J . Phys. Chem. 1988, 92, 4222-4226
4222
only behavior which one might associate with concentration quenching as previously defined. At the LC-SC transition point a small recovery in intensity was noted, indicating that at least portion of the nonfluorescent chlorophyll is reorganized to fluorescent form. As was seen in Figure 6, this was accompanied by the appearance of two-component fluorescent decay indicating two new environments for the pigment. The short component shows a very high pressure sensitivity over a very small range of area change. From our previous studies in DOL, it is unlikely that this may be assigned simply to changes in the Forster energy-transfer rate, facilitating quenching by a set number of traps in the medium; rather it would appear more likely that one is observing a sharp increase in the number of trapping sites present which become available to the fluorescent population of chlorophyll over a small area/molecule range. Because one could not expect the total number of traps in the system to increase significantly over such a short concentration region, the observed decreases in lifetime may well indicate diminishing barriers to energy transfer across phase boundaries in this region of pressure. Somewhat more surprising-and perhaps the most interesting finding of this study-is the appearance in this region of the long-lifetime component whose magnitude is essentially that of unquenched chlorophyll in a nonaqueous medium. At this point one may only speculate on the type of organization which insulates this population from either concentration quenching or interaction
with traps. It may be essentially free, Le., ejected from the lipid layer either into the subsurface region or into the region above the lipid layer. Alternatively, although less likely, regions which lock monomeric chlorophyll into small concentrations or into orientations highly unfavorable to energy transfer may be formed. The interplay between alterations in local dye concentration and crystalline domain growth as observed by Miller and Mohwald may play a role in creating some microenvironments of isolated chlorophyll as the transition from LC to S C takes place.25 Any explanation of such behavior must also take into account the observed reversibility as pressure is lowered. It may be seen, from both these and previous studies, that Chl a emission parameters respond markedly to changes in the lipid monolayer environment. While earlier studies in DOL indicated that self-quenching via a Forster transfer mechanism governed fluorescence behavior and was essentially independent of lipid packing within the single DOL LE phase, the present results have focused on the discontinuous changes in both lifetime and intensity that can accompany phase changes within the lipid matrix. These data, with those already available, may provide a basis for building more complex monolayer models involving other components of the photosynthetic apparatus from which an expanded understanding of pigment luminescence and its dependence on environment may be obtained. Registry No. DPL, 2644-64-6; Chl a, 479-61-8
A Method for Obtaining Multicomponent Diffusion Coefficients Directly from Rayleigh and Gouy Fringe Position Data Donald G . Millert Chemistry and Materials Sciences Department, Lawrence Livermore National Laboratory, Livermore, California 94550, and Dipartimento di Chimica, University of Naples, 801 34 Naples, Italy (Received: November 3, 1987; In Final Form: January 28, 1988)
A nonlinear least-squares (NLLS) method for analyzing three-component Rayleigh fringe position data for diffusion, described briefly in previous papers, is extended here to four-component data. By a simple transformation of equations for the Gouy method, an analogous NLLS procedure can be applied directly to three- and four-component Gouy fringe position data. The only complication is that the Gouy NLLS is subjected to a condition equation. Results are given for the free-diffusion case, but the procedure can be extended to other boundary conditions.
I. Introduction Optical methods are the most precise for determining diffusion coefficients in binary and ternary systems. Of these, the Rayleigh and Gouy methods with free-diffusion boundary conditions seem to be the best.’J For ternary systems, an advantage of Rayleigh data has been that the fringe positions can be analyzed directly by a nonlinear least-squares (NLLS) procedure. This Rayleigh method, due to Miller, Eppstein, and Albright,3 has been briefly described by Miller et al.4 and Rard and Miller,5 and noted still earlier by Miller6 and Albright and SherrilL7 This straightforward procedure is easily extended to four components, as described below. Full details will be published elsewhere. The current procedures for analyzing ternary Gouy data8,9do not use fringe positions directly; instead, a number of auxiliary quantities are used. Some, described below, are defined in section 111; others are defined in the appropriate references. One such quantity, C,, is obtained from extrapolation of a function of fringe positions from each exposure of an experiment. The C, values Address correspondence to Lawrence Livermore National Laboratory.
