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On the determination of concentration of dimers in luminescent solutions
JOURNAL
ON THE
OF LUMINESCENCE 5 (1972) 372-378 @ North-Holland Publishing Co.
DETERMINATION
OF CONCENTRATION
IN LUMINESCENT
OF DIMERS
SOLUTIONS
C. BOJARSKI Institute
qf Physics, Technical
Received
University, Gdatisk, Poland
27 February
1972
A new method for determining the concentration of dimers and the equilibrium constant in the process of dimer formation, as well as the critical concentration for excitation energy transfer from monomers to dimers, has been presented. Their magnitudes may be determined on the basis of the measurements of the concentration dependence of emission anisotropy and the quantum efficiency of photoluminescence.
1. Introduction In concentrated luminescent solutions dimers very often appear, as well as associates of higher orders’). The concentrations C” of dimers may be determined by different methods’-‘), among others from the family of absorption curves corresponding to various solution concentrations. There is a significant error in the values of C” determined in this way in the case of systems with small equilibrium constant K of dimerization, as well as in the case when the absorption spectra of dimers and monomers are not significantly different from each other. The method presented below for determining the concentration c” of dimers reduces the whole problem solely to the measurements of the concentration dependence of emission anisotropy (degrees of polarization) and quantum yield of photoluminescence of the systems investigated, provided the solutions are rigid or viscous enough to eliminate rotational depolarization and the concentration quenching in the system is due to the formation of dimers. 2. Description of the method Recently, within the framework of the general concentration on the luminescence of solutions, has been obtained for the quantum yield’, lo): rl
VO
_
l-.f 1 372
elf
theory of the influence of the following expression
(1)
ON THE
and for emission
DETERMINATION
anisotropy’
IN
LUMINESCENT
SOLUTIONS
373
’ ):
(2)
where*
f
E
f(y)
=
TC+ y exp(y2) [ 1 - Tt 2 1 evC-yi)di],
(3)
and
c’ and C” denote concentrations respectively, where
of monomers
D and dimers D,, in solution,
c ‘+ 2cl’ = (‘.
(6)
c is the analytical concentration of the luminescent molecules in solution. The quantity c(~ denotes the constant, independent of concentration, which characterizes the quenching of excitation energy during its transfer between the monomers, while y10is the absolute photoluminescence yield of monomers when c” --f 0. The quantities cb and c6 signify the critical concentrations, that is,14, lo), cb = 5.18 x 1O-‘o n2 (5’)2
[I,
qo]-*
mole lK’,
(7)
where M is the refractive index of the medium, (3’) denotes the mean value of the wave number in the overlapping area of the absorption and emission spectra, while I,- is, in I mole-’ cm- ‘, the overlap integral of these spectra for monomers. In order to compare the empirically determined values of ~1 q0 and rlro with eqs. (1) and (2) it is necessary to know the values of y corresponding to particular concentrations of c. Because of (5) and (6) there is the following relation between the quantities y and c:
* The function
f(y)
also appears
in FGrster’s
theory”,
13).
C.
