J. Phys. Chem. 1995,99, 13909-13916
13909
Group Contribution-Additivity and Quantum Mechanical Models for Predicting the Molar Refractions, Indices of Refraction, and Boiling Points of Fluorochemicals Thao D. Le* and Jeffry G. Weers Advanced Product Research, Alliance Pharmaceutical Corporation, 3040 Science Park Road, San Diego, Califomia 92121 Received: May 22, 1995; In Final Form: July IO, 1 9 9 P
Group contribution-additivity (GCA) and quantum mechanical (QM) models for estimating the molar refractions of fluorochemicals are proposed. The GCA model was developed using experimental refractive indices and densities, and the Lorentz-Lorenz electromagnetic correlation. The QM model was developed using the distortion polarizability (a)term obtained from quantum mechanical calculations based on the PM3 Hamiltonian. The GCA model provides molar refractive contributions for 23 heterogeneous groups and atoms (e.g., RF, RH,C, N, 0, F, C1, Br, I, C-0, NH2, OH, CH-C, CH=CH, CH2=C, CH=CFp, CF-CF2) and yields an absolute average error of less than 2% ( n = 100). The QM model, on the other hand, offers unlimited groups but requires knowledge of the a descriptor. Methods for acquiring a are provided. Both models have been successfully applied to the predictions of refractive index (nD) and the normal boiling point ( T b ) of fluorochemicals, with average errors of less than 1% and 6%, respectively. The higher method error observed in the Tb model is attributed to errors introduced by Tt,measurements at pressures slightly below 760 mmHg. The T b model, nonetheless, provides accurate predictions of boiling point in the range between 30 and 230 "C.
for other purposes and thus have very few FC classes represented in their basis set. A number of semiempirical relationships have been proposed Recognizing the limitations of current R, estimation methods, for estimating the physicochemicalproperties of nonpolar liquids the present study proposes a group contribution-additivity(GCA) such as the standard heat of formation,' solubility ~ a r a m e t e r , ~ . ~ model and a quantum mechanical (QM) model that can surface tension? dielectric constant,5 critical constants (i.e., accurately estimate R, for a variety of FCs. The GCA and volume, temperature, and pressure)? critical solution temperQM models have been developed and tested using diverse ature: index of r e f r a c t i ~ n ,and ~ , ~boiling point.Io Some of these classes of FCs in the basis set to achieve the optimal accuracy. correlations have been successfully expanded to include fluoAccurate estimates of R, are critical since Rm contributes to rochemicals (FCs). the method errors of models used for predicting physical FCs have been extensively researched for use as antihypoxic properties. Both the GCA and QM models have been tested in agents in surgical and liquid ventilation procedures, and as the predictions of refractive index (n) and boiling point (Tb) of contrast agents in X-ray, MRI, and ultrasound, because of their FCs to show their applicability and to explore their limitations. extraordinary capacity for dissolving blood gases, their low surface tension, excellent biocompatability, and other special Experimental Section features.".12The ability to predict these unique chemical properFluorochemical Selection. Refractive indices (n) and densities would aid in the development of novel FCs for specific ties (e) of 160 fluorochemicals were obtained from various biomedical applications. source^'^-^^ for this study. Values of the refractive index are Despite the fact that there are a number of semiempirical affected by changes in temperature, pressure, and