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The infinite dilution binary diffusion coefficients (D12) of acetylferrocene and 1,1′-diacetylferrocene were measured in supercritical carbon dioxide (scCO2) at (313, 353, 363 and 373) K and at pressures from (11 to 26) MPa using the chromatographic impulse response method, and in atmospheric-pressure liquid organic solvents such as acetone at 303 K, methanol at (313, 323, and 333) K and ethanol at (323 and 333) K using the Taylor dispersion method. The D12 values for both solutes were approximately consistent under all of the investigated measurement conditions and were well represented by the hydrodynamic equation, in which D12/T is expressed as a function of solvent viscosity, over almost the entire range of investigated viscosities ranging from the viscosities of liquid organic solvents to that of scCO2. At 313 K and 11 MPa, the D12 values for both the organometallic solutes deviated from the correlation of the D12 values with the solute molecular mass for nonmetallic organic compounds reported in the literature.

The diffusion coefficients of chromium(III) acetylacetonate (Cr(acac)3) were measured at infinite dilution in supercritical carbon dioxide and in liquid ethanol. The diffusion measurements in supercritical carbon dioxide were carried out at five different temperatures varying from T = (308.15 to 343.15) K and at pressures up to 40.00 MPa by using the chromatographic impulse response (CIR) method. And the measurements in liquid ethanol were also made by the Taylor dispersion method at five temperatures in the range from T = (299.15 to 333.15) K and 0.10 MPa. The determined diffusion activation energies in supercritical carbon dioxide were Ea = (22.9–11.2) kJ·mol−1 at p = (13.00–35.00) MPa, respectively, and that in liquid ethanol was 15.9 kJ·mol−1 at 0.10 MPa. All the diffusion coefficients of Cr(acac)3 in supercritical and in liquid states can be correlated by the hydrodynamic equation as a function of temperature and fluid viscosity with average absolute relative deviation (AARD) of 4.6% over a wide viscosity range from η = (3.036·10−5 to 1.113·10−3) Pa·s for 85 data points.

The infinite dilution binary diffusion coefficients D12 for aluminum acetylacetonate in supercritical CO2 were determined at 308.15–333.15 K and at 7.80–40.00 MPa by the chromatographic impulse response (CIR) method. And, the diffusion measurements in liquid ethanol were carried out at 300.15–333.15 K and at 0.10 and 30.00 MPa by the Taylor dispersion method. It was found that the D12 values measured in supercritical CO2 show slowing down in the region of near critical point. The determined activation energies of diffusion were 23.0, 13.7, 12.0, 11.3 kJ/mol at 12.00, 20.00, 25.00, 35.00 MPa in supercritical CO2 and 19.1 kJ/mol at 0.10 MPa in liquid ethanol, respectively. All determined 90 diffusion data in this study can be correlated with the equation of D12 [m2/s] = 1.558 × 10−14 T [K] η [Pa s]−0.761 with average absolute relative deviation (AARD) of 5.6% over a wide fluid viscosity range from 2.462 × 10−5 to 1.258 × 10−3 Pa s.

Infinite dilution binary diffusion coefficients (D12) of two dyestuffs, disperse blue 14 (DB14) and disperse orange 11 (DO11), were measured in supercritical CO2 at 310–330 K and up to 35 MPa by the chromatographic impulse response (CIR) method and in liquid methanol and ethanol at 308–328 K and atmospheric pressure by the Taylor dispersion method. The retention factors (k) were also measured by the CIR method. The diffusion coefficients were well represented by a hydrodynamic equation, as a function of temperature and solvent viscosity, over a wide range of solvent viscosity from scCO2 to liquid solvents with average absolute relative deviations (AARDs) of 1.7% and 2.6% for 63 and 93 data points of DB14 and DO11, respectively. The solubility data for the two dyestuffs reported in the literature were well correlated with solute retention factor measured in the present study and solvent density with AARD of 7.05% for DB 14 of 109 data points and that of 15.4% for DO11 of 43 data points.