0022-3654f 88/2092-4222$01.50/0
from all the exposures are then linearly extrapolated to get the quantity D, for that experiment, as well as the corrected time. A second quantity, Qo, is obtained from the area under the Q deviation function;* this involves the fringe positions and C,’s for this same experiment. From two different experiments there are two D,’s and two which are sufficient to calculate the four diffusion coefficients D,,. Typically, four or more experiments are done to provide better statistics. Variant methods, due to Revzin’O and Dunn and Hatfield,” use the deviation function
eo’s,
( I ) Dunlop, P. J.; Steel, B. J.; Lane, J. E. In Physical Methods of Chemisfry;Weissberger, A,, Rossiter, B. W., Eds.; Wiley: New York, 1972; Vol. 1, Chapter IV. (2) Tyrrell, H.J. V.; Harris, K. R. Diffusion in Liquids; Butterworths:
Loddon,.1984. (3) Miller, D. G.; Eppstein, L. B.; Albright, J. G., in preparation. (4) Miller, D. G.; Ting, A. W.; Rard, J. A,; Eppstein, L. B. Geochim. Cosmochim. Acta 1986, 50, 2397-2403. ( 5 ) Rard, J . A.; Miller, D. G . J . Phys. Chem. 1987, 91, 4614-4620. (6) Miller, D. G.J. Solution Chem. 1981, 10, 831-846. (7) Albright, J. G.; Sherrill, B. C. J. Solution Chem. 1979, 8 , 201-215. (8) Fujita, H.; Gosting, L. J. J . Phys. Chem. 1960, 64, 1256-1263. (9) Woolf, L.A,; Miller, D. G.; Gosting, L. J. J . A m . Chem. SOC.1962, 84, 317-331. (10) Revzin, A. J. Phys. Chem. 1972, 76, 3419-3429.
0 1988 American Chemical Society
Multicomponent Diffusion Coefficients directly but still use extrapolations to get C, and D,. The usual extrapolation to get C, is based on a power series e x p a n s i ~ n . ’ ~ However, -~~ disturbing uncertainties in C, and thus DA arise from the subjective decisions of whether to use one or two terms of the series and how many of the lowest numbered fringe positions should be used. A set of powerful new programs by Albright” has eliminated this dependence on extrapolation procedures. Now C, can be obtained directly from a NLLS of all fringe positions of an exp u r e using the complete theoretical functional form for a ternary. Finally, D, and Qo for each exposure can be calculated and then averaged over all exposures. These programs provide improved values of DA and Qo for input to the Fujita-Gosting procedure, as well as extensive diagnostics. Although the Fujita-Gostingss9 procedure and other Gouy procedureslOJ1are satisfactory for three-component systems, the emphasis on DA and Qo (or 0)has led to serious difficulties for four-component systems. Here, there are nine DY)s, and clearly three (or more) experiments are needed. However, the three DA’s and three Qis, which are easily obtained, give only six of the nine quantities needed. Therefore, some additional quantity from each experiment, or else some combination of other experiments, is required. This issue was discussed at length by Kim.16J7 He found no suitably accurate third quantity, and the combination of Gouy with Rayleigh, diaphragm cell, etc., has not been successful so far. Several years ago, the author explored the idea of using Q1, the area under the curve of the first moment of Q , as the third set of quantities for the four-component case. Fujita18 had previously suggested Q1 as an alternative to Qo for the ternary case, and Revzin prepared a computer program to use it.18,19However, the extention of Ql to four-component systems is very complicated algebraically and has not yet been completely developed. Since the Rayleigh method can be easily extended to four components, Rayleigh data alone suffice; consequently, combined experiments are unnecessary. However, in principle there should be an analogous method for Gouy data alone. In this paper, we sketch the extension of the Rayleigh procedure to four components and provide a straightforward, analogous new method for Gouy for both three and four components. No extrapolations or deviation functions are needed in this new Gouy method, and the NLLS is done directly on the fringe positions, as in the Rayleigh method. Moreover, the least-squares variables and the equations for the D , in terms of them are the same for both Gouy and Rayleigh methods. It is surprising that this simple Gouy method was not found earlier. This new method does not eliminate the necessity of Albright’s Gouy programs, which are essential for the analysis of individual experiments. It is interesting that the fringe position variables for both Rayleigh and Gouy methods are “reduced” variables which are independent of the time of the exposure. Therefore, these variables can be averaged over all exposures for a given fringe j , and these average values for all j form a time-corrected ”representative pattern”. This averaging will be essential when automated data collection provides 100-1000 scans, each equivalent to a single photographic exposure in a given experiment. 11. Rayleigh Nonlinear Least-Squares We summarize the ternary Rayleigh below because the four-component Rayleigh and three- and four-component Gouy procedures are analogous to it. Dunn, R. L.; Hatfield, J. D. J . Phys. Chem. 1965, 69, 4361-4364. Albright, J. G.; Miller, D. G. J . Phys. Chem. 1980, 84, 1400-1413. Gosting, L. J.; Onsager, L. J. Am. Chem. SOC.1952.74.6066-6074. Edwards, 0. W.; D u m , R. L.; Hatfield, J. D.; Huffman, E. 0.; , K. L. J . Phys. Chem. 1966, 70, 217-226. Albright, J. G.; Miller, D. G., submitted for publication in J . Phys.
The Journal of Physical Chemistry, Vol. 92, No. 14, 1988 4223 The expression for the reduced fringe number f(j)of a freediffusion Rayleigh experiment for a ternary system in terms of experimental quantities is6,7J7
with
yj =
(4)
Xj/(2t1/2)
where hi are the eigenvalues of the diffusion coefficient matrix and are functions only of the Dij, ui are their reciprocals as used by Fujita and Gostings!zo(denoted there by u+ and u-), xi is the position in the cell corresponding to fringe j , t is the corrected time, J is the total number of fringes, and cyi is the refractive index fraction given by cy. I
=
RjAcj RlAc1 R ~ A C Z
+
(5)
The Ri are determined from the J and the Aci of two or more experiments from a least-squaring of
J = RlAc1
+R~Ac~
(6)
The a and b are functions of the Dij and Ri. Expressions for sl, sz, a, and b were given by Miller6 and are presented in Appendix
B. Moreover, it has been shownz0 that r, + rz= 1
(7) (8)
Cq+ff2=1
Hence, eq 1 can be written as f(’j) =
rl erf ( s l y j )+ ( 1 - I’,)
erf (szvj)
(9)
and therefore we have
f G ) = ( a + ba,) erf ( s l y j )+ ( 1 - a - b a l ) erf ( s g j )
(10)
In the above expressions, the quantities (’j, xj, t ) are the experimental quantities for each fringe of an exposure in a given experiment, and the experimental quantities ( J , cyi) are characteristic of that entire experiment. To obtain the four Dij, four quantities common to all the experiments are necessary. Suitable ones are sl, sz, a, and b. Although least-squares fitting of eq 9 to all the fringes of a single experiment will yield values of (r1, sl, s,), these values provide only three of the four quantities necessary to calculate Dij. It requires at least two experiments at different a1values to separate a and b. After a NLLS of eq 10 for two or more experiments, one can obtain the Dij from the resulting (sl, sz, a, b ) using eq 13-16 of Miller6 (also found in Albright and Sherrill’ and in Appendix B). It is also possible to do a NLLS on eq 9 for two or more experiments, obtaining a common sl, sz for all experiments but a separate r, for each experiment. Then a and b are obtained from the set of rl’sby a linear least-squares fit of eq 3. This has the advantage of indicating an occasional bad experiment in the typical set of four, but has the serious disadvantage of increasing the number of least-squares variables and therefore their uncertainties. The NLLS of eq 10 in terms of (sl, sz, a, b), temporarily denoted by gi, can be done by using a Taylor’s series expansion in terms of the four gi:
Chem. (16) Kim, H . J . Phys. Chem. 1966, 70, 562-575. (17) Kim, H.J . Phys. Chem. 1969, 73, 1716-1722. (18) Revzin, A,, Ph.D. Thesis, University of Wisconsin, 1969 (University Microfilms 70-3676). (19) Revzin, A., private communication, 1981.