374
ROJARSKI
For C” < c, which covers a large range of concentrations, y is a linear function of c (also when cb = 2~;). The relation may be determined to a satisfactory approximation from (7) as follows: (9) if q. is known*. In table I values of c, q 1‘lo, r 1r. and y are listed (columns 1-4) for the system of rhodamin 6G in water-glycerol solution taken from refs. IO and 15). In column 5 there are values of y determined on the basis of relation (9). These may be compared with those in column 4 determined from relation (5). For c > 8 x 10m3 M the values of y from column 5 are greater than those presented in column 4, following the increase in concentration c”. It is obvious that this occurs because of the increase in concentration c” with an increase in C, as well as with regard to C; 1ci - 2 < 0 [see eq. (g)]. However, the above difference in y does not exceed I5 per cent. The estimated values of y and their corresponding experimental ones q 1qo, as well as r I ro, may be compared with theory. In fig. I eqs. (1) and (2) are presented for three values of the equilibrium constant’“* K, for the process of dimer formation, and with r. as a parameter. A correct description of the empirical results by eqs. (I ) and (2) imposes in an unequivocal way the acceptance of K, = 0.067 and c(~ = 0.9. For these parametric values of K, and rxo a satisfactory agreement is obtained between theory and experiment (cf. fig. 2). In order to determine the concentration of dimers C” we shall first deal with values yu,, for the respective quantities y. From eqs. (1) (4) and (5) we have
yD and c’ (q. and cb are known). y0, ,, from eq. (5) we determine Finally, we find C” by means of relation (6). The values of 8 determined in this way are presented in column 6. For comparison, in column 7 values of c” taken from ref. IO and determined by spectroscopic investigations are also given. With a knowledge of C” and y,,, , it is possible to determine from
Knowing
* Since the determination of the values of II’, r, and fi -L. in (9) does not present any difficulties, ?,a may be determined by matching the experimental values I.;POcorresponding to the particular values of c (for the concentration range where ?,~Y/ostill remains constant) 0.6 for the system discussed. with the theoretical curve (2). In this way r/o ** The equilibrium where
constant
ON THE
DETERMINATION
3’
”
IN LUMINESCENT
SOLUTIONS
I
m-2
lo-'
1
315
,.-;1 -._
IO
Fig. 1. Theoretical curves of the quantum yield q 1,/o and emission anisotropy mined by eqs. (I) and (2) for various values of K,, and ~0: ~corresponds to no = 0.9; ‘- - - - - to LLO= 0.8.
~1~0 deterto ao = 1;
to I 0.8 0.6
$a4 h ..
-%2
0 -3
W"
ill” T-~
-
Fig. 2. Quantum yield ~170 and emission anisotropy rlro of the photoluminescence of rhodamin 6G in water-glycerol solutions: 0, 0 experimental points; theoretical curves.
x
x x x x x
2
3 4 5 7 8
10-2 10-a 10-z 10-a 10-a
IO-”
10m3 10-a
10-3
10-j 10 1 lolo--’ 10-s 10-a 10 3 10-a
0.993 0.928 0.836 0.820 0.736 0.661 0.579 0.514 0.487 0.405 0.416 0.478 0.455 0.455 0.510 0.529 0.579
2
1
of dimers
0.013 0.045 0.134 0.178 0.222 0.442 0.659 0.870 1.086 1.710 2.118 4.086 5.976 7.818 9.636 13.14 14.88
4
“i
c” of rhodamin
0.013 0.043 0.129 0.173 0.216 0.432 0.648 0.864 1.080 1.728 2.160 4.320 6.480 8.640 10.80 15.12 17.28
5
6G in water-glycerol
1
0.0 0.0 0.0 0.0 0.0 0.0 0.08 0.18 0.28 0.46 0.73 0.19 0.49 1.04 x x x x x x x x
6
solutions:
10-2 10-2 10-s
IO-3
10-a IO-” 10m3 10-a
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.05 0.48 0.90 0.25 0.46 0.71 1.02 1.53 1.95
x x x x x x x x x
7
10-a 10-a lo-” 10-2 IO-’ lo-:’ 10-2 IO-” 10-2
7.2 3.6 3.6 6.2 1.0 0.04 0.08 0.14 0.22 0.51 0.75 0.24 0.45 0.70 0.97 1.56 1.88
x x x x x x x x x x x x x x x x x
8
in M, )/~,a K -mm5.3 poise
cx (mole 1 r)
concentrations
IO-” IO-” 10-P 10-a IO-” 10-T
10-3
10-a
IO-3
lo-” lo-”
IO-3
lo-* 10-r lo-6 IO-” 10-5
for I/O = 0.6 as well as CO’and CO”equal to 3.06 x lo-” M and 2.37 x 1O-3 M, respectively, on the basis of values CO’and CO”are different from those given in ref. 10, since CO and CII” have been computed there for 7,~ ~ 0.5. Let us note that CO’ N )IO~: [cf. (7)].
