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X-ray study of the orientation distribution of crystallites
1906
A. SH. GOIKHMAI~
11. M. Z. GAVRILOV and I. N. YERMOLENKO, Zh. prikl, spektroskopii 6: 197, 1967 12. Ch. N. R. RAO, Elektronnye spektry v khimii (Electron Spectra in Chemistry). Izd. "Mir", 1964 13. I. N. YERMOLENKO and M. Z. GAVRILOV, Reports on I Conf. on Use of Methods of Molecular Spectral Analysis (Russian), publ. by Byeloruss. State Univ., 1958 14. I. N. YERMOLENKO and M. Z. GAVRILOV, Vestnik Akad. Nauk BSSR, set. fiz.-mat. n., No. 4, 126, 1966
X-RAY
S T U D Y OF T H E O R I E N T A T I O N
DISTRIBUTION
OF C R Y S T A L L I T E S * A. SH. GOIKHMAN
Kiev Branch of the All-Union Synthetic Fibre Research Institute (Received 6 June 1966)
X-RAY scattering in a crystallite system with ideal uniaxial orientation and statistical azimuthal distribution of the crystallites (a texture with complete rotation) produces a pattern like the X-ray diagram of a rotating crystal. In this case a certain direction in the crystal, e.g. normal to hkl, makes a quite definite angle with the texture axis, and this angle is the same for all crystallites. I n a concrete case, for instance in partially oriented fibres and films, there is a certain dispersion of crystallites relative to the texture axis and the points in the X-ray diagram of the texture are blurred into arcs ("sickles") along D e b y e Sherrer rings. It is thought that the cryst~llite distribution relative to the texture axis can be estimated from the intensity distribution along these ares. Let us consider the connection between the intensity distribution for the interference of hkl with respect to the azimuthal angle J (read off the meridian in the X-ray picture of the fibre) and the distribution of the crystallites. Let us say the position of the crystallite is the direction normal to hk/--one of the many crystallographic axes. We know [1] that the integral reties intensity in an X-ray texture diagram is connected with the intensity of the same reflex in the X-ray rotation diagram of a single crystal b y the relationship
tox =Irotn/2
(1)
where Itex, trot are the integral intensities in the texture diagram and rotation diagram respectively; ~ is the number of crystallites making a contribution to "~tex °
* Vysokomol. soyed. A9: No. 8, 1693-1698, 1967.
X - r a y study of orientation distribution of crystallites
1907
I f ~ is the angle between normal to hkl and the texture axis, and the density of the normals is I(~), then n--~2u S L(~) sin q d q .
(2)
o
Then the crystallites with normals lying within limits of ~ and q-J-dtt scatter with an intensity of d l in the differential element of a Debye ring with a length of dl: dI=XrotL (~) sill qd~. (3) Using the expression for the integral reflex intensity in a rotation diagram we obtain d I = A 2L(9) sin 9 sin 0 dq, ~/sin ~ a--cos ~ 20
(4)
where 0 is the Wolf-Bragg angle; A is a factor which is unimportant in our case (it is independent of ~); a is the angle between the scattered r a y and the texture axis. Using the known relationships [2]: cos ~ = c o s 0.cos g and cos a----2 cos q. •sin 0, we obtain dI----AL (~) dS . (5) This expression (5) shows t h a t the density of the intensity distribution in regard to azimuthal angle 5 is proportional to the density of the distribution of normals in regard to q. When the plane under consideration is diatropie the intensity distribution in regard to 5 gives the distribution of the crystallite axes. Generally a crystallite axis makes an angle of ?, with the normal. B y crystallite axis we mean the direction in the crystallite which during orientation is set parallel to the elongation axis (the fibre axis). I n most polymeric fibres the crystallite axis coincides with the direction of the molecular chains. Since as a rule the distribution of axes is not coincident with the distribution of normals, the expression (5) cannot be used to find the axes distribution: we must find the relationship between the function L (q) and the density of the axes Q (q). The solution of this problem gives the following relationship [3]: ~/2
0
fl and W are connected b y the relationship sin fl=eos ~.eos ?,--sin ~.sin ?,.cos N,
(7)
where fl is an angle supplementary to the angle between a cryst~llite a d s and the fibres axis fi----~/2--9', (the designation ~', is necessary as ~' #{a).
