Langmuir-Derived Model for Diffusion- and Reaction-Limited

Environ. Sci. Technol. 1997, 31, 2754-2760

Langmuir-Derived Model for Diffusion- and Reaction-Limited Adsorption of Organic Compounds on Fractal Aggregates JORDI DACHS* AND JOSEP M. BAYONA Department of Environmental Chemistry, CID-CSIC, Jordi Girona 18-26, E-08034, Barcelona, Catalunya, Spain

Natural colloids, aerosols, and aggregates generated in wastewater treatments have fractal geometries, and so not all the surface sites of these aggregates possess the same collision probability. It is this property that modifies the classical Langmuir adsorption isotherm, since it describes the equilibrium of the rate of collisions that adhere with the rate of desorption. In this paper, the growth site probability distribution is computed for aggregates with different fractal dimensions in the two- and three-dimensional models. These distributions are then used to find the adsorption isotherms depending on a model parameter η. This parameter is related to the reactivity of the adsorbate, and its relationship to solubility and molecular size has tentatively been proposed and studied. The low concentration range of the adsorption isotherms can be fitted with the Freundlich isotherm for description purposes. When there is low affinity between adsorbate and matrix (reaction-limited adsorption), linear adsorption isotherms are obtained; conversely, isotherms are less linear when adsorbate reactivity increases. Furthermore, the increase of adsorbate solubility led to lower partition coefficients, in agreement with the results reported in the literature. Therefore, the isotherms obtained with the Langmuir-derived equation take into account the fractal geometry of aggregates and provide a good description of the linear and nonlinear isotherms of organic compounds in terms of their solubility, reactivity, and molecular size.

Introduction Sorption and partitioning between the dissolved, colloidal, and particulate phases is one of the fundamental processes affecting the fate of organic pollutants in the aquatic environment, in which either the organic matter or the mineral surfaces can play an important role (1). The nature of adsorption processes between dissolved and particulate phases is complex, and the adsorption isotherms are often nonlinear following the Freundlich law. On the other hand, the behavior of submicrometer colloids is poorly understood, even though they represent a major fraction of particles in seawater (2, 3). Few studies on colloids in natural waters have been carried out due to difficulties in their separation, characterization, and modeling (4, 5). During the last decade, fractal analysis has been developed and applied to the study of aggregates and surfaces (6, 7) and now can be applied to environmental colloids. A fractal is * Corresponding author present address: Department of Environmental Sciences, Rutgers University, P.O. Box 231, New Brunswick, NJ 08903; e-mail: [email protected].

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TABLE 1. Fractal Dimensions of DLA Clusters Generated for Adsorption Simulations

a

Fa

two dimensions

three dimensions

1 0.1 0.05 0.01 0.001

1.67 ( 0.01 1.76 ( 0.01 1.93 ( 0.01 1.94 ( 0.01 1.96 ( 0.01

2.35 ( 0.03 2.55 ( 0.03 2.63 ( 0.04 2.71 ( 0.05 2.89 ( 0.03

Fraction of collisions that adhere.

an object whose statistical properties are invariant when its scale is changed. This property is a class of symmetry called scale invariance or self-similarity. There are two regimes of colloid aggregation; the first, when no repulsive force is present between particles, and the second, when a certain degree of repulsion is present. These two pathways are called diffusionand reaction-limited aggregation (DLA and RLA, respectively). They are universal and independent of the particularities of the colloid system (9). To date, fractal aggregates and surfaces have been detected in a wide variety of materials such as charcoal, soils, aerosols, aquatic colloids, marine snow, and humic acids (9-17). Although natural fractal aggregates are ubiquitous in the environment, few practical studies on their implications in environmental physicochemical processes have been performed (18, 19). The purpose of this paper is to elucidate the role of fractal geometry in the adsorption of organic compounds on fractal aggregates in equilibrium conditions. We provide a mechanism of adsorption based on the Langmuir adsorption isotherm, taking into account the different sticking probabilities of the active points on the aggregate. These distributions of collision probabilities have a major role in the adsorption isotherms since Langmuir isotherms are derived by accounting for the equilibrium of rates of collisions that adhere to the rate of desorption. Relationships between the isotherms, model parameters, and aggregate fractal dimensions are presented. Furthermore, the contributions of the different adsorption processes, limited by diffusion or reaction, are elucidated.

