Deconvolution in Single Photon Counting (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29)
Ch. Soret, C. R . Acad. Sci., faris, 91, 289 (1880). Ch. Soret, Ann. Cbim. fbys. [5], 22, 293 (1888). G. Lipmann, Compt. Rend., 145, 104 (1907). M. Aubert, Ann. Cbim. fbys. [a], 20, 145, 551 (1912). H. Freundlich, "Kapillarchemie", 2nd ed, Akademische Verlagsgesellschaft, Leipzig, 1922 p 371; ibid., Vol. 1, 3rd ed., 1930. B. V. Derjarguin and G. Sidorenkov, Doki. Acad. Sci. USSR, 32, 622 (1941). H. P. Hutchinson, I. S.Nixon, and K. G. Denbigh, Discuss. Faraday Soc., 3, 86 (1948). H. F. Winterkorn, "Proceedings of the 27th Annual Meeting of the Highway Research Board", 1947, p 433. R. Haase, Z. Pbys. Chem. (Frankfurt am Main), 21, 244 (1959). R. Haase and C. Steinert, Z. fbys, Cbem. (Frankfurtam Main), 21, 270 (1959). R. Haase and H. J. De Greiff, Z. fbys. Cbem. (Frankfurt am Main), 44, 301 (1965). R. Haase, Z. fbys. Cbem. (Frankfurt am Main), 51, 315 (1966). R. Haase, H. J. De Greiff, and P. Buchner, Z. Naturforscb. A , 25, 1080 (1970). R. Haase and H. J. De Greiff, Z . Naturforsch. A , 20, 1773 (1971). C. W. Carr and K. Sollner, J . Electrochem. SOC.,109, 616 (1962). R. P. Rastogi, Blokhra, and R. K. Agarwal, Trans. Faraday Soc., 60, 1386 (1964). R. P. Rastogl and K. Singh, Trans. Faraday Soc., 62, 1754 (1966). R. P. Rastogi, K. Singh, and P. C. Skukla, Proc. Int. Symp. Fresh Water Sea, 3rd, 4, 203 (1970). R. P. Rastogl, K. Singh, and P. C. Skukla, Indian J. Cbem., 9, 1372 (1971). Y. Kobatake and H. Fujita, J. Cbem. fbys., 41, 2963 (1964). N. Riehl, Z. Elektrochem., 49, 306 (1943). H. Voellmy and P. Lauger, Ber. Bunsenges. Phys. Cbem., 70, 165 (1966). M. S. Dariel and 0.Kedem, J . fbys. Cbem., 79, 336 (1975). W. E. Goldstein and F. H. Verhoff, AICbE J . , 21, 229 (1975). H. Vink and S.A. A. Chishti, J . Membr. Sci., 1, 149 (1976). F. S.Gaeta, Pbys. Rev., 182, 289 (1969). F. S.Gaeta, Sommerschule "Physik des flussigen Zustandes", St. Georgen, Sept 18-29, 1972, F. Kohler and P. Weinzierl, Ed., Universitiit Wien, IV, pp 9-29.
