J. Chem. SOC., Faraday Trans, 1, 1985, 81, 2067-2082 Molecular Diffusion in Monolayer Films of Water Adsorbed on a Silica Surface BY JONATHAN W. CLARK AND PETER G. HALL Department of Physical Chemistry, University of Exeter, Exeter EX4 4QD AND ALAN J. RDDUCK Materials Centre, Director of Quality Assurance, Royal Arsenal East, London SE18 6TD AND CHRISTOPHER J. WRIGHT*? Materials Physics & Metallurgy Division, AERE Harwell, Didcot, Oxfordshire OX1 1 ORA Received 1 1 th October, 1984 The dynamics of water adsorbed in near-monolayer films at the surface of silica has been measured and characterized by quasielastic neutron scattering. The analysis shows the coexistence of two phases of sorbed water molecules with different dynamics. One component is immobile on the experimental timescale, while the other has a two-dimensional diffusion coefficient of ca.6 x 10-lo m2 s-l and a residence time during reorientation of 4 x s. The precise details of the reorientation process in the mobile phase are still unclear. Modelling the surface structure of microcrystalline or amorphous solids and the structure and dynamics of their adsorbates is an important goal in surface science. One such material is amorphous silica, and the interactions of its surface with water have been investigated by numerous workers because of the widespread use of this material as a drying agent, catalyst support and filler. Much of the reported experimental evidence concerning the adsorption of water is consistent with the concept that at low partial pressures individual water molecules adsorb at pairs of surface hydroxyl groups.Differential heats of adsorption of water on various silicas are ca. 50 kJ mol-1 at ON* = 0.5, which can be shown to be generally consistent with the double hydrogen bonding of sorbed water molecules to pairs of surface OH groups of favourable geometry. Inelastic-neutron-scattering experiments2 have shown that the librations of adsorbed water can also be explained on this basis. The near-infrared spectra of water adsorbed on silicas at low coverages show only small departures from the spectrum of critical water at 5300 cm-l, the position of the unperturbed v,+ v2 combination band.3q4 As a consequence, therefore, water must adsorb so that its oxygen atoms form new hydrogen bonds with the surface.At higher coverages, but still sub-monolayer, the same evidence suggests that some water molecules bond through their hydrogen atoms as water clusters begin to form. Direct measurements of the dynamical behaviour of adsorbed water at a silica surface are, however, incomplete. Translational diffusion coefficients, DT, of adsorbed water at room temperature have been measured by spin-echo magnetic resonance techniques and at low coverages, 1.0 > ON* > 0.5, were found5 to be Wycombe, Buckinghamshire HP12 3QR. -f Present address: RHM Research Ltd, The Lord Rank Research Centre, Lincoln Road, High 2067 68-22068 DIFFUSION OF WATER ON SILICA (3-5) x loplo m2 s-l. In contrast unequivocal and quantitative information about the reorientation of such molecules has not been obtained although an attempt has been made to obtain their dipole correlation function by analysing the width of the v, + v2 band in the infrared spectrum of adsorbed water.In this paper we report results obtained with quasielastic-neutron-scattering techniques, since they provide insight into the translational and rotational diffusion behaviour of adsorbed molecules leading to important structural information relevant to the surface-bonding models discussed earlier. The experiments have been performed on a well characterized, high-surface-area silica called Spherisorb. This material and its interactions with water have been investigated by inelastic2 and quasielastic6 neutron scattering and by diffra~tion,~ in addition to measurements of its pore and surface properties.