Model Development Fractal Aggregates. A number of aggregation models have been proposed (6, 7). The most simple is that of diffusionlimited aggregation (DLA), which is a particle-cluster model in which all the collisions between particles and aggregates are effective. Conversely, reaction-limited adsorption (RLA) takes place when only a fraction F of collisions are effective due to a certain degree of repulsion between particles and a cluster leading to denser aggregates (Figure 1). These aggregates can be characterized by their fractal dimensions (Table 1) (6, 7) (Appendix I; this and all subsequently cited appendices are available as Supporting Information, see paragraph at end of paper). Even though other cluster-cluster aggregation (CCA) models have been proposed (6, 7), our work is focused on particle-cluster DLA and RLA (Appendix II). Growth-Site Probability Distribution (GSPD). The growth of a DLA cluster is governed by the probability distribution pk, (i.e., the probability that the next particle sticks on the surface site k of a cluster in a two- or three-dimensional lattice). The growth-site probability distribution (GSPD), also called the harmonic measure, is a highly complex system function with great spatial variability. In fact, the GSPD is a multifractal measure with an infinite number of singularities (20) in which points of high and low growth probability can

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for the summation of ui,j for all the adjacent sites to the cluster:

B)

1

∑u

(2)

η i,j

This method for computing the GSPD is identical to those used for the formation of fractal aggregates with the Dielectric Breakdown Model (DBM) (23, 24) (Appendix IV). To provide an example, in Figure 2 are shown the 1000, 5000, and 9000 surface sites with higher aggregation probabilities of the aggregate depicted in Figure 1A. Due to different GSPD, these surface sites have different aggregation probabilities depending on the parameter η. The sites with high probabilities are mainly located in the outer parts of the aggregates (Figure 2A) while lower pi values are held by inner surface sites (Figure 2B,C). The parameter η does not have a generalized physical meaning but could be due to a nonlinear response to the local value of the field u (23). However, the parameter η is very useful for explaining differences in the aggregation behavior, and in this study, this parameter is related to several phenomena. For η equal to 1, the aggregate is consistent with a DLA model, whereas in the case of values of η lower than 1 the fractal dimensions obtained (24; Appendix I) are consistent with the RLA model. This suggests that η might be related to the molecular reactivity of the cluster or the sorptive capacity of the chemical. Note, that in cases where η is lower than unity, the growth-probability distribution is flatter (i.e., the differences between high and low probability points will be smaller since aggregation may be limited by reaction), permitting diffusion to the inner zones. In the case η ) 0, the collision probability will be constant over the entire aggregate surface. Conversely, the growth probabilities in the inner parts of the aggregates may be even lower than those obtained from u when the η value is higher than unity (i.e., corresponding to GSPD with greater differences between the high and low probability surface sites, see Figure 2). Furthermore, η is suspected to be related to other influences. For example, compounds with a large molecular size cannot react or associate with all the surface sites due to steric effects, permitting them to diffuse to inner sites (assuming pore and cavity sites larger than the molecular size), leading to a decrease in the value of η. The negative correlation between molecular size, for example, the total surface area (TSA) and the solubility logarithm (25), is well known. Compounds with higher solubilities would tend to associate with outlier active sites, have a very low tendency to go inside the aggregate, and therefore have values of η higher than unity. The different kinds of contributions to η may be considered as

(3)

