case of rapidly moving molecules.
2. The general case In the vapour phase the energy transfer occurs at the encounter distance “during the collision time”. The values obtained for the rate constants [17,181 are of the order of l0b0_~l00 !mol~ s’. depending, among other factors, on the values of the overlap integral for the transfer for a dipolar or exchange mechanism. In the liquid phase one must also have a finite value for the rate constant at the encounter and then one must use a more general boundary condition. If we assume that the flux which tends to form the pairs at the encounter equals the rate of reaction at the. encounter, we obtain the so-called radiation boundary condition [10] (16)
4~DR~(~)=k~U(R..t).
where k~(in cm3 s 1) represents the rate constant for the disappearance of the pairs in a thin spherical shell. If we make the substitutions =
~
(17)
U=vr”~2 in (4) we obtain the modified Bessel equation
(18)
2v dv 2 2d —~+z~——(~+z )v=O
(19)
whose solution, under conditions (16) and U(r
I ~ F(l/4) +
~
K
—~ ~.
t)=
1. is
1/4
K_ 3/4(z~)
F( 1/4) I~.l/4(zQ)
—
4$U(z (a/D)’ 0) 3~2z7/4I— 2 3/4(Z~)
(~)k
I ~
(20)
J
where /3=k~/4irDR~
(21)
is a parameter that takes into account the possibility that the encounter pair has a probability of reaction (y) different from one.
/
f.M. G. Martinho, J. C. Conte
Then, for z
445
Solution of the diffusion equation
we obtain
=
3’~4z”~’4
U(z )= 2 ° ~ F(l/4)
~2z ‘
(2zO)3~2L
1
\3/2 °‘
3/4(zO)+$(a/D)’~
(22)
—
4I,/ 4(zO)
Z0
The value of k~ is now easily obtained from eq. (3). Applying Gauss’ theorem to this equation, GOsele [4] obtained for steady state conditions / ~dU\
k~ = lim 4irD1 r ~ r-~ u~j
(23)
Hence 13/4(zO)
= 41rDRF~
I_3/4
Z0
4 2/3(a/D)~
+ ~~[(2zo)3/2J
3/4(zo)+$(a/D)~4JI/4(zo)]
Kl/4(zO)L3/4(zO)+K.3/4(zO)I1/4(zO)
(24)
.
I_3/4(zO)
Since for Bessel functions [19] Kl/4(zO)L3/4(z0)+K3/4(zO)II/4(z0)=
I/z0
(25)
the following relationship is obtained 13/4(zO) kw=4ITDRF
j —
3/4 ~
4
26
2$(a/D)~
+ ~~zOI
3/4(zO)[(2zO)J3/4(zO)+$(a/D)Il/4(zO)]
.
(
)
From the general expression of k~ (26) the two particular cases already mentioned can be obtained: if $ = 0 we obtain case (ii) and if /3 is large (fiRe>> I) the Smoluchowski boundary condition is obtained and then case (i) is recovered. In fig. 1 the variation of the effective distance for the energy transfer process Ret = k~,/4irD with ~ for Re = 7.0 A and different values of /3 is shown. The values of ~ vary between the two particular cases already reported, according to the /3
446
f.M. G. Martinho, J. C. Conte
/
Solution of the diffusion equation
4
/ ~i7—~ 0
.7’
I 1.0
I — ~zo
2.0
Fig. I. Theoretical variation of the effective radius (R (1) /3 (5) /3
= =
011) with ~fzo, from eq. (26) with R~= 7.0 A. lOx lO’~cm~ (2) j3 = lOx 10~cm~ (3) /3 = 2.Ox lO~cm ‘~ (4) /3 = lOx l0~cm~: 5.Ox 106 cm (6) /3 = lOx 106 cm (7) /3 = 0.
value. From fig. 1 we can distinguish three cases according to the values of z0. If z0>> 1.0, Ret1 does not depend on the /3 value and is given by RF (9) in accordance with the results already proposed by different authors [5,14.15]. However for z0 ~ 1 R eff — ~ +$R~ $R~R ~ — — 1 kc/4ITDRe + kc/4~DReR
e
( 27 )
which is the result obtained on the study of partially diffusion-controlled reactions [10]. For values of z0 between these two particular cases (z0 1.0) the values of RCff are obviously given by eq. (26). The value of /3 is related to the transfer processes at the encounter. If it is assumed that $R~is only related to the probability of reaction of the pair at
J. M. C. Martinho, J.C. Conte
/
Solution of the diffusion equation
447
the encounter, then $Re=
(28)
1~.
Under these circumstances the possibility of re-encounters, that are particularly important for values of y ~ 1, is not considered. On the contrary if this possibility is considered, then, assuming the equivalence of the treatment of Noyes [20] based on particle pair approach and the concentration gradient approach of Collins and Kimball [10] it is possible to obtain a “new” relationship [21,22], viz. (29)
$Re=~Y P0 Po
where Po is the total reencounter probability (with or without reaction).
3. Transient effects The treatment developed is only valid under steady state conditions. The steady state is only achieved for values of ~ >> R~~~/irD. Gösele [4,5] proposed for the case of z0>> 1 an approximate time-dependent equation, based on the known solutions for long and short times k(t)=41TDRF+r~/~.
(30)
This eq. (30) seems to give results just as good as the more complicated relationships presented by Yokota and Tanimoto [14] and Allinger and Blumen [15]. In the transfer between cresol violet and azulene this equation was shown [23] to be valid for times greater than 1 ps. However the more general equation k(t) = 4ITDRet1[l + ~
J
(31)
proposed by Gösele [4,5] is also a good approximation for any value of z0 as shown by Butler and Pilling [24] using a numerical integration method. This equation is reduced to the known equation for the partially diffusion-controlled reactions, when z0 << 1. For z0>> I it gives the same results as eq. (30) within 13%. So it is easy to understand that the set of equations presented in this work should be a good approximation for all cases. 4. Conclusions The relationships developed in this paper take into account the fact that the probability of reaction at the encounter (y) may be less than one. This
448
f.M. G. Martinho, f.C. Conte
/
Solution of the diffusion equation
probability is given by ~
where kte is the transfer rate constant at the encounter and kD is the rate constant for the “dissociation” of the pair. They constitute a general set of equations incorporating the known and already verified cases obtained by other authors and should be of help when the dipole—dipole interaction is the dominant process. These equations are relevant for low values of k1~and/or large values of kD. However, for systems with low values of R1~(R0
< 20 A) other mechanisms of transfer (multipole or exchange processes) may become important
[25].
The application of these equations to experimental results will be left to future publications.
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