Linking Monte-Carlo Simulation and Target
March 18, 2005
Time:
06:37pm
ps13000.tex
Physica Scripta. Vol. T115, 912–914, 2005
Linking Monte-Carlo Simulation and Target Transformation Factor Analysis: A Novel Tool for the EXAFS Analysis of Mixtures A. Rossberg* and A. Scheinost Institute of Radiochemistry, Forschungszentrum Rossendorf, D-01314 Dresden, Germany The Rossendorf Beamline, ESRF, F-38043 Grenoble, France Received June 26, 2003; accepted June 28, 2004
pacs number: 6110HT
Abstract Aqueous metal complexes are important chemical species in the geosphere, since they often determine the mobility of metals in the environment. While EXAFS spectroscopy is the method of choice to determine the structure of such species in environmental matrices, it often fails when several complexes coexist. In this case, all present complexes contribute to the EXAFS signal, and the deconvolution into single contributions is difficult. We have developed a new algorithm to determine the spectral contributions of the single in mixtures of species, based on a combination of Monte-Carlo Simulation and Target Transformation Factor Analysis (MCTFA). This novel approach requires solely the structure of the interacting ligand as input data, enabling the determination of complexes even in “messy” environmental samples.
1. Introduction Many attempts have been made in the past to determine structure, chemical behavior and relative distribution of aqueous metal complexes at a given set of geochemical conditions (defined by pH; Eh; concentrations of aqueous ions, solids, gasses; temperature; etc.). EXFAS spectroscopy is one of the most versatile and powerful techniques to determine the often low concentrations in the presence of complex backgrounds typical for geochemical settings. However, if more than one metal complex exists, then the EXAFS signal is the superposition of the contributions of the backscattering atoms of all metal complexes, and only average structural parameter (i.e. radial distances, coordination numbers) can be obtained. In case the EXAFS spectra of isolated complexes can be measured, one can derive the identity and quantity of the complexes in mixtures by Target Transformation Factor Analysis (TFA) and/or Linear Combination Fits (LCF) [1, 2]. Especially for uranyl-organocomplexes, however, species may remain undetected because they cannot be prepared (and measured) as pure species. Under these conditions, the EXAFS spectra of the pure metal complexes may be obtained by Iterative Target Transformation Factor Analysis (ITFA) [3–6]. Starting point for ITFA is a series of EXAFS spectra measured in dependence of one ore more physicochemical parameters, which influence the speciation (relative concentration distribution of species) of the metal complexes. Although ITFA does not depend on EXAFS spectra of pure species, it still depends on speciation data, which may often be provided by thermodynamic data bases and models. If even the data of the relative species distribution are missing, however, ITFA must fail. Therefore, we propose another approach, where only the structure of the metal-binding ligand is required. Our approach varies the position of the metal in relation to the ligand by MonteCarlo Simulation (MC). For each structural configuration, the theoretical EXAFS spectrum is then calculated and submitted as * e-mail: [email protected]
Physica Scripta T115
