Regular Solution Theories Are Not Appropriate for Model Compounds

Jan 12, 2011 - The solubility, melting point temperature, and enthalpy of fusion of a series of model compounds for heavy petroleum fractions were mea...
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Energy Fuels 2011, 25, 737–746 Published on Web 01/12/2011

: DOI:10.1021/ef101405t

Regular Solution Theories Are Not Appropriate for Model Compounds for Petroleum Asphaltenes Greg P. Dechaine,* Yadollah Maham, Xiaoli Tan, and Murray R. Gray Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta T6G 2V4, Canada Received October 14, 2010. Revised Manuscript Received December 7, 2010

The solubility, melting point temperature, and enthalpy of fusion of a series of model compounds for heavy petroleum fractions were measured experimentally in n-heptane, toluene, dichloromethane, and mixtures thereof. Two porphyrins (meso-tetraphenylporphyrin and octaethylporphyrin), the vanadyl analogues of the same two porphyrins, and one model asphaltene (4,40 -bis-(2-pyren-1-yl-ethyl)-[2,20 ]bipyridinyl) were selected to test the efficacy of the regular solution and the Flory-Huggins theories in predicting the solubility behavior of these complex polyaromatic molecules in organic solvents. These simple solubility models were not capable of qualitatively or quantitatively describing the solubility behavior of these highmelting solid compounds, even with the solubility parameter as an adjustable parameter. Various different methods of estimating the differential heat capacity, ΔCP, improved the fits in certain cases, although no universally acceptable method was identified. Several group contribution theories were also tested on the current measurements, and none of the methods tested was able to quantitatively match the measured solubility parameters or the melting point properties of these compounds. density, and in turn, the solubility parameter of SARA fractions. These methods rely on the extrapolation of correlations obtained with model compounds to predict the pure component properties of the larger molecules based on molecular weight. Unfortunately, the extrapolation of these correlations to high-molecular-weight fractions has not been validated against real model compounds of comparable size. Successful modeling of asphaltene solubility behavior has the added benefit of enabling the modeling and design of vanadium separation processes because the majority of vanadium (in the form of vanadyl porphyrins6) is contained within this asphaltene fraction.7,8 Vanadium leads to deactivation of both desulfurization and cracking catalysts.9 As well, the presence of vanadium compounds in product coke leads to the formation of vanadium pentoxide during combustion, posing a toxicity concern10-12 if emitted directly to the environment and a corrosion concern for turbines when used in power generation applications.13 Because asphaltenes are defined by solubility (or lack thereof), it is very likely that the inclusion of the vanadium compounds within the asphaltene fraction is driven by solubility behavior. In fact, the work by Freeman et al.14 indicates that the solubility parameter of

1. Introduction By definition, asphaltenes constitute a solubility class defined solely by their solubility in toluene and lack of solubility in paraffins. This class of compounds represents one of the more troublesome fractions of petroleum streams because of their propensity to precipitate and foul process equipment. The concentration of this fraction in bitumen and heavy oils is generally much higher, which poses a problem for the economical upgrading of these feedstocks. These problems are relevant because the world reserves of conventional light oils are dwindling and being replaced by an increasing amount of heavier feedstocks. Accurately predicting and modeling the solubility behavior of this asphaltene fraction is therefore important for the accurate and effective design of petroleum production and processing units. Several investigators1-5 have successfully modeled the precipitation behavior of asphaltenes using the regular solution and Flory-Huggins solution models. One of the main drawbacks of these types of solution models is that the results are highly sensitive to the values of the solubility parameters of the solute. Unfortunately, these values are neither well-known nor characterized for asphaltenes and various other saturates, aromatics, resins, and asphaltenes (SARA) fractions. This shortcoming is usually overcome by applying distribution functions and/or equations of state to predict the heat of vaporization,

(6) Dechaine, G. P.; Gray, M. R. Energy Fuels 2010, DOI: 10.1021/ ef101050a. (7) Barwise, A. J. G.; Whitehead, E. V. Prepr.;Am. Chem. Soc., Div. Pet. Chem. 1980, 25, 268–279. (8) Reynolds, J. G. Liq. Fuels Technol. 1985, 3, 73–105. (9) Branthaver, J. F. ACS Symp. Ser. 1987, 344, 188–204. (10) Zychlinski, L.; Byczkowski, J. Z.; Kulkarni, A. P. Arch. Environ. Contam. Toxicol. 1991, 20, 295–298. (11) Cooper, R. G. Indian J. Occup. Environ. Med. 2007, 11, 97–102. (12) Occupational Safety and Health Administration (OSHA). U.S. Occupational Safety and Health Guideline for Vanadium Pentoxide; OSHA, U.S. Department of Labor: Washington, D.C., www.osha.gov (accessed on Jan 7, 2010). (13) Power Plant Engineering by Black & Veatch; Drbal, L. F., Boston, P. G., Westra, K. L., Eds.; Springer: New York, 1996. (14) Freeman, D. H.; Swahn, I. D.; Hambright, P. Energy Fuels 1990, 4, 699–704.

*To whom correspondence should be addressed. Telephone: (780) 492-1107. Fax: (780) 492-2881. E-mail: [email protected]. (1) Akbarzadeh, K.; Alboudwarej, H.; Svrcek, W. Y.; Yarranton, H. W. Fluid Phase Equilib. 2005, 232, 159–170. (2) Akbarzadeh, K.; Dhillon, A.; Svrcek, W. Y.; Yarranton, H. W. Energy Fuels 2004, 18, 1434–1441. (3) Alboudwarej, H.; Akbarzadeh, K.; Beck, J.; Svrcek, W. Y.; Yarranton, H. W. AIChE J. 2003, 49, 2948–2956. (4) Yarranton, H. W.; Masliyah, J. H. AIChE J. 1996, 42, 3533–3543. (5) Cimino, R.; Correra, S.; Sacomani, P. A.; Carniani, C. Thermodynamic modelling for prediction of asphaltene deposition in live oils. Proceedings of the Society of Petroleum Engineers (SPE) International Symposium on Oilfield Chemistry; San Antonio, TX, Feb 14-17, 1995; pp 499-512. r 2011 American Chemical Society

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Figure 1. Chemical structures of the model compounds used in this study.

the solvent does play a significant role in the solubility of vanadyl porphyrins. The objective of this study was to determine the equilibrium solubility of a model asphaltene compound as well as some model porphyrins at ambient (20 °C) conditions. These data were then used to test the validity of various modeling methods (regular solution theory, Flory-Huggins theory, and group contribution theory) for predicting the solubility of such large polyaromatic and heteroatomic compounds.