by the relationships available for predicting these much needed properwavelength of monochromatic light used in the measurement. ties, most cannot be used because they require input parameters, Typically, n is measured at 20 "C using the yellow light of the such as molar refraction (R,), that are not commonly available. sodium D line, with an emission wavelength at 589 nm. The Primarily, this is because R, is a value acquired from a refractive body of literature also included data obtained at other temperindex (nD) measurement, which is not widely measured for atures and wavelengths. To obtain a uniform basis set, fluorochemicals. compounds that had measured values of nD20 and g20 were Literature searches show four group contribution methods that retained. Compounds with refractive indices and densities can provide rough R , estimates for fluorochemicals. Upon within f 5 "C from the desired temperature were also kept to rigorously testing them for applicability and accuracy, they were maintain a sizable basis set (n = 100). The molecular structures found to be unusable for our purpose due to inaccuracy or lack of the remaining fluorochemicals are shown in Table 1 and of essential functional group slat om^.^^ Methods by VogelI4 and identified as FC 1-100 for the convenience of discussion. All Ei~enlohr,'~ for instance, were found to underestimate R, of compounds contain the fluorine (F) atom, hence the name fluorochemicals by as much as 27% (n = 100). Methods by fluorochemicals (FCs). In nearly all cases, the F atoms are not Fainberg et a1.I6 and Van Der Puy,Io on the other hand, were shown so that special structural features and substituted hetfound to be reasonably accurate (14%) but have very limited eroatoms can clearly be displayed. FCs in this basis set contain group contributions. All in all, these methods were developed between 9 and 71% fluorines (by total atoms) and have molecular weights in the range of 96-700 glmol. * To whom correspondence should be addressed. Abstract published in Advance ACS Absrrucrs, August 15, 1995. Computational Methods. All molecular calculations were
Introduction
~~~
~
@
0022-365419512099- 13909$09.00/0
0 1995 American Chemical Society
Le and Ween
13910 J. Phys. Chem., Vol. 99, No. 38, 1995 TABLE 1: Fluorochemicals Used in This Study" FC molecular structure FC molecular structure FC molecular structure 26
2
27
-
FC molecular structure
-CFzH
52
LI
53
-Br
54
-Br
55
-Br
CI
3 -
28 CIJ
4 -
29
c'7 CI
30
5-
C
31
I
X
clq
56 &r
CI
32
" I n CI
33
)a-
CI
CI
35
"'Y'r"'
36
c'Pc' CI
31
c
CI
CI
CI
l
p
l
62 38
cl&cl CI
03
-
59
CI
l4
A/v'
CI
34
lo
57
CI
39
63
CI
T
7-
"'XY"' CI CI
CI
40 A
41
I
cl&cl CI
42
C
CI
CI
"": CI CI CI
43
e,
44
&IC
45
cn3.
46
Eln2 CH3YCH3 CI
47
C~,cn~cH3
69
-0'
CI CI CI
97
22 23
b
48 CI,
2
C H ~cn3
73
-0-
Cl+CnbF2 CI CF3
Group Contribution and Quantum Mechanical Models
J. Phys. Chem., Vol. 99,No. 38, 1995 13911
TABLE 1 (Continued) --FC molecular structure FC molecular structure FC molecular structure 49
&
24
""C,"
x
74
CI
75
FC molecular structure
c ~ ~ n ~ ~ " ' ~ w99~ ~ c,-~~ccF2 " ' ~ u l
CF3
CI
CH3
CI
50
kH*P
CI\
*,;on
100
p' CH~""~H*~~*CF~
'?&
CH3
-
All compounds in this table contain between 9% and 71% fluorines (by total atom). The RP groups are represented by their molecular skeletons; fluorine (F)atoms are not shown.
I
1
I
Mokculir Partition (inlluaoe on h)
i
I
Molecular Simulation
effectiveness of the CB algorithm to minimize the method error and to distribute any foreseeable error propagation evenly throughout the entire basis set. The algorithms detailed below represent the best results in two groups of successive trials.