where fG)is obtained from the middle expression of eq 1 . The (20) Fujita, H.; Gosting, L. J. J . Am. Chem. SOC.1956, 78, 1099-1106.
4224
The Journal of Physical Chemistry, Vol. 92, No. 14, 1988
analytical expressions for f(j)and [8f(j’)/8gi] are obtained from eq 10. The superscript 0 means these analytical expressions are evaluated by using the current values of the g,. A linear leastsquares of eq 11 yields Agi, the corrections to the current g,. The corrected gj are then used as the new current gi, and this process is iterated until Agj/gjvalues are smaller than 0.00001. Because the error function and its derivatives are well-behaved, this NLLS procedure ordinarily converges in five to seven iterations. Suitable initial values of (s,, s2, a, b) are (DI-’I2,D2-’i2,0, I ) , respectively, where D , and D2 are the pseudobinary D at a , = 1 and a i= 0, respectively. The D , and D2 are obtained from the respective ternary experiment by treating it as if it were a binary. These four starting values result from assuming cross terms are zero and D,, is less than D22. Other NLLS methods can be used. The elaboration of the above procedure to include base line corrections and to delete outlying points is summarized in Miller et aL4 and will be described in more detail elsewhere., This technique has been in use at LLNL since 1972 and is very straightforward. Other Rayleigh methods based on series expansion have been proposed for three-component but the method presented here4-7 is less subject to error.6 This idea is easily extended to the four-component system. In this case Kim’s e q u a t i o n ~ l ~can , ’ ~ be expressed as
fo)= T I erf (sly,) + r2erf (stvj) + ( 1 - r, - r,) erf (s3y,) (12)
r, = a l a , + aza2 +
(13)
with the obvious extensions of eq 5-8. Equations 13 and 14 have been written in symmetric form in order to get a more compact expression for the D,. Equation 12, with eq 13 and 14 substituted in, is the appropriate least-squares equation. It is the analogue of eq 10. To obtain the nine D,, nine suitable quantities are (s,, s2, $3; a,, 02, 03; b l , b,, b,). Clearly, at least three experiments with different a1and a2are required to separate the a, and b, in rl and r2.The NLLS procedure is the same as with the ternary case, and suitable initial D2-1/2, starting values of (sl,s2,s3;a,, a2,a3;b,, b2,b,) are D3-li2;1, 0,O; 0, 1,0), respectively, where D, are the pseudobinary D at a, = 1. The D , expression in terms of the nine (s,, a,, b,), obtained from Kim’s equation^'^^'^ after considerable algebra, is
Miller Rayleigh analysis in the preceding section, and it uses the same least-squares constants and equations to calculate D,,. For any Gouy system, the height/area ratio DA is given byl.2.12.13,21
DA = P X 2 B 2 / ( 4 ~ C f 2 t )
(17)
where J is the total number of fringes, X the wavelength of the light, B the optical distance from the center of the cell to the camera plane, and t the corrected time. The quantity C, is the limiting value of C , at z, = 0, where
c, = y//(exP(-z,2))
(18)
Here 5 is the position of Gouy fringe j at the camera plane, and z, is obtained by iteration from the expression for f(z,): f(z,) = erf (z,) - ( ~ z , / T , / ~exp(-z,2) )
(19)
The numerical value off(z,) is obtained from the experimental quantities j and J by = ZCi,J)/J
(20)
+
The function Z(j’,J)is approximately (j’ 3/4) for fringe minima; its value depends on the approximation to the wave theory of Gouy fringes.I2J3 The traditional procedure to obtain C, is by extrapolation of Cl, vs f ( ~ , ) , / ~to f(z,)*/, = 0. For the ternary system, Fujita and Gosting* have shown that