target test vector to TFA. We have tested this approach using a series of uranyl acetate complexes at varying pH. 2. Experimental Eight samples were prepared in the pH range 0.10–4.48 at an ionic strength of 1.2 mol/L. For all samples [U(VI)] was 0.05 mol/L and the acetic acid concentration [ac] was 1 mol/L except for the sample at pH 4.48, where [U(VI)] = 0.025 mol/L and [ac] = 0.5 M. The U LIII -edge EXAFS spectra of the pH series were measured at the Rossendorf Beamline (ROBL) at the ESRF (Grenoble, France) in transmission mode at room temperature. Crystallographic data of sodium tris(acetato)dioxouranate [7] were used to create the ligand structure for the MCTFA and to calculate the theoretical phase and amplitude functions for EXAFS data analysis (FEFF6). 3. Monte-Carlo Simulation Target Transformation Factor Analysis (MCTFA) MCTFA combines the advantages of modeling and statistics. The MC simulation allows to model the three dimensional structure of the complexes by using the experimental XAS spectra and the structure of the interacting ligand as constraints [8–10]. TFA allows to check whether the resulting EXAFS spectrum of the calculated complex is a component of the series of measured EXAFS spectra [1, 3, 11]. In the following capital bold letters are matrixes, small bold letters are vectors, and italic letters are scalars. Starting point of TFA is the calculation of the abstract (abs) factor solution for the abstract reproduction of the data matrix D(r,c) = Rabs(r,c) Cabs(c,c) (D accumulates the c EXAFS spectra of complex mixtures at r measuring points, Rabs and Cabs contains eigenvectors) by using Eigenanalysis. The orthogonal factor solution consists of c eigenvectors. Only n (n – number of main components) eigenvectors are necessary to reproduce D. In the second step of TFA, n is determined using the eigenvalues in and the indicator function IND [11]. After the factor compression the dimension of Rabs is (r, n) and that of Cabs is (n, c). In the third step a EXAFS spectrum is subjected as a test vector xtest and gives a predicted (pred) vector xpred = Rabs −1R abs xtest ( stands for transponse). The spectrum xtest is a component of the spectra of the complex mixtures if it is in agreement with the predicted spectrum xpred . The spectrum xtest we calculated using MC technique. The steps of the whole algorithm are as follows: (1) Fit of a spectrum of the series to determine Debye-Waller factors i for the ligand atoms and the energy shift E0 (2) Eigenanalysis of the EXAFS spectra of mixtures, factor compression. (3) Set up a cube and insert the ligand molecule. (4) Put the metal atom at a random C Physica Scripta 2005
March 18, 2005
Time:
06:37pm
ps13000.tex
Linking Monte-Carlo Simulation and Target Transformation Factor Analysis
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Fig. 1. (a) Speciation for [U(VI)] = 0.05 mol/L, [ac] = 1.0 mol/L, I = 1 mol/L (complex stability constants from [12]); (b) (c) content of data matrix D and their abstract reproduction using abstract spectra Rabs and (d) abstract concentrations Cabs .
position in the cube. (5) Calculate the distances Ri between metal atom and the ligand atoms. (6) Calculate the theoretical EXAFS spectrum xtest using i , Ri , and E0 . (7) Calculate xpred using the TFA procedure. (8) Determine 2 between xtest and xpred and normalize 2 to the variance xpred xpred , save the best normalized 2 . (9) Go to step 4 and repeat m times. (10) Put the metal atom at the position of the lowest normalized 2 , divide edge length of the cube by a factor, shift the center of the cube to the position of the metal atom, go to step 4 if the edge length of the cube is greater than the expected uncertainty in atomic distances otherwise stop. (11) The metal atom has reached the optimum position towards the ligand. The algorithm has two advantages: (1) The TFA identifies also the spectrum of the pure complex xtest as a component of the spectral mixtures, therefore the algorithm is sensitive to the pure metal complexes. (2) Only the metal atom is moved during the MC simulation, which drastically reduces the computing time in relation to methods which move the whole ligand. This is especially important when large ligands are investigated.
Fig. 2. Structures of uranyl hydrate and the U(VI)/ac complexes.