Results and Discussion
(CAChc WorkSystmT
4 Molecular Descriptor Gmerstlon
(a.V B A)
I GCA Model
(R, estimation)
1 Empirical Relationships (a&%R. &G)
1 QM Model
(R estimation)
I
I
I
t
Group Contribution-Additivity (GCA) Model. Examination of the fluorochemicals in the basis set revealed notable influences on molar refraction due to CF2 and CH2 groups in cyclic arrangements and only negligible influences due to branching. Thus, structural influences in these two groups were accounted for by their structural classifications, CF2 (aliphatic), CF2 (cyclic), and CH2 (aliphatic). There was no CH2 (cyclic) group represented in the data population, and therefore it is not included. Other groups that had significant influences on R, were C, CF, C S H , CH, CH3, N, 0, F, C1, Br, I, C=O, NH2, and OH. Double bonds in FCs used for pharmaceutical applications are mostly in vinyl and allylic forms, hence those groups were developed to represent all the forms typically found in highly fluorinated compounds. Generally, a group contribution method is developed by comparing between two classes of compounds with one varying molecular fragment, and then, after a series of comparisons, averaging the differences in physical property contributed by the fragment. Unlike the conventional method adopted in previous models, the present study implemented a CB I algorithm which utilizes linear algebra and least squares-residual minimization techniques to delineate group contributions. This combination was designed to minimize bias in the assignment of contributory grouplatomic influence and to provide a uniform distribution of propagated errors over the entire range of the basis set. Some of these procedures are outlined below. A 24 x 100 rectangular array (Au) consisting of the 23 molecular fragments and corresponding R, values was setup for the 100 fluorochemicals shown in Table 1. Molar refractions (R,) were calculated using the Lorentz-Lorenz relationship based on electromagnetic theory:29
I Utility of Q
(bmand TI)
Figure 1. Flow diagram outlining the steps involved in the development of the GCA and the QM models. (a) V and A are needed in the estimation of Tb. (b) Checkhalance (CB) loop; each loop is cycled until the method error is minimized and the errors due to propagation are proportionately distributed.
performed on the CAChe WorkSystem (CAChe Scientific, Beaverton, OR). The PM3 Hamiltonian25 was selected to optimize the molecular structures because predicted heats of formation were closer to published values, and torsional and dihedral angles were better represented. Polarizabilities were calculated from the heats of formation using an algorithm developed by Kurtz et al.26 This calculation is available on the molecular orbital software package (MOPAC)27 provided by CAChe. Isopotential surfaces were calculated by an algorithm for contouring multidimensional objects28 in the Tabulator software also available on CAChe. Statistical analysis and least-squares residual minimization were performed on a 486DX2-66 MHz computer using the statistical software package MicroMath Scientist (MicroMath, Inc., Salt Lake City, UT). Design Considerations in Model Development. Figure 1 is a flow diagram outlining the sequence of steps involved in the development and testing of the GCA and QM models. As illustrated, each model has a checkhalance (CB) algorithm designed to cycle until the optimal performance is reached. The success or failure of each model depended critically on the
where a is the distortion polarizability, N A is Avogadro's number, M, is the molecular weight, is the density, and TZDis the index of refraction. The array was subsequently divided into groups according to similar molecular fragments to yield at least as many equations as unknowns. In most cases, the number of equations far exceeded the number of unknowns providing ample opportunities for improving the accuracy of the evaluation method. Equations 2-4 are the mathematical equivalents of the Au array:
Le and Weers
13912 J. Phys. Chem., Vol. 99, No. 38,1995
XU/;;= a l l x l + a12x24- ... + a1yj= R ,
(2)
i= 1 n
+
C a / ; ,= ~~~x~ a,,x,
+ ... + a2,xj = R,
0
(3)
i= 1
...
where ag is the number of a specific molecular fragment, xj is the molecular fragment, and R; is the molar refractive contribution. The subscripts i a n d j represent the elements in the array. This collection of nonhomogeneous linear dependency equations (nonzero solutions) can be simplified to a set of linear combinations
Transcribing eq 5 yields a coefficient matrix, a column of unknowns, and a column of constants.
Equation 6 can be further simplified to a single augment matrix in 53' space.
