1350
J. Q. UMBERGER
Solution Kinetics via Fluorescence Quenching-Transient and Solvent Effects by J. Q. Umberger Holmdel, New Jersey
07733
(Received October 23, 1967)
The Smoluchowski-Sveshnikoff diffusion treatment is modified in accordance with new experimental data particularly for high concentrations of quencher. The data indicate that proton transfer from the solvent to the photoexcited dye, in polarized association with the quencher, might be the quantum event leading t o quenching of uranin.
Introduction
or
The Smoluchowski' treatment of collision processes in solution was generalized by Umberger and La Mer2--following Debye3-for diffusion in a potential field as in the case of ions. I n applying this Fick's law calculation to their fluorescence-quenching measurements, Umberger and La Mer included the transient by which Sveshnikoff rationalized departures from Stern-Volmer kinetics. At the suggestion of Kimball, however, the subsequent co~orkers5-~of La Mer abandoned the transient. Later, in a theoretical paper, Kimball and Collins* reconsidered and generalized Smoluchowski's boundary condition ; this vindicated transient diffusion for fluorescence quenching, in general, and the original Smoluchowski boundary condition for rapid quenching, in particular. The above diffusion treatments emphasize the influence of solvent viscosity and dielectric strength, but Umbergerg*'o has recently shown that solvent proticity, or H-bond donor strength, also can influence fluorescence quenching. The general equation of Collins and Kimball has proven unwieldy in interpreting solvent and transient effects;" l 2 thus simpler boundary condition and equation are presented and tested against new quenching data.
Theory Calculation of the Quenching Constant, k q o , for Low Concentration of Quencher. I n the calculation of Sveshnikoff,4 the excited dye molecule is made the center of diffusion, but the calculation is easier when the yuencher is made the center-provided the individual quencher molecules are far apart. At the photostationary state, the average concentration, N , of excited dye molecules at distance r from the central quencher is independent of time, i.e. -4.
bN bN - = Q = DV.at
br
- N- + constant ro
excess of loss of formation in-diffusing excited dye of excited over out- molecules dye molecules diffusing by fluoresby light excited dye cence/cc absorption/ molecules/ sec cc sec ca sec
The Journal of Physical Chemistry
(1)
o = -D- d2(rN) - -N+ - No r
dr2
r0
r0
The boundary conditions are: (a) when r = 0 0 , N = No, where No is the concentration of excited dye molecules in the absence of quencher; (b) when r = R , N = a N o , where a is a constant less than unity. Quenching starts at this center-to-center distance between dye and quencher: R = 1*d rq 6 , i.e., R exceeds, by small amount 6, the sum of the radii of dye andquencher. The solution of eq 2 n4th boundary conditions (a) and (b) is
+ +
Equations 3 and 4 apply when r >= R. It is assumed that quenching occurs within the spherical shell of thickness 6 and volume v = 4nR26,Le., between r = R and r = R - 6. The shell is assumed sufficiently thin that the concentration of excited dye molecules within is maintained at a N o by the diffusion (1) M. V. Smoluchowski, Ann. Physik, 48, 1103 (1915); 2. Physik. Chem., 92, 129 (1917). (2) J . Q. Umberger and V. K. La Mer, J . Am. Chem. Sac., 67, 1099 (1 945). (3) P. Debye, Trans. Electrochem. Sac., 82, 265 (1942). (4) B. Sveshnikoff, Acta Physicochim. U.R.S.S., 3, 257 (1935). (5) B. Williamson and V. K. La Mer, J . Am. Chem. Sac., 70, 717 (1948). (6) K . C. Hodges and V. K . LEIMer, {hid., 7 0 , 722 (1948). (7) J . C. Rowel1 and V. K . La Mer, i h i d . , 73, 1630 (1951). (8) F. C . Collins and G. E. Kimball, J . Colloid Sci., 4, 425 (1949); 5, 499 (1950); Ind. Eng. Chem., 41, 2551 (1949). (9) (a) J . Q. Umberger, J . Phys. Chem., 71, 2054 (1967); (b) ihid., 71, 4588 (1967). (10) (a) J. Q. Umberger, Phot. Sci. Eng., 11, 385 (1967); (b) ibid., 11, 392 (1967). (11) (a) R . IT.Noyes, J . Am. Chem. Soc., 86, 4529 (1964); (b) ibid., 79, 551 (1957); (e) J . Phys. Chem., 69, 3182 (1965); (d) Progr. Reaction Kinetics, 1, 131 (1961). (12) W. R. Ware and J. S. Novros, J . Phys. Chem., 70, 3246 (1966).