1.004 1.008 1.018 0.992 1.020 0.930 0.835 0.720 0.621 0.439 0.329 0.089 0.037 0.018 0.010 0.004 0.003
3
‘1Ip/o
The values of ;I have been calculated the data given in ref. 10. The above
x x x x x x x x x x x
6 2 6 8 1 2 3 4 5 8 1
I’ )‘”
of 7, and the concentrations
c
The values
TABLE
x
8 $
0
:
ON THE
DETERMINATION
IN LUMINESCENT
377
SOLUTIONS
eq. (5) the critical concentration for excitation transfer from D* to D,,, and then the equilibrium constant K = c”jcr2 by means of eq. (9’). In table 2 values of c& KY and K, obtained as described above, are compared, as well as values of the same magnitudes determined on the basis of spectroscopic investigations taken from ref. IO (this time calculated for y10 = 0.6). 3. Discussion A comparison of columns 6 and 7 in table I indicates a satisfactory agreement of values c” determined according to the method presented with those evaluated from spectroscopic investigations. It should be emphasized that the new method provides better results than the former one in the low concentrations range (3 x 10m3-5 x 10m3 M). The values of C” collected in column 8 and determined on the basis of the law of mass action for K = 10.4 I mole-’ may be evidence for this. For the largest concentrations (c 2 4 x IO-’ M) a direct determination of c” by means of the proposed method is useless on account of the following: (a) the experimental values q 1u], within this range of concentration carry a considerable error; (b) the values of y determined from eq. (9) are too high (cf. columns 4 and 5). The second component cannot be neglected on the right-hand side of eq. (8) in the case of large c. A comparison of the values c& KY and K collected in table 2 proves that the proposed method allows the determination of the TABLE
Critical concentrations
2
and dimerization constants for the system as in table 1 KY
According to the new method From spectroscopic* investigations
* Values
RI” and KY computed
2.12 2.37
0.068 0.067
K (1 mole-l) 10.4 11.1
for PJO = 0.6.
above values with the right accuracy. The values of cg and K listed in table 2 are the mean values of these quantities computed for six finite values of c” from column 6, with the omission of c” = 1.04 x IO-’ and 8 x 1O-5 M. The first omitted one represents a value of c” which is considerably too high, while the other is sufficiently accurate (cf. also column 8), but its corresponding magnitude yD,, [necessary for computing c& see eq. (5)] cannot be determined from eq. (10) with satisfactory accuracy within the range of small c.
378
C. BOJARSKI
The determination of the concentration c”, as well as the constants ci, K, and K, requires quite simple measurements of the concentration dependence of emission anisotropy rl r. and of the quantum efficiency of photoluminescence q 1‘lo. We hope that the method presented may be useful in the case of luminescent systems for which the concentration dependence of the absorption spectra is insignificant on account of a small value of the equilibrium constant K, or a poor separation of the absorption spectra of monomers and dimers. A further verification of the method seems to be necessary. Acknowledgements I should like to express my gratitude help.
to Mrs A. Sodolska
for her technical
References I) Th. Forster, Fluoreszenz organidler Verhintlun~en (Vandenhoeck und Ruprecht, Gottingen, 1951). 2) Th. Forster and E. Konig, 2. Elektrochem. 61 (1957) 344. 3) V. L. Levshin and J. G. Baranova, Optika i Spektrosc. 6 (1959) 55. 4) V. L. Levshin and L. V. Krotova, Optika i Spektrosc. 13 (1962) 809. 5) K. Bergmann and C. T. O’Konski, J. Phys. Chem. 67 (1963) 2169. 6) W. West and S. Pearce, J. Phys. Chem. 69 (1965) 1894. 7) K. Sauer, J. R. Lindsay Smith and A. J. Schultz, J. Am. Chem. Sot. 88 (1966) 2681. 8) A. R. Monahan and D. F. Blossey, J. Phys. Chem. 74 (1970) 4014. 9) C. Bojarski and J. Domsta, Acta Phys. Hung. 30 (1971) 145. 10) C. Bojarski, J. KuSba and G. Obermiiller, 2. Naturf. 26a (1971) 255. 11) C. Bojarski, J. Luminescence, in press. 12) Th. Forster, 2. Naturf. 4a (1949) 321. 13) Th. Forster, Disc. Faraday Sot. 27 (1959) 7. 14) Th. Forster, Ann. Phys., Leipzig 2 (1948) 55. 15) C. Bojarski and J. Dudkiewicz, Z. Naturf. 26a (1971) 1028.