q -- rlslexp(-sI2y,2) _
+ r2s2e~p(-s,~y,2)
r,s, + r2s2 i / ~ , 1 / 2= r,sl + r2s2
(21)
f(zj) = rd(slyj) + rzf(sgj)
(23)
c,
(22)
where y j in eq 21 must be determined from eq 23, using the numerical value offlz,) from eq 20 together with the known values of ri and si. This also requires iterations. The key idea for the simple Gouy formulation which follows is the recognition that, by means of eq 17, DA and C, will cancel in the combination of eq 21 and eq 22. Thus, from eq 17, 21, and 22, we obtain for each fringe j at time t
YJt1/2 = K [ F , s , exp(-s12y,2) + r2s2exp(-s>y?)]
(24)
where K is a numerical factor for each experiment which depends only on apparatus constants and J , namely K = JXB/(~TI/~)
If we use eq 3 for
rl and define y,*
(25)
y/* as
= yyl2
(26)
we obtain the important result yj* =
where
K [ ( a + b a l ) s , exp(-sI2y?)
+ (1 - a - ba,)s2exp(-s,y,2)] (27)
subject to the condition (Le., y j to be obtained from) k =
I l i - 3i2 - 4
-.L
f(zj) = ( a
1=6-i-k
The appropriate ternary derivatives for eq 11, and the four-component derivatives for the four-component analogue of eq 11, are given in Appendix A. 111. A Direct Method of Least-Squaring Gouy Fringe Data
As noted in the Introduction, current methods for determining ternary Du from Gouy data require extrapolations and integrations, and the previously proposed ideas for four-component systems are even more indirect and as yet unsuccessful. However, Albright’s ideal5 of using fringe position data directly to analyze an individual experiment can be extended to analyze all the data of a set of experiments. The resulting procedure is very similar to the
+ ballf(slYj) + (1 - a - b a l l f ( s g j )
(28)
Equation 27 is quite simple and contains no C, or DA. It looks similar to eq 10, has exactly the same least-squares variables, uses the same initial starting values, and uses the same equations for D, (Appendix B) as in the Rayleigh analysis6x7in terms of (s,, s2, a, b). The only more complicated aspect of this Gouy NLLS compared to the Rayleigh is that the Gouy is a NLLS with a condition, namely, eq 28. This condition can be included in the NLLS in two ways. One is by calculating y j from eq 28 and the starting values by iteration and then substituting these y j in the appropriate derivatives. The second and better way adopted here is to recognize that y j is also (21) Dunlop, P.J.; Gosting,
L.J. J . Am. Chem. SOC.1955, 77, 5238-5249.
Multicomponent Diffusion Coefficients
The Journal of Physical Chemistry, Vol. 92, No. 14, I988
a function of a, b, sl, and s2. Consequently, the ayj/t3gi included in derivative expressions of aq*/agi from eq 27 can be eliminated by using the dyj/agi obtained from eq 28. It is also possible to NLLS eq 28 using eq 27 as the condition. The extension to four-component systems is now as easy as for the Rayleigh analysis. From Kim’s four-component e q u a t i o n ~ ’ ~ * ’ ~ and the above analyses, we have
subject to
Equations similar to eq 27 and 28 are obtained by substituting eq 13 and 14 in eq 28 and 29. The NLLS is then exactly analogous to that for eq 12, with the same least-squares variables (sl, s2, s3;a l , a2, a3;b l , 62, b3),the same initial starting values, and the same expressions for Dij. Again, the only complication is that this is a NLLS subject to the condition given by eq 30. The NLLS derivatives for both ternary and four-component Gouy procedures are given in Appendix A.
