Chapter 35
Role of Reactive-Surface-Area Characterization in Geochemical Kinetic Models Art F. White and Maria L. Peterson
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U.S. Geological Survey, Water Resources Division, Mail Stop 420, 345 Middlefield Road, Menlo Park, CA 94025
Modeling of kinetic and sorption reactions requires estimates of the surface area of natural substrates which are often difficult to measure directly. A synthesis of data indicates that BET measurements on fresh surfaces exceed geometric estimates by a mean roughness factor of 7 over a wide range in particle sizes. Surface roughness factors for naturally weathered silicates are shown to approach 200 and are strongly dependent on mineral composition. Fractal analysis indicates a dimension of 2.0, and a self similarity comparable to a smooth spherical geometry. Estimates of reactive surface areas are commonly related through transition state theory to the surface defect density. Data indicate, however, that the actual surface dislocation density is lower than commonly assumed and the number of surface dislocations that actually represent potential reaction sites must be extremely low. A compilation of available kinetic models indicates that reactive surface areas commonly are one to three orders -of-magnitude lower than physical surface areas, with closer fits for geochemical systems having short residence times.
Major advances in incorporating reaction kinetics into geochemical models have occurred since the publication of the last Proceedings of the Symposium on Chemical Modeling in Aqueous Systems, ten years ago. These advances include greater understanding of reaction mechanisms, determination of experimental reaction rates, and development of computational methods linking kinetic expressions with reaction path and coupled transport processes. A review of the recent literature, however, indicates that advances in applying and validating kinetic models for natural systems have proceeded much more slowly. The difficulty in applying reaction kinetics to natural systems is inherent in the integrated rate equation, M = —kt, (1) m s where the mass transfer, M (moles/kg solution), is equal to the product of the ratio of reactive surface area, A (m ), to solution mass, m (kg), the kinetic rate constant, k (moles/m /s), and reactiontime,t (s). For most natural systems, M is known from solution analyses, k is assumed equal to experimentally determined rate constants, and t is estimated from fluid residence times using age dating techniques or hydraulic head and conductivity measurements. The surface area to mass ratio in such models usually is calculated from geometric or surface area measurements (1-3). This chapter not subject to U.S. copyright Published 1990 American Chemical Society
Melchior and Bassett; Chemical Modeling of Aqueous Systems II ACS Symposium Series; American Chemical Society: Washington, DC, 1990.
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CHEMICAL MODELING OF AQUEOUS SYSTEMS II
Because these parameters are linearly related in Equation 1, the statistical errors introduced by incorrect estimates are comparable for each term. The greatest cumulative error in many kinetic models lies in the estimating of natural systems' reactive surface areas. This paper will review previous work, present additional data, and provide an estimate of the errors involved in using surfacearea parameters in geochemical models.
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CHARACTERIZATION OF PHYSICAL SURFACES The simplest method of estimating mineral reactive surface area is to equate it to the physical surface area, usually reported in m /gm of substrate. Specific surface area, A/V (m /cm of substrate), will be used here for directly comparing surface areas of minerals of differenulensities. Geochemical models commonly define the physical surface area in terms of solution mass (m ), as in Equation 1, rather than solid phase volume (V ). The two ratios are related by the expression
s
m
3
where ρ is the solution density (cm /kg), η is the percent porosity, and f is the volume percent of a reacting phase in the host rock. The definition and magnitude of the physical surface area is closely associated with the scale and type of the measurement technique. Geometric measurement includes size fractionation by sieving or settling, optical methods including microscopy, particle counters, photocorrelation spectroscopy (£), and small angle neutron scattering (SANS) (5). Thesetechniquesprovide minimum, maximum or average dimensions which then employ a geometric model, i.e. sphere, cube, plane, and so forth, to calculate the surface area. Such geometric estimates are often the only available approach for some complex systems, such as fracture surface areas. The validity of such estimates depends on the correctness of the assumed macroscopic shape, accuracy of the particle size measurements, and microscopic smoothness of the surfaces. Without detailed descriptions of individual particle geometry, mineral grains are normally characterized by length or diameter, 9, which represents a range of lengths. The specific surface area is then defined as oo
J-J-pp)da.
