1692
J. Phys. Chem. 1990, 94, 1692-1699
TABLE XII: Charge Distribution (Qx) and Overlap Population ( P O Hof ) the OH Bond in the Trimers
clustera Qo Qn [Si,H,Si,Si] -0.550 0.323 0.258 [Si,H,AI,Si] -0.526 [Si,H,P,Si] 0.396 -0.543 [AI,H,Si,Si] 0.255 -0.525 [AI,Si,H,Si] 0.313 -0.548 [Si,H,AI,P] 0.264 -0.531 0.347 -0.484 [Si,AI,H,P] 0.351 -0.497 IP,H,ALPl 0.322 -0.516 [AI,H,P,All [ AI,H,Si,AI] 0.239 -0.536 0.405 -0.540 [P,H,Si,Pl 'See Table VI11 for clusters notation.
POH
0.269 0.268 0.270 0.265 0.271 0.268 0.270 0.270 0.27 1 0.266 0.265
conclusions and restrictions analogous to those put forward following the dimeric clusters calculations.
Conclusions Ab initio molecular orbital calculations on model clusters simulating silicate, aluminosilicate, and silicoaluminophosphate dimeric and trimeric entities have enabled us to obtain information on the relative stability of the Si-0-A1, Si-0-P, and AI-0-P
bridges and on modifications generated by the presence of phosphorus. In particular, our work suggests the following tentative conclusions which should be supported by complementary investigations on larger systems: 1. The A1-O-P bridge is more stable than Si-O-A1 or Si-0-P linkages, which is in agreement with the high stability of framework aluminum phosphate structures. 2. Phosphorus has a tendency to replace preferentially Si rather than AI in aluminosilicate structures, because of the high stability of the AI-0-P linkage. 3. Incorporation of Si in aluminum phosphates should occur preferentially by replacement of P, in agreement with recent experimental results. 4. Bridging hydroxyl groups in aluminum phosphates (originating from nonstoichiometry, defects, or Si incorporation) should show acidic properties. If associated to the incorporation of Si, the acidic proton prefers to be associated with a Si*-AI rather than an AI-0-P bridge. Hence, acidic protons in silicoaluminophosphates could be more localized than in aluminosilicates.
Acknowledgment. We acknowledge J. M. Andrt for useful comments and suggestions. J.G.F. thanks the Scientific Affairs Division of NATO for a fellowship in the area of their International Intersectorial Exchanges in Oriented Research.
Diffusion of Small Molecules in Glassy Polymer Thin Films Studied by Waveguide Raman Techniques N. E.Schlotter Bell Communications Research, Red Bank, New Jersey 07701 (Received: May 23, 1989)
Waveguide Raman spectroscopy (WRS) allows one to monitor the spectrum of a polymer film in contact with a liquid phase. A unique advantage of WRS is its ability to monitor as a function of time all of the molecular species involved in the diffusion
process simultaneously with minimal interference from the bulk solvents or substrate. Combining the linear Raman response with the integrated optical field in the polymer waveguide allows the mass sorption to be measured. If an adequate model is available, diffusion profiles, at any particular time in the experiment, can be calculated. The system studied in the present work consists of a polystyrene waveguide on a fused quartz substrate in contact with a binary solvent mixture of ethanol and perdeuteriotoluene. In this case the diffusion takes place over a distance of approximately 1 pm. Finally, experimental results will be compared to several diffusion models.
Introduction The diffusion of small molecules into a polymer matrix is a complex problem with many applications. Technologically, diffusion plays an important role in the fabrication of integrated circuits.' Photolithography relies on the differences in the diffusion behavior of small molecules in the exposed regions relative to the virgin film material. To a lesser extent the drying behavior of the polymer film is also important in resist processing. As integrated circuit fabrication pushes toward higher densities the resist behavior becomes more critical. Rates of penetration, anisotropic swelling, response to small concentrations of contaminants, etc. all form part of a complex problem that can limit resolution, An analogous diffusion problem is found in the biological membrane." Diffusion of molecular species through a biological membrane is controlled by similar molecular interactions as well ( I ) Manjkow, J.; Papanu, J. S.; Soong, D.S . ; Hess, D. W.; Bell, A. T.J . Appl. Phys. 1987, 62, 682. (2) Pace R. J.; Chan, S. I. J . Chem. Phys. 1982, 76, 4241. (3) Frey, W.; Schneider, J.; Ringsdorf, H.; Sackmann, E. Macromolecules 1987, 20, I 3 12. (4) Ringsdorf, H.; Schmidt, G.; Schneider, J. Thin Solid Films 1987, 152, 207.
0022-3654/90/2094- 1692$02.50/0
as active mechanism^.^*^ Improved understanding of diffusion could have impact on the molecular biology of cell recognition and drug delivery. Finally, polymers are extensively used in science and technology. Improvements in our control of mass transport into and out of polymers used as barrier materials and coatings would be important in many packaging technologies.6 Likewise, thin film electrodes rely on the mass transport of ions. Crack healing, polymer/polymer diffusion, effects of plactizers, and polymer bonding also rely on diffusion processes, but will not be addressed in this paper. Current knowledge of the mechanism of diffusion is often empirical, at best. This phenomenological approach tends to impose constraints from observations of bulk behavior on the diffusion models. For example, surface concentration phenomena which modify boundary conditions and the resulting solutions to the diffusion equation, polymer swelling and crack behavior, kinetics, and molecular interaction through Flory-Huggins x parameters have been incorporated into diffusion models without (5) Terner, J.; Campion, A.; El-Sayed, M. A. Proc. Natl. Acad. Sci. 1977, 7 4 , 5212. ( 6 ) Aronhime, M. T.; Neumann, S.; Marom, G. J . Marer. Sci. 1987, 22, 2435.