Tbe Journal of Physical Chemisfry, Vol. 83, No. 17, 7979 2285 (30) F. S.Gaeta and A. Di Chiara, J . Polym. Sci., Pbys. Ed., 13, 163 (1975). (31) F. S.Gaeta and N. M. Cursio, J. Poiym. Sci. A- 1, 7 , 1697 (1969). (32) F. S. Gaeta, A. Di Chiara, and G. Perna, Nuovo Cimento 6 , Ser. X . 06. 260 (1970). (33) G.' Brescia, E. Grossetti, and F. S. Gaeta, Nuovo Clmento 6 , Ser. XI, 8, 329 (1972). (34) F. S.Gaeta, G. Brescia, A. Di Chiara, and G. Scala, J. Polym. Sci., fbys. Ed., 13, 177 (1975). (35) F. S.Gaeta, G. Perna, and G.Scala, J. Polvm. Sci., fbys. Ed., 13, 203 (1975). (36) F. S.Gaeta, D. G. Mita, G. Perna, and G. Scala, Nuovo Cimento 6, Ser. X I , 30, 153 (1975). (37) F. S.Gaeta, "Proceedings of the IVth International Winter School on the Biophysics of Membrane Transport", WlsYa, Poland, Feb 19-28, 1977, part 111, pp 67-106. (38) B. V. Derjarguin, Nature (London), 138, 330 (1936). (39) E. Forslind, Acta Polytecbi., 115, 9 (1952). (40) E. Forslind, "Proceedings of the Second Internatlonal Congress on Rheology", Butterworths, London, 1953. (41) B. V. Derjarguin and A. S.Titjevskaya, Proc. Int. Congr. Surf. Act., 2nd. 1. 211 119571. (42) B. V.Derjarghn and V. V. Karassev, Proc. h t . Congr. Surf. Act., Znd, 3, 531 (1957). (43) . . M. S.Metsik, "Research in the Flekl of Surface Forces", Vol. 11, Nauka Press, Moscow, 1964, p 138. (44) N. V. Churaev, B. V. Derjarguin, and P. P. Zolotarev, Dokl. Acad. Nauk USSR, 183, 1139 (1968). (45) J. Clifford, Water, Compr. Treat., 5, 75-132 (1975). (46) F. S.Gaeta and D. G. Mita, J . Membr. Sci., 3, 191 (1978). (47) F. Belluccl, M. Bobik, E. Drbll, F. S. Gaeta, D. G. Mita, and G. Orlando, Can. J. Cbem. Eng., 30, 698 (1978). (48) F. Belluccl, E. Drioli, F. S.Gaeta, D. G. Mita, N. Pagliuca, and F. G. Summa, J. Cbem. Soc., Faraday Trans. 2 , 75, 247 (1979). (49) A. H. Emery, Jr., and H. G. Drickamer, J . Cbem. Phys., 23, 2252 (1955). (50) G. Langhammer, Naturwissenschafien, 41, 525 (1954). (51) 6. Langhammer and K. Quitzsch, Makromol. Cbem., 17, 74 (1955). (52) G. Langhammer, H. Pfenning, and K. Quitzsch, Z. Electrochem., 62, 458 (1955).
Applications of Fast Fourier Transform to Deeonvolution in Single Photon Countingt J. C. Andre, Laboratoire de Cbimie Generale, E.R.A. No. 136 du C.N.R.S.,E.N.S.I.C., 52042 Nancy, Cedex, France
L. M. Vincent, L.S.G. C.-E.N.S.I.C., 54042 Nancy, Cedex, France
D. O'Connor, and W. R. Ware* The Photochemistry Unit?Department of Chemistry, University of Western Ontario, London, Ontario, N6A 587. Canada (Received August 28, 1978; Revised Manuscript Received April 30, 1979)
A technique, based on the fast Fourier transform (FFT), for the deconvolution of fluorescence decay curves is proposed. To use this technique the exact functional form of the fluorescence decay law need not be assumed. The difficulties usually encountered with the Fourier transforms of noisy data are greatly reduced by choosing a limit v1 in the frequency domain and extrapolating the transformed function for v > v1 by the Fourier transform of an exponential. Inverse transformation leads in most cases to a deconvoluted decay law practically free of unwanted oscillations.
