Fractal properties of polymer crystalsTetsuya Ogawa, Satoru Miyashita, Hideki Miyaji, Shoji Suehiro, and Hisao Hayashi Citation: The Journal of Chemical Physics 90, 2063 (1989); doi: 10.1063/1.455997 View online: http://dx.doi.org/10.1063/1.455997 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/90/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Compact Polymers on Fractal Lattices AIP Conf. Proc. 899, 598 (2007); 10.1063/1.2733339 The fractal properties of internet AIP Conf. Proc. 574, 85 (2001); 10.1063/1.1386821 Fractal properties and swelling behavior of polymer networks J. Chem. Phys. 100, 9181 (1994); 10.1063/1.466673 Theory of branched polymers on fractal lattices J. Chem. Phys. 93, 7471 (1990); 10.1063/1.459421 Optical properties of fractal clusters AIP Conf. Proc. 160, 455 (1987); 10.1063/1.36738

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Fractal properties of polymer crystals Tetsuya Ogawa, Satoru Miyashita, and Hideki Miyaji Department of Physics, Faculty of Science, Kyoto University, Kyoto 606, Japan

Shoji Suehiro and Hisao Hayashi Department of Polymer Chemistry, Faculty of Engineering, Kyoto University, Kyoto 606, Japan

(Received 20 June 1988; accepted 20 October 1988)

Small angle x-ray scattering is performed on the crystals of polyethylene and polytetraftuoroethylene up to Porod's region. The scattering intensity I obeys a power law for the magnitude of the scattering vector q for large values of q: 1-q - a • The exponent a is smaller than 4 given by Porod's law. If it is assumed that the folded chain crystals and the extended chain crystals have the fractal surface, the fractal dimension is from 2.2 to 2.8 depending on the crystallizing condition.

I. INTRODUCTION

Depending on the crystallizing condition, polymer crys­tals have three types of morphology, thin folded chain crys­tals (Fee) of the order of 10 nm in thickness in usual crys­tallization, extended chain crystals (Eee) as thick as 1 p,m in melt crystallization at high pressures, and spherical no­dules of the order of several tens nm in diameter in melt crystallization at large supercoolings.

In small angle x-ray scattering (SAXS), the informa­tion about the surface structure can be obtained by measure­ment of the intensity I(q) in the relatively large scattering angle region known as Porod's region, where q = 417" sin ( t/J/2 ) I A is scattering vector with scattering angle t/J and the wavelength A. In this region, the scattering intensi­ty decreases rapidly with increasing q; I( q) - q-4 for smooth and sharp boundary: Porod's law. 1

Since the size of polymer crystals is from 10 nm to 1 p,m, SAXS can determine the surface smoothness. In the present paper, the two types of morphology, Fee and Eee, estab­lished by electron microscopy are examined by SAXS in the

100 7

a )

• b 5.6 <

4.2

2.B

1.4

wide range of the scattering angle. The deviation from Por­od's law is reported and discussed in the light offractals.

II. EXPERIMENTAL

High density polyethylene (PE) used is Sholex 6009 (MN = l.4x 104

, MwlMN = 8.4). Three specimens of the PE were examined: the folded chain crystals (Fee) crystal­lized by cooling the melt at a rate of 3 K/min, the extended chain crystals (Eee) crystallized at 620 MPa and 530 K for 20 min and cooled at a rate of 3 K/min, and the Eee etched by fuming nitric acid (FNA) at 333 K for a week. Poly tetra­ftuoroethylene (PTFE) was also examined since this rigid polymer is crystallized to form Eee even at atmospheric pressure.

The SAXS experiment was performed at the High In­tensity X-ray Laboratory of Kyoto University. The SAXS camera was Franks-type point-focusing camera.2 The sam­ple to detector distance (SOD) was either 650 or 1650 mm. The detector was a two dimensional position sensitive pro­portional counter which has active area of 128 X 128 mm2

)

~C

FIG. 1. SAXS curves for extended chain crystals of poly ethel ene. (a) CuKa at 1650 mm of sample to detector distance; (b) CuKa at 650 mm; (c) MoKa at 650 mm.

&~ __ ~~ __ ~ __ ~~~-L ____ L-~~ ____ ~ __ ~~~-L __ ~

-2.3 -1.82 -1. 34 -&.86 -8.38 &.1

100 q

J. Chern. Phys. 90 (3),1 February 1989 0021-9606/89/032063-05$02.10 @ 1989 American Institute of Physics 2063

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2064 Ogawa et at: Fractal properties of polymer crystals

log 7

I J I I I

:...