0 1990 American Chemical Society
Diffusion of Small Molecules in Thin Films a detailed knowledge of the underlying chemistry and physics of the molecular interactions. The result is a plethora of models that give results applicable primarily to specific cases. Yet, it is known that the diffusion behavior is very sensitive to the molecular species involved. This suggests that diffusion studies can be used to study molecular interactions if our understanding of the diffusion process can be improved. In glassy polymers observations of a sharp boundary at the penetrant front are well do~umented.~-" Diffusion with a sharp boundary that also moves with a constant velocity is referred to as Case I1 d i f f ~ s i o n . ~ ~ ~Similarly, ~ ' l - ~ ~ a solvent's ability to diffuse in a polymer can be drastically modified by trace amounts of other molecule^.^^'^-'^ Additionally, there is evidence that surface concentrations are not simple continuations of adjacent bulk c o n c e n t r a t i ~ n s . l ~ J ~Various -'~ forms of "anomalous" diffusion have been noted11J6~18~20-21 Often this refers to deviations from a simple diffusion model in which the "Fickian" solution shows a linear mass sorption increase relative to time to the one-half power. The existence of pores, or cracks, has been suggested.13J6922-25In glassy polymers there are also relaxation phenomena occurring that modify diffusion and are in turn modified by the penetrant c o n ~ e n t r a t i o n . l ~ ~In ~ ~some J ~ ~ ~cases ~ this is incorporated in models by introducing a concentration-dependent diffusion ~ o e f f i c i e n t . ~Another ~ , ~ ~ approach has been to incorporate the polymer's mechanical response to the osmotic pressure due to the solvent at the diffusion At present no model has been completely successful in predicting all the observations of diffusion.33 In Case 11 diffusion the stress between the glassy core and the solvent swollen surface can control the dimensional behavior of the In extreme cases the pressure differences established by the diffusing material is enough to lead to fractureI2J3 and stress-induced o r i e n t a t i ~ n . ~A~ similar . ~ ~ result occurs under
(7) Hartley, G. S . Trans. Faraday Sac. 1949, 45, 820. (8) Thomas, N. L.; Windle, A. H. Polymer 1977, 18, 1195. (9) Thomas, N.; Windle, A. H. Polymer 1978, 19, 255. (10) Klier, J.; Peppas, N. A. Polym. Bull. 1986, 16, 359. (1 I ) Mills, P. J.; Kramer, E. J. J . Mater. Sci. 1986, 21, 4151. (12) Alfrey, T. Chem. Eng. News 1965, 43(41), 64. (13) Alfrey, Jr., T.; Gurnee, E. F.; Lloyd, W. G. J. Polym. Sci.: Part C: 1966, 12, 249. (14) Thomas, A. G.; Muniandy, K. Polymer 1987, 28,408. (15) Tonyali, K.; Rogers, C. E.; Brown, H. R. Polymer 1987, 28, 1472. (16) Park, G. S. Trans. Faraday Sac. 1951, 48, 11. (17) Crank, J. J. Polym. Sci. 1953, 11, 151. (18) Long, F. A.; Richman, D. J . Am. Chem. Sac. 1960,82, 513. (19) Hui, C.-Y.; Wu, K.-C.; Lasky, R. C.; Kramer, E. J. J . Appl. Phys. 1987,61, 5129. (20) Mandelkern, L.; Long, F. A. J . Polym. Sci. 1951, 6, 457. (21) Peterlin, A. Polym. Lett. 1965, 3, 1083. (22) Michaels, A. S.; Bixler, H. J.; Hopfenberg, H. B. J . Appl. Pofym.Sci. 1968, 12, 991. (23) Garcia-Fierro, J. L.; Aleman, J. V. Eur. Polym. J . 1985, 21, 753. (24) Scherer, J. R.; Bailey, G. F.; Kint, S.;Young, R.; Malladi, D. P.; Bolton, B. J . Phys. Chem. 1985, 89, 312. (25) Turner, D. T. Polymer 1987, 28, 293. (26) Hui, C.-Y.; Wu, K.-C.; Lasky, R. C.; Kramer, E. J. J . Appl. Phys. 1987, 61, 5137. (27) Korsmeyer, R. W.; Von Meerwall, E.; Peppas, N. A. J . Pofym. Sci.: Polym. Phys. Ed. 1986, 24, 409. (28) Korsmeyer, R. W.; Lustig, S. R.; Peppas, N. A. J . Polym. Sci.: Polym. Phys. Ed. 1986, 24, 395. (29) Petropoulos, J. H.; Roussis, P. P. J . Membr. Sci. 1978, 3, 343. (30) Thomas, N.; Windle, A. H. Polymer 1980, 21, 613. (31) Gostoli, C.; Sarti, G. C. Polym. Eng. Sci. 1982, 22, 1018. (32) Thomas, N. L.; Windle, A. H. Polymer 1982, 23, 529. (33) Korsmeyer, R. W.; Peppas, N. A. Pofym. News 1984, 9, 359. (34) Hopfenberg, H. B.; Frisch, H. L. Pofym. Lett. 1969, 7, 405. (35) Thomas, N. L.; Windle, A. H. Polymer 1981, 22, 627.
