Hildebrands Solubility Parameters

The Hildebrand solubility parameter, 6, is a semi-quantitative entity related to the thermodynamic properties of dense gases (supercritical fluids) and solutions.t The solubility parameter in calories per cubic centimeter is calculated from the equation  

Based on the semi-empirical calculations as shown in Ref 18, the r liq) is given by  

From Eqs. 1-3, the following relationship can be derived to relate the Hildebrand solubility parameter with the density of supercritical fluid. 

Equation 4 was used to calculate the Hildebrand solubility parameters for various supercritical fluids. Here p is the density of the supercritical fluid which is related to the pressure and temperature as described earlier. 

The enhancement of solvent power obtained by compressing a gas into its critical region can be demonstrated dramatically. An estimate of the solubility of a solid in an SCF solvent can be made using the following expression  

Another old-fashioned measure is the Hildebrand solubility parameter, 8, which is defined as the square root of the cohesive pressure [2a]  

Similarly, the correlations of C with (8po, — 8liq)2 for the sets of Ph(CH2) H liquids [160] and the liquid ketones, esters and ethers [177, 178] produce a set of four essentially parallel lines (Fig. 52) given approximately by  

Since the reported 8n for a given Z(CH2)nH series usually represent less than four of the first six members of that series, the rest of the needed 8n data for the first 12 members were calculated [161-164, 177, 178] by the method of additive contribution of molecular components to the cohesive energy density of the sorbed liquid [27] as noted in Tables 4-7. Despite the uncertainty in identifying 8po  

The calculated Xn is correlated with n for each Z(CH2) H series in Fig. 59, which shows that Xn increases with the difference n — n, in accordance with Eq. 39  

---

The Hildebrand Solubility Parameter. This parameter, 4 can be estimated (10) based on data for a set of additive constants, E, for the more common groups ia organic molecules to account for the observed magnitude of the solubiHty parameter d = EE/V where Erepresents molar volume. SolubiHty parameters can be used to classify plasticizers of a given family ia terms of their compatibihty with PVC, but they are of limited use for comparing plasticizers of differeat families, eg, phthalates with adipates. 

Numerous reports of comparable levels of success in correlating adhesion performance with the Scatchard-Hildebrand solubility parameters can be found in the literature [116,120-127], but failures of this approach have also been documented [128-132J. Particularly revealing are cases in which failure was attributed to the inability of the Scatchard-Hildebrand solubility parameter to adequately account for donor-acceptor (acid-base) interactions [130,132]. Useful reviews of the use of solubility parameters for choosing block copolymer compatibilizers have been prepared by Ohm [133] and by Gaylord [134]. General reviews of the use of solubility parameters in polymer science have been given by Barton [135], Van Krevelen [114], and Hansen [136]. 

Cohesive energy density is the energy of isothermal vaporization per unit volume to the ideal-gas state. It is the square of the Hildebrand solubility parameter. 

Lohse et al. have summarized the results of recent work in this area [21]. The focus of the work is obtaining the interaction parameter x of the Hory-Huggins-Stavermann equation for the free energy of mixing per unit volume for a polymer blend. For two polymers to be miscible, the interaction parameter has to be very small, of the order of 0.01. The interaction density coefficient X = ( y/y)R7 , a more relevant term, is directly measured by SANS using random phase approximation study. It may be related to the square of the Hildebrand solubility parameter (d) difference which is an established criterion for polymer-polymer miscibility ... 

Various models of SFE have been published, which aim at understanding the kinetics of the processes. For many dynamic extractions of compounds from solid matrices, e.g. for additives in polymers, the analytes are present in small amounts in the matrix and during extraction their concentration in the SCF is well below the solubility limit. The rate of extraction is then not determined principally by solubility, but by the rate of mass transfer out of the matrix. Supercritical gas extraction usually falls very clearly into the class of purely diffusional operations. Gere et al. [285] have reported the physico-chemical principles that are the foundation of theory and practice of SCF analytical techniques. The authors stress in particular the use of intrinsic solubility parameters (such as the Hildebrand solubility parameter 5), in relation to the solubility of analytes in SCFs and optimisation of SFE conditions. 

The promising approach taken by Vandenburg et al. [37,489] is to use initially a solvent with a Hildebrand solubility parameter several MPa1/2 different from the polymer (i.e. a poor , nonswelling solvent for the polymer) to determine experimentally the maximum... 

Vandenburg et al. [37,489] have described the use of Hildebrand solubility parameters in a simple and fast solvent selection procedure for PFE of a variety of polymers. Hildebrand parameters for several common solvents and polymers are presented in Tables 3.2 and 3.34, respectively. When the proper solvent mixture for the polymer was determined, PFE resulted in essentially the same recoveries as the traditional extraction methods, but used much less time and solvent. PFE can be used to give very fast extractions and appears to offer the greatest flexibility of solvents and solvent mixtures. The method is ideal for a laboratory which analyses a large number of different polymers. 

Limited method development (Hildebrand solubility parameters for solvent choice)... 

The use of the Hildebrand solubility parameter approach to aid solvent selection with a few simple experiments, starting from the liquid solvents used in traditional extraction methods, limits the efforts needed in method development. As for other extraction... 

Fig. 3.4 Plot of calculated vo-2ot values, (in kcal/mole)2, versus Hildebrand solubility parameters 8, in MPa2, for the molecules given in Table 3.3. The linear correlation coefficient and standard deviation are 0.930 and 1.9 MPai, respectively.

Figure 3.4 shows a fair correlation between vo-2ot and the Hildebrand solubility parameter 8 (linear correlation coefficient = 0.930) which makes intuitive sense. The Hildebrand parameter, which is often used to characterize liquids, is defined as the square root of the cohesive energy density (Barton 1991), while vcr2o( can be viewed as reflecting how strongly a molecule interacts with others of the same kind (Murray et al. 1994). 