ON THE
OF LUMINESCENCE 5 (1972) 372-378 @ North-Holland Publishing Co.
DETERMINATION
OF CONCENTRATION
IN LUMINESCENT
OF DIMERS
SOLUTIONS
C. BOJARSKI Institute
qf Physics, Technical
Received
University, Gdatisk, Poland
27 February
1972
A new method for determining the concentration of dimers and the equilibrium constant in the process of dimer formation, as well as the critical concentration for excitation energy transfer from monomers to dimers, has been presented. Their magnitudes may be determined on the basis of the measurements of the concentration dependence of emission anisotropy and the quantum efficiency of photoluminescence.
1. Introduction In concentrated luminescent solutions dimers very often appear, as well as associates of higher orders’). The concentrations C” of dimers may be determined by different methods’-‘), among others from the family of absorption curves corresponding to various solution concentrations. There is a significant error in the values of C” determined in this way in the case of systems with small equilibrium constant K of dimerization, as well as in the case when the absorption spectra of dimers and monomers are not significantly different from each other. The method presented below for determining the concentration c” of dimers reduces the whole problem solely to the measurements of the concentration dependence of emission anisotropy (degrees of polarization) and quantum yield of photoluminescence of the systems investigated, provided the solutions are rigid or viscous enough to eliminate rotational depolarization and the concentration quenching in the system is due to the formation of dimers. 2. Description of the method Recently, within the framework of the general concentration on the luminescence of solutions, has been obtained for the quantum yield’, lo): rl
VO
_
l-.f 1 372
elf
theory of the influence of the following expression
(1)
ON THE
and for emission
DETERMINATION
anisotropy’
IN
LUMINESCENT
SOLUTIONS
373
’ ):
(2)
where*
f
E
f(y)
=
TC+ y exp(y2) [ 1 - Tt 2 1 evC-yi)di],
(3)
and
c’ and C” denote concentrations respectively, where
of monomers
D and dimers D,, in solution,
c ‘+ 2cl’ = (‘.
(6)
c is the analytical concentration of the luminescent molecules in solution. The quantity c(~ denotes the constant, independent of concentration, which characterizes the quenching of excitation energy during its transfer between the monomers, while y10is the absolute photoluminescence yield of monomers when c” --f 0. The quantities cb and c6 signify the critical concentrations, that is,14, lo), cb = 5.18 x 1O-‘o n2 (5’)2
[I,
qo]-*
mole lK’,
(7)
where M is the refractive index of the medium, (3’) denotes the mean value of the wave number in the overlapping area of the absorption and emission spectra, while I,- is, in I mole-’ cm- ‘, the overlap integral of these spectra for monomers. In order to compare the empirically determined values of ~1 q0 and rlro with eqs. (1) and (2) it is necessary to know the values of y corresponding to particular concentrations of c. Because of (5) and (6) there is the following relation between the quantities y and c:
* The function
f(y)
also appears
in FGrster’s
theory”,
13).
C.
