5538
J. Phys. Chem. 1996, 100, 5538-5540
Relationship between Measured Diffusion Coefficients and Calculated Molecular Surface Properties Peter Politzer,* Jane S. Murray, and Pa1 r Flodmark Department of Chemistry, UniVersity of New Orleans, New Orleans, Louisiana 70148 ReceiVed: NoVember 9, 1995; In Final Form: January 8, 1996X
We show that the diffusion coefficients in both dry and swollen gelatin of a group of organic molecules can be expressed analytically in terms of computed molecular quantities: surface areas and measures of positive and negative electrostatic interaction tendencies, obtained by ab initio SCF calculations. The molecules’ intrinsic mobilities are found to vary inversely with size and to be enhanced in these media by weakly positive and strongly negative surface electrostatic potentials.
Introduction In a recent series of applications of our general interaction properties function [GIPF], we have shown that a variety of liquid, solid, and solution properties that reflect molecular interactions can be represented analytically in terms of quantities computed for the individual molecules. Such relationships have now been developed for pKa values,1-3 supercritical solubilities4,5 and enhancement factors,6 boiling points and critical constants,7 partition coefficients,8,9 heats of vaporization,10 liquid and solid densities,11 surface tensions,11 and impact sensitivities.12 In the GIPF approach, we express the property of interest in terms of some subset of a group of eight quantities, most of which are related to the electrostatic potentials on molecular surfaces and reflect various aspects of the molecules’ capacities for noncovalent interactions.13 Our objective is to develop relationships which provide insight into the factors that govern these molecular interactions. In the present work, we extend this approach for the first time to a transport property, diffusion. Specifically, we seek an analytical representation of the diffusion coefficients in gelatin of a group of organic molecules. The latter are photographic developing agents, the function of which is to react with silver halide particles dispersed in the gelatin, reducing Ag+ to Ag.14,15 The intrinsic mobility of the organic molecule in this medium, as reflected in its diffusion coefficient D, is an important factor in determining the rate of the development process.14-17 Methods Optimized geometries were computed for the molecules 1-10 with the code GAUSSIAN 9218 at the ab initio HF/STO-3G level, which has been shown to be quite satisfactory for this purpose.19 These structures were used to calculate HF/STO5G molecular electrostatic potentials, from which were obtained the related quantities of interest, defined below. The electrostatic potential V(r) that is created at any point r by the nuclei and electrons of a molecule is given rigorously by eq 1. ZA is the charge on nucleus A, located at RA, and
ZA
X
m
2 σ+ ) (1/m)∑[V+(ri) - V h S+]2
(2)
i)1 n
2 σ-
) (1/n)∑[V-(rj) - V h S-]2
(3)
j)1
h S- are their averages: of V(r) on the surface, and V h S+ and V + m n 2 2 + h S ) (1/n)∑j)1 V-(rj). σ+ and σV h S ) (1/m)∑i)1V (ri) and V indicate the variabilities of the positive and negative potentials; because they involve squared terms, they emphasize the extrema in these regions. We have found that these two quantities, 2 2 and σ, plus the computed molecular surface area, suffice σ+ to permit accurate analytical representations of the diffusion coefficients of 1-10 in gelatin. The SAS statistical analysis program was used for this purpose.24 Results 2 2 and σfor Table 1 presents the calculated surface area, σ+ each of the molecules 1-10. Also listed are their experimentally determined diffusion coefficients in both dry and swollen gelatin. The larger are their magnitudes, the greater is the intrinsic mobility of the molecule in the medium. We found that the measured diffusion coefficients can be 2 2 and σby eqs 4 and 5. expressed in terms of surface area, σ+ Dry gelatin:
2 + D × 107 (cm2 s-1) ) 533.5(area)-1 - 0.03168σ+
F(r′) dr′
V(r) ) ∑ -∫ |r′ - r| A |RA - r|
F(r) is the electronic density. V(r) was computed on each molecular surface, the latter being defined as the 0.001 au contour of the electronic density, as suggested by Bader et al.20 The conclusions reached with the GIPF technique are not sensitive to the exact contour chosen to represent the surface, within the range 0.0005-0.002 au.21-23 We have found that the variances of the positive and negative potentials on the surface of a molecule are effective quantitative measures of its tendencies for positive and negative electrostatic interactions.8,9 These are statistical quantities, defined by eqs 2 and 3. V+(ri) and V-(rj) are the positive and negative values
2 - 1.620 (4) 0.01425σ-
(1)
Abstract published in AdVance ACS Abstracts, March 1, 1996.
