J. Phys. Chem. 1992, 96, 9074-9079
9074
Stochastic Treatment of Fluorescence Quenching in Monodisperse Micellar Systems with Exchange of Probes and Quenchers. 2 A. V . Barzykin Department of Photochemistry, Institute of Chemical Physics, Russian Academy of Sciences, Chernogolovka, Moscow Region 142432 Russia (Received: April 10, 1992; In Final Form: June 25, 1992)
A stochastic model for migration-assisted fluorescence quenching in monodisperse micellar systems is extended to account for the micelle limited solubilization capability. An exact solution to the stochastic master equation for the excited probe
survival probability including one-particle migration of both probes and quenchers between micelles during the excitation lifetime is derived for two-component quenching and concentration self-quenching using the generating function technique. One of the basic assumptions accepted in the model is that the equilibrium distribution of solubilizates among micelles obeys binomial statistics. The validity of this assumption is discussed using the formalism of classical statistical mechanics.
Introduction Fluorescence quenching has been extensively used for probing dynamics and structure of micelles and similar self-aggregated molecular structures.'-3 The basic kinetic model for fluorescence quenching in micellar solutions has been proposed by Infelta et a1.4 and Tachiya5assuming that the excited probe remains in the original micelle during its lifetime and the quenchers migrate between micelles via the bulk aqueous phase. The model applies to small monodisperse micelles, for which intramicellar quenching follows pseudo-first-order kinetics2and the distribution of solubilizates among micelles obeys Poisson statistics. Intermicellar exchange is regarded to occur via a one-particle mechanism with the rate constant for exit of a quencher from a micelle being proportional to and the entrance rate of quenchers independent of the number of quenchers in the micelle. The model has been extended by several authors to account for static quenching and migration of quenchers through collisions between micelles: polydispersity,'V8 and effects of micelle shape9J0 as well as micelle limited solubilization capability." Almgren et al.I2 have developed an approximate model including general migration possibilities, where both probe and quencher may change their host confinement during the excitation lifetime either via a one-particle mechanism or via fusion-fission processes. The approach is based on the assumption that the variance of the distribution of quenchers over the micelles with excited probe is linearly related to the average. Tachiya" has analyzed the problem of fluorescence quenching with migrating probes and immobile quenchers using the Laplace transform method and obtained the exact expression for the decay function as an infinite series of exponentials. Gehlen et al.I4 have derived the equivalent result in the form of a Neumann series using the generating function technique and the integral equation formalism. Recently, fluorescence quenching in monodisperse micellar systems with exchange of both probes and quenchers via a oneparticle mechanism has been treated by using the standard generating function technique in the framework of a stochastic model.1s316The derived exact solution for the fluorescence decay has been shown to reduce to the Infelta-Tachiya equation in the case of an immobile probe and to the Tachiya-Gehlen result for an immobile quencher. The decay functionflt) obeys the following Volterra's integral equation of the convolution type flt) = h(t)
+ k-ph(t) @ f i t )
(1)
where @ denotes the convolution product, k-P represents the generalized first-order rate constant for probe migration including one-particle exchange both through the bulk medium and via micellar collisions, and h(t) is given by h(t) = exp(-A2t + A3[exp(-A,t) - 11) where
(2)
+ kqk_si(kq+ k-)-I A3 = Ak,2(kq + k-)-2 A4 = kq + k-
A2 = k-P
-
(4)
(5)
where f i is the average number of quenchers per micelle and kq and k- are the intramicellar quenching and the quencher oneparticle migration first-order rate constants, respectively. The experimentally observable fluorescence decay is obtained by multiplyingflt) by the lifetime decay, exp(-t/T). 7 is the probe excited-state lifetime and determines the relevant time scale on which the fluorescence quenching dynamics has to be studied. Equation 1 is most conveniently resolved by using the Laplace transform technique At) = h(t)[l - k-Ph(6)I-I (6) whereAe) and h(t) denote the Laplace transforms of the fun$ions At) and h(t), respectively, and e is the Laplace variable. h(e) is expressed as the following Poisson weighted infinite series