The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 1693 e
d f
Figure 1. Waveguide Raman liquid sampling cell. Shown are the high-index coupling prisms (b) and quartz liquid flow cell (d), all pressure mounted (c, e) on the asymmetric slab waveguide sample (f, g). The support structure is made of stainless steel (a) and the pressure mounts (c, e) are spring loaded to compensate for thermal expansion mismatches. The waveguide sample consists of a 1 by 2 in. fused silica plate (g) with a spun polystyrene film (f) on one side.
conditions of applied stress, in that crack behavior and failure times are influenced by solvent c ~ n c e n t r a t i o n . ' ~It* ~should ~ be noted that above the glass transition temperature, and in the melt, diffusion is well behaved compared to that in glassy material^.^^"^ Techniques to follow diffusion include the use of chemical labels detected ~ptically,'-~J~mass labels detected by using Rutherford backscattering (RBS),"J9,26 Raman microprobe on sections,1° rotating-polarizer ellipsometer', capillary column inverse gas chromatography,S' weighing,6~l4~l8,2O~22.23~2S,27~37,38,40,41,48,SO,S2,53 mechanical measurement of thickness changes as a function of s ~ e l l i n gbirefringen~e,~',~~, ,~~~~~~ NMR,27UV a d s ~ r p t i o nand ,~~ the optical density of thin layers.49 An alternate approach has been to measure steady-state transport across a polymer membrane, thereby extracting diffusion constants.39 One method used in this catagory relies on the permeation of argon gas through thin films.22 In this work a new technique for studying diffusion will be presented. It combines waveguide Raman sampling (WRS), using an asymmetric slab waveguide, with a liquid/solid interface. It has several novel features including the simultaneous monitoring of all the molecular species involved in the experiment and the collection of in situ. "real time" data. Experimental Section A central component of this study is a liquid/solid cell developed for use with an asymmetric slab waveguide structure. A schematic of this cell is shown in Figure 1. It basically consists of a fused quartz cavity approximately 1 mm deep with lateral dimensions
(36) Truong, V. T.; Williams, D. R. G.; Allen, P. E. M. Eur. Polym. J. 1987, 23, 41. (37) Prager, S.; Long, F. A. J . Am. Chem. SOC.1951, 73, 4072. (38) Kokes, R. J.; Long, F. A. J . Am. Chem. Sac. 1953, 75, 6142. (39) Aitken, A.; Barrar, R. M. Trans. Faraday Soc. 1955, 51, 116. (40) Hayes, M. J.; Park, G. S . Trans. Faraday SOC. 1955, 51, 1134. (41) Fujita, H.; Kishimoto, A.; Matsumoto, K. Trans. Faraday Soc. 1960, 56, 424. (42) Green, P. F.; Mills, P. J.; Palmstrom, C. J.; Mayer, J. W.; Kramer, E. J. Phys. Rev. Lett. 1984, 53, 2145. (43) Kramer, E. J.; Green, P.; Palmstrom, C. J. Polymer 1984, 25, 473. (44) Mills, P. J.; Green, P. F.; Palmstrom, C. J.; Mayer, J. W.;Kramer, E.J. Appl. Phys. Lert. 1984, 45,957. (45) Green, P. F.; Palmstrom, C. J.; Mayer, J. W.; Kramer, E. J. Macromolecules 1985, 18, 501. (46) Koszinowski, J. J . Appl. Polym. Sci. 1986, 32, 4765. (47) Sada, E.; Kumazawa, H. J. Appl. Polym. Sei. 1986, 32, 5567. (48) Iwai, Y.; Kohno, M.; Akiyama, T.; arai, Y. Polym. Eng. Sci. 1987, 27, 837. (49) Ito, T.; Seta, J.; Urakawa, H. Colloid Polym. Sci. 1987, 265, 557. (50) Kalachandra, S.; Turner, D. T. Polymer 1987, 28, 1749. (51) Pawlisch, C. A.; Marcris, A,; Laurence, R. L. Macromolecules 1!?87, 20, 1564. (52) Turner, D. T.; Abell, A. K. Polymer 1987, 28, 297. (53) Czanderna, A. W.; Thomas, T. M. J. Vac. Sci. Technol. 1987, 5, 24 12.
1694
The Journal of Physical Chemistry, Vol. 94, No. 4, 1990
of 5 mm by 7 mm. The top and bottom surfaces are polished to an optical finish. Opposite the cell cavity are inlet and outlet ports for pumping liquids through the cavity. These ports are quartz tubes with outside threads that connect to Teflon tubing with Teflon connectors. (Starna Cells, Inc. custom fabricated this cell.) The cell is mounted directly on the waveguide surface with a thin film of silicone grease (index matched to the quartz cell is preferable, but silicon stopcock grease will work in some cases) and held in place with a spring-loaded clamping system. The binary solvent mixture is pumped through the cell using a syringe pump driving a gas-tight syringe. The polymer waveguides were formed by spin casting. A Headway resist spinner is used at approximately 2000 rpm to spin polystyrene films from benzene solutions. The solution concentrations are adjusted to give viscosities that will produce dry films 1-2 pm in thickness. The polystyrene samples were all narrow M J M , distributions from Pressure Chemical with ratios near 1.06. The 4000 and 9000 molecular weight samples were prepared with 4 g in 20 mL of benzene. The 17 500,35000,and 50000 molecular weight samples were prepared with 4 g in 30,35, and 40 mL of benzene, respectively. The 4000-35 000 molecular weight samples were filtered through Anotec's 0.2 pm Anotop syringe filters while the 50000 molecular weight sample had to be filtered through a 0.45-pm Nalgene PTFE syringe filter. The benzene used was glass distilled spectroscopic grade solvent. The fused quartz substrates were cleaned by using basic cleaners such as Alconox followed by immersion in hot 5:l concentrated sulfuric acid and 30% hydrogen peroxide. The substrates were then rinsed thoughly in 18 Mohm deionized water and stored at 80 OC in an oven purged with dry nitrogen. Spinning was done by first rinsing the substrate with benzene and spinning it dry several times followed by covering the complete substrate with the polymer solution and spinning for several minutes to an initial dryness. The coated substrates were then returned to the oven at 80 OC to dry overnight. When the humidity is above about 40% R H it is necessary to raise the oven temperature to 120 OC to avoid having the polymer films delaminate from the quartz when exposed to the solvent during the experiment. The binary solvent solution was prepared from Aldrich perdeuteriotoluene (T-d8) 99+ atom % D and Quantum Chemicals (US1 division) 200 proof ethyl alcohol (EtOH). With regard to the ethanol source it is important to check on the denaturing status. In many cases glass distilled, spectroscopic grade ethanols will have as much as 15% of other solvents. The binary solvent mixtures were prepared with 2, 4,and 8% T-d8 to EtOH by volume. These mixtures were pumped across the film by using a Sage Instruments Model 355 syringe pump. Pumping rates were on the order of 1 mL/h. The Raman system consists of a Coherent 20W argon ion laser, f/2 double achromat collection lens system, scrambler, Spex Triplemate, and a modified PAR intensified OMA. The data collection software was written in house and runs on a DEC PDP 11/84 computer. The PAR Model 1420 intensified OMA has been modified to improve its low light characteristics and is described in ref 54. The spectra were obtained with 4880-Alaser excitation with approximately 500 mW of power incident on the coupling prism. Collection times ranged from 15 s to I min depending on the polymer film's optical quality. Collection times were short compared to the time of the overall experiment which could run for several days. The waveguide mount has been designed to position the film in the focus of the collection optics and then to couple the laser beam into the film's guided modes without disturbing the position of the waveguide in the viewing field of the collection optics. The laser was coupled to the waveguide via the transverse electric (TE) modes (polarization of the laser in the film plane of the waveguide). The viewing field of the collection optics is inside the liquid cell, approximately centered between the coupling prisms. Approximately 1 mm of the propagating laser beam path in the guide (54) Schlotter, N. E.; Schaertel, S.A.; Kelty, S. P.; Howard, R. Appl. Spectrosc. 1988, 42, 146.