I908
A. S m GoIx~S.A~r
Thus
dI
2A f Q(fl)d~.
(8)
o
On examining equation (8) we m a y arrive at a rather unexpected conclusion. Since for different subintegral functions the value of a particular integral m a y be the same, different functions of the axes distribution m a y correspond to one function of the azimuthal distribution of intensity F (6). Thus Q (fl) cannot be unambiguously determined from the data for the azimuthal distribution of interference intensity for hkl in the X-ray diagram; t h i s w a s first pointed out by :Hermans [4]. The only possible method is to t r y different functions of Q (fl), and to compare the experimental and theoretical distribution curves. Several authors use gaussian distribution [5, 6]. Thus in [5] a highly-oriented case (where sin q ' = ~ ' ) is investigated. Obviously the point of interest is the general case where the condition sin q' = q ' is not fulfilled, e.g. in investigation of fibres with the multiplicity of elongation v; varying from v = l (isotropic fibre) to Vmax. In this case it is desirable t h a t the distribution function Q (fl) should be one in which v figures directly. Gaussian distribution is "inconvenient" in this case, as it is not based on a concrete model of the deformation process during stretching, and contains the value of v in the form of a "hidden" parameter. I n studying the deformation of swollen cellulose films K r a t k y [7] assumed a distribution of the axes of particles ("miceHes") based on the following model: the particles are suspended in a medium which is considered to be completely isotropic, and the forces of cohesion of the particles and m e d i u m are large in comparison with their weight and inertia, so t h a t under deformation the medium does not flow round the particles, which are not deformed under elongation b u t merely turn; for the particles the system is dilute, steric hindrances do not impede the turning of the particles. These ideas can be transferred to the process of deformation (elongation) of synthetic fibres of low crystallinity t h a t have crystalline regions of small dimensions [8]. The K r a t k y density distribution of axes takes the form:
Q (fl)= Qo v3/[1+ (v 3 - 1 ) cos ~ilia/2,
(9)
where Q0 is the axes density when v = 1 (the isotropic case); fl is the same as in formula (7). Taking (9) into account, formula (8) m a y be written as .Y' ( ~ ) = ~ - f
Q°vSd~
[1 -t- (v8 - 1 ) cos z ilia/2 •
(10)
o
Considering only the paratropic interferences of hko (or hol in a monoclinic
X - r a y s t u d y of orientation distribution of crystallites
system) when
r=n/2,
1909
then according to (7) cos s fl-= 1--sin s q cos s ~ .
(11)
Putting (11) in (10) and transposing, we get ~12
d~/
2AQov-s/z f
;
v3 - 1
oJ (1-
sinsq"c o s 2 ~)312"
(12)
Interpolating b2= v a - 1 sin 2 va
then ~/2
(1 - - k 2 cos 2 ~)3/2" o
Substituting ~ = - - - e 2
we obtain ~/2
2AQ v -3/2 f & F@= - j (1--k2 sin2 t) s/2"
(13)
o
The integral of (13) is in the table (9) and is reduced to the form
_P(~)=~( 1-k2) E kl where E
1~
,
(14)
is a complete elliptical integral of the second type; k : s i n q.
• ~/(va - 1)/v3. Remembering the relationship of ~ and J, it is easy to calculate F(J)
if we know the value of v. When v = l (the isotropie ease) k=O, E(O,, 2)
x 2
and F ( J ) = A o Q0=const. Assuming that 2AQo/x=C, let us consider F (J) = F (J)/C. Figure I shows curves calculated from equation (14) for different elongation multiplicity factors. We must also consider the integral intensity ~/2
o
to find its relationship to v.