2.3. Compositional Analysis. The composition of each liquid solution was measured using ultraviolet-visible (UV-vis) spectroscopy. The tubes were removed from the tube roller and allowed to settle for a minimum of 1 day prior to analysis. A sample of the saturated liquid was taken with a 100 μL gastight syringe, being careful not to disturb the solids on the bottom of the tube. This sample was diluted such that the absorbance was within the linear calibration range for the analyte. The dilution ratio was quantified both volumetrically and gravimetrically, and the measured concentrations represent the mean of these two methods. The diluted solutions were scanned in a 10 mm cuvette at 20 ( 2 °C using a SI-Photonics (Tucson, AZ) model 440 spectrophotometer. A deuterium light source was used from 200 to 460 nm, and a tungsten light source was used from 460 to 950 nm. Quantitative measurement of the porphyrin concentration used a baseline correction method similar to that by Freeman et al.14 to correct for a sloping and/ or variable non-zero baseline. Details of the calibration curves for all analytes are summarized in the Supporting Information. 2.4. Temperature and Enthalpy of Melting Measurements. The melting points and enthalpies of fusion were measured using a TA Instruments (New Castle, DE) model Q1000 differential scanning calorimeter (DSC). The DSC was calibrated with indium, tin, and zinc standards. The samples were prepared by weighing 8-15 mg of solid sample into the aluminum pan of the instrument and hermetically sealing it to limit the exposure of the sample with air during the measurements. All measurements were performed using a heating rate of 5 °C/min.

2. Experimental Section 2.1. Chemicals. The four model porphyrin compounds (see Figure 1) used in this study were obtained from Sigma-Aldrich: meso-tetraphenylporphyrin (Sigma-Aldrich 247367, abbreviated as H2TPP), octaethylporphyrin (Sigma-Aldrich 252409, abbreviated as H2OEP), vanadyl meso-tetraphenylporphyrin (SigmaAldrich 283649, abbreviated as VOTPP), and vanadyl octaethylporphyrin (Sigma-Aldrich 363715, abbreviated as VOOEP). Both H2TPP and H2OEP were used as received, with stated purities of 99.9 and 97.3%, respectively. VOTPP and VOOEP were both purified by flash chromatography on silica gel columns to >99.5% purity. The model asphaltene compound, 4,40 -bis-(2-pyren-1-yl-ethyl)-[2,20 ]bipyridinyl (PBP; see Figure 1), was synthesized as per literature methods.15 The solvents were all HPLC-grade and used as obtained from Fisher Scientific. 2.2. Solubility Measurements. The solubility of each compound was measured by placing an excess of solid in a 16  100 mm glass tube with a Teflon-lined screw cap. The tube was filled with 5 mL of the desired solvent mixture and capped tightly. The tubes were subjected to simultaneous heating and sonication to speed up the dissolution process. Toluene þ n-heptane solutions were sonicated at 50-60 °C, while dichloromethane (DCM) þ n-heptane mixtures were sonicated at 35 °C, both for a 6-8 h duration. The tubes were then cooled to ambient temperature (20 ( 2 °C) and continuously agitated in a tube roller to allow the solids to slowly recrystallize. This heated sonication treatment was repeated at least 3 times for each tube to ensure that the solution had reached equilibrium. The tubes were given a minimum of 1 week at ambient temperature following the final heated sonication treatment prior to sampling and analysis of the liquid composition. The sampling and analysis procedure (see section 2.3) was carried out following each course of heated sonication to test whether or not the liquid solution had reached its saturation point. If the measured composition deviated significantly from the previous measurement, then another course of heated sonication was performed, and the process was repeated until the liquid composition remained constant.

3. Analysis of Data 3.1. Solvent Properties. The solubility of the aforementioned model compounds was tested in two series of solvents: toluene þ n-heptane mixtures and dichloromethane (DCM) þ n-heptane mixtures. The solubility parameters of the solvent mixtures, δm, were calculated using the volume average mixing rule P P L φi δi xi vi δi i i δm ¼ P ¼ P ð1Þ φi xi vLi i

i

where φi is the volume fraction of component i, xi is the mole fraction of component i, vLi is the liquid molar volume of component i, and δi is the solubility parameter of component i. The solubility parameters of the pure solvents were estimated using the definition of the solubility parameter16 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΔHv, i - RT δi ¼ ð2Þ vLi

(15) Tan, X.; Fenniri, H.; Gray, M. R. Energy Fuels 2008, 22, 715–720.

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where ΔHv,i is the enthalpy of vaporization of component i, R is the ideal gas constant (8.314 J mol-1 K-1), and T is the system temperature. The molar volume and enthalpy of vaporization for each pure solvent were obtained from the correlations given by Design Institute for Physical Properties (DIPPR)17 (see the Supporting Information for details). In the case of the binary solvent mixtures, the solvent will be treated as a pseudo-single component with properties calculated using molar averages. Equation 1 is used to calculate the mixture solubility parameter, while the mixture liquid molar volume (vLm) is calculated using eq 3. X xi vLi ð3Þ vLm ¼

the solution. Equation 6 assumes that the normal melting point is not far removed from the triple point, so that the enthalpy of fusion and temperature of fusion at the triple point (ΔHt and Tt) can be replaced by the values at the normal melting point16 (ΔHm and Tm). To apply eqs 5 and 6 to solubility data, it is necessary to estimate the activity coefficient of the solute in the liquid phase, γ2. According to the Scatchard-Hildebrand regular solution theory, the activity coefficient of the solute in the liquid phase can be estimated from pure component properties16 ln γ2 ¼

i

ð7Þ

where vL2 is the molar volume of the subcooled liquid solute at temperature T, δ1/δ2 is the solubility parameters of the solvent and solute, respectively, and φ1 is the volume fraction of the solvent. The activity coefficient of the solute can also be predicted using the Flory-Huggins theory for polymer solutions. This theory was developed for use with asymmetric polymer solutions, where the solute molecules are much larger than the solvent molecules. According to this theory, the activity coefficient of the solute is20 ! ! vL2 vL2 vL ln γ2 ¼ ln L þ φ1 1 - L þ 2 ðδ1 - δ2 Þ2 φ1 2 ð8Þ vm RT v1

