Ind. Eng. Chem. Res. 2005, 44, 7345-7351
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Micronization of Phenanthrene Using the Gas Antisolvent Process: Part 2. Theoretical Study Yousef Bakhbakhi, Sohrab Rohani,* and Paul A. Charpentier* Department of Chemical and Biochemical Engineering, University of Western Ontario, London, Ontario, Canada N6A 5B9
In the second part of our study, a rigorous mathematical model was developed and simulated using Parsival for the gas antisolvent recrystallization process using phenanthrene-toluenecarbon dioxide as a model system, This model accounts for the governing physical phenomena, i.e., the thermodynamics of near-critical solutions, and the particle formation process controlled by primary and secondary nucleation, and crystal growth. Simulations were performed for changes in the main operating parameters, i.e., the antisolvent addition rate and saturation level. The simulations were performed at a process temperature of 25 °C, while the antisolvent addition rate (QA) was varied between 1 and 100 mL/min, and the initial solute concentration was varied between 25% and 100% of the concentration ratio. The model was successfully able to predict/represent the experimental observations phenomenologically. It was shown that the simulation findings were consistent with the experimental results, and good quantitative agreement was achieved. 1. Introduction The manufacturing of particles with controlled size and size distributions has attracted significant interest in the scientific and industrial communities with applications for pharmaceuticals, food, nutraceuticals, chemical, paint/coating, and polymer industries.1-6 The important properties of these products are a narrow particle size distribution (PSD), a uniform morphology, and enantiomeric purity.7,8 The employment of supercritical fluid techniques has attracted considerable interest as an emerging “green” technology for the formation of particles in these size ranges.9,10 Particle formation using supercritical fluids (SCFs) can be performed using to several different techniques, including antisolvent techniques such as the gas-antisolvent (GAS) process. Antisolvent techniques exploit the low solubility of most compounds in the antisolvent, in particular CO2, which has to be miscible with the organic solvent.11,12 In the GAS process, high-pressure CO2 is injected into the liquid-phase solution, which causes a sharp reduction of the solute solubility in the expanded liquid phase. As a result, precipitation of the dissolved compound occurs. The potential advantages of the GAS recrystallization process lies in the possibility of obtaining solvent-free, micrometer and submicrometer particles with a narrow size distribution.13 By varying the process parameters, the particle size, size distribution, and morphology can be “tuned” to produce a product with desirable qualities. This makes the GAS technique attractive for the micronization of high-valued products, such as pharmaceuticals.6 Unfortunately, only a limited number of theoretical studies have been undertaken to develop the understanding and to achieve a quantitative description of the GAS process. The theoretical studies that have investigated the antisolvent processes primarily have consisted of phase * To whom correspondence should be addressed. Tel.: (519) 661-3466 (P.A.C.); (519) 661-4116 (S.R.); Fax: (519) 661-3498. (P.A.C.; S.R.) E-mail:
[email protected] (P.A.C.);
[email protected] (S.R.).
equilibria calculations. Muhrer et al.,14 to rationalize the precipitation kinetics in the GAS process, presented a model that accounts for solution thermodynamics and particle formation and growth. The model describes and explains the effect of the antisolvent addition rate on the average particle size and the particle size distribution. The developed model was constructed under the assumption of instantaneous phase equilibrium of the vapor and liquid phases upon antisolvent addition (i.e., there is no mass-transfer resistance). Their results showed that the model correctly predicts the variation of particle size and particle size distribution with the main operating parameter, the antisolvent addition rate. Their findings also demonstrated the possibility of adjusting the antisolvent addition rate in accordance with the final product specifications. Elvassore et al.15 proposed a population balance model to study the kinetics involved in the GAS process. The developed model accounts for particle nucleation, growth, aggregation, and settling. Nucleation