374
ROJARSKI
For C” < c, which covers a large range of concentrations, y is a linear function of c (also when cb = 2~;). The relation may be determined to a satisfactory approximation from (7) as follows: (9) if q. is known*. In table I values of c, q 1‘lo, r 1r. and y are listed (columns 1-4) for the system of rhodamin 6G in water-glycerol solution taken from refs. IO and 15). In column 5 there are values of y determined on the basis of relation (9). These may be compared with those in column 4 determined from relation (5). For c > 8 x 10m3 M the values of y from column 5 are greater than those presented in column 4, following the increase in concentration c”. It is obvious that this occurs because of the increase in concentration c” with an increase in C, as well as with regard to C; 1ci - 2 < 0 [see eq. (g)]. However, the above difference in y does not exceed I5 per cent. The estimated values of y and their corresponding experimental ones q 1qo, as well as r I ro, may be compared with theory. In fig. I eqs. (1) and (2) are presented for three values of the equilibrium constant’“* K, for the process of dimer formation, and with r. as a parameter. A correct description of the empirical results by eqs. (I ) and (2) imposes in an unequivocal way the acceptance of K, = 0.067 and c(~ = 0.9. For these parametric values of K, and rxo a satisfactory agreement is obtained between theory and experiment (cf. fig. 2). In order to determine the concentration of dimers C” we shall first deal with values yu,, for the respective quantities y. From eqs. (1) (4) and (5) we have
yD and c’ (q. and cb are known). y0, ,, from eq. (5) we determine Finally, we find C” by means of relation (6). The values of 8 determined in this way are presented in column 6. For comparison, in column 7 values of c” taken from ref. IO and determined by spectroscopic investigations are also given. With a knowledge of C” and y,,, , it is possible to determine from
Knowing
* Since the determination of the values of II’, r, and fi -L. in (9) does not present any difficulties, ?,a may be determined by matching the experimental values I.;POcorresponding to the particular values of c (for the concentration range where ?,~Y/ostill remains constant) 0.6 for the system discussed. with the theoretical curve (2). In this way r/o ** The equilibrium where
constant
ON THE
DETERMINATION
3’
”
IN LUMINESCENT
SOLUTIONS
I
m-2
lo-'
1
315
,.-;1 -._
IO
Fig. 1. Theoretical curves of the quantum yield q 1,/o and emission anisotropy mined by eqs. (I) and (2) for various values of K,, and ~0: ~corresponds to no = 0.9; ‘- - - - - to LLO= 0.8.
~1~0 deterto ao = 1;
to I 0.8 0.6
$a4 h ..
-%2
0 -3
W"
ill” T-~
-
Fig. 2. Quantum yield ~170 and emission anisotropy rlro of the photoluminescence of rhodamin 6G in water-glycerol solutions: 0, 0 experimental points; theoretical curves.
x
x x x x x
2
3 4 5 7 8
10-2 10-a 10-z 10-a 10-a
IO-”
10m3 10-a
10-3
10-j 10 1 lolo--’ 10-s 10-a 10 3 10-a
0.993 0.928 0.836 0.820 0.736 0.661 0.579 0.514 0.487 0.405 0.416 0.478 0.455 0.455 0.510 0.529 0.579
2
1
of dimers
0.013 0.045 0.134 0.178 0.222 0.442 0.659 0.870 1.086 1.710 2.118 4.086 5.976 7.818 9.636 13.14 14.88
4
“i
c” of rhodamin
0.013 0.043 0.129 0.173 0.216 0.432 0.648 0.864 1.080 1.728 2.160 4.320 6.480 8.640 10.80 15.12 17.28
5
6G in water-glycerol
1
0.0 0.0 0.0 0.0 0.0 0.0 0.08 0.18 0.28 0.46 0.73 0.19 0.49 1.04 x x x x x x x x
6
solutions:
10-2 10-2 10-s
IO-3
10-a IO-” 10m3 10-a
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.05 0.48 0.90 0.25 0.46 0.71 1.02 1.53 1.95
x x x x x x x x x
7
10-a 10-a lo-” 10-2 IO-’ lo-:’ 10-2 IO-” 10-2
7.2 3.6 3.6 6.2 1.0 0.04 0.08 0.14 0.22 0.51 0.75 0.24 0.45 0.70 0.97 1.56 1.88
x x x x x x x x x x x x x x x x x
8
in M, )/~,a K -mm5.3 poise
cx (mole 1 r)
concentrations
IO-” IO-” 10-P 10-a IO-” 10-T
10-3
10-a
IO-3
lo-” lo-”
IO-3
lo-* 10-r lo-6 IO-” 10-5
for I/O = 0.6 as well as CO’and CO”equal to 3.06 x lo-” M and 2.37 x 1O-3 M, respectively, on the basis of values CO’and CO”are different from those given in ref. 10, since CO and CII” have been computed there for 7,~ ~ 0.5. Let us note that CO’ N )IO~: [cf. (7)].