These two mixing rules assume ideal mixing of the two components. Because the excess molar volumes of toluene þ n-heptane mixtures and DCM þ n-heptane mixtures do not exceed 0.118 and 0.8%,19 respectively, the use of ideal mixing to estimate solvent mixture properties is appropriate. The solvent mixtures are specified and blended by volume fractions, and the corresponding mole fractions are determined as follows: φ =vL ð4Þ xi ¼ P i i L φi =vi i

This equation also assumes ideal mixing, which, as discussed above, is a reasonable approximation for these solvent mixtures. 3.2. Solubility Modeling. The equilibrium solubility of a solid (component 2) in a liquid (component 1) is described by the following thermodynamic relation:16 f2s ¼ f2solution ¼ γ2 x2 f2°

vL2 ðδ1 - δ2 Þ2 φ1 2 RT

where vL1 is the liquid molar volume of the solvent and vLm is the liquid molar volume of the mixture. At the limit of low solute solubility (x2 , 1), the mixture molar volume approaches that of the solvent and the solvent volume fraction approaches 1 (φ1 ≈ 1), yielding the form used by Yarranton and co-workers.2-4 ! vL2 vL vL ln γ2 ¼ ln L þ 1 - 2L þ 2 ðδ1 - δ2 Þ2 ð9Þ v1 v1 RT

ð5Þ

where fs2 is the fugacity of the pure solid solute, γ2 is the activity coefficient of the solute in the liquid phase referenced to f2°, x2 is the mole fraction of the solute in the liquid phase, and f°2 is the standard state fugacity to which γ2 refers. It should be noted that eq 5 assumes that the solvent is not soluble in the solid phase, and therefore, the activity coefficient and mole fraction of the solute in the solid phase are unity. The fugacity ratio, f s2/f °, 2 resulting from a rearrangement of eq 5, can be determined using an appropriate thermodynamic cycle16 "   #   f2° ΔHm Tm ΔCP Tm Tm -1 þ ln þ1 ð6Þ ln s ¼ f2 RTm T R T T

4. Results and Discussion 4.1. Melting Point Data. To apply the regular solution and Flory-Huggins theories for solubility modeling, the melting point properties of the solutes are required. The normal melting point temperature and the enthalpy of fusion for each solid were determined and are summarized in Table 1. The measured melting point temperature for H2TPP compares well to the values reported by Rothemund and Menotti25 (450 °C; unknown experimental technique) and Bergstresser and Paulaitis26 (444 °C; DSC). The tests performed herein with DSC and thermogravimetric analysis (TGA) showed signs of decomposition upon melting, which was also noted by Rothemund and Menotti.25 This behavior

where ΔHm is the enthalpy of fusion of the solid, Tm is the absolute normal melting temperature, ΔCP = CLP - CSP is the difference between the heat capacity of the liquid and solid at the normal melting point, and T is the absolute temperature of the solution. Equation 6 was derived by choosing the standard state fugacity, f°, 2 to be the fugacity of the pure subcooled liquid solute at the same temperature and pressure as

(20) Walas, S. M. Phase Equilibria in Chemical Engineering; Butterworth Publishers: Boston, MA, 1985. (21) Fleischer, E. B.; Miller, C. K.; Webb, L. E. J. Am. Chem. Soc. 1964, 86, 2342–2347. (22) Drew, M. G. B.; Mitchell, P. C. H.; Scott, C. E. Inorg. Chim. Acta 1984, 82, 63–68. (23) Lauher, J. W.; Ibers, J. A. J. Am. Chem. Soc. 1973, 95, 5148–5152. (24) Molinaro, F. S.; Ibers, J. A. Inorg. Chem. 1976, 15, 2278–2283. (25) Rothemund, P.; Menotti, A. R. J. Am. Chem. Soc. 1941, 63, 267– 270. (26) Bergstresser, T. R.; Paulaitis, M. E. ACS Symp. Ser. 1987, 329, 138–148.

(16) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid Phase Equilibria, 2nd ed.; PrenticeHall, Inc.: Englewood Cliffs, NJ, 1986. (17) Design Institute for Physical Properties (DIPPR). DIPPR Project 801;Full Version; DIPPR, American Institute of Chemical Engineers (AIChE): New York, 2008. (18) Holzhauer, J. K.; Ziegler, W. T. J. Phys. Chem. 1975, 79, 590– 604. (19) Bissell, T. G.; Okafor, G. E.; Williamson, A. G. J. Chem. Thermodyn. 1971, 3, 393–399.

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Table 1. Crystal Density, Melting Point Temperature, and Enthalpy of Fusion for the Model Compounds Studied solid

molecular weight (g/mol)

H2TPP VOTPP H2OEP VOOEP PBP a

crystal density (g/cm3) a

614.74 679.66 534.78 599.70 612.77

1.34 1.31b 1.19c 1.25d 1.34e

melting point, Tm (°C)

enthalpy of fusion, ΔHm (J/g)

453 511 343 352 233

62.4 60.9 71.0 63.7 85.1

From ref 21. b From ref 22. c From ref 23. d From ref 24. e From ref 15.

Table 2. Measured Equilibrium Solubility of Vanadyl Porphyrins and PBP at 20 ( 0.2 °C equilibrium solubility at 20 °C, S (g/L)a

solvent compositionb (vol %)

δ20 °Cc (MPa1/2)

PBP

H2TPP

H2OEP

VOTPP

VOOEP

n-heptane 20% toluene 40% toluene 60% toluene 80% toluene toluene 40% DCM 60% DCM 80% DCM DCM

15.4 16.0 16.6 17.2 17.8 18.4 17.4 18.5 19.5 20.6

0.0110 ( 0.0006 0.089 ( 0.006 0.240 ( 0.002 0.480 ( 0.004 0.60 ( 0.01 1.19 ( 0.02 0.86 ( 0.01 1.82 ( 0.06

0.029 ( 0.001 0.0629 ( 0.0004 0.198 ( 0.001 0.533 ( 0.001 1.34 ( 0.01 2.68 ( 0.02 0.879 ( 0.004 2.212 ( 0.007 3.99 ( 0.03 5.48 ( 0.04