The Laplace transform can be readily inverted numerically by using the Stehfest a1gorithm.l' The above model is based on the assumption of a random distribution of solubilizatesamong micelles obeying Poisson statistics. Strictly speaking, the Poisson law holds only in the case of dimensionless and noninteracting particles.I8 The assumption of infinite micelle capacity (dimensionless particles) is justified at sufficiently low solubilizate concentrations. At higher concentrations, when the average number of particles per micelle, A, is comparable to the maximum available number of solubilization sites in a micelle, N, the overall intermicellar distribution is well described by the binomial
where [M,]is the concentration of micelles containing n molecules and [MI the total micelle concentration. The binomial distribution, eq 8, reduces to the Poisson distribution at N =, Assuming the following dependence of the rate constant for entry of a molecule into a micelle on the number of molecules already contained in that micelleI9 k,,t(n) = (1 - n / N ) k + (9) that leads to the binomial distribution of solubilizates among micelles, Tachiya has analyzed in detail the effect of micelle limited solubilization capability on fluorescence quenching with migrating quenchers and immobile probes by using the generating function technique." In this paper, we generalize the Tachiya model to
0022-3654/92 , /2096-9074S03.00/0 , 0 1992 American Chemical Society -I
(3)
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The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 9075
Fluorescence Quenching in Micelles the case where both probes and quenchers migrate during the excitation lifetime. The obtained exact solution for the fluorescence decay is shown to reduce to the result derived under the assumption of infinite micelle capacity in the limit of N 01, In all the above models it is assumed that the efficiency of excitation of a probe is sufficiently low, so that one micelle cannot contain more than one excited probe and is independent of the number of quenchers in the micelle with the probe. The distribution of quenchers over the micelles with excited probe immediately after pulse excitation (initial condition) then appears to be the same as the equilibrium distribution of quenchers among micelles. In a one-component system, where the ground-state probes act as quenchers of the excited state (concentration selfquenching), the photoexcitation efficiency is proportional to the number of molecules in a micelle. This leads, in general, to the initial condition different from that for a two-component system. Recently, we have analyzed the peculiarities of fluorescence concentration self-quenching in monodisperse micellar systems with exchange of both excited- and ground-state probes on the basis of Almgren’s approach extended to the case where the number of solubilization sites in a micelle is limited.’O In this paper, the exact solution to the problem is derived via the generating function technique. The limiting fluorescence decay behavior at N = is shown to coincide with that for a two-component system.
-
-
Stochastic Model for Fluorescence Quenching in Monodisperse Micelles with Limited Solubilization Capability Two-Component System. Consider an ensemble of fluorescent probes and quenchers confined to small monodisperse micelles, and assume the binomial distribution of quenchers among micelles (eq 8). In the original Tachiya model” the probes are implicitly assumed to be dimensionless. In what follows we accept this assumption, and generalization to finite-sized probes would be straightforward. Following Tachiya we also assume that the excitation efficiency of a probe is sufficiently low, so that one micelle cannot contain more than one excited probe and is independent of the number of quenchers in the micelle with the probe. The initial distribution of quenchers among the micelles with excited probe after &pulse excitation then also obeys binomial statistics (eq 8). The deactivation of the excited probes occurs through the following transitions n = 0, ...,N where M,* denotes a micelle with excited probe and n quenchers and 7 and k, are defined as above. Following Almgren et a1.I’ we consider the intermicellar exchange processes as transitions where the excitation is conserved
kwU,nfl
M,* Mj* (1 1) The transition frequencies for different migration mechanisms can be readily obtained following Almgren’s treatment with eq 9 taken into account. For the migration of quenchers one gets
(n
+ ti - %)k-w,(j,n,N)
=
where 6 is the Kronecker delta. The migration of excited probes leads to the following equation for the transition frequency
