low-angle scattering in polymers

Low-Angle Scattering in Polymers

0. KRATKY, Institute for Theoretical and Physical Chemistry of the Uninersity of Graz, Austria

DISCONTINUOUS LOW-ANGLE SCATTERING

THE DIFFRACTION of x-rays under very small angles is called x- ray low-angle scattering.’ The diffraction occurs either in the form of sharp interferences (discontinuous) or as diffuse scattering (continuous).

Discontinuous low-angle interferences were first observed by American and British investigators with single crystals of soluble proteins.2 The very large molecules of these substances necessitate large unit cells and large lattice constants; therefore, according to Bragg’s law, the x-ray reflections must occur a t very small angles. * These reflections can be employed for structure determinations using well- known methods. A determination of the molecular size is comparatively easy, but the completion of a real structure analysis is extremely difficult because of the large number of atoms in the unit cell. To date, there has been onIy one successfully completed Fourier ana l~s is .~ It seems that with protein Gbers the relations between the x-ray diagram and the structure are easier to comprehend. An experiment along these lines will be discussed.

Low-Angle Interferences from Protein Fibers and Molecular Structure

.The protein fibers (collagen, keratin, myosin, and silk fibroin) are polycrystals with fiber structure. In the 1930’s, i t was discovered in the Anglo-Saxon countries that these substances show low-angle diagrams with pronounced interferences in addition to the familiar wide-angle pattern^.^ The lengths of the periods that had been calculated from the wide-angle diagrams discovered by R. 0. Herzog and co- workers are fractions of the much larger, true identity periods, which, in. turn, give

1 Review articles: J. D. H. Donnay and C. G. Shull, Am. Soc. X-Ray Electron Diffraction 1, iii (1946). See also W. Nowacki, Schweiz. Chem. Ztg. u. Tech. Id., 29, No. 14-15 (1946).

2 G. L. Clark and K. E. Corrigan, Phys. Rev., 40,639 (1932). I. Fankuchen, J. Am. Chem. Soc., 56, 2398 (1934). J. D. Bernal and D. M. Crowfoot, Nature, 133, 794 (1934). D. M. Crow- foot, ibid., 135, 591 (1935); 140, 149 (1937).; 144, 1011 (1939); Proc. Roy. Soc. London, A164, 580 (1938); Review article: E. J. Cohn, I. Fankuchen, J. L. Oncley, H. B. Vickery, and B. E. Warren, Ann. N. 3’. Aead. Sci., 41,77 (1941).

* Bragg’s well-known law states that nX = 2 0 sin 6, where n is the order of reflection, X is the wave length of the x-rays used, D is the lattice plane spacing, and 6 is half the angle of deflection of the x-ray.

Chem. Revs., 28, 215 (1941).

Note that large values of D result in small values for 6. 3 J. Boyes-Watson and M. F. Perutz, Nature, 151, 714 (1943). 4 R. W. G. Wyckoff, R. B. Corey, and J. Biscoe, Science, 82, 175 (1935). R. B. Corey and

R. W. G. Wyckoff and R. B. Corey, Proc. R. W. G. Wyckoff, J. Biol. Chem., 114, 407 (1936). Soc. Ezptl. Biol. Med., 34, 285 (1936). I. MacArthur, Nature, 152, 38 (1943).

Volume 3, No. 2 (1948) 195

0. K R A T K Y

rhe to the low-angle diagrams. We are in the main interested in the p e r i d along the fiber axis, because these are apparently most intimately related to the molecular structure of the polypeptide chains that are parallel to the fiber axis.

Following the investigations of silk firoin by *11,6 Meyer and Mark6 ca,me to the conclusion that the ordinary f i e r period of 7 A. corresponds to the length of the glycyl residue plus alamine residue and that, to a fist approximation, the structure of the substance can be thought of as a polypeptide chain consisting of those two building stones which alternate along the fiber axis. Subsequently, Astbury: in a series of comprehensive investigations, endeavored to establish the connection between the results of the chemical analysis and the length of the x-ray periods along the fiber axis. Astbury proposed that each fiber protein is built according to an ideal structural plan, in which the various amino acid residues in the polypeptide chain along the fiber axis are supposed to be arranged in a regular sequence, and their periodicity (the number of residues containing one residue of the kind under consideration) is given by a number of the type 2”3”, in accordance with the rule of Bergmann and Niemann.* Deviations from this ideal regularity may occur, through interchanges of individual amino acid residues, without modifying the fundamental structural plan. A similar situation is encountered with the silicates, in which silicon atoms may be replaced by various cations without loss of the original crystal structure. The largest periods found experimentally, and on which Astbury’s discussion was based, are of the order of 100 A.

A decisive enlargement of the experimental basis was accomplished during the war, when even greater periods were found and accurately measured. On different collagen fibers Bearn and, indepentently, the author together with Sekora,lo measured periods of the order of 600 A. Investigating the kangaroo tail tendon, the latter authors found a basic period along the fiber axis of 642 A. In addition, nu- merous higher orders of this distance are observed, up to the 220th order, as shown in Table I. This high order corresponds to a distance of 642/220 or 2.89 A., which, according to Astbury,” is the length of a single amino agd residue in collagen. Therefore, it appears certain that the giant period of 642 A. contains a group of 220 residues; with an average weight of about 94 per residue in collagen, this group must have a “moleculir weight” of 220 X 94 or about 20,000.

Figures 1 and 2 show photometric curves of x-ray diagrams of the kangaroo

6 R. Brill, Ann., 434,204 (1923). 6 I(. H. Meyer and H. Mark, Ber., 61,1932 (1928). 1 W. T. Astbury, Kolloi&Z., 69, 340 (1934); J. I A r n . Soc. Leather Trades’ Chemists, 24,

69 (1940); Trans. Faraclay Soc.. 36,871 (1940); Chemistry d Industry, 60,491 (1941); J . Chem. SOG., 1942,337. Ann. Repts. Progress Chem. Chem. Soc. London, 39, 337 (1942). W. T. Astbury and H. J. Woods, Tmns. Roy. Soc. London, A232,333 (1933). W. T. Astbury and W. A. Sisson, Pm. Roy. Soc. London, A150, 533 (1935). W. T. Astbury and S. Dickinson, ibid., B129. 307 (1940).

* M. Bergmann and C. Niemann, J. Biol. Chem., 110, 471 (1935); 115, 77 (1936); 118, 301 (1937); Science. 36,187 (1937).

* R. S. Bear, J. Am. Chem. Soc., 64, 727 (1942); 65, 1784 (1943); 66, 1297,2043 (1944); 67, 1625 (1945).

