P a r t i c l e i n

S c a t t e r i n g

R a y l e i g h

F a c t o r s

S c a t t e r i n g

E d w a r d F. C a s a s s a Department of Chemistry, Carnegie-Mellon University, Pittsburgh, PA, USA

A. Introduction: General Relations for a Homogeneous Solute B. Scattering Factors for Various Molecular Models C. Effects of Dispersion in Molecular Weight D. Determination of Molecular Weight and Radius of Gyration E. Calculation of Scattering Factors F. Particle Scattering Factors and Dissymmetries Table 1. Dissymmetries for Monodisperse Systems Table 2. Dissymmetries for Monodisperse and Polydisperse Coils G. References

VII-629

where M is the molecular weight of the solute, c is the concentration in units of mass/volume, Ai is the second virial coefficient, and q is an angle variable

VII-630 VII-631

(A2)

VII-632 VII-633 VII-633 VII-634 VII-634 VII-635

A. INTRODUCTION: GENERAL RELATIONS FOR A HOMOGENEOUS SOLUTE (1,2)

h denoting the refractive index of the solution, and A the vacuum wavelength of the incident radiation. In the optical constant (assuming incident radiation polarized perpendicular to the plane containing angle 9) (A3) dh/dc is the specific refractive index increment at constant pressure and temperature, and NA is Avogadro's number. The intensity R(q, c) is that due to the solute; the (relatively weak) background solvent scattering is assumed to be that measured for the pure solvent. The first term on the right-hand side of Eq. (Al) contains the intramolecular interference function, or particle scattering factor, defined by (4,5)

When electromagnetic radiation interacts with any fluid medium, inhomogeneities in refractive index (dielectric constant) arising from microscopic thermal fluctuations in density and composition cause scattering of a small fraction of the incident radiation. If scattering elements are spatially correlated over distances of the order of magnitude of the wavelength of the exciting radiation, the scattering is subject to optical interferences that affect the angular intensity distribution. This distribution can provide information on the size and shape of macromolecular solutes or suspended colloidal particles: in principle, the molecular radius of gyration is determinable unambiguously, and calculated scattering distributions for idealized models can be compared with experimental data to infer an acceptable molecular conformation. However, an experimental scattering distribution does not permit a unique model assignment in the absence of additional information or supposition. The limiting dilute-solution behavior of the reduced intensity R(q, c) (Rayleigh's ratio) of light scattered through an angle 9 is given by (3)

the superscript zero denoting the limit of infinite dilution. Thus the intensity distribution P(q) is normalized to unity at zero scattering angle. Unless particle dimensions are infinitesimally small compared with A, P(q) is less than unity for q < 0; and for particles of less than colloidal dimensions, it is almost always a monotonously decreasing function of q throughout the accessible range of q with visible light. The second term of Eq. (Al) contains the effects of pairwise interactions between solute molecules (and, implicitly, of interactions involving solvent). The dimensionless angle factor, Q(q), is a function generally unknown, dependent on both intermolecular and intramolecular correlations (3,6). Like P(q) it is normalized to unity at 9 = 0, but usually the approximation is made that Q(q) is unity at all angles so that

(Al)

(A5)

(A4)

replaces Eq. (Al). In all that follows, Eq. (A5), which is Zimm's "single contact" approximation (4), is assumed to hold for any macromolecular species in solution. At 6 = 0, Eqs. (Al) and (A5) revert to a thermodynamic equation of a state in virial form. A scattering entity (polymer molecule, particle) may be represented by n optically isotropic point scattering centers whose spatial arrangement in any given conformation is specified by n(n — l)/2 « n2 J2 vectors r,y, separating pairs of scattering centers ij. The scattering function P(q) is then

where (B4) is Dawson's integral, for which tabulations are available (9). The variable u is defined as in the first equality in Eq. (B2), so that in this case u — 2q2R2ori since the meansquare radius R\sin% is nb2/12. For flexible regular-star branched molecules with / identical arms (/ random-flight chains starting from a common node), P(q) is given by (10,11)

(A6) The vector q is the difference between vectorial wave numbers of incident and scattered rays such that |q| — 3 (Ref. 34):