η ) ηrηTSAηsolηx

FIGURE 1. DLA and RLA of 10 000 particles obtained for different values of F: (A) F ) 1, (B) F ) 0.1, and (C) F ) 0.01 in a twodimensional space. lie close together (21, 22). GSPD have been computed as described elsewhere (23, 24) (see Appendix III). The growth probabilities pk of the cluster surface sites are computed from the u values obtained by solving the Laplace equation around the aggregate, which are proportional to the probability of finding a diffusive particle in the point k, adjacent to the cluster (eq 1)

pk ) -Bui,jη

(1)

where η is a model parameter and B is a normalization factor

where ηr, ηTSA, ηsol, and ηx are related to cluster-molecule reactivity, steric effects, solubility, and any other unexplained effect, respectively. Note that η is always positive, with values either higher or lower than unity. The values of ηr and ηst are lower than unity while ηsol is positive and might be assumed to have values positively correlated with the logarithm of solubility in order to be consistent with the relationship between TSA and solubility. Indeed, ηsol contains the hydrophobic character of the compound, which has been recognized as a principal factor in explaining the partitioning of a wide range of organic compounds (25, 26). Adsorption Model. This model is based on the Langmuir adsorption isotherm (27, 28), although some modifications are considered taking into account the effect of geometry on the rate of collisions. The assumptions are as follows: (i) there is no interaction between the adsorbed molecules, this means that when one molecule adheres, the GSPD does not have to be recomputed; (ii) adsorption in all the sites is due

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bate concentration C in the medium (gas or liquid) and to local growth probabilities in the zone of the aggregate where adsorption is being considered (i.e., the GSPD). Furthermore, since the extent of the adsorption is less than a monomolecular layer, we should only consider the adhering probabilities of the noncovered points. Therefore, the rate of adsorption is given by Nss

∑p Φ

ra ) kaC

i

i

(4)

i)1

where ka is the adsorption constant containing the proportionality of ra with C; Φi is equal to zero when the surface site i is occupied or one if it is not; and Nss is the number of surface sites of the aggregate. Equation 4 gives the fraction of collisions that are active when considering the adhering probabilities of these empty surface sites. The probabilities pi are calculated following the method described by eqs 1-3. Similarly, the rate of desorption is proportional to the fraction of the surface (θ) covered by the adsorbate, assuming that the desorbing probabilities are constant. At equilibrium, the rate of desorption rd equals the adsorption rate, so the equilibrium equation for adsorption on a surface with nonhomogenous growth probabilities is given by Nss

kaC

∑p Φ ) k θ i

i

d

(5)

i)1

where kd is the desorption constant. When the geometry of the aggregate or surface is smooth without heterogeneities, for example, a sphere, all the pi are equal and eq 5 reduces to the Langmuir isotherm. Equation 5 shows that, on average, the sites with high probabilities will be covered first; therefore, the apparent fraction of noncovered sites, which is the sum in eq 5, decreases faster than in the classical Langmuir case. After that, a proportionally higher concentration will be necessary to cover the surface sites with lower probabilities. This adsorption model was applied to fractal aggregates of 10 000 and 5000 particles in two and three dimensions, respectively (see Appendix V for size effects). These aggregates were obtained with different values of F, resulting in a wide range of fractal dimensions (D values from 1.7 to 3, see Figure 1, Table 1, and Appendixes VI and VII), similar to those dimensions of natural occurring colloids and surfaces with D values ranging from 1.3 to 2.9 (8-18).

Results and Discussion

FIGURE 2. Surface sites depending on aggregation probabilities. (A) 1000 surface sites with the highest probabilities, for example, pk > 3 × 10-4 for η ) 0.5 and pk > 2 × 10-5 for η ) 2. (B) 5000 surface sites with pk > 2 × 10-5 and pk > 10-9 for η ) 0.5 and 2, respectively. (C) 9000 surface sites with pk > 10-6 and 8 × 10-14 for η ) 0.5 and 2, respectively. The aggregate is a DLA of 10 000 particles as shown in Figure 1A. to the same mechanism; (iii) the extent of the adsorption is less than a monomolecular layer on the surface; and (iv) all the surface or active sites are energetically uniform even though some differences in adsorption activity are present due to different growth or collision probabilities. The rate of adsorption ra is equal to the rate of collisions rc multiplied by F, the fraction of collisions that adhere. The number of adhering molecules is proportional to the adsor-

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Influence of the GSPD and Fractal Dimension. The parameter η used in eqs 1 and 2 can modify the distributions of growth probabilities, thus affecting the adsorption isotherms given by eq 5. The adsorption isotherms obtained following the method described in this paper are shown in Figure 3 for two- and three-dimensional DLA clusters. The lower the η value, the higher the similarity between the fractal adsorption and the classical Langmuir isotherm (η ) 0). The isotherms show a power law behavior at low concentrations and with fractions of covered sites lower than approximately 0.1; above this value of θ, the effect of the monolayer limitation becomes important, and there is an isotherm saturation (Figure 3). The adsorption isotherms at low concentration were compared with Freundlich isotherms

