J . Chem. SOC., Faraday Trans. 1, 1982, 78, 3163-3175 Heterogeneity Analysis of the Silica Surface by Gas Adsorption BY CHRISTINE A. LENG* AND ALEC T. CLARK Unilever Research, Port Sunlight Laboratory, Quarry Road East, Bebington, Wirral, Merseyside L63 3JW Received 18th November, 198 1 In recent years computerised analysis methods have been developed for the determination of the distribution of adsorption energies on the surfaces of powders and precipitates from highly accurate experimental adsorption isotherms. An apparatus for the measurement of such isotherms has been constructed at Port Sunlight, and the iterative analysis method HILDA of House and Jaycock has been applied to the investigation of surface modifications of silica. The samples studied, Gasil I and TK800, are IUPAC surface-area standards and are available from the National Physical Laboratory.Results are given for a series of krypton isotherms, and the technique has been extended by the use of the nitrogen gas probe in the low-pressure rangeplp, z lo-*- 1 OW3, enabling the detection of the very high-energy regions of the surface. The high-energy heterogeneity analysis approach is shown to be sensitive to the surface changes of Gasil I induced by heat treatment in the range 423-1 170 K, and yields distinctive surface-energy distribution functions for the xerogel and Aerosil silica types. Many powders and precipitates important in industrial processing have non-uniform surfaces. Non-uniformities are due to the presence of composite structures such as catalysts, local variations in the structure of amorphous materials, non-uniform distributions of surface groups, defects, impurities, edges and corners and different types of crystal planes.' Characterisation of these surfaces is a difficult but compelling problem owing to the interest in the materials and their applications. A considerable body of work has accrued which, at times, has a highly empirical content.In the absence of more rigorous methods, conventions and rules are sought to classify experimental data, and to reveal trends without achieving absolute values. This is the case in the application of gas-adsorption techniques to the investigation of hetero- geneous samples. For uniform surfaces, the calculations of Langmuir,2 Brunauer, Emmett and Teller,3 Hill,4 de Boer,5 and Fowler and Guggenheim6 provide theoretical models for the adsorption isotherm. These models assume that the surface has a single adsorption potential energy, U, and have been applied to surfaces approaching this condition such as graphitised carbons.However, the models have also been widely applied to heterogeneous samples which clearly violate the above condition, resulting in a purely empirical description of the adsorption process. This approach, nevertheless, has yielded much useful information for a wide range of gas-solid systems, particularly in showing trends from a series of measurements such as nitrogen B.E.T. surface areas. Progress has been made in the last decade with the development of computerised methods of analysis of the adsorption isotherm.*.Such methods explicitly include surface heterogeneity, and determine the distribution of adsorption energy,f( U ) , over the sample surface. Ross and Morrisong demonstrated the narrow, single-peaked energy distributions resulting from the isotherms of nitrogen and argon on graphitised carbon and boron 31633164 HETEROGENEITY ANALYSIS OF THE si SURFACE nitride, and presented a very interesting series of energy distributions showing the increasing uniformity of the carbon black surface as the graphitisation process proceeds. Very detailed adsorption isotherms for krypton adsorbed onto silver iodide were analysed by Sidebottom et d.l0 to give energy distributions showing a series of sharp peaks; the peaks were discussed in terms of the crystal planes present and the technique was able to distinguish between four different preparative methods for silver iodide.Further applications include an investigation of annealing of precipitated sodium chloride crystals,ll and the analysis of the data of Aristov and KiselevI2 for a silica ~amp1e.l~ This type of computer analysis requires very high quality data; the vast quantity of isotherms in the literature are not suitable for analysis owing to limitations of accuracy, number of points and the pressure range of the data. For this reason, developments of the analysis methodP have been more rapid than applications of the technique. We have constructed a gas-adsorption apparatus for the measurement of accurate, extensive and detailed submonolayer isotherms. This first investigation deals with modifications of silica surfaces.Previous accurate heterogeneity studies have involved ionic crystals or graphitised carbon