* It has a very narrow pore-size distribution created by the close packing of spherical particles.As a consequence, therefore, the micropore volume fraction is very small and all the water molecules in the first monolayer can be assumed to be adsorbed at a single surface where the adsorption potential is unperturbed by that of any neighbouring surface. EXPERIMENTAL MATERIALS The silica samples were of the Spherisorb type and their physical properties have been documented elsewhere.27 The preparations of the degassed samples and those equilibrated with water were also identical to those which have been described before., Details specific to the samples examined in this work are collated in table 1. The coverage, 8oH, is the ratio of adsorbed water molecules to surface OH groups given a surface OH group concentrations of 4.6 nm-2 and an N, B.E.T.surface area of 232 m2 g-1.2 8Hz0 is the coverage related to the water B.E.T. monolayer concentration of 0.021 & 0.001 g g-l obtained from water isotherm measurement at 302 Ks This B.E.T. plot was linear over the range of PIP, = 0.05 + 0.4 with a c constant of 12&4. ON* is the proportion of the N, B.E.T. surface area covered by water assuming each water molecule occupies 0.108 nm2. The adsorption experiments could not be performed in situ because of the time (ca. 24 h) required for equilibration to take place. Consequently the sample cans were selected to be sufficiently well matched that the sample volumes were identical to within 3%. In all cases the time between sample preparation and irradiation was sufficient for equilibrium to be attained.NEUTRON DIFFRACTION Diffraction experiments were performed on silica samples encapsulated in 12 mm diameter vanadium cans on the 'guide-tube' diffractometer at AERE Harwell using an incident wavelength of 4.7 A. QUASIELASTIC NEUTRON SCATTERING The quasielastic-scattering data were obtained at the Institut Laue-Langevin, Grenoble using the time-of-flight spectrometer INS. The instrument was used with an incident wavelength of 6.1 A and an elastically scattered energy resolution with f.w.h.m. of 80 peV. Each sample was positioned in transmission geometry at an angle of 35 f 1" to the incident beam in a helium-filled container at room temperature. The detectors were grouped in sixes such that the angular range covered by each group was 4" in 28.The mean 28 values of each group were 30, 44.3, 63.5, 83.5, 104 and 128". The raw IN5 data were initially treated by subtracting the container and degassed silica scattering (Is+c) from that of the sorbed water samples (Is+w+c) according to Iw = Is+,+, - 4 + c T,+w+,/T,+cTable 1. Details of the preparation of Spherisorb samples quantity outgassing of water coverage conditions scattering (%) adsorbed mean thickness equilibrium, sample /g g-l OoH &IzO ONz T/K t/h of sample/mm total fromH,O P/POC - 1 0 0 0 0 3 53 20 2.97 8.5" - - 2 0.020 0.6 1.0 0.3 353 20 2.96 12.0" 3.5" 5.2b 0.20 3 0.029 0.9 1.4 0.45 353 20 3.04 13.4" 5.1" 6.6b 0.38 4 0.040 1.3 1.9 0.6 353 20 3.00 15.2" 6.7" 8.0b ca.0.55 "Calculated ; bobserved ; ccalculated from isotherm. X r F s ?2070 DIFFUSION OF WATER ON SILICA where the variables T,,,,, and T,,, were the theoretical transmissions of the samples with and without sorbed water, respectively. This was performed on data which had been normalised to incident neutron flux and corrected for counter efficiency, using the ILL program CROSSX. A 2 mm thick slab of vanadium was used to record the experimental resolution function. THEORY For water, the incoherent scattering from lH so dominates the cross-section that only the incoherent contribution to the scattering intensity needs to be considered. Therefore we have where fiQ is the neutron momentum transfer, Q is defined as k,-k, k, and k are the incident- and scattered-neutron wavevectors, fiw is the neutron energy transfer, agC is the incoherent cross-section of the protons, S(Q, cu) is the symmetrised scattering law and is an experiment-independent quantity related to the self-correlation function for diffusive motions, GJr, t), by Fourier transformation in space and time:9 Sinc(Q, 0) = - JI G,(r, t) exp [i(Qr - cut)] dr dt.(3) 2z As is usual in quasielastic neutron scattering, the Q range and quality of data is insufficient to allow calculation of Gs(r, t) by numerical transformation. Hence quasielastic data are interpreted by comparing Sinc(Q, LO) derived from experimental data from isotropic powder systems using eqn (2) with theoretical forms derived from models for G,(r, t). Rotational and translational diffusive motions of lH nuclei will both contribute to the quasielastic broadening. For analysis to be tractable, it is assumed that both these two motions are uncorrelated. GJr, t ) is then factorisable, and the total quasielastic scattering law can be expressed as a convolution product: (4) where exp ( - Q2( u2)), the Debye-Waller factor, and B, the level of cu-independent background, represent the contribution from vibrational motions, and 3( u2) is the total mean-square vibrational amplitude of lH nuclei if the vibrations are isotropic.Sinc(Q7 0 ) = ~ X P (- Q2(u2>) [Xk?YQ7 0) x sfnotC(Q, W ) + BQ21 