Langmuir 2002, 18, 7089-7091
Conformational Entropy Drives Slow Sorption of Organic Chemicals into Fractal Sorbents Jordi Dachs* Department of Environmental Chemistry, IIQAB-CSIC, Jordi Girona 18-26, 08034 Barcelona, Catalunya, Spain Received April 10, 2002. In Final Form: June 25, 2002
Introduction Adsorption is a key process in a number of industrial procedures and is one of the major processes affecting the fate and transport of organic pollutants in natural environments including groundwater, estuaries, and other aquatic systems, and its study is a central issue of research in many scientific fields.1-5 However, slow sorption is not well understood even though it is a key but complex process controlling the final sink and bioavailability of organic pollutants in the environment and in a number of remediation and engineering procedures.4,6 Retarded sorption arises from the chemical and geometric heterogeneities of sorbents, the latter significantly decreasing the effective diffusion into the inner sites.7 Nevertheless, the role that these heterogeneous geometrical structures of aquatic colloids and particles play in the adsorption of organic compounds is poorly understood. During the past decade, a wide range of environmental colloids and particles has been successfully characterized using the fractal geometry formalism. Although fractal aggregates and particles are ubiquitous in the environment,8-14 there are very few practical studies on the influence of fractal geometry on environmental physical processes12,15,16 and none on its role on slow sorption. Adsorption onto fractal surfaces and aggregates has received considerable attention during the past years after the pioneering work of Avnir and co-workers, who assessed multilayer adsorption onto fractal surfaces.8,17,18 Recently, * To whom correspondence should be addressed. E-mail:
[email protected]. Telephone: (34)-93-4006118. Fax: (34)-932045904. (1) Masel, R. I. Principles of adsorption and reaction on solid surfaces; John Wiley & Sons: New York, 1996. (2) Jaroniec, M.; Madey, R. Physical Adsorption on Heterogeneous Solids; Elsevier: New York, 1998. (3) Schwarzenbach, R. P.; Gschwend, P. M.; Imboden, D. M. Environmental Organic Chemistry; John Wiley & Sons: New York, 1993; pp 255-341. (4) Lully, R. G.; Aiken, G. R.; Brusseau, M. L.; Cunningham, S. D.; Gschwend, P. M.; Pignatello, J. J.; Reinhard, M. M.; Traina, S. J.; Weber, W. J.; Westall, J. C. Environ. Sci. Technol. 1997, 31, 3341-3347. (5) Dachs, J.; Eisenreich, S. J. Environ. Sci. Technol. 2000, 34, 36903697. (6) Pignatello, J. J.; Xing, B. Environ. Sci. Technol. 1996, 30, 1-11. (7) Wu, S.-C.; Gschwend, P. M. Water Resour. Res. 1988, 24, 13731383. (8) Avnir, D.; Farin, D.; Pfeifer, P. Nature 1984, 308, 261-263. (9) Buffle, J.; Leppard, G. G. Environ. Sci. Technol. 1995, 29, 21762184. (10) Ismail, M. K. I.; Pfeifer, P. Langmuir 1994, 10, 1532-1538. (11) Kuo, L.-C.; Hardy, H. H. R.; Owili-Eger, A. S. C. Org. Geochem. 1995, 23, 29-42. (12) Johnson, C. P.; Li, X.; Logan, B. E. Environ. Sci. Technol. 1996, 30, 1911-1918. (13) Rice, J. A.; Lin, J.-S. Environ. Sci. Technol. 1993, 27, 413-414. (14) Logan, B. E.; Wilkinson, D. B. Limnol. Oceanogr. 1990, 35, 130136. (15) Dachs, J.; Bayona, J. M. Environ. Sci. Technol. 1997, 31, 27542760. (16) Dachs, J.; Bayona, J. M. Ecol. Modell. 1998, 107, 87-92. (17) Avnir, D.; Farin, D.; Pfeifer, P. J. Am. Chem. Soc. 1983, 79, 3558-3562.