Introduction Fluorescence decay curves of electronically excited molecules that are pumped by flashes of light of short duration are distorted by the finite width of the excitation pulses and by the limited frequency response of the This is publication no. 227 of t h e Photochemistry Unit,
OO22-3654/79/2083-2285$01 .OO/O
e1ectronics.l Because of the linear nature of the fluorescent system and the detecting apparatus one can write the COnvolution equation f(t) = e(t)*d(t)*r(t)
(1)
in which f(t) is the observed decay curve, e(t) is the true time profile of the exciting pulse, r(t) is the time response 0 1979 American Chemical Society
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The Journal of Physical Chemistry, Vol, 83, No. 17, 1979
of the electronics, and d(t) is the fluorescence decay law, or the response of the fluorescent system to a Dirac 6 pulse. Similarly the observed time profile of the exciting pulse, g(t), may be written g(Q = e(t)*r(t) (11) When the decay curve is measured in a single photon counting experiment a number of techniques may be used to recover d(t) from the observed time profiles f(t) and g(t). Among these the following may be mentioned: (i) At the convenient wavelength of 337 nm an air lamp has a relatively high intensity. However, it has been shown2 that the time profile of such an excitation source is wavelength dependent and therefore the e(t) in eq I cannot be derived from a measurement of g(t) at the emission wavelength. It is necessary to assume that r(t) in eq I and I1 is wavelength independent. An alternative approach is the assumption that changing wavelength leads only to a small shift in the time base of e(t) while it leaves the overall profile unchanged. (ii) A less simple technique, when an air lamp is used for excitation, is the measurement of a decay curve of a molecule, the decay time of which is known exactly and the emission of which is in the same wavelength region as the fluorescence of the sample. One observes fo(t) = e(t)*do(t)*r(t) (111) The zero subscript refers to the reference molecule. e(t)* r(t) may now be ~ a l c u l a t e d . ~ (iii) Hydrogen (or deuterium) flash lamps lack the intensity of air lamps but have a time profile, e@),that is virtually independent of wavelength. g(t) can therefore be measured a t the emission wavelength. (iv) Synchrotron radiation may be used as the source of excitation. The pulse shape, which is wavelength independent, is Gaussian and can be calculated e ~ a c t l y .Eq ~ I and 11, when combined, yield f(t) = g(t)*d(t) (IV) If g(t) can be measured, or is known, it is possible, in principle, to determine d(t) from the measurement of f(t) and g(t). For the recovery of d(t) from eq IV a number of mathematical techniques have been proposed: (i) iterative re~onvolution~ (g(t) is convoluted with a chosen function d,(t). Adjustable parameters in d,(t) are varied until the best fit between the reconvoluted and the observed f(t) is obtained. This method is usually used when the functional form of d(t) is known to be a single or a double exponential); (ii) the method of moments6 and the Laplace transform method,'^^ (iii) the method of modulating functions, which is applied especially to multiexponential decay^;^ (iv) deconvolution, in which no a priori assumptions about the functional form of d(t) are made. Two different techniques have been proposed FourierS1l transformation, and the exponential series method.lbJ2 The advantages and disadvantages of these various techniques, as applied to the analysis of double exponential decays, have been discussed re~ent1y.l~ However, if it is not possible to express d(t) by a simple exponential or sum of exponentials (as, for example, in Smoluchowski decay, kinetics associated with the competition between diffusion and chemical reaction in exciplex formation14), the determination of d(t) can be achieved with a classical nonlinear least-squares procedure or with a deconvolution technique which involves no a priori assumptions. When d(t) has been calculated, the decay parameters may be obtained by numerical fitting. However, the use of this technique in two steps reduces the time of calculation with a computer (by a factor of 10 a t least).
We have attempted to develop a mathematical technique, easy to use and of reasonable cost, for the analysis of complex decays. The difficulties, in general, are as follows: (i) For deconvolution with the Laplace transform method, one must calculate integrals of the type
which are time consuming and difficult to calculate to the necessary precision when large values of s are (ii) We have also found that in the analysis of very complex decays or of instrumental functions the exponential series method proposed by Ware et al.lb is not always a satisfactory route to the determination of d(t). For these reasons we have attempted to devise a technique by using the fast Fourier transform (FFT). Although this general technique has been applied in many areas of research (see, for example, ref 15-19), it has, to date, received little attention as a means of analyzing fluorescence decay curves.10i28The theory of its application to multiexponential decays, and the effect of Poissonian noise on the results, has been discussed by Provencher.20,21 In a recent publication Wild et a1.l0 proposed a new technique for analyzing with Fourier transforms fluorescence decay curves measured with the single photon counting technique. The decay curves considered had single exponential decay laws. Since the decay parameters are determined in Fourier space, the technique may be regarded as intermediate between the reconvolution and deconvolution approaches. It is particularly suited to the instrumental method used by Wild et al.1° However, the mathematical technique consists essentially in considering only a part of the Fourier coefficients. In the transition from Fourier space to real space this leads to a transfer function, dl(t), that contains spurious oscillations. We wished to develop an extension of the method that should eliminate this effect. The FFT calculation of the Fourier coefficients employs a well-known algorithm that leads to a discrete number of coefficients (the frequency domain extends from the limit v = 0 to the limit u = yo = 1 and is divided into N channels). 11. Effects of Noise on FFT With the Fourier transforms
F(v) = l , f ( t ) e-2nivtdt (VI) G(v) = l _ g ( t ) e-2aivtdt we can, in principle, by using FFT techniques discussed in the literature (cf. for example, ref 23-27), obtain D(v), the Fourier transform of d(t) by means of the equation D(v) = F(v)/G(v) (VII) However, it is well known that measurements performed with the single photon counting technique introduce Poissonian noise into the data, which leads to an uncertainty, nL1/2,in the number of counts, nj,a t a particular time, t,. When d(t) is calculated by the FFT technique without any assumptions about the functional form or the physical nature of d(t), the presence of this noise may give rise to a mathematically exact result that has no physical significance. Such a result is illustrated in Figure l a and lb. It was obtained by using, without modification, a standard FFT deconvolution technique and agrees with the results published by Wild et al.1° These authors show that the standard deviations of the Fourier coefficients
Deconvolution in Single Photon Counting
The Journal of Physical Chemlsity, Vol. 83, No. 17, 1979 2287 VIR, UII.