E. CALCULATIONS OF SCATTERING FACTORS Figure 2 and Tables 1 and 2 illustrate some quantitative comparisons among scattering factors for several molecular models and the effects of dispersion in molecular weight for Gaussian coils. Tables of P{q) and Z45 in the older literature are much more extensive and reflect the needs of a time without widely available computer technology. Today, for careful work to compare experimental R(q, 0) data with theoretical scattering functions, investigators will find it preferable to return to the mathematical functions and write computer routines to do calculations for data analysis rather than to routinely interpolate in tables. For those who may wish to use a traditional programming language such as FORTRAN, the following rational approximation formulas will facilitate accurate calculation of the special functions that appear in the various expressions for P(q). The sine integral Si(x) for x < 1 (Refs. 32,33):

Modern mathematics programs such as Mathematica and Maple constitute higher level languages that make possible direct operations with special functions on a personal computer (35,36). F. PARTICLE SCATTERING FACTORS AND DISSYMMETRIES

sphere disk ring

llP{q)

4-star

coil rod

The Bessel function Jx(x) for 0 < x < 3 (Ref. 34):

Figure 2. Reciprocal scattering factors yP{q) versus q2R2 for monodisperse systems of the models indicated. R2 is: nb2/6 (linear Gaussian coil); 5nb2/48 (4-branch regular star); nb2/M (flexible ring); L2/12 (thin rod); D2/8 (disk); 3D2/20 (sphere). References page VII-635

TABLE 1. DISSYMMETRIES FOR MONODISPERSE SYSTEMS Z45 and P- 1 C^o) R%h/ X

Coil

4-Starfl

Rod

Disk

Sphere

0.00 0.01 0.02 0.03 0.04

1.000 (1.000) * 1.004(1.003) 1.015(1.011) 1.034(1.024) 1.060(1.043)

1.000 (1.000) 1.004(1.003) 1.015(1.011) 1.034(1.024) 1.061(1.043)

1.000 (1.000) 1.004(1.003) 1.015(1.011) 1.034(1.024) 1.060(1.043)

1.000 (1.000) 1.004(1.003) 1.015(1.011) 1.034(1.024) 1.061(1.043)

1.000 (1.000) 1.004(1.003) 1.015(1.011) 1.034(1.024) 1.062(1.043)

0.05 0.06 0.07 0.08 0.09

1.094 (1.067) 1.136(1.097) 1.186(1.133) 1.245(1.175) 1.311 (1.224)

1.095 (1.067) 1.139(1.098) 1.191(1.135) 1.253(1.178) 1.325 (1.229)

1.093 (1.067) 1.137(1.097) 1.187(1.133) 1.245(1.176) 1.311 (1.225)

1.097 (1.068) 1.143(1.099) 1.200(1.138) 1.268(1.183) 1.351 (1.237)

1.098 (1.068) 1.146(1.100) 1.204 (1.139) 1.277 (1.186) 1.365 (1.242)

0.10 0.11 0.12 0.13 0.14

1.386(1.280) 1.469(1.342) 1.559(1.412) 1.657(1.490) 1.761(1.575)

1.407(1.287) 1.500(1.353) 1.604(1.428) 1.720(1.512) 1.848(1.607)

1.384(1.280) 1.463(1.343) 1.546(1.412) 1.632(1.488) 1.717(1.570)

1.448(1301) 1.564(1.374) 1.700 (1.459) 1.859(1.558) 2.044(1.671)

1.473(1.308) 1.604(1.386) 1.765(1.477) 1.963(1.584) 2.208(1.710)

0.15 0.16 0.17 0.18 0.19

1.872(1.669) 1.987 (1.771) 2.108(1.881) 2.231 (2.001) 2.357 (2.129)

1.987(1.712) 2.138 (1.828) 2.299(1.957) 2.470 (2.098) 2.650 (2.253)

1.801(1.659) 1.879 (1.753) 1.951(1.853) 2.013 (1.956) 2.067 (2.063)

2.258(1.800) 2.504 (1.949) 2.783(2.120) 3.096 (2.315) 3.439 (2.538)

2.514(1.858) 2.901 (2.034) 3.397(2.242) 4.045 (2.490) 4.906 (2.789)

0.20 0.21 0.22 0.23 0.24

2.485(2.266) 2.613 (2.413) 2.741(2.568) 2.868 (2.733) 2.993(2.906)

2.837(2.422) 3.030 (2.606) 3.277(2.806) 3.427 (3.022) 3.627(3.254)

2.110(2.171) 2.145 (2.279) 2.173(2.387) 2.195 (2.493) 2.212(2.597)

3.807(2.791) 4.190 (3.079) 4.574(3.403) 4.942 (3.768) 5.278(4.174)