1351
SOLUTION KINETICSvia FLUORESCENCE QUENCHING
70
7,
diffusion of excited dye molecules into the shell
loss of excited dye molecules from the shell by quenching
loss of excited dye molecules from the shell by fluorescence
formation of excited dye molecules in the shell by light absorption
(5)
The effect of possible association between quencher and dye, prior to excitation, is considered later. The quenching constant at low concentration of quencher is k," = avro/rq,where 7 , is the average time required for quenching of excited dye molecules within the shell. On inserting the value of a obtained by eliminating dN/drIR from eq 4 and 5 , k," becomes
k,"
=
+
[4aRroD(1 R / d r T ) ____ 1 [4nRroD(l R/drq)
+
+
+ +
V]
v]~q/(~ov)
(6)
The three ternis in the numerator of eq 6, uiz., 4aRrOD, 4 a R 2 d 7 3 , and v, depend, respectively, on the first power, the one-half power, and the zeroth power of the diffusion constant D. These terms might be regarded as arising, respectively, when dye and quencher are originally far apart, at medium distance requiring less diffusion for quenching, and in the quenching shell requiring no diffusion. This last term represents static quenching. Equation 6 can be simplified if v is small and if 7 , is replaced by 6/lc, where k is the specific quenching rate of Collins and IGmball* k,"
=
+
Equation 7 can be confirmed by the alternate calculation where the excited dye molecule is made the center of diffusion. The correct expression for the quencher flux, @*, is8 9 , = 4aR2D(g)
R
4aDCo[R
=
-p
In eq 10 when r = R, C = aCo; when r = , C = Co; when t = a ,C = Co[l - (1 - a ) R / r ] ; when t = 0, C = Coexcept at r = R. These represent the boundary conditions for actual quenching except that the initial concentration of quencher within the thin quenching shell at r = R actually should be Co rather than aCo. This difference can be allowed for if, on excitation, the central dye molecule is assumed to react instantaneously with a fraction, (1 - a), of the original amount of quencher, vC0, in the shell. I n other words, very rapid statio quenching is assumed to reduce the quencher concentration in the shell to UNObefore significant diffusion occurs. With eq 10 and corrected boundary condition at r = R, Sveshnikoff's method of calculation now yields 10[1
4nRroD(l R / d a ) (7) 1 (1 R I ~ ~ I D I ~ R
+ +
When rearranged and expanded, eq 7 can be placed in a form agreeing exactly with eq 9. Calculation of the Quenching Constant, k,, f o r A n y Concentration of Quencher. Though eq 8 must be integrated exactly to obtain k , for all, including high, concentrations of quencher, a good approximation for moderate concentrations is obtained if the following expression is employed for C as a function of time and distance from the central excited dye molecule
+ (R - P)2eDt/" P
P
where
27ef
QI
=
-
(1 - a)vCo]/I = (1
+ Lyro)6
~ T R D N A [ Q ]( ~a)/1000;
dT-2- S'e"'
dz]; y
=
1/6
(11) =
1
(R/d?rD)ac/(a
0
-
+
l/r0)"*; [Q] = concentration of quencher in moles per liter; N A = Avogadro's constant. At low concentrations of quencher, eq 11 correctly reduces to an expression for k," identical with eq 6. At somewhat higher concentrations of quencher and when v is small, eq 11 can be expanded in a form useful for interpreting experimental data a--2 L
4-
Q
This, when employed in the Sveshnikoff-type calculation, yields
where the symbols are those of Collins and Kimball. Equation 8 can be integrated a t low concentrations of quencher, Co, for then the complicated exponential can be simply expanded. I n this way, eq 9 is ultimately obtained.
where the dimensionless parameter g = r0D/R2. This predicts an almost linear plot of k,/k," us. k," [Q] with slope[(a - 2 ) / n d g / 2 ] / ( 1 There should, however, be slight downward curvature as predicted from the small negative value of the second derivative, [ 2 ( a - 3)/a - g/4]/(1 dg)3, and from the requirement that k , must approach the value 4rR2kro at large
+
+ di)2.
+
Volume 72,Number
April 1968
J. Q. UMBERGER
1352 Table I : k , us. Solvent Proticity for Fluorescein Dianion (Uranin) kqb
Relative viscositya
Dieleotrio constant
Solvent
at 2 5 O
at 2 5 O
Water Methanol Ethanol Dimethylformamide (DMF)
1 7 . 1 sec 13.4 24.1 16.2
79 33 24 38
kq
(iodide quenching) a t 27O
(aniline quenching) a t 27O
H-bond donor strengths
15 1.5
28 19 10 5
Strong Med. strong Medium Weak
... 0.0
Drainage time of pure solvent through vacuum-jacketed glass capillary. IC, = ((Io/I)- 1)/[&] l./mole: IO is fluorescence of 1OyoHzO by volume; I is fluorescence of same but 0.10 &I in KI or 0.05 M in aniline; materials were analytical reagent where available. a
10-6 144 uranin in 90y0 organic solvent
+
concentrations of quencher. The slope of the plot should be greater than predicted by eq 12 if the quenching shell thickness, 6, becomes appreciable compared with R , then 8-2
lCq/kq0
=
1 3. [ L g
di
+T+ ; .
"1
g T G
kq"
[&I (13)
Experimental Section I n measuring the effect of transient diffusion on quenching, IC, was determined for: (a) M fluorescein with NaOH added to about pH 12 and employing aniline hydrochloride (plus NaOH to neutralize to pH 12) as quencher in 0.02, 0.04, 0.08, and 0.16 M concentrations; (b) loF5 M fluorescein in presence of 0.002 M HClOe and employing KI as quencher in 0.02, 0.04, 0.08, and 0.16 M concentrations; and (c) 10-6 M fluorescein with NaOH added to pH 12 and employing I