1910
A. S m GOrKH~A~
The integral intensity I m a y be written as follows: - s/21
=/2 ~/2
v
v s-
1
X'/ 1
vs
(
V3__I
l-~2A-~Q°f f 0 0
1
sin S ~.sin S
va
d~aa
)
sin S
(15)
or, according to (14) i 2 v-S/2 E ( k 1 2 ) s i n q~d ¢ \ (1 --k ~) x/cos z 0--cos 2
I'=
(16)
0
The values of I or I', m a y be calculated from (16) using the Gauss]an method of calculating q u a d r a t u r e s - a n d taking the number of terms in the Ganssian formula as n----3, we found the following values of I ' for different values of v (when 0 = 10°10'): v 1.0 2.0 2.5 3.0 8.5 4.0 I',qon9entional units 2.3386 1.9808 1.9250 1.8593 1.7660 1.6431 These data show t h a t the integral intensity is not independent of v, b u t decreases with increase in the elongation multiplicity factor. Thus the areas beneath the curves in Fig. 1 are not equal, and in cases where the azimuthal 40
8-
3'5
-
~
-
30
x
2.5 4-
20
I
90
I
70
"
I
I
I
L
t
50 30 Polo/" anq/e, ~o
I.
lO
FIG. 1. Density distribution in respect of angle ~ ealeulaf~d from equation (14). Figures
by curves are values of v. distribution of intensity is described b y formula (14) it is not possible to normalize the experimental curves b y equalizing the squares. EXPERIMENTAL
Polycaproamide fibre m a y be used to verify K r a t k y distribution as it has low crystallinity [10] and small crystallite dimensions [11]. These are the proper conditions for the K r a t k y model.
X-ray study of orientation distribution of crystallites
1911
Unoriented fibre ( v = l ) was obtained under normal production conditions. I t is not completely isotropic owing to spinneret stretching, and does not contain a n y appreciable a m o u n t of the monoelinic ~-form, or rather it contains only the nuclei of this crystalline modification, statistically oriented, so t h a t it m a y be regarded as isotropic for these crystallites. The stretching was accomplished on a special testing structure with a heating element, the element temperatures being 20, 100, 180 and 200 °, and the rate 70 m/rain on a dynamometer at 20 °, at a rate of 11 ram/rain. The elongation multiplicity factor v was found from the ratio of the rates of the take-in and feed devices on the testing structure and the d y n a m o m e t e r scale of elongation. The azimuthal curves of intensity distribution for the interference (200) were recorded on an ~tS-50II~I diffractometer (Ni-filtered CuK~ radiation) using a device that enabled the sampie to be slowly turned in a plane perpendicular to the beam (a complete revolution takes 72 rain) with a fixed counter (20 =20°20'). The intake slit was the widest possible in order to take in the whole width of interference. Because of the marked widening of the lines there is some overlapping of adjacent interferences (200) and (002) and (202). After restoring the profile of the line b y means of Gaussian distribution it can be shown t h a t the optimum slit width is 2 ram. The equatorial reflex (002) (202) is the superimposition of two interferences so t h a t it m a y not be used to verify distribution (14). S
S2 ~I
gO
I
f
x
70
50
30
" I
t~
~ °2
I0
Azirnuthal ang/e ,
FxG. 2. Intensity components in the azimuthal curve of distribution. For comparison with the experimental values q must be converted ¢o 6 in (14) according to the relationship cos ~ = c o s 0 cos 6. For a definite value of 0 a certain region close to q = 0 is forbidden, namely the region of 0>@~0, as in this region cos 6 > 1 , which is impossible. It is extremely difficult to draw a background line below the azimuthal distribution curve for X-ray diagrams of polycaproamide fibres as analysis of the area below the diffractometric curve
1912
A. SH. GOIKH~AI~
into crystalline and amorphous components cannot be reliably accomplished when we do not know the shape of the curve of scattering b y the amorphous component. Because of this the following method was used. Figure 2 shows that for a certain value of 6 (for instance when 6 = 9 0 °) the overall intensity
i
c
90
90
70
70
50
50
30
30
10 90
\.
o.3
I
I
I
[
70
50
30
10
gO 70 50 30gO' Azimuthal angle, o~
70
50
30
Fie. 3. Calculated curves (unbroken lines) of azimuthal distribution in respect ot7 anglo
(0= 10°10') for different values of v and experimental results (points) for the interference (200) of polycaproamide. Values of v: a--2.0; b--2-5; c--3.0; d--3-5; e--4.0. Stretching on a bench, and on a dynamometer at temperatures of (°C): 1--20; 2--100; 3--180; 4--200.