The infinite dilution binary diffusion coefficients D12 of lithium acetylacetonate were measured in supercritical carbon dioxide by the chromatographic impulse response (CIR) method at 308.15–333.15 K and pressures up to 36.00 MPa. The D12 values were also measured in liquid ethanol at 298.15–333.15 K, and at 0.10, 10.00 and 30.00 MPa by the Taylor dispersion method. The effects of temperature, pressure and density of fluid on D12 values were studied. The D12 data increased from 1.140 × 10−9 m2 s−1 in liquid ethanol at 298.15 K and 30.00 MPa to 2.218 × 10−8 m2 s−1 in supercritical carbon dioxide at 333.15 K and 14.00 MPa. It was observed that the D12 data in supercritical carbon dioxide at 15.00, 20.00 and 30.00 MPa can be well correlated as a function of temperature using an Arrhenius type equation, which provided the diffusion activation energies of 17.5, 13.4 and 11.0 kJ mol−1; that in liquid ethanol at 0.10, 10.00, 30.00 MPa gave the diffusion activation energies of 14.0, 14.2 and 13.8 kJ mol−1, respectively. It was found that the proposed correlation based on hydrodynamic approach can well represent all the measured D12 data of lithium acetylacetonate in supercritical carbon dioxide and in liquid ethanol with the average absolute relative deviation (AARD) of 1.7% for 88 data points.

The effects of temperature and density on retention of platinum(II) 2,4-pentanedionate in supercritical fluid chromatography were investigated at temperatures of 308.15-343.15K and pressure range from 8 to 40MPa by the chromatographic impulse response method with curve fitting. The retention factors were utilized to derive the infinite dilution partial molar volumes of platinum(II) 2,4-pentanedionate in supercritical carbon dioxide. The determined partial molar volumes were small and positive at high pressures but exhibited very large and negative values in the highly compressible near critical region of carbon dioxide.

Here is presented the first experimental data for the infinite dilution partial molar volumes of ibuprofen in supercritical carbon dioxide. The measurements were carried out by supercritical fluid chromatography using chemically bonded polyethylene glycol and carbon dioxide as stationary and mobile phases, respectively. The chromatographic retentions of ibuprofen at infinite dilution in supercritical carbon dioxide were measured at six supercritical isotherms of 308.15, 313.15, 323.15, 333.15, 343.15 and 353.15 K and at a pressure range of 8.50–40.00 MPa. The partial molar volumes were determined from the partial derivatives of the retention factors with respect to the densities and were in the range of −2.364 × 10−3 to 1.680 × 10−4 m3/mol. A reverse behaviour was observed, where the partial molar volumes decreased monotonically with decreasing pressure from positive to negative values in all temperature regions investigated. It was also found that the infinite dilution partial molar volumes showed a sharp decrease near the critical point of carbon dioxide. The experimental partial molar volumes were correlated with isothermal compressibilities and densities of carbon dioxide. The correlation proposed in this study could predict the experimental data with an average absolute relative deviation of 4.6% for 80 data points, which is better than that previously reported in the literature.

Binary diffusion coefficients (D12) and retention factors (k) of platinum(II) acetylacetonate at infinitesimal concentration in supercritical (sc) carbon dioxide (CO2) were measured by the chromatographic impulse response method with a poly(ethylene glycol) coated capillary column at temperatures from (308.15 to 343.15) K and pressures from (8.5 to 40.0) MPa, and D12 in liquid ethanol at temperatures from (298.15 to 333.15) K and atmospheric pressure by the Taylor dispersion method. As has been seen for our previously reported data on other metal complexes measured in sc CO2 and organic solvents, the D12 data in sc CO2 and liquid ethanol were represented by a function of temperature and solvent viscosity. The D12 values for metal complexes were not related to the solute molecular weights. The k values in sc CO2 were expressed by a function of temperature and CO2 density.

The use of chromatographic impulse response (CIR) method with a coated open tubular capillary column has potential advantages in supercritical fluid chromatography. In this review, applications of the CIR method to measuring the thermodynamic properties such as diffusion coefficients, solubilities and partial molar volumes are presented. This survey gives the theoretical backgrounds for the CIR method with linear adsorption and nonlinear adsorption models. Furthermore, the brief theoretical backgrounds for using retention factors to determine solubilities and partial molar volumes are also provided. In addition, the data sources for the diffusion coefficients with an emphasis on the results published after 2004 and for the partial molar volumes in supercritical carbon dioxide are presented.