Table I. EXAFS Structural parameters for the 1 : 3 U(VI)/ac complex (top) and MCTFA result (bottom; bold – determined, ∗ constrains). Atom
R
N
2 ∗ 103
1 : 3 U(VI)/ac complex
Oax Oeq C1 C2
1.78 2.47 2.87 4.39
2 6.0 3.1 3.1
1.4 8.5 3.8 3.8
U(VI)-carboxylate (MCTFA)
Oax Oeq C1 C2
1.78∗ 2.46 2.84 4.34
2∗ 4∗ 2∗ 2∗
1.7∗ 9.5∗ 4.0∗ 4.0∗
4. Results and discussion In the pH interval 0.10-4.48, the U(VI)/ac system contains four complexes (Fig. 1a). Our previous study using ITFA [3, 5] has shown that only two spectroscopic components (i.e. factors, Fig. 1 b–d) are required to reproduce all spectra: the spectra of uranyl hydrate (UO2 (H2 O)2+ ) and of the 1 : 3 U(VI)/ac complex, while the 1 : 1 and the 1 : 2 complexes can be reconstructed from them (Fig. 2). From a structural point of view, these two spectroscopic units correspond to uranyl coordinated to a water molecule and uranyl C Physica Scripta 2005
R – atomic distance [Å], N – coordination number, 2 – Debye-Waller factor [Å2 ]. Physica Scripta T115
March 18, 2005
914
Time:
06:37pm
ps13000.tex
A. Rossberg and A. Scheinost and accounted for the twofold degenerated 4-legged multiple scattering path of the uranyl chain in our calculation. Using these constraints, we achieved the structure of the uranyl-carboxylate unit in 110.000 iterations. The resulting U-Oeq distances are identical to those found by conventional EXAFS data analysis of the 1 : 3 U(VI)/ac complex considering an error of 0.02 Å (Table I, Fig. 3). Thus, we are confident that MCTFA may be successfully applied to solve the local structures of a wide range of complexes based solely on the ligand structure, which is often known. Thus, the newly developed method could be extremely useful to decipher the structure of metal complexes in environmental samples where a wide range of organic ligands, from simple organic acids to large biopolymers may be present. Acknowledgement We thank Tobias Reich for the preparation of the U(VI)/acetic acid samples and for fruitful discussions during the ITFA investigation.
Fig. 3. MCTFA result. Brighter balls indicate U positions with the highest likelihood (best fits), darker balls indicate positions resulting in poorer fits. Structural data for the optimum position (black dot) are given in Table I.
coordinated to a carboxylic group in a bidentate arrangement. For all complexes only the number of this two structural units vary and the short-range structure as determined by EXAFS remains invariable (Table I). Note that this result was obtained by ITFA [3, 5]. In the following we demonstrate, that we can achieve the same uranyl-carboxylate structure by MCTFA, even when we use a much smaller pH range of 0.10–2.69, which does not include the pure end-member spectrum, and with a more restricted chi-range of 2.9–12.3 Å−1 , representing the typically poorer data quality of environmental samples. For the calculation of the theoretical EXAFS spectrum, we used the distance U-Oax , 2 of Oax and Oeq , and E0 determined from the spectrum at pH 2.69. Furthermore, we kept the distances of the two axial oxygen atoms constant,
Physica Scripta T115
References 1. Wasserman, S. R., J. Phys. IV France 7, C2 (1997). 2. Scheinost, A. C., Kretzschmar, R., Pfister, S. and Roberts, D. R., Environmental Sci. Technol. 36, 5021 (2002). 3. Roßberg, A., Doctoral Thesis, Technical University Dresden, (Dresden 2002). 4. Roßberg, A., Reich, T. and Bernhard, G., Analytical Bioanalytical Chem., online publication 6/17/2003 (2003). 5. Reich, T. and Roßberg, A., Physica Scripta this issue (2003). 6. Scheinost, A. C., et al., Physica Scripta this issue (2003). 7. Templeton, D. H., Zalkin, A., Ruben, H. and Templeton, L. K., Act. Cryst. C41, 1439 (1985). 8. McGreevy, R. L., J. Phys.: Condens. Matter 13, 877 (2001). 9. Winterer, M., J. Appl. Phys. 88, 5635 (2000). 10. Benefatto, M., Congiu-Castellano, A., Daniele, A. and Della Longa, S., J. Synchr. Rad. 8, 267 (2001). 11. Malinowski, E. R., “Factor Analysis in Chemistry” (John Wiley & Sons, New York 1991). 12. Ahrland, S., Acta Chem. Scand. 5, 199 (1951).