0.016 ( 0.001 0.042 ( 0.002 0.073 ( 0.002 0.124 ( 0.001 0.171 ( 0.001 0.202 ( 0.002 0.513 ( 0.002 1.13 ( 0.01 1.541 ( 0.008 1.76 ( 0.01

0.0008 ( 0.0001 0.0045 ( 0.0006 0.0167 ( 0.0008 0.0538 ( 0.0002 0.154 ( 0.002 0.355 ( 0.003 0.194 ( 0.002 0.627 ( 0.003 1.36 ( 0.03 2.12 ( 0.03

0.0112 ( 0.0003 0.0547 ( 0.0007 0.139 ( 0.001 0.32 ( 0.01 0.639 ( 0.005 1.20 ( 0.02 7.58 ( 0.07 19.2 ( 0.3

4.4 ( 0.7

a

The uncertainty values represent the standard deviations of multiple measurements (minimum of 4). b The composition indicates the percentage of the strong solvent, with the balance made up by n-heptane. c This is the solubility parameter of the mixture calculated using the volume average mixing rule (eq 1). The solubility parameters of the pure solvents were estimated using the correlations for molar volume and enthalpy of vaporization given by DIPPR.17

makes extracting the value of enthalpy of melting from the DSC curve difficult and, therefore, leads to additional uncertainty in the measured value for ΔHm. As was the case for H2TPP, the DSC and TGA results with VOTPP showed signs of decomposition upon melting, making it difficult to extract an accurate value for the enthalpy of melting from the DSC curve. Unfortunately, no values were available in the literature for a comparison of either the temperature or enthalpy of fusion for VOTPP. The measured melting point temperature for H2OEP is higher than the values reported by Whitlock and Hanauer27 (324-325 °C; unknown experimental technique) and Eisner et al.28 (318 °C; unknown experimental technique). Unfortunately, purity is not stated in either of those studies, and impurities could result in a reduced melting temperature. No values were available in the literature for a comparison of either the temperature or enthalpy of fusion for VOOEP. Unlike the meso-tetraphenylporphyrin samples, the DSC and TGA results for both octaethylporphyrins did not show signs of decomposition upon melting, and therefore, the values obtained for ΔHm are more reliable. No values were available in the literature for a comparison of either Tm or ΔHm for PBP. The DSC and TGA results showed no signs of decomposition upon melting, and therefore, the measured value for ΔHm is considered reliable. 4.2. Solubility Data. The measured solubilities of the five compounds in various solvent mixtures are summarized in Table 2. Each data point represents the mean of at least four replicates of the entire sampling, dilution, and analysis procedure. For all of the compounds, the solubility in toluene was several orders of magnitude higher than in n-heptane, which indicates that all five of these compounds meet the

operational definition of asphaltenes: they are “soluble” in toluene and “insoluble” in n-heptane. The solubility measurements obtained in this work for H2TPP are compared to data reported in the literature in Figure 2A. Some of the polar solvents tested by other investigators29 (acetic acid, acetone, and 1,4-dioxane) showed anomalously low solubilities, although given the drastically different physical nature of these solvents, it is difficult to draw any concrete conclusions about these data points. The data exhibit the classical maxima for a plot of solubility as a function of the solubility parameter, and the measured solubility in benzene matches very well with the current measurements in toluene, indicating that the current measurements are valid. The current solubility measurements for VOOEP are compared to the values reported by Freeman et al.14 in Figure 2B. The equilibrium solubilities measured by Freeman et al. in two chlorinated solvents (DCM and chloroform) are both lower than those measured in this work for a mixture of DCM and n-heptane. The values reported by Freeman et al. are definitely higher than the solubility in toluene, but the magnitude of the increase is consistent with the observations with H2OEP. The solubility measurements performed for VOOEP in DCM mixtures are likely biased high and may not be reliable. The solubility data for VOOEP at DCM concentrations >60 vol % were very high and required excessive amounts of solid solute. As such, they were not included in this study. 4.3. Solubility Parameter Calculations for a Comparable Test Substance: Pyrene. For both of the solution theories outlined in section 3.2, if the properties and solubility of the solute (ΔHm, Tm, ΔCP, vL2 , and x2) and the properties of the

(27) Whitlock, H. W., Jr.; Hanauer, R. J. Org. Chem. 1968, 33, 2169– 2171. (28) Eisner, U.; Lichtarowicz, A.; Linstead, R. P. J. Chem. Soc. 1957, 733–739.

(29) Koifman, O. I.; Berezin, B. D.; Zelov, V. V.; Nikitina, G. E. Zh. Fiz. Khim. 1978, 52, 1782–1784. (30) Berezin, B. D.; Koifman, O. I.; Zelov, V. V.; Nikitina, G. E. Russ. J. Phys. Chem. 1978, 52, 1281–1282.

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Figure 2. Comparison of the current solubility measurements to data reported in the literature: (A) H2TPP (n-alcohol data from Berezin et al.30 and aromatics and polar solvents from Koifman et al.29) and (B) VOOEP (data reported by Freeman et al.14).

solvent (vL1 and δ1) are known, then the only remaining unknown is the solubility parameter of the solute, δ2. Given solubility measurements as a function of the solubility parameter of the solvent, it is possible to determine δ2 by nonlinear regression. Before analyzing the measured solubility data for the model compounds, this regression methodology was tested on a solid solute for which all of the pertinent data are available: pyrene. Because pyrene is a relatively high melting compound (Tm = 150.7 °C31) with a fused-ring aromatic structure, its behavior should be a good representation of the behavior of the model compounds in this study. The solubility of pyrene in toluene þ n-heptane mixtures was measured experimentally by Ali et al.32 at 293 K (see the Supporting Information), and these data were used to test the regression scheme. The pure component properties of pyrene are summarized in Table 3. Experimental values were used where available; however, because pyrene is a solid at room temperature with a melting point far removed from 20 °C, the value of the subcooled liquid volume (vL2 ) was an estimate rather than an actual measurement. A value of vL2 = 0.178 m3/kmol was chosen because it coincides with the value given by Wakeham et al.,33 as well as the value obtained by assuming that the volume change upon melting, Δvm = vL - vS, remains constant between the normal melting point and 20 °C (Δvm = 18.9 cm3/mol34). This estimate should provide an upper limit for vL, with the solid molar volume providing a lower limit. If this value of vL2 and the enthalpy of vaporization listed by Roux et al.35 are combined with eq 2, then the predicted value for δpyrene is 22.1 MPa1/2.

