- Email: [email protected]
Simulation of low angle scattering of linear polarized light by anisotropic polymeric bodies
2674
Yu. V. BRESTKIN and B. M. GINZBURG
CONCLUSIONS For the "normal" binary copolymerizing system (reactivity of co-monomers is constant) analytical expressions (13) and (17) were obtained for integral and differential weight functions of distribution respectively, according to composition. These functions being known, integral and differential weight curves of copolymer distribution according to composition for different degrees of conversion can be plotted from well known copolymerization constants. A comparison of curves thus plotted with experimental curves facilitates conclusions conceming the mechanism of copolymerization, e.g. can distinguish the "normal" system from the "anomalous" one. Translated by E. SEMERE
REFERENCES 1. S. Ya. FRENKEL', Vvedenie v statistieheskuyu teoriyu polimerizatsii (Introduction to the Statistical Theory of Polymerization). Izd. "Nauka", 1965 2. V. A. MYAGCHENKOV, V. F. KURENKOV and S. Ya. FRENKEL', Dokl. AN SSSR 181, 147, 1968 3. V. A. MYAGCHENKOV, S. Ya. FRENKEL', Uspekhi khimii 37: 2245, 1968 4. F. M. LEWIS and F. It. MAYO, J. Amer. Chem. Soc. 66: 1594, 1944 5. J. SKEIST, J. Amer. Chem. Soe. 68: 1781, 1946 6. L. M. GINDIN, A. D. ABKIN and S. S. MEDVEDEV, Zh. fiz. khimii 21: 1969, 1947 7. V. E. MEYEIt and G. G. LOWRY, J. Polymer Sci. A3: 2843, 1965 8. V. A. MYAGCHENKOV, V. F. KUItENKOV and S. Ya. FItENKEL', Vysokomol. soyed. A10: 1740, 1968 (Translated in Polymer Sci. U.S.S.R. 10: 2013, 1968) 9. F. T. WALL, J. Amer. Chem. Soc. 66: 2050, 1944
SIMULATION OF LOW ANGLE SCATTERING OF LINEAR POLARIZED LIGHT BY ANISOTROPIC POLYMERIC BODIES* YU. V. BRESTKIN and B. ]~. GINZB~RG Institute of High Molecular Weight Compounds, U.S.S.R. Academy of Sciences
(Received 4 October 1968) SIMULATIO~ techniques are often used in diffractometric studies of po]ymer structure. They are widely used, particularly in eases when the diffraction p a t t e r n is poor and the Fourier synthesis is difficult [1, 2]. To select a model which represents the structure of a lens, all possible irLdireet information is used and the diffraction of this model is compared with the p a t t e r n of seatteriug experimentally observed; agreement between these patterns is the criterion for selection of the model, although it does not exclude the possibility of ambiguity. A diffraction pattern can either be obtained b y calculation, or directly using a model, the shape which is geometrically similar to the structure proposed; the wavelength of the * Vysokomol. soyed. A l l : No. 10, 2351-2354, 1969.
Simulation of low angle scattering of linear polarized light
2675
radiation should be varied in proportion to the dimensional ratio used [1-3]. I n the latter ease cumbersome calculations can be avoided. F o r optical simulation of X - r a y scattering by quasi-planar objects normally situated in the direction of an incident wave [1, 3] the instrument diagrammatically shown in Fig. 1 is used successfully. We used a similar apparatus for the simulation of low-angle scattering of linearly polarized light by optically anisotropic plane objects. This simulation is apparently the first of its kind and involves certain specific methods, which will be described.