Models for Strans(Q, cu) and Fot(Q, cu) can now be considered. TRANSLATIONAL DIFFUSION The simplest model for translational diffusion is that of continuous isotropic diffusion which leads to a Lorentzian-shaped scattering law: 1 - Y(DQ2, 0).1 DQ2 ~ c u ~ + ( D e ~ ) ~ z qE:ns(Q, W) = - The translational diffusion coefficient D can be obtained by plotting the width of the quasielastic broadening against Q2 at small Q values (i.e. for QL < 1, where L is the diffusion step length). Another model which leads essentially to the same result for small Q is the isotropic jump-diffusion model,lo where the diffusion coefficient is related to the jump distance L by D = L2/6ti, where ti is the mean time between jumps. At higher Q (such that QL 2 1) the Lorentzian broadening reaches a limit at a f.w.h.m. of 2fi/t;. For two-dimensional jump diffusion the form of the scattering law will alsoJ. W. CLARK, P. G. HALL, A. J. PIDDUCK AND C. J.WRIGHT 207 1 approximate to a Lorentzian if its f.w.h.m. is small compared with the resolution width. In this case the f.w.h.m. is 2?i/ti [ l -Jo(QL)],ll where Jo is the cylindrical Bessel function of zeroth order. The true form of the scattering law for powder averaged continuous diffusion in two dimensions is $i:ns(Q, LO) = - i,” LY(D2D sin2 0, LO) sin 0 d 0 (6) 27r which is more peaked around LO = 0 than a Lorentzian. LOCALIZED ROTATIONAL MOTION The scattering law for a localized rotational process can be written generally as12 1 N-1 qgk(Q, 0) = Ao(Qa) &LO) + C An(Qa) (G1, 0) (7) n-1 where, for powder-averaged, uniaxial, jump rotation I N A,(@) = _f_ C jo[2Qa sin (xP/N)] cos (2nnP/N) N P-1 t0 sin2(7r/N) t , = 1 - [cos (27r/N)] sin2(nn/N) ‘ N is the number of residence sites of the proton on the circle of radius a describing the motion and to is the proton residence time.The delta function in eqn (7) arises from the bound nature of the motion. Its relative intensity, A,(Qa), the elastic incoherent structure factor (e.i.s.f.), has a form which depends on the geometry of the rotation. The series of Lorentzians in eqn (7) have widths which vary with n, reflecting the higher-order harmonics of the motion which have correlation times t,. A model of powder-averaged, uniaxial, jump rotation is a plausible model of the uncorrelated rotation of water protons about the axis of a fixed hydrogen bond, such as those in solid or sorbed water. N can lie between 2 (two-fold uniaxial rotation) and (30 (uniaxial rotational diffusion), but it has been pointed out previously12 that the scattering laws calculated from eqn (7)-(9) for N < 6 and at Qa < 3 are virtually indistinguishable.The computation with N = 6 thus gives a satisfactory model scattering law for uniaxial rotational diffusion with D,, the rotational diffusion coefficient, given by l/tl. RESULTS COHERENT SCATTERING Fig. 1 (A) shows the diffraction data recorded on the Harwell diffractometer for a degassed silica sample and two further samples containing 0.003 and 0.015 g g-l water. The instrumental background is also shown. Subtractions showing the contribution to the scattering from the adsorbed water are shown in fig. 1 (B). The principal features of the dry silica diffraction pattern are the intense small-angle scattering at Q 5 0.3 A-1 and the broad Bragg maximum near Q = 1.55 A-l, neither of which is affected by the presence of sorbed water.The scattering enhancement from the water is virtually isotropic for 1.7 > Q/A-l > 0.4, and as a consequence it was concluded that in this Q region elastic and quasielastic scattering from the water could be assumed to be completely incoherent for the purpose of subsequent data analysis.2072 8000 h m U c 8 2 f 2 '? c) .- e N & a m 4000 c 3 * 8 v A r: E Y m .- * .3 0 DIFFUSION OF WATER ON SILICA . . . . It.:. . . . . ' . .'. ,.. 0 . .. e m p t y can .-. bac k g r o u n d -.._ ., . . .".*a. ..-%--.a. " . - -~.~.-.~-.~..'...-,...~,.,~~ I I I I I I I I 1 L ) I 0 3 0 5 0 . 7 5 1 0 1 ' 5 1 7 Q1A-l I , I B I I I 0 . 