J. Phys. Chem. 1985,89, 3474-3477
3474
Infrared Spectrum of VltrHled Llquld Water. A Comparison wlth the Vapor Deposited Amorphous Form Erwin Mayer Institut fur Anorganische und Analytische Chemie, Universitat Innsbruck, A 6020 Innsbruck, Austria (Received: April 8, 1985)
The infrared spectrum of vitrified liquid water is reported from 4000 to ZOO0 cm-'. The vitrified samples were prepared by rapid cooling of an aqueous aerosol containing 11 mol % HOD in H20, on a solid cryoplate. Spectral distortion by the solidified aerosol droplets was corrected by deposition of argon or by deposition of water vapor. A comparison with vapor deposited amorphous solid water shows two differences: first, in the vitrified aerosol the peak maximum of the decoupled 0-D stretching transition at 16 K is at lower wavenumbers, at 2416 cm-I, and second, in the coupled 0-H stretching band region an additional shoulder is observed at approximately 3120 cm-I. These spectral differences are related to structural differences, implying that the amorphous forms of water made from the liquid and the vapor have different structures.
Introduction It has been debated recently whether or not the amorphous forms of water made from the liquid state and from water vapor have the same From investigations of vitrified liquid water experimental evidence for3 and against it4 has been presented. Investigations of vitrified liquid water have until now been limited and some results been somewhat ambiguous because a liquid cryomedium had to be used for heat transfer and rapid cooling. This had either to be removed by evaporation3v4or, for larger samples, could not be removed at all without partial devitrification." Therefore, a new method for the vitrification of pure liquid water was developed which works as the only one without liquid cryomedium for heat transfer: rapid cooling of aqueous aerosol droplets on a solid c r y ~ p l a t e . For ~ this method the only impurity is some codeposited water vapor. The infrared spectrum of vitrified liquid H 2 0 and of the decoupled 0 - D oscillator in excess H 2 0 is a first application of this method and is reported in this paper. For comparison the infrared spectrum of the vapor deposited form of H 2 0 , H20(as), is given as well. This comparison shows spectral differences which are related to structural differences, implying that the amorphous forms of water made from the liquid and the vapor have different structures. Giguere and Harvey have already reported the infrared spectrum of vitrified pure liquid water for 1-pm layers between AgCl windows.6 This could not be reproduced by others.' A comparison of their spectra with the spectrum of vitrified aerosol is problematic because Gigusre and Harvey had published only a survey spectrum. However, they reported that "essentially the same absorption spectra were obtained at all temperatures even down to that of liquid air".6 This is clearly different from what has been observed in this work for example, the principal maximum of the 0-H stretching band is in liquid water near 3490 cm-I at 298 K,8 and is in the vitrified aerosol shifted by more than 200 cm-' to lower frequencies. Experimental Section The vitrification of water by rapid cooling of aqueous aerosol droplets is described elsewhereSand is only summarized; aqueous aerosol was produced with an ultrasonic nebulizer operating at (1) Sceats, M. G.; Rice, S. A. In 'Treatise on Water"; Franks, F., Ed.; Plenum Press: New York, 1982; Vol. 7, Chapter 2. (2) Johari, G. P. Philos. Mag. 1977, 35, 1077. (3) Dubochet, J.; Adrian, M.; Vogel, R. H. Cryo-Letters 1983, 4, 233. (4) Mayer, E.; Brliggeller, P. J. Phys. Chem. 1983, 87, 4744. ( 5 ) Mayer, E. J. Appl. Phys., in press. (6) Gigu&e, P. A.; Harvey, K. B. Can. J. Chem. 1956, 34, 798. (7) Angell, C. A. In 'Treatise on Water"; Franks, F., Ed.;Plenum Press: New York, 1982; Vol. 7, Chapter 1, p 21. (8) Eisenberg, D.; Kauzmann, W. "The Structure and Properties of Water"; Clarendon Press: Oxford, 1969; p 200.