and growth were represented by the McCabe model. To evaluate their findings, they developed an ultraviolet-visible (UVVis) spectroscopy-based technique to study the precipitation kinetics of a biodegradable polymer poly(L-lactide) acid (PLA) from dichloromethane using supercritical CO2. Their theoretical findings showed that their model gave a correct physical interpretation of the experimental data and reasonably predicted the particle size distribution of the precipitated PLA microparticles. The objective of this work was to provide a theoretical framework for the interpretation of the experimental results presented in Part 1 of this study. For a better understanding of the particle formation dynamics of the GAS process, the effect of process parameters, namely, the antisolvent addition rate and solute concentration on the particle size and size distribution of phenanthrene, was examined. Furthermore, a mathematical model was developed to describe the elementary phenomena involved in the GAS crystallization process, i.e., the thermodynamics and the kinetics of particle forma-
10.1021/ie0502077 CCC: $30.25 © 2005 American Chemical Society Published on Web 08/20/2005
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tion that govern the process. This model was found to be useful to describe particle design and should be useful for tailoring PSDs by the GAS process. 2. Theory The aim of the mathematical modeling of the GAS process is to acquire a fundamental understanding of the crystallization mechanisms governing this unconventional crystallization technique, to determine how nucleation (primary and secondary) and growth occur during the expansion process, and to determine how this affects particle size and size distribution. To determine the nature of the crystallization kinetics in the GAS process, the developed model has to relate the volumetric expansion of the liquid phase to the dynamics of particle formation. A proposed model by Muhrer et al.14 was employed to fill the gap between the experimental results obtained from the GAS crystallization of phenanthrene from toluene using CO2 as the antisolvent, and the theoretical understanding of the particle formation mechanisms and the influence of the process parameters in the GAS process. 2.1. Expansion of the Liquid Phase. In the GAS process, the supersaturation of the solute is created by dissolving a pressurized gas as an antisolvent into the liquid phase solvent. One of the most important process stages is the volumetric expansion of the liquid phase, which can be simulated using the Peng-Robinson equation of state (PR-EOS). With this equation of state, a large number of important process parameters can be calculated, such as the equilibrium mole fractions of solvent and antisolvent, the molar volumes and the molar fluxes of the liquid, and the gas phase, as a function of the pressure and temperature. The PR-EOS is given by
P)
a RT v - b v2 + 2vb - b2
(1)
The values of parameters a and b in the equation of state are related to the corresponding pure component quantities through two mixing rules that are valid for multicomponent mixtures:16
a)
∑i ∑j xixjaij
(2)
aij ) (1 - δij)xaiaj b)
∑i ∑j xixjbij
bij ) (1 - ηij)
(
(3) (4)
)
bi + bj 2
(5)
where δij and ηij are binary interaction coefficients and ai and bi are related to the pure component critical properties.
{ [ ( ) ]}
0.45724R2Tc2 T 1+R 1ai ) Pc Tc
1/2
2
0.07780RTc bi ) Pc
(6) (7)
R ) 0.37464 + 1.54226ωi - 0.26992ωi
2
(8)
where ω is the acentric factor, whereas Tc and Pc are the pure component’s critical temperature and pressure, respectively. 2.2. Crystallization Kinetics. The driving force for crystallization is the supersaturation of the solute. In the GAS process, supersaturation increases because of the antisolvent action and decreases because of the precipitation. The nucleation and growth of the particles in the GAS process is described using the population balance equation, which describes the evolution of the PSD with time. The population balance approach to the analysis of crystallizers was formalized and presented by Randolph and Larson.17 This technique parallels other balance approaches such as material. However, what is different in the population balance is the accounting for both the size and number of particles. A number balance over a size range L and L + dL over an increment of time dt is given as
∂n ∂(nG) n + + ) B(L) - D(L) ∂t ∂L τ
(9)