(7)
where % . is a matrix and superscript n is the matrix size. A total of 23 matrices of this type were evaluated and reevaluated until the objectives of the CBI algorithm are achieved. The results are molar refractive (R;) contributions for 23 heterogeneous groups and atoms shown in Table 2. Molecular fragments are arranged in order of increasing R; values. These contributions are intended for use only in estimating the molar refractions of fluorinated materials. How To Use R; Values. Twenty diverse fluorochemicals from the basis set were selected as representative compounds to illustrate the use of R; contributions. These examples are presented in Table 3. The purposes for the examples are to show how similar compounds should be calculated and the errors associated with each molecular fragment type. The positive and negative signs of the relative errors indicate over- and underestimations to the experimental results. These rf: deviations were intentionally done to effectively distribute propagated errors. Otherwise, the method error (dR,) would, in a worst case scenario, increase as a function of molecular size, viz.30 n
dR,
%
&dR; i= I
=aldR,
+ a,dR, + ... + andRn
TABLE 2: Molar Refractive Contributions for 23 Heterogeneous GroupdAtoms group/atom Ria group/atom
(8)
where ai is the number of molecular fragments, and dRi is the
F
N C OH CH CF CFzH CHz (aliphatic) CF2 (cyclic) CF2 (aliphatic) CF3 a
2.051 2.589 2.838 3.303 3.659 4.082 4.308 4.3 14 4.601 4.616 5.310 5.318
c1 CH3 c=o NHz Br CHz=C CH=C CH-CH CH=CFz CF=CF2
I
Ria
5.512 5.521 5.899 4.492 1.122 8.534 8.122 9.359 9.410 10.068 12.443
Contributions are arranged in increasing order of molar refractivity.
error introduced by each grouplatomic contribution. The method error, as shown in Table 3, remains constant for a variety of molecular sizes (or number of inputs). The largest compound in the group (FC-22) has 16 inputs, yet it yields an error of only 2%. Smaller compounds with fewer inputs, like FC-100, yield nearly the same error. These results c o n f i i the effectiveness of the CBI algorithm to distribute propagated errors. Quantum Mechanical (QM) Model. Unlike the GCA model, the quantum mechanical (QM) model has inherently broader applications. Instead of relying on the availability of Ri contributions, the QM model uses the distortion polarizability (a)as an input parameter. Because a can be acquired from a group contribution method3' or commercially available molecular modeling software packages, the QM model was developed as an alternative means for calculating R , should the present GCA model not have the desired groupslatoms. The first step in developing the QM model was to define an empirical relationship between experimentally determined polarizability and that obtained from quantum mechanics, viz.
a = 1.461%
+ 1.775
(9) where a is the distortion polarizability determined from eq 1 and Q is the polarizability determined from quantum calculations. This empirical relationship (6 = 0.999) is necessary because present molecular modeling programs do not accurately describe the polarizabilities of fluorochemicals. Equations 1 and 9 were then combined to relate molar refractivity to Q: R, = nN,( 1.948% 2.367)lk (10)
+