5.6 '-

;...

".2 r-

2.8

1. .. ~ '-

& .1 1 1

& &.12 &.24 &.36

q**Z (A"-Z)

with resolution of 1.0XO.5 mm2; flowing gas is a mixture of

Ar and ethane or Xe and ethane. The fine-focus x-ray gener­ator equipped to RU-lOOO (Rigaku Corp. Japan) was oper­ated at 3.5 and 2.2 kW for the Cu and Mo target, respective­ly. The scattering experiment was done at 25 ·C. The scattering intensity was subtracted by the intensity in the background and was circularly averaged since the scattering pattern is isotropic azimuthally.

III. RESULTS Figure 1 shows the SAXS intensity I(q) in log-log plot

for the ECC ofPE with MoKa radiation at 650 mm ofSDD

log 1

J I

-

-'

--'

--'

FIG. 2. Log Ivs q2 plot of the SAXS curves for ECC of PE. The solid line is subtracted

-: to get I, (q).

-'

..

.. -'

.1

Ii. 48 &.6

and CuKa at 650 and 1650 mm. The three scattering curves were superposed by vertical shifts to get the best fit each other at the overlapped region of the scattering angle. The gradual increase in I( q) at large angles comes from the tail of the amorphous halo, multiple scattering and the density fluctuation with long wave length. Following the method of Ruland,3 we have subtracted the contribution at the large angles; in the plot oflog I (q) vs q2, the tail at the large angles is assumed to decrease linearly with decrease in q2 (Fig. 2). The subtracted intensity Is (q) is shown in log-log pot (Fig. 3). The power-law scattering, Is(q)-q-a, is observed in the range from 10-2 to 10 - 0.5 of q. The exponent a is deter-

8~~~--~~--~T---~~~-r~~~~~~~~----~~--'

6

4

Z

-ZL-__ -L ____ L-__ ~ ____ L_~~ ____ L_ __ _L ____ ~ __ _L __ ~

-2.3 -1.92 -1.54 -1.16 -&.78 -&.4

log q

J. Chern. Phys., Vol. 90, No.3, 1 February 1989

FIG. 3. Log-log plot of the SAXS curve for ECC of PE. The slope is - 3.2.

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Ogawa et al.: Fractal properties of polymer crystals 2065

mined to be 3.2 by the least square method. The same procedure was applied to the ECC of PTFE

and the ECC ofPE etched by FNA (Fig. 4). The ECC of PTFE and the FNA-etched ECC ofPE show the power-law scattering in a wide range of q with the exponents of 3.3 and 3.8, respectively. The FCC ofPE shows the scattering peak around 0.02 of q; the thickness of the folded lamellar crystals is 25 nm. For q larger than 0.1, the power-law scattering is observed with the exponent of 3.8.

All the polymer crystals examined show the power-law scattering in SAXS for the large values of q. The exponents are not integers and smaller than 4.

log I

IV. DISCUSSION

Polymers crystallize to form thin lamellar crystals (FCC) or thick bands (ECC). Both crystals have wide sur­faces normal to the chain axis. Therefore, the surfaces con­tribute to the scattering in Porod's region.

The Porod's law requires that the boundary must be sharp and there exists no long range order; namely there is no correlation between two points separated widely enough. In the present specimens examined, the stacking of FCC or ECC has no long range order since the scattering maxima at large values of q vanish. The boundary of the polymer crys-

8~--~--~~~~~--~~---r~--~~--r---~~--~--~

6

4

2

-2~ __ ~~ __ ~~ __ L-~~ __ ~~~~~~~~ __ ~~ __ ~~~

-2.3 -1.92 -1.54 -1.16 -8.78 -8.4 log q

log 9~---r----r----r----~---r~~~~~~~~~~~~

7

5

3

1

-1. 92 -1.54 -1.16 -8.78 -8.4 log q

J. Chern. Phys., Vol. 90, No.3, 1 February 1989

FIG. 4. Log-log plots of the SAXS. (a) ECC ofPE etched by fuming nitric acid; (b) ECCofPTFE; (c) FCCofPE.

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2066 Ogawa et a/.: Fractal properties of polymer crystals

log I 8r-~~----~--~~--~~---r~--~~--r-~~--~~--~

6

. . .. 4

FIG. 4. (continued).