Table 3. Pure Component Properties for Pyrene at 293 K property

value

source

Tm ΔHm ΔCP vL vS ΔHv, 298 K δ293 K

423.81 K 17.36 kJ/mol -23.39 J mol-1 K-1 0.178 m3/kmol 0.1591 m3/kmol 89.4 kJ/mol 22.1 MPa1/2

a a a b c d e

a From ref 31. b Estimated; see the Results and Discussion. c From ref 36. d From ref 35. e Estimated; see the Results and Discussion.

Because eqs 5, 7, and 8 cannot be linearized in δ2, the fitting procedure is iterative. The solver function in Microsoft Excel was used to vary the value of δ2 to minimize the following objective function for the pyrene modeling: X meas 2 objective function ¼ ðxcalc Þ ð10Þ 2 - x2 The results of the best fits obtained using the pure pyrene properties in Table 3 are shown in Figure 3, and the resulting solubility parameters are summarized in Table 4. Both models (RS and FH) are able to qualitatively capture the solubility trend, although the data exhibit some anomalous scatter between 16 and 16.5 MPa1/2. A similar sensitivity analysis was performed to assess the impact of the value of the differential heat capacity, ΔCP, on the regression results. One assumption that is commonly made when applying these equations is to assume that the differential heat capacity is negligible and can be ignored.16 The calculations for pyrene were also performed with ΔCP = 0, and the results are included in Table 4. Neglecting this term entirely results in an increase in the calculated solubility parameter with a corresponding decrease in the quality of the fit (as indicated by R2). The value of the solubility parameter for pyrene at 20 °C predicted by the solubility modeling varied between 20.8 and 21.6 MPa1/2. These values do not compare well to the value listed in Table 3, which was calculated directly from eq 2 (22.1 MPa1/2). Although an error of 1-1.3 MPa1/2 may seem like a relatively small discrepancy, the predicted solubility

(31) Wong, W. K.; Westrum, E. F., Jr. J. Chem. Thermodyn. 1971, 3, 105–124. (32) Ali, S. H.; Al-Mutairi, F. S.; Fahim, M. A. Fluid Phase Equilib. 2005, 230, 176–183. (33) Wakeham, W. A.; Cholakov, G. S.; Stateva, R. P. J. Chem. Eng. Data 2002, 47, 559–570. (34) McLaughlin, E.; Ubbelohde, A. R. Trans. Faraday Soc. 1957, 53, 628–634. (35) Roux, M. V.; Temprado, M.; Chickos, J. S.; Nagano, Y. J. Phys. Chem. Ref. Data 2008, 37, 1855–1996.

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weight of the solute (g/mol). Equation 12 assumes that the solution is dilute (x2 , 1) and that the solute has a negligible effect on the molar volume of the mixture (i.e., vLm = vL1 ). 4.4.1. meso-Tetraphenylporphyrins (H2TPP and VOTPP). The results of modeling the solubility of the two mesotetraphenylporphyrins using both the regular solution (RS) and Flory-Huggins (FH) theories are shown in Figure 4, and the resulting solubility parameters are listed in Table 5. Neither theory does a good job of completely capturing the observed solubility behavior for these two compounds when the heat capacity difference is assumed to be zero (ΔCP ≈ 0; base case for the modeling). The FH theory is able to capture the trend in toluene solutions but overpredicts the solubility in DCM by an order of magnitude. The inability of the regular solution theory to capture the solubility trends of these compounds is a result of the relatively high values for both the melting point temperatures and the enthalpies of fusion. The ideal solubility for a solute occurs when the activity coefficient is unity, yielding the following equation for mole fraction solubility: !   1 ΔHm Tm -1 ð13Þ ln ideal ¼ RTm T x2

Figure 3. Results of modeling the pyrene solubility data by Ali et al.32 using both the regular solution (RS) and Flory-Huggins (FH) theories. The regression curves represent the least-squares best fit of the data using the pyrene properties in Table 3 (base case in Table 4). Table 4. Sensitivity Analysis on the Effect of Assuming a Negligible Differential Heat Capacity, ΔCP = 0 and Assuming vL ≈ vS on the Solubility Parameter Regression Results for Pyrene regular solution (RS)

Flory-Huggins (FH)

method

base

ΔCP = 0

v ≈v

base

ΔCP = 0

vL ≈ vS

δ (MPa1/2) R2

20.8 0.952

21.3 0.914

21.0 0.964

21.1 0.907

21.6 0.868

21.2 0.939

a

a

L

S

For H2TPP and VOTPP, the ideal solubilities obtained from = 8.33  10-5 and 2.40  10-5, respectively. eq 13 are xideal 2 These two values are lower than the majority of the measured solubilities. To explain the measured solubilities, the activity coefficient must be less than unity (γ2 < 1), which is not possible for the RS theory.16 This same discrepancy does not apply for the FH theory, where activity coefficients of less than unity are possible. One major source of error in the aforementioned solubility calculations was the assumption that the ΔCP term was negligible. A positive value of ΔCP would lead to an increase in the ideal solubility and improve the ability of the RS theory to fit the results. In the case of meso-tetraphenylporphyrins, where the system temperature is far removed from the melting point (Tm/T = 2.5), assuming ΔCP ≈ 0 is not a good approximation.38,39 Another common approximation is to assume that the differential heat capacity is equal to the entropy of fusion38 (i.e., ΔCP ≈ ΔSm ≈ ΔHm/Tm). The addition of more flexible side chains and/or bridges to the core of fused polyaromatic structures increases the ability of a molecule to absorb heat, thus making ΔCP ≈ ΔSm a better estimate than ΔCP ≈ 0.40 Wu and Yalkowsky37 examined various methods for predicting ΔCP and developed a simple method for predicting the value of ΔCP from the molecular structure. The effects of two values of ΔCP on the solubility modeling were tested further: ΔCP ≈ ΔSm ≈ ΔHm/Tm and ΔCP estimated from the correlation of Wu and Yalkowsky.37 The results of these calculations are included in Table 5 and Figure 4. In the case of H2TPP, the two methods yield very similar values for ΔCP, and therefore, the regression results are comparable. The quality of the fit improved significantly over the base case (ΔCP ≈ 0) for the RS theory,