IV. Discussion In successive exposures, the distances between fringe positions xj of the Rayleigh increase and those between the E;. of the Gouy decrease with time. However, the y j of eq 4 and ?* of eq 26 are independent of corrected time for a given value of j in a given experiment. Consequently, they can be averaged and the outliers rejected. The average value for each j taken over all j of an experiment can be used as a representative pattern in the leastsquares, thereby reducing the amount of computer time. This is useful with small computers and, as suggested independently by Dr. R. Sartorio,22will be particularly important when automated data collection gives data sets equivalent to a large number of photographic exposures. Although the new Gouy NLLS procedure uses all the U,* data from all the experiments, this method should be used in conjunction with a previous experiment-by-experiment analysis using the new Albright t e c h n i q ~ e s . ’ ~ As noted for the ternary Rayleigh, it is possible to least-square the ternary Gouy for the common sI and s2 as well as a rl for each experiment. Then afterward these rl’scan be least-squared to get (a, b). This procedure has the same advantages and disadvantages as the analogous Rayleigh case. It is also possible to use the Albright programs to get an average sl, s2,and rl for each experiment, average these average si's and sis over all experiments, and least-square the rl’sas above. That procedure is less suitable because experiments with some values of rl give less precise values of all three constants. There is an additional advantage to the similarity of the new NLLS procedures for Gouy and Rayleigh. It provides a more equivalent comparison of Gouy and Rayleigh D, when they are calculated from data taken alternately on a diffusiometer capable of being switched from one optical system to the other. For ternary systems, there may be no particular advantage to the new Gouy method over the Fujita-Gosting DA, Qomethod or the methods of Revzin’O or Dunn and Hatfield.” However, there is some value to the avoidance of extrapolations. The decisive advantage of this new Gouy procedure appears to lie in applications to systems with four or more components, since no extrapolations, combination experiments, or complex algebraic expressions are required. It is now possible to do the four-component Gouy case directly from the fringe position data. Probably this will be the only workable Gouy method for systems of more than three components. Experiments are planned to test the four-component Rayleigh and three- and four-component Gouy procedures using the high-precision Gosting diffusiometer at LLNL. Results will be reported elsewhere. (22) Sartorio, R., private communication, May 1987.
4225
Our discussion has been concerned with free diffusion boundary conditions. However, other boundary conditions lead to the same Rayleigh equations as eq 9 or eq 12, except that the error functions are replaced by other functions of the si which are characteristic of the specific boundary condition^.^^,^^ Similarly, the Gouy equations, eq 24 or eq 29, have the exponential functions replaced by other functions of the si. Consequently, the variables and procedures are exactly the same with different boundary conditions, except that the derivatives in Appendix A must be changed to correspond to the different functions replacing the error and exponential functions. Finally, all the equations and solutions of this paper are for systems whose diffusion coefficient matrix has distinct eigenvalues. However, there can exist systems with some or all eigenvalues equal. In these circumstances the equations have different forms, as has been shown for special ternary cases.2e26 The general case will be discussed elsewhere. Acknowledgment. Most of this work was performed at the Dipartimento di Chimica, University of Naples. The author is very grateful for their hospitality during his stay in Naples in the spring of 1987. The ideas expressed here arose out of conversations with Prof. Roberto Sartorio (Naples) and Prof. John Albright (Texas Christian University). Dr. J. A. Rard is thanked for his helpful comments. Portions of this work were performed under the auspices of the U S . Department of Energy, Office of Basic Energy Sciences (Geosciences), at Lawrence Livermore National Laboratory under Contract No. W-7405-ENG-48. Appendix A: Derivatives for NLLS of Rayleigh and Gouy Fringe Data A . Rayleigh Derivatives. For the ternary system ( i = 1, 2)