The solubilities of organometallic complexes in supercritical CO 2 were determined from retention factors measured by the chromatographic impulse response (CIR) method, in which a solute was pulse-injected into a supercritical CO 2 flowing in a polymer coated open capillary column, and the response curve, i.e. time change of solute concentration, was monitored at the exit of the column. The retention factor and binary diffusion coefficient were simultaneously obtained as two parameters so that the fitting error between response curves of solute species measured and calculated was minimized. The validities of the method and the apparatus employed in this study were verified by determining the solubilities of naphthalene, which have extensively been reported in the literature. Then, the solubilities of organometallic complexes such as ferrocene and cobalt (III) acetylacetonate in supercritical CO 2 were determined. The data determined in this study were in good agreement with the literature data. The solubilities were also correlated with the Chrastil equation including the three adjustable parameters.

Infinite dilution binary diffusion coefficients, D12, of ferrocene, 1,1′-dimethylferrocene and ethylferrocene in hexane, cyclohexane and ethanol at 313.2 K and pressures from 0.2 to 19 MPa, in acetonitrile at 298.2–333.2 K and 0.2 MPa, and various metallic acetylacetonate, acac, complexes such as Co(acac)3, Ru(acac)3, Rh(acac)3, Pd(acac)2 and Pt(acac)2 mainly in ethanol at 313.2 K and 0.2 MPa were measured by the Taylor dispersion method. The D12 values in m2 s−1 for the three ferrocenes in the present study and those of ferrocene and 1,1′-dimethylferrocene in supercritical carbon dioxide in our previous studies were represented by the modified hydrodynamic equation over a wide range of viscosity: M0.5D12/T = 1.435 × 10−13η−0.8446 with average absolute relative deviation of 2.40% for 316 data points, where M is the solute molecular weight, T is the temperature in K, η is the solvent viscosity in Pa s. Although the D12 values for the acac complexes were roughly represented by the above hydrodynamic equation, the accuracies were lower because they were dependent on not solute molecular weight but the number of acac ligand in the complex molecules.

Binary diffusion coefficients, D12, of the metal acetylacetonates, palladium(II) acetylacetonate and cobalt(III) acetylacetonate, were measured from 308.2 to 343.2 K over the pressure range from 9 to 40 MPa at infinite dilution in supercritical carbon dioxide using the chromatographic impulse response method. The effects of pressure, temperature, density, and viscosity on D12 values were examined. It was observed that the D12 values of palladium(II) acetylacetonate and cobalt(III) acetylacetonate were larger than those of lipids with similar molecular weights, such as arachidonic acid and monoolein, respectively. Furthermore, the measured D12 data of each metal acetylacetonate were well correlated by the hydrodynamic equation D12/T as a function of carbon dioxide viscosity.

Binary diffusion coefficients D12 and retention factors k of ferrocene and 1,1′-dimethylferrocene at infinite dilution in supercritical (sc) carbon dioxide were measured by the chromatographic impulse response (CIR) method at temperatures from (308.2 to 323.2) K over a pressure range from (8.0 to 40.3) MPa for ferrocene and from (8.2 to 40.1) MPa for 1,1′-dimethylferrocene. The D12 data for ferrocene were also measured by the Taylor dispersion method with two injection procedures of ferrocene: ferrocene dissolved both in sc CO2 and in liquid hexane. The D12 values of ferrocene were not affected by the injection methods and agreed with those measured by the CIR method. As has been seen for various organic compounds measured in our previous reports, the D12 data for both metal complexes were correlated with the hydrodynamic equation D12/T = αηβ, where η is the CO2 viscosity and α and β are the constants determined by each solute. The retention factors were also correlated with CO2 density.

Binary diffusion coefficients D12 and retention factors k for dibenzo-24-crown-8 and 15-crown-5 ethers at 308.18 and 313.20 K, and vitamin K1 at 313.20 K were measured in supercritical carbon dioxide by the chromatographic impulse response technique, and the effects of molecular shapes on the D12 values were studied. At 313.2 K and 11.0 MPa the D12 value of 15-crown-5 was higher than that predicted from the D12 correlation with molecular weight MW over the range from 32 of methanol to 1138 of trinervonin reported in our previous studies, while the D12 value of dibenzo-24-crown-8, disk shape molecule with MW = 448.5, was almost in agreement with that of vitamin K1, long chain molecule with MW = 450.7, and with those predicted from the correlation.