TABLE 11: Pyrene Fluorescence Quenched by Biacetyl (0.05M ) in a Mixture of Cyclohexane (0.25)and Cyclohexanol (0.75)at 22 "Ca a b lo-' *1 ns- 1 / 2 .x channels ns-' 0.003 104 4.1 6.68 5.50 5.0 0.004 57 5.45 5.0 47 0.008 5.50 38 5.0 0.014 5.40 5.1 33 0.030 6.70 4.3 28 0.036 7.19 24 4.95 0.070 6.9 4.95 19 0.080 12.7 4.45 15 0.10 16.2 6.09 11 0.20 2.7 31.6 7 0.40 1.39 33.8 4 0.50 d(t) is filled with exp(-at - b t ' / ' ) , a and b are calculated with an iterative convolution program: a = 5.0 x lo-' ns-112, ns-I , b = 5.5 x
constant^.^^^^^ An example of this type of decay, the quenching of the fluorescence of pyrene by biacetyl in a mixture of cyclohexane and cyclohexanol, was studied. In this case, in a second step, a fitting method is satisfactory for recovering the parameters from d(t) and one can save computer time by comparison with a classical non-linear least-squares procedure (iterative convolution) in one step. IV.2.2. Effect of vl. It is clear that a choice of a value of u1 close to vo will lead to a high level of noise in the values of d(t). On the other hand, if v l is too close to zero a substantial loss in information will result. In Figure 7 and in Table I1 we present a number of illustrations of the effect of the position of vl on the calculated decay laws. A concentrated solution of pyrene, becawe of excimer formation, exhibits a nonexponential fluorescence decay
curve. In the analysis of this decay the position of u1 has a large influence on the value of U(v,) and, hence, on the shape of the extrapolated exponential tail in Fourier space. However, it has little effect on the shape of the decay or the quality of the reconvolution. For the experiment corresponding to the fluorescence quenching of pyrene by biacetyl (cf. section IV.2.1) we have studied the influence of the position of v1 on the values of the coefficients a and b. The results are shown in Table 11. We obtained them by choosing in Fourier space a level x,which was a certain fraction of the maxima of F(v) and G(v). Of the four frequency limits v1 resulting from this choice, the smallest was retained for the subsequent calculations. By examining the data of Table I1 we may conclude that x must be about 2 or 3% in order to obtain a decay that is both sufficiently free of oscillations and accurately representative of the physical phenomenon. IV.3. Instrumental Functions. As was discussed in the introduction, the observed lamp profile, g(t), is a convolution of the true exciting pulse, e(t), with the response of the photomultiplier at the analyzing wavelength, r(t). If e(t) is known exactly (= eo(t)),as it is for the pulses derived from synchrotron radiation (e.g., in the ACO machine at Orsay, Paris), the measurement of g(t)(= go@)) may be used to calculate r(t) goW = edt)*rr(t) Such a measurement was made by Ware et al.32with an EMR 561F-09-13-M1 photomultiplier on the ACO synchrotron at Orsay. eo(t)was an exactly calculable Gaussian. By using the FFT technique to analyze this go(t), we calculated the curve presented in Figure 8a, the time response of the EMR photomultiplier to a Dirac 6 pulse. r(t) of this photomultiplier was now known, and therefore an unknown e(t) could be calculated from a measurement of g(t). The EMR photomultiplier was used to measure the g(t) of a hydrogen flash lamp. Figure 8c shows the e(t) of this lamp calculated with the FFT technique. It can be seen that the full-width at halfmaximum (fwhm) (at 2200 A) is very short, about 1.41 ns. We have found that the fwhm is almost independent of
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Andre et al.