Schlotter
Figure 2. Schematic of the waveguide Raman sampling experiment. The quartz cell (e) in Figure 1 is mounted such that the laser focus (9) is at the front edge of one of the coupling prisms which is also coincident with the rotation axis of the coupling angle, cp, adjustment. The incident laser direction is varied through the angle cp to find the allowed angles for beam coupling. The waveguide sample (a, b) is held fixed relative to the spectrometer. The out-coupled light (c) can be analyzed for mode structure. The light is in- and out-coupled with high-index prisms (d) made of a Schott glass LaSF5. The Raman signal is collected through the cell and directed into the spectrograph system (f). I ~
I
1
4
Polystyrene
1 f IO
600
WAVENUMBERS
Figure 3. Spectra of the individual molecular components and the solvent mixture. The polystyrene spectrum is from the bare guide while the solvents and mixture are taken from bulk samples in cuvette cells.
was collected. The Raman signal was collected at right angles to the waveguide film plane through the fused quartz of the liquid/solid cell as shown in Figure 2. Results Figure 3 shows the spectrum of a bare polystyrene waveguide film 1-2 pm thick. This spectrum was collected with 4880-A excitation at 500 mW for 5 s. The pure ethanol spectrum in Figure 3 was collected from a bulk liquid sample in a cuvette (4880A,
The Journal of Physical Chemistry, Vol. 94, No. 4 , 1990 1695
Diffusion of Small Molecules in Thin Films
TABLE I material fused silica polystyrene toluene ethanol
refractive index' 1.463 1.605
1.507 1.366
'The values are from tables in the American Institute of Physics Handbook and were measured at 0.486 pm (line F) and 20 O C , or interpolated to these values.
cm
z w c
71 2 657 577 425
z
z
9
B
299 212 114 62 6 7-
600
1600 WAVENUMBERS
Figure 4. A selection of spectra taken from the waveguide Raman liquid sampling cell over time. The numbers to the right are the times of collection, in minutes. from the start of the experiment. These spectra have been scaled to the 1009 wavenumber band of polystyrene. Growth of the main ethanol band at 890 wavenumbers and the main perdeuteriotoluene band at 965 wavenumbers can be clearly seen. The other solvent bands are also present, but much more weakly.
200 mW, IO s). Similarly, the perdeuteriotoluene (4880 A, 200 mW, 5 s) and a 5% T-d8 in EtOH mixture (4880 A, 200 mW, 10 s) were recorded, The conditions of collection were adjusted to give peak intensities of approximately equal magnitudes for the main peak of each spectrum. The spectra are plotted on the same scale with offsets. Figure 4 shows a set of spectra from the liquid/solid cell waveguide system. The waveguide is a 17 500 M , polystyrene film spun on quartz. The binary solvent used is the 4% T-d8 in EtOH mixture. The spectra were recorded with 500 mW of 4880-A excitation with exposures of 15-30 s. All spectra have been scaled to have the polystyrene band, at 1009 wavenumbers, equal in magnitude and are plotted on the same scale with offsets. The time that the film has been in contact with the solvent system is given in minutes to the right of each spectrum. The syringe pump was driven at about 0.8 mL/h. The EtOH band a t 890 cm-l and the T-d8 band at 965 cm-* are clearly seen to grow into the spectra with time. No molecular weight dependence has been observed for diffusion in the range of polystyrene molecular weights used. Also, the general nature of the diffusion, other than the rate, seems to be independent of the toluene concentration.
Discussion The laser input beam is injected into the polystyrene waveguide by using prism c o ~ p l i n g . ~ *Similarly, -~~ the propagating light is outcoupled by another prism after traversing the waveguide. Input and output coupling is subject to the physical structure of the guide and the optics of the system. The parameters that affect the coupling are the refractive indexes of the materials, wavelength of the laser, angle of laser input into the prism, film thickness, and prism geometry. For film thicknesses that are on the order of a few wavelengths of the laser wavelength there will be only a few discrete guided modes possible and these will correspond to distinct coupling angles.55 ( 5 5 ) Rabolt, J. F.; Santo, R.; Schlotter, N . E.; Swalen, J. D. I E M J . Res. Dev. 1982, 26, 209.
( 5 6 ) Rabolt. J. F.;Schlotter, N. E.;Swalen, J. D.; Santo, R. J. Polym. Sci.: Polym. Phys. Ed. 1983, 21, 1. (57) Schlotter, N. E.; Rabolt, J. F. J . Phys. Chem. 1984, 88, 2062.
