pubs.acs.org/Langmuir © 2010 American Chemical Society
Correlation between Fractal Dimension and Surface Characterization by Small Angle X-ray Scattering in Marble Manlio Favio Salinas-Nolasco*,† and Juan Mendez-Vivar†,‡ †
Laboratorio de Fisicoquı´mica, Escuela Nacional de Conservaci on, Restauraci on y Museografı´a-INAH, Calle General Anaya 187 Col. San Diego, Churubusco 04120 (Coyoac an) M exico, D.F., M exico. ‡ Permanent address: Departamento de Quı´mica, Universidad Aut onoma Metropolitana Iztapalapa, A.P.55-534, 09340, M exico, D.F. Received March 2, 2009. Revised Manuscript Received December 11, 2009
Among several analysis techniques applied to the study of surface passivation using dicarboxylic acids, small angle X-ray scattering (SAXS) has proved to be relevant in the physicochemical interpretation of the surface association resulting between calcium carbonate and the molecular structure of malonic acid. It is possible to establish chemical affinity principles through bidimensional geometric analysis in terms of the fractal dimension obtained experimentally by SAXS. In this Article, we present results about the adsorption of malonic acid on calcite, using theoretical and mathematical principles of the fractal dimension.
Introduction The current trends in conservation of the cultural heritage made of limestone and marble have to overcome the challenge of maintaining their structure taking into account their continuous exposition to environmental agents. Faccades, sculptures, stuccoes, and mortars based on sedimentary (limestone) or metamorphic (marble) calcium carbonate exhibit high solubility when they are exposed to acid rain, a common phenomenon in urban zones and in the vicinity. The first attempts to protect limestone and marble have been the use of layers deposited on the surface in order to provide an effective covering, experimenting with all kinds of materials, ranging from those of biological origin (oils, waxes, resins, proteins, etc.) up to those synthesized in the lab (epoxy, vinyl, acrylic, etc.). Exhaustive studies1 have demonstrated that a film deposited on top of a mineral substrate has to be permeable to moisture but also has to maintain the original morphology of the stone (porosity, grain size, crystallinity, roughness, etc.). Using as a criterion the application of nonaggressive and reversible interventions, some researchers have developed new compounds2 and techniques3 in order to produce thin porous protective films while maintaining the characteristic pore size distribution of the materials. Based on the intrinsic physicochemical properties of mineral calcium carbonate, which possess a high affinity to organic molecules, it has been proposed the term surface passivation, that implies an intermediate protective status between a consolidation or covering with permanent films and the clean status of the original structure. Passivation suggests the factibility of a surface association of organic compounds that can be bonded to calcium carbonate, in a thermodynamic system characterized by a chemical equilibrium provided by a few adsorbed monolayers. This *To whom correspondence should be addressed. Telephone: (525) 55604 5188 Ext. 4531. Fax: (525)55604 5163. E-mail
[email protected]. mx. (1) Didymus, J. M.; Oliver, P.; Mann, S.; De Vries, A. L.; Hauschka, P. V.; Westbroek, P. J. Chem. Soc., Faraday Trans. 1993, 89, 2891. (2) Van Capellen, P.; Charlet, L.; Stumm, W.; Werns, P. Geochim. Cosmochim. Acta 1993, 57, 3505. (3) D€obias, B. Surfactant Adsorption on Minerals Related Flotation; Elsevier Series: New York, 1998.
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preferential adsorption phenomenon exhibited by organic molecules is quite common in nature and has originated a great number of biogenic compounds, giving as a result structurally stable associations, that is, bones, snails, sea shells, and exoskeletons, among others. In previous papers,4,5 we have proved the high affinity of malonic acid to geological calcium carbonate, whose physicochemical surface association generates thermodynamic equilibria that lower the solubility in acidic pH, in addition to maintaining the original substrate morphology. Those findings were derived from experimental measurements done by small angle X-ray scattering (SAXS), that provided information about the particle shape, pore size distribution, and surface electronic density before and after impregnation of malonic acid solutions on calcium carbonate surfaces. In order to clarify the adsorbate distribution on top of the mineral surface and to determine the adsorption mechanisms that dictate such association, the fractal dimension values obtained by SAXS provides the theoretical elements to draw some conclusions about the effectiveness of the process, same as the impregnation conditions that give as a result the effective conservation of any calcium carbonate cultural heritage object.