Base case: ΔCP and vL from Table 3.

varies exponentially with the square of the terms containing the solubility parameter. Therefore, even small changes in the value for the solubility parameter can result in large discrepancies in the predicted solubility. 4.4. Analysis of Current Solubility Data. Now that the methodology has been tested on a solute with well-known properties, it will be applied to the solubility data listed in Table 2. Because the solubilities of the current compounds span several orders of magnitude, the objective function for the nonlinear regression was modified to eq 11 to ensure that the regression results are not as heavily influenced by the high solubility results. 2 ! !32 X 1 1 4ln - ln meas 5 ð11Þ objective function ¼ x2 xcalc 2 As mentioned in the pyrene analysis, the subcooled liquid molar volumes of the current compounds are not known, and therefore, the solid molar volume will be used as an approximation for this analysis (i.e., vL ≈ vS; see Table 1 for values). As well, the differential heat capacity (ΔCP) is not known, and therefore, all terms including this value will be neglected unless otherwise stated. The measured solubilities were determined using UV-vis calibrations in molar concentrations, and therefore, these values must be converted to mole fractions using the following equation: x2 ¼

S2 vL1 MW2

ð12Þ

(37) Wu, M.; Yalkowsky, S. Ind. Eng. Chem. Res. 2009, 48, 1063– 1066. (38) Neau, S. H.; Bhandarkar, S. V.; Hellmuth, E. W. Pharm. Res. 1997, 14, 601–605. (39) Pappa, G. D.; Voutsas, E. C.; Magoulas, K.; Tassios, D. P. Ind. Eng. Chem. Res. 2005, 44, 3799–3806. (40) Neau, S. H.; Flynn, G. L. Pharm. Res. 1990, 7, 1157–1162.

vL1

where S2 is the solubility of solute (g/L), is the molar volume of the solvent (L/mol), and MW2 is the molecular (36) Baxter, G. P.; Hale, A. H. J. Am. Chem. Soc. 1936, 58, 510–515.

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Figure 4. Comparison of the effect of different values of ΔCP on the solubility modeling of meso-tetraphenylporphyrins (H2TPP and VOTPP). (‡) Wu and Yalkowsky.37

4.4.2. Octaethylporphyrins (H2OEP and VOOEP). The results of modeling the solubility of the octaethylporphyrins using both the RS and FH theories are shown in Figure 5, and the resulting solubility parameters obtained by regression were included in Table 5. In the case of H2OEP, the RS theory does a reasonable job of describing the overall solubility trend with ΔCP ≈ 0, although it is not capable of capturing the discontinuity in solubility that occurs when switching from toluene to DCM as the strong solvent. The FH theory qualitatively captures the step change in solubility between toluene and DCM, although it is underpredicting the toluene solubility and overpredicting the DCM solubility by orders of magnitude. In the case of VOOEP, both theories do a reasonable job of describing the solubility trend in toluene with ΔCP ≈ 0, although both show signs of deviation at one end or the other of the range. As was the case for H2OEP, the FH theory qualitatively captures the discontinuity in solubility that occurs when switching from toluene to DCM as the strong solvent, while the simpler RS theory does not. The two other ΔCP estimation schemes were tested as well, and the results are included in Table 5 and Figure 5. For both theories (RS and FH), the quality of the fit worsened when the non-zero estimate of ΔCP was used. The base method (ΔCP ≈ 0) combined with the RS theory worked the best of all of the methods, although this base method was not able to account for the step change in solubility between toluene and DCM. 4.4.3. 4,40 -Bis-(2-pyren-1-yl-ethyl)-[2,20 ]bipyridinyl (PBP). The results of modeling the solubility of PBP using both the RS and FH theories are shown in Figure 6, and the resulting solubility parameters obtained by regression were included

Table 5. Sensitivity Analysis on the Effect of Different Values of the Differential Heat Capacity, ΔCP, on the Regression Results for the Compounds in This Study δ (MPa1/2) ΔCP (J mol-1 K-1)

regular solution (RS)

Flory-Huggins (FH)

case

2a

3b

1c

2a

3b

1c

2a

3b

H2TPP VOTPP H2OEP VOOEP PBP

52.8 52.7 61.6 61.1 103

47.8 27.8 63.5 43.7 46.4

18.9 19.6 20.4 19.5 19.3

21.4 22.2 22.2 21.4 20.9

21.2 21.1 22.3 20.9 20.0

19.9 20.9 21.6 20.7 20.4

22.4 23.1 23.0 22.3 21.9

22.2 22.2 23.1 21.9 21.2

a Case 2: ΔCP ≈ ΔSm ≈ ΔHm/Tm. b Case 3: ΔCP estimated using the method by Wu and Yalkowsky.37 c Case 1 (base): ΔCP = 0.

particularly for the toluene solutions. However, this simple theory still cannot describe the entire series of data. In the case of the FH theory, the quality of the fits declined significantly when a non-zero heat capacity was used. In the case of VOTPP, the quality of the fits for the RS theory improved significantly when using either estimate of ΔCP. The ΔCP value by Wu and Yalkowsky37 combined with the RS theory provided the best fit of all of the methods and, for the first time, was able to describe the entire series of solubility data, including the DCM measurements. In the case of the FH theory, the quality of the fits declined significantly when a non-zero value of ΔCP is used. The base method (ΔCP ≈ 0) worked the best with the FH theory, although this base method was only able to account for the behavior in the toluene mixtures and overpredicted the solubility in DCM by an order of magnitude. 743