θ ) kFCn

(6)

where kF and nF are the Freundlich constant and exponent, respectively. By increasing the value of η, the power law zone becomes much smaller, even though it can still be fitted by the Freundlich isotherm (eq 6) over 3 orders of magnitude with r 2 > 99%. In Figure 4A, we report the Freundlich

FIGURE 4. Fitted Freundlich parameters nF (A) and kF (B) for threedimensional adsorption.

FIGURE 3. Adsorption isotherms for different values of η obtained from adsorption in a three-dimensional (A) and two-dimensional space (B). exponents and constants for several chosen values of η for three-dimensional aggregation, fitting eq 6 to the isotherms obtained for different DLA and RLA aggregates over 3 orders of magnitude of concentrations (from 0.0001 to 0.1). The results show that the Freundlich equation fit is good (r 2 > 99% in all cases), with Freundlich exponents (nF) between 0.86 and 1 depending on the aggregate dimensionality and the parameter η. However, the low concentration zone of the isotherms shown in Figure 3A could also be fitted with the linear adsorption model even though the confidence of fit is lower. In the case of two-dimensional aggregation, the behavior is less linear, with values of nF ranging from 0.65 to 1 (Appendix VIII), indicating that the surface adsorption processes in the inner parts of the aggregates could easily lead to the nonlinear shape of the isotherms. Conversely, the apparent Freundlich constant decreases in a nonlinear way when the parameter η is increased. Two statistical equations for predicting the Freundlich parameters were obtained from the fitted data, simulating the aggregation with different values of D and η. These equations were obtained by least-square fitting, testing the linear and nonlinear relationships between the Freundlich constant and exponent with the fractal dimension D, and the probability distribution factor η. The equations of best fit were

nF ) a + bη + cηD

(7)

log kF ) d + eη + fηD

(8)

where a-f are the parameters obtained from least-squares fitting (see Appendix IX). In fact, other variables were tested, for example, fractal dimension D and the nonlinear terms of D and η, but these were not significant. The parameter that explains most of the variability (more than 76%) was the probability distribution factor η and to a lesser extent the fractal dimension D interacting with η. The effect of increasing the fractal dimension of the aggregate is similar to the effect of reducing the parameter η for dense aggregates (i.e., with adsorption isotherms similar to the Langmuir type when increasing the fractal dimension to 2 and 3 in the two- and three-dimensional cases). However, while this effect is pronounced in the two-dimensional process, in the case of three-dimensional aggregation the differences in the Freundlich exponents are small if the parameter η is kept constant. In fact, the variable η explains 92.55% and 76.24% of variability for three- and twodimensional cases, respectively, while the interaction ηD is less important, only explaining 2.61% of the variability for 3-D aggregation. This suggests that the aggregate fractal dimensions play a secondary role, at least directly. The interaction ηD shows that there is an increase in the Freundlich exponent when the effective collisions that adhere decrease from 1 to 0.001, giving higher ranges with linear adsorption behavior in the case of dense aggregates. From the interpretation of η given above with the different contributions from ηr, η st, and ηsol (eq 3), it can be suggested that, when η < 1, adsorption is limited by reaction and/or site steric effects. Consequently, the diffusing molecules are allowed to enter the inner surface sites of the aggregate, since there are a more homogenous growth probability distribution (Figure 2 and Appendix X). Therefore, the adsorption isotherm is close to that described by the Langmuir case. Conversely, when diffusion, solubility, and other nonlinearities play an important role (η < 1), the collision probabilities in the inner active sites are very low (Figure 2 and Appendix


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TABLE 2. Values of Solubilities, TSA, and Freundlich Exponents (nF) Given in the Literaturea C satw (mg/L)

compd

TSA (A2)

matrix

nF

range Cb

peat, mulch soil soil, peat soil sediments shale (tertiary) shale (Jurassic) peat lignite bituminous anthracite humics soil soil soil soil, peat, humics soil, peat, humics soils