6.080(3.149) 7.724 (3.589) 10.112(4.130)

0.25 0.26 0.27 0.28 0.29

3.116(3.089) 3.236(3.280) 3.353 (3.481) 3.466 (3.690) 3.576 (3.908)

3.826(3.502) 4.022(3.768) 4.214 (4.051) 4.400 (4.350) 4.579 (4.667)

2.226(2.698) 2.238(2.796) 2.249 (2.892) 2.259 (2.985) 2.268 (3.076)

5.566(4.622) 5.797(5.112) 5.970 (5.640) 6.089 (6.202) 6.163 (6.788)

a b

Regular star with four arms. Entries in parentheses indicate P~l(qg0) values.

TABLE 2.

DISSYMMETRIES FOR MONODISPERSE AND POLYDISPERSE COILS*

P-(^)

Z45

p =l

/t> = 1 . 2

and ( ^

(*^ 2 )

/7 = 1 . 5

p = 2.0

p = 5.0

1.00 1.02 1.04 1.06 1.08

1.000(O)6 1.014(23) 1.028 (33) 1.042 (40) 1.056(46)

1.000(0) 1.014(21) 1.028 (30) 1.042 (37) 1.056(43)

1.000(0) 1.014(20) 1.028 (29) 1.042 (35) 1.056(40)

1.000(0) 1.014(19) 1.028 (27) 1.042 (33) 1.056(38)

1.000(0) 1.014(17) 1.028 (24) 1.042 (30) 1.056(35)

1.10 1.12 1.14 1.16 1.18

1.070 (51) 1.085(56) 1.099(61) 1.113(65) 1.128(69)

1.070 (48) 1.085(53) 1.099(57) 1.114(61) 1.129(65)

1.071 (45) 1.086(49) 1.100(53) 1.115(57) 1.130(61)

1.071 (43) 1.086(47) 1.101(51) 1.116(54) 1.131(58)

1.071 (39) 1.087(43) 1.102(47) 1.116(50) 1.133(53)

1.20 1.22 1.24 1.26 1.28

1.142(72) 1.157(76) 1.171(79) 1.186(82) 1.201(85)

1.143(68) 1.159(71) 1.173(75) 1.189(78) 1.204(80)

1.145(64) 1.161(67) 1.176(71) 1.191(73) 1.207(76)

1.146(61) 1.162(64) 1.177(67) 1.193(70) 1.210(73)

1.148(56) 1.164(59) 1.180(62) 1.196(65) 1.213(68)

TABLE 2. conf'c/ P ^ 9 0 ) and ( ^ Z45

P= I

p = 1.2

(Rl)1J^

p = 1.5

p = 2.0

p = 5.0

1.30 1.40 1.50 1.60 1.70

1.215(88) 1.290(102) 1.366(113) 1.444(124) 1.524(134)

1.219(83) 1.297(97) 1.377(109) 1.460(119) 1.546(129)

1.222(79) 1.303(92) 1.387(104) 1.473(115) 1.564(125)

1.225(76) 1.308(88) 1.394(100) 1.484 (111) 1.578(121)

1.229(70) 1.314(82) 1.404(93) 1.498(104) 1.597(114)

1.80 1.90 2.00 2.10 2.20

1.607 (144) 1.693(152) 1.781(161) 1.874(169) 1.969(178)

1.636 (139) 1.729(149) 1.826(158) 1.927(167) 2.032(175)

1.659 (135) 1.758(145) 1.861(154) 1.971(163) 2.084(172)

1.677 (131) 1.782(141) 1.890(150) 2.007(160) 2.128(169)

1.703 (125) 1.815(134) 1.932(144) 2.058(154) 2.190(163)

2.30 2.40 2.50 2.60 2.70

2.069 (185) 2.174(194) 2.283 (201) 2.397 (209) 2.517 (217)

2.143 (184) 2.260(192) 2.382 (201) 2.510 (209) 2.645 (218)

2.205 (181) 2.332(190) 2.466 (199) 2.607 (209) 2.757 (218)

2.257 (178) 2.393(188) 2.538 (198) 2.691 (207) 2.855 (217)

2.332 (173) 2.484(183) 2.645 (193) 2.817 (203) 3.003 (214)

2.80 2.90 3.00 3.50 4.00

2.643 2.776 2.916 3.755 4.948

2.788 2.939 3.099 4.070 5.488

2.916 3.085 3.264 4.375 6.044

3.029 3.215 3.414 4.665 6.601

3.202 (225) 3.416 (236) 3.648 (247) 5.145 (312) 7.588 (397)

11 b

(225) (233) (240) (283) (333)

(227) (235) (244) (292) (349)

(227) (236) (246) (299) (364)

(227) (237) (247) (305) (377)

Polydisperse systems characterized by the Schulz-Zimm distribution: p — A/ w /A/,,. Entries in parentheses denote weight-average value of R2%.