z
,
IViT
cz b c
d
eL I f L z
g
FIG. l. L a y o u t of apparatus for observing Frauenhofer diffraction from flat objects: a - - l i g h t source, b--light filter, c--condenser, d - - d i a p h r a g m of ~0-05 m m diameter, situated in the focal plane of a long focus lens L~, e--diaphragm, f - - o b j e c t ; g - - i m a g e plane of the diffraction pattern, coinciding with the focal plane of long focus lens L~. Let us examine an beam of light, assuming conditions indicated the isotropie object is given
instantaneous pattern of scattering by the object of a parallel t h a t the instantaneous electric feld strength is unity. U n d er the amplitude of the wave scattered at low angles by a flat optical by the formula [4]: A = k J" p ' c o s g'ds 8
where 27~
&=--(~..~) A
(l)
Here k is a constant, which is of no further interest; p is the density of scattering elements in the element of cross section ds; A is the wavelength; ~*=-s0---~', So and ~ are unit vectors the direction of which coincide with the incident and scattered waves; 9 is a vector drawn from the arbitrarily selected centre of the object to the element of cross section ds; & is the phase difference between the beam coming from the cross section element ds in direction ~' and the beam going in the same direction from the centre. The amplitude of the wave scattered at low angles by an optically anisotropie plane object when exposed to linearly polarized light is determined as [5]
A =k ~ p (M'O) cos 5.dt
(2)
8
Here .37/ is the dipole moment induced in the cross-sectional element ds by the electric fieId of the incident wave: () is the unit vector perpendicular to the polarization plane o f the analyser. The other symbols are the same as irL formula (1). The scalar product (3/1.0) in formula (2) may be either positive or negative. Let us express equation (2) as:
A = k f p'. cos d " d s 8
(3}
2676
Y u . V . BRESTKII~ a n d B. M. GINZBURG
where
p'=p I ( i 6)1
6
, =|2~F~
~
~,-I
-.
Comparing (1) and (3) it is easy to see that the isotropic object m a y be used as a model of the anJsotropie object, if the following conditions hold good: 1) the isotropie object should have a distribution density of scattering elements determLrmd b y the function p] ( ~ / . O ) [ ; 2) a n additional phase difference should be formed in the half-waves between the beams from parts of the object, to which the positive and negative products ( M ' O ) * correspond.
FIG. 2. a - - N e g a t i v e of the diffraction p a t t e r n of two closely situated round openings, b-same with matehing of one opening with phase inverting filter. When changing from a to b interference maxima and m i n i m a vary at certain points. The second requirement can be fulfilled by arranging phase (more exactly phase rotating) filters in the space between lens L~ (Fig; 1) and the object and b y combining them with parts of the object, to which values (M O) of a certai_~ sign correspond (selection is indifferent since it m a y only affect the sign of the amplitude, which is of no importance in the p a t t e r n of scattering). The thickness of transparent plates h, which can be used as phase rotating filters, is related to their refractive index n b y (n--1).h=z').-F)~/2 ,
where
z ~ 0 , 1, 2...
I f the source is not sufficiently monochromatic z should not be greater t h a n 1-2. We used mica plates as phase filters. The thickness was controlled b y observing the diffraction from two closely placed circular apertures. One opening was combined with the filter tested; the plate was regarded suitable when the diffraction p a t t e r n shown in Fig. 2 was transformed. These considerations were used for designing a fiat (disc type) spherulite model of uniform density, the cross-sectional elements of which have both radia] (~) and tangential (~) components of polarization. * Similar considerations concerning the phase difference between the beams from different parts of anisotropic objects accounted for one of the possible explanations of a central spot on Hv-diffraction curves of spherulites [6].