C Physica Scripta 2005
Time:
06:37pm
ps13000.tex
Physica Scripta. Vol. T115, 912–914, 2005
Linking Monte-Carlo Simulation and Target Transformation Factor Analysis: A Novel Tool for the EXAFS Analysis of Mixtures A. Rossberg* and A. Scheinost Institute of Radiochemistry, Forschungszentrum Rossendorf, D-01314 Dresden, Germany The Rossendorf Beamline, ESRF, F-38043 Grenoble, France Received June 26, 2003; accepted June 28, 2004
pacs number: 6110HT
Abstract Aqueous metal complexes are important chemical species in the geosphere, since they often determine the mobility of metals in the environment. While EXAFS spectroscopy is the method of choice to determine the structure of such species in environmental matrices, it often fails when several complexes coexist. In this case, all present complexes contribute to the EXAFS signal, and the deconvolution into single contributions is difficult. We have developed a new algorithm to determine the spectral contributions of the single in mixtures of species, based on a combination of Monte-Carlo Simulation and Target Transformation Factor Analysis (MCTFA). This novel approach requires solely the structure of the interacting ligand as input data, enabling the determination of complexes even in “messy” environmental samples.
1. Introduction Many attempts have been made in the past to determine structure, chemical behavior and relative distribution of aqueous metal complexes at a given set of geochemical conditions (defined by pH; Eh; concentrations of aqueous ions, solids, gasses; temperature; etc.). EXFAS spectroscopy is one of the most versatile and powerful techniques to determine the often low concentrations in the presence of complex backgrounds typical for geochemical settings. However, if more than one metal complex exists, then the EXAFS signal is the superposition of the contributions of the backscattering atoms of all metal complexes, and only average structural parameter (i.e. radial distances, coordination numbers) can be obtained. In case the EXAFS spectra of isolated complexes can be measured, one can derive the identity and quantity of the complexes in mixtures by Target Transformation Factor Analysis (TFA) and/or Linear Combination Fits (LCF) [1, 2]. Especially for uranyl-organocomplexes, however, species may remain undetected because they cannot be prepared (and measured) as pure species. Under these conditions, the EXAFS spectra of the pure metal complexes may be obtained by Iterative Target Transformation Factor Analysis (ITFA) [3–6]. Starting point for ITFA is a series of EXAFS spectra measured in dependence of one ore more physicochemical parameters, which influence the speciation (relative concentration distribution of species) of the metal complexes. Although ITFA does not depend on EXAFS spectra of pure species, it still depends on speciation data, which may often be provided by thermodynamic data bases and models. If even the data of the relative species distribution are missing, however, ITFA must fail. Therefore, we propose another approach, where only the structure of the metal-binding ligand is required. Our approach varies the position of the metal in relation to the ligand by MonteCarlo Simulation (MC). For each structural configuration, the theoretical EXAFS spectrum is then calculated and submitted as * e-mail: [email protected]
Physica Scripta T115