0022-3654/96/20100-5538$12.00/0
correlation coefficient ) 0.990 standard deviation ) 0.09 © 1996 American Chemical Society
Diffusion Coefficients and Surface Properties
J. Phys. Chem., Vol. 100, No. 13, 1996 5539
TABLE 1: Diffusion Coefficients of Developing Agents in Gelatin Layera and Computed Molecular Surface Properties 107D, dry layer (cm2 s-1)
predicted 107D, dry layer
107D, swollen layer (cm2 s-1)
predicted 107D, swollen layer
area (Å2)
2 σ+ (kcal/mol)2
2 σ(kcal/mol)2
1
1.85
1.81
16.7
16.3
191
35.5
123.5
2
1.69
1.61
15.2
14.5
142
69.9
118.6
3
1.58
1.55
14.3
14.0
163
50.0
103.9
4
1.42
1.49
12.8
13.5
156
72.4
141.1
5
1.20
1.27
10.8
11.4
184
46.7
103.0
6
1.05
0.97
9.46
8.70
226
39.8
104.3
7
0.962
1.05
8.66
9.44
217
29.9
81.1
8
0.69
0.79
6.2
7.12
137
90.9
98.0
9
0.42
0.34
3.8
3.08
143
93.1
82.9
10
0.32
0.31
2.9
2.78
157
83.6
82.8
developing agent
a
Levenson, G. I. P. In The Theory of the Photographic Process, 4th ed.; Macmillan: New York, 1977; Chapter 16.
Swollen gelatin: 2 D × 107 (cm2 s-1) ) 4815(area)-1 - 0.2860σ+ + 2 - 14.60 (5) 0.1283σ-
correlation coefficient ) 0.990 standard deviation ) 0.83 The values predicted for the diffusion coefficients from eqs 4 and 5 are also in Table 1. Overall, they are in good agreement with the experimental data, as is shown in Figure 1. Discussion Equations 4 and 5 show that the diffusion coefficients vary inversely with molecular area. This is in agreement with theoretical analyses, which predict an inverse dependence of D upon the size of the molecule.25,26 However, the area is only one of three quantities which make contributions of the same order of magnitude to D. Thus, 2 and 9 have essentially the same areas but very different diffusion coefficients; the same is true of 4 and 10. 2 2 and σThe effects of the medium can be seen in the σ+ terms in eqs 4 and 5. Gelatin is a mixture of water-soluble proteins27 and consequently contains a large number of amide groups, >N-C(dO)-, which are the peptide linkages. There are strongly negative electrostatic potentials associated with the lone pairs on the amide nitrogen and oxygen, which can interact with positive sites on the diffusing molecules, 1-10, such as hydroxyl and amine hydrogens, and inhibit diffusion. Accord-
Figure 1. Plot of predicted vs experimentally measured diffusion coefficients of the developing agents in Table 1 in a dry gelatin layer. The linear correlation coefficient is 0.990. 2 ingly diffusion should be promoted by σ+ being small in magnitude, indicating a low tendency on the part of the molecule 2 being large, for interaction through positive regions, and by σsuggesting repulsive interactions with the gelatin. For example, 1 has the highest diffusion coefficients in Table 1 despite its 2 2 . and high σrelatively large size, because of its low σ+ Conversely, 8-10 are among the smallest of the molecules in Table 1 but have the lowest D values because of their relatively 2 2 and low σ, both of which impede diffusion in these high σ+ media. These considerations are reflected in the signs of the 2 2 are term being negative in eqs 4 and 5, while the σσ+ positive.