Theoretical Background Mineral surfaces due to their morphological characteristics of porosity and roughness can be considered theoretical subjects of fractal geometry. Several works have proposed simulating models that determine the typical structure associated to the mineral surfaces through the fractal theory.6 An approaching model that allows comparing the fractal similarity to a one-dimension profile of a mineral surface7 is the so-called Koch polygonal curve (Figure 1). It can be seen in the Koch polygonal curve that in every change of magnitude step the curve length increases in a quantitative ratio. The (4) Salinas-Nolasco, M. F.; Mendez-Vivar, J.; Lara, V. H.; Bosh, P. J. Colloid Interface Sci. 2004, 274, 16. (5) Salinas-Nolasco, M. F.; Mendez-Vivar, J.; Lara, V. H.; Bosh, P. J. Colloid Interface Sci. 2005, 286, 68. (6) Laurini, R.; Thompson, D. Fundamentals of spatial Information Systems; Academic Press: New York, 1999. (7) Harrison, A. Fractals in Chemistry; Oxford University Press: New York, 1995.
Published on Web 02/17/2010
DOI: 10.1021/la903835m
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Figure 1. Koch polygonal curve showing the self-similarity property for fractal surface profiles, similar to mineral surfaces.
Figure 2. Hypothetical schematic representation of surface deposition of covering layers and subsequent morphology modification on top of a polygonal Koch surface.
equivalence of this model and the observation of real mineral surfaces allow establishing that the morphology and the length depend on the resolution of the measurement instrument; that is, the size of an object makes sense only in the range in which the measurement is done. On the other hand, the self-similarity principle establishes the morphological norm of the fractal recognition. To use only a portion of a fractal entity to determine its magnitude is not appropriate, due to the similarity and equivalence among them. Using the scaling and self-similarity criteria, it is possible to relate them to fractal dimension: D ¼
logðnÞ logðnÞ ¼ logð1=rÞ logðsÞ
ð1Þ
where D is the fractal dimension, n is the number of fragments or self-similar objects, r is the self-similarity quotient, and s is the magnifying factor or scale magnitude. The calculation of D for smooth geometric objects using eq 1 will give as a result an integer number, corresponding to the Euclidean dimension of the analyzed object (dot, line, surface, or volume). In fractal geometry, D is always a fractional value, larger than the Euclidean one. Specifically, for surfaces 2 < D < 3. As a generalization of the definition of the surface model considering the fractals, it is interesting to study as a specific case thin film deposition on top of substrates. Figure 2 illustrates the modification of the Koch polygonal curve in the case of the formation of a film adhered to a theoretical substrate. The interest of studying the adsorption on fractal surfaces is that several porous materials exhibit a scaled behavior of the covering as a function of the adsorbate particle size.8 The fractal symmetry can be tested in these adsorption processes over a range given by the precision of the analytical experimental technique. Quasielastic neutron scattering (QENS) and small angle X-ray Scattering (SAXS) allow one to obtain the fractal dimension value D experimentally.9 (8) Pfeifer, P.; Stella, A. L.; Toigo, F.; Cole, M. W. Europhys. Lett. 1987, 3, 717. (9) Pernyeszi, T.; Dekany, I. Colloid Polym. Sci. 2003, 281, 73.
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Figure 3. SAXS dispersion curve showing the Guinier and Porod regions as theoretical limits to determine the fractal dimension D.
Regarding SAXS, the technique is based on the dispersion phenomena and it is defined in terms of the law of reciprocity that provides an inverse relationship between the particle size and the dispersion angle. The result of the interaction phenomenon is the SAXS dispersion curve (Figure 3), where in the intermediate region (between the Guinier and Porod regions) it is possible to determine D as the slope.10 The size range for the geometric characterization of the surface is given by the limits defined by 1/ξ , q , 1/r0. The model used to determine the particle size (molecules of size r required to cover a surface) is a function of r, as follows: Nm ¼ cr -D
ð2Þ
where Nm is the minimum number of adsorbed molecules on the surface (monolayer) and c is a constant. In those cases where adsorption occurs as a multilayer, the theoretical study can consider that the substrate is a small perturbation of the film thermodynamic properties.11 Considering that the effective film thickness d can be scaled to small distances on the substrate surface, Pfeifer et al. proposed a relationship that allows determining the overall covering N provided by the adsorbate molecules:8 3 -D d N ¼ kNm r
ð3Þ
where k ¼
21 -D ΓðD þ 1ÞΓð3=2Þ ΓðD -1=2Þ
ð4Þ
It is important to consider the interaction and chemical equilibrium factors in order to perform an analysis of the film over a fractal surface. In the present paper, the model of Pfeifer et al. is considered strictly as theoretical background in order to establish comparative criteria of adsorption under several experimental conditions. The N and D values are known, and they were derived from potentiometric analysis and SAXS, respectively,4 for the experimental system consisting of malonic acid/calcium carbonate. Taking into account these results, eq 3 can be written (10) Glatter, O.; Kratky, O. Small Angle X-Ray Scattering; Academic Press: New York, 1982. (11) Cheng, E.; Cole, M. W.; Stella, A. L. Europhys. Lett. 1989, 8, 537.