where “n” represents the population density of the crystals at time “t” of a given size “L” per unit mass of solvent per unit size, G represents the growth rate of particles, B(L) represents the birth function of particles of size L at any instant by means of agglomeration and breakage, and D(L) is the death function representing the death of particles of size L by means of breakage at any instant. The equation for a semibatch crystallizer, in which particle agglomeration and breakage are not included, and particle shapes are uniform, and the sizeindependent growth rate is given as
∂(n) n d(NLvL) ∂n +G + )0 ∂t ∂L NLvL dt
(10)
The material balance on the antisolvent in the crystallizer is given by
d(NLxA + NVyA) ) QA dt
(11)
where QA is the molar flow rate of antisolvent, NL the molar holdup of the liquid phase in the crystallizer, NV the molar holdup of the gas phase in the crystallizer, xA the mole fraction of the antisolvent in the liquid phase, and yA the mole fraction of the antisolvent in the gas phase. The material balance on the solvent in the crystallizer is given by
d(NLxS + NVyS) )0 dt
(12)
where xS and yS are the mole fractions of the solvent in the liquid and the gas phase, respectively. The material balance on the solute, which relates the change in solute concentration to changes in magma density, is given by the following equations:
d(NLxP + NP) )0 dt NP )
NLvLkVm3 vP
(13) (14)
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where NP, xP, kV, vP, and m3 are the molar holdup of the solute in the solid phase, the mole fractions of the solute in the liquid phase, the volume shape factor, the molar volume of the solid solute, and the third moment of the population density function, respectively. The third moment of the population density is related to the total weight of crystals in the crystallizer and is given as
mi )
∫0L
max
Lin(L) dL
(for i ) 3)
(15)
The boundary condition for the population balance equation is the ratio of nucleation to growth rate at size zero:
n(t,0) )
B G
(for t > 0)
(16)
where B is the nucleation rate and G is the growth rate, which are given by the following equations:
B ) B′ + B′′ B′ ) 1.5D(cPNA)7/3
(if S > 1; else B ) 0)
( ) [
x
γ vP kT NA exp
B′′ )
dM4
xP vL
(19)
[ ( ) ]
γdM2 2 1 exp -π kT ln S
(20)
aV ) kam2
(21)
kT 2πηdM
(22)
D)
3
dM ) G ) kg(S - 1)g
( ) ( )( ) ]
-16π γ 3 vP 1 2 (18) 3 kT NA ln S
cP ) R′′aVD
(17)
x
vP NA
(23)
(if S > 1; else G ) 0)
(24)
where B′ is the primary nucleation rate, B′′ the secondary nucleation rate, cP the solute concentration, k the Boltzman constant, γ the interfacial tension, NA Avogadro’s number, R′′ the secondary nucleation rate effectiveness factor, aV the specific surface area, ka the surface shape factor, m2 the second moment of the population density function, D the solute diffusion coefficient, η the dynamic viscosity, dM the molecular diameter, and S the supersaturation. The supersaturation is given as
S)
sol° fsol P ) fP exp
fliq P
(25)
fsol P
[
]
vP(P - P0) RT
liq fsol° P ) fP (P0, T, x0)
(26) (27)
3. Results and Discussion 3.1. Simulation. The population balance model was implemented in Parsival, which is a dynamic simulation
Table 1. Thermodynamic and Kinetic Parameters for Model Simulations parameter
value
ref
vP interaction parameter δija CO2-toluene, δ12 CO2-phenanthrene, δ23 toluene-phenanthrene, δ13 interaction parameter ηij Boltzmann constant, k Avogadro’s number, NA interfacial tension, γ diffusion coefficient, D dynamic viscosity, η
1.512 × 10-4 m3/mol
19
0.09 0.12 0 0 1.38 × 10-23 J/K 6.022 × 1023 mol-1 1.75 × 10-2 J/m2 3.46 × 10-9 m2/s 3 × 10-4 Pa s
19 19 19 19 14 14 14
a
14
From the Peng-Robinson equation of state (PR-EOS).
package that solves complex sets of partial integrodifferential equations using a numerical algorithm of a finite-element type known as “Galerkin h-p method” with a time discretization of Rothe’s type.18 The simulations were performed for the GAS crystallization of phenanthrene from toluene, as a model system, using carbon dioxide as the antisolvent. The process temperature was kept constant at 25 °C for all of the simulations. The antisolvent addition rate, QA, was varied between 1 and 100 mL/min. The initial solute concentration was varied between 25% and 100% of the concentration ratio. The final pressure, where the antisolvent addition was terminated, is 60 bar. Particles were assumed to have a spherical shape (i.e., the volume shape factor, kV ) π/6 and the surface shape factor, ka ) π). The physical properties of solid phenanthrene employed in the implemented model are given in Table 1. The interaction parameters (δij and ηij), the Boltzmann constant (k), Avogadro’s