3 MHz (LKB instruments, Model 108) and transferred with gaseous nitrogen as carrier gas through an electron microscope aperture with 200-pm diameter into a high-vacuum system containing the low-temperature infrared cell. Carrier gas flow rate was 3 L m i d ; pure liquid water nebulized to aerosol at 0.5 mL m i d or lower. The aerosol was led through a tubing (1-m long, 23" diameter) to the aperture. To reduce the amount of water vapor it was cooled to approximately 273 K. The aerosol was deposited on a KBr window attached to a Displex cryotip (Air Products, Model CS-208 L), using 45O geometry. The distance between KBr plate and orifice was 40 mm, vacuum during deposition 2 X mbar. Deposition temperatures were mostly 50-55 K, but in two experiments were 77 and 100 K with identical results. To obtain high cooling rates it is essential to have uninterrupted flow of the aerosol in the vacuum system from the orifice exit to the cryoplate (i.e., to use no radiation shield5)and low base pressure during deposition. The infrared cell was constructed from a DN-100-EO-K vacuum flange (Balzers) adapted with various ports. For the high-density aerosol, deposition time generally was between 6 and 10 s. This gave absorbances of the 0 - H stretching band principal maximum of 1. The temperature was measured on the lower end of the cryotip. The temperature of the KBr cryoplate was measured in addition with a calibrated Fe-CuNi thermocouple down to 77 K and was found to agree with the cryotip temperature to within f 2 K. To obtain sufficient heat transfer the KBr sample holder was constructed from OFHC copper, and the KBr window embedded in the holder with indium seals. To reduce spectral distortion by the solidified droplets the deposit was covered with Ar at 16 K in discrete pulses, following a procedure described by Perutz and T~rner.~ The spectra were recorded in absorbance on a Pye Unicam SP 3-300 instrument with S P 3-050 data processing system, and calibrated with CO, or polystyrene film. Resolution was 7 cm-'. The peak maximum of the decoupled 0-D stretching transition is believed to be accurate to f 2 cm-'. In addition to KBr a sapphire disk was used in several experiments as cryoplate. This gave identical spectra over the limited range accessible with sapphire windows. Therefore, it is probable that neither deposition of the aerosol nor distortion of the spectra is influenced by the KBr cryoplate used in most experiments.
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Results and Discussion In Figures 1 and 2 infrared spectra of vitrified aqueous aerosols are shown from 4OOO to 2000 cm-',the 0 - H and 0-D stretching band region, and compared with vapor deposited amorphous solid water. For most experiments a solution containing 11 mol % HOD in H,O was used. For this HOD concentration OD-OD inter(9) Perutz, R. N.; Turner, J. J. J. Chem. Soc., Faraday Trans. 2 1973.69, 452.
0022-3654/85/2089-3474$01.50/00 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 16, 1985 3475
Vitrified Liquid Water I
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3500
3000
2500
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Figure 1. Infrared spectra of the 0 - H and 0-D stretching transition region of (a) vitrified aqueous aerosol before (solid line) and after (broken line) coating with Ar, (b) vitrified aqueous aerosol, using a lower aerosol droplet density for preparation, and (c) vapor deposited dilute HDO in H20(as) (for all three experiments: starting solution 11 mol 5% HOD in H 2 0 , deposition temperature 50-55 K, recorded at 16 K).
I
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Figure 2. Infrared spectra of the decoupled 0 - D stretching transition: the 0-D stretching transition region in Figure 1 is fivefold expanded at higher gain, using the same numbering.
actions contribute somewhat to the bandwidth of the decoupled 0-D stretching transition.I0 Therefore, a solution with 6 mol % HOD in H20was used in a few experiments and found to give identical results. For identical aerosol density the infrared spectra were reproducible: for example, the peak maximum value of 241 1 cm-’ in Figure 2a (solid line) has beem observed in 10 experiments to within f l cm-l. The main problem in this investigation was the distortion of the infrared spectra of the vitrified aerosol droplets by the Christiansen effect. This type of distortion is observed only for absorption bands with rapidly changing refractive indices, and leads to band symmetry and even shift of band maxima.”-I5 In (10) Haas, C.; Hornig, D. F. J. Chem. Phys. 1960, 32, 1763. (1 1) Duyckaerts, G. Amulysr 1959, 84, 201. (12) Harvey, M. R.;Stewart, J. E.; Achammer, B. G . J . Res. Natl. Bur. Srand. 1956,56, 225.