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as follows:
d logðNÞ ¼ logðkNm Þ þ ð3 -DÞlog r
Article Table 1. Surface Adsorption Parameters of Malonic Acid on Calcite and Fractal Dimension Values Obtained by SAXS
ð5Þ
Taking into account that (d/r) ≈ constant for similar experimental conditions (same concentration or pH), eq 5 can be rewritten as follows: logðNÞ ¼ ξ2 þ ð3 -DÞξ1
ð6Þ
The amounts ξ1 and ξ2 are obtained from the linear adjustment of several N and D values obtained under the same experimental conditions, being the slope and the y intersection value of eq 5, respectively. It is posible to calculate the minimum number of adsorbed molecules Nm and the length value dθ, that indicates the thickness value of the effective layer compared to the size of the adsorbate particles. 10ξ2 k
ð7Þ
dθ ¼ 10ξ1
ð8Þ
Nm ¼
To determine quantitatively the effective covering of the calcium carbonate surface with the malonic acid molecules, fθ is defined as the covering fraction; fθ > 1 indicates a deposition larger than the minimum one (multilayer). Consequently, fθ < 1 indicates a partial covering. fθ ¼
N Nm
ð9Þ
It is important to consider the fractal dimension change as a parameter that determines the change in the substrate surface due to the adsorption. Considering that the geometric partition n of the surface defined by eq 1 is constant during the measurement along the adsorption process, the relationship between those fractal dimensions can be arranged as logðsf Þ Di ¼ logðsi Þ Df
ð10Þ
The magnifying factor s can be correlated to the surface area value of calcium carbonate Ac obtained by the BET method (2.7 ( 0.6 m2/g) and the surface area obtained by SAXS, before (Ao) and after the adsorption (Aa): si ¼
Ac Ao
ð11Þ
sf ¼
Ac Aa
ð12Þ
Replacing eqs 11 and 12 in eq 10, it is posible to obtain a relationship between the areas measured by SAXS under the experimental conditions: Aa ¼ jAo
ð13Þ
j ¼ ðAc Þ1 -Di =Df
ð14Þ
where
When j = 1, the surface area value in both processes is maintained; j > 1 indicates that the surface area is larger than the uncovered area, whereas j < 1 indicates that the adsorbed area is Langmuir 2010, 26(6), 3889–3893
pHexp
N (μmol/m2)a
Db
Di/Df
jc
5 10-2 M
6.00 7.00 8.00
7.9739 8.2253 8.9739
2.95 2.74 2.68
1.01 0.94 0.92
1.00 1.08 1.08
5 10-3 M
6.00 7.00 8.00
0.8742 0.9295 0.9435
2.74 2.84 2.95
0.94 0.97 1.01
1.08 1.04 1.00
concentration
5 10-4 M
6.00 0.0821 2.84 0.97 1.04 7.00 0.1081 2.88 0.99 1.00 8.00 0.1195 2.94 1.01 1.00 calcium carbonate 2.92 a Adsorbed amounts of malonic acid. b Fractal dimension values obtained by SAXS. c Proportionality factor defined by eq 14.
smaller than the adsorbent surface. For a passivation process, it is desirable that the surface area can be maintained without changes; in this way, it can be assured that the general pore size distribution and surface morphology are maintained.
Experimental Section The calcium carbonate was a polycrystalline precipitate of marble powder from Conservator’s Emporium (Reno, NV), highly pure with a surface area value of 2.7 ( 0.6 m2/g, calculated by the Brunauer-Emmett-Teller (BET) method by nitrogen adsorption. The impregnations were done by dispersing 5.0 g of calcium carbonate into 50.0 mL of malonic acid solutions at three concentrations (5 10-2, 5 10-3, and 5 10-4 M) and using three pH values (6.00, 7.00, and 8.00) using a high-speed stirrer, enough to keep the suspension homogenized for 72 h at room temperature (293 K). The SAXS curves (I(h) vs h, where I(h) is the intensity and h = 4π sin θ/λ, with λ as the wavelength and 2θ the scattering angle) were measured using a Kratky camera coupled to a Cu anode tube and a Ni filter. The data, collected with a proportional linear counter, were the input of the program ITP92 to calculate the fractal dimension values.