samples; here we test the sensitivity of the technique for an amorphous sample. Silicas are naturally abundant, and are manu- factured by two main processes to produce types of silica known as xerogels, resulting from the mixing of acid and silicate, and pyrogenic or high-temperature silicas such as Aero~i1.l~ The silica samples used in this study, Gasil I (xerogel) and TK800 (Aerosil), are surface area standardsI6 and are available from NPL. The Gasil I surface was modified by heat treatment in the temperature range 423-1 170 K. Heating alters the surface structure by the removal of hydroxy groups,17 and hence changes the powder properties.Adsorption isotherms were measured in the range 10-5-1 Torr* using krypton and nitrogen adsorbates; the isotherms were analysed using the HILDA method of House and Jaycock* to derive the surface adsorption energy distributions. In the case of krypton, the isotherms were measured from the lowest pressures attainable up to the completion of the B.E.T. monolayer. The use of the nitrogen gas probe in the same pressure range is a very different proposition, and represents a significant extension of the application of this technique. For nitrogen, the available p / p o range extends from lop8 to ( p o is the saturation vapour pressure), so that at the highest pressures measured, the surface coverage is well below that of the monolayer. Attention is focused on the very low pressure range of the isotherm, and thus information is provided on the very high energy regions of the surface.This is a particularly interesting part of the energy distribution, as the higher-energy sites are expected to be very sensitive to surface changes. Energy distributions are reported for Gasil I heated at 423,623 and 870 K using krypton as the adsorbate, and for Gasil I heated at 423, 673, 870 and 1170 K with nitrogen as the adsorbate. Classification of the different types of silica is an important problem, and towards this end we have determined the energy distribution of the Aerosil sample, TK800, to enable a comparison with the Gasil I results. EXPERIMENTAL The gas-adsorption apparatus, shown schematically in fig. 1, utilises the volumetric method.Gas is leaked from the dose region through valve A into the sample tube, distributing itself in the gas phase in the warm and cold parts of the sample tube, and adsorbing onto the sample. * 1 Torr = 133.33 Pa.C. A. L E N G A N D A. T. C L A R K 3165 The change in pressure recorded by the pressure gauge P, gives the amount of gas entering the sample region, (the relevant volumes are calibrated), and the equilibrium pressure above the sample is measured by P,. The amount of gas adsorbed per gram of sample is calculated from these readings, taking in to account thermal transpiration. lH He I I I A I x c r---------’ I vapo u r dose 6 A I I I I I -----I pressure momet er sample I J 1 i q N2/02 T1 FIG. 1 .-Schematic diagram of the volumetric adsorption apparatus.TI denotes the sample temperature (77.8 K in this work). The dashed box indicates a cabinet thermostatted at a temperature T2 slightly above room temperature. ‘ x ’ marks the position of a Springham greaseless vacuum valve with a Viton A diaphragm. The ‘dose’ region between taps A and B supplies the adsorbate for a given run. The bulb below valve C is of known volume and enables the calibration of experimental volumes required for the calculation of the amount of gas adsorbed, and further provides flexibility in the isotherm measurements by increasing the dose volume. IG indicates an ionisation gauge; an Edwards Diffstak with a 63 mm diffusion pump utilising Santovac 5 fluid and an EDM2 rotary pump provides the pumping system. The pressure is measured by Baratron transducers with ranges 1 0-4- 10 and 1 Torr.The use of two pressure gauges, rather than one, gives a considerable improvement in accuracy by eliminating cumulative errors. The system is free from mercury or tap grease contamination utilising an Edwards ‘Diffstak’ with Santovac 5 fluid and Springham valves with Won A diaphragms. Temperature control is particularly important and a nitrogen vapour pressure thermometer is used to maintain the sample temperature constant to within f O . O 1 K. The glassline is housed in a thermostatted cabinet. Research grade gases are supplied by B.O.C. A silver iodide sample was studied initially as a calibration. The krypton adsorption isotherm for this sample has been very carefully measured,’O and shows three steps in the submonolayer region.The steps indicate the condensation of a two-dimensional gas-like layer of krypton on parts of the sample with very similar energies, to a two-dimensional liquid-like film, and provide a stiff test of experimental reproducibility. ANALYSIS A straightforward extension of