Figure la the infrared spectrum of a vitrified aerosol sample from 11 mol % HOD in H20 is shown in absorbance which contains the features typical for the Christiansen effect (solid line): abnormally low absorbance [i-e., high transmittance) on the highfrequency side of the 0-H stretching transition region, and asymmetrical shape of the band. The 0-D stretching transition region from decoupled HOD is expanded at higher intensity in Figure 2a, again with abnormally low absorbance on the highfrequency side. Since the whole value of this work depends on whether it is possible to eliminate the spectral artifacts, this aspect has been investigated in great detail. Spectral distortions were observed only in samples with deposited aerosol droplets; they were never observed in amorphous deposits of water vapor, H,O(as), investigated for comparison. The aerosol experiences in the vacuum system supersonic flow conditions and the droplets are flattened considerably on the KBr disk before solidification simply by the mechanical impact,I6 comparable with the splat-cooling of liquid metals and alloys. This should give a deposit with curved surfaces The background loss of light from the vitrified aerosol deposits is almost independent of frequency over the whole range investigated except through an absorption band. The solidified droplets therefore seem to behave more as reflectors than as scatterers, and the amount of light reflected from the beam changes greatly only when the refractive index of the sample changes, as it does through an absorption band.15 The usual method to eliminate the Christiansen effect distortion is by embedding the absorbing medium in a solid with a similar index of refraction.I1-l3 The refractive index of liquid water as a function of frequency in the infrared region has been calculated by Ki~lovskii,~J~ and as expected it varies strongly in the vicinity of an absorption band, with values between 1.2 and 1.5 for the 0-H stretching region. The refractive index of the embedding material should therefore be within this range. To reduce or eliminate the spectral distortions, the following two methods have been used: (i) deposition of a layer of argon on the aerosol deposit (n(Ar) = 1.29 a t 20 KI8), and (ii) codeposition of increasing amounts of water vapor, giving H20(as) in addition to vitrified liquid water. Constant absorbance (Le., disappearance of the abnormally low absorbance) on the high-frequency side of the 0-H and 0-D stretching bands can be taken as indicator that spectral distortion has been eliminated. Figures l a and 2a demonstrate the effect of a layer of Ar on the infrared spectrum. Here the broken lines, Le., the spectrum after Ar deposition, should be compared with the solid lines, the spectrum of the same aerosol deposit before treatment with Ar. In Figure la the peak at 3115 cm-I is reduced in intensity to a shoulder, and the overall intensity below 3000 cm-I is decreased strongly. After Ar treatment the principal maximum is at 3250 cm-I. In both Figures l a and 2a the distortion by low absorbance on the high-frequency side of the bands is reduced or even eliminated. In Figure 2a the peak maximum is shifted from 241 1 to 2416 cm-’, and a weak shoulder at about 2440 cm-I becomes visible. In the second method for reducing the spectral distortion an aerosol with a lower droplet density had been used and therefore increasing amounts of H,O(as) had been formed simultaneously with the solidified droplets. The infrared spectrum of a loweraerosol-density deposit is shown in Figure lb, and the 0-D stretching transition region again expanded at higher intensity in Figure 2b. These spectra are very similar with the Ar-coated spectra (broken lines), showing in Figure l b again a shoulder at approximately 3 120 cm-I, and in Figure 2b the peak maximum (13) Price, W. C.; Tetlow, K. S. J. Chem. Phys. 1948, 16, 1157.
(14) Laufer, G.; Huneke, J. T.; Royce, B. S. H.; Teng, Y. C. Appl. Phys. Lett. 1980, 37, 517. (15) Bertie, J. E.; Whalley, E. J. Chem. Phys. 1964, 40, 1637. (16) Flattening of the droplets before solidification follows from own unpublished density measurements. If the droplet shape would have been preserved during solidification,a much lower density would have been obtained. (17) Kislovskii, L. D. Opt. Spectrosc. 1959, 7, 201. (18) Hallam, H. E.; Scrimshaw, G. F. “Vibrational Spectroscopy of Trapped Species”; Hallam, H. E., Ed.; Wiley: New York, 1973; Chapter 2.
3476 The Journal of Physical Chemistry, Vol. 89, No. 16, 1985
Mayer
at 2416 cm-'. The shoulder a t 2440 cm-l is assigned to the decoupled 0-D stretching transition of HDO in H20(as), the intensity being obviously higher in part b of Figure 2 than in part a. Further lowering of the droplet density in the aerosol gives deposits whose spectra resemble more and more the spectrum of HDO in H20(as). For example, the peak maximum of the decoupled 0-D stretching transition, observed at 2416 cm-'in Figure 2b, shifts with decreasing droplet densities in the aerosol to higher wavenumbers, with 2440 cm-', the literature value for HOD in H20(as), as limiting value' (marked by the vertical line in Figure
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2).