where Ra is the molar refractivity as a function of polarizability, N A is Avogadro's number, and k is a conversion constant ( A3 = 1 cm3). Error Analysis of the GCA and QM Models. As noted earlier, performances of the GCA and QM models were measured by the effectiveness of the error distribution/ minimization. Figure 2 is a chart showing the error profiles of the GCA and QM models. Particularly interesting about these profiles is the symmetrical and uniform error distribution about the zero axis (true value), providing evidence that the CB1+2 algorithms adequately account for propagated errors. Statistical analysis revealed that the GCA and QM models yield average absolute errors of 1.8% and 3.4%, respectively. These errors are quite good considering the many varieties of classes and sizes of FCs used. Fluorochemicals in the basis set have molecular weights and molar refractions in the ranges 96-700 @mol and 18-71 cm3, respectively. Refractive Index Estimation Using the R , Term. The GCA and QM models were tested in the prediction of nD*O. The performances of these two models were evaluated by
J. Phys. Chem., Vol. 99, No. 38, 1995 13913
Group Contribution and Quantum Mechanical Models
TABLE 3: Sample Calculations of 20 Representative Fluomchemicals
.. 3
-
1.4
388.1
r ,
CFa CH, CH;
1 4.314 3 13.821 1 5.521
Q 1 U U 174.1 -1.3
400.1 CF, 3 15.954 ~ ~ ~ ( ~ y4 e18.704 i ) 500.1
31.77
total
48.88
z
m
47.582
368.1
40.38
-2.7
CF, CF, (aliph)
1 5.318 4 21.240 C F , ( C ~ C ~ ) 5 23.380
D14.308 550.1
53.83
54.246
0.8
10 46.760 6 25.848 total 72.608
2.4
total
319.0
43.21
CF,(+) 686.1
70.93
19.73
CF, CF, (aliph) CH
2 10.636 4 21.240 I 4.082
Q H L M
total
39.617
CF, CF,(aliph) CH, CH’ Br
1 5.318 2 10.620 1 4.607 1 5.521 1 7.722
CEQI5&&? total 39.687
-1.8
I 1 1
CF CH; CI
21.164
CF,
CF CH,
3 14.028 1 4.308 1 4.607
total
28.464
(CYC~)
m 1 m 174.1
28.22
39.51
0.9 166.2
38.90
-1.9
4.2
7.3
1 1 1
5.318 5.310 4.308 1 5.521 2 11.024
akC1m 261.0
-2.4
5.318 5.310 4.607
m1zeze
total CF, CF,(diph)
a
562.1
31.023
CF, CF,(aliph) CH, 149.1
CF, 3 15.954 C F ~ ( C ~ C I4) 18.704 5 21.540 total 56.198 57.22
total
total
40.203
CF, CH, CH,
I 5.318 4 18.428 1 5.521
G€kl3 1 total 38.626
1.7
-0.7
C F ~ 1 5.318 CFz(diph)
CF 56
58
2
-
509.9
446.0
51.28
43.74
5 26.550 I 4.308
kr ZLL44!l
total
51.620
CF, CF,(diph)
1 5.318 5 26.550
I 11 &?!I2
total
44.311
CF, CF, (aliph)
3 15.954 5 26.550
N 1 2 . w
471.1
333.1
45.36
33.35
total
45.342
F CF,(cycl) CF
5 23.380
B
total
1
CF, CF>(diph) C C1
0.7
-1w 265.0
34.05
210.1
0.0
29.99
m
33.115
34.365
CFI CF,(&ph) CH,
1 5.318 2 10.620 1 5.521
0.9
!xE€1u?A total 29.993
0.0
CH, 2 11.042 CH 2 8.164 BR 1 7.722
4.308
L
total
1.3
2.589
I
I 5.318 1 5.310 1 3.303 2 11.024
i
-0.7
comparing estimated refractive indices to measured values. The index of refraction was calculated using the following relationship.
where Vm is the molar volume (At,,,/@), and eq 11 is another form of eq 1. As shown in Figure 3, the GCA model provides slightly better n~~~predictions as exemplified by 9 (0.99 vs 0.97) and standard deviation (0.058 vs 0.056). Additional
100
217.0
37.77
total
io.wa
36.996
-2.0
evidence is provided by the error profiles of the n~~~estimations using the two models (Figure 4). It can clearly be seen that the GCA model yields a tighter error range as compared to itsQM counterpart, which translates to lower average error, 0.5% vs 0.9%. This is expected since more opportunities were available in the GCA model for minimizing contributory errors. Both models, nonetheless, prove to be quite reliable in estimating nD20,having standard deviations of only f0.06for a wide range of fluorinated compounds. Boiling Point Estimation Using the R , Term. In a previous
Le and Weers
13914 J. Phys. Chem., Vol. 99, No. 38, 1995
TABLE 4: Isopotential Surface Contributions
t
l5
eroudatom
0
0
Madcl Legend
1
n Avg. 8Rm ( % )
,
,
40
60
.30 ..
20
0
80
Rm (cm') Figure 2. Error analysis of the GCA and QM models for predicting the molar refractions of FCs. The 100 FCs used have molecular weights and molar refractions in the ranges 96 5 M , 5 700 and 18 5 R, 5 7 1, respectively.
2
1.50
1
I +
f
isopotential surface vol (A?