af -0’) - erf ( s l y j )- erf ( ~ ~ v i ) aa
where
r2= 1 - rl
(A-4)
For the four-component system ( i = 1, 2, 3)
a- -m - ai[erf ( s l y j )- erf ( S I ~ ~ ) ]
(A-5)
dfci) - - ai[erf ($9,) - erf ( S I ~ ~ ) ]
(A-6)
aai abi
af 0’) -ri 2Yj asi 71t2
-=
e~p[-(s$~)~]
(‘4-7)
where
r3= 1 - rl - r2
(‘4-8)
In eq A-3 and A-7, r i has been used to simplify the notation. However, in the least-squares calculations, ri should be replaced by their expressions in terms of (a, b, cy1) or ( a , b , ai),respectively, from eq 3 and A-4 and from eq 13, 14, and A-8, respectively. B. Gouy Derivatives. As noted earlier, the y j in eq 27 are determined from eq 28, which involves a, b, sl, and s2. Therefore, the preferred way to obtain any derivative of eq 27 should include the derivative of y, as well. This, however, can be obtained from (23) (24) (25) (26)
Toor, H.L. AIChE J . 1964, 10, 448-455. Toor, H . L. AZChE J . 1964, 10, 460-465. Kirkaldy, J. S. Can. J . Phys. 1958, 36, 899-906. Sundelof, L.-0.; Sodervi, I. Ark. Kemi 1963, 21, 143-160.
J. Phys. Chem. 1988, 92. 4226-4231
4226
the differentiation of eq 28. Since the numerical value off(zj) is obtained from eq 20, which does not involve a, b, sl, and s2, the derivative of the left-hand side is zero. Therefore, we can solve for t h e y derivative and substitute this back into the derivative of eq 27. A similar procedure is used for eq 29 and 30. After some algebra we obtain the following results: For the ternary system (i = 1, 2) a?* - -
As with the Rayleigh derivatives, the T i should be replaced in the actual least-squares equations by their expressions in terms of (a, b, ai)for ternary and ( a , bi, ai) for four-component systems.
Appendix B: Summary of Ternary Equations For completeness, we present the expressions for a, b, si, s2in terms of Dij, and conversely.6
d/2K[erf ( s l y j )- erf ( s t v j ) ] 2Yj
aa
a?*
aI;* aa
CUI----
(A-10)
- - - mi e ~ p [ - ( s p ~ ) ~ ]
(A-11)
-=
db
a yj* asi
a =
('4-9)
(
s12D- DZ2-
[D(s12- s22)] (B-1)
For the four-component system (i = 1, 2 , 3) a?* - - d/2Kai[erf (slyj) - erf (s3yj)]
aai
2Yj
a?* - -
(A-12) 03-71
d/2Kai[erf (stvj) - erf ( s g j ) ]
abi
2Yj
-a?*-
asi
- mi e ~ p [ - ( s p ~ ) ~ ]
(A- 13)
(A-14)
D22 = [ ( a + b)(l - a)s12 - ~ ( -l u - b ) ~ 2 ~ ] / ( b ~ I ~ (B-9) ~2~)
Effect of Dye Aggregation on the Photogeneration Efficiency of Organic Photoconductors Kock-Yee Law Xerox Webster Research Center, 800 Phillips Road, 01 14-390, Webster, New York 14580 (Received: November 30, 1987)
The photoconductivities of a soluble vanadyl phthalocyanine dye, ~-Bu,,~VOPC, and model squaraines bis[4-(dimethylamino)phenyl]squaraine(1) and bis(4-methoxypheny1)squaraine (2) have been studied in single-layerand bilayer photoreceptor devices, respectively, by xerographic photodischarge technique. The aggregational behavior of these materials was studied by absorption spectroscopy and X-ray diffraction. Results show that t-Bul,4VOPccan exist as a glassy phase I and a crystalline phase I1 in polystyrene matrix and that 1 and 2 form different aggregates in solid. Xerographic results showed that the phase I1 of ~ - B U ~ , ~ VisO>300 P C times more sensitive than the phase I and the aggregate of 1 is >lo0 times more sensitive than that of 2. These results are attributable to crystallization effect and aggregational effect, respectively. Analysis of the data on the xerographic properties of 4-[p-(dimethylamino)phenyl]-2,6-diphenylthiapyrylium perchlorate (4) reveals that, despite the wide structural variation among the phase I1 of ~ - B U ~ , ~ V Oaggregate PC, of 1, and aggregate of 4, these high-efficiency organic photogenerators share remarkable similarity in electronic and solid-state properties. The use of these properties as a guide for the design and the synthesis of future high-performance organic photoconductors is recommended.