1.004 1.008 1.018 0.992 1.020 0.930 0.835 0.720 0.621 0.439 0.329 0.089 0.037 0.018 0.010 0.004 0.003
3
‘1Ip/o
The values of ;I have been calculated the data given in ref. 10. The above
x x x x x x x x x x x
6 2 6 8 1 2 3 4 5 8 1
I’ )‘”
of 7, and the concentrations
c
The values
TABLE
x
8 $
0
:
ON THE
DETERMINATION
IN LUMINESCENT
377
SOLUTIONS
eq. (5) the critical concentration for excitation transfer from D* to D,,, and then the equilibrium constant K = c”jcr2 by means of eq. (9’). In table 2 values of c& KY and K, obtained as described above, are compared, as well as values of the same magnitudes determined on the basis of spectroscopic investigations taken from ref. IO (this time calculated for y10 = 0.6). 3. Discussion A comparison of columns 6 and 7 in table I indicates a satisfactory agreement of values c” determined according to the method presented with those evaluated from spectroscopic investigations. It should be emphasized that the new method provides better results than the former one in the low concentrations range (3 x 10m3-5 x 10m3 M). The values of C” collected in column 8 and determined on the basis of the law of mass action for K = 10.4 I mole-’ may be evidence for this. For the largest concentrations (c 2 4 x IO-’ M) a direct determination of c” by means of the proposed method is useless on account of the following: (a) the experimental values q 1u], within this range of concentration carry a considerable error; (b) the values of y determined from eq. (9) are too high (cf. columns 4 and 5). The second component cannot be neglected on the right-hand side of eq. (8) in the case of large c. A comparison of the values c& KY and K collected in table 2 proves that the proposed method allows the determination of the TABLE
Critical concentrations
2
and dimerization constants for the system as in table 1 KY
According to the new method From spectroscopic* investigations
* Values
RI” and KY computed
2.12 2.37
0.068 0.067
K (1 mole-l) 10.4 11.1
for PJO = 0.6.
above values with the right accuracy. The values of cg and K listed in table 2 are the mean values of these quantities computed for six finite values of c” from column 6, with the omission of c” = 1.04 x IO-’ and 8 x 1O-5 M. The first omitted one represents a value of c” which is considerably too high, while the other is sufficiently accurate (cf. also column 8), but its corresponding magnitude yD,, [necessary for computing c& see eq. (5)] cannot be determined from eq. (10) with satisfactory accuracy within the range of small c.
378
C. BOJARSKI
The determination of the concentration c”, as well as the constants ci, K, and K, requires quite simple measurements of the concentration dependence of emission anisotropy rl r. and of the quantum efficiency of photoluminescence q 1‘lo. We hope that the method presented may be useful in the case of luminescent systems for which the concentration dependence of the absorption spectra is insignificant on account of a small value of the equilibrium constant K, or a poor separation of the absorption spectra of monomers and dimers. A further verification of the method seems to be necessary. Acknowledgements I should like to express my gratitude help.
to Mrs A. Sodolska
for her technical
References I) Th. Forster, Fluoreszenz organidler Verhintlun~en (Vandenhoeck und Ruprecht, Gottingen, 1951). 2) Th. Forster and E. Konig, 2. Elektrochem. 61 (1957) 344. 3) V. L. Levshin and J. G. Baranova, Optika i Spektrosc. 6 (1959) 55. 4) V. L. Levshin and L. V. Krotova, Optika i Spektrosc. 13 (1962) 809. 5) K. Bergmann and C. T. O’Konski, J. Phys. Chem. 67 (1963) 2169. 6) W. West and S. Pearce, J. Phys. Chem. 69 (1965) 1894. 7) K. Sauer, J. R. Lindsay Smith and A. J. Schultz, J. Am. Chem. Sot. 88 (1966) 2681. 8) A. R. Monahan and D. F. Blossey, J. Phys. Chem. 74 (1970) 4014. 9) C. Bojarski and J. Domsta, Acta Phys. Hung. 30 (1971) 145. 10) C. Bojarski, J. KuSba and G. Obermiiller, 2. Naturf. 26a (1971) 255. 11) C. Bojarski, J. Luminescence, in press. 12) Th. Forster, 2. Naturf. 4a (1949) 321. 13) Th. Forster, Disc. Faraday Sot. 27 (1959) 7. 14) Th. Forster, Ann. Phys., Leipzig 2 (1948) 55. 15) C. Bojarski and J. Dudkiewicz, Z. Naturf. 26a (1971) 1028.