.
isopotential surface area ~ 4
4.15 7.73 5.68 4.76 7.51 6.10 10.56 18.04 10.87 11.70 15.36 15.33 20.18 12.62 16.82 10.82 16.75 16.96 11.60 16.87 24.25 27.47 18.18 1 1.43 10.57 13.27
F" N C OH CH CF CF2H NH2 CHI (aliphatic) CF2 (cyclic) CF2 (aliphatic) CF3
c1
CH3
c=o Br CHz=C" CH=CR CH-CH" CHPCFf CF=CF2" I CF= CH= CFH
2 )
10.40 17.30 7.09 - 1.25 16.68 4.70 12.88 32.33 20.67 17.75 25.74 25.64 37.51 22.47 27.50 16.78 25.03 26.39 16.26 25.68 43.88 49.32 28.25 22.75 15.63 19.57
.L
'Delineated in this study using the CBI algorithm. Otherwise, isopotential surface areas and volumes are obtained from our previous study3' using another method of evaluation.
e,
a W
1.30 -
1
0"-*-
/
v
1.20 1.20
QM
I
1.28
I
o
I
1
I .44
1.36
I
/10010.05610.971 I
1
1.52
I
1.60
Estimated nD20
Figure 3. Plot of experimental ~ D ' O vs nD2' estimated from eq 11 using R, values determined either by the GCA (Table 2) or QM (eq 10) method. 10
5 0 h
5P
0
100
200
300
O
Lo
Figure 5. Plot of experimental Tb vs estimated Tb. The boiling point
Model Legend Avg. Bn,(%)
-5
GCA
e
0.5
data range from 29 to 237 OC.
QM
0
0.9
relationship
-10
1.20
1.30
1.40
1.50
1.60
Experimental nDZo nDZo estimations using the GCA and QM models. Statistical analysis reveals slightly better error distribution in the GCA model.
Figure 4. Errors resulted from
study, we proposed a GCA model for estimating the normal boiling point (Tb) of FCs using molecular descriptors such as isopotential surface volume (V), isopotential surface area (A), and polarizability (a).3' Parameters V and A were used to account for the various molecular shapes, and a was used to account for the attractive forces between molecules, as described by the London equation for dispersion forces. Since the a0 term has successfully been related to R , (eq lo), attempts to estimate Tb based on R , are warranted. Reworking the boiling point model to include the molar refractivity term instead of a establishes the following empirical
~ ~ ( o= c )306.1 - 452.ge (-V/AlA)(0.0300R~+0.1963)
(12) where r b is the boiling point, V is the isopotential surface volume, and A is the isopotential surface area. To test the Tb model (eq 12), the compounds in Table 1 were used once again. Unfortunately,not all compounds in that table have published values for normal boiling point (760 " H g ) . To maintain a sizable basis set without severely sacrificing accuracy, FCs with Tb measurements within 10 " H g from the standard pressure were kept. To efficiently use all remaining FCs, new isopotential surface contributions for groups CH-C, CH%H, CH2=C, CH=CFz, and CF=CF2 were developed using the GCA method described in Figure 1. These new contributions and those developed in the previous study are presented in Table 4. Figure 5 is a correlation between the experimentally determined Tb and values estimated using equation 12. For a boiling point range between 29 and 237 "C, the method yields an
J. Phys. Chem., Vol. 99, No. 38, I995 13915
Group Contribution and Quantum Mechanical Models
TABLE 5: Summary of the Test Results of the Models Developed in This Study physical property
Rm nDZo Tb
available model GCA QM GCA QM GCA QM
input parameter molecular structure
a, Rm, M w , Q
Rat, Mw Q
V,A RaC,V ,A Rm,
applicability
RF,RH,C, N, 0, F, C1, Br, I, C-0, NH2, OH, CH=C, CH-CH, CH2%, CHzCF2, CFECFz same as aboveb same as above same as aboveb same as above plus CF=, CH=, CFH same as above plus CF-, CH=, CFH
na
av 6Em (%)
100
1.8
100 100 100
50
3.4 0.5 0.9 5.76
50
7.56
a Number of fluorochemicals used in model development. May be applicable to FCs containing groupslatoms beyond what is shown. Additional testings are required, however, before it could be used outside the realm that has been tested here. Molar refractive contribution was calculated using a. The higher error is introduced by Tb data measured at slightly reduced pressures (c760 mmHg).