n
Figure 8. PM tube response and time profile of excitation pulse at 170 nm. (a) v, = ( 1 9 1 2 5 6 ) ~[r(t)]. ~ (b) Comparison by reconvolution. (c) Time profile of excitation pulse e(t) obtained from r(t) and g(t). vi = ( 2 3 1 2 5 6 ) ~ (d) ~ . Time profile of excitation pulse e(t) obtained from f(t), fo(t), eo(t). v, = ( 2 2 1 2 5 6 ) ~ ~ .
wavelength between 1700 and 2200 A.32 As a consequence it may be inferred that the assumption that e(t) is wavelength independent, made in most analyses of decay curves when the time profile of the excitation pulse is measured at the emission wavelength, is a reasonable one. The function e(t) may also be calculated directly with the equations
G(v)= E(v)R(v)
Go(v) = E ~ ( vR(v) )
(XXII)
whence
Inverse transformation yields e(t ) . Figure 8d illustrates the result of this calculation, which is seen to be in good agreement with the less direct procedure (Figure 8c). IV.4. Discussion. We always observe oscillations, generally of small amplitude, in the deconvoluted d(t). These oscillations do not interfere appreciably with the calculation of kinetic parameters from d(t); but they do significantly affect the determination of a d[d(t)]/dt vs. t plot. To overcome this difficulty we have used a number
of smoothing subroutines, the most satisfactory of which is a subprogram of CII (Compagnie Internationale d’ informatique) that achieves a smoothing of the data points by the method of orthogonal polynomials.
V. Conclusion The deconvolution technique proposed in this paper allows one to determine the &pulse response of electronically excited molecules without any a priori assumptions about the functional form of the decay law. In addition, it may be used to determine complex decay laws that are not easily recoverable by means of iterative reconvolution techniques. It is worthy of note that an exact knowledge of fluorescence lifetimes, which may be obtained, for instance, with synchrotron radiation could be used to calculate g ( t ) , by deconvolution with the FFT technique, at a number of different wavelengths. Experiments with this calculation as their object are contemplated. A comparison, using real data, of the FFT technique with the other techniques mentioned in the Introduction, for the deconvolution of exponential and more complex decay laws, is also currently in progress. Some results have already
The Journal of Physical Chemistry, Vol. 83, No. 17, 1979 2293
Deconvolution in Single Photon Counting Chart I
D(n) =
FR(n)GR(n) - FI(n)GI(n) GR2(n) + GI%)
i fluorescence
= DR(n)
g(t)
+ iDI(n)
+
FR(n)GI(n) f FI(n)GR(n) GR2(n) + Gl2(n) (IIIA)
With the assumption that the real and imaginary parts of D(n) are independent, we use the relation
to obtain D(v) = G(v)/F(v) J.
1
calculation of tables of uDR(u) a n d uDI(uj‘ by use of relation IVA
[
-
of uDR(v) t o DR(u) lead to of u D I ( u ) to DI(u)
’lR]
VlI
J. u , = least
for u > u 1 D(v) =
{[
of u l R , v I I
+
GR2(n) i- G12(n)
and
B
A t 2niu
n t
been p u b 1 i ~ h e d . l ~ ~ Acknowledgment. The authors thank Professor M. Niclause and Professor J. Bordet for helpful discussions. W.R.W. and D.O.C. acknowledge the support of the National Research Council of Canada and ARO-Durham.