5 1 . o 1 .5 Q1A-l Fig. 1. (A) Diffraction traces recorded for silica samples containing (a) 0.0 15 and (b) 0.003 g g-l of water and (c) degassed silica. (B) Subtracted spectra showing the contribution to the scattering from the adsorbed water. MOLECULAR VIBRATIONS Vibrational spectra, in the form of an amplitude-weighted density-of-states function G( Q, m), have been reported previouly.2 The vibrational frequencies were interpreted in terms of the development of a stable network of doubly hydrogen-bonded water molecules attached to surface hydroxyl groups. Eqn (4) indicates that mean vibrational amplitudes can be calculated from the attenuation of quasielastic peak area (q.e.p. area) with increasing Q. Plots of In (q.e.p. area) against Q2 are shown in fig. 2. The lowest Q point appears to be systematically 7 4 % higher than expected, and so this point was omitted from the least-squares measurement of the slopes.The slopes are, at x = 0.020 g g-l, -0.07(0) k2; at x = 0.029 g g-l, -0.06(1) k2; and at x = 0.040 g g-l, -0.09(4) k2. They corre- spond to r.m.s. thermal cloud radii, (3(u2)):, of 0.46, 0.43 and 0.53 A, respectively, which are comparable with previously reported values for ice.13 QUASIELASTIC SCATTERING Typical examples of quasielastic-scattering laws obtained at three different scattering angles are shown in fig. 3. The narrow component in each spectrum has a width equal to that of the resolution function. They correspond to the sample with a water coverage of 0.29 g g-l. In view of the fact that the surface hydroxyl groups cover < 50% of the available surface area of the silica it could have been expected that these groups might be undergoing rotational diffusion, which would cause quasielasticJ.W. CLARK, P. G. HALL, A. J. PIDDUCK AND C. J. WRIGHT 2073 - 0 . 7 I I I 0 -0.9 h $ - 1 ' 1 iJ + p! 3 - 1 . 3 v c I - l o I i j 3 . 0 L . 0 Q2/A-' Fig. 2. Plots of the area of the quasielastic peaks against momentum transfer squared, Q2: 0, silica plus 0.02 g g-' H,O; +, silica plus 0.03 g g-l H,O; 0, silica plus 0.04 g g-' H,O. Fig. 3. Typical examples of experimental scattering laws at a coverage of 0.29 g g-l. The continuous curves are the two components which result from fitting the data with eqn (1 1). The difference function is also shown on a x 10 scale. Values of S/" and Q/w-l: (a) 128.0,1.858; (b) 103.9, 1.628; (c) 83.5, 1.376; ( d ) 63.5, 1.008; (e) 44.3, 0.779; (f) 30, 0.535.scattering. Analysis of the scattering from the degassed silica, however, showed no detectable broadening, which implies that no further account need be taken of the hydroxyl group rotation in the subsequent analysis. ANALYSIS OF QUASIELASTIC SCATTERING BY CONVOLUTION TECHNIQUES Model scattering laws were convoluted with the instrumental resolution function and tested against the experimental data using an iterative procedure. To imitate the2074 DIFFUSION OF WATER ON SILICA experimental process, the convolution was performed on scattering laws expressed in time-of-flight using the relationship where zo and zi are the incident time-of-flight and the time-of-flight in channel i, respectively, Smodel(Q, z) is the model scattering law converted to time-of-flight units and Rj(r) is the experimental resolution in time-of-flight units.Rj(z) is defined over a range of 2K channels, which are sufficient to include its complete width and where channel K contains its centre of gravity. 24 channels were used, corresponding to 3 f.w.h.m. The least-squares minimisation program V A ~ S A D ~ ~ was used to obtain the values of variables which lead to the best agreement between a model and the experimental data. In addition to the variables required to calculate the model scattering law, a further two parameters, Am and B, were needed. Am was the fractional difference between the channels containing the centres of gravity of the model and experimental scattering laws, and BQ2 was the level of the one-phonon inelastic scattering which was added to the convoluted, model scattering law.Data were fitted over a range of energy transfer, from -0.75 to 1.0 meV, which was more than sufficient to include the observed broadenings. Account was taken of the small variation of Q in the energy transfer across the quasielastic peak. The delta function in the model was represented by allocating the intensity to the two channels closest to zero energy transfer, in the proportions of their mean energy transfers. This results in a negligible perturbation of the resolution function after c o n v ~ l u t i o n . ~ ~ A Simpson’s rule integration was used to calculate the Lorentzian broadenings when they approached the magnitude of a channel width.16 Finally, note that the neutron-counting statistics do not vary symmetrically across the quasielastic S(Q, m) peaks.Graphical outputs have been used to supplement values of 02 (averaged over the data points of the area-normalised scattering laws) for assessing the quality of a fit. QUASIELASTIC DATA ANALYSIS MODEL-INDEPENDENT ANALYSIS A useful model scattering law, for a preliminary