average error of 5.7% (n = 50). This is slightly higher than the error (3%) in the previous model.3' Since the basis set included Tb data obtained at slightly reduced pressures, the increased error may in part be due to errors in Tb measurements. Much effort has gone into reducing propagated errors during the conversion process from a to Ra,hence it is unlikely that error propagation is the cause of the increase. The fact that higher boiling compounds (with higher Mw) yield nearly the same errors as the lower boiling compounds provides evidence that the increased error is not due to error propagation. Applicability of Models. In general, the limitation of any given model is characterized by the type and physical properties of the compounds used in the basis set, and this holds true for the GCA and QM models. The best results will be obtained when applied to liquid FCs and FC-HC diblocks that have similar molecular structure and physical characteristics as those listed in Table 1. Compounds containing between 9% and 71% fluorines (by total atom) with molecular weight, molar refractivity, index of refraction, and boiling point in the ranges 97-700 @mol, 18-71 cm3, 1.2400-1.5198, and 30-230 "C, respectively, can be expected to have similar results as those noted earlier. Applications and limitations of the present models are summarized in Table 5 . It is important to reemphasize that the group contributions for normal alkanes (e.g., CH2 and CH3) were developed specifically for FC-HC diblocks like those shown in Table 1 and should be used only on such compounds. Larger diblocks without heteroatoms have not been tested (due to lack of data) may result in larger errors. In addition, since the models are based on the principle that the London dispersion forces are the dominate molecular attractive forces in fluorochemicals, these models should not be applied to compounds where other forces (e.g., dipole-dipole interaction or hydrogen bonding) become increasingly important.
Summary A group contribution-additivity (GCA) model and a quantum mechanical (QM) model have been developed for predicting the molar refractions of a wide range of fluorochemicals. The GCA model provides quantitative molar refractive contributions for 23 heterogeneous groups and atoms, such as RF,RH,C, N, 0, F, C1, Br, I, C-0, NHz, OH, CH=C, CH%H, C H 2 4 , CH=CF?, CF=CF*, and yields an average absolute error of 1.8% (n = 100 compounds). The QM model, on the other hand, offers unlimited groups but requires knowledge of the polarizability descriptor. Polarizability can be obtained from a group contribution-additivity model developed in a previous study3' or from commercially available quantum mechanical software packages. The GCA and QM models have been successfully applied to the prediction of for a variety of FCs, with an average absolute error of less than 1%. Higher errors (5.7%) were observed in a predictive boiling point (Tb) model. This
increase in error is introduced by measurements of Tb at slightly reduced pressures. The Tb model, nonetheless, provides accurate predictions of boiling point as evidenced by the good correlation between experimental and estimated boiling points in Figure 5 .
Symbols Used number of molecular fragments isopotential surface area a 24 x 100 rectangular array consisting of 23 molecular fragments and corresponding R, for the 100 compounds shown in Table 1 a polarizability (A3) a polarizability (A3) computed from quantum mechanics using the PM3 Hamiltonian method error (%) method error in calculating (%) method error in calculating R, (%) method error in calculating Tb (%) fluorochemical fluorocarbon-hydrocarbon diblock compound group contribution-additivity (model) hydrocarbon conversion factor, 1024 A3 = 1 cm3 molecular weight (&mol) index of refraction measured at 20 "C using the yellow monochromatic light of the sodium D line (d = 589 nm) Avogadro's number (6.02205 x lou) quantum mechanics density (g/cm3) molar refraction as a function of polarizability fluorine-saturated alkyl fragment molar refractive contribution (cm3) molar refraction (cm3) molecular fragment normal boiling point (760 mmHg) isopotential surface volume
References and Notes (1) Krishnakumari, B.; Naseem, S.;Dutt, N. V. K. Ind. J. Technol. 1992, 30, 375.
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Le and Ween
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