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Figure 5. Comparison of the effect of different values of ΔCP on the solubility modeling of octaethylporphyrins (H2OEP and VOOEP). (‡) ΔCP estimated from the correlation by Wu and Yalkowsky.37

in Table 5. Neither theory does a good job of describing all of the observed solubility behavior for this compound when ΔCP ≈ 0. The FH theory qualitatively matches the toluene data trend but deviates by orders of magnitude for the DCM solubility. The poor performance of both models is related to the high values for the temperature and enthalpy of fusion, = 1.22  10-4. This which give an ideal solubility of xideal 2 value is lower than many of the measured solubilities and explains why the base model (ΔCP ≈ 0) cannot adequately describe the behavior. The two other ΔCP estimation schemes were tested as well, and the results are included in Table 5 and Figure 6. In the case of the RS theory, the quality of the fit improved when using either estimate of ΔCP. In the case of the FH theory, the quality of the fits declined when a non-zero value of ΔCP is used. The base method (ΔCP ≈ 0) worked the best with the FH theory, although this base method was only able to account for the behavior in the toluene mixtures and overpredicted the solubility in DCM by an order of magnitude. 4.4.4. General Discussion. The regression analysis of the pyrene solubility data indicates that the simple theories used herein (regular solution and Flory-Huggins) can qualitatively describe the solubility behavior of pyrene in n-heptane þ toluene mixtures if the solubility parameter of the solid is used as an adjustable parameter. When an estimated value for the solubility parameter is used, the predictions from both theories are not accurate and lead to a significant underprediction of the solubility. These discrepancies are not entirely surprising considering that the values for ΔHv and vL used to predict the solubility parameter are hypothetical values, which in turn means that the solubility parameter is itself a hypothetical value. It cannot have any

Figure 6. Comparison of the effect of different values of ΔCP on the solubility modeling of 4,40 -bis-(2-pyren-1-yl-ethyl)-[2,20 ]bipyridinyl (PBP). (‡) Wu and Yalkowsky.37

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physical meaning at these low-temperature conditions (Tm/ T = 1.45) because pyrene cannot exist as a liquid at this low temperature. Therefore, trying to estimate δ using thermodynamic quantities according to its definition (eq 2) is neither meaningful nor accurate. The same argument is even more applicable to the porphyrins and PBP because their melting point temperatures are even higher than pyrene. In the case of the solubility parameters obtained by the regression method, they are nothing more than curve-fitting parameters, which do not hold any physical meaning. These best fit parameters can be used to perform engineering design calculations (interpolate solubilities in n-heptane þ toluene mixtures); however, drawing specific conclusions regarding the thermodynamics of the solutes from these values is not meaningful. Also, extrapolating the regressed solubility parameters to different solvent systems is inherently risky because intermolecular forces could drastically change the nature of the solvation process and lead to significant errors in prediction (see Figure 2). One of the main advantages of using the RS and FH theories is the ability to predict solubility and solution behavior from pure component properties without the need for actual solubility data. If a regression of solubility data is required to determine values for δ, then this advantage is no longer relevant because any suitable activity coefficient method (e.g., NRTL, UNIQUAC, WILSON, etc.) could just as easily be fit to this type of data. Because the more advanced activity coefficient models are much better equipped to handle nonideal systems, it is anticipated that these models will do a much better job of describing the solubility data than the very simple RS and FH models. 4.5. Group Contribution Modeling. Numerous investigators have attempted to apply group contribution methods for estimating the solubility parameters of asphaltene-type molecules.41,42 A small cross-section of these methods will be tested on the model compounds in this study to determine if they are appropriate for the type of molecules expected in asphaltene fractions. Jaffe et al.42 applied the group contribution method by Fedors43 to estimate the solubility parameters for hypothetical asphaltene structures. The method by Fedors43 estimates the energy density (Δe = ΔHv - RT) and the liquid molar volume (vL) using a group contribution methodology. This method was intended for use with polymers but was extended by Jaffe et al.42 to asphaltenes. The results predicted by this method for some of the model compounds are summarized in Table 6. Rogel41 also used the method by Fedors to predict the solubility parameter for hypothetical asphaltene structures. These solubility parameters were then correlated to the hydrogen/carbon (H/C) ratio for the compounds, and a linear fit was obtained. This linear fit was used to predict the solubility parameters for the model compounds in this study, and the results are included in Table 6. The values for VOTPP and VOOEP were not included in the analysis because these group contribution methods do not include vanadium or vanadyl as a group. The “best fit” values included in Table 6 are the values that corresponded to the best fit of all of the methods applied based on the highest R2 value. These values were chosen because they best

Table 6. Comparison of Group Contribution Predictions of Solubility Parameters to the Best Fit Values for the Model Compounds δ (MPa1/2) compound

Fedors43

Rogel41

best fit

H2TPP H2OEP PBP pyrene

26.3 22.8 22.4 22.8

28.7 22.4 28.6 29.3

21.2a 20.4b 20.9c 21.1d

a δbest fit from RS theory þ ΔCP from Wu and Yalkowsky.37 b δbest fit from RS theory þ ΔCP = 0. c δbest fit from RS theory þ ΔCP ≈ ΔSm ≈ ΔHm/Tm. d δbest fit from base RS theory.

capture the solubility behavior of the model compounds and, as such, represent the best value for the solubility parameter. The exception was pyrene, where the base fit was chosen because this value was obtained with known pure component properties, and as such, this value of δ is a true representation of the full RS theory. Neither of these group contribution methods is able to adequately match the best fit results obtained in the preceding solubility analysis. With the exception of H2OEP, the results obtained from the method by Rogel41 are not even close to the best fit values. As discussed previously, even small differences in the value of δ can lead to large discrepancies in the predicted solubility as a result of the exponential dependence of solubility upon the square of the solubility parameter. Therefore, the values obtained by the method by Rogel are not recommended. The method by Fedors performed slightly better, although they are still much higher than the best fit values and, as such, are not recommended. The fact that the method by Fedors did not perform well is not surprising, considering the simple nature of this method. This method does not have any explicit means/groups for accounting for fused aromatic structures. Because all of these model compounds contain significant aromatic character, this is a major weakness for this method when applied to asphaltenes. Several other more advanced group contribution methods were considered for modeling the solubility parameters of these model compounds.44-47 However, all of these models lacked the necessary groups to correctly describe the structures of the model compounds. In particular, these methods do not have explicit groups to take into account fused aromatic structures nor do they have groups to take into account aromatic nitrogen structures, such as pyridine or pyrrole, both of which are abundant in the current model structures. Consequently, these models could not be used. One group contribution method that does take into account fused aromatic structures as well as various forms of aromatic nitrogen is the method by Marrero and Gani.48 This method uses a three-level group contribution approach, which corrects for larger molecular structures (e.g., fused aromatics and isomers). Therefore, this model should be able to represent the structures being considered herein. Unfortunately, this model does not include the liquid molar volume or the solubility parameter explicitly, and therefore, it cannot (44) Constantinou, L.; Gani, R. AIChE J. 1994, 40, 1697–1710. (45) Constantinou, L.; Gani, R.; O’Connell, J. P. Fluid Phase Equilib. 1995, 103, 11–22. (46) Stefanis, E.; Constantinou, L.; Panayiotou, C. Ind. Eng. Chem. Res. 2004, 43, 6253–6261. (47) Stefanis, E.; Panayiotou, C. Int. J. Thermophys. 2008, 29, 568– 585. (48) Marrero, J.; Gani, R. Fluid Phase Equilib. 2001, 183-184, 183– 208.