Simulation of low angle scattering of linear polarized light
2677
For this object, according to a former study [5]
Here fl is wave and the I t follows ization with a
(M "0) ~ (er--~) sin ;6 cos fl
with
Ho-polarization
(4)
( M . O ) , . ~ cos ~ fl+er sin~ fl
with
Vo-polarization
(5)
the angle between the vector direction of the electric field of the incideat line joining the cross-sectior~M element with the spherulite centre. from equation (4) that a disc m a y be used as spherulite model with H~-polardenMty distribution of scattering elemer~ts p ~ I siI~ .~ c o s .81
(6)
Hi,
Hv
/IIQ//
5
FIG. 3. Negatives of models and diffraction patterns for spherulites (1, 2), deformed (3, 4) a n d annular (5) spherulites. I n the simulatior~ of V v diffraetior~ of spherulites it was assume(1 that :(t>>~r. The diffraetiori pattern of armular spkerulite is eor~siderably reduced, compared with the others. Asterisks indicate the figures which correspond to diffraetion from models 1 a n d 3, b u t without usiilg phase filters: 1--object, 2--polarization, 3--model, 4--diffraetiom
2678
Yu. V. BRESTKIN and B. M. GINZBUI~G
I n view of the fact that function (4) changes sign with variation in angle B, phase filter~ are required for simulation of Hv diffraction of spherulite. These were arranged in the path of beams falling on the sectors of the model with angular dimensions of fl ranging from ~/2 to z~ and from 3zt/2 to 2zc. According to equation (5) for V~ polarization a disc with density distribution p ~ cos 2 B+~rl~t sin 2 fl,
(7)
is the spherulite model, since function (5) has a constant sign and phase filters are not required. Photographs of models which show the density distribution of scattering elements approximately corresponding to functions (6) and (7) are shown in Fig. 3. This Figure also illustrates the corresponding diffraction curves, which are very close to the scattering patterns of linear polarized light from real polymer films containing spherulites*. Optical models of deformed and armular spherulites were designed b y a similar method. Structural informatior~ [5, 6, 8] available in the literature was used. Photographs of models and corresponding diffraction curves arc also shown in Fig. 3. Scattering observed in the centre of the diffraction curve is very typical of spherulites elongated at a n angle of 45 ° to one of the planes of polarization; a similar effect is observed in real polymer systems [6]. Since optical simulation enables diffraction curves to be obtained without calculation, this method m a y be very useful i n rapid interpretation of real scattering patterns. I t is possible t h a t it will be used for the study of supermolecular re-organization of polymers and other transparent optically anisotropie objects. Finally, the authors would like most sincerely to t h a n k S. Ya. Frenkel for fruitful discussions.
CONCLUSIONS (1) I t is shown that low-angle scattering of linear polarized light with anisotropio quasi-flat objects can be imitated by scattering of non-polarized light with two-dimensional isotropic objects. These should not only have corresponding density distribution of scattering elements, b u t should partly be matched by phase-inversion plates. (2) Optical models of quasi-flat spherulites were proposed for H v and Vv polarizatior~ and models of a n n u l a r and deformed spherulites for H v polarization. Translated by E. SEMERE
REFERENCES 1. R. HOSEMANN and S. N. BAGCHI, Direct Analysis of Diffraction by Matter. North Holland Publ. Co., Amsterdam, 1962 2. R. BONART and R. HOSEMANN, Makromolek. Chem. 39: 105, 1960 3. C. A. TAYLOR, Europ. Polymer J. 2: 279, 1966 4. B. K. VAINSHTEIN, Diffraktsiya rentgenovskikh luchei na tsepnykh molekulakh, Izd. AN SSSR, 1963 5. It. S. STEIN and M. B. RHODES, J. Appl. Phys. 31: 1873, 1960 6. Yu. V. BRESTKIN, B. M. GINZBURG, I. T. MONEVA, S. Ya FRENKEL', Fizika tverdogo tela 10: 3130, 1968 7. V. G. BARANOV, Optika i spektroskopiya 21: 610, 1966 8. R. S. STEIN, P. E R H A R D T , g. g. VAN AARTSEN, S. CLOUGH and M. B. RHODES, J. Polymer Sci. C13: 1, 1966 * The similarity of diffraction curves from a spherulite model and a film containing o, large n u m b e r of spherulites is due to irregular arrangement in real specimerm. I t is also well known that the pattern of scattering from the three-dimensional spherulites which are rmrreally, contained b y a polymer film differs little from the p a t t e r n of scattering from flat spherulites [7].