F=Fcr-~Fbk where Fbk=Famorph~Fcompton-t-FaL r. Drawing a line 01S1 from a point in the curve 01 corresponding to 6=01, we m a y write
(0) where F~ (0) is a certain unknown part of the intensity of For ; J1 is any selected value of 6. As it is possible to use equation (14) to find the theoretical values of F'cr (61) and For (0) for definite values of v and 0, the value of F~ (0) is easily found from
For
(o).
Combining in this w a y two points in the experimental and theoretical curves we now proceed to plot the whole curve, normalizing the experimental values of F (61) b y dividing them b y the coefficient of K = F ~ (O)/F~r (0)*. Figure 3 shows the results of comparing the experimental normalized values with the theoretical relation (14) for different elongations and different temperatures and rates of stretching. * When doing so, it is assumed the amorphous components to be tmorien~ed.
Thermodynamic functions of methacrylic acid
1913
CONCLUSIONS
(l) T h e a z i m u t h a l distribution of intensity o f p a r a t r o p i c interferences as a function o f t h e degree o f elongation has been calculated using K r a t k y ' s distribution of crystallite axes. (2) I t has been shown t h a t t h e integral reflex i n t e n s i t y calculated from' t h e a x i m u t h a l i n t e n s i t y distribution, decreases with increase in elongation multiplicity. (3) The calculated i n t e n s i t y distributions agree with the e x p e r i m e n t a l results for the interference (200) of p o l y c a p r o a m i d e . Translated by R. J. A. HZN'DRY
REFERENCES
1. L. G. KAZARYAN and D. Ya. TSVANKIN, Vysokomol. soyed. 5: 1963, 976 (Translated in Polymer Sci. U.S.S.R. 5: 1, 29, 1964) 2. B. K. VAUNSTEIN, Difrakteiya rent genovskikh luchei net tsepnykh molekulakh (Diffractiort of X-rays by Chain Molecules). Izd. Akad. Nauk SSSR, 1964 3. O. KRATKY, G. POROD and E. TRIEBER, Kolloid.Z. 121: 1, 1959 4. $. ,I. HERMANS, P. H. HERMANS, D. VERMAAS and A. WEIDINGER, Recuell. tray. chim. Pays.-Bas. 65: 427, 1946 5. A. I. SLUTSKER and A. I. GROMOV, Physics of solids 5: 2185, 1963 6. H. KR,I.SSIG and W. KITCHEN, J, Polymer Sci. 51: 123, 1961 7. O. KRATKY, Kolloid-Z. 64: 213, 1933 8. J. CERNY, Textil, No. 4, 141, 1961 9. I. S. G R A D S H T E I N and I. M. RYZIfl~, Tablitsy intcgralov, summ, ryadov i proizvedennii (Tables of IategraLq Sums, Series and Products). 170, Fizmatgiz, 1962 10. I. L N O V A K and V. I. V E T T E G R E N ' , Vysokomol. soyed. 7: I027, 1965 (Translated in Polymer Sci. U.S.S.R. 7: 6, 1027, 1965) I. G. U R B A N C Y K , J. Polymer Sci. 59:215 1962
THE HEAT CAPACITY AND THERMODYNAMIC FUNCTIONS OF METHACRYLIC, POLYMETHACRYLIC AND ISOBUTYRIC ACIDS*t I. ]3. RA~ZNOVlCH,
B. V. L E B E D E V
and T. I. M E L E N T ' E V A
Chemical Research Institute of the N. I. Lobachevskii University, Gorki (Received 9 June 1966)
MEASUREMENT of the h e a t c a p a c i t y o f p o l y m e r s in t h e l o w - t e m p e r a t u r e region has a l r e a d y been r e p o r t e d on in several p a p e r s (for e x a m p l e [1-3]) as a m e a n s of calculating t h e m a i n t h e r m o d y n a m i c characteristics of these systems. The same * Vysokomol. soyed. All: No. 8, 1699-1705, 1967. 3rd Report in the series "Thermodynamics of monomers and polymers of the vinyl ~rios".