In this case the asymmetric slab waveguide mode structure is modified by the jump boundaries in the refractive index that occur at the edges of the liquid/solid cell. There are two transition types: air/polymer/quartz to quartz/polymer/quartz and quartz/ polymer/quartz to binary solvent/polymer/quartz. This set of transitions occurs both on entering the cell and exiting the cell. These jump boundaries have the effect of mixing the modes. This mode mixing occurs because the effective angles for each mode are slightly shifted at each boundary and the projections of the previously populated modes are mapped onto the new modes. Each boundary then tends to enhance the distribution of the mode intensities, leading to an approximately even illumination of the modes. The even illumination corresponds to approximating the sum of the Fourier squared terms of the mode intensity distributions with a square wave. It should be noted that the propagating wave solutions correspond to a geometric optics picture in which the light bounces from interface to interface at an angle such that the sum of the phase shifts at the interfaces plus that of the round trip equals an integer multiple of 27. The resulting standing wave solutions are similar to particle-in-the-box wave functions. The addition of finite energy barriers as box walls, which allows tunneling, completes the picture and corresponds to the exponentially decaying evanescent fields outside the waveguide. For the purposes of this paper the optical field intensity will be taken to be a uniform field from the substrate/polymer interface to the polymer/liquid interface. A further modification of the waveguide optics can occur due to the penetration of the solvents into the polymer. If there is a significant uptake of solvents it could be plausible to consider the solvent gradient as forming a gradient index structure. This would focus more of the light into the polymer waveguide and away from the liquid/solid interface much the way a gradient index optical fiber self-focuses into its core. This is more likely to occur in cross-linked rubbers that could swell and have substantial density changes. The glassy, low molecular weight, polystyrenes used in this study are expected to dissolve before there is a significant density and refractive index change. These changes cannot be entirely discounted and calculations are in progress to model this type of structure. It is not believed to be a large effect for this system, but may be an important factor in a quantitative study. The binary solvent system was chosen for several reasons. The solvents are of lower refractive index than the polymer as shown in Table I. This is a necessary criterion for the maintenance of the guided wave in the polymer film. Ethanol and toluene are also miscible over a large range of concentrations. Ethanol is a nonsolvent for polystyrene while toluene is a good solvent for polystyrene. Perdeuteriotoluene was used to shift the Raman spectrum relative to that of polystyrene. This aids the analysis since the combination of EtOH, T-d8, and polystyrene gives a spectrum that has distinct Raman bands, corresponding to the individual molecular components, with minimal band overlaps. The spectra shown in Figure 4 indicate that several interesting phenomena are occurring. When the same experiment is done with EtOH alone there is no growth of the EtOH bands with time. The polymer uptake of EtOH from the binary solution indicates a changing structure or interactions presumably induced by the small toluene concentration present. Additionally, the concentrations of EtOH and T-d8 in the polymer are dependent on the external concentration of T-d8. In experiments where the concentration of T-d8 is changed the spectral intensities of both of the solvents track the T-d8 concentration changes. By comparing
1696 The Journal of Physical Chemistry, Vol. 94, No. 4, 1990
Schlotter
EtOH / PS
T-d8
/
EtOH
1
0.120,
1.000 0 01
2
2 01 m
I
0.750
"._"" TIME (mi.)
/
PS
I
".VT"
I
0
500
250
750
TIME (min)
Figure 6. Ratio of the main band intensity of perdeuteriotolueneto the main band of polystyrene. Intensities as simple peak intensities. The smooth curve is a regression fit to the data.
the band intensity ratios of the 965-cm-I line of T-d8 and the 890-cm-' line of EtOH in the bulk mixture spectrum (Figure 3) to the waveguide experiment (Figure 4) it is seen that the T-d8 concentration has been enhanced in the polymer relative to the EtOH concentration. These concentration effects can be more clearly Seen in Figures 5-7 which plot simple band intensity ratios. The 890/1009 ratio in Figure 5 compares the EtOH concentration to that of polystyrene while the 965/1009 ratio in Figure 6 compares the T-d8 concentration to the polystyrene. The plot in Figure 7 shows the ratio of T-d8 to EtOH. The bulk solvent ratio for the 965 to 890 cm-' bands is approximately 0.25. The curves plotted on Figures 5-7 are nonlinear least-squares fits. The curves are primarily guides for the eye. The curve used in Figure 5 for EtOH/PS is an eighth-order polynomial which gave the best fit with a correlation coefficient of 0.99. Similarly, in Figure 6 for T-d8/PS the best fit is fifth order with a correlation coefficient of 0.97 and in Figure 7 an eighth-order fit with a correlation coefficient of 0.98 was found. The plots of EtOH/PS and Td8/EtOH show similar behavior in the sense that an initial slow change in the band ratios is followed by a transition to a faster rate of change. The T-d8/PS plot may have a hint of a break in the same region as the other two but is mostly a smooth rise to saturation. Clearly there are molecular changes taking place in the polymer film. At this point it is helpful to look at various models for the diffusion of small molecules in glassy polymers. In order to interpret the changes in the Raman spectra as a function of time, expected results from several models for the diffusion of small molecules in glassy polymers were calculated. Most theories of diffusion start with Fick's second law d c = D -a2c at
250
500
750
TIME (min)
Figure 5. Ratio of the main EtOH band intensity to that of the main band of polystyrene. Intensities as simple peak intensities. The smooth curve is a regression fit to the data. T-d8
0
ax2
where C is the concentration of the diffusing material, x is the space coordinate normal to the surface, and D is the diffusion
Figure 7. Ratio of the intensities of the main bands of perdeuteriotoluene to that of ethanol. Intensities as simple peak intensities. The smooth curve is a regression fit to the data.