analysis of data where the broadenings are small, is that composed of a Lorentzian of variable f.w.h.m. AE, and a delta function of variable relative intensity A :I7 Physically, the delta function represents the e.i.s.f. in a rotational model, but it also takes account of any molecules which are immobile on the timescale dictated by the resolution width.In the limit of poor resolution, the Lorentzian is appropriate for diffusion in 2 or 3 dimensions and for a rotation of undefined periodicity.l* Using this model four parameters ( A , A E , B and Am) were refined whilst the exponent (- Qz (1.2)) was obtained from the slopes of fig. 2. The values of A and AE obtained from the refinements are plotted in fig. 4 against momentum transfer and momentum transfer squared, respectively. The complete set of fits obtained at the coverage of 0.029 g g-l is shown in fig. 3. Since the quality of the fits is good, the model and the data points being nearly coincident, the model scattering laws are shown withJ. W. CLARK, P. G. HALL, A. J. PIDDUCK AND C. J. WRIGHT 2075 \ ( a ) \ 1.0 - 0.21 + + a" o v - i 0 1 Q21A-2 Fig.4. Values of (a), A , the elastic fraction, and (6) AE, obtained from fitting eqn (1 1) to the experimental data. The straight line in (b) is the relationship between AE and Q2 for pure water at 25 "C. 0, silica+0.02 g g-l H,O; A, silica+0.029 g g-l H,O; 0, silica+0.04 g g-l H,O; +, values of A , from eqn (12); (----) predicted e.i.s.f. for a = 0.90 A. the difference function (data -model) multiplied by a factor of 10. Typical values of 02 are given in table 2. Inspection reveals a systematic tendency for the model to overestimate the intensity in the wings of the quasielastic peaks between 0.1 and 0.2 meV. Differences in the elastic region are cu. 1 % of peak height, reflecting the counting statistics. The values of A in fig. 4 show a tendency to decrease with both increasing Q and coverage.The e.i.s.f. for uniaxial rotation with a radius of 0.90 A, corresponding to the radius of motion of one hydrogen atom in a water molecule rotating around a fixed hydrogen bond containing the other hydrogen atom of the same molecule, is shown as a dotted line. Whilst the Q dependence is indicative of rotation, the quantitative agreement with the e.i.s.f. is poor. The Lorentzian broadening increase2076 DIFFUSION OF WATER ON SILICA Table 2. Values of fit factor, 02, for best fits to data at an adsorbed water coverage of 0.03 g g-l model 0.53 0.148 0.146 0.136 0.066 0.068 0.78 0.179 0.172 0.155 0.088 0.086 1.09 0.105 0.092 0.079 0.042 0.035 1.38 0.178 0.156 0.142 0.088 0.087 1.63 0.132 0.1 13 0.105 0.082 0.078 1.86 0.179 0.148 0.159 0.150 0.132 a Eqn (1 1) (fits shown in fig.3). Eqn (1 1) with Lorentzian replaced by summation in eqn Eqn (1 1) with Lorentzian replaced by integral Eqn (1 3) (7) with N = 6 (uniaxial rotational diffusion). in eqn (6) (two-dimensional continuous diffusion). with an additional delta function to represent an immobile concentration of 0.01 g g-l. Eqn (1 3) (fits shown in fig. 5). with increasing Q and are virtually independent of coverage. They do not however extrapolate to AE = 0 at zero Q, as would be expected for translation. The relation- ship between AE and Q2 for bulk water is shown as a solid line. The results are therefore inconsistent with either simple translation or uniaxial rotation about a single axis. This conclusion was confirmed by further fits in which the Lorentzian was replaced both by the correct scattering law for continuous two-dimensional diffusion [eqn (6)] and by that for uniaxial rotational diffusion [eqn ( 7 ) with N = 61.In the first case the new refined values of A were lower at all angles and coverages by between 0.03 and 0.05 and the values of 2fiDZDQ2 were larger than AE at all angles and coverages by a factor of between 1.3 and 1.5. In the second case virtually identical values of A were obtained and the refined values of 2?i/t, were lower than AE in fig. 4 at the highest Q points by a maximum of 15 % . The modified scattering laws resulted, in most cases (see table 2) with a slight improvement in the quality of the fit, most notably at the two highest angles. ANALYSIS ASSUMING A BOUND LAYER The main differences between the refined parameters obtained so far for the different data sets is that the elastic fraction decreases with increasing coverage.Conversely, the values of AE and the form of the plots of A against Q are similar, suggesting that the magnitude and the mechanism of diffusion is independent of