Results The amounts of malonic acid adsorbed on the surface of the marble powder under several impregnation conditions (concentration and pH) were reported in a previous publication elsewhere,4 and the corresponding fractal dimension values obtained by SAXS are given in Table 1, in addition to the increasing area results. In Figure 4, it can be seen how the fractal dimension of marble powder can be determined from the SAXS curve data obtained between the Guinier and Porod regions that are the theoretical limits of the fractal dimension D. The modification of the SAXS results due to the change on the concentrations of the adsorbate can be seen in Figure 5. According to these results, the SAXS technique is sensitive to the influence of the adsorbate deposited on the calcite surface. Figure 6 shows the linear correlation between the adsorbed amounts and the corresponding fractal dimensions according to eqs 5 and 6. Figure 7 shows the corresponding results for the pH variation. These correlations allowed us to determine Nm, dθ, and fθ, defined by eqs 7, 8, and 9, respectively. The results appear in Table 2 for concentration and pH.
Discussion In Table 1, it is shown that the proportionality parameter j defined by eq 14 for all the experimental samples is larger or equal DOI: 10.1021/la903835m
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Figure 4. Fractal dimension determination of marble powder from the scattering SAXS curve.
to the unit value. This can be interpreted as the maintaining or increasing of the surface area after the adsorbate deposition. This increasing effect can be considered appropriate, supposing that the adsorbed layer is a potential surface of molecules whose particles contribute to generate a sinuous surface of partial covering. The increase is not significant at the macroscopic scale, and it is only a small proportion (approximately 8.0%) in the study range, which is the SAXS scale. These results indicate that using the fractal geometry it is possible to prove the deposition of malonic acid on the surface of calcium carbonate. In this regard, while determination of the pore dimensions by adsorption analytical techniques (Hg, N2) is complementary to those determinations by SAXS, the theoretical proposal of Pfeifer et al. studies this phenomenon geometrically, according to the properties of fractal objects and surfaces; so the consistency between experimental results and the correlation between the theory of multilayer adsorption with the fractal dimension obtained by SAXS is an approach that allows the interpretation of results from the viewpoint of surface adsorption. In a similar way, N2 adsorption can be correlated with the model of Pfeifer et al.; however, the scales would correspond to a fraction of the surface in the monolayer region, not the complete filling of pores, according to the isotherm. The linear regression of the adsorbed amount with the fractal dimension in terms of eq 5 shows an adequate value correlation, referred to the concentrations (Figure 4), whereas that does not occur for the different pH conditions (Figure 5). In this latter case, the experimental results appear significantly dispersed, far away from an acceptable statistical regression. From these results, using the pH of reference does not correlate properly in terms of the model of Pfeifer et al. On the other hand, the concentrations as a function of the pH present an acceptable correlation coefficient. Regarding the correlation for the concentrations (Table 2), there is a trend to the diminishing of the number of adsorbed particles Nm, the specific thickness value dθ, and the covering fraction fθ as the concentration diminishes. From this, it is important to refer to a specific amount of adsorbate in order to establish an optimum covering of the total mineral surface. Then, it is possible to determine a covering factor fθ = 1.00, that is, a total covering, for concentrations of malonic acid in aqueous solutions between 5.0 10-2 and 5.0 10-3 M. It is interesting to note the differences among the adsorbate quantities Nm (monolayer) in all cases, because it was expected 3892 DOI: 10.1021/la903835m
Figure 5. SAXS curves for three concentrations of malonic acid on calcite surface: (a) 5.0 10-2 M ((0.014), (b) 5.0 10-3 M ((0.016), and (c) 5.0 10-4 M ((0.020).
that the substrate adsorbed the same amount, with this one being a constant value, particularly because it is composed of identical chemical species and has a specific surface area. Working in different pH regions implies several ionic species in the solutions. Malonic acid in aqueous solutions has been characterized as a weak diprotic acid, implying the coexistence of all the chemical species in different amounts, and depending on their chemical structure and the solvent they could appear completely dissociated, partially dissociated, or nondissociated. In addition, it has been demonstrated that the malonic acid molecule exhibits the tautomerism phenomenon.12 This adds new possible adsorption molecular structures. Overall, every chemical species in the (12) Asciutto, E.; Sagui, C. J. Phys. Chem. A 2005, 109, 7682. (13) Thompson, D. W.; Ponwall, P. G. J. Colloid Interface Sci. 1989, 131, 74.