coefficient (length2/time). This can be modified to incorporate diffusion constants that depend on concentration, inhomogeneous media, and a host of boundary conditions which can assume many forms. In many cases these modifications result in complicated solutions even for relatively simple models. Further, these empirical models have not been fully successful in modeling experiments using glassy polymers. This is why it is also popular to incorporate empirical models of glassy polymer relaxation processes into the diffusion models. A discussion of these approaches is given by WindleS6 and the references contained in that paper. It is helpful, and common, to divide the experimental diffusion observations according to the following definitions: (1) Case I diffusion or Fickian diffusion occurs when the rate of diffusion is much less than the relaxation rate of the polymer. For diffusion into a semiinfinite medium from an infinite solvent source mass sorption is proportional to time to the 0.5 power (initial medium concentration is zero and surface concentration is constant). However, for a semiinfinite planar sheet in an infinte solvent source the time to the 0.5 relation holds at short times, at best.s9 (2) Case I1 diffusion: the diffusion is very rapid compared to the polymer relaxation process. Frequently a sharp solvent front which propagates into the polymer at a constant velocity is associated with Case I1 diffusion. The resulting mass sorption is then directly proportional to time.I2 It has been assumed that the solvent front demarcates a rubbery shell from a glassy core, and when the mechanical properties of the two states are large, fracture can occur in the glassy core. (3) Case 111 diffusion, non-Fickian, or anomalous diffusion occurs when the diffusion and relaxation rates are similar. Basically all cases that cannot be accounted for by Cases I and I1 are collected as Case 111. As an indicator for anomalous diffusion some descriptions &@est mass sorption is related to time raised to a power between 0.5 and 1.O. Exceptions occur, such as at short times when this is not a reliable indicator due to the sensitivity of the solutions of the diffusion equation to boundary and initial conditions. Fickian diffusion and Case I1 diffusion models provide extremes that will be used in this paper to compare to the observed results. It should be noted that often the Boltzmann transformation on Fick's second law is cited as evidence that the concentration penetration is proportional to tl/* without noting the limited conditions of appli~ability.~~ Specifically, C must be a function of x / 2 t 1 / *only and the boundary conditions must also be functions of ~ / 2 t I only.59 /~ The Boltzmann transformation is mainly a technique for changing a partial differential equation into an ordinary differential equation. For the Fickian diffusion case a model can be developed from a symmetric structure corresponding to a free-standing film im(58) Windle, A. H. Case I1 Sorption. In Polymer Permeability; Comyn, J., Ed.; Elsevier Applied Science: London, 1985; Chapter 3, p 75. ( 5 9 ) Crank, J. The Mathematics of Dij/usion, 2nd ed.;Oxford University Press: London, 1975.
The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 1697
Diffusion of Small Molecules in Thin Films
1 .o
1 .o
8
0
I
0 \
\
0
I
0.0 1 .o
0.0
X / L Figure 8. Calculated concentration profiles from the solution in the text to Fick's second law with a time-dependent surface concentration. In this case the surface concentration rises to C(0) nearly instantaneously (a = 300) and is therefore representative of concentration profiles for the standard problem of diffusion in a semiinfinite slab from an external solution with no dependence of D on concentration. The concentration is normalized to C(O), the surface of the plane in contact with the solution is at 1 .O, and distance is normalized to the thickness of the slab. Time has also been scaled by D / L 2 for ease of calculation.
mersed at t = 0 in an infinite solution. These results are applicable to having only one face in contact with the liquid phase and an impermeable barrier at the center of the film (noting the reduction in the thickness), Le., the waveguide structure. This is equivalent to the free film because in both cases there is no net mass transport through particular planes, that is, the plane at the center of the free film and the plane at the polymer/quartz interface in the waveguide. At the same time the effects of a time-dependent surface concentration can be incorporated into the model in a parametrized form. In this case the surface concentration behavior is incorporated into the boundary condition^.^^ The boundary conditions are C = C(O)(l C=0
- exp(+t)),
for x
< -L and x > L
C/C(O) = 1 - exp(-aDt/L2) cos (c~'/~x/L)/cos( d / 2 ) 4 (-l)n exp(-(2n I)*a2Dt/4LZ) an=O
/ '1
2.0
Figure 9. Mass sorption curves calculated for the time-dependent surface concentration diffusion problem. Normalized mass is plotted versus scaled time. Rates of increase in the surface concentration, a = 300 being the fastest, are used to generate a family of curves showing the uptake rate versus time. The plot for a = 300 corresponds to classical Fickian response. 1.0
1
T and T'" Figure 10. Plot of the mass sorption results given in Figure 9 for a = 300, representative of Fickian diffusion, versus time and time to the 1/2 power.
for t = 0, for -L Ix 5 L
j3 controls the rate at which the concentration goes to C(O),the initial, or bulk concentration. Large values of represent immersing the film in solution and having the interface very rapidly, or nearly instantaneously, approach the equilibrium concentration. Smaller values of j3 have a slower growth of the surface concentration with time and could represent a wetting process. The general solution for this system is
-E
Dt
0.0
1.3
0
0 \ 0
+
(2n
+ 1)[1 - (2n + 1)2r2/4a] cos [(2n + l)nx/2L]
where j3 has been scaled as j3 = aD/L2. Mass transport, or sorption, versus time is proportional to the Raman response versus time and is derived from the concentration by integrating C from -L to L giving
-
M/ZLC(O) = 1 - ( Y - I / ~ exp(-aDt/L2) tan ( o r ' / 2 ) 8 exp(-(2n 1)2n2Dt/4L2) -a2En a g (2n + 1)2[1 - (2n + 1)2n2/4a]
+
The solution corresponding to Fickian behavior can be obtained from the above equations when a gets large. Calculated concentration profiles for large a (=300)are shown in Figure 8. The corresponding mass sorption curve is shown in Figure 9 (for CY = 300). Figures 10 and 11 show the concentration profiles corresponding t o small values of a (=2, 5 ) . The corresponding mass sorption curves (for a = 2, 5 ) are shown in Figure 9 as well as the additional mass sorption curves for (Y = 1.0 and 0.5.
