Drude equations

My interest at that time revolved around evaluating optical rotary dispersion data [12]. The paired values of optical rotation vs. wavelength were used to fit a function called the Drude equation (later modified to the Moffitt equation for William Moffitt [Harvard University] who developed the theory) [13]. The coefficients of the evaluated equation were shown to be related to a significant ultraviolet absorption band of a protein and to the amount of alpha-helix conformation existing in the solution of it. 

The influence of the solvent on chiroptical properties of synthetic polymers is dramatically illustrated in the case of poly (propylene oxide). Price and Osgan had already shown, in their first article, that this polymer presents optical activity of opposite sign when dissolved in CHCI3 or in benzene (78). The hypothesis of a conformational transition similar to the helix-coil transition of polypeptides was rejected because the optical activity varies linearly with the content of the two components in the mixture of solvents. Chiellini observed that the ORD curves in several solvents show a maximum around 235 nm, which should not be attributed to a Cotton effect and which was interpreted by a two-term Drude equation. He emphasized the influence of solvation on the position of the conformational equilibrium (383). In turn, Furakawa, as the result of an investigation in 35 different solvents, focused on the polarizability change of methyl and methylene groups in the polymer due to the formation of a contact complex with aromatic solvents (384). 

Section II, 1. Theoretical aspects of asymmetric polymerization have been discussed by Fueno and Furdkawa [T. Fueno, J. Furukawa J. Polymer Sci., Part A, 2, 3681 (1964)]. 1-phenyl-l,3-butadiene has been polymerized using (R)-2-methyl-butyl-lithium or butyl-lithium complexed with menthyl-ethyl-ether, yielding optically active polymers with [a] f, referred to one monomeric unit, between +0.71 and —1.79. Optical rotation dispersion between 589 m u and 365 mft is normal and the Drude equation constant is comprised between 255 raft and 280 raft [A. D. Aliev, B. A. Krenisel, T. N. Fedoiova Vysokomol. Soed. 7, 1442 (1965)]. 

McCrackin, F. L., Colson, J. P. Computational techniques for the use of the exact drude equations in reflection problems, in Ellipsometry in the Measurement of Surfaces and Thin Films (eds.) Passaglia, E., Stromberg, R. R., Kurger, J., p. 61, NBS Miscellaneous Publication 256, Washington, D. C., Superintendent of Documents, U. S. Government Printing Office 1964... 

Ellipsometry measures the relative attenuation and phase shift of polarized light reflected from a polymer-coated surface. The Drude equations (Drude, 1889a,b, 1890 Stromberg et ai, 1963 McCrackin and Colson, 1964) relate the attenuation and phase shift to the average refractive index and thickness tel of an equivalent homogeneous film. Interpretation of fel in terms of the actual refractive index distribution or the polymer distribution [Pg.189]

The analysis is carried out using the Drude equations this leads to a combination of the ellipsometric thickness and the refractive index Increment. These characteristics of the adsorbate cannot be unambiguously separated. Conversion of the refractive index increment into the composition of the adsorbate layer is usually done by assuming drt/dx to be the same as in a fluid of composition x for 0 not too high this is usually allowed, but problems may arise when the adsorbate differs substantially from the solution, for Instance because of alignment of adsorbed chain molecules. The result obtained is not unique, in the sense that different profiles may lead to the same pair of ellipsometric parameters. Therefore, normally totally adsorbed amounts are presented. For accurate measurements a good optical contrast between adsorbate and solution is mandatory. 

Ellipsometric data are usually interpreted in terms of one equivalent homogeneous film with a refractive index n and an ellipsometric thickness d . This homogeneous layer is defined as a layer that gives the same reflected intensity and phase shift as the actual polymer layer with a z-dependent concentration. The parameters n " and can be extracted from the experimental data using the Drude equations (sec. 1.7.10b) usually, a numerical iteration is required. 

The rotary dispersion of collagen, like that of most of the a-helical proteins, is of the simple type (over the region 400-700 mu) and may be fitted to a one-term Drude equation ... 

The observed dispersions of many substances in the visible spectrum obey the simple Drude equation. 

Fig. 2. Graphical treatment of dispersion data in Fig. 1 according to the simple Drude equation (11). The disordered forms of the synthetic polypeptides, poly-y-benzyl-L-glutamate (PBG, O — O) and poly-n-glutamio acid (PGA, A — A), give linear plots, which specify their dispersions as simple. The dispersions of bovine serum albumin (BSA) in both its native ( ---- ) and denatured ( — ) forms are...

As an illustration, the rotatory dispersions shown in Fig. 1 have been replotted in Fig. 2 in a manner that yields linear behavior when the simple Drude equation is valid. It is seen that the disordered forms of the three substances do yield straight lines conforming with Eq. (10). The native form of bovine serum albumin does likewise, but the two purely helical structures, in contrast, exhibit distinct curvature. 