(3)
0
m where P(9) is the probability of a mineral grain having a diameter between 3 and (9 + d3). Two commonly used statistical functions for describing the probable distribution frequency of d are the Gaussian and log-normal distributions. The commonly observed skewed distribution with tailing toward finer particles isfittedby the latter log-normal law (£). Κ is an empirical factor used to correct for deviations from the spherical or cubic form. For natural rounded sand particles, Κ was found to equal 6.1 (2), close to the theoretical value of 6.0. For crushed quartz, Κ ranges from 14 to 18 (8). If the particles are of the same size or are assumed to be all of a mean size diameter, θ , the specific surface area is — = — . (4) V a m 0
o
w
Methods for micro-measurement of surface areas include the Brunauer, Emmett, and Teller (BET) method (2), which relies on the adsorption of monolayers of gas, commonly nitrogen or argon, the adsorption of organic molecules such as ethylene glycol and ethylene glycol monoethyl ether (EGME) (10). and the use of infrared internal reflectance spectroscopy (11) which characterizes bonding of sorbed water. These last twotechniqueshave been confined principally to surface areas of clay minerals.
Melchior and Bassett; Chemical Modeling of Aqueous Systems II ACS Symposium Series; American Chemical Society: Washington, DC, 1990.
Downloaded by CORNELL UNIV on October 31, 2016 | http://pubs.acs.org Publication Date: December 7, 1990 | doi: 10.1021/bk-1990-0416.ch035
35.
WHITE & PETERSON
Reactive-Surface-Area Characterization
463
In BET measurements, the physical surface area is commonly assumed to be covered by nitrogen atoms, of radius 1.62 nm (12). When using surface areas in geochemical modeling, this area is assumed to be comparable to the physical interface existing between aqueous solution and the mineral surface. For the literature data plotted in Figure 1 (13-161 water BET surface areas are comparable to or exceed those calculated for nitrogen. A number of additional studies involving water sorption found that adsorption isotherms did not fit the BET equation because of surface chemical interactions. Excessively high BET water surface areas, such as exhibited by the smectite data (14). result from the fact that water, with its larger dipole moment (ID and higher bonding strength, can penetrate into expandable clay lamellae and smaller pore spaces than can nitrogen (16). In such expandable clays, van Olphen (IS) estimated that BET water vapor surface areas may be in error between 50 and 100 percent. Except for such clays, the assumption that the commonly employed BET surface area is representative of the substrate physically available to water molecules appears valid. The measured BET surface area reflects the sum of both the internal and external surfaces. Internal surfaces are composed of pores which may vary greatly both in size and shape. The International Union of Pure and Applied Chemistry (£) classified pores according to their average width: micropores 50 nm. These ranges reflect, in part, the methods by which internal surfaces areas are measured. Internal surfaces in the micropore and mesopore range are calculated from deviations from the BET isotherm in the form of differing hysteresis loops between the adsorption and desorption isotherms (12)· The distribution of macropores are usually determined by mercury porosimetry (2Q). Because of specific pore morphology, the BET surface area may not reflect the actual surface area involved in chemical reactions (21, 22)· Internal pore spaces may exhibit reaction mechanisms controlled by local equilibrium and diffusional transport at the same time as the external surface is controlled by surface reactions. A lack of correlation between reactivity and BET surface area, for example, is shown for biogenic marine carbonates in which the dissolution rates are more accurately estimated from the grain morphology and pore microstructure (23). QUANTITATIVE RELATIONSHIP BETWEEN GEOMETRIC AND BET SURFACES A number of individual studies have concluded that BET surface areas are greater than geometric estimates of surface areas (24-27). However no systematic attempt has been made to correlate geometric and BET surface areas over a wide range of mineral types and particle sizes to determine if measurement methods are interchangeable in kinetic models. The logarithmic plot in Figure 2 shows such a synthesis of about 40 references (28-67) in which both grain size ranges and BET values are reported. The ranges in particle size, shown along the horizontal axis, vary from sand to clay size. Data are divided into six general compositions and include both natural and synthetic minerals. Biogenic material was excluded on the premise that surface characteristics were controlled by processes different than those for inorganic substrates. As expected, the regression fit lies above the geometric line indicating that measured BET surface areas are almost always greater than the geometric estimates. However a striking and somewhat unexpected feature is the parallel nature of the two lines indicating that the differences in BET and geometric surface areas are consistent over a wide range in grain size. The classical treatment of particle surface area and surface irregularity has been to regard it as a deviation from the ideal reference, i.e. Equations 3 and 4, or to model it as a superposition of a regular periodic function using Fourier transforms (£8). In geochemical models, the roughness factor, R, is most often employed (£2). R is defined as the ratio of the actual surface, assumed in this case to be the BET surface, and the estimated geometric surface. Reported roughness factors frequently range from close to unity (1.08 for mica (70)) to relatively large values such as 15.0 for biogenic carbonate sediments (22). Based on the data in Figure 2, the best estimate of the mean roughness of reported minerals is the ratio of the y-intercepts of the two lines, or 7.