coverage. The increase in A with decreasing coverage can be accounted for if a constant number of immobile molecules are present, such as those which might be held at particularly active sites filled at the lower vapour pressures. The word immobile is used with respect to the timescale dictated by the resolution function [lOfi/AR(w) = s]. The variation of A with coverage is given by A , = A M ( l - x , / x ) + x , / x (12) where x, is the surface concentration of immobile molecules and AM is the elastic fraction due to the motion (i.e.the e.i.s.f.). A plot of A , against l / x will be of slopeJ. W. CLARK, P. G. HALL, A. J. PIDDUCK AND C. J. WRIGHT 2077 Fig. 5. Fits of eqn (13) to the experimental scattering laws. Details as for fig. 3. xB( 1 -A,) and intercept A,. Linear regression analysis of these plots at each Q gave values for xB which lay in the narrow range from 0.008 to 0.01 1 g g-l, although the individual correlation coefficients at each individual scattering angle were poor. The values of A , are shown in fig. 4. A , appears to be almost independent of Q, with the exception of the lowest Q point. ANALYSIS ASSUMING UNIAXIAL ROTATION A previous study of the libration frequencies of adsorbed water on Spherisorb silica concluded that the adsorbate is largely composed of doubly hydrogen-bonded molecules at low coverages.2 Basing further analysis of the present data on this model it is possible to imagine that diffusion of these molecules occurs as a result of the consecutive rupture of the two bonds.Rupture of one bond leaves the molecule free to rotate, with an unspecified periodicity, around the axis of the remaining bond, which may or may not be fixed in space. If sufficient thermal energy is then available to break the second bond, a translational jump can occur. A suitable model scattering law for such a process, assuming that the motions are uncorrelated and that uniaxial rotation around a hydrogen bond is the major rotational motion, is then given by where AER 9 AET.This model, which involves one additional variable over those specified above, was fitted to the data to give the results shown in fig. 5 and 6. A significant improvement in the quality of fit was obtained at all values of Q, as evidenced both by the plots shown in fig. 5 and the values of 02 given in table 2. Comparing fig. 5 with fig. 3 shows that much of the systematic deviation in the wings of the elastic peak has now disappeared. Just as significant are the virtually constant refined values of AE, (all values lay in the range 0.285k0.5 meV) and the more pronounced decays of A with increasing Q.2078 4 Q 0 . 1 1 , I1 DIFFUSION OF WATER ON SILICA \ 9‘ + A I I I ; .o 1 .5 2.0 0.21 0.5 Q1A-l $ 0.012 --- h € 4 0.008 3 i v t $ 0.004 .The values of A again show a smooth tendency to decrease with increasing coverage, and applying eqn (12) to these values shows that xB lies in the range 0.009f0.001 g g-l, which is close to the value obtained previously. The data were refined further with a model which included an additional delta function to represent this 0.010 g g-l of bound water. Fig. 7 shows the results of these fits, which showed expected behaviour in that the values of A were similar at all coverages and close to the values of A , in fig. 6. The values of AET were approximately twice those corresponding to the highest coverage in fig. 6 and fairly similar at the three coverages. Though the fitted values of AET do not extrapolate to zero at zero Q, they do pass through a maximum in the range 0.5 < Q/A-l < 2.5.Since J,(QL) exhibits a shallow maximum near QL = 4, this allows us to estimate the jump distance, L, to be ca. 3.0 A. This value is physically reasonable since it is close to the diameter of a water molecule. The maximum values of AET at each coverage lay between 0.024 and 0.040 meV, giving a value for th of (4.5-8) x s.J. W. CLARK, P. G. HALL, A, J. PIDDUCK AND C. J. WRIGHT 2079 1 I 0 0 . 5 1 .o 1 .s 2 .o 0 . 3 2 2 0 . 2 8 0 . 2 4 E --. e 5 0 . 2 0 3 5 0 1 6 Q 0 . 1 2 Q c- 0 . 0 8 0 . 0 4 0 1 2 3 Q21A-2 0 . 5 , I O .:o' 0 . 5 1 .o 1 . 