In order to apply this general relationship to the empirical data of helical polymers, Moffitt and Yang (1956) then developed a phenomenological equation in which the above sums are replaced by single terms and thus derived an expression that is analogous to the simple Drude equation. 

Inasmuch as this form of rotatory dispersion is in principle characteristic of the interaction of like chromophores, one would wish to employ it in the investigation of ordered polypeptide structures. But, as will become evident in the experimental use to which Eq. (18) has been put, it possesses in addition the distinct practical advantage that Xo of helices is close to Xc for disordered pol3q)eptide chains, that is, about 220 m/j, so that one equation can characterize mixtures of disordered chains and helices and at the same time yield a parameter, 6o, uniquely related to helical content. This feature is highlighted by a lack of similar adaptability displayed by alternative empirical equations proposed to describe helical dispersion. Yang and Doty (1957) have fitted data for helical poly-y-benzyl-L-gluta-mate to an abbreviated two-term Drude equation. 

The polypeptides in Table I all exhibit simple dispersion in the random coil, so that their optical rotation is given by the simple Drude equation at any wavelength within the range of its validity. 

The origin of nonzero bo values in these cases is not entirely clear, but it can be formally traced to the difference between Xo and values for the simple dispersion of random coils and hence to a failure of the assumption that Xo equals Xo. If K equals Xo, then the first term of the Moffitt equation will of course be the same as the simple Drude expression known to describe the data and, there being no necessity for a second term, bo will vanish. However, if Xo differs from Xo, the Moffitt plot may still be linear but with a nonvanishing slope. Thus dispersion data that are simple when referred to one dispersion constant may appear complex when plotted against another by a form that sees matters as complex, thereby generating what may be properly suspected as pseudocomplexity. The Moffitt equation was initially intended to describe the complex dispersion of polypeptides for which the simple Drude equation is inadequate, but, as will be seen, its form is also applied to protein dispersions which can be expressed equally well by either formula. It is therefore important to examine more fully the relation of the two equations for cases in which both fit the data. 

The general relation of X to the parameters of the Moffitt equation has been stated by Downie (1960). If a given set of dispersion data obey the simple Drude equation, Xo may be obtained from the slope of [m ]xX plotted against [m ], d([m ]xX )/d[m ]x (see Section II, B). If each value of [w ]x can also be accommodated by the Moffitt equation, a condition which is often satisfied for measurements in the spectral region 350-600 m, then Eq. (18) can be substituted into this slope and the appropriate differentiation carried out with the following result. 

On the other hand, the dispersions obtained by Yang and Doty (1957) for poly-y-benzyl-L-glutamate in a mixture of /3-forms and coil forms did not obey the simple Drude equation, and Imahori s analysis of these data yielded positive Moffitt slopes and intercepts after corrections had been made for the intrinsic residue contribution (Imahori, 1960). Imahori has moreover found that denatured bovine serum albumin and ovalbumin in solution display positive ba and corrected ao values in sharp contrast to the negative slopes characteristic of the native proteins and has in addition been able to correlate the positive slopes with the /3-form in the protein precipitates by infrared spectra. Wada et al. (1961) have recently carried forward this suggestion that the (8-conformation displays complex disper-... 

The optical rotatory dispersions of all native globular proteins thus far measured have obeyed the simple Drude equation in the spectral range usually encompassed, 350-600 m/i. In most cases, K values vary from 230 to 280 m/i, although a number of proteins have values below 230 m/i and some as low as 180 m/ . Optical rotations, which are usually reported as the specific rotations at 589 m/t, range from about —20 to —80°. Upon de-naturation, all dispersions remain simple, but values shift to the more narrow range of 210-230 m/i and the specific rotation becomes more levorotatory to a range of about —80 to —120°. 

The proposal to assimilate the rotatory dispersions of proteins to those of synthetic polypeptides is, in barest form, the recommendation to treat these same data by the Moffitt expression despite the fact that they can be described by the simple Drude equation. Thus one proceeds to plot [m ] -(X — Xo)/Xo against Xo/(X2 — Xo) with Xo set to 212 m/j, and thereby obtains an intercept, a , and a slope, which can be given conformational interpretations through Eqs. (35) and (36). These parameters, which are in general cited simply as Oo and foo when their experimental origin is... 

Fio. 14. Hypothetical mixtures of dispersion data for the helical and randomly coiled forms of poly-L-glutamic acid according to the simple Drude equation, together with experimental points for the two pure conformations, as shown previously in Figs. 1 and 2. The curvature characteristic of complex helical dispersion clearly emerges only as the proportion of helix increases above 40%. The dashed line indicates points corresponding to 589 m/i. (Yang and Doty, 1957.)... 

---