adsorption onto a homogeneous surface is to consider a collection of such surfaces with different energies, which are of sufficient size to neglect boundary interactions. For this system, the adsorption process is described by the following equation:19 At constant temperature, OT,T(p) is the fractional coverage of gas over the total surface at a pressure p of adsorbate above the sample, and 8 ( p , U ) is the fractional coverage on a uniform region of the surface with energy U.The distribution function f(U) denotes the frequency of regions of energy U per unit energy interval, i.e. the fraction of surface with energies in the range U and U+ dU; f ( U) is normalised.3166 HETEROGENEITY A N A L Y S I S OF THE sl SURFACE Analytical transforms have been found for eqn (1) to link a homogeneous isotherm 0 (e.g. Langmuir) to an empirical equation for the total isotherm OTOT such as the Dubinin-Radushkevich or Freundlich isotherms, via an energy distribution function.20 Such investigations are very satisfying in providing a basis for these often observed isotherm shapes; however, the transform methods severely limit the complexity of the functions involved. Ross and Olivier21 chose a Gaussian, or a sum of Gaussians, to describe the energy distribution; for a range of adsorbants this is a good approximation. Using the Hill- de Boer model for 0, they calculated total isotherms from eqn (1) which compared well, in many cases, with experiment.Samples were characterised by heterogeneity parameters such as the position and width of the Gaussian. The method used in this work was the iterative scheme initiated by Adamson and Lingz2 and refined and computerised by House and Jaycock.8 0 ( p ) is calculated using a homogeneous adsorption model, and a trial function f o ( U ) is used to calculate O&,,(p) from the equation. The latter value is compared with the experimental result at each point, and iterations continue until fn( U ) gives @GOT( p ) in agreement with experiment.The extent of agreement is measured by the root mean-square deviation ( o ~ , ~ . ~ . ) : where O%& is the experimental result, and the isotherm has N points. Equations of this type are difficult to handle,z3 and care is required with their numerical solution. The numerical stability of the analysis necessitates high-quality data with many experimental points and low scatter. Given good data, the iterative method is straightforward if the initial function f o ( U ) is close to the exact value so that convergence is rapid. It is desirable to aim for this situation, and this aspect of the analysis supports the use of krypton gas at liquid-nitrogen temperatures. Two-dimensional condensation takes place in this case, and O(p) approaches the step-function form assumed in the calculation offo( U ) .s Experience is needed to clarify the importance of this consideration with respect to other factors. Data-handling is an important feature of the analysis, and specialised computer- graphics programs have greatly assisted in this respect. The first step of the numerical procedure is the reduction of the scatter of the experimental points by a smoothing routine. In general it is a difficult task to follow data closely without reproducing, or even magnifying, irregularities in the data. The method currently used is a double rn-point, least-squares quadratic fit, with rn between 5 and 13. For rn = 5 this smoothing routine takes the first 5 data points (points 1-5), finds the best quadratic curve through the points, and calculates one fitted point.Points 2-6 are similarly fitted, and the procedure is continued over the 70-100 points of the isotherm, and then repeated throughout once more. The smoothing results must be closely checked for each set of data. A second smoothing option performs this operation on the energy distribution function between iterations. Parameters specifying the result of the analysis include the number of data points, N , the number of iterations, n, required for convergence (minimum r.m.s.), the computer C.P.U. time taken, smoothing factors indicating the quality of the smoothing of the raw data (SMl), and of the energy distribution on the final iteration (SM2), and the r.m.s. value.C. A. LENG AND A. T. CLARK 3167 RESULTS Kr/GAsIL I : 423, 623 AND 870 K Three krypton adsorption isotherms were measured on a sample of G a d I which was outgassed at 423 K, then heated at 623 K and further at 870 K in a vacuum.Outgassing at 423 K removes the physically adsorbed water without chemically modifying the surface, while calcination at higher temperatures removes surface hydroxy groups.17 The adsorption isotherms were measured at a temperature of 77.8 K in all cases; this value was calculated from the vapour-pressure measurement using loglop = a+b/T+cT