10 0. Kratky and A. Sekora. J . makromol. Chem., 1,113 (1943). 11 W. T. Astbury, J. Intern. 5ix. Leather Trades’ Chemists, 24,69 (1940).

196 Journal of Polymer Sciencr

L O W - A N G L E S C A T T E R I N G I N P O L Y M E R S

%,.I. ~- 642 321 214.5 160.5 128.3 107.8 90.7 79.4 70.7 58.3

n b

1 2 3 4 5 6 7 8 9

11

642/n

642 321 214 160.6 128.4 107 91.7 80.3 71.3 58.3

TABLE I LATTICE PLANE SPACINGS~

Intensity

Very strong Very strong Very strong Weak Weak Very strong Weak Strong Strong Very strong

53.4 36.6 32.2 24.2 21.6 (8.00) 4.52 4.00 3.51 2:88

nb

12 18 20 26 30 80

140 160 180 220

53.5 35.7 32.1 24.7 21.4 (8.03) 4.58 4.01 3.57 2.92

Weak Weak Medium strong Very weak Very weak Weak Very weak Weak Very weak Strong

0 Measurements taken along the fiber axis in kangatoo tail tendon (according to Kratky

b Orders of at least medium intensity emphasized by boldface. and Sekora).

I I \ I

Fig. 1. -Photometer curve of the meridian of an x-ray diagram of the kangaroo tail tendon; medium resolution. The figures indicate the order of the reflection of D = 642 b. Scale of reproduction: 4; distance specimen-to-film: 104.5 mm.

Fig. 2. Photometer curve of the meridian of an x-ray diagram of the kangaroo tail tendon; higher resolution. The figures indicate the order of the reflection of D = 642 d. Scale of reproduction: 5.6; distance specimen-to-am: 104.5 mm.

tail tendon of medium and of high resolution. We gather from these figures as well as from Table I that the lst, Znd, 3rd, 6th, 8th, 9th, l l th , 20th, and 220th orders of the giant period are distinguished by their high intensities. It has been shown,lQ.l2 and it will be explained below, that the appearance of individual promi- nent reflections yields an important clue to the arrangement of lhe amino acid residues. Let us make the simple assumption that some particular amino acid residue of high weight, i.e., of high scattering power for x-rays, is arranged regularly (equidistantly),

le 0. Kratky, Monatsk., 77, 224 (1948).

197

0. K R A T K Y

e.g., that it occurs in the cycle of 220 residues 20 times and, therefore, occupies every 11th position. In this case, the giant period would be subdivided by this particular amino acid 20 times, which should result in the 20th order being an intense reflection. In this argument it was assumed that the equidistant arrangement of two consecutive amino acids of the kind under consideration is maintained beyond the

0 0 0 0 0 0 0

e e [ j

1 0 0 0 0

2

-0

-

-0

0 0

-

0 0

0 0 0 0 0 0 0 0 q 0

-0 3 -0

0

0 0 0 0 0 O b

Fig. 3. The indicated dis- tribution of 7 designated elements (full circles) in a total period of 23 elements m a y lead, among others, to a di- vision into 6 or 8 parts and, as a result, to the appearance of the 6th or 8th order of the basic period.

boundaries of one cycle, i.e., that we have a true, over-all periodicity of 642/20 A. It is possible to have this case realized when the frequency (i.e., the number of residues of one particular kind among the total 220 residues) is a divisor of 220. Other- wise, such a true period cannot exist. If the frequency is not a divisor of 220, then the equidistant arrangement must be limited to one cycle and will start all over again in the next cycle. Figure 3 illustrates this point. In the cycles shown, which consist of 23 or 19 units, there is one particular kind (filled- in circles) with a frequency 7. Even in such an arrangement certain orders of the basic period D will have high intensities, e.g., the 8th order in Figure 3a and the 6th order in Figure 3b. A further variation is encountered when the cycle actually represents a chemical moleculeit may, just as well, be a peri- odically repeated section of a very large molecule in the chemical sense-and when there is a gap between adjoining molecule ends which perturbs and modifies all periodicities.

We shall now investigate whether the kangaroo tail tendon shows any obvious connectidn between the frequency of the amino acid residues and the (medium or highly) incense orders of the large period. This will be done with the aid of Table 11. This table lists the frequency of the amino acids contained in the collagen, where frequency is defined, as above, as the number of residues of the individual amino acids contained in the basic period of 220 residues. These frequencies have been calculated from the values of the analysis by Bergmann and Niemann.8 Of course, these figures include analytical errors; we may certainly assume that these values are in reality integers. If all the amino acids

were arranged regularly, then the deviations of the weights of the residues from the average of 94 would probabIy represent a measure of the x-ray scattering power. However, it appears quite unlikely that glycine, for instance, with its high frequency of 70, can be arranged regularly without seriously disturbing the other frequencies. But, if glycine (possibly together with proline) occupies all those places which are left over after the regular spacing of the less abundant amino

198 Journal of Polymer Science


Amino acid

Histidine Aspartic acid Lysine Leucine, isoleucine Arginine Alanine H ydrox yproline Proline G1 ycine

acids has been completed, then for a first qualitative examination the proper measure of the scattering power is the excess of the individual residue weight over the weight of glycine, 57 (see column 2, Table 11).

Reaidue weight ! Fresuency Order of intense minus 57 reflsctiom

90 1.2 1,293 58 5.24 6 71 8 . 3 899

11

20 20.4 22.8 :”!

56 99 14 56 40 35.2

0 70 .5

TABLE I1 MERIDIAN REFLECTIONS OF COLWLGEN~

As a result, arginine, with an excess weight of 99 and a frequency of 11, would be the most effective residue in the chain. An over-all periodicity might exist, since 11 divides evenly into 220. Indeed, the 11th order is intense-as shown in Figs. 1 and 2 as well as in Table 11. This reflection is also listed in the last column of Table 11. Next in effectiveness is histidine, with an excess weight of 90 and a frequency of 1. It is conceivable that the basic period is accented by other factors as well, for instance, by the end of the molecule, and it is, therefore, not too surpris- ing that, besides the 1st order, the 2nd and 3rd orders also possess considerable, though diminishing, intensities. Furthermore, lysine, with an excess weight of 71, should be expected to contribute to the diffraction pattern. The intense 8th and 9th orders may be connected with its frequency of 8.3 (actually 8 or 9). In this case, no over-all periodicity is possible. To aspartic acid, with an excess weight of 58 and a frequency of 5, we may ascribe the 6th order reflection. Leucine and isoleucine, with an excess weight of 56 and a frequency of 11, again contribute to the 11th order reflection. Hydroxyproline, again with an excess weight of 56, might be connected with the 20th order reflection.