transformed in a usual way to the first-order partial differential equation aG at
+
aG k, as
k-n
+ (S - l)[k, + k-] + -(sN
-G[k-P-kfl(S
- 1)’
- l)] + k-pG(l,t,N)
for the generating function, G(s,t,N), defined by N
G(s,t,N) = Cd[Mj*l(t)/[M*l(O) j=O
(15)
Equation 14 is solved by the method of characteristic curves which are expressed as follows ds k-ti k, + (S - l)(k, + k-) + -(s - 1)’ (16) dt N sl - cs2 exp[(s, - sz)kjit/Nl s = l + 1 - c exp[(s, - s,)kjit/Nl (17) where c is a constant and s1 and s2 are given by (-(kq + k-) f [(k, + k-)’ - 4kqkA/N11/’)N/2k-A (18) SI,’ Here subscript 1 corresponds to plus and 2 to minus. Along the characteristic curves G(s,t,N) satisfies the ordinary differential equation dG + G(kP - k-ti[s(c,t) - 111 = dt
which solution is straightforward. Using this solution, it can be readily shown that the quantity of interest,f(t) = G(l,t,N), obeys the Volterra’s integral equation of the convolution type, eq 1, with h(t) given by
where A2 k-P - k-irs, = NSl(1 + S’fi/N)/(SI A4 = (SI - s,)kA/N
(21) (22) 443 - s2) (23) Equation 1 is resolv@ by using the Laplace transform technique yielding eq 6 with h(c) expressed as the following binomial weighted series N &E)
(:)[$]’I -
= j.0
$]”(E
+ A , + jA4)-’
(24)
-
It can be readily verified that eqs 20-23 reduce to eqs 2-5 and eq 24 reduces to eq 7 at N 01, In the case where only the quenchers migrate and the probes are immobile (k-P = 0), the obtained exact solution, when multiplied by the lifetime decay, exp(-t/r), reduces strictly to the decay function derived by Tachiya.l I The Laplace transform is readily inverted numerically by using the Stehfest algorithm.” However, following TachiyaI3 one can derive the exponential series expansion for the fluorescencedecay in the time domain from eqs 6 and 24 N
Here k- and k-P are as defined above and represent one-particle exchange both through the bulk and via micellar collisions. In what follows we eliminate the lifetime decay from the problem and explore pure fluorescence quenching assisted by intermicellar migration. The experimentally observable fluorescence decay function then will be a product of exp(-t/.r) andflt), whereflt) = [M*](t)/[M*](O) with [M*](t) being the total concentration of micelles with excited probes. Using eqs 10-13 one can derive an associated set of coupled rate equations for the time evolution of [M,*](t), which can be
At) = C B j exp(-@jt)
(25)
3’0
where
ajare the roots of equation 1 k-Ph(t)
-
=0
(27)
9076 The Journal of Physical Chemisrry, Vol. 96, No. 22, 1992
such that sign(j)[A2 + (j - 1)A4] > a, > - ( A 2+ jA4)
j = 0, ..., N
(28) Bj = -a,and so are all positive. The roots have to be calculated numerically. The role of micelle limited solubilization capability in influencing the fluorescence decay behavior is illustrated in Figure 1. The effect of different exchange mechanisms, as well as the relationship between the rate constants for migration and intramicellar quenching, has been analyzed previously,16so here we only consider the case where quenchers and probes migrate equally fast and k- = k-P = O.lk, at A = 2. The initial stage of the decay is quenching-controlled and independent of both intermicellar migration and micelle limited solubilization capability. The decay constant is given by tik, (note that self-decay is excluded). At longer times, the smaller the value of N the faster is the decay. The asymptotic long-time exponential decay constant Bo along with the corresponding amplitude Bo are most sensitive to the variation of N, as shown in Figures 2 and 3. At low quencher concentrations, bo is proportional to ii and independent of N Bo = iik,(k- k-p)(k- k-P + kq)-I (29)
+
1
-71 + k- + k-P
+ iik'7 7
+ k- + k-P + k,
where [M,*](O) stands for the initial concentration of micelles with excited probe and n ground-state probes (0 In IN - l), [M*](O) is the total initial concentration of micelles with excited probe. The deactivation of excited probes is given by eq 10 with MH1 in the right-hand side (rhs) and n = 0, ..., N - 1. The exchange processes occur via transitions in eq 11, where the transition frequencies are written as followszo
0.4 I
I
(
= 1 - - k-*/3(j,N,n)
0.0
0
1
-
3
2
n
Figure 2. Asymptotic long-time exponential decay constant bo as a function of ii for two-component quenching (---) and concentration with k-* = k-P = k- = O.lk for different valuts of self-quenching N = 3, 5 , 10, m, 10, 5 , 3 (top to bottom). Thelull-drawn curve corrcsponds to the case of micelle infinite solubilization capability where self-quenching is indistinguishable from two-component quenching. (Note that Bo does not include the self-decay.)