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Figure 8. Representation of the pore geometric parameters considering the steps: (a) initial, (b) filling with a monolayer or partial heterogeneous filling, (c) moderated, and (d) semisaturated.
Figure 6. Linear regression defined by eq 5 for different concentrations at the three pH values.
always a function of the thickness of the multilayer film. As it can be seen in Figure 8b, it could occur the deposition of a monolayer. This process is characterized by values dθ =1 and fθ =1. Intermediate processes characterized by thick layers and low covering could be represented by Figure 8c and d. These proposed models can be applied in the selective adsorption processes, particularly in the physical chemistry related to the formation of biogenic materials.13 The SAXS technique identifies the geometric shapes derived from the adsorption for the case of malonic acid deposited on top of calcium carbonate, as it has been reported in ref 4. Referring to the pH variations and taking into account the uncertainty of the results, it is particularly interesting to look at the results obtained for pH=6.00. This was the only condition that produces a significant amount of adsorbed molecules. Calcium carbonate has an isoelectric point at pH=6.20, as it was reported in ref 4. This condition represents the electric equilibrium state and includes all the ions presents in the solution, in a condition of null electrophoretic mobility over the substrate. An equilibrium state is dictated by the intrinsic mineral properties in the vicinity of the isoelectric point, where the maximum adsorption occurs.
Conclusions
Figure 7. Linear regression defined by eq 5 for the three pH values employed at the experimental concentrations. Table 2. Correlation between the Fractal Dimension and the Adsorption Capacity of Malonic Acid on Calcite at Different Concentrations concentration
ka
ξ1b
ξ2c
Nmd
dθe
fθf
5.0 10-2 M 1.0260 0.1620 0.8855 7.4876 1.45 1.12 5.0 10-3 M 1.0232 -0.1657 -0.0053 0.9655 0.68 0.95 5.0 10-4 M 1.0167 -1.4125 -0.8031 0.1548 0.04 0.67 a Constant defined by eq 4. b Regression parameter defined by eq 6. c Regression parameter defined by eq 6. d Amount of malonic acid adsorbed as a monolayer (μmol/m2) defined by eq 7. e Specific thickness of the adsorbed layer defined by eq 8. f Covering fraction defined by eq 9.
solution provides a specific particle size as an adsorbate. In this way, the adsorbate deposition on the mineral surface cannot be considered as composed of particles of the same size when working under different conditions. For high concentrations, we suppose the presence of small size structures (a big amount of chemical species deposited per surface unit), whereas the opposite occurs for low concentrations. Confirmation of this reasoning is still pending. Regarding the total or partial filling of the surface pores, according to the dθ and fθ values, it is possible to propose geometric models to explain the pore filling (Figure 8). The covering is not Langmuir 2010, 26(6), 3889–3893
The study of surface phenomena and adsorption processes in minerals using fractal geometry provides adequate interpretation basis, explaining the deposition mechanisms and the relevance of the experimental conditions on those processes. The SAXS is an important analysis technique that in addition to providing the fractal dimension value generates parameters that allow one to verify the morphology and distribution of the substrate pores by using geometric models. In this Article, we proved the effective adsorption of malonic acid on marble powder (calcium carbonate) in amounts that do not modify significantly the surface area of the native mineral. The conformation of the surface at the submicroscopic scale is considerable and can be quantified in the SAXS range. We found that the conditions that favor an almost total covering occur at malonic acid concentrations ranging between 5.0 10-2 and 5.0 10-3 M at pH=6.00, which is close to the isoelectric point of calcium carbonate. The filling correlations and the thickness of the adsorbed film exhibit a maximum, based on the concentrations analysis, that do not occur when the analysis is done under the pH conditions. It is possible to establish a filling geometric model that contributes to explain the scale definitions given by the fractal dimension concept, in addition to the criteria of pore shape given by the Kratky diagrams. In this way, the SAXS technique provides the theoretical elements to characterize entirely a surface. Finally, the model of Pfeifer et al. represents a consistent approximation in the interpretation of the fractal dimension results of adsorptive systems. Acknowledgment. The authors acknowledge CONACYT for the financial support provided. DOI: 10.1021/la903835m
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