0.0 1 .o
0.0 X / L Figure 11. Calculated concentration profiles similar to those in Figure 8, but with a = 5.
Figure 8 shows a set of concentration profile curves that plot normalized concentration (concentration/solution concentration) versus normalized penetration depth (depth/total thickness). The family of curves are plots done at different values of time (scaled by OIL2). These curves correspond to concentration profiles that would be obtained for a plane sheet in contact with solvent on one side, or half of a plane sheet dipped in solvent, if diffusion follows Fick's second law. The normalized mass sorption curve for a = 300 in Figure 9 is the result of integrating the family of curves in Figure 8. The mass sorption is also plotted versus the
1698 The Journal of Physical Chemistry, Vol. 94, No. 4, 1990
Schlotter
............_...____. .........._. ..............
8 I
\
\
2
>-----
-------
0.0 I 10
X / L
0.0
Figure 12. Calculated concentration profiles similar to those in Figure 8. but with a = 2.
1
1. I
0.0 1 .o
X / L
0.0
Figure 13. Schematic concentration profiles expected for pure Case I1 diffusion. Concentration is normalized to the bulk concentration and depth normalized to the total thickness of the slab. In this case time has been normalized to the total time required to saturate the film with solvent. This also shows the constant velocity of penetration observed for Case I1 diffusion.
same scaled time as used in Figure 8. Plotting the mass sorption versus time was done for ease of comparison with the experimental data, also linear in time. If the diffusion in the polymer waveguide is Fickian it would be expected that the Raman signal should behave similarly to the mass sorption curve in Figure 9 with a = 300. When the mass sorption (for CY = 300) is plotted versus ( D Z / L ~ )the ' / ~curve is approximately linear up to ( D t / L 2 ) 1 /= 2 0.75 as expected for Fickian diffusion. This is shown in Figure 10.
Figures 1 I and 12 show families of concentration profiles for = 5 and 2, respectively. These figures demonstrate the effect of having a surface concentration that rises to the bulk solution concentration over some time period. The corresponding mass sorption curves are given in Figure 9 as well as mass sorption curves for CY = 1 and 0.5. Clearly, the diffusion behavior has moved away from classical Fickian diffusion as defined above (although this is simply a variation of Fick's second law with a modified boundary condition). Physical evidence of surface effects have been detected"J7-19 and a surface interaction is very plausible as part of the explanation of anomalous diffusion. A problem with this approach is determining what the surface effect is and how to model it. Further, it cannot be used to model the observations that have been grouped under Case I1 behavior. When Case I1 diffusion dominates, the rearrangement of the polymer structure has been considered to control the penetration of the solvent molecules. Experimentally it has been observed as a sharp boundary between solvated polymer and dry polymer.35 CY
0.0 00
T
/
Tf
i 1 .o
Figure 14. The resulting Case I1 mass sorption from the integration of the concentration curves in Figure 12. As before the sorbed mass is
normalized to the total mass at saturation. In this case time is normalized rather than scaled as in Figure 9. Schematic concentration profiles expected for a Case I1 polymer/solvent system are sketched in Figure 13. In Figure 13 normalized concentration is plotted versus normalized distance from the polymer surface. The series of curves are for increasing increments of time normalized to the time needed to reach saturation of the polymer film. The transition region for each curve is shown with small rounding of the break to reflect a finite region of transition rather than an instantaneous change. This is assumed to be true, but the exact nature and size of the transition region is not known. The corresponding mass sorption is given in Figure 14 and the Raman response from the waveguide should track the mass sorption and show a linear increase in signal strength with time. With these models of diffusion in a glassy polymer in mind, the data in Figures 5-7 can be reevaluated. In Figure 5, the EtOH/PS curve, a slow induction period is followed by a rapid increase that is plotted to near-saturation value (saturation is at a ratio of 1.4 out to 1800 min). The slow induction period may represent a Fickian response for the diffusion of the ethanol into the initial polystyrene composition. About 400 min into the experiment the uptake of ethanol begins to increase linearly with time as might be expected for Case I1 diffusion. One possibility is that the toluene has diffused into the polystyrene and at some concentration level the T-dI/PS composition is modified to a form that is compatible with EtOH, yet is still glassy in nature. However, it is more plausible that the T-d8/PS composition is really rubbery and the EtOH uptake is not linear with time but is showing some modified Fickian response. At this point it will require more resolution to distinguish the functional form of the sorption. But it should be noted that Figure 7, which is the ratio of the T-d8 to EtOH peak intensities, shows a similar induction time followed by an increase in the EtOH concentration relative to the T-d8 concentration. At the same time Figure 6 shows uptake of T-d8 that is much more Fickian in behavior and no induction period is present. This supports the idea that polymer composition is changed by the toluene and makes it plausible that one is really seeing two different diffusion regimes for the EtOH in the T-d8/PS composition. It also points out an important feature of this experiment. By simultaneously monitoring all the molecular species involved cooperative phenomena can be detected. It may be possible with quantitative improvements in the method to see changes in the polymer morphology during diffusion. The binary solvent/nonsolvent system is an explicit part of this experiment and several speculations can be made concerning how it functions. As noted above the EtOH/PS curve shows a large break in the curve while the T-d8 shows a mild break at best. All the breaks in slope occur at the same time. Essentially the polystyrene seems to be converted to a T-d8 solvated environment, masking the polystyrene from the EtOH, and thereby allowing the EtOH to enter. Finally, the T-dB/EtOH curve shows an initial
J . Phys. Chem. 1990, 94, 1699-1702 enhancement of the T-d8 concentration relative to EtOH (bulk solution has a value of =0.25) followed by a near-linear decrease after the break. Returning to the Fickian solution and using the slower turn on of the surface concentration could represent the EtOH diffusion into the polystyrene. Figure 9 with a = 1 would be indicative if an induction period was included. That the EtOH is constrained from significantly penetrating the polystyrene until it has been modified by the T-d8 could be understood in terms of wetting phenomena. This would lower the EtOH concentration at the surface and give rise to a delay in the buildup of the EtOH concentration. In general these experiments show no significant evidence for simple Case 11 diffusion behavior. There may be several reasons for this result. The distance scale is significantly less than that used in most sorption experiments. The measurements are done in “real time”. A true three-component chemical system has been studied in contrast to many studies that have two components and a “marker” component that is treated as an inert material. And, finally, the fused silica substrate may introduce some compensating strain into the polymer. Some of these questions will be addressed in future work.