Melchior and Bassett; Chemical Modeling of Aqueous Systems II ACS Symposium Series; American Chemical Society: Washington, DC, 1990.
464
CHEMICAL MODELING OF AQUEOUS SYSTEMS II 3.0
τ
I
I I I | I I I I |
I
I I I I I 1 I
I I I
11/1
2.5 Ε
eu
A^
2.0 h
A
Φ
S 1-5 h
A A
Downloaded by CORNELL UNIV on October 31, 2016 | http://pubs.acs.org Publication Date: December 7, 1990 | doi: 10.1021/bk-1990-0416.ch035
D CO
H ^m 1.0
Θ
/ /
/
A
Ô
/
eu x
/
/
-silica Q3) 0 -silica (15) A-marine sediments Q6) H -smectite (14)
0.5 ι ι ι I ι ι ι ι I ι ι
0
0.5 Ν
1.0 2
I
II 1.5
I I I
Iιιι ιIιιι ι
2.0
- B E T surface area, m g m " 2
2.5
3.0
1
Figure 1. Comparison of surface areas calculated from BET sorption isotherms for nitrogen and water. T I I I I I I I Γ~| I Γ 1 I ι ι ι ι ι ι ι ι ι I I I I I Γ
Ε ο 03
ω
S
CD O 03
ι
k_ 13 CO O
-oxides ~ ν -clays - Δ -quartz Θ -feldspar [- • -other silicates s -calcite ' ' ' ' I' ' ' ' I' ' -7 Log particle size, cm
Figure 2. BET surface areas as a function of particle diameter. The dashed line is solution to Equation 4 (K=6) assuming spherical geometry, with slope and intercept of -1.00 and -3.22 respectively. The solid line is the linear regression fit to reported BET data (28-67L with R-squared coefficient of 0.86 and respective slope and intercept of -0.99 and -2.38. Melchior and Bassett; Chemical Modeling of Aqueous Systems II ACS Symposium Series; American Chemical Society: Washington, DC, 1990.
35.
WHITE & PETERSON
Reactive-Surface-Area Characterization
465
Downloaded by CORNELL UNIV on October 31, 2016 | http://pubs.acs.org Publication Date: December 7, 1990 | doi: 10.1021/bk-1990-0416.ch035
Mandelbrot (24) has suggested a different approach to surface irregularity by using fractal dimensions. A recent proliferation of studies has substantiated the hypothesis of self similarity for a number of natural systems (71-73). By this approach, surface irregularity scaling is given by the fractal dimension D, whose range is defined as 2 < D < 3 and which is related to the surface area by the proportionality A-3 . (5) D equals 2 when the scaling of A is defined by a simple geometric relationship (Equation 4). As D approaches 3, the volume dimension, the surface area is no longer dependent on the diameter of the particle. This latter limit implies that the surface irregularity is so great that it completely fills any volume defined by D. The fractal dimension is then an intensive parameter which describes the change in surface topography or roughness as a function of grain size. Extensive roughness and pronounced pore structure, as in zeolites for example, can still yield D values close to 2.0 Q4). The fractal dimension manifests itself in the size distribution of roughness rather than the presence of roughness. This approach has been applied to the surface analysis of a small number of geologic materials over a limited particle size range including synthetic periclase (3-22μπι, D=2.0), crushed quartz (0.5-12.0μπι, D=2.0 to 2.2), and crushed limestone (50-2600μπι, D=2.1 to 2.8) (Zl). As demonstrated by Pfeifer (74), the range of self similarity defining the fractal dimension is f< f