A. SH. GOIKHMAI~
11. M. Z. GAVRILOV and I. N. YERMOLENKO, Zh. prikl, spektroskopii 6: 197, 1967 12. Ch. N. R. RAO, Elektronnye spektry v khimii (Electron Spectra in Chemistry). Izd. "Mir", 1964 13. I. N. YERMOLENKO and M. Z. GAVRILOV, Reports on I Conf. on Use of Methods of Molecular Spectral Analysis (Russian), publ. by Byeloruss. State Univ., 1958 14. I. N. YERMOLENKO and M. Z. GAVRILOV, Vestnik Akad. Nauk BSSR, set. fiz.-mat. n., No. 4, 126, 1966
X-RAY
S T U D Y OF T H E O R I E N T A T I O N
DISTRIBUTION
OF C R Y S T A L L I T E S * A. SH. GOIKHMAN
Kiev Branch of the All-Union Synthetic Fibre Research Institute (Received 6 June 1966)
X-RAY scattering in a crystallite system with ideal uniaxial orientation and statistical azimuthal distribution of the crystallites (a texture with complete rotation) produces a pattern like the X-ray diagram of a rotating crystal. In this case a certain direction in the crystal, e.g. normal to hkl, makes a quite definite angle with the texture axis, and this angle is the same for all crystallites. I n a concrete case, for instance in partially oriented fibres and films, there is a certain dispersion of crystallites relative to the texture axis and the points in the X-ray diagram of the texture are blurred into arcs ("sickles") along D e b y e Sherrer rings. It is thought that the cryst~llite distribution relative to the texture axis can be estimated from the intensity distribution along these ares. Let us consider the connection between the intensity distribution for the interference of hkl with respect to the azimuthal angle J (read off the meridian in the X-ray picture of the fibre) and the distribution of the crystallites. Let us say the position of the crystallite is the direction normal to hk/--one of the many crystallographic axes. We know [1] that the integral reties intensity in an X-ray texture diagram is connected with the intensity of the same reflex in the X-ray rotation diagram of a single crystal b y the relationship
tox =Irotn/2
(1)
where Itex, trot are the integral intensities in the texture diagram and rotation diagram respectively; ~ is the number of crystallites making a contribution to "~tex °
* Vysokomol. soyed. A9: No. 8, 1693-1698, 1967.
X - r a y study of orientation distribution of crystallites
1907
I f ~ is the angle between normal to hkl and the texture axis, and the density of the normals is I(~), then n--~2u S L(~) sin q d q .
(2)
o
Then the crystallites with normals lying within limits of ~ and q-J-dtt scatter with an intensity of d l in the differential element of a Debye ring with a length of dl: dI=XrotL (~) sill qd~. (3) Using the expression for the integral reflex intensity in a rotation diagram we obtain d I = A 2L(9) sin 9 sin 0 dq, ~/sin ~ a--cos ~ 20
(4)
where 0 is the Wolf-Bragg angle; A is a factor which is unimportant in our case (it is independent of ~); a is the angle between the scattered r a y and the texture axis. Using the known relationships [2]: cos ~ = c o s 0.cos g and cos a----2 cos q. •sin 0, we obtain dI----AL (~) dS . (5) This expression (5) shows t h a t the density of the intensity distribution in regard to azimuthal angle 5 is proportional to the density of the distribution of normals in regard to q. When the plane under consideration is diatropie the intensity distribution in regard to 5 gives the distribution of the crystallite axes. Generally a crystallite axis makes an angle of ?, with the normal. B y crystallite axis we mean the direction in the crystallite which during orientation is set parallel to the elongation axis (the fibre axis). I n most polymeric fibres the crystallite axis coincides with the direction of the molecular chains. Since as a rule the distribution of axes is not coincident with the distribution of normals, the expression (5) cannot be used to find the axes distribution: we must find the relationship between the function L (q) and the density of the axes Q (q). The solution of this problem gives the following relationship [3]: ~/2
0
fl and W are connected b y the relationship sin fl=eos ~.eos ?,--sin ~.sin ?,.cos N,
(7)
where fl is an angle supplementary to the angle between a cryst~llite a d s and the fibres axis fi----~/2--9', (the designation ~', is necessary as ~' #{a).