1 (42, 43) 0.69-0.89 (44) 0.8 (45) 0.68-0.89 (44) 1 (46) 0.96 (32) 0.92 (32) 0.88 (32) 0.88 (32) 0.82 (32) 0.8 (32) 1 (47) 1 (48) 1 (48) 1 (48) 0.89-0.92 (49) 0.80-0.93 (49) 0.63-0.88 (50)

1 2 4 2 3 4 4 4 4 4 4 2 1 1 1 3 na 3

benzene 1,4-DCB

1434 (36) 125 (37)

109.5 142.7-144.7

1,2,4-TCB HCB TCE

33 (37) 0.0034 (37) 1092 (36)

158.3-162.2 203 110

PCB-18 PCB-153 naphthalene phenanthrene atrazine prometon phenylureas

0.814 (38) 0.00072 (38) 25 (39, 40) 1.6 (39) 30 (41) 750 (41) 5-3850 (41)

218 267 155.8 198 na na na

a The references are in parentheses. na, data not available. isotherms.

b

Number of orders of magnitude of water-phase concentration for fitting Freundlich

X). Simultaneously, the probability aggregations to the outlier sites are much higher, leading to an apparent fraction of covered surface that is much higher than the real fraction of adsorbed sites. Therefore, inner surface sites could only be reached at higher adsorbate concentrations in solution giving adsorption isotherms that are described by Freundlich exponents lower than unity. Sorption of Organic Compounds. In order to determine whether or not the adsorption prediction obtained with this model are realistic, certain conclusions were derived from the isotherms. First, we evaluate the effect of eq 8 relating apparent partitioning coefficient and the physicochemical properties of adsorbates by means of parameter η. Also, the relationships between η and physicochemical parameters including molecular size and solubility assumed in the model are tested. Let η be linearly related to TSA and to the logarithm of aqueous solubility (C satw) for a given matrix with fixed dimension D

ηTSA ) KTSATSA + constant ηsol ) Ksol log C

sat w

+ constant

(9) (10)

where the rates of proportionality KTSA and Ksol are negative and positive, respectively. In the case where the aggregate is mainly composed of organic matter, then kF could be considered as the Kom or Koc factors used in the literature (1). Therefore, for a given aggregate with a fractal dimension D, eq 8 holds, and together with eq 10 we get

log Kom ) -K′sol log C satw + constant

(11)

where the proportionality between the apparent partition coefficient Kom and the solubility is given by K′sol ) -(e + fD)Ksol, where Ksol is positive and the sum of the fitting parameters (e + fD) is negative (Appendix IX). Equation 11 describes a well-known relationship in environmental chemistry (1), though it is based on the constant character of Kom or Koc and here we are considering nonlinear adsorption isotherms. However, if the linear adsorption isotherms are fitted by least squares, the linearity of log Kom to η is also obtained, leading to an analogous expression of eq 11 (Appendix XI). Similarly, this model could also account for similar correlations found in sorption on mineral surfaces (29). Direct experimental verification of eq 11 due to adsorption is difficult to carry out due to limitations on shape determinations of three-dimensional particles and because,