The chromatographic impulse response (CIR) method with curve fitting was employed for measuring binary diffusion coefficients D12 at infinite dilution for polar compounds such as benzoic acid and C1 to C3 mono-alcohols in supercritical (SC) carbon dioxide at 313.2 K and pressures higher than 9.5 MPa. The measured D12 values for the polar compounds were expressed with two correlations which well represent binary diffusion coefficients for non-polar and weak polar compounds, i.e. the correlation of D12/T as a function of CO2 viscosity and the Schmidt number correlation. Moreover, the slope was −0.5 in logarithmic plot of D12 versus solute molecular weight over the range of molecular weight from 32 to 1138 at 313.2 K and 11.0 MPa.

Infinite dilution binary diffusion coefficients D12 and retention factors k for γ-linolenic acid were measured by chromatographic impulse response technique in supercritical carbon dioxide over the ranges of temperature from 308 to 343 K and pressure from 9 to 30 MPa, together with those for the methyl and ethyl esters at 313 and 343 K and pressures from 8 to 16 MPa. In comparison with literature values for C18 free fatty acids and the esters, the differences in D12 data were almost indiscernible for each isomer group: α- and γ-linolenic acid, oleic and elaidic acids, oleic and elaidic acid methyl esters, and their ethyl esters. The number of carbon double bonds did not affect the D12 values significantly. Moreover, the differences in the values between free fatty acid and the ester forms were also small, whereas the D12 values increased in order of acid to ethyl ester to methyl ester. The validities of the predictive correlations we had proposed such as the D12/T–CO2 viscosity and the Schmidt number correlation were verified. The retention factor k values for the acid form were much higher than those for the methyl and ethyl esters. The k values were correlated with temperature and CO2 density for each compound, and the solute partial molar volumes were determined from the k–CO2 density correlation.

Infinite dilution binary diffusion coefficients D12 and retention factors k have been measured in supercritical carbon dioxide for the long-chain triglyceride triarachidonin at 313.2 K at (10 to 30...

To determine binary diffusion coefficients from tracer response measurements with a coiled capillary column (curved tube) at relatively high flow rates, a conversion formula to correct for the secondary flow effect was derived from the second central moment of the impulse response curve. Its practical expression was obtained analytically by the perturbation method up to λ10, where λ is the ratio of the tube radius to the coil radius. The reliability of the correction was evaluated by an alternative series of λ. As a result, the effective range of DeSc1/2

A noise elimination technique was applied to the determination of binary diffusion coefficients D 12 from the response curves having extremely low absorbance intensities in impulse response methods under supercritical conditions of carbon dioxide. The effectiveness of this technique was experimentally examined for the analyses of response curves through both the curve-fitting and the moment methods in two cases: the chromatographic impulse response method for phenol and β-carotene with a polymer-coated capillary column, and the Taylor dispersion method for acetone with an uncoated capillary column. Unreliable D 12 values were obtained from the moment method of the response curves at lower absorbance intensities, even treated with noise elimination. The curve-fitting method with the noise elimination treatment was quite effective for determining the D 12 values accurately, and was valid at the lowest absorbance intensities, on the order of 10 −4 absorbance unit of UV-Vis multi-detector, corresponding to the smallest quantity of the solute, i.e. 6×10 −5 , 6×10 −6 , and 5×10 −2 μ mol for phenol, β-carotene, and acetone, respectively, under conditions studied. Infinite dilution regions for binary diffusion coefficients were obtained by injecting various amounts: the binary diffusion coefficients showed constant values at concentrations less than 0.6, 0.004, and 0.08 mol m −3 for phenol, β-carotene, and acetone, respectively, in supercritical carbon dioxide at 313.2 K and 16–18 MPa.

Infinite dilution binary diffusion coefficients and retention factors for the oleic acid family such as oleic acid, methyl- and ethyl esters, and mono-, di-, and triglycerides at 313.21 K and 8–30 MPa, and for other 13 lipids consisting of C18–C22 unsaturated fatty acid constituents at 313.21 K and 11 MPa were measured in carbon dioxide by a chromatographic impulse response method with a polymer coated capillary column. The binary diffusion coefficients and the retention factors for each compound decreased with increasing pressure. The binary diffusion coefficients could be represented by a Schmidt number correlation and by an empirical equation written in terms of temperature and CO 2 viscosity. The retention factors for each solute could be expressed in terms of CO 2 density. While the binary diffusion coefficients at 313.21 K and 11 MPa decreased with solute molecular weight, those for compounds having the same carbon numbers in each group, such as methyl ester, ethyl ester, and triglyceride, decreased with increasing the number of C–C double bonds.