the relationship with constants given by Friedman and White.26 Fig. 2 shows details of the experimental results. Within experimental error, the isotherms are generally smooth; this very amorphous sample does not show the isotherm steps present for AgI. The experimental points were measured in groups of 5-20 per run. It is important to demonstrate agreement of points from different groups, as errors between runs are greater, unless care is taken, than the variation of points of a single experiment.For this reason, measurements were taken between pairs of points of a previous run, and considerable overlap ensured as points were taken throughout the pressure range. The reproducibility of the isotherms is very good; over most of the pressure range the error is of the order of The experimental points were smoothed using a double 9-point least-squares quadratic fit to give points lying on the curve shown with the raw data (*) in fig. 2. The smoothed data were analysed to give the energy distributions shown in fig. 3, without further smoothing during the numerical calculations.The Hill-de Boer isotherm was used for 8 ( p , T ) . The factor A , relating the measured pressures to the adsorption energy U s has the value 2.373 x lo6 Torr. Termination of the energy distribution at the high-energy limit corresponds to the lowest pressure measureable; the low-energy limit is given by the B.E.T. monolayer capacity. Table 1 gives details of the B.E.T. analysis. The latter value is a reasonable and well recognised choice, although arbitrary, as is common for krypton B.E.T. plots.25 The main requirement for a meaningful comparison of the energy distribution as a function of heat treatment is a consistent termination limit. The continued rise off( U ) at low energies seen in fig. 3 indicates that, in this case, multilayer adsorption intervenes before the completion of the monolayer.Results for the 623 and 870 K samples were taken at lower pressures than for the 423 K sample, and fig. 4 compares the resulting wider-ranging energy distributions. The general form of the krypton/Gasil I energy distribution is roughly exponential. This result may be compared with the energy distribution for the adsorption of argon onto a fully hydroxylated silica;13 the measurements of Aristov and Kiselev,12 were not taken for this purpose so that the energy distribution has lower resolution than those shown in fig. 3 and 4; however, the overall shape is in agreement. The 423 K Gasil I energy distribution shows a maximum at 8.16 kJ mol-l, and a ‘hump’ at 9.62 and 10.3 kJ mol-l. The 623 K energy distribution shows more structure, with peaks at 8.41, 8.62 and 9.08 kJ mol-’ and an indication of structure at 10.3 kJ mol-l.The 870 K result is almost linear. As the calcination temperature increases, the concavity of the energy distributions decreases, with a reduction of low-energy regions in the range 8.0-8.9 kJ mol-l, and an increase of intermediate energies in the range 8.9- 1 1.3 kJ mol-l. Fig. 4shows that the higher-energy regions, 11.3-12.6 kJ mol-l, have decreased on heating from 623 to 870 K. 1 %.3168 HETEROGENEITY ANALYSIS OF THE si SURFACE 0.70 r 0.65 - 0.60 - - 0 . 5 5 - 0.50 - E I w 0.05 I I I I 0 .ooo 0.005 0.01 0 0.015 0.020 FIG. the I I I I 0 0.1 0.2 0.3 0-4 pressure/Torr 2.-Krypton adsorption isotherms on the silica sample Gasil 1 heated at 423,623 and 870 K; (a) shows low-pressure region and (b) shows the full isotherms.*, Experimental points; (-) smoothed data. In terms of the analysis, the general smoothness of the energy distribution functions is a clear indication of the quality of the data, and the stability of the numerical methods. Table 2 shows that convergence of the iterative analysis is rapid, computer time low, and the r.m.s. values are of the order of Further discussion on the stability of the energy distributions is given in the Discussion. N,/GASIL I : 423, 673, 870 AND 1170 K Nitrogen is the standard gas used for silica B.E.T. surface-area measurements which involve the portion of the isotherm near the completion of the monolayer (in the rangeC. A. LENG A N D A. T.CLARK 0.4 0.0 3169 - - I I I I3170 1 . 4 - 1 . 6 - h s? 1 . 2 - + 1.0 0.8 0.6 0 . 4 0.2 0.0 HETEROGENEITY ANALYSIS OF THE si SURFACE 5 . -. - - - - - .'.--\ ---__ J I I I TABLE 1 .-Kr/GAsIL I B E T . ANALYSIS samplea S.A.b/m2 g-l Vmb/m2 g-l Cb 423 K 178.9 34.1 14.3 623 K 179.8 34.3 14.0 870 K 182.8 34.9 15.1 a The sample is identified by the temperature of heat treatment. S.A. represents the krypton surface area, V, the monolayer capacity and the B.E.T. c parameter is also given. po was taken to be 2.46 Torr and the value of 19.5 A2 was used for the krypton molecular area. The trend is similar to that shown by the nitrogen and argon B.E.T. results for Gasil I reported by Rouquerol et aZ.24 TABLE ~.