81
/1/!
-lnBo
n Figure 3. Amplitude Bo for the asymptotic exponential decay as a
function of iifor two-component quenching (---) at k-p = k- = O.lk, and N = 3, 5 (top to bottom), and for concentration self-quenching (-) at k-* = k- = O.lk and N = 3. The full-drawn curve corresponds to the case of micelle indnite solubilization capability. way as in eqs 12 and 13. S i in a omcomponent system excited probes and ground-state quenchers are the same molecules, it would be reasonable to regard their migration rate constants as equal, &-• k-. However, here we keep the general notation in order to follow the influence of different exchange processes on the resulting excited-state deactivation dynamics. By means of a generating function defined by
-
for the migration of the ground-state probes and
- i)k-*w*0.,n.M
Figure 1. Time evolution of the excited probe survival probabilityflt) for two-component quenching with k-p = k- = O.lk, and ii = 2 for different values of N. (Note that self-decay is excluded.)
(-a)
(31)
C ) I I & O ~ System. Consider an ensemble of luminophores distributed among small monodisperse micelles according to the binomial law, eq 8. Assume that the excitation efficiency of a probe is sufficiently low, so that one micelle cannot contain more than one excited probe, and is projmtional to the number of probes in a micelle. The initial distribution of ground-state probes among the micelles with excited probe after &pulse excitation is then given byZo
(1
i
+
The steady-state observable 'p/cpo, the ratio of the fluorescence quantum yield in the presence and in the absence of quenchers, respectively, can be calculated in accord with P/% = (1/7)A1/7) (30) The plot of (po/'p versus ii at several values of N is displayed in Figure 4. In agreement with the results reported in Figure 1, the smaller the value of N the stronger is the fluorescence quenching efficiency. The observed behavior appreciably deviates from the linear Stem-Volmer dependence even at relatively low quencher concentrations. When ti > VM,where M is the total number of micelles in solution. Once the system is in dynamic equilibrium, the probability for the micelle to contain n probes can be written as
9078 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992
Barzykin
where
2.0
,
3i
t
F
ZtrV e x ~ [ @ ( ~ + h ) l
-
1 "
(43)
Z , is the translationalpartition function, j3 = (ken-' with kBbeing the Boltzmann's constant and T the absolute temperature, I( is the chemical potential and h the hydrophobic potential or the n is the grand thermodynamic potential energy of ~olubilization~~ providing the grand canonical partition function, exp(-@a), and Z,, is the configuration integral 0.0
(44)
where uij = u(lri - ril) and u is the intermolecular interaction potential. We are trymg to understand in a general way the effect of micelle limited solubilization capability, so consider the hard-sphere potential: u(r) equals infinity at r C ro,and equals zero ar r > ro. In the van der Waals approximation the hardsphere configuration integral is given by
2:
N
P ( 1 - n/N)"
4
,
I
0
-2
4
2
Ph Figure 6. Average number of solubilized molecules pcr micelle A as a function of the hydrophobic potential h at [C]/[M] = 2 for Poisson (-), binomial (---), and hard-sphere (-.) distributions. In the two latter cases N = 8. 0.5 0.4
(45)
4
=e
0
2
A I\
ii=5
where N = 3V/2*ro3 is the limit to the number of probes in a micelle. The grand partition function is determined from the normalization condition (the sum of P,, over all possible values of n equals unity) NE"
exp(-@n) = ,,.on!-( 1 - n / N ) "
(46)
The average of the distribution is given by
a = CnP,,= --an N
a=O
aP
(47)
If a is known, one can find from eq 47 and then calculate the resulting distribution {P,,]. The distribution function for the bulk phase can be derived in a similar way. The number density of the probe molecules in this phase pa is usually low, pa