Appendix I We start with the standard deviations U G and ~ q2of G(v) and F(v) that result from the Poissonian noise in the observed data, i.e. f(t) = fo(t) f [f~(t)ll’~ g(t) = go(t) f [ g ~ ( t ) l ” ~(IN (The subscript zero indicates the true value.) By considering separately the real and imaginary parts of the Fourier transform, we may make use of the expressions of Wild et al.1° to obtain for the nth channel in Fourier space between 1 and N: gFR2(n)= (1/2)[FR(1) -I-FR(2n + I)] u F ? ( ~ ) = (1/2)[FR(l) - FR(2N + l)] (W U G R ~ ( ~=) (1/2)[GR(1) + GR(2n + I)] U G ? ( ~ ) = (1/2)[GR(1) - GR(2N + l)] We may also write
Appendix I1 We start with the values of f(t) and g ( t ) ; the major steps of our program are described in Chart I. References and Notes (1) (a) Cf., for example, W. R. Ware, “Transient Luminescence Measurements” in “Creation and Detection of the Excited State”, Vol. l A , A. A. Lamola, Ed., Marcel Dekker, New York, 1971, p 213. (b) W. R. Ware, L. J. Doemeny, and T. L. Nemzek, J . Phys. Chem., 77, 2038 (1973). (2) A. E. W. Knight and B. K. Selinger, Austr. J . Chem., 26, 1 (1973). (3) P. Wahl, J. C. Auchet, and B. Donzel, Rev. Sci. Insfrum., 45, 28 (1974). (4) (a) R. Lopez-Delgado, private communication. (b) R. Lopez-Delgado, A. Tramer, and 1, H. Munro, Chem. Phys., 5 , 72 (1974). (5) A. E. W. Knight and 6.K. Selinger, Specfrochim. Acta, far? A , 27, 1223 (1971); 0. Monod-Herzen, L. Langouet, and J. Philippe, C.R. Acad. Sci., Pan‘& 264, 1679 (1967); A. Ginvatd and I.Z.Steinberg, Anal. Biochem. J., 15, 263 (1975); A. Grinvald and J. Z. Steinberg, Anal. Biochem., 59, 583 (1974); programs in use in both laboratories (Nancy and London). (6) I. Isenberg and R. D. Dyson, Biophys. J., 9, 1337 (1969); P. Wahl and H. Lami, Biochim. Biophys. Acta, 133, 233 (1967). (7) W. P. Helman, Int. J. Rad&. phys. Chem.,3, 283 (6971); M,Alqren, J . Chem. Soc., 145 (1973). (8) A. Gafni, R. L. Modlin, and L. Brand, Biophys. J., 15, 263 (1975).
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M. C. Collins and W. F. Ramirez
(9) B. Valeur and J. Moirez, J. Chlm. Phys., 70, 500 (1973); J. Loeb and G. Cahen, Automatisme, 3, 479 (1963). (10) U. Wild, A. Holzwarth, and H. P. Good,Rev. Sci. Instrum., 48, 1621 11977). (11) D. G. brdner, J. C. Gardner, G. Laush, and W. W. Meinke, J. Chem. Phys., 31, 978 (1959). (12) M. La1 and E. Moore, int. J . Num. Methods Eng., 10, 979 (1976). (13) A. E. McKinnon, A. G. Szabo, and D. R. Miller, J: Phys. Cheh., 81, 1564 (1977). (14) (a) M. H. Hui and W. R. Ware, J. Am. Chem. Soc., 98, 4712 (1976); (b) J. C. Andre, D. O'Connor, and W. R. Ware, J . Phys. Chem., 83, 1333 (1979). (15) T. Nishikawa and K. Someno, Anal. Chem., 47, 1290 (1975). (16) K. W. K. Yim, T. C. Miller, and L. R. Faulkner, Anal. Chem., 49, 2069 (1977). (17) T. A. Maldacker, J. E. Davis, and L. B. Rodgers, Anal. Chem., 46, 637 (1974). (18) H. P. Larson and U. Fink, Appl. Spectrosc., 31, 386 (1977). (19) Y. G. Biraud, Astron. Astrophys., 1, 124 (1969). (20) S. W. Provencher, J. Chem. Phys., 64, 2772 (1976).