1699
Conclusions It has been demonstrated that it is possible to monitor all the chemical species involved in a diffusion process and that it is important to do so. Waveguide Raman sampling allows real time sampling to be done and is applicable to many polymer/solvent systems and potentially polymer/polymer systems. The assumption that the initial materials and concentrations remain unchanged is not valid in this case and probably questionable in most cases of chemical diffusion. The length scale for this experiment was on the order of 1 pm with submicrometer resolution possible. Although the results indicate that the solvents EtOH and T-d8 are controlled by different parameters, it is probably not possible to decide what model of diffusion is correct. Even further work to quantify these results may not be able to separate the models due to their empirical and adjustable natures. What does seem clear from these results and those in the literature is that molecular interactions must be incorporated from the start and that a microscopic model of diffusion is needed. Registry No. EtOH, 64-1 7-5; polystyrene, 9003-53-6; toluene-d8, 2037-26-5.
Primary Electron Transfer in Photosynthetic Reaction Centers Eberhard v. Kitzing and Hans Kuhn* Max- Planck- Institut fur biophysikalische Chemie, Am Fassberg, Gottingen, F.R.G. (Received: April 26, 1989)
A simple model is presented for the primary step in the photoinduced electron transfer in the photosynthetic reaction centers of Rps. uiridis and Rb. sphaeroides. The interaction of the chromophore system (consisting of photoexcited donor P, conduction intermediate BL, and acceptor HL) with the environment is assumed to be negligible until vibronic deexcitation takes place resulting in a stochastically perturbed adiabatic electron transfer. This process constitutes a three-level problem. It is shown that this problem, in the present case, can be approximated by a two-level problem which can simply be solved. The energy level of IBL-) is found to be 0.06 eV above the level of (P*).The unidirectionalityof the electron flow is explained by coincidence of energy levels in the L branch due to evolutionary constraint.
Introduction In the primary electron-transfer step in the photosynthetic reaction center of purple bacteria the electron donor P, the special pair, a bacteriochlorophyll dimer, is photoexcited to P* and the electron is transferred within some picoseconds from P* (via BL the accessory bacteriochlorophyll) to the spectroscopically resolvable intermediate electron acceptor HL a bacteriopheophytin (for a recent review see ref 1). Fleming et aL2 have found an increase in electron-transfer rate k with decreasing temperature which is stronger in Rps. viridis than in Rb. sphaeroides. The rate of electron transfer from P* via BL to HL along the L-branch is at least 10 times larger than from P via BM to HM3along the M-branch, which structure is nearly symmetric to that of the L-branch, and this leads to the unidirectionality of the charge separation. The rate is of the order of a reciprocal picosecond at 10 K. The electron transfer is given by a single rate constant, and a transient bleaching in the BL absorption region is a b ~ e n t . These ~ findings ( I ) Michel, H.; Deisenhofcr, J. Bull. Inst. Pasteur 1988, 86, 37. (2) Fleming, G . R.;Martin, J. L.; Breton, J. Nature 1988, 333, 190. (3) Michel-Beyerle, M. E.; Plato, M.; Deisenhofer, J.; Michel, H.; Bixon, M.; Jortner, J. Eiochim. Eiophys. Acta 1988, 932, 52. Plato, M.; M6bius, K.; Michel-Beyerle, M. E.; Bixon, M.; Jortner, J. J . Am. Chem. SOC.1988, 110..~ 7279. (4) Breton, J.; Martin, J. L.; Fleming, G . R.;Lambry, J. C. Biochemistry 1988, 27, 8276.
0022-3654/90/2094-1699$02.50/0
are evidence against mechanismss in which BL- is a distinct, kinetically resolvable intermediary electron acceptor between P* and HL-. It is difficult to rationalize the results on the basis of the conventional theory for nonadiabatic electron transfer. A nonadiabatic superexchange-mediated electron transfer from P* via BL to HL has been prop~sed.~ The unidirectionality was explained as being due to a delicate difference in the electronic interaction terms between the two b r a n ~ h e s .The ~ approach is based on the conventional assumption that the chromophores are strongly coupled to the environment, as described by the Marcus equation6 or its quantum mechanical analogues. In the present approach the interaction with the environment is treated differently, and this leads to a different judgment of the role of specific structural asymmetries in inducing this unidirectionality. In contrast to the model in ref 3, here the difference in energies of HL- and HM-is sufficient to explain the unidirectionality in electron transfer. The results by Fleming et aL2 can be rationalized. By its simplicity this approach should be of interest focusing on some important features. Studies on sitespecific mutants and on other organisms’ should shed light on the (5) Shuvalov, V. A,; Amerz, J.; Duysens, L. N. M. Eiochim. Eiophys. Acta 1986,851, 327. Marcus, R. A. Chem. Phys. Lett. 1987, 133, 471. (6) Marcus, R.A. J . Chem. Phys. 1956, 24, 966. (7) Yeates, T. 0.;Komiya, H.; Chirino, A.; Rees, D. C.; Allen, J. P.; Feher, G . Proc. Natl. Acad. Sci. USA 1988, 85, 1993.
0 1990 American Chemical Society