In this manner, we have obtained all the intense reflections and only those which actually occur, once we disregard the most abundant amino acids with very h3gh frequencies (glycine and proline). The reader is referred to the last column of Table 11, listing the intense reflections required by the amino acids. See, on the other hand, Table I and Figures 1 and 2.

This agreement is undoubtedly not just coincidental, and it is certainly reasonable, when setting up a structure, to pay attention to the regularities of the amino acid sequence, whether they be of the over-all type or confined to one cycle.

If different periods coexist, i t may happen that the same place is “claimed” by two different amino acids. If, as a result, one of the two amino acids skips that location,

One contingency will make the setting up of a structure more diffkult.

Volume 3, No. 2 (1948) 199

0. K R A T K Y

its periodicity is thereby disturbed. Nevertheless, in accordance with the precepts of crystal structure analysis, the reflection associated with that period may be observed with considerable intensity, just as a crystal reflection will occur even though the lattice is perturbed by holes (faults). Such irregularities are instances of deviations from the simple “perfect structure,” as Astbury discussed in great detail.

Needless to say, further considerations in this direction will have to include guuntitatiue calculations of the interference intensities. However, it seems that such an investigation will be justified only when more accurate analytical data become available. If we conclude these considerations without’having formulated a specific structure, it is hoped, nevertheless, that the reader will have gained the im- pression that our simple considerations have brought us much closer to this goal. Almost nowhere else in crystal structure analysis does there exist such a simple cor- respondence between the reflections and individual building stones of the lattice, as is shorn in this case.

Table 111, which was made available to the author by C. Wolpers, compares the x-ray results reported above with electron microscope observation^.'^ We fkd that the periods produced by the regular arrangement of the amino acids are also found in the histological structure. Thus there is a direct correlation betweeo molecular and histological structure. The information obtainable through x-ray low-angle methods and by use of the electron microscope already overlap in part; and in studying this involved protein structure we have apparently come near understanding the structure range from the elementary atom to the macroscopic body.

TABLE III X-RAY DATA AND ELECTRON MICROSCOPE ME A S U R E M E ~ S ON COLLAGEN”

x-ray data for kangaroo tail tendon b

Amino add

Histidine

Aepartic acid

Lysine Arginine, leucine, isoleucine

Alanine, arginine

0 According to Wolpers. * Kratky and Sekora.

Wolpers.

Order

6

(9“ 11

20 220

A. 642 321 214.5 107.8 79.4 70.7 58 .3

32.2 2.88

Electron miomsa, measurementa on human and &ne tendon0

D-part (greatest vafue) 600 A. D-part (smallest value) 300 A.

6-disks 110-130 1. (6, or g b ) exterior lamellae 70-80 A.

-interior lamellae 45-60 R. (61 or 6,)

(62 or 83)

Limit of resolution

Mention should be made of a second case in which low-angle methods and the electron microscope appear to measure the same quantity Investigating myosin fibers, the author, together with Sekora and Weber,14 found equatorial x-ray inter-

Is C. Wolpers, Arch. path. Anal. Physiol. Virchow’s, 312, 292 (1944); Klin. Wochschr.. 22, 1 (1943).

l4 0. Kratky, A. Sekora, and H. H. Weber, Natururissemchaften, 31, 91 (1943).



ferences corresponding to distances of 32, 42, and 66 A., while electron microscope measurements by von Ardenne and Weber15 yielded a particle width of 50-60 8.

CONTINUOUS LOW-ANGLE SCATTERING The Two Limiting Cases

Every substance containing 'colloidal particles gives rise to diffuse low-angle scattering. This effect was Grst observed on cellulose fibers by Mark and Hengsten- berg16 and on powders by Krishnamurti, l7 Hendricks, and Warren,I9 and was cor- rectly interpreted as being associated with the colloidal structure of these substances. Quantitative studies were initiated by GuinierZo and by the author,21 and from different points of view. Guinier considered the scattering from the individual particle. The author, on the other hand, in connection with quantitative measurements on cellulose fibers, started out with the similarity between these systems and a liquid made up of very large particles of very uneven dimensions-in other words, the main emphasis was placed on the spatial'arrangements. Soon it became clear2? that the approach of Guinier can be applied to the case of dilute systems (i.e., when the distances between the colloidal particles are large compared with the particle dimensions), while the author's approach is applicable in the case of densely packed systems.

HosemannZ3 particularly has used Guinier's formulas for densely packed systems, namely, solid cellulose. Since these formulas were not applicable, results having the wrong order of magnitude were obtained. The inadmissibility of this procedure and the significance of the two limiting cases for the treatment of law- angle scattering were pointed out a number of years ago.32 24 In what follows only the dilute systems will be treated, while the densely packed systems will be discussed in a forthcoming paper.

Theory of Dilute Systems

Continuing Guipier's train of thought, we were able to show that Debye's theory of scattering of molecular gasz6 is an excellent tool for an understanding of the effect of the form factor on scattering.26 With that theory i t is easy to compute

16 M. yon Ardenqe and H. H. Weber, Kolloid-Z., 97, 322 (1941). 16 See H. Mark, Physik und Chemie der Zellulose. 17 P. Krishnarnurti, Indian J . Phys., 5 , 473 (1930). 18 S. B. Hendricks, 2. Krist. Mineral. Petrog., Abt. A , 83, 303 (1932). 19 B. E. Warren, J . Chem. Phys., 2, 551 (1934); Phys. Rev., 49, 885 (1936). 10 A. Guinier, T h h a Sdrie A, Nr. 1854 (1939), Nr. d'Ordre 2721; Compt. rend., 204, 1115

I1 0. Kratky, Ndururhsenschuften, 26, 94 (1938); 30, 542 (1942). 22 0. Kratky, remarks concerning a lecture by R. Hosemann, 2. Elektrochem., 46, 550

0. Kratky, Kolhidchemisches Tuschenbuch., Akadem. Verlagsgesellschaft, Leipzig, '1943,

$8 R. Hosemann, 2. Physik, 113, 751 (1939); 114, 133 (1939); 2. Elektrochem., 46, 535

24 0. Kratky, A. Sekora, and R. Treer, 2. Elektrochem., 48, 587 (1942). 25 P. Debye, Physik. Z.', 31, 348 (1930). *6 0. Kratky and A. Sekora, Nutumissemchuften, 31, 46 (1943); comprehensive presenta-

ion by 0. Kratky, Monatsh.. 76, 325 (1946).