I908
A. S m GoIx~S.A~r
Thus
dI
2A f Q(fl)d~.
(8)
o
On examining equation (8) we m a y arrive at a rather unexpected conclusion. Since for different subintegral functions the value of a particular integral m a y be the same, different functions of the axes distribution m a y correspond to one function of the azimuthal distribution of intensity F (6). Thus Q (fl) cannot be unambiguously determined from the data for the azimuthal distribution of interference intensity for hkl in the X-ray diagram; t h i s w a s first pointed out by :Hermans [4]. The only possible method is to t r y different functions of Q (fl), and to compare the experimental and theoretical distribution curves. Several authors use gaussian distribution [5, 6]. Thus in [5] a highly-oriented case (where sin q ' = ~ ' ) is investigated. Obviously the point of interest is the general case where the condition sin q' = q ' is not fulfilled, e.g. in investigation of fibres with the multiplicity of elongation v; varying from v = l (isotropic fibre) to Vmax. In this case it is desirable t h a t the distribution function Q (fl) should be one in which v figures directly. Gaussian distribution is "inconvenient" in this case, as it is not based on a concrete model of the deformation process during stretching, and contains the value of v in the form of a "hidden" parameter. I n studying the deformation of swollen cellulose films K r a t k y [7] assumed a distribution of the axes of particles ("miceHes") based on the following model: the particles are suspended in a medium which is considered to be completely isotropic, and the forces of cohesion of the particles and m e d i u m are large in comparison with their weight and inertia, so t h a t under deformation the medium does not flow round the particles, which are not deformed under elongation b u t merely turn; for the particles the system is dilute, steric hindrances do not impede the turning of the particles. These ideas can be transferred to the process of deformation (elongation) of synthetic fibres of low crystallinity t h a t have crystalline regions of small dimensions [8]. The K r a t k y density distribution of axes takes the form:
Q (fl)= Qo v3/[1+ (v 3 - 1 ) cos ~ilia/2,
(9)
where Q0 is the axes density when v = 1 (the isotropic case); fl is the same as in formula (7). Taking (9) into account, formula (8) m a y be written as .Y' ( ~ ) = ~ - f
Q°vSd~
[1 -t- (v8 - 1 ) cos z ilia/2 •
(10)
o
Considering only the paratropic interferences of hko (or hol in a monoclinic
X - r a y s t u d y of orientation distribution of crystallites
system) when
r=n/2,
1909
then according to (7) cos s fl-= 1--sin s q cos s ~ .
(11)
Putting (11) in (10) and transposing, we get ~12
d~/
2AQov-s/z f
;
v3 - 1
oJ (1-
sinsq"c o s 2 ~)312"
(12)
Interpolating b2= v a - 1 sin 2 va
then ~/2
(1 - - k 2 cos 2 ~)3/2" o
Substituting ~ = - - - e 2
we obtain ~/2
2AQ v -3/2 f & F@= - j (1--k2 sin2 t) s/2"
(13)
o
The integral of (13) is in the table (9) and is reduced to the form
_P(~)=~( 1-k2) E kl where E
1~
,
(14)
is a complete elliptical integral of the second type; k : s i n q.
• ~/(va - 1)/v3. Remembering the relationship of ~ and J, it is easy to calculate F(J)
if we know the value of v. When v = l (the isotropie ease) k=O, E(O,, 2)
x 2
and F ( J ) = A o Q0=const. Assuming that 2AQo/x=C, let us consider F (J) = F (J)/C. Figure I shows curves calculated from equation (14) for different elongation multiplicity factors. We must also consider the integral intensity ~/2
o
to find its relationship to v.