target test vector to TFA. We have tested this approach using a series of uranyl acetate complexes at varying pH. 2. Experimental Eight samples were prepared in the pH range 0.10–4.48 at an ionic strength of 1.2 mol/L. For all samples [U(VI)] was 0.05 mol/L and the acetic acid concentration [ac] was 1 mol/L except for the sample at pH 4.48, where [U(VI)] = 0.025 mol/L and [ac] = 0.5 M. The U LIII -edge EXAFS spectra of the pH series were measured at the Rossendorf Beamline (ROBL) at the ESRF (Grenoble, France) in transmission mode at room temperature. Crystallographic data of sodium tris(acetato)dioxouranate [7] were used to create the ligand structure for the MCTFA and to calculate the theoretical phase and amplitude functions for EXAFS data analysis (FEFF6). 3. Monte-Carlo Simulation Target Transformation Factor Analysis (MCTFA) MCTFA combines the advantages of modeling and statistics. The MC simulation allows to model the three dimensional structure of the complexes by using the experimental XAS spectra and the structure of the interacting ligand as constraints [8–10]. TFA allows to check whether the resulting EXAFS spectrum of the calculated complex is a component of the series of measured EXAFS spectra [1, 3, 11]. In the following capital bold letters are matrixes, small bold letters are vectors, and italic letters are scalars. Starting point of TFA is the calculation of the abstract (abs) factor solution for the abstract reproduction of the data matrix D(r,c) = Rabs(r,c) Cabs(c,c) (D accumulates the c EXAFS spectra of complex mixtures at r measuring points, Rabs and Cabs contains eigenvectors) by using Eigenanalysis. The orthogonal factor solution consists of c eigenvectors. Only n (n – number of main components) eigenvectors are necessary to reproduce D. In the second step of TFA, n is determined using the eigenvalues in and the indicator function IND [11]. After the factor compression the dimension of Rabs is (r, n) and that of Cabs is (n, c). In the third step a EXAFS spectrum is subjected as a test vector xtest and gives a predicted (pred) vector xpred = Rabs −1R abs xtest ( stands for transponse). The spectrum xtest is a component of the spectra of the complex mixtures if it is in agreement with the predicted spectrum xpred . The spectrum xtest we calculated using MC technique. The steps of the whole algorithm are as follows: (1) Fit of a spectrum of the series to determine Debye-Waller factors i for the ligand atoms and the energy shift E0 (2) Eigenanalysis of the EXAFS spectra of mixtures, factor compression. (3) Set up a cube and insert the ligand molecule. (4) Put the metal atom at a random C Physica Scripta 2005
March 18, 2005
Time:
06:37pm
ps13000.tex
Linking Monte-Carlo Simulation and Target Transformation Factor Analysis
913
Fig. 1. (a) Speciation for [U(VI)] = 0.05 mol/L, [ac] = 1.0 mol/L, I = 1 mol/L (complex stability constants from [12]); (b) (c) content of data matrix D and their abstract reproduction using abstract spectra Rabs and (d) abstract concentrations Cabs .
position in the cube. (5) Calculate the distances Ri between metal atom and the ligand atoms. (6) Calculate the theoretical EXAFS spectrum xtest using i , Ri , and E0 . (7) Calculate xpred using the TFA procedure. (8) Determine 2 between xtest and xpred and normalize 2 to the variance xpred xpred , save the best normalized 2 . (9) Go to step 4 and repeat m times. (10) Put the metal atom at the position of the lowest normalized 2 , divide edge length of the cube by a factor, shift the center of the cube to the position of the metal atom, go to step 4 if the edge length of the cube is greater than the expected uncertainty in atomic distances otherwise stop. (11) The metal atom has reached the optimum position towards the ligand. The algorithm has two advantages: (1) The TFA identifies also the spectrum of the pure complex xtest as a component of the spectral mixtures, therefore the algorithm is sensitive to the pure metal complexes. (2) Only the metal atom is moved during the MC simulation, which drastically reduces the computing time in relation to methods which move the whole ligand. This is especially important when large ligands are investigated.
Fig. 2. Structures of uranyl hydrate and the U(VI)/ac complexes.