Springer, Berlin, 1932, p. 139.

(1937).

:1940). 3p. 132-150.

,1940).

r'olums 3, No. 2 (19488) 201

0. K R A T K Y

the scattering power of any body that consists of spheres. Since the scattering power is not too sensitive to form effects, it is possible, in sufficiently good approxi- hation, to work with substitute models composed of spheres instead of using the involved scattering formulas of nonspherical bodies. However, the form influence is still sufficiently marked to permit the determination of an axial ratio, for instance, of elongated or flattened particles. Besides, the experiment will give information on the absolute sizechanges in the absolute dimensions which leave the shape intact will merely produce an abscissa transformation of the scattering curve- but tied up in a defhite manner with the information on shape. Quantitatively, the situation may be treated as follows. For a particle, that is composed in some arbitrary fashion of atoms, Debye's scattering theory for a molecular gas leads to the formula:

sin m i = n k = n

I- C C f i f k p i = l k = l m

4a1, sin I? x m =

I is the diffracted intensity, n the number of atoms in the molecule, f stands for the diffracting power of the atoms, I , , is the distance between the centers of the ith and the kth atom, 9 is one-half the angle of deflection, and h the wave length of the x-rays used. As indicated in this formula, every atom is to be combined'with every other atom; for each such combination we must obtain the value of the expression .fifk(sin m/m), and then the summation is to be carried out over all possible combinations of i and k. Since i and k are to assume all values from 1 to n, the summation includes the combination of every atom with itself, whenever i equals k.

In that case, lin becomes zero, m vanishes too, and the fraction ~ approaches

unity. In addition to those terms that refer to pairs of nonidentical atoms, we thus obtain for each atom one term that is simply the square of the diffracting power of that atom.

In applying equation (1) to our problem, we shall consider a t first only systems consisting of spheres of equal size. The form factor of a sphere, the interior of which we may assume has a constant electron density, is given by:

f =

sin m m

(3) sin p - p cos p

P3

(4) 4 ~ r sin I?

x P =

where r is the radius of the sphere, while the remaining symbols have the same significance as in equations (1) and (2).

It is obvious that, for r>>X and for ZBk, i.e., for particles whose dimensions are large compared with the wave length, intensity I becomes already negligibly small for angles a t which the difference between sin 8 and 8 (in radians) may be still



disregarded. Hence, we may replace sin d in equation (2) and (4) by 9 itself, with the resulting simplifications:

Equations (1) and (5 ) , (3) and (6) provide the basis for computing every scattering curl . What remains to be done i s to express the Z t x values in terms of r and then to carry out the summation (1).

In plotting the curves, i t is convenient to represent I as a function of p or, preferably, as a function of x:

This quantity has been selected so that all values of m can be expressed in terms of 5 as simply as possible. To obtain the curve I = I (&) for any value of r , we must apply an appropriate scale to the abscissa, by choosing for each value of x or 1.1 the corresponding value of 6, in accordance with equation (7) .

A number of cases have been calculated in this fashion. As an example, we shall treat here the straight- line chains formed of spheres of equal size, which can serve as substitute models for bodies of ellipsoidal or cylindrical shape. In particular, we shall demonstrate lor

the method of calculating the l,, values, both their Fig- ’. Straight chain Of

magnitudes and the frequency of their occurrence, by the example of a chain consisting of 6 spheres (Figure 4). ‘The scattering function I obviously contains the following terms.

,

- 4r

1 6 r 8r

* c - 0 c

spheres.

(1) 6f”. (2) 10 times the term with I = 2r. To understand the number 10, we must count how

often the distance I = 2r appears as the distance between the centers of a pair of spheres. Spheres numbered 1 and 6 have one neighbor each a t that distance, spheres 2 and 5, respectively; but spheres 2, 3, 4, and 5 have two neighbors each a t a distance of 2r, their immediate right-hand and left-hand neighbors. Each distance has been counted twice, i.e., starting with each of the two spheres involved. This procedure is in accordance with equation (I), which specifies that both i and k assume all possible values.

8 times the term with I = 4r. The pairs having this distance are (1,3), (2,4), (3,1),

6 times the term with 1 = 6r, since exactly one such distance originates a t each sphere. 4 times the term with I = 8r. Such distances originate only a t the four spheres num-

2 times the term with I = lor, i.e., the pairs (1,6) and (6,l).

All told, we arrive a t the count of 1 + 1 + 2 + 2 + 2 + 2 = 10.

(3)

(4) (5)

(6)

It is now possible to write immediately the formula for a chain of n spheres.

(3-5)- (4,2), (4,6), (5,3), and (64) .

bered 1, 2, 5, and 6.

If we introduce again x = 2p, as in equation (3, this formula becomes:

Volume 3, No. 2 (1948) 203

0. K R A T K Y

sin (n - 2 ) x + sin (n - l ) x (n - 1)x + (n - 2 ) s

If this formula is applied to chains consisting of 2,3,4,6,10, and 12 spheres, one obtains the curves for the relative scattering intensity which are shown in Figure 5.27 Also shown is the intensity for the single sphere (-jz).

0 1 2 3 4 5 6 7 8 X- X-

Fig. 5. Scattering curves for straight Fig. 6. Scattering curves of straight chains of spheres and of the single sphere “altered” by using slits.

chains of spheres, compared with the scattering from a single sphere.

These scattering functions represent the dependence of the intensity on the angle of deflection in the actual x-ray diagram only if the object is being irradiated with an extremely narrow pencil of x-rays which has a cylindrical cross section. Actually, in order to shorten the time of exposure, i t is customary to employ an x-ray beam that converges in a plane, i.e., the primary ray is delimited by two slits. As a result, all the scattering curves are altered due to the use of slits in a manner which must be carefully determined. The curves of Figure 5 must be replaced by the set shown in Figure 6. In what follows, we shall use these curves, which were experimentally determined with slit-defmed x-ray beams.

In order to be able to compare the shapes of the curves, i t is convenient to carry out such an abscissa transformation that all the scattering curves pass through the same point H . This point H is defined as the point at which the scattering intensity I has exactly half its maximum ,value. As a result of this transformation, we obtain Figure 7. The abscissa now no longer represents the quantity x , which is defined by equation (7), but a quantity 2‘:

where fn is a transformation coefficient that depends on the number of spheres in the chain, n, and that has been chosen so that all the points, H , for which the ordi-

n The numerical values for ~ have been taken from the tables by J. Sherman, 2. Krist.