-K~/GAsIL I T/K N m SMl SM2 nu r.m.s. C.P.U. time/min 423 69 9 0.0025 - 15 L O X 10-9 2.5 623 73 9 0.0057 - 15 L O X 10-9 2.8 870 86 9 0.0071 - 15 4 .2 ~ 2.8 a These analyses were set to terminate at 15 iterations. All isotherms were measured at 77.8 K determined from the nitrogen vapour pressure measurement and the relationship of Friedman and White.2s The calculations were carried out on a Harris series 500 computer. 40-300 Torr or p/po : 0.05-0.35). In the pressure range 1 0-5- 1 Torr @/p0 : l O-s- l 0-3), the entire measured isotherm is well below the monolayer capacity, and we seek to determine the effect of the heat treatment on the very high-energy regions of the Gasil I surface. The isotherms for this gas probe were again smooth within experimental error. In contrast to the krypton results, analysis of the nitrogen data after double 9-point smoothing yielded a noisy distribution function with many sharp peaks.TheC . A . LENG A N D A. T. CLARK 3171 additional smoothing option was used between iterations to produce the energy distributions shown in fig. 5, again using the Hill-de Boer model for adsorption on a uniform region of the surface. Considerable detail of structure is shown, and the effects of heat treatment are clearly observable. For nitrogen, A , = 1.390 x lo6 Torr.8 At 423 K we see a very peaked energy distribution, showing that nitrogen is preferentially adsorbed onto some parts of the surface with respect to others. More structure is expected than for the krypton adsorbate as the quadrupole moment of nitrogen enhances the polar contributions to the adsorption energy. f ( U ) has sizeable values throughout the energy range, indicating the presence of significant regions of the Gasil I surface with these energies.After heating at 673 K the surface has a much smoother energy distribution. The surface is no longer very specific to nitrogen adsorption, and the fraction of surface with a given energy decreases almost linearly with energy. The result for 870 K is also smooth but no longer linear; the downward convexity shows a significant loss of surface with energies in the range 10.8-14.2 kJ mol-l. At 1170 K a marked change occurs: there is a return of surface specificity with peaks at 8.8 and 9.6 kJ mol-l, and a further substantial reduction of high-energy surface. The structure at 1170 K is very similar to that at 423 K with a strong damping of f ( U ) with increasing U ; comparing peak positions, there is agreement at 8.8 and 9.6 kJ mol-l, a slightly altered structure in the range 10.4-12.1 kJ mol-l, and a peak at 12.5 kJ mol-1 in both cases.The integral off( U ) over an energy interval is proportional to the fraction of surface present with energies in that interval. Thus we can monitor the reduction of surface with energies in the range, say, 10.8-14.2 kJ mol-l compared with that in the interval 8.3-10.8 kJ mol-l, by calculating the ratio of the respective areas A , and A , , under thef(U) curve. These values are shown in table 3. TABLE ~.--N,JGASIL I 423 0.41 673 0.32 870 0.25 1170 0.19 TABLE ~.-N,/GAsIL I T/K N rn SM1 SM2 n r.m.s./10-3 C.P.U. time/min 423 76 9 0.0091 0.0011 52 1.53 14.5 673 94 9 0.0058 0.0010 52 1.30 15.3 870 88 9 0.0034 0.0014 52 1.92 14.7 1170 78 9 0.0014 0.0097 52 1.41 15.1 The parameters shown in tables 2 and 4 confirm the predictions based on the analysis details.The nitrogen analysis results in a higher r.m.s., longer computing time and more iterations. However, these values are well within practical limits, and the nitrogen gas probe in this high-energy range should provide very interesting results over a range of samples.3172 HETEROGENEITY ANALYSIS OF THE si SURFACE I . 6 H 1.6 Ll . 1 . 6 - 1.4 1.2 h 1.0 b - - - 0.8- 0 . 6 - 0 . 4 0.2 0.0 - - 10 12 U/kJ mol-’ FIG. 5.-High-energy nitrogen energy distributions for Gasil I heated at (a) 423, (b) 623, (c) 870 and ( d ) 1170 K.C. A. LENG A N D A.T. CLARK 3173 N2/TK800: 423 K Results were taken for the Kr/TK800 system, but the energy distribution was not very different from that of the Gasil I sample. The nitrogen probe in the higher-energy range, however, gave a distinctive energy distribution for TK800. This result is shown in fig. 6; a series of 4 closely spaced peaks occurs between 8.3 and 10.2 kJ mol-I, with a further peak at 1 1.1 kJ mol-l and slight structure at higher energies. This spectrum is very different from that for Gasil I outgassed at the same temperature; the general shape of the energy distribution function is more like that of the Gasil I sample heated at 673 K. This result is in agreement with i.r. measurements which show a lower density of hydroxy groups for Aerosils than xerogels outgassed at 423 K.DISCUSSION We now discuss some aspects of