5540 J. Phys. Chem., Vol. 100, No. 13, 1996 A striking feature of the experimentally determined diffusion coefficients in Table 1 is that the values for swollen gelatin are all 9.0 or 9.1 times as great as those for dry gelatin; as a result, each coefficient in eq 5 is exactly 9.0 times as great as its counterpart in eq 4. This remarkably consistent enhancement of diffusion in swollen gelatin may be due to a dilution effect. Acknowledgment. This work was supported in part by the Eastman Kodak Co. References and Notes (1) Brinck, T.; Murray, J. S.; Politzer, P.; Carter, R. E. J. Org. Chem. 1991, 56, 2934. (2) Brinck, T.; Murray, J. S.; Politzer, P. J. Org. Chem. 1991, 56, 5012. (3) Politzer, P.; Lane, P.; Murray, J. S.; Brinck, T. J. Phys. Chem. 1992, 96, 7938. (4) Brinck, T.; Murray, J. S.; Politzer, P. Int. J. Quantum Chem. 1993, 48, 73. (5) Politzer, P.; Murray, J. S.; Lane, P.; Brinck, T. J. Phys. Chem. 1993, 97, 729. (6) Murray, J. S.; Lane, P.; Brinck, T.; Politzer, P. J. Phys. Chem. 1993, 97, 5144. (7) Murray, J. S.; Lane, P.; Brinck, T.; Paulsen, K.; Grice, M. E.; Politzer, P. J. Phys. Chem. 1993, 97, 9369. (8) Brinck, T.; Murray, J. S.; Politzer, P. J. Org. Chem. 1993, 58, 7070. (9) Murray, J. S.; Brinck, T.; Politzer, P. J. Phys. Chem. 1993, 97, 13807. (10) Murray, J. S.; Lane, P.; Politzer, P. J. Mol. Struct. (THEOCHEM) 1995, 342, 15. (11) Murray, J. S.; Brinck, T.; Politzer, P. Chem. Phys., in press.
Politzer et al. (12) Murray, J. S.; Lane, P.; Politzer, P. Mol. Phys. 1995, 85, 1. (13) Murray, J. S.; Brinck, T.; Lane, P.; Paulsen, K.; Politzer, P. J. Mol. Struct. (THEOCHEM) 1994, 307, 55. (14) The Theory of the Photographic Process, 4th ed.; Macmillan: New York, 1977. (15) Kirk-Othmer Encyclopedia of Chemical Technology, 3rd ed.; WileyInterscience: New York, 1982; Vol. 17. (16) Jaenicke, W. J. Photogr. Sci. 1972, 20, 2. (17) Brown, E. R.; Tong, L. K. J. Photogr. Sci. Eng. 1975, 19, 314. (18) Frisch, M. J.; Trucks, G. W.; Head-Gordon, M.; Gill, P. M. W.; Wong, M. W.; Foresman, J. B.; Johnson, B. G.; Schlegel, H. B.; Robb, M. A.; Replogle, E. S.; Gomperts, R.; Andres, J. L.; Raghavachari, K.; Binkley, J. S.; Gonzalez, C.; Martin, R. L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A. GAUSSIAN 92; Revision C; Gaussian, Inc.: Pittsburgh, PA, 1992. (19) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley-Interscience: New York, 1986. (20) Bader, R. F. W.; Carroll, M. T.; Cheeseman, J. R.; Chang, C. J. Am. Chem. Soc. 1987, 109, 7968. (21) Murray, J. S.; Brinck, T.; Politzer, P. Int. J. Quantum Chem., Quantum Biol. Symp. 1991, 18, 91. (22) Murray, J. S.; Brinck, T.; Grice, M. E.; Politzer, P. J. Mol. Struct. (THEOCHEM) 1992, 256, 29. (23) Brinck, T.; Murray, J. S.; Politzer, P. Mol. Phys. 1992, 76, 609. (24) SAS; SAS Institute Inc.: Cary, NC 27511. (25) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954. (26) Levine, I. N. Physical Chemistry, 3rd ed.; McGraw-Hill Book Co.: New York, 1988. (27) The Merck Index, 10th ed.; Windholz, M., Ed.; Merck & Co.: Rathway, NJ, 1983.
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