5 2-0 QIA-' Fig. 7. Values of (a) A , the translationally broadened elastic fraction, (b) AET and (c) AER, obtained by fitting eqn (13), together with an additional delta function representing 0.01 g g-l of bound water, to the experimental data.Key as for fig. 4. The two lower hatched regions (model a) represent predicted values of A for a = 0.90 and 0.77 A in the interval N 2 2. The uppermost hatched region (model p) represents the predicted value for N 2 2 for a = 0.90 A and for 25% of the protons remaining stationary during the reorientation. DISCUSSION It has been found that a two-component model of the dynamics of the adsorbed water produces a good fit to the experimental scattering laws (see table 2 and fig. 5). One component, which has been shown to have a concentration of 0.01 g ggl at all coverages, is immobile on the experimental timescale of s, whilst the other component undergoes both translational and rotational diffusion. Surface hydroxyl groups have been shown to be immobile on this timescale so that the mobile component can be fully ascribed to adsorbed water molecules.Similar observations2080 DIFFUSION OF WATER ON SILICA of two-phase behaviour have been made for water intercalated between clay lamellae, although in the clays the phenomenon of ion hydration provides a complicating factor which is absent in the si1i~a.l~ The refined variables from the best fits to the experimental scattering laws are shown in fig. 7. These were obtained with a model in which the mobile fraction undergoes consecutive uniaxial rotation and two-dimensional jump translation with t , = 4.4 x s. Thus approximately ten reorientations take place between each translational jump. The translational jump distance has been shown above to be ca.3 x m, so that the two-dimensional diffusion coemcient, D,,,, for the mobile phase is ca. 6 x 10-lo m2 s-l, a factor of four less than the diffusion coefficient of bulk water at the same temperature. This value of D,, for the mobile film agrees well with the diffusion coefficients measured by spin-echo n.m.r. techniques for complete monolayers of water adsorbed upon silica^.^ s and t, z 6 x GEOMETRY OF REORIENTATION The values of A derived from the fits to the experimental scattering laws have been compared in fig. 4, 6 and 7 with predictions of this quantity based upon a model in which all the water protons execute N-fold uniaxial rotation about an axis which is 0.9 A removed from both hydrogen atoms. This distance corresponds to the separation between one hydrogen atom in a water molecule and the ‘pseudo C3v’ axis of the doubly hydrogen-bonded water molecule, which lies on one of these hydrogen bonds.The comparisons shown in fig. 7 are unsatisfactory and there is a tendency for the results to be systematically lower than the predictions for A , especially at lower angles. In the introduction we discussed the currently accepted model for adsorption at the silica surface in which most of the ‘first down’ oxygen atoms in the adsorbed water molecules act as double donors of electron density to form hydrogen bonds. As a consequence there are two alternative uniaxial rotation models that can be postulated ; the one that has been described above and one in which both hydrogen atoms rotate about the CZv axis of the water molecule.In this latter model the distance between the axis of rotation and the hydrogen atoms is 0.77 A, and A , for this model is shown in fig. 7. Motion about the CZv axis requires the rupture of both hydrogen bonds, whereas the rupture of a single hydrogen bond followed by either 360” rotation or the formation of a hydrogen bond to another -OH group might be expected to have a higher probability. At higher coverages of water there is an increasing probability that hydrogen-bond formation occurs through the hydrogen atoms of the water molecules and in the limit that all orientations of the adsorbed water have equal probability there will be a 25% chance that one proton will remain stationary during the reorientation.In this limit the e.i.s.f. will equal 0.75A0(Q) + 0.25, and this prediction is plotted for a 0.9 A radius of rotation, with N 2 2, as model B in fig. 7. It is compared with the standard model in which all hydrogen atoms diffuse during the reorientation, designated model a in fig. 7. Fig. 7 shows that the standard model, with a radius of rotation of 0.9 A, is in marginally better agreement with the data than the other models. At the same time it is clear that none of the fits to the e.i.s.f. is very good and that the experimental data do not provide strong support for any of the different geometric models that have beer, considered. STRUCTURE OF THE IMMOBILE PHASE It has been found that 0.01 g g-l of water is adsorbed into an immobile phase at the surface of the silica, which is a quantity equivalent to a,, = 0.3 and OHIO z 0.5.The specific mass of this immobile water remains constant at the three coverages investigated