Binary diffusion coefficients D12 and partition ratios k at infinite dilution for β-carotene and α-tocopherol in supercritical carbon dioxide were measured at temperatures from 308.15 to 333.15 K and pressures from 9 to 30 MPa by a tracer response technique with a poly(ethylene glycol) coated capillary column. Both parameters, simultaneously determined by fitting the calculated response curve to that measured experimentally, were well represented with the correlations: the D12/T values were correlated with CO2 viscosity, and the k values were expressed with a function of temperature and CO2 density. However, the partial molar volumes obtained from the k values were not well consistent with those estimated using equation of states having the interaction parameters kij reported in the literature.

A tracer response technique with a poly(ethylene glycol) coated capillary column was employed to measure binary diffusion coefficient and partition ratio for the ω3 group of long chain unsaturated fatty acids such as docosahexaenoic acid, eicosapentaenoic acid and α-linolenic acid at infinite dilution in supercritical (SC) carbon dioxide at temperatures from 308.15 to 343.15 K and pressures from 9 to 30 MPa. The binary diffusion coefficients for each solute were expressed with the Schmidt number correlation as well as the correlation with temperature and CO2 viscosity. The partition ratios were correlated with temperature and CO2 density. Moreover, the partial molar volumes were obtained from the partition ratios for each solute.

The binary diffusion coefficient D12 and partition ratio k were measured for two ω6 group compounds, linoleic acid and arachidonic acid, at infinite dilution in carbon dioxide at temperatures from 308.15 K to 343.15 K and pressures from 9 MPa to 30 MPa by a tracer response technique with a poly(ethylene glycol)-coated capillary column. The measured D12 values were well represented by a correlation of D12/T versus CO2 viscosity and by the Schmidt number correlation for each solute, and the k values were well represented as a function of temperature and CO2 density. Partial molar volumes were also obtained from the k values.

Infinite-dilution binary diffusion coefficient D12 and partition ratio k for ubiquinone CoQ10 in carbon dioxide were measured by a tracer response technique with a poly(ethylene glycol)-coated capillary column at temperatures from 308.15 to 333.15 K and pressures from 8.5 to 30 MPa. D12 and k values were simultaneously determined from a measured response curve by the curve-fitting method. The fitting error between measured and calculated response curves increased with decreasing pressure, as had been the case in our previous studies on various solutes in supercritical CO2 by the Taylor dispersion method. The D12 values were correlated with temperature and CO2 viscosity and the k values with temperature and CO2 density. Moreover, partial molar volumes of CoQ10 in supercritical CO2 were obtained from the k values.

Binary diffusion coefficients ( D 12 ) for low volatile compounds such as Vitamin K 3 , ubiquinone (UQ-10), α-tocopherol, β-carotene, and phenol in supercritical carbon dioxide were measured by the Taylor dispersion method with two columns connected in series: a poly(ethylene glycol) (PEG) coated capillary column followed by a stainless steel uncoated capillary column, at 313.15 K and pressures from 9 to 30 MPa. The tracer species dissolved with a common organic solvent such as hexane or chloroform was injected to the diffusion column as a delta shot through an ordinary HPLC injector. The response curve measured at the exit of the first column, coated with PEG, was regarded as an input signal, and that measured at the exit of the second column made of the uncoated stainless steel tubing as the output signal. The tracer species and the organic solvent were separated during flowing in the first column. The output response signal was calculated from the input signal and the inverse Laplace transform of the transfer function through the convolution integral. The D 12 values were determined by fitting the calculated output curve to that measured experimentally. The measured D 12 values were fully consistent with those measured with the PEG coated capillary column for one point measurement reported [Ind. Eng. Chem. Res. 39 (2000) 4462]. This two-point measurements with the two columns were adequate for measuring D 12 values for low volatile compounds in supercritical fluids.