Yu. V. BRESTKIN and B. M. GINZBURG
CONCLUSIONS For the "normal" binary copolymerizing system (reactivity of co-monomers is constant) analytical expressions (13) and (17) were obtained for integral and differential weight functions of distribution respectively, according to composition. These functions being known, integral and differential weight curves of copolymer distribution according to composition for different degrees of conversion can be plotted from well known copolymerization constants. A comparison of curves thus plotted with experimental curves facilitates conclusions conceming the mechanism of copolymerization, e.g. can distinguish the "normal" system from the "anomalous" one. Translated by E. SEMERE
REFERENCES 1. S. Ya. FRENKEL', Vvedenie v statistieheskuyu teoriyu polimerizatsii (Introduction to the Statistical Theory of Polymerization). Izd. "Nauka", 1965 2. V. A. MYAGCHENKOV, V. F. KURENKOV and S. Ya. FRENKEL', Dokl. AN SSSR 181, 147, 1968 3. V. A. MYAGCHENKOV, S. Ya. FRENKEL', Uspekhi khimii 37: 2245, 1968 4. F. M. LEWIS and F. It. MAYO, J. Amer. Chem. Soc. 66: 1594, 1944 5. J. SKEIST, J. Amer. Chem. Soe. 68: 1781, 1946 6. L. M. GINDIN, A. D. ABKIN and S. S. MEDVEDEV, Zh. fiz. khimii 21: 1969, 1947 7. V. E. MEYEIt and G. G. LOWRY, J. Polymer Sci. A3: 2843, 1965 8. V. A. MYAGCHENKOV, V. F. KUItENKOV and S. Ya. FItENKEL', Vysokomol. soyed. A10: 1740, 1968 (Translated in Polymer Sci. U.S.S.R. 10: 2013, 1968) 9. F. T. WALL, J. Amer. Chem. Soc. 66: 2050, 1944
SIMULATION OF LOW ANGLE SCATTERING OF LINEAR POLARIZED LIGHT BY ANISOTROPIC POLYMERIC BODIES* YU. V. BRESTKIN and B. ]~. GINZB~RG Institute of High Molecular Weight Compounds, U.S.S.R. Academy of Sciences
(Received 4 October 1968) SIMULATIO~ techniques are often used in diffractometric studies of po]ymer structure. They are widely used, particularly in eases when the diffraction p a t t e r n is poor and the Fourier synthesis is difficult [1, 2]. To select a model which represents the structure of a lens, all possible irLdireet information is used and the diffraction of this model is compared with the p a t t e r n of seatteriug experimentally observed; agreement between these patterns is the criterion for selection of the model, although it does not exclude the possibility of ambiguity. A diffraction pattern can either be obtained b y calculation, or directly using a model, the shape which is geometrically similar to the structure proposed; the wavelength of the * Vysokomol. soyed. A l l : No. 10, 2351-2354, 1969.
Simulation of low angle scattering of linear polarized light
2675
radiation should be varied in proportion to the dimensional ratio used [1-3]. I n the latter ease cumbersome calculations can be avoided. F o r optical simulation of X - r a y scattering by quasi-planar objects normally situated in the direction of an incident wave [1, 3] the instrument diagrammatically shown in Fig. 1 is used successfully. We used a similar apparatus for the simulation of low-angle scattering of linearly polarized light by optically anisotropic plane objects. This simulation is apparently the first of its kind and involves certain specific methods, which will be described.