2’ = fnX (9)

sin x z

Mineral. Petrog., Abt. A , 85, 404 (1933).


L O W - A N G L E S C A T T E R I N G I N P O L M Y E R S

n f* n/f: n

1 0.286 42.9 6 2 0.400 31.2 10 3 0.472 28.6 12 4 0.535 26.2

nate assumes half its maximum value have the abscissa value of one. The values of f n are, therefore, the reciprocal abscissa values of points H in Figure 6. The magnitude of these factors is shown in Table IV and in Figure 8. In that figure, the

f” df

0.647 22.2 0.840 16.9 1.23 6.45

100

80

t 6o 5 40

20

0 I 3 4 5 f x -

40

.30

“,‘ 20 3

10

* O 2 4 6 8 10 n,. -

Fig. 7. Comparison of the shape of “altered” Fig. 8. Relationship between scattering curves of straight chains of spheres the axial ratio n of chains of and the single sphere. sphereeand e in the case of disk-

like particles-and the corresponding form factor ./.:. The upper curve refers to chains of spheres, the lower one to disks.

points have been connected into a curve, even though f n , properly speaking, is defined only for integral values of axis ratio n; however, it appears permissible to interpret the continuous curve, in sufficient approximation, as the relationship between a continuously variable axis ratio n and the corresponding factor f n defined above. If we substitute (9) in equation (7) , we obtain:

x’ 8Tr,tY x = - = - f n x

and, applying this relation to point H with half the maximum value of‘ the ordinate:

1 87rrn& x==--=- f n x

It follows that:

Velome 3, No. 2 (1948)

x f n 8 7 ~ 8 ~

rn = -

0. K R A T K Y

If we now substitute (10) into the expression for the volume, Vn, of the scattering chain of spheres:

v,, = - n.4?rr3 3

X3 n 1 384?r2f: 9;

we immediately obtain: v , , = - - - and, for X = 1.54 s.:

Since the molecular weight is given by :

(d = density, NL = Loschmidt's number) substitution of equation (Ila) results in:

ao This relationship then presumes that the particles can certainly be approximated by chains of spheres. For purposes of evaluation, it is necessary to ascertain in the scattering curve the abscissa of point H , where the scattering intensity has dropped to one-half its maximum value, and

0 5 10 15 to determine the corresponding f i H . If the scattering curve is not known for sufficiently small angles,

60

t 40 >

20

0-

the determination of SH may cause difficulties. A case in point is shown in Figure 9. The scattering curve for a solution of insulin, shown in that figure,

Fig. 9. Experimental scattering curve of an insulin golution.

was actually measured only up to point P. Unfortunately, this substance was available only at a time when the low-angle method was

not yet sufficiently developed and, as a result, the experimental material is defective. We feel, though, that the experimentally obtained scattering curves are more suitable for demonstrating an evaluation, with all its difficulties, than would be fictitious curves. Moreover, these curves furnish the first information, although uncertain in some points, about the low-angle scattering of insulin solution.

Even though the general type of these curves enables us to assume that they will bend and intersect the ordinate at right angle, the maximum value of the ordinate nevertheless remains uncertain. It is not possible to make a decision between the two sketched curves, 1 and 2, and all intermediate curves. (This difficulty illustrates the necessity of extending measurements into the immediate vicin- ity of the primary beam even when scattering extends over a considerable range of angle.) ,As shown in Figure 9, the abscissa of the half-value point H depends on the maximum value, which is reached at 6 = 0. If we choose curve 1, we find:

lpa = 5.51 x 10-3

206 Journal of Polymer Scienco


If this value is substituted in equation (12), we obtain:

n M = 4520 3 f n

i.e., we obtain a functional dependence of the molecular weight on the form factor n/f:. In other words, to each axial ratio belongs, in accordance with Figure 8 and the relation shown above, a particular molecular weight and vice versa. The molecular weight is very 8o

frequently known from some other kind of measurement (ultracentrifuge, 60

the determination of the angle ir, im- 40 osmotid measurements) ; in that case

mediatey leads to the axial ratio n with the help of equation (12).

Actually, i t is necessary to dis- 20

I 3

cuss other shapes in the same manner 0 I 2 3 4 as well, e.g., thin plates. Figure 10 f x - shows the exDerimenta1 scattering Fig- 10- Comparison of shape of “a1- -

for disks (using slits described tered” srattering curves for the sphere and, for disks consisting of 7, 13, and 19 spheres, above), where the abscissas have

already been normalized so that the respectively.

abscissa of point N equals unity. Shown are the curves for platelets consisting of 7, 13, and 19 spheres compared again with the single sphere. Table V fur- nishes the corresponding values of f n and of n/j:, where n is the number of spheres, while e is the appropriately defined axial ratio. The lower curve in Figure 8 shows again the relationship between e, now considered a continuously variable parameter, and n,!f:. The value of n/fi, which belongs t: a particular molecular weight M according to equation (12), will, therefore, with thehelp of Figure 8, lead to different axial ratios, de- pending on whether we assume a chain (elongated shape) or a disk (flattened shape).

TABLE V DISKS OF SPHERES

Otherwise, relations (111, ( l la), and (12) remain unchanged.

1

7 13 19

0.286

0.637 1.755 2.105

42.9 31.0‘ 27.3 2 4 . 0 20.5

Interpolated value.

Until now, we have made no use of the actual shape of the scattering curves. We may do so by adapting the scattering curve to the sets of curves of Figure 7 and 10 through a coordinate transformation (Le., by making the maximum ordinate 100

Volume 3, No. 2 (1948) 207

0. K R A T K Y

and by making the abscissa of the point H equal to unity). Figure 11 shows, be- side a few curves from Figure 7 and 10, the scattering curve of insulin solution, cor- rected as described above; the crosses refer to curve 1 from Figure 9 and the circles to curve 2 from the same figure. We fhd that the theoretical curve, d3, for disks with an'axial ratio of 3, and even more curve r2, for chains with an axial ratio of 2, fit the two experimental curves quite well. It was not possible, though, to fmd a theoretical curve that would fit the experimental curve perfectly both above and

100

80

t 6o

0.5 0

XI-

zom Fig. 11. Comparison between the experimental scattering curve of in-

sulin -(crosses and circles) and the theoretical curves for the sphere (r l ) , for chains of 2 and 3 spheres, respectively (rz, ra), as well as for a disk with an axial ratio 6 = 3 (&).

below H . (Defects of an experimental nature may well cause this failure.) Through such a comparison, the determination of the shape dan thus be accomplished. Its accuracy depends primarily on the degree of accuracy of the experiment. Only when a very high degree of perfection has been achieved in this respect will it be reasonable further to improve the theory of the scattering curves, i.e., to carry out comparisons with a greater variety of theoretical scattering curves.