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in addition, partition coefficients also depend on the ka/kd ratio and partitioning into organic matter. Data Compilation of Freundlich Exponents. In order to validate eq 7 correlations, the results were compared with the Freundlich exponents described in the literature (Table 2). If the model described in this work has some validity, the exponent nF should be described, at least qualitatively, using several parameters such as molecular size, solubility, and the “average” reactivity of the adsorbate with the mineral surfaces, humics, colloids, and organic matter. The reactivity of the adsorbates to matrix is difficult to determine due to the complex nature of their chemical structure. For example, the chemical functional groups of fulvic acids have been elucidated by NMR, showing major contributions of aliphatic and oxygenated carbon protons while the aromatic protons have a lower presence (30). However, in other samples, the relative contribution of the different functional groups can be quite distinct leading to differences in binding behavior (31). For this work, a general interpretation of reactivity arises, taking into account the association of adsorbate with both reactive and ionizable surface groups. In fact, in this simplified model, reaction means that when the molecule sticks to the matrix, it is adhered with a probabily F, as described before, which will be near unity for compounds of high “reactivity” and near zero for very stable nonpolar compounds or those with a very low dipole-dipole interaction. For example, the results for nF given in Table 2 for benzene and chlorobenzene species indicate certain trends that confirm ideas in our discussion regarding adsorption on fractal aggregates [i.e., the Freundlich exponents of di- and trichlorobenzene (DCB and TCB) differ slightly from the value of unity shown by benzene and hexachlorobenzene (HCB)]. A possible explanation of this might be that the di- and trisubstituted species have relatively higher solubilities in water than HCB (see TSA and solubilities in Table 2). Furthermore, HCB is more chemically stable, with correspondingly higher values of η for DCB and TCB. Therefore, for similar fractal dimensions of the matrix, their isotherms are less linear with lower values of nF. On the other hand, benzene has a relatively stable chemical structure; therefore, its reactivity parameter ηr may be near zero leading to Langmuir-like isotherms that are linear at low concentrations. Furthermore, trichloroethylene (TCE) showed an increasing affinity with organic matter when increasing the degree of coalification (32), while the corresponding isotherms showed a decrease of nF that may be in agreement with eq 7. Other compounds with linear

isotherms, such as polychlorinated biphenyls and polycyclic aromatic hydrocarbons (PCBs and PAHs, respectively), have a relatively high TSA, low solubility, and a high stability, also with low dipole moments, leading to a reaction-limited adsorption described by near-linear isotherms. Note that with the influence of steric and solubility effects, the less volatile PAHs, such as pyrene and benzofluoranthenes, may be adsorbed in the inner parts of the aggregates, where they are more protected and less degraded as observed in the marine environment (33, 34); these adsorption processes may take place either in series to or parallel with the partitioning processes into organic matter described for this hydrophobic pollutants (1). This behavior contrasts with that of the nonlinear isotherms shown by triazines and phenylureas. For these chemicals, in addition to high solubility, their chemical structure suggests high reactivity with functional groups of humics, which in turn suggests a control of adsorption by diffusion. Indeed, the quite high joint η value results in adsorption isotherms that are far from linear. In addition to the processes studied in this work, a nonnormal distribution of aggregation energies, as implicitly assumed in this model, could lead to a modification of isotherm shapes (35). However, further work is necessary in order to determine quantitatively the importance of the phenomena affecting adsorption and the relative contributions of adsorption and partitioning as well as to understand the detailed kinetics of adsorption.

Acknowledgments Thanks to Dr. S. J. Eisenreich for his comments on the manuscript. J.D. acknowledges doctoral and postdoctoral fellowships from the Spanish Research Council (CSIC) and the Spanish Ministere of Education and Culture. Financtial support was obtained from the Spanish Plan for Research (AMB 96-0926).

Glossary B C C satw D F ka kd kF Koc Kom Ksol K′sol KTSA Nss nF pk ra rc rd u uij

normalizing factor for growth probabilities concentration of adsorbate in the liquid or gas phases solubility in water fractal dimension of the aggregate fraction of collisions that adhere adsorption constant desorption constant Freundlich constant organic carbon partition coefficient organic matter partition coefficient constant of proportionality between η and log (C satw) constant of proportionality between log Kom and log (C satw) constant of proportionality between η and total surface area number of surface sites of the aggregate Freundlich exponent growth or collision probability on the surface site k rate of adsorption rate of collisions rate of desorption probability of finding a diffusing molecule at a given point, continuum case probability of finding a diffusing molecule at the grid point i,j

TSA η ηr ηTSA ηsol θ Φi

total surface area exponent used for computing pi from uij contribution of reaction on η contibution of site steric effects on η contribution of solubility on η fraction of covered sites by adsorption function indicating whether or not the surface site i is covered

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Received for review December 9, 1996. Revised manuscript received June 13, 1997. Accepted June 19, 1997.X ES961021P X

Abstract published in Advance ACS Abstracts, August 1, 1997.

Langmuir-Derived Model for Diffusion- and Reaction-Limited

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