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monolayer adsorption onto fractal aggregates has been studied by means of a Langmuir-derived model.15,19 This study has shown that Freundlich- and Tempkin-like adsorption isotherms arise as a result of geometrical heterogeneities of the sorbent at low and mid coverages, respectively.15,19 This work is consistent with and complementary to other recent studies on adsorption onto fractal surfaces, a field that is rapidly evolving.20,21 However, there is a need for studies relating statistical thermodynamics to sorption onto fractal surfaces and aggregates.21 Furthermore, the kinetic aspects of adsorption onto fractal sorbents, the spatial distribution of sorbate molecules on the surface, and the mechanisms that drive them have not been studied in detail. The objective of the present study is to elucidate the role of fractal geometries of aggregates and surfaces in the slow sorption of organic compounds into aquatic particles and colloids. In particular, we aim to assess whether the entropic or enthalpic contribution to the adsorption Gibbs free energy is driving slow sorption processes. Therefore, a previously described mechanistic Langmuir-derived model is applied to fractal aggregates, and intra-aggregate diffusion and adsorption processes are assessed in terms of entropic changes due to different spatial distributions of sorbate molecules. Indeed, estimation of the entropy associated with the adsorbate spatial distribution on the aggregate surface for different equilibration times will allow for the elucidation of the role of entropy-driven processes in slow sorption of organic compounds into colloids and particulate matter. Model Development. Briefly, adsorption on fractal aggregates can be described by means of a Langmuirderived model, taking into account the different sticking probabilities of the active sites on the aggregates.15 These site-specific sticking probabilities (pk) are estimated from the potential (u) values obtained by solving the Laplace equation around the aggregate (eq 1) and correcting by a model parameter η that depends on a number of physicochemical parameters such as adsorbate size, affinity to the surface, and equilibration time (eq 2).15,19,22
∇2u ) 0 pk ) -
ukη
∑ ukη
(1) (2)
The Laplace equation provides the concentration around the surface; indeed, uk is proportional to the concentration of the sorbate found at the surface site k around the aggregate. Adsorption isotherms are obtained by equaling the rates of adsorption and desorption on the surface of the aggregates. The rate of adsorption is proportional to the concentration of the adsorbate in the solvent (C), the adsorption constant (ka), and the number of sticking collisions on the empty sites of the surface. Conversely, the rate of desorption is proportional to the fraction of surface covered by the adsorbate θ and the desorption constant,15 (18) Sery-Levy, A.; Avnir, D. Langmuir 1993, 9, 3067-3076. (19) Dachs, J.; Eisenreich, S. J. Langmuir 1999, 15, 8686-8690. (20) Ehrburger-Dolle, F. Langmuir 1997, 13, 1189-1198. (21) Rudzinski, W.; Lee, S.-L.; Yan, C.-C. S.; Panczyk, T. J. Phys. Chem. B 2001, 105, 10847-10856. (22) Vicsek, T. Fractal Growth Phenomena; World Scientific: Singapore, 1992; Part II.
10.1021/la025823b CCC: $22.00 © 2002 American Chemical Society Published on Web 08/07/2002
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Notes
Nss
kaC
piφi ) kdθ ∑ i)1
(3)
where Nss is the number of active surface sites on the aggregate and φi is equal to 0 when the surface site i is occupied or 1 if it is not. Adsorption is simulated by randomly choosing a surface site and then generating an adsorption probability (pad) from a set of uniformly distributed numbers between 0 and pmax, where pmax is the maximum pk value for all the empty sites. If pad < pk, the surface site k is covered; otherwise, another surface site is randomly chosen.15 This adsorption process is repeated until a monolayer of adsorbate molecules is obtained. This approach allows us to know the pk value associated to each adsorbate molecule and to know the spatial distribution of the adsorbate on the surface. After these simulations are run, the adsorption isotherms are obtained from eq 3. The effective diffusivity abruptly decreases inside fractal aggregates, and uk and pk values are several orders of magnitude lower at the inner than at the outer sites.23,24 Therefore, the adsorbate requires a long equilibration time to reach these inner sites. The detailed study of the adsorption isotherms obtained with this model shows that Freundlich isotherms are obtained for high values of η, while the classical Langmuir isotherm is obtained when η ) 0. Since η depends on the equilibration time, the adsorption isotherms are dependent on the criteria that equilibrium has been reached. While, for short-term to midterm equilibration times, these isotherms may be consistent with the Freundlich isotherms, on the longterm basis, all the isotherms will be close to the Langmuir isotherm and thus linear in the low coverage range. Adsorption kinetics can be adequately assessed by means of the sticking probability distribution.1 The sticking probability (S(θ)), the ratio of the number of molecules that stick over the number of molecules that impinge on the surface, is given by the summation of the probabilities of the empty sites and should be normalized by the sticking probability at zero coverage (S(0)).