For further evaluation, we shall proceed in the opposite direction than when we knew only the value of Sa. We now substitute in equation (12), the values of n/fi (corresponding to the shapes ascertained) along with the value of Sx, and compute the molecular weight. In our case, we obtain, if we use curve 1, a molecular weight:

141,000 for the chain n = Z(r2). . M={ 123,000 for the disk e = 3(d3).

If we use curve 2:

M = { 170,000 for the chain n = 2(rz). 149,000 for the disk e = 3(d3).



Experiments that could be carried out today without difficulty would permit a decision between the two curves 1 and 2, and very accurate measurements even between rods and disks could be obtained.

Knowledge of the molecular weight obtained by other methods may be used afterward for a check or for reaching a decision between alternative possibilities. For insulin, for instance, the molecular weight in solution was found to be 41,000.28 Our values, which are so much larger, lead us to suspect that association is involved, Le., our molecular weight should be a multiple of 41,000. Now 123,000 equals three times, and 170,000 about four times, 41,000. For the time being, it is im- possible to decide between these two cases-and all others which result from scattering curves lying between curves 1 and 2. Naturally, from this kind of shape comparison one cannot obtain an exact axial ratio of, say, 2. It might just as well be 1.8, or 2.2, and the molecular weight is, therefore, subject to a corresponding uncertainty. By basing the computation on a particular value of the molecular weight (or the basic molecular weight), one would, by reversing our procedure, be able to determine the exact corresponding shape. There is no question that moder- ately accurate experiments are sufhient to permit a rough determination of weight and shape even without previous knowledge, but if the weight were already known, a quite satisfactory determination of the shape is possible. Since there are several good methods known to determine the molecular weight, but no general method for the determination of shape, the application of the low-angle method is likely to become particularly important. In contrast to measurements by ultracentrifugations, when the form factor determination is afilicted with the uncertainty of solvation, in low-angle scattering the solvate envelope (which in x-ray diffraction should hardly be distinguishable from the remainder of the solvent) does not apparently interfere to any marked extent. Naturally this new method will require additional develop- ment, checking, and many applications before its ultimate usefulness can be judged.

An interesting example is furnished by the computation of the chain consisting of an infinite number of spheres. This chain is to be substituted for the scattering power of very long cylindrical or nearly cylindrical particles. The count can be taken by focusing attention on some one sphere and combining it with all the others. It is easy to see that the resulting expression is proprotional to the total scattering power. The computation results immediately in the following expression:

+ ...)I [ (si; x sin 22 sin 32 I - - f " l f 2 -+-+- 22 X

n = m sinnx I - - f 1 + 2 c - [ ( n = l nx )I The expression within the brackets has the value :

WJ The Svedberg and K. 0. Pedersen. Die Ultrazentrijuge. Steinkopff, Dresden, 1940, p. 370.

Volume 3, No. 2 (1948) 209

0. K R A T K Y

inside the interval from x = 2za and x = (22 + 2 ) H, where z equals either zero or some integer.

I t is discontinuous at x =

2 ~ , 4 ~ , 6 ~ . . . ; the jump at the first discontinuity is from l / ~ to 3/2, at the second discontinuity from 3/4 to 5/4, etc., so that the curve approaches asymptotically the straight line paralleling the x axis at a distance of one. Multiplication of this func-

The curve for this expression is shown in Figure 12.

X-

Fig. 12. Concerning the scattering curve of an infinite straight chain of spheres. The upper curve

n= 1 represents the expression 1 4- 2 sin -, while

the lower curve shows this expression, multiplied- by fa, the square of the form factor of the single sphere.

n x

n = m n x

tion byf2, in accordance with equation (13), leads to a rapid drop of the scattering curve with increasing z (Fig. 12, lower curve), but of course the discontinuities remain. It is possible to correlate the locations of these jumps with a well-known regularity. According to equation (13), the locations of the jumps are at x = n.2?r, where n is an integer; if we now substitute for x according to equation ( 7 ) :

8 ~ r sin 6 x x =

we obtain: nX = 2(2r) sin 6

i.e., Bragg’s law, with 2r as the distance between lattice planes. In other words, the scattering function of the infulite chain of spheres jumps to higher values at those angles at which a set of lattice planes in a crystal would reflect if they were spaced the same as the spheres are in the chain. Thus, the periodicity in the structure of the chain of spheres manifests itself in the jumps. We may generalize this result and claim that a periodicity in any structure (however complex), that is extremely elongated, will result in a step, or at least in a hump, in its gcattering curve, which otherwise falls off gradually.



If, in an actual case, we meet with a genuine string of spheres-e.g., the result of threadlike association of spherical particles-then, at least, the first and largest step may become noticeable. However, if we investigate a body without such a marked periodicity, e.g., a virus particle that is approximately cylindrical in shape, then we should expect an over-all agreement with the theoretical curve of Figure 12, but not, of course, the appearance of steps.

Because of alteration due to the use of slits (mentioned previously), the step in Fig. 12 is transformed into a hump, whose maximum will be observed a t the abscissa of the original step. The behavior of the curve at small values of 5 may also be taken from Figures 5 and 7 .

Some Tentative Measurements on Protein Solutions

The original fdms and photometer curves unfortunately were lost because of the events dur-

In investigations of diffuse scattering, it is essential that the diagrams are ob- To this end, Ross’s method

ing the war, so that only the curves which already are redrawn for intensity can be shown.

tained by the use of strictly monochromatic radiation.

I I I 0 0.5 1.0 1.5

8-

Fig. 13. Scattering curve of a solution of edestin (the triangles), compared with the theoretical curve of a spherical particle with a molecular weight of 310,000.