1910
A. S m GOrKH~A~
The integral intensity I m a y be written as follows: - s/21
=/2 ~/2
v
v s-
1
X'/ 1
vs
(
V3__I
l-~2A-~Q°f f 0 0
1
sin S ~.sin S
va
d~aa
)
sin S
(15)
or, according to (14) i 2 v-S/2 E ( k 1 2 ) s i n q~d ¢ \ (1 --k ~) x/cos z 0--cos 2
I'=
(16)
0
The values of I or I', m a y be calculated from (16) using the Gauss]an method of calculating q u a d r a t u r e s - a n d taking the number of terms in the Ganssian formula as n----3, we found the following values of I ' for different values of v (when 0 = 10°10'): v 1.0 2.0 2.5 3.0 8.5 4.0 I',qon9entional units 2.3386 1.9808 1.9250 1.8593 1.7660 1.6431 These data show t h a t the integral intensity is not independent of v, b u t decreases with increase in the elongation multiplicity factor. Thus the areas beneath the curves in Fig. 1 are not equal, and in cases where the azimuthal 40
8-
3'5
-
~
-
30
x
2.5 4-
20
I
90
I
70
"
I
I
I
L
t
50 30 Polo/" anq/e, ~o
I.
lO
FIG. 1. Density distribution in respect of angle ~ ealeulaf~d from equation (14). Figures
by curves are values of v. distribution of intensity is described b y formula (14) it is not possible to normalize the experimental curves b y equalizing the squares. EXPERIMENTAL
Polycaproamide fibre m a y be used to verify K r a t k y distribution as it has low crystallinity [10] and small crystallite dimensions [11]. These are the proper conditions for the K r a t k y model.
X-ray study of orientation distribution of crystallites
1911
Unoriented fibre ( v = l ) was obtained under normal production conditions. I t is not completely isotropic owing to spinneret stretching, and does not contain a n y appreciable a m o u n t of the monoelinic ~-form, or rather it contains only the nuclei of this crystalline modification, statistically oriented, so t h a t it m a y be regarded as isotropic for these crystallites. The stretching was accomplished on a special testing structure with a heating element, the element temperatures being 20, 100, 180 and 200 °, and the rate 70 m/rain on a dynamometer at 20 °, at a rate of 11 ram/rain. The elongation multiplicity factor v was found from the ratio of the rates of the take-in and feed devices on the testing structure and the d y n a m o m e t e r scale of elongation. The azimuthal curves of intensity distribution for the interference (200) were recorded on an ~tS-50II~I diffractometer (Ni-filtered CuK~ radiation) using a device that enabled the sampie to be slowly turned in a plane perpendicular to the beam (a complete revolution takes 72 rain) with a fixed counter (20 =20°20'). The intake slit was the widest possible in order to take in the whole width of interference. Because of the marked widening of the lines there is some overlapping of adjacent interferences (200) and (002) and (202). After restoring the profile of the line b y means of Gaussian distribution it can be shown t h a t the optimum slit width is 2 ram. The equatorial reflex (002) (202) is the superimposition of two interferences so t h a t it m a y not be used to verify distribution (14). S
S2 ~I
gO
I
f
x
70
50
30
" I
t~
~ °2
I0
Azirnuthal ang/e ,
FxG. 2. Intensity components in the azimuthal curve of distribution. For comparison with the experimental values q must be converted ¢o 6 in (14) according to the relationship cos ~ = c o s 0 cos 6. For a definite value of 0 a certain region close to q = 0 is forbidden, namely the region of 0>@~0, as in this region cos 6 > 1 , which is impossible. It is extremely difficult to draw a background line below the azimuthal distribution curve for X-ray diagrams of polycaproamide fibres as analysis of the area below the diffractometric curve
1912
A. SH. GOIKH~AI~
into crystalline and amorphous components cannot be reliably accomplished when we do not know the shape of the curve of scattering b y the amorphous component. Because of this the following method was used. Figure 2 shows that for a certain value of 6 (for instance when 6 = 9 0 °) the overall intensity
i
c
90
90
70
70
50
50
30
30
10 90
\.
o.3
I
I
I
[
70
50
30
10
gO 70 50 30gO' Azimuthal angle, o~
70
50
30
Fie. 3. Calculated curves (unbroken lines) of azimuthal distribution in respect ot7 anglo
(0= 10°10') for different values of v and experimental results (points) for the interference (200) of polycaproamide. Values of v: a--2.0; b--2-5; c--3.0; d--3-5; e--4.0. Stretching on a bench, and on a dynamometer at temperatures of (°C): 1--20; 2--100; 3--180; 4--200.