(41) Rogel, E. Energy Fuels 1997, 11, 920–925. (42) Jaffe, S. B.; Freund, H.; Olmstead, W. N. Ind. Eng. Chem. Res. 2005, 44, 9840–9852. (43) Fedors, R. F. Polym. Eng. Sci. 1974, 14, 147–154.

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Table 7. Comparison of Group Contribution Predictions of Melting Point Parameters by Marrero and Gani to the Measured Values for the Model Compounds measured compound

predicted

ΔHm(kJ/mol)

Tm (K)

38.4

726

a

H2TPP H2TPPb H2OEPa H2OEPb PBP pyrene a

38.0

616

52.1 17.4

506 424

ΔHm(kJ/mol)

percent error (%)

Tm (K)

percent error (%)

58.2 69.1 63.9 37.1 63.3 17.3

52 80 68 -2.4 21 -0.6

549 579 467 521 562 421

-24 -20 -24 -15 11 -0.7

Assumes an aromatic structure for the porphyrin backbone. b Assumes a cyclic structure for the porphyrin backbone.

be used to predict these values. It is capable of predicting the melting point properties, and these values are summarized in Table 7. Once again, VOTPP and VOOEP were not included because vanadium and vanadyl groups are not included in the method. Within the framework of the method by Marrero and Gani, the porphyrin backbone can be described in two ways: as a large fused aromatic structure or as a cyclic structure. Both of these predictions are included in Table 7. This method does a very good job of describing the pyrene properties, which is not surprising because pyrene is a fundamental building block within the method and its properties are included explicitly in the regression of the group contributions. Therefore, this method has been optimized for pyrene and should yield good results. In the case of porphyrins, this method significantly underpredicts the melting point temperatures, with the aromatic formalism yielding better results in both cases. As for the enthalpy of fusion, this method seems to overpredict the value by quite a large margin, with the exception of the aromatic formalism for H2OEP, which was very close to the actual value. The fact that this method has difficulty describing the porphyrins is not surprising. The highly conjugated porphyrin backbone obeys H€ uckel’s rule for aromaticity and can exhibit several resonance forms.49 This macrocyclic aromaticity implies that the behavior of the porphyrin backbone will be more than the sum of its individual parts, and as such, simple group contribution methods, by definition, will not suffice. In fact, the power of the method by Marrero and Gani over other methods lies in the use of second- and third-level groups to take into account such macrocyclic, molecular level effects. The absence of such a group for the unique porphyrin macrocycle, therefore, makes predicting its properties difficult. Unless the method takes this into account explicitly (which it does not), then it is unlikely that it will be able to accurately model its behavior. In the case of PBP, this model does a better job than it did for the porphyrins. Again, this is expected because pyrene and its various substitutions have been accounted for directly by this method. However, the properties of PBP were still only predicted to within an order of magnitude [ΔTm = þ56 °C; Δ(ΔHm) = 11.2 kJ/mol], and therefore, this method cannot be relied upon for anything more than orders of magnitude estimates for asphaltene molecules.

operational definition of an asphaltene (“soluble” in toluene and “insoluble” in n-heptane), and hence, any discussion of native petroporphyrins is intrinsically tied to that of asphaltenes. The relatively simple RS and FH theories, which are so prevalent in the literature for modeling the solution behavior of asphaltenes, had difficulty capturing the observed solubility behavior of the pure compounds in this study. This is not surprising because the main parameter used to capture the behavior, the solubility parameter, is a hypothetical construct for these molecules at the conditions considered. With the addition of this to the fact that the reference point (the normal melting point) is far removed from the actual system conditions, these models quickly degrade. The use of the hypothetical subcooled liquid as the reference state for these systems represents a huge extrapolation and is likely not a very appropriate choice for such high melting compounds. The results were no better when group contribution methods were employed to attempt to predict either the solubility parameters or the melting point properties of these complex structures. Very few group contribution methods contain enough detail to adequately capture the structures that were present in the present compounds (in particular, the aromatic and pyrrole structures). Of the methods that did contain sufficient detail or have been applied in the past to asphaltene structures, none of the methods tested were capable of predicting the properties of these compounds. The combination of all of these factors indicates that these simple models are not recommended as a rigorous treatment for the solution behavior of these model compounds or any such complex, high melting compounds (e.g., asphaltenes). The best fit solubility parameters obtained herein represent an empirical correlation that may be adequate for engineering design calculations, provided that all of the other parameters are kept the same as used in the fitting procedure (i.e., ΔCP, ΔHm, and vL). However, to attempt to draw any specific conclusions from the solubility parameters obtained herein or to extend these parameters to other models would not be recommended because these values do not necessarily have a sound thermodynamic basis.

On the basis of the equilibrium solubility in toluene and n-heptane, the simple model porphyrins in this study fit the

Acknowledgment. The authors are grateful for the financial support of the Centre for Oil Sands Innovation for funding this project and also the Natural Sciences and Engineering Research Council of Canada (NSERC) in the form of a scholarship for Greg Dechaine. We thank Brenden Boddez for his help with the chromatographic separations and the UV-vis calibration work. We also thank Tuyet Le, Dr. Cindy Yin, Andree Koenig, Dr. James Dunn, and Dr. Cornelia Bohne for their insightful technical discussions.

(49) Smith, K. M. General features of the structure and chemistry of porphyrin compounds. In Porphyrins and Metalloporphyrins; Smith, K. M., Ed.; Elsevier Scientific Publishing Company: Amsterdam, The Netherlands, 1975; pp 1-28.

Supporting Information Available: UV-vis calibrations for porphyrins and PBP, solvent property correlations, and pyrene solubility data. This material is available free of charge via the Internet at http://pubs.acs.org.

5. Conclusions

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