With traditional solvents, the solvent power of a fluid phase is often related to its polarity. Compressed C02 has a fairly low dielectric constant under all conditions (e = 1.2-1.6), but this measure has increasingly been shown to be insufficiently accurate to define solvent effects in many cases [13], Based on this value however, there is a widespread (yet incorrect ) belief that scC02 behaves just like hexane . The Hildebrand solubility parameter (5) of C02 has been determined as a function of pressure, as demonstrated in Figure 8.3. It has been found that the solvent properties of a supercritical fluid depend most importantly on its bulk density, which depends in turn on the pressure and temperature. In general higher density of the SCF corresponds to stronger solvation power, whereas lower density results in a weaker solvent. 

Being able to change the density, via either changes in pressure or temperature, is the key difference in SFC over GC and LC separations. Typical density ranges are from 0.3 to 0.8g/ml for pure carbon dioxide. Table 16.2 shows data obtained from ISCO s SF-Solver Program for the calculation of density (g/ml), Hildebrand Solubility Parameter and a relative equivalent solvent for pure carbon dioxide at a constant pressure of 6000 psi, approximately 408 atm. 

Calculation of density, Hildebrand solubility parameter using ISCO s SF solver program at constant pressure of 6000 psi (408 atm)... 

A gradient that runs with 30-80% methanol or acetonitrile is not uncommon. This amount of modifier is generally not needed in supercritical fluid chromatography to affect the same separation. Typical modifier composition in SFC is 1.0-10% and would achieve higher Hildebrand Solubility Parameter adjustment overall than the broader gradients found in LC. 

The parameters a and p indicate the capacity of a solvent to donate or accept a hydrogen bond from a solute, i.e., the solvent s hydrogen bond acidity or basicity. % is intended to reflect van der Waals-type solute-solvent interactions (dipolar, dispersion, exchange-repulsion, etc.). Equation (43) was subsequently expanded to include a term representing the need to create a cavity for the solute (and thus to interrupt solvent-solvent interactions).188 For this purpose was used the Hildebrand solubility parameter, 5, which is defined as the square root of the solvent s energy of vaporization per unit volume.189 Thus Eq. (43) becomes,190... 

Once the local parameters have been fitted to a limited set of data then solubilities can be calculated in a representative set of solvents. Plotting the experimental and predicted data against the Hildebrand solubility parameter of the solvent gives a veiy good indication of behaviour with solvent type, figure 19. The application of the SoluCalc method to Cimetidine is briefly presented in Section 6. 

First a database of solute-solvent properties are created in SoluCalc. The database needs the melting point, the enthalpy of fusion and the Hildebrand solubility parameter of the solute (Cimetidine) and the solvents for which solubility data is available. Using the available data, SoluCalc first prepares a list of the most sensitive group interactions and fits sequentially, the solubility data for the minimum set of group interaction parameters that best represent the total data set. For a small set of solvents, the fitted values from SoluCalc are shown in Table 9. It can be noted that while the correlation is very good, the local model is more like a UNIQUAC model than a group contribution model... 

The mixture is going to be identified by its ability to not mix with water (total immiscibility), normal boiling point (each compound in the mixture has a Tb above 350 K so the mixture will be a liquid), normal melting point (each compound in the mixture has a Tm below 250 K so the mixture will be a liquid), the Hildebrand solubility parameters of each of the compounds should be between 18-22 MPa172 (so the two compounds are mutually miscible). 

In this respect, the solvatochromic approach developed by Kamlet, Taft and coworkers38 which defines four parameters n. a, ji and <5 (with the addition of others when the need arose), to evaluate the different solvent effects, was highly successful in describing the solvent effects on the rates of reactions, as well as in NMR chemical shifts, IR, UV and fluorescence spectra, sol vent-water partition coefficients etc.38. In addition to the polarity/polarizability of the solvent, measured by the solvatochromic parameter ir, the aptitude to donate a hydrogen atom to form a hydrogen bond, measured by a, or its tendency to provide a pair of electrons to such a bond, /, and the cavity effect (or Hildebrand solubility parameter), S, are integrated in a multi-parametric equation to rationalize the solvent effects. 

Carbon dioxide is a non-polar solvent characterized by a low polarizability per volume, a low Hildebrand solubility parameter, and a low dielectric constant. The dielectric constant of CO2 as a fimction of pressure is shown in Fig. 6 [27]. 

The three kinds of forces described above, collectively known as the cohesive forces that keep the molecules of liquids together, are responsible for various properties of the liquids. In particular, they are responsible for the work that has to be invested to remove molecules from the liquid, that is, to vaporize it. The energy of vaporization of a mole of liquid equals its molar heat of vaporization, Ay//, minus the pressure-volume work involved, which can be approximated well by Rr, where R is the gas constant [8.3143 J K" mol" ] and T is the absolute temperamre. The ratio of this quantity to the molar volume of the liquid is its cohesive energy density. The square root of the cohesive energy density is called the (Hildebrand) solubility parameter of the liquid, 8 ... 

Table 6.2 presents data showing the effect of various CMOS on the activity coefficient or mole fraction solubility of naphthalene, for two different solvent/water ratios. To examine the cosolvent effect, Schwarzenbach et al. (2003) compare the Hildebrand solubility parameter (defined as the square root of the ratio of the enthalpy of vaporization and the molar volume of the liquid), which is a measure of the cohesive forces of the molecule in pure solvent. 

Lee, S.H. and Lee, S.B., The Hildebrand solubility parameters, cohesive energy densities and internal energies of l-alkyl-3-methylimidazolium-based room temperature ionic liquids, Chem. Commun., 3469, 2005. 

---