z
,
IViT
cz b c
d
eL I f L z
g
FIG. l. L a y o u t of apparatus for observing Frauenhofer diffraction from flat objects: a - - l i g h t source, b--light filter, c--condenser, d - - d i a p h r a g m of ~0-05 m m diameter, situated in the focal plane of a long focus lens L~, e--diaphragm, f - - o b j e c t ; g - - i m a g e plane of the diffraction pattern, coinciding with the focal plane of long focus lens L~. Let us examine an beam of light, assuming conditions indicated the isotropie object is given
instantaneous pattern of scattering by the object of a parallel t h a t the instantaneous electric feld strength is unity. U n d er the amplitude of the wave scattered at low angles by a flat optical by the formula [4]: A = k J" p ' c o s g'ds 8
where 27~
&=--(~..~) A
(l)
Here k is a constant, which is of no further interest; p is the density of scattering elements in the element of cross section ds; A is the wavelength; ~*=-s0---~', So and ~ are unit vectors the direction of which coincide with the incident and scattered waves; 9 is a vector drawn from the arbitrarily selected centre of the object to the element of cross section ds; & is the phase difference between the beam coming from the cross section element ds in direction ~' and the beam going in the same direction from the centre. The amplitude of the wave scattered at low angles by an optically anisotropie plane object when exposed to linearly polarized light is determined as [5]
A =k ~ p (M'O) cos 5.dt
(2)
8
Here .37/ is the dipole moment induced in the cross-sectional element ds by the electric fieId of the incident wave: () is the unit vector perpendicular to the polarization plane o f the analyser. The other symbols are the same as irL formula (1). The scalar product (3/1.0) in formula (2) may be either positive or negative. Let us express equation (2) as:
A = k f p'. cos d " d s 8
(3}
2676
Y u . V . BRESTKII~ a n d B. M. GINZBURG
where
p'=p I ( i 6)1
6
, =|2~F~
~
~,-I
-.
Comparing (1) and (3) it is easy to see that the isotropic object m a y be used as a model of the anJsotropie object, if the following conditions hold good: 1) the isotropie object should have a distribution density of scattering elements determLrmd b y the function p] ( ~ / . O ) [ ; 2) a n additional phase difference should be formed in the half-waves between the beams from parts of the object, to which the positive and negative products ( M ' O ) * correspond.
FIG. 2. a - - N e g a t i v e of the diffraction p a t t e r n of two closely situated round openings, b-same with matehing of one opening with phase inverting filter. When changing from a to b interference maxima and m i n i m a vary at certain points. The second requirement can be fulfilled by arranging phase (more exactly phase rotating) filters in the space between lens L~ (Fig; 1) and the object and b y combining them with parts of the object, to which values (M O) of a certai_~ sign correspond (selection is indifferent since it m a y only affect the sign of the amplitude, which is of no importance in the p a t t e r n of scattering). The thickness of transparent plates h, which can be used as phase rotating filters, is related to their refractive index n b y (n--1).h=z').-F)~/2 ,
where
z ~ 0 , 1, 2...
I f the source is not sufficiently monochromatic z should not be greater t h a n 1-2. We used mica plates as phase filters. The thickness was controlled b y observing the diffraction from two closely placed circular apertures. One opening was combined with the filter tested; the plate was regarded suitable when the diffraction p a t t e r n shown in Fig. 2 was transformed. These considerations were used for designing a fiat (disc type) spherulite model of uniform density, the cross-sectional elements of which have both radia] (~) and tangential (~) components of polarization. * Similar considerations concerning the phase difference between the beams from different parts of anisotropic objects accounted for one of the possible explanations of a central spot on Hv-diffraction curves of spherulites [6].