number (NA), and the values of interfacial tension γ, the solute diffusion coefficient D, and the dynamic viscosity η, are taken from the literature and provided in Table 1, along with the corresponding references. In the employed population balance model, the secondary nucleation rate effectiveness factor (R′′) was shown to be the only parameter controlling the qualitative effect of the process variables, i.e., the antisolvent addition rate, on the particle size, and the size distribution of the final precipitate.14 The secondary nucleation rate parameter R′′ is defined as a measure of the effectiveness of the physical events leading to secondary nucleation and ultimately determines which of the two phenomena between primary and secondary nucleation is predominant. It is worth noting that the functional form of the growth rate is not a determining factor in the model behavior. Moreover, R′′ is the only parameter in the relationships describing nucleation mechanisms that cannot be estimated based on physical properties of the model system. With the only exception of R′′, kg, and g, the same set of parameters that are reported in Table 1 were used in the simulations. The optimization algorithm varies the secondary nucleation rate effectiveness parameter R′′, the growth rate parameter kg, and the growth exponent g in the model, until the error between simulated and experimental variables is minimized. It is worth noting that the final time PSD is the only variable included in the optimization objective function. Therefore, the objective is to find the value of the mentioned parameters (R′′, kg, and g) that minimizes the difference between the measured and simulated PSD. A correct match between the experimental and simulated PSDs is an indication
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Figure 1. Comparison between the experimental and simulated particle size distributions (PSDs), based on volume percentage, at antisolvent addition rates of 100, 50, and 1 mL/min. The antisolvent addition rates are indicated by an arrow for the experimental and simulation results (T ) 25 °C, concentration ratio ) 100%). Table 2. Simulation Results parameter
estimated value
R′′ kg g
8.11 × 10-17 4.49 × 10-5 m/s 1.87
of an accurate estimation of the primary and secondary nucleation rates. Experimental investigations of the GAS process have accentuated the importance of the qualitative influence of process parameters such as antisolvent addition rate and initial solute concentration in determining the particle size and size distribution of the final precipitate. In Part 1 of our study of the crystallization of phenanthrene from toluene using CO2 as the antisolvent, we found that increasing the addition rate and decreasing the solute concentration led to smaller-sized particles and possibly also to unimodal PSDs. Comparisons between the measured and simulated PSDs (in terms of volume percentage), as a function of the antisolvent addition rate (100, 50, and 1 mL/min) (100% solute concentration ratio) are displayed in Figure 1. It is apparent that the predicted and experimental PSDs match reasonably well, although small discrepancies can be observed. The estimated secondary nucleation rate parameter R′′, within the investigated level of antisolvent addition rate of 1 mL/min, as presented in Table 2, is in the range of 8.11 × 10-17. An important means of testing any model is to use it for prediction of data not used to estimate the unknown model parameters, e.g., by changing certain experimental conditions such as the antisolvent addition rate (50 and 100 mL/min), and the initial solute concentration (25% solute concentration ratio). Failure of the model (within experimental error) to predict the full PSD indicates an incorrect/incomplete kinetic scheme. Figure 2 shows the highest value attained for both the primary and secondary nucleation rates at the antisolvent addition rates of 1, 50, and 100 mL/min. At an addition rate of 100 mL/min, the maximum primary nucleation rate, B′, attained a higher order of magnitude (9.9 × 1012 m-3 s-1) than that for the secondary nucleation rate, B′′ (1.5 × 109 m-3 s-1). Thus, at this level of antisolvent addition rate, primary nucleation is much faster than secondary nucleation, whose contribution to the final unimodal particle size distribution
Figure 2. Highest value attained for both primary and secondary nucleation rates at different antisolvent addition rates; the labels A1, A2, and A3, refer to antisolvent addition rates of 1, 50, and 100 mL/min, respectively.