and so it is reasonable to assign this water to the molecules which wereJ. W. CLARK, P. G. HALL, A. J . PIDDUCK AND C. J. WRIGHT 208 1 the first to adsorb at each surface. Infrared evidence shows that for a silica surface prepared under similar conditions to our own, none of this water is involved in a rehydroxylation process and that the concentration of non-hydrogen-bonded, ' free', Si-0-H groups initially decreases uniformly with the concentration of adsorbed water. At the same time a,, = 0.3 is close to the value (0.26) previously determined8 for the concentration of non-hydrogen bonded hydroxyl groups at the surface of Spherisorb and we propose, therefore, that these immobile water molecules are those 'first down' water molecules which each form new hydrogen bonds to two surface hydroxyl groups, at least one of which was previously non-hydrogen bonded.STRUCTURE OF THE MOBILE PHASE As adsorption proceeds it is generally thought that clusters of water molecules can form where each newly arriving molecule has, on average, a smaller number of hydrogen bonds than the 'first down' molecules. This accounts for the commonly observed decrease in adsorption enthalpy, AHad, as adsorption proceeds. Initial values Of AHad are in the range 60-70 kJ mol-l, decreasing to near 50 kJ mol-l at ON* z 0.5'. It is probable that this reduction in adsorption enthalpy is associated with the enhanced mobility of the water molecules adsorbed at higher coverages, as has been observed in this paper.Linear and cyclic clusters are consistent both with the inelastic-neutron-scattering evidence, which shows these mobile molecules to be doubly hydrogen bonded, and with the observation that their dynamical behaviour is independent of coverage at concentrations up to 1 molecule per OH group. MACROSCOPIC DIFFUSION Diffusion within the mobile phase has been characterised by a two-dimensional diffusion coefficient D,,, = 6 x m2 s-l. In contrast a calculation of the Knudsen diffusion constant for the transport of water through the 90 A diameter pores of Spherisorb, assuming a zero residence time at the walls of the pores, is given by20 D = = 4.6 x lo-' m2 s-l where T is the absolute temperature, M is the molecular weight, V is the pore volume, S is the surface area, p is the bulk density and Fis the tortuosity factor.The residence time at the walls, however, will not, in practice, be equal to zero but will be given t" = t: exp QIRT where Q is the enthalpy of adsorption of a water molecule and t,N is the inverse frequency of the perpendicular vibration of a water molecule against the silica surface. In this case by 1 3t" +- - 1 - DKnudsen d2 where d is the pore diameter. For a vibration frequency of 100cm-l and Q = 50 kJ mol-1 D = 1.8 x m2 s-l. Since this value is substantially less than the surface diffusion coefficient, the bulk of the mass transport along the pores at partial pressures > 0.2 must be via surface diffusion.2082 DIFFUSION OF WATER ON SILICA CONCLUSIONS This work has shown the coexistence of two phases of sorbed water molecules with different dynamics in monolayer and near-monolayer films on the surface of Spherisorb silica.One component (0.01 g g-l, or 1 H,O per 3 OH groups) is immobile on the experimental timescale (< s). The other component, at concentrations between 0.01 and 0.03 g g-I, gives quasielastic broadening, which is fitted well by a model of consecutive uniaxial rotation (tR = 4 x lo-’, s) and two-dimensional jump translation (D2D x 6 x m2 s-l). The e.i.s.f. obtained from this fit is not satisfactorily accounted for by any of the models that have been considered and the geometry of the reorientation process in the mobile phase is still unclear.We propose that initial adsorption at the SiO, surface results in formation of the immobile phase. ‘First down’ water molecules interlink pairs of surface OH groups, one of which is non-hydrogen bonded and both of which are rotationally immobile on the experimental timescale. Subsequent adsorption on this network forms mobile, molecular clusters, which must be doubly hydrogen bonded to be consistent with librational frequencies reported previously2 and with near-infrared and adsorption measurements on related systems.’? 3 9 Comparison of the surface and Knudsen diffusion coefficients shows that most of the mass transport occurs via surface diffusion. Yu. Babkin and A. V. Kiselev, Russ. J. Phys. Chem., 1963, 37, 118. P. G. Hall, A. Pidduck and C. J. Wright, J. 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