S(θ) S(0)
Nss
)
piφi ∑ i)1
(4)
Further details on the Langmuir-derived model and the adsorption simulation procedure have been described elsewhere.15,19,24 The model was applied to model fractal aggregates. The fractal aggregates and surfaces considered in the present study are those obtained by diffusion- and reaction-limited aggregation, DLA and RLA, respectively. These aggregates are generated by introducing a seed particle to the lattice origin, and then, a second particle is launched far from the aggregate and is allowed to follow a diffusive walk until it reaches a site adjacent to the seed particle. Then a third particle is launched until it collides with the aggregates and so forth. Details can be found elsewhere.15,19,22 The mass fractal dimensions of the fractal three-dimensional aggregates used as model surfaces range from 2.35 to 2.89. Additionally, the approach presented here has also been applied to two-dimensional DLA and RLA with mass fractal dimensions ranging from 1.67 to 2. Simulations on two-dimensional aggregates have been done for illustrative purposes, since three-dimensional aggregates, and thus adsorption on them, cannot be represented in two-dimensional figures. (23) Mandelbrot, B. B.; Evertsz, C. J. G. Nature 1990, 348, 143-145. (24) Dachs, J.; Eisenreich, S. J. Langmuir 2001, 17, 2533-2537.
Figure 1. Sticking probabilities versus coverage for different equilibration times (parameter η) onto an aggregate with a fractal dimension of 2.55.
Results and Discussion Figure 1 depicts the sticking probabilities versus coverage for adsorption onto a three-dimensional aggregate with a mass fractal dimension of 2.55 and for different values of η. The sticking probabilities decrease with coverage faster than predicted by the classical Langmuir model because of geometric heterogeneities.19 This fast decrease of S(θ) occurs even at low coverage (Figure 1). Since longer equilibration times are related to lower values of η, the sticking probability distribution of the active adsorption sites tends to be that of the classical Langmuir model (η ) 0) after long equilibration periods. Indeed, after the adsorbate has had time to diffuse into the aggregate (η ) 0), the site specific sticking or adsorption probabilities (pk) will be equal for all the adsorption sites. Changes in adsorption probability distribution with time will also lead to different adsorption isotherms after the adsorbate diffuses inward into the aggregate and there is a reordering of the adsorbate positions on the surface. While, for short-term or midterm equilibration times, adsorption isotherms are consistent with the Freundlich and Tempkin isotherms at low coverage and midcoverage, respectively,15,19 on the long term, the adsorption isotherms are closer to the Langmuir isotherm and, thus, linear in the low coverage range. While S(θ) values and adsorption isotherms are strongly dependent on η, the influence of fractal dimension is secondary in the studied range (2.352.9), even though it is important to consider that fractal geometry occurs, since, for an Euclidean sorbent (for instance, a sphere with mass dimension of 3), the results are considerably different and, indeed, are those predicted by the classical Langmuir model.15,19 This explanation for Freundlich-like adsorption isotherms is complementary to the other two existing explanations: first, that Freundlich-like isotherms arise from the occurrence of energetic heterogeneities and, second, that they are the result of geometrical heterogeneities.21 Furthermore, experimental evidence of a modification of adsorption isotherms with time can be found elsewhere,6,25 and linear isotherms have been interpreted as the result of the filling of pores by the adsorbate after long equilibration times. The positions of the adsorbate molecules on the surface show important differences depending on the value of η and, thus, on the equilibration time. For short equilibration times (η > 1), the adsorbate occupies the outer, high (25) Xia, G.; Pignatello, J. J. Environ. Sci. Technol. 2001, 35, 84-94.