Fig. 14. Scattering curve of a tobacco mosaic virus solution. The dots shown represent the experimental data, while the curve has been computed for an iniin!te chain of spheres with 2r = 167 A.

of differential fdtering29 is used in a slightly modified arra11gement.3~ With copper radiation, a nickel and a cobalt filter are placed side by side behind the primary ray stop (zero ray stop)-their thickness adjusted according to the requirements of the Ross method-so that one-half of the radiation field passes through nickel, the other through cobalt. The photometer curves, redrawn for x-ray intensity and one subtracted from the other, yield the practically monochromatic scattering distribu-

29 See E. 0. Wollan, Phys. Rat., 43, 955 (1933). 30 0. Kratky, Natumissenschufkn, 31, 325 and Heft 37/38 (1943).

Volume 3, No. 2 (1948) 211

0. K R A T K Y

tion. Further details of the experimental method have been described elsewhere, so that it will be s&cient merely to refer to these papers.logZ4

Edestin Solution. mestin,* according to SvedbergY3l has a molecular weight of 310,000, and the molecules are probably approximately spherical in shape. Figure 13 shows how well the theoretical curve computed for this case agrees with the experiment.

29- -2.9

Fig. 15. Scattering curve of hemocyanin solution (curve l), the pure solvent (curve 2), and their difference, which results from the solute particles alone (curve 3).

Tobacco Mosaic Virus So1ution.t According to the well-known investigations of the crystallized virus by Bernal and F a n k u ~ h e n , ~ ~ this virus “moleculeyy is to be considered a very long six-sided prism, the short edge of which has a length of 75 8. The substitute model having the same volu5e is an infinite series of spheres, the individual spheres having a diameter of 167 A. Figure 14 shows the comparison between the theoretical curve computed for this case and the experimental curve. Ekcept for the hump, which is, of course, missing on the experimental scattering curve, the fit is satisfactory.

* We wish to express our appreciation to Dr. Bucher of the Kaiser Wilhelm-Institut fiir Zellphysiologie at Berlin-Dahlem for furnishing us with this substance.

31 The Svedberg and K. 0. Pedersen, Die Ultrazedrijuge. j We wish to express our appreciation to Dr. Friedrich-Freksa of the Kaiser Wilhelm-

32 J. D. Bernal and I. Fankuchen, J. Gen. Physiol., 25, 11, 147 (1941).

Steinkopff, Dresden, 1940.

Institut fiir Biologie (now at Ti ib i en ) for furnishing us with this substance.



Solution of Hemocyanin. This protein is contained in the blue blood pigment of Helix pomatia. Figure 15, taken from the joint paper by the author with Sekora and Friedri~h-Freksa,~~ shows that the scattering curve has a marked hump, which indicates a period of 260 A. This apparently explains a contradiction that had re- sulted from the investigation of this protein by different methods. From ultracentrifuge measurements, one obtains a molecular weight of 8.91 X 0106.34 If the shape is spherical, this weight corresponds to a diameter of 276 A. Actually, Friedrich-Freksa succeeded in observing particles of that size under the electron microscope. * On the other hand, under certain conditions, solutions of hemocyanin exhibited flow birefringence, which would indicate very elongated particles. Our observation apparently explains this inconsistency. It seems plausible to conclude that the primary particles are really spherical in shape and that they have a diameter equal to 260 A., but that they are capable of forming threadlike aggre- gates responsible both for flow birefringence and for the hump in the x-ray scattering diagram. With this interpretation, the molecular state of hemocyanin may not yet be fully described. There remains a high factor of asymmetry of 1.45, which has been measured with the ultracentrifuge; furthermore, the basic particle may de- compose into halves, quarters, or eights, which also complicates matters.36

Insulin. In the discussion of the general method of evaluation we reported one measurement on an insulin solution. The measurements were carried out a t different temperatures (4-36 "C.) both in weak acid and weak alkaline solutions. Because of the reasons that are mentioned above, we wish to state with reservation that in general the particles were apparently associated and that, at 11°C. in alkaline solution, one aggregate apparently comprised between 6 and 9 basic molecules, while at a higher temperature, according to the interpretation which we consider the most likely one, they have three times the basic molecular weight. A check will be necessary. We consider the question of change of particle weight as settled beyond doubt, and also the discovery of a marked deviation from spherical shape. A renewed investigation with the means now available, will undoubtedly make it possible to more accurately delimit weight and shape of the association, as well as its geometry.

Synopsis

The diffraction of x-rays a t small angles occurs in the form of sharp interferences (discontinuous) or as diffuse scattering (continuous). The first effect is observed mostly in crystallized proteins and in protein fibers. New measurements on collagen were obtained by the author in collaboration with Sekora, and independently by Bear. Using these data, an attempt has been made to establish a connection between the orders of the intense reflections belonging to the giant period of 642 A. along the fiber axis and the frequency of the amino acid residues (frequency is defined as the number of amino acid residues of a particular kind among the total 220 residues which constitute the basic period of 642 A. along the fiber axis). With the exception of the most abnn-

a3 0. Kratky, A. Sekora, and H. Friedrich-Freksa, Ant. Akad. Wiss. Wien Math. dun^.

8' Brohult, Nova Acta Reg& Soc. Sci. Upsaliensis, 12, Series 4, No. 4 (1940). Klasse (March 7,1946).

* Unpublished measurements by H. Friedrich-Freksa carried out on specimens carefully dried with simultaneous freezing.

86 See also Trurnit and Bergold, Kolloid-Z., 100, 177 (1942).

213 Volume 3, No. 2 (1948)

0. K R A T K Y

dant amino acids, glycine and proline, it is possible to assign to each frequency a corresponding reflection, and vice versa. This indicates a high degree of regularity in the arrangement of these amino acid residues. But the relationships are apparently more complex than would be expected from the Bergmann-Niemann rule. For example, the frequencies of 11 and 20 (corresponding t o an identical residue at each 220/11 or 20th location or a t each 220/20 or 11th location, respectively) appear to play a role. Fundamentally, however, we arrive a t a picture in accordance with the comprehensive investigations and results of Astbury. It has not yet been possible to arrive at a detailed structure.

In the theory of diffuse low-angle scattering, it is necessary to distinguish between densely packed systems and dilute systems. For the latter case, molecular scattering is computed in accordance with Debye’s theory of scattering. It is shown how size factor and form factor may be separated. Both of these properties can be approximated without making previous assumptions, though the accuracy of the form factor determination can be improved appreciably if the particle weight is known. Preliminary trial nieasurements on solutions of edestin, tobacco mosaic virus, and hemocyanin are in agreement for the fist two substances mentioned. The measurements on insulin indicate associated particles that deviate strongly from spherical shape (however, these measurements were made a t an earlier date and require checking). Those on hemocyanin indicate an association similar to a string of pearls, a result which probably explains the contradiction of the substance appearing as spherical particles under the electron microscope, while exhibiting flow birefringence when in solution. When this method is developed further it will undoubtedly become a valuable tool for the determination of form and size of polymer niolecules and other colloidal particles.