Since the aim of this investigation is a comparison with H20(as), the infrared spectrum of the vapor deposited amorphous form, again prepared from 11 mol % HOD in HzO, is shown in Figure IC, and the 0-D stretching region expanded at higher gain in Figure 2c. Recently it-has been reported that the infrared spectrum of H2O(as)Ig and of the decoupled 0-D stretching transition of HOD in H20(as)" depend on temperature and rate of deposition. Therefore, the spectra are from samples deposited at the same temperature as the aerosols. Spectral distortion allows to differentiate even in devitrified samples between aerosol and vapor deposits. The infrared spectrum of Figure 2a after devitrification is shown as Figure 3a, and the spectrum of Figure 2c as 3b. This comparison shows that the sample used for Figure 2a contained very little crystalline material. The spectrum of the devitrified droplet deposit (Figure 3a) shows again the features typical for the Christiansen effect, whereas the vapor deposited material (Figure 3b) produces a spectrum without distortion. The peak maximum in Figure 3a is lowered to 2409 cm-I, whereas the maximum in Figure 3b is identical with the literature value for 20 K.' The full width at half maximum (fwhm) of the band in Figure 3b is 27 cm-I, in Figure 3a 36 cm-' due to the band asymmetry. A fwhm of 25 cm-I results for the band in Figure 3a if only the left side of the band is taken for evaluation. Coating the crystalline aerosol deposit with Ar again reduces the spectral distortion, and most significantly, shifts the peak maximum from 2409 to 2413 or 2414 cm-I. This might be important, because the same shift to higher wavenumbers was observed for vitrified droplets coated with Ar (Figure 2a, solid and broken lines). And it gives greater confidence that the peak maximum value of 2416 cm-' in Figure 2a,b is really the value of the decoupled 0-D stretching transition, corrected for spectral distortion, because the spectrum of the devitrified sample can be regarded as internal standard. Elimination of spectral distortion by Ar coating has its drawbacks because the layer of solid Ar causes frequency-dependent scattering. As alternatives Xe and N2have been tested. However, these offer no advantages, because their effect is comparable to Ar, only accompanied by even more scattering. If it is accepted that the spectral distortions in the infrared spectra of the vitrified aerosols have been reduced or eliminated sufficiently, the following spectral differences emerge from a comparison with HOD/H,O(as) at 16 K: (i) a shift of the peak maximum of the decoupled 0-D stretching transition from 2440 to 2416 cm-' (Figure 2), and (ii) an additional shoulder at 3 1 15-3 120 cm-' in the spectrum of the coupled 0-H stretching transition region of the vitrified aerosols. A comparison of intensities at 31 15 cm-', marked in Figure 1 by a vertical line, shows that the absorbance in part c of Figure 1 is much lower than in part a or b. The bandwidth of the 0-H stretching band in Figure la,b is clearly larger than the width of the band in IC. For Figure I C a fwhm of 305 cm-I is obtained, somewhat larger than the value reported by Hagen et al. for 50 K deposition t e m p e r a t ~ r e . 'An ~ exact evaluation of fwhm for Figure 1, part a (broken line) or b, is not possible because the tailing a t low frequencies gives problems what base line should be used, and only an approximate fwhm value of 400 cm-' is given. The fwhm of the decoupled 0-D (19) Hagen, W.; Tielens, A. G. G.; Greenberg, J.
M. Chem. Phys. 1981,
56, 361. (20) Mayer, E.: Pletzer, R. J . Chem. Phys., submitted for
publication.
I \
0
c -
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n L 0 VI
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Figure 3. Infrared spectra of the decoupled 0-D stretching transition after devitrification, (a) using the sample shown in Figures la and 2a, vitrified aerosol, and (b) corresponding to Figures IC and 2c, HDO in H,O(as) (devitrified at 160 K, recorded at 16 K).