Copyright: © 2014 Kong CY. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. A supercritical fluid (SCF) is an adjustable solvent which is reached at a temperature and pressure higher than its critical point. It can diffuse through solids like a gas, and dissolve materials like a liquid. In addition, close to the critical point, small changes in pressure or temperature result in large changes in density or viscosity, allowing many properties to be tuned. SCFs are suitable as a substitute for organic solvents in a range of industrial processes. Furthermore, there is no surface tension in a SCF, as above the critical temperature there is no phase transition. The applications of SCF is still expanding in the fields of extraction [1,2], food [3], pharmaceutical [4,5], material [6-8] and chemical reaction [9]. The quantitative knowledge of mass transport phenomena such as diffusion coefficients of various compounds in SCF is of considerable important in the design and efficient operation of the newly proposed SCF processes. Experimental methods, compounds and SCFs employed to determine the diffusion coefficients are highlighted in this editorial.

Binary diffusion coefficients, D 12 , and partition ratios, k, for the poly(ethylene glycol) (PEG) layer to supercritical carbon dioxide for acetone and some solid solutes such as phenol, α-tocopherol, and β-carotene were measured with a PEG-coated capillary column by a tracer response technique. The D 12 values for acetone with the PEG-coated column were consistent with those measured by the Taylor dispersion method in which an uncoated capillary column was employed. The D 12 and k values for all of the solutes decrease simply with increasing pressure, and the D 12 values were represented by the Schmidt number correlation.

The diffusion coefficients D(12) of phenylbutazone at infinite dilution in supercritical CO(2) were measured by the chromatographic impulse response (CIR) method. The measurements were carried out over the temperature range from 308.2 to 343.2 K at pressures up to 40.0 MPa. In addition, the D(12) data of phenylbutazone at infinite dilution in ethanol were also measured by the Taylor dispersion method at 298.2-333.2K and at atmospheric pressure. The D(12) value of phenylbutazone increased from 4.45×10(-10) m(2) s(-1) at 298.2 K and 0.1 MPa in ethanol to about 1.43×10(-8) m(2) s(-1) at 343.2 K and 14.0 MPa in supercritical CO(2). It was found that all diffusion data of phenylbutazone measured in this study in supercritical CO(2) and in ethanol can be satisfactorily represented by the hydrodynamic equation over a wide range of fluid viscosity from supercritical state to liquid state with average absolute relative deviation of 5.4% for 112 data points.

The binary diffusion coefficients, D(12), and retention factors for myristoleic acid and its methyl ester at infinite dilution were measured by the chromatographic impulse response technique in supercritical carbon dioxide at temperatures of 313.2, 333.2 and 343.2 K and pressures from 9.2 to 30 MPa for the acid, and from 8.0 to 14 MPa for the ester. Although the D(12) values were represented by the two correlations, the D(12)/T vs. CO(2) viscosity and the Schmidt-number correlations, which are valid for more than 40 compounds that we have measured so far, significant temperature dependences were observed for the ester. Moreover, the D(12) values for the ester at 313.2 K downward deviated from the background values around 400 kg m(-3), where the partial molar volumes, obtained from the correlation between the retention factors measured and CO(2) densities, showed large negative values.

The chromatographic impulse response technique with a polymer coated capillary column was applied to measurements of infinite dilution binary diffusion coefficients D and retention factors k in supercritical carbon dioxide by injecting a hexane solution dissolving a mixture of three unsaturated fatty acids such as alpha-linolenic acid, eicosapentaenoic acid, and docosahexaenoic acid. The coefficients were simultaneously estimated by the curve fitting analysis even from partially overlapping response curves with the resolution of 0.8. The D and k values for each solute were able to be so obtained as accurately as those determined by individually injecting a single component solution. Almost no effect of the interaction among the components in the mixture was found from various approaching ways for curve fitting and the consecutive injection of the mixture at a certain interval.

The theoretical basis of a Gaussian-like approximate solution was applied to a chromatographic impulse response technique with curve fitting for measuring binary diffusion coefficients and retention factors using a polymer-coated capillary column. The formulae were derived for evaluating both the accuracy of the approximate solution and the sensitivity of the parameters. The validity of the solution also was confirmed experimentally for pulse injection of phenol in acetone into supercritical carbon dioxide flowing at 313.15 K and 11.6-28.6 MPa. Potential sources for experimental errors of the method are discussed.