2

8

-2~ __ ~~~~~~~~~~~~ __ ~~ __ ~w-__ ~ ____ ~ __ ~

-2.3 -1.92 -1.54 -1.16

100 q

tals has been argued not to be sharp but to be continuous since Blunde1l4 proposed a detailed profile of SAXS for a model with continuous boundary. In fact, Vonks showed the effect of continuous boundary for a polymer; at large q, gradual deviation from Porod's law was observed. However, in the present experiment, especially for ECC, we have shown that in the wide range of q, the power law scattering is observed. The power law behavior is expected for the materi­al with fractal surface. 6.

7 The density correlation function P( r) of materials with fractal shape can be scaled by a simple power law of distance r; P( r) = 1 - A,P - D where d is the dimension of the material, D is the fractal dimension, and A is a constant. For surface fractals, the scattered intensity I(q) is given by a power ofthe magnitude ofthe scattering vector q as follows:

I(q)_q-<2d-Dl. (1)

Since in three-dimensional materials 2 < D < 3, the exponent a of the power-law scattering [I(q) _q-a] takes a value between 3 and 4. Porod's law is the limit of D to 2 in Eq. (1) for three-dimensional materials (d = 3). Therefore, we can

TABLE I. The scattering exponent a and the fractal dimension D of the polymer crystals.

a D

ECCofPE 3.2 2.8

FNA-etched 3.8 2.2 ECCofPE

PTFE 3.3 2.7

FCCofPE 3.8 2.2

-&.78 -&.4

assume that the exponent smaller than 4 corresponds to the fractal properties. Using the exponents we can deduce the fractal dimension D from Eq. (1) (TableI).

ECC of PE and PTFE shows a considerable deviation from Porod's law. It is reasonable that etching of the surface ofECC ofPE gives rise to decrease in the fractal dimension; the rough surface approaches to smooth one by the etching. Recently, Hikosaka8 developed a nucleation theory for crys­tallization of ECC and FCC, and suggested that sliding dif­fusion of the chains along the chain axis plays an important role for the formation of ECC; the distribution of the thick­ness of the crystals is wider for ECC than for FCC.

FCC ofPE also shows a deviation from Porod's law and therefore rough surface. A number of authors9

-12 have theo­

retically treated the problem of the roughness of the surface in FCC. They developed the theories on fluctuations of fold period (thickness of the lamellar crystals) on the basis of kinetics of the crystallization. On the other hand, Seto 13

pointed out the possibility of roughening transition of the surface at 100 K for PE single crystals on the basis of the theory of Burton, Cabrera and Frank. 14 Recently in crystal­lization from the melt, it has been suggested that thickening of the lamellar crystals occurs during the crystallization 15;

thickening process may cause the surface to be rough.

ACKNOWLEDGMENTS

The authors express their thanks to the committee of HIXL of Kyoto University for the SAXS system available. This work was partly supported by a Grant-in-Aid for Scien­tific Research from the Ministry of Education, Science and Culture, Japan.

'G. Porod, Kolloid Z. 124, 83 (1951). 2H. Hayashi, F. Hamada, S. Suehiro, N. Masaki, T. Ogawa, and H. Miyaji,

J. Chern. Phys., Vol. 90, No.3, 1 February 1989

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Ogawa et al.: Fractal properties of polymer crystals 2067

J. Appl. Crystallogr. 21, 330 (1988). 3W. Ruland, Pure Appl. Chern. 49, 505 (1977). 4D. J. Blundell, Acta Crystallogr. Sect. A 26, 472 (1970). sC. G. Vonk, J. Appl. Crystallogr. 6, 81 (1973). 6B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Fran­cisco, 1982).

7J. E. Martin and A. J. Hurd, J. Appl. Crystallogr. 20, 61 (1987). 8M. Hikosaka, Polymer 28, 1257 (1987). 9p. C. Frank and M. Tosi, Proc. R. Soc. London Ser. A 263,323 (1961). IOJ. I. Lauritzen, Jr. and E. Passaglia, J. Res. Nat!. Bur. Stand. A 71, 261

(1967). liE. P. Price, J. Chern. Phys. 35, 1884 (1961). 12J. I. Lauritzen, Jr., E. A. DiMarzio, and E. Passaglia, J. Chern. Phys. 45,

4444 (1966). I3T. Seto, Physical Properties of Polymers (Kagaku Dojin, Kyoto, Japan,

1965), Chap. 7, in Japanese. 14W. K. Burton, N. Cabrera, and F. C. Frank, Philos. Trans. R. Soc. Lon­

don Sect. A 243,299 (1951). ISJ. Martinez-Salazar, P. J. Barham, and A. Keller, J. Mater. Sci. 20, 1616

( 1985).

J. Chern. Phys., Vol. 90, No.3, 1 February 1989

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