is much smaller. It is evident that the primary nucleation rate is more responsive to the antisolvent addition rate than the secondary nucleation rate. At the very beginning of the process, when particles do not yet exist, the dynamics of the GAS process is dominated by primary nucleation and antisolvent addition competition. The faster the addition rate, the higher the volumetric expansion rate (i.e., the supersaturation magnitude, where nucleation is initiated). Therefore, the higher the primary nucleation rate, the larger the number of nuclei and particles produced. Moreover, the larger the antisolvent addition rate, the shorter the time required for the primary nucleation to occur. As a consequence, a large number of fast growing nuclei are formed, which consume the solute concentration and reduce the level of supersaturation, and, at the same time, build up the surface area necessary to make secondary nucleation possible. The contrary phenomena happened at the low antisolvent addition rate of 1 mL/min. At this level of addition rate, primary nucleation had a lesser effect. In this case, secondary nucleation reaches higher rates than primary nucleation and, thus, has a larger role in determining the final shape of the PSD. At this level of the antisolvent addition rate, the maximum secondary nucleation rate B′′ attained a higher order of magnitude (5.8 × 106 m-3 s-1) than that for the primary nucleation rate B′ (8.9 × 105 m-3 s-1). Therefore, the primary nucleation burst forms enough particles and enough surface area to trigger secondary nucleation, whose rate, under these conditions, is large enough that the secondary nucleation burst forms more particles than the primary one. However, the two nucleation bursts produce a more-similar number of particles. Nevertheless, the two sets of particles (i.e., those formed during the first and the second burst of nucleation) are born at different times, and these grow for a longer and for a shorter time, respectively. As a consequence, the final PSD is distinctively bimodal, as shown in Figure 1. At the intermediate level of antisolvent addition rate of 50 mL/min, the difference between the maximum values attained for both primary and secondary nucleation rates is far less, in comparison to the higher addition rate (i.e., 100 mL/min). Moreover, the particles formed
Ind. Eng. Chem. Res., Vol. 44, No. 19, 2005 7349 Table 3. Simulation Results simulation runa A1 A2 A3 C1
Moments m0 (m-3) m1 (m/m3) 9.52 × 109 5.95 × 109 6.52 × 1010 4.78 × 1010
1.47 × 106 2.52 × 107 1.29 × 108 1.16 × 108
particle size distribution mean, L h (µm) 153.99 42.35 19.79 24.43
a Each simulation run is characterized by a label that indicates the type of the experiment according to the operating condition. The labels “A1”, “A2”, and “A3” correspond to the experimental runs with antisolvent addition rates of 1, 50, and 100 mL/min, respectively. The label C1 denotes the solute concentration ratio (25%).
Figure 3. Comparison between the experimental and simulated PSDs (based on volume percentage) at a solute concentration ratio of 25% (T ) 25 °C, 50 mL/min).
Figure 5. Comparisons between experimental and simulated volume-percentage-based average particle sizes.
Figure 4. Highest value attained for both primary and secondary nucleation rates at the initial solute concentration ratio of 25%.
during the second burst of nucleation have a shorter time to grow than the primary nucleation generated particles (i.e., to a smaller and to a larger size, respectively). Therefore, the produced final PSD is broader than that of the 100 mL/min addition rate, as illustrated in Figure 1. However, the maximum primary nucleation rate B′ attained a higher order of magnitude (6.1 × 1010 m-3 s-1) than that for the secondary nucleation rate B′′ (0.2 × 109 m-3 s-1), whose role in the shaping of the final unimodal PSD is moderately smaller. Comparison between the measured and simulated PSDs (based on volume percentage) at 25% solute concentration ratio (50 mL/min antisolvent addition rate) is displayed in Figure 3. It is apparent that the predicted and experimental PSDs are in a good agreement. Figure 4 shows the highest value attained for both the primary and secondary nucleation rates at the initial solute concentration ratio of 25%. At this solute concentration ratio, the maximum primary nucleation rate B′ attained a higher order of magnitude (0.95 × 1012 m-3 s-1) than that for the secondary nucleation rate B′′ (0.89 × 109 m-3 s-1), whose contribution to the final unimodal PSD is much smaller. It is apparent that the primary nucleation rate is less sensitive to the initial solute concentration than the secondary nucleation rate, because the maximum primary nucleation rate attained
a lower order of magnitude (6.1 × 1010 m-3 s-1) at a solute concentration ratio of 100%. At higher solute concentrations, the supersaturation profile has a tendency to get closer to the saturation line quickly, initiating a primary nucleation burst, and thus allows a longer time for the particles formed during the first burst of nucleation to grow (i.e., the growth mode dominates and superimposes to secondary nucleation, and thus, larger size particles with broad size distributions are produced). As shown in Table 3 of the simulation results, the PSD of the produced particles were analyzed in terms of the moments of the distribution given by eq 15. The mean of the distribution, L h , provides the average particle size:
L h)
m1 m0
(28)
Comparisons between the measured and simulated volume-percentage-based average particle size for the studied cases of the antisolvent addition rate (1, 50, and 100 mL/min at a solute concentration ratio of 100%) and a solute concentration ratio of 25% (50 mL/min antisolvent addition rate) are displayed in Figure 5. It is apparent that the predicted and experimental results match reasonably well. 4. Conclusions In this work, a theoretical model was implemented to describe the elementary phenomena involved in the gas antisolvent (GAS) crystallization process, through the thermodynamics and kinetics of particle formation
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that govern the process. It was demonstrated that the size and size distribution of the precipitated particles can be strongly influenced in the GAS process through manipulation of the process parameters, antisolvent addition rate, and concentration. It was also shown that the simulation results were consistent with the experimental results, and good quantitative agreement was achieved. The theoretical findings showed that the observed pattern of behavior could be explained by invoking differences in the relative weight of the primary and secondary nucleation rates. Furthermore, the obtained simulation results demonstrated the importance of the antisolvent addition rate and the initial saturation level to control the final particle size distribution and to tune it in accordance with the product specifications. In fact, if possible (i.e., in the case when primary nucleation is dominant or when primary and secondary nucleation are of similar importance), the average particle size of the product may be adjusted over a relatively wide range by selecting both the initial saturation level and the antisolvent addition rate accordingly. Acknowledgment We wish to acknowledge Dr. Gerhard Muhrer at Novartis Pharma AG for fruitful correspondence on the GAS modeling, as well as Dr. Michael Wulkow and Dr. Albert Mo¨nig at the enterprise CiT GmbH for the simulation package Parsival (demo-version). The authors acknowledge the financial support of the Natural Sciences and Engineering Council (NSERC) of Canada, the Canadian Foundation for Innovation (CFI) and the University of Western Ontario Academic Development Fund (UWO-ADF). Nomenclature a ) parameter in the PR-EOS [J m3 mol-2] aV ) specific surface area of the particles in the dispersion [m2/m3] b ) parameter in the PR-EOS [m3/mol] B ) nucleation rate [m-3 s-1] ci ) molar concentration of component i in the liquid phase ci ) xi/vL [mol/m3] cv ) coefficient of variation of the PSD dM ) molecular diameter [m] D ) diffusion coefficient of the solute in the liquid phase [m/s] Fi ) molar flow rate of component i from the vapor to the liquid phase [mol/s] fL2 ) pure subcooled liquid fugacity of solute [Pa] fS2 ) pure solid fugacity of solute [Pa] g ) exponent in the growth model G ) growth rate [m/s] k ) Boltzmann constant; k ) 1.38 × 10-23 J/K ka ) surface shape factor kg ) rate constant in the growth model [m/s] kij ) interaction parameter in the PR-EOS kV ) volume shape factor lij ) interaction parameter in the PR-EOS L ) particle characteristic length [m] L h ) mean of the PSD [m] mj ) jth order moment of the PSD [mj/m3] n ) population density function [m-4] NA ) Avogadro’s number; NA ) 6.022 × 10-23 mol-1 NR ) molar hold-up in the R phase [mol] P ) pressure [N/m2] PR-EOS ) Peng-Robinson equation of state
Qi ) molar flow rate of component i [mol/s] R ) universal gas constant; R ) 8.3144 J mol-1 K-1 S ) supersaturation T ) temperature [K] Tm ) melting point temperature [K] t ) time [s] vR ) molar volume of the R-phase [m3/mol] V ) volume of the precipitator [m3] xi ) mole fraction of component i in the liquid phase yi ) mole fraction of component i in the vapor phase ′ ) primary nucleation ′′ ) secondary nucleation Greek Letters R′′ ) secondary nucleation rate parameter γ ) interfacial tension [J/m2] η ) dynamic viscosity of the liquid phase [Pa s] ω ) Pitzer acentric factor Superscripts and Subscripts 0 ) reference or initial state A ) antisolvent c ) critical parameter i, j, k ) running parameters L ) liquid phase P ) product, and product (solid) phase S ) solvent V ) vapor phase
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(17) Randolph, A.; Larson, M. A. Theory of Particulate Processes, 2nd Edition; Academic Press: San Diego, CA, 1988. (18) Wulkow, M.; Gerstlauer, A.; Nieken, U. Modeling and simulation of crystallization processes using Parsival. Chem. Eng. Sci. 2001, 56, 2575-2588. (19) Dixon, D. J.; Johnston, K. P. Molecular thermodynamics of solubilities in gas antisolvent crystallization. AIChE J. 1991, 37, 1441-1449.
Received for review February 20, 2005 Revised manuscript received June 21, 2005 Accepted July 12, 2005 IE0502077