the stability of the energy distributions shown in fig. 3-6. The experimental reproducibility is incorporated into the data as each isotherm is a compilation of several experiments, and care was taken to represent measurement errors evenly throughout the pressure range. Repetition of the whole isotherm for a given sample would merely give a different arrangement of the data points within experimental error. We consider here the possibility that this arrangement might affect the energy distribution, with perhaps particular groupings of points on the isotherm giving rise to structure on the energy distribution. For example, could the peak at 9.08 kJ rnol-' on the Kr/Gasil I 623 K energy distribution be dependent on the positions of a few points? We answered this question by using a random-number generator to move the data points within the experimental error, 1 %.The energy distributior, from the isotherm generated in this way was unchanged from that in fig. 4 except for very slight changes in the shape, but not position, of the small peaks at 8.41 and 8.62 kJ mol-l. The peak at 9.08 kJ mol-I and the remainder of the fine structure were unaltered. The highly structured energy distribution for the N2/Gasil I 423 K system was similarly tested, and no change resulted. We conclude from these tests, and from the values of the analysis parameters, that the energy distribution functions reported are very stable and that the details of these spectra may be considered closely.The differences in the krypton energy distributions are not large, but are well outside the errors involved in the experiment and the analysis. The energy distributions before and after randomisation are in agreement because the smoothed data points from both isotherms are in agreement. The smoothing routine is thus successfully eliminating experimental scatter. Uncertainty about the validity of a peak in the energy distribution occurs when the resolution of the peak is of the same order as that of the data. The smoothing level used here is such that the information extracted from the data is of a lower resolution than that of the measurements; thus the features present in the energy distribution are very stable. This also suggests that valid fine structure may appear from a lower level of smoothing, particularly in the case of the nitrogen results which were calculated using the two smoothing options, and are stable to randomisation of the data in a range greater than the experimental error.Investigation of alternative smoothing methods is an important aspect for future work. At this stage, where relatively few systems have been investigated by heterogeneity analysis, we feel that the highly stable features provide sufficient information, and present our results in this form. We conclude that the heterogeneity analysis technique is sensitive to the surface modifications of Gasil I due to the effects of heat treatment, and shows distinct energy distributions for an Aerosil and a xerogel. Two probes have been investigated.The3174 1 . 6 1 . 4 1.2 h 1.0 HETEROGENEITY ANALYSIS OF THE si SURFACE 0.6 0 . 4 0.2 0.0 10 12 U/kJ mol-' 14 FIG. 6.-High-energy nitrogen energy distribution for the Aerosil sample TK800 outgassed at 423 K. use of krypton optimised the conditions for rapid and stable analysis for the energy distribution. Generally, krypton enables the investigation of very low surface-area materials and is particularly useful for samples which have ' patchy ' non-uniformity , such as different crystal planes. For these amorphous samples, the nitrogen-gas probe at low pressures yielded structured energy distribution for high-energy surface regions. The analysis was less rapid than for krypton, but remains well within practical limits. We are currently investigating the effect of different isotherm models for 8 on the form of the energy distribution, and of particular interest is the analysis of these results using the approach recently reported by Sacher and Further experimental study is directed towards the consideration of sample variability.Many thanks are due to Mr F. R. Morgan for his considerable efforts in the construction and development of the apparatus. We thank Dr M. J. Jaycock for consultations on the technique, Dr A. L. Love11 for useful discussions on silica and Mrs D. A. Patchett for graphics programming. M. J. Jaycock and G. D. Parfitt, Chemistry of Interfaces (Ellis Horwood, Chichester, 1981), chap. 4. S. Brunauer, P. H. Emmett and E. Teller, J . Am. Chem. SOC., 1938, 60, 309. T. L. Hill, Introduction to Statistical Thermodynamics (Addison-Wesley, Reading, Mass., 1960).J. H. de Boer, The Dynamical Character of Adsorption (Clarendon Press, Oxford, 1953). R. Fowler and E. A. 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