stretching band in vitrified aerosols varied between 70 and 80 cm-I, probably depending on the amount of codeposited vapor. The fwhm values for part a of Figure 2 are 70 and 73 cm-' (broken line), and for part b 80 cm-I. The fwhm of the band in Figure 2c was 93 cm-', identical with the fwhm reported recently for vapor deposition at 50 K.20 The librational region of the infrared spectra of vitrified aerosol and H,O(as) (not shown in the figures) is very similar. There is a small shift of the peak maximum of the broad band from 780 cm-I in H20(as) to 800 cm-' in vitrified aerosol. The value of 780 cm-' for H20(as) is for a sample deposited at 50 K and cooled subsequently to 16 K, and is identical with the literature value for these conditions.2' Since the principal maximum of H20(as) is strongly dependent on temperature and thermal history, varying between 760 and 810 cm-' (ref 21; Figure 3), it is not meaningful to overinterpret the difference between vitrified aerosol and H20(as). However, the principal maximum for the vitrified aerosol is clearly at lower wavenumbers than the maximum for crystalline ice, observed after devitrification at 835 cm-' (16 K).2' The spectral differences between HzO(as) and vitrified liquid water seem to be related to differences in the intrinsic properties of the two amorphous materials. However, as pointed out by one of the reviewers, it is at least possible that the cooling of the aerosol droplets leads to strains which broaden the infrared spectrum so that while it is plausible that the two forms of amorphous solid water obtained from the vapor and the liquid are different, it is still possible that, in fact, they are the same. The widths of the decoupled 0-H (0-D) stretching transitions in both ice II5 and H,0(as)',22 have been interpreted by a static structural property, a distribution of nearest-neighbor oxygenoxygen distances due to proton disorder, this resulting in a 0-H (0-D)stretching force constant distribution. In keeping with this, the shift of the peak maximum of the decoupled 0-D stretching transition from 2440 in HOD/H,O(as)' to 2416 cm-l in vitrified liquid water indicates a decrease of the average 0-0separation. This corresponds to about 0.02 A, applying the known correlation for 0-D stretching frequency and average 0-0separation, which is dloo/dRoo = 1362 cm-' A-' for Roo = 2.76 A.'323
(21) Hagen, W.; Tielens, A. G. G. Spectrochim. Acra, Part A 1982,38A,
1089.
(22) Madden, W. G.; Bergren, M. S.; McGraw, R.: Rice, S. A. J . Chem. Phys. 1978, 69, 3491.
J . Phys. Chem. 1985,89, 3411-3482 No attempt is made to correlate the infrared spectrum of vitrified liquid water with the spectrum of water a t room temperature and to determine and interpret temperature dependencies of the band frequencies and widths. A comparison seems to be problematic because the bandwidths of the vitrified aerosol spectra are unreliable due to spectral distortion, and the infrared spectrum of supercooled water, necessary for a detailed discussion, has to my knowledge not yet been investigated in the 0-H and 0-D stretching transition region. Raman spectra should be free of the type of spectral distortion observed in this work.14 Therefore, it (23) Falk, M. “Proceedings of the Electrochemical Society, Symposium on Chemistry and Physics of Aqueous Solutions, Toronto, 1975”; Electrochemical Society; New York, 1975.
3477
is better to investigate first the Raman spectrum of vitrified aqueous aerosol, and only then make a comparison with the careful Raman spectroscopic investigations of supercooled ~ a t e r . ~ ~ - * ~
Acknowledgment. This work was supported by the “Fonds zur Forderung der wissenschaftlichen Forschung” of Austria. I am very grateful to Dr. R. Abermann for his help in constructing the low-temperature infrared cell. Registry No. H 2 0 , 7732-18-5; HOD, 14940-63-7. (24) Krishnamurthy, S.;Bansil, R.; Wiafe-Akenten, J. J . Chem. Phys. 1983, 79, 5863. (25) D’Amgo, G.; Maisano, G.; Mallamace, F.; Migliardo, P.;Wanderlingh, F. J . Chem. Phys. 1981, 75, 4264.
Interfacial Reaction Dynamicst R. D. Astumian* and P. B. Chock Laboratory of Biochemistry, National Heart, Lung, and Blood Institute, National Institutes of Health, Bethesda, Maryland 20205 (Received: April 10, 1985)
Bimolecular reactions in which one of the reactants is localized at an interface while the other reactant is initially molecularly dispersed in the homogeneous phase can occur by two paths. One involves the direct interaction of a homogeneous reactant with its interfacially localized reaction partner, and the other proceeds by initial adsorption of the homogeneous reactant and subsequent surface diffusion to reaction. A branching method for the surface dynamics is developed, and allows for the calculation of the specific rate of reaction, and the relative contributions of the two paths. It is found that although rate enhancement over the analogous homogeneous reaction can be expected only for very low interfacial concentrations of the localized reactant, the two-dimensional surface diffusion mechanism accounts for an appreciable portion of the total reactivity within a wide range of circumstances.