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Figure 2. Example of sorption onto/into a fractal sorbent and location of the sorbate molecules: (A) two-dimensional DLA aggregate with a fractal dimension of 1.67; (B) first 2000 adsorbate molecules adsorbed onto the aggregate surface for η ) 2 (short equilibration time); (C) first 2000 adsorbate molecules adsorbed onto the aggregate surface for η ) 0.25 (longer equilibration time).
collision probability sites of the aggregate surface. However, after diffusion to the inner sites occurs, porous and active sites into the aggregate are occupied (Figure 2). For sufficiently long equilibration times (η ) 0), there would be sorption indistinctly on the inner and outer active surface sites, since all these have the same adsorption probability. Therefore, slow sorption is related to a modification of the distribution of sticking or adsorption probabilities onto the surface and, thus, to a remobilization of the sorbate on the surface. These changes lead also to changes of the entropy associated with the conformational or spatial distribution of adsorbate molecules onto the surface and/or particulate matter. In fact, the entropy difference (∆S) due to geometric heterogeneities on the surface was estimated using eq 5,26 Nss
∆S ) R
∑ (pk ln pk - qk ln qk)
(5)
k)1
where R is the universal gas constant, pk is the probability to find the adsorbate at the surface site k, and qk is the probability to find the adsorbate at surface site k for long equilibration times (η ) 0) when all the sites have equal adsorption probabilities. The estimations of entropy increase for the adsorption on a fractal aggregate for different values of η, as shown in Figure 3. ∆S is 0 for η ) 0 (long equilibration time or Langmuir case), since this is the reference state that we choose. However, ∆S increases when increasing the value of η, showing that the entropic contribution to the Gibbs free energy is a factor favoring and maybe even driving the slow sorption of chemicals into fractal sorbents and other porous matrixes. Estimations of T∆S, assuming T ) 298 K, are in the range 0-5 kJ mol-1 for aggregates with fractal dimensions (26) Wannier, G. H. Statistical Physics; Dover Publications: New York, 1987; Chapter 5.
Figure 3. Increase of entropy versus η for a three-dimensional aggregate with a fractal dimension of 2.55.
in the range between 2.35 and 2.9. These entropic contributions to Gibbs free energy are low in comparison to those of physical-sorption enthalpies, which are in the range 30-60 kJ mol-1. However, entropy may have a significative role either when the interaction of the adsorbate with the matrix is weak or for long time dynamics, such as slow sorption due to diffusion of the adsorbate to the inner sites of the surface. Therefore, assuming that the physical and chemical characteristics of the sorbent do not change with time, we determine that the slow kinetic rates responsible for slow sorption processes would be the result of entropic changes due to redistribution of the adsorbate onto the surface. Entropy increase will also lead to changes in the pseudoequilibrium adsorption constants with time4 which is consistent with their dependence on η, as described elsewhere.15,19 Briefly, the detailed analysis of changes in sticking probabilities and the associated redistribution of the adsorbate molecules onto the surface suggests that conformational entropy increase may drive slow sorption processes. Much research is needed to understand the quantitative relationship between entropy and slow sorption rates as well its interactions with other variables such as changes in chemical reactivity, dynamic aggregation-desegregation processes, and so forth. Acknowledgment. Prof. Steven J. Eisenreich (Rutgers University) is acknowledged for many insightful discussions. This work was funded in part by the Spanish Ministry of Science and Technology through the PRODELTA project. LA025823B