G. REFERENCES 1. H. Yamakawa, "Modern Theory of Polymer Solutions", Harper and Row, New York, 1971, Ch. 5. 2. E. F. Casassa, G. C. Berry, in: P. E. Slade Jr., (Ed.), "Polymer Molecular Weights", Marcel Dekker, New York, 1975, C. 5. 3. A. C. Albrecht, J. Chem. Phys., 27, 1014 (1957). 4. B. H. Zimm, J. Chem. Phys., 16, 1093 (1948). 5. P. Doty, R. F. Steiner, J. Chem. Phys., 18 1211 (1950). 6. E. F. Casassa, J. Polym. Sci., Polym. Phys. Ed., 17, 2077 (1979). 7. P. Debye, J. Phys. Colloid Chem., 51, 18 (1947). 8. E. F. Casassa, J. Polym. Sci. A, 3, 605 (1965). 9. M. Abramovitz, I. A. Stegun (Eds.), "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables", National Bureau of Standards Applied Mathematics Series 55, US Government Printing Office, Washington DC, 10th printing, 1972. 10. H. Benoit, J. Polym. Sci., 11, 507 (1953). 11. E. F. Casassa, G. C. Berry, J. Polym. Sci. A-2, 4, 881 (1966). 12. H. Benoit, Compt. Rend., 245, 2244 (1957). 13. O. B. Ptitsyn, Zh. Fiz. Khim., 31, 1091 (1957). 14. A. J. Hyde, J. H. Ryan, F. T Wall, J. Polym. Sci., 33, 129 (1958). 15. P. J. Flory, R. L. Jernigan, J. Am. Chem. Soc, 90, 3128 (1968). 16. A. Peterlin, in: M. Kerker, (Ed.), "Electromagnetic Scattering", Pergamon Press, New York, 1963, p. 357.

17. T. Neugebauer, Ann. Phys., 42, 509 (1943). 18. O. Kratky, J. Porod, J. Colloid Sci., 4, 35 (1949). 19. M. Kerker, "The scattering of Light and other Electromagnetic Radiation", Academic Press, New York, 1969, Ch. 8. 20. B. H. Zimm, J. Chem. Phys., 16, 1099 (1948). 21. J. G. Kirkwood, R. J. Goldberg, J. Chem. Phys., 18, 54 (1950). 22. W. H. Stockmeyer, J. Chem. Phys., 18, 58 (1950). 23. H. Benoit, J. Polym. Sci., 11, 507 (1953). 24. A. H. Dautzenberg, C. Ruscher, J. Polym. Sci. C, 16, 2913 (1967). 25. K. Kajiwara, W. Burchard, M. Gordon, Brit. Polym. J., 2, 110(1970). 26. W. Burchard, Macromolecules, 10, 919 (1977). 27. M. Goldstein, J. Chem. Phys., 21, 1255 (1953). 28. E. F. Casassa, J. Am. Chem. Soc, 78, 3980 (1956). 29. W. H. Beattie, C. Booth, J. Phys. Chem., 64, 696 (1960). 30. P. Becher, J. Phys. Chem., 63, 1213 (1959). 31. W. H. Beattie, C. Booth, J. Polym. Sci., 44, 81 (1960). 32. M. Abramovitz, I. A. Stegun (Eds.), "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables", National Bureau of Standards Applied Mathematics Series 55, US Government Printing Office, Washington DC, p. 232. 33. C. Hastings, Jr., "Approximations for Digital Computers", Princeton University Press, Princeton, 1955, pp. 197, 199.

34. M. Abramovitz, I. A. Stegun (Eds.), "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables", National Bureau of Standards Applied Mathematics Series 55, US Government Printing Office, Washington DC, p. 370.

35. S. Wolfram, The Mathematica Book, Wolfram Media, Champaign, Illinois, 1996. 36. B. W. Char, K. O. Geddes, K. H. Gonnet, B. L. Leong, M. B. Monagan, S. M, Watt, "First Leaves: ATutorial Introduction to Maple V", Springer, New York, 1992.