Simulation of low angle scattering of linear polarized light
2677
For this object, according to a former study [5]
Here fl is wave and the I t follows ization with a
(M "0) ~ (er--~) sin ;6 cos fl
with
Ho-polarization
(4)
( M . O ) , . ~ cos ~ fl+er sin~ fl
with
Vo-polarization
(5)
the angle between the vector direction of the electric field of the incideat line joining the cross-sectior~M element with the spherulite centre. from equation (4) that a disc m a y be used as spherulite model with H~-polardenMty distribution of scattering elemer~ts p ~ I siI~ .~ c o s .81
(6)
Hi,
Hv
/IIQ//
5
FIG. 3. Negatives of models and diffraction patterns for spherulites (1, 2), deformed (3, 4) a n d annular (5) spherulites. I n the simulatior~ of V v diffraetior~ of spherulites it was assume(1 that :(t>>~r. The diffraetiori pattern of armular spkerulite is eor~siderably reduced, compared with the others. Asterisks indicate the figures which correspond to diffraetion from models 1 a n d 3, b u t without usiilg phase filters: 1--object, 2--polarization, 3--model, 4--diffraetiom
2678
Yu. V. BRESTKIN and B. M. GINZBUI~G
I n view of the fact that function (4) changes sign with variation in angle B, phase filter~ are required for simulation of Hv diffraction of spherulite. These were arranged in the path of beams falling on the sectors of the model with angular dimensions of fl ranging from ~/2 to z~ and from 3zt/2 to 2zc. According to equation (5) for V~ polarization a disc with density distribution p ~ cos 2 B+~rl~t sin 2 fl,
(7)
is the spherulite model, since function (5) has a constant sign and phase filters are not required. Photographs of models which show the density distribution of scattering elements approximately corresponding to functions (6) and (7) are shown in Fig. 3. This Figure also illustrates the corresponding diffraction curves, which are very close to the scattering patterns of linear polarized light from real polymer films containing spherulites*. Optical models of deformed and armular spherulites were designed b y a similar method. Structural informatior~ [5, 6, 8] available in the literature was used. Photographs of models and corresponding diffraction curves arc also shown in Fig. 3. Scattering observed in the centre of the diffraction curve is very typical of spherulites elongated at a n angle of 45 ° to one of the planes of polarization; a similar effect is observed in real polymer systems [6]. Since optical simulation enables diffraction curves to be obtained without calculation, this method m a y be very useful i n rapid interpretation of real scattering patterns. I t is possible t h a t it will be used for the study of supermolecular re-organization of polymers and other transparent optically anisotropie objects. Finally, the authors would like most sincerely to t h a n k S. Ya. Frenkel for fruitful discussions.
CONCLUSIONS (1) I t is shown that low-angle scattering of linear polarized light with anisotropio quasi-flat objects can be imitated by scattering of non-polarized light with two-dimensional isotropic objects. These should not only have corresponding density distribution of scattering elements, b u t should partly be matched by phase-inversion plates. (2) Optical models of quasi-flat spherulites were proposed for H v and Vv polarizatior~ and models of a n n u l a r and deformed spherulites for H v polarization. Translated by E. SEMERE
REFERENCES 1. R. HOSEMANN and S. N. BAGCHI, Direct Analysis of Diffraction by Matter. North Holland Publ. Co., Amsterdam, 1962 2. R. BONART and R. HOSEMANN, Makromolek. Chem. 39: 105, 1960 3. C. A. TAYLOR, Europ. Polymer J. 2: 279, 1966 4. B. K. VAINSHTEIN, Diffraktsiya rentgenovskikh luchei na tsepnykh molekulakh, Izd. AN SSSR, 1963 5. It. S. STEIN and M. B. RHODES, J. Appl. Phys. 31: 1873, 1960 6. Yu. V. BRESTKIN, B. M. GINZBURG, I. T. MONEVA, S. Ya FRENKEL', Fizika tverdogo tela 10: 3130, 1968 7. V. G. BARANOV, Optika i spektroskopiya 21: 610, 1966 8. R. S. STEIN, P. E R H A R D T , g. g. VAN AARTSEN, S. CLOUGH and M. B. RHODES, J. Polymer Sci. C13: 1, 1966 * The similarity of diffraction curves from a spherulite model and a film containing o, large n u m b e r of spherulites is due to irregular arrangement in real specimerm. I t is also well known that the pattern of scattering from the three-dimensional spherulites which are rmrreally, contained b y a polymer film differs little from the p a t t e r n of scattering from flat spherulites [7].