Introduction Reactions at interfaces often take place at significantly different rates than analogous reactions occurring in a homogeneous phase. There can be many reasons for the variance in the specific rates, R* and R , observed in heterogeneous and homogeneous systems, respectively, which may be separated into energetic and “othern reasons. The first group involves changes in the potential energy surface of the reaction, and are reflected in the activation energy, E,. The other group encompasses changes in essentially geometric factors governing the reaction, such as the relative arrangement, accessibility, and orientation of reactants and the dimensionality of the space in which they move. These are expressed through the frequency factor in the Arrhenius equation. While the activation energy can be theoretically calculated only for a few, simple homogeneous gas-phase reactions, geometric factors influencing the dynamics of reaction can be analyzed by using elementary theories. In a previous paper’ it was shown that a primary geometric effect of loading one of the reactants of a bimolecular reaction onto a neutral surface is a decrease in the rate. This effect is particularly pronounced when the reaction in question is diffusion controlled. On the other hand, Adam and Delbriick have shown that if the surface is very sparsely covered by the localized reactant, thus allowing for nonspecific binding and subsequent surface diffusion of the incoming ligand, a dramatic rate enhancement is possible. They postulated that this effect might contribute to the efficiency of some interfacial biochemical reactions a t low concentrations, effectively overcoming the barrier of diffusion control. Also, Richter3 and Eigen4 investigated diffusion control in nonspherical geometries, and concluded that if surface diffusion is sufficiently rapid, the longest ‘This paper is dedicated to Professor Tatsuya Yasunaga on the occasion of his retirement from Hiroshima University.
This article not subject to US.Copyright.
linear dimension of an ellipsoidal molecule can become decisive for the overall diffusion-controlled rate. This sort of mechanism has become quite well-known in studies on the tremendous efficiencies found for binding between regulatory proteins and specific control sites on polynucleotide^.^-^ In order to analyze the contribution of surface diffusive mechanisms to the overall interfacial rate, it is necessary to separate terms arising out of direct interaction of bulk ligand with surface reactant from those due to initial nonspecific binding of ligand to the surface followed by two-dimensional diffusion leading to reaction. Thus, in the present work we utilize a branching method*-12 for the calculation of diffusion-controlled interfacial reaction rates. From this point of view, the total reaction velocity is given by the rate at which reactants enter into a given, arbitrary configuration, multiplied by the product of the probabilities of the subsequent (1) Astumian, R. D.; Schelly, Z,.A. J . Am. Chem. SOC. 1984, 106, 304-308.
(2) Adam, G.; Delbrlick, M. In “Structural Chemistry and Molecular Biology”; Rick, A., Dauidson, N., Eds.;Freeman and Co.: San Francisco, CA, 1968.
(3) R.ichter, P. H.; Eigen, M.Biophys. Chem. 1974, 2, 255-263. (4) E;gen, M. In “Quantum Statistical Mechanics in the Natural Sciences ;Kursunoglu, B., Minte, S. L., Widmayer, S.M., Eds.; Plenum Pres: New York, 1974. ( 5 ) Riggs, A. D.; Bourgeois, S.; Cohn, M. J . Mol. Biol. 1970, 53, 401. (6) Berg, 0. G.; Winter, R. B.; von Hippel, P.H. Biochemistry 1981.20, 6929-6948. (7) Jack, W. E.;Terry, B. J.; Modrich, 1982, 79,4010-4014.
P.Proc. Natl. Acad. Sci. W.S.A.
( 8 ) Northrup, S. H.; Hynes, J. T. J . Chem. Phys. 1979, 71, 871-883. (9) Berg, H. C.; Purcell, E. M. Biophys. J. 1977, 20, 193-219. (10) Northrup, S.H.; Allison, S.A.; McCammon, J. A. J. Chem. Phys. 1984,80, 1517-1524. (11) Noyes, R. M. J . Chem. EHys. 1953, 22, 1349-1359. (12) Noyes, R. M. J . Am. Chem. SOC.1956, 78, 5486-5490.
Published 1985 by the American Chemical Society