(21) S. W. Provencher, Biophys. J., 16, 27 (1976). (22) J. Schlesinger, Nucl. Instrum. Methods, 106, 503 (1973). (23) R. Bracewell, "The Fourier Transform and Its Applications", McGraw-Hill, New York, 1965. (24) J. W. Cooley and J. W. Tukey, Math. Comput., 19, 297 (1965). (25) W. T. Cochran, J. W. Cooley, D. L. Favin, H. D. Helms, R. A. Kaenel, W. W. Lang, G. C. Maling, Jr., D. E. Nelson, C. M. Rader, and P. D. Welch, proc. rfEE, 55, 1664 (1967). (26) B. R. Hunt, I€€€ Trans. Audio Electroacoust., No. AU-20, 94 (1972). (27) H. F. Silverman and A. E. Pearson, If€€ Trans. Audio Electroacoust., NO. AU-21, 112 (1973). (28) F. J. Harris, Proc. I€€€, 66, 5 1 (1978). (29) D. V. O'Connor and W. R. Ware, J . Am. Chem. Soc., 98, 4706 (1976). (30) J. C. Andre, M. Niclause, and W. R. Ware, Chem. Phys., 28, 371 (1978). (31) J. C. Andre, M. Bouchy, and W. R. Ware, Chem. Phys., 37, 107, 118 (1979). (32) J. C. Andre, R. Lopez-Delgado, R. Lyke, and W. R. Ware, Appl. Opt., in press.
Mass Transport through Polymeric Membranes Michael C. Collins and W. Fred Ramirez* Deparlment of Chemical Engineering, University of Colorado, Boulder, Colorado 80309 (Received November 3, 1977) Publication costs assisted by the University of Colorado
Mass transport through three polymeric membranes (Cuprophan, poly(acrylonitrile),and a poly(acry1onitrile) membrane with an adsorbed protein layer) is studied. A series of eight compounds ranging in molecular weight from 60 (urea) to 1355 (vitamin BIZ)were investigated. A complete set of transport properties are reported including the permeability coefficient, sieving coefficient, pressure-filtrationcoefficient, and frictionalcoefficients representing the interactions of solute-membrane, solute-solvent, and solvent-membrane. Solute permeability for diffusive transport correlates well for all membranes studied with solute molal volume except for sulfobromophthalein (BSP). Because of strong membrane-solute frictional interactions, BSP has a reduced permeability. The sieving coefficient, which characterizesconvective transport, also correlated with solute molal volume except for BSP and the lipophilic compound thiopental, The reduced convective transport of BSP is again due to its high frictional interactions with the membranes. The sieving coefficient for thiopental is markedly reduced although its permeability was normal. The membrane tortuosity factors for thiopental were also reduced. These facts indicate the existence of multiple diffusive pathways for thiopental and its exclusion from some solvent pathways in the membranes studied.
Introduction The purpose of this study is to evaluate the mechanisms of mass transport through polymeric membranes. Membranes are important separation devices extensively used in application such as renal dialysis and reverse osmosis. Although the theory of membrane transport is well established, there are few experimental studies of membrane transport mechanisms. The work of Kauffman and Leonard7y8and Ginzberg and Katchalsky6 are notable exceptions. In order to test theoretical concepts a wide range of solutes and two membranes often used in renal dialysis were chosen for study. Membranes are usually characterized by the following three parameters of the solute: permeability coefficient, w , the pressure filtration coefficient, L,, and the reflection coefficient, u. Often the sieving coefficient, S = 1 - u, is used instead of the reflection coefficient. The flux equations written in terms of these coefficients are (Keedem and K a t ~ h a l s k y ) ~ N , = wRTAC, + (1 - g)C8Jv (1) Jv = L,(AP - uRTAC,) (2)
flows. However, these three practical parameters provide little information about the detailed mechanisms of membrane transport. Spieglerlg proposed a frictional model to use with the principles of irreversible thermodynamics to give physical meaning to the phenomenological coefficients. He used the law of friction that the frictional force that hinders the motion of one object is proportional to the relative velocity between two objects (3) F , = -fij(ui - Uj) where f i j is the frictional coefficient between the ith and jth species. Kedem and Katchalsky'O related the frictional coefficients to the measurable macroscopic membrane transport parameters of permeability, filtration, and sieving by
(4)
where N , is the solute flux and Jv the total volume flux. These three coefficients of permeability, pressure filtration, and sieving can all be determined experimentally and suffice to characterize membrane transport at low volume 0022-3654/79/2083-2284$0l .OO/O
0 1879 American Chemical Society