F=Fcr-~Fbk where Fbk=Famorph~Fcompton-t-FaL r. Drawing a line 01S1 from a point in the curve 01 corresponding to 6=01, we m a y write
(0) where F~ (0) is a certain unknown part of the intensity of For ; J1 is any selected value of 6. As it is possible to use equation (14) to find the theoretical values of F'cr (61) and For (0) for definite values of v and 0, the value of F~ (0) is easily found from
For
(o).
Combining in this w a y two points in the experimental and theoretical curves we now proceed to plot the whole curve, normalizing the experimental values of F (61) b y dividing them b y the coefficient of K = F ~ (O)/F~r (0)*. Figure 3 shows the results of comparing the experimental normalized values with the theoretical relation (14) for different elongations and different temperatures and rates of stretching. * When doing so, it is assumed the amorphous components to be tmorien~ed.
Thermodynamic functions of methacrylic acid
1913
CONCLUSIONS
(l) T h e a z i m u t h a l distribution of intensity o f p a r a t r o p i c interferences as a function o f t h e degree o f elongation has been calculated using K r a t k y ' s distribution of crystallite axes. (2) I t has been shown t h a t t h e integral reflex i n t e n s i t y calculated from' t h e a x i m u t h a l i n t e n s i t y distribution, decreases with increase in elongation multiplicity. (3) The calculated i n t e n s i t y distributions agree with the e x p e r i m e n t a l results for the interference (200) of p o l y c a p r o a m i d e . Translated by R. J. A. HZN'DRY
REFERENCES
1. L. G. KAZARYAN and D. Ya. TSVANKIN, Vysokomol. soyed. 5: 1963, 976 (Translated in Polymer Sci. U.S.S.R. 5: 1, 29, 1964) 2. B. K. VAUNSTEIN, Difrakteiya rent genovskikh luchei net tsepnykh molekulakh (Diffractiort of X-rays by Chain Molecules). Izd. Akad. Nauk SSSR, 1964 3. O. KRATKY, G. POROD and E. TRIEBER, Kolloid.Z. 121: 1, 1959 4. $. ,I. HERMANS, P. H. HERMANS, D. VERMAAS and A. WEIDINGER, Recuell. tray. chim. Pays.-Bas. 65: 427, 1946 5. A. I. SLUTSKER and A. I. GROMOV, Physics of solids 5: 2185, 1963 6. H. KR,I.SSIG and W. KITCHEN, J, Polymer Sci. 51: 123, 1961 7. O. KRATKY, Kolloid-Z. 64: 213, 1933 8. J. CERNY, Textil, No. 4, 141, 1961 9. I. S. G R A D S H T E I N and I. M. RYZIfl~, Tablitsy intcgralov, summ, ryadov i proizvedennii (Tables of IategraLq Sums, Series and Products). 170, Fizmatgiz, 1962 10. I. L N O V A K and V. I. V E T T E G R E N ' , Vysokomol. soyed. 7: I027, 1965 (Translated in Polymer Sci. U.S.S.R. 7: 6, 1027, 1965) I. G. U R B A N C Y K , J. Polymer Sci. 59:215 1962
THE HEAT CAPACITY AND THERMODYNAMIC FUNCTIONS OF METHACRYLIC, POLYMETHACRYLIC AND ISOBUTYRIC ACIDS*t I. ]3. RA~ZNOVlCH,
B. V. L E B E D E V
and T. I. M E L E N T ' E V A
Chemical Research Institute of the N. I. Lobachevskii University, Gorki (Received 9 June 1966)
MEASUREMENT of the h e a t c a p a c i t y o f p o l y m e r s in t h e l o w - t e m p e r a t u r e region has a l r e a d y been r e p o r t e d on in several p a p e r s (for e x a m p l e [1-3]) as a m e a n s of calculating t h e m a i n t h e r m o d y n a m i c characteristics of these systems. The same * Vysokomol. soyed. All: No. 8, 1699-1705, 1967. 3rd Report in the series "Thermodynamics of monomers and polymers of the vinyl ~rios".