Table I. EXAFS Structural parameters for the 1 : 3 U(VI)/ac complex (top) and MCTFA result (bottom; bold – determined, ∗ constrains). Atom
R
N
2 ∗ 103
1 : 3 U(VI)/ac complex
Oax Oeq C1 C2
1.78 2.47 2.87 4.39
2 6.0 3.1 3.1
1.4 8.5 3.8 3.8
U(VI)-carboxylate (MCTFA)
Oax Oeq C1 C2
1.78∗ 2.46 2.84 4.34
2∗ 4∗ 2∗ 2∗
1.7∗ 9.5∗ 4.0∗ 4.0∗
4. Results and discussion In the pH interval 0.10-4.48, the U(VI)/ac system contains four complexes (Fig. 1a). Our previous study using ITFA [3, 5] has shown that only two spectroscopic components (i.e. factors, Fig. 1 b–d) are required to reproduce all spectra: the spectra of uranyl hydrate (UO2 (H2 O)2+ ) and of the 1 : 3 U(VI)/ac complex, while the 1 : 1 and the 1 : 2 complexes can be reconstructed from them (Fig. 2). From a structural point of view, these two spectroscopic units correspond to uranyl coordinated to a water molecule and uranyl C Physica Scripta 2005
R – atomic distance [Å], N – coordination number, 2 – Debye-Waller factor [Å2 ]. Physica Scripta T115
March 18, 2005
914
Time:
06:37pm
ps13000.tex
A. Rossberg and A. Scheinost and accounted for the twofold degenerated 4-legged multiple scattering path of the uranyl chain in our calculation. Using these constraints, we achieved the structure of the uranyl-carboxylate unit in 110.000 iterations. The resulting U-Oeq distances are identical to those found by conventional EXAFS data analysis of the 1 : 3 U(VI)/ac complex considering an error of 0.02 Å (Table I, Fig. 3). Thus, we are confident that MCTFA may be successfully applied to solve the local structures of a wide range of complexes based solely on the ligand structure, which is often known. Thus, the newly developed method could be extremely useful to decipher the structure of metal complexes in environmental samples where a wide range of organic ligands, from simple organic acids to large biopolymers may be present. Acknowledgement We thank Tobias Reich for the preparation of the U(VI)/acetic acid samples and for fruitful discussions during the ITFA investigation.
Fig. 3. MCTFA result. Brighter balls indicate U positions with the highest likelihood (best fits), darker balls indicate positions resulting in poorer fits. Structural data for the optimum position (black dot) are given in Table I.
coordinated to a carboxylic group in a bidentate arrangement. For all complexes only the number of this two structural units vary and the short-range structure as determined by EXAFS remains invariable (Table I). Note that this result was obtained by ITFA [3, 5]. In the following we demonstrate, that we can achieve the same uranyl-carboxylate structure by MCTFA, even when we use a much smaller pH range of 0.10–2.69, which does not include the pure end-member spectrum, and with a more restricted chi-range of 2.9–12.3 Å−1 , representing the typically poorer data quality of environmental samples. For the calculation of the theoretical EXAFS spectrum, we used the distance U-Oax , 2 of Oax and Oeq , and E0 determined from the spectrum at pH 2.69. Furthermore, we kept the distances of the two axial oxygen atoms constant,
Physica Scripta T115
References 1. Wasserman, S. R., J. Phys. IV France 7, C2 (1997). 2. Scheinost, A. C., Kretzschmar, R., Pfister, S. and Roberts, D. R., Environmental Sci. Technol. 36, 5021 (2002). 3. Roßberg, A., Doctoral Thesis, Technical University Dresden, (Dresden 2002). 4. Roßberg, A., Reich, T. and Bernhard, G., Analytical Bioanalytical Chem., online publication 6/17/2003 (2003). 5. Reich, T. and Roßberg, A., Physica Scripta this issue (2003). 6. Scheinost, A. C., et al., Physica Scripta this issue (2003). 7. Templeton, D. H., Zalkin, A., Ruben, H. and Templeton, L. K., Act. Cryst. C41, 1439 (1985). 8. McGreevy, R. L., J. Phys.: Condens. Matter 13, 877 (2001). 9. Winterer, M., J. Appl. Phys. 88, 5635 (2000). 10. Benefatto, M., Congiu-Castellano, A., Daniele, A. and Della Longa, S., J. Synchr. Rad. 8, 267 (2001). 11. Malinowski, E. R., “Factor Analysis in Chemistry” (John Wiley & Sons, New York 1991). 12. Ahrland, S., Acta Chem. Scand. 5, 199 (1951).
C Physica Scripta 2005