Introduction Organic pigments, which have a small band gap and absorb strongly in the visible, are often found to be useful as photoconductors for xerographic photoreceptor and organic solar cell applications. Useful classes of organic photoconductors are metallophthalocyanines,'-" squaraines,12-21thiapyrylium salts,22-24 (1) Loutfy, R. 0.;Hsiao, C. K.; Hor, A. M.; Baranyi, G. D. J . Imaging Sci. 1985, 29, 148. Loutfy, R. 0.;Hor, A. M.; Rucklidge, A. J. Imaging Sci. 1986, 31, 31. (2) Loutfy, R . 0.;Hor, A. M.; Baranyi, G. D.; Hsiao, C. K. J. Imaging Sci. 1985, 29, 116. (3) Kakuta, A.; Mori, Y . ;Takano, S.;Sawada, M.; Shibuya, I. J . Imaging Techno!. 1985, I ! , 7. (4) Yanishita, T.; Ikegami, K.; Narusawa, T.; Okuyama, H. IEEE Trans. Ind; Applr 1984, lA-20,-1642. (5) Arishima, K.; Hiratsuka, H.; Tate, A,; Okada, T. App1. Phys. Lett. 1982, 40, 219.
0022-3654/88/2092-4226$01.50/0
p e r y l e n e ~ , azo ~ ~ - compound^,^^*^*-^^ ~~ etc. Among these organic photoconductors, metallophthalocyanines and squaraines are ~~~~~~~
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(6) Grammatica, S.; Mort, J. Appl. Phys. Lett. 1981, 38, 445. (7) Minami, N.; Sasaki, K.; Tsuda, K. J . Appl. Phys. 1985, 54, 6764. (8) Loutfy, R. 0.;McIntyre, L. F. Can. J . Chem. 1983, 61, 72. (9) Dodelet, J. P. J . Appl. Phys. 1982, 53, 4270. (10) Martin, M.; Andre, J. J.; Simon, J. Nouu. J. Chim. 1981, 5 , 485. (11) Loutfy, R. 0.;Sharp, J. H. J . Chem. Phys. 1979, 71, 1211. (12) Law, K. Y.; Bailey, F. C. J . Imaging Sci. 1987, 31, 172. (13) Tam, A. C.; Balanson, R. D. I E M J . Res. Deu. 1982, 26, 186. (14) Wingard, R. E. IEEE Ind. Appl. 1982, 1251. (15) Tam, A. C. Appl. Phys. Lett. 1980, 37, 978. (!6) Melz, R. J.; Champ, R . B.; Chang, L. S.;Chiou, C.; Keller, G. S.; Liclican, 1.C.;Neiman, R. B.; Shattuck, M. D.; Weiche, W. J. Photogr. Sci. Eng. 1977, 21, 73. (17) Loutfy, R. 0.;Hsiao, C. K.; Kazmaier, P. M. Photogr. Sci. Eng. 1983, 27, 5. (18) Morel, D.L. Mol. Cryst. Liq. Cryst. 1979, 50, 127.
0 1988 American Chemical Society