RBsum B

La diffraction des rayons X aux petits angles se present en forme d’interfkrances aigues (interrompues) ou comme la dispersion diffuse (continue). Le premier effet est observe pour la plupart dans les protkines crystalliskes et dans les fibres des protkines. Des mesures nouvelles du collagbne ktaient obtenues par l’auteur en collaboration avec Sekora, e t independamment par Bear. En utilisant ces donnees, un essai avait fait ktablir une correiation entre les ordres des r6flexions intenses, qui appartiennent B la pkriode gigantesque de 642 A. au long de I’axe du fibre et de la frequence des restes d’acides amino. (La frkquence peut &re d k h i e comme le nombre des restes d’acides amino d‘une sorte particuliere parmi les restes totales de 220, qui constituent la pkriode basique de 642 A. au long de l’axe de fibre.) Avec l’exception des acides amino le plus abondants : glycine, proline, il est possible d’assigner B chaque frequence m e reflexion correspond- ante et vice versa. Ceci dknote un haut dkgrk de la rkgularite dans l’arrangement de ces restes d’acides amino. Mais les relations solit kvidemment plus complexe qu’on attendrait de la rhgle de Bergmann-Nieman. Par exemple, les frkquences de 11 et 20 (qui correspondent au reste identique B chaque 220/11 ou vingtibme emplacement ou chaque 220/20 ou onzibme emplacement, respec- tivement) paraissent jouer une rBle. Fondamentalement, nous arrivons au tableau, qui s’accorde avec les recherches Btendues et les resultats d’hstbury. Nous ne sommes pas encore arrivks B la structure d6taUe. Dans le t h h i e de la dispersion diffuse de I’angle bas, il est nkcessaire de distinguer entre les systbmes BtanchBifiBs et les systkmes dilub. Dans le dernier cas, la dispersion mol&ulaire est calcul6e en concordance avec la thbr ie de la dispersion de Debye. I1 est dkmontrk comment on peut skparer le facteur du grosseiir et le facteur de la forme. Sans m e supposition preliminaire, toutes les deux propriktes peuvent &re approximkes, quoiqu’on puisse perfectionner considkrablement la prMsion de la dktermination du facteur de la forme, si le poids de la particule est connu. Les mesures d’essai pr8iminaire pour les solutions de I’kdestin, du virus mosaique du tabac, et de la hkmocyanine s’accordent pour les deux premieres substances mentionkes. Les mesures pour I’insuline denotent les particules associkes, qui dhvient fortement de la forme sphkr- ique. (Cependant, ces mesures 6taient faites B une ancienne date et exigent d’ktre vkrifikes.) Celles de hemocyanine denotent une association pareille B m e chaine, un resultat qui probable- ment explique la contradiction de la substance qui apparait comme les particules sphkriques sous le microscope 6lectronique en meme temps montrant la birkfringence courante en solution. Si cette mkthode est encore plus dBvelop$e, elle deviendra, peut &e, un outil prhieux pour la dkter-



mination de la forme et pour le grosseur des molecules polymeriques e t des autres particules col- loidales.

Zusammenfassung

Die Beugung von Riintgenstrahlen unter kleinen Winkeln kann in Form von scharfen Interferenzen (diskontinuierlich) oder als diffuse Streuung (kontinuierlich) erfolgen. Der estere Effekt tritt vor allem bei kristallisierten Eiweisstoffen und Eiweissfasern auf. An Hand neuerer Messungen an Kollagen, wie sie unabhHngig einerseits der Verfasser gemeinsam mit Sekora, andererseits Bear durchgefuhrt hatten, wird versucht, einen Zusammenhang zwischen den Ordnuc- gen intensiv auftretender Reflexe der in der Faserachse verlaufenden Riesenperiode von 642 A. under der Zaligkeit der Aminosaurereste herzustellen. Als Zlhligkeit wird die Anzahl der Aminosaurereste einer bestimmten Art bezeichnet, wtlche auf die 220 Reste entfallt, die, entlang der Faserachse angwrdnet, eine Grundperiode (642 A.) ergeben. Es zeigt sich, dass, abgesehen von den haufigsten Aminosluren Glycin und Prolin, jeder Zahligkeit ein entsprechender Reflex zugeordnet ist und umgekehrt, was auf eine hohe Regelmassigkeit in der Anordnung der betreffen- den Aminosaurereste hinweist. Die Znsammenhlnge scheinen komplizierter zu sein, als nach der Regel von Bergmann und Niemann zn erwarten ist, wie sich z.B. daraus ergibt, dass die Zahligkeiten 11 und 20 (also Besetzung jeder 220/11 = 20ten bezw. 220/20 = l l t en Stelle) offenbar eine Rolle spielen. Im Prinzip kommen wir aher doch zu einem Bild, wie es den weit ausholenden Untersuchungen und Erkenntnissen von Astbury entspricht. Ein genauer Bauplan konnte noch nicht aufgestellt werden.

In der Theorie der diffusen Kleinwinkelstreuung muss man zwischen dichtgepackten und verdunnten Systemen unterscheiden. Fiir die letzteren wird auf Basis der Debye'schen Streu- theorie des Molekulgases die Streuung berechnet und gezeigt, wie man Grossen- und Formeinfluss trennen kann. Es ist eine voraussetzungslose annahernde Bestimmung beider Eigenschaften moglich, doch steigt die Genauigkeit der Formbestimmung sehr erheblich, wenn das Teilchenge- wicht bekannt ist. Vorllufige orientiereude Messungen an Liisungen von Edestin, Tabakmosaik- Virus, Insulin und HHmocyanin ergeben im ersten und zweiten Fall Ubereinstimmung mit den bekannten Ergebnissen, bei Insulin fiihren sie auf assoziierte Teilchen mit starken Abweichungen von der Kugelform (diese Messungen sind allerdings llteren Datums und bediirfen einer Uberpru- fung) und bei Hamocyanin auf eine perlenschnurartige Assoziation; welche Feststellung wahr- scheinlich einen Widerspruch aufkl'art : diese Substanz zeigt namlich im Elektronenmikroskop kugelfiirmige Teilchen, wlhrend andererseits Liisungen mit Striimungsdoppelbrechung gefunden wurden.-Die Methode stellt in ihrer weiteren Entwicklung zweifellos ein wertvolles Werkzeug zur TJntersuchnng von Form und Grosse von polymeren Molekiilen und sonstigen kolloiden Teilchen dar.

Received April 12, 1947

Volume 3, No. 2 (1948) 215

low-angle scattering in polymers

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