Temperature dependence of expansion of crystalline lattices of some flexible chain polymers

Polymer Science U.S.S.R. Vol. 2~, 1~o. 8, pp. 1837-1844, 1982 Printed in Poland

TEMPERATURE LATTICES

DEPENDENCE OF

SOME

0052-3950/82 $7.50+.00 ~ 1983 PergamonPress Ltd.

OF EXPANSION

FLEXIBLE

CHAIN

OF CRYSTALLINE POLYMERS*

G. DADOBAYEV a n d A. I. SLUTSKER Ioffe Physico-Technical Institute, U.S.S.R. Academy of Sciences

(Received 30 January 1981) The temperature shifts of the equatorial and meridional X-ray reflexions of PE, polycaproamide and PVA have been measured in the region 5-400 K. Gradual excitation of the torsional and bending vibrations has been established leading to transverse expansion and longitudinal compression of the crystallite lattice. The characteristic temperatures of the vibrations have been evaluated. The expansion coefficients are compared for the polymers considered with different intermoleeular interaction. STUDY b y t h e m e t h o d of large-angle X - r a y diffraction of crystalline p o l y m e r s over a wide t e m p e r a t u r e r a n g e gives i n f o r m a t i o n on t h e d y n a m i c s of the straightened portions of the chain molecules. Such i n f o r m a t i o n is necessary, in particular, in dealing w i t h the problems of the t h e r m o - f l u c t u a t i o n m e c h a n i s m of r u p t m ' e o f stressed (and hence more or less straighte.md) linear macromolecules, w h i c h nnderlies t h e process o f disruption of m a n y oriented polymers. T h e present w o r k looks at the d a t a given b y X - r a y diffraction on t h e r m a ! e x p a n s i o n in t h e crystallites of three linear polymers: P E , p o l y c a p r o a m i d e (PCA) a n d P V A . I n these p o l y m e r s w i t h a similar s t r u c t u r e of the c a r b o n skeleton o f t h e molecules the side g r o u p s differ considerably apd, as a result, the interm o l e c u l a r interactions. The investigations were carried out over the t e m p e r a t u r e r a n g e f r o m liquid helium to close to the melting points of the polymers. The conditions of X-ray diffraction measurements in the temperature region from 5 to ~400 K arc described in [1]. To determine the temperature changes of the interplane distances d we used change in the position of the reflexions characterized by the angle ~m, corresponding to the centre of the intensity of the reflexion. The relative expansion of the lattice e----Ad/do (do is a constant of the lattice for T-~0) was estimated on the basis of the Wolf-Bragg equation from the expression

e(T)-.~-~ cot ~m(T) A~m(T),

(1)

2

where A~m(T): ~m(T)--~m(0); ~m(T) and era(0) are the positions of the reflexions at the eurremt and zero temperature, respectively. The test objects were oriented PA, PGA and PVA films, 100-200 ~m thick. The dispersion of the texture over the c-axes amounted to several degrees; the dispersion was greater over the other axes. The linear dimensions of the crystallites were those usual for oriented polymers and were within the range ~ 10--20 nm (evaluation from the diffraction width of the reflexion for two orders of reflexion with use of CuK a and MoK a radiation). * Vysokomol. soyed. A24: No. 8, 1616-1622, 1982. 1837

1838

G. DADOBAYEV and A. I. SLUTSKER

For PE we also studied samples obtained by annealing under high pressure of initially oriented polymer [2]. * The dimensions of the crystallitcs in them were eonsiderably larger and reached ~ 100 nm. In the crystalline lattice of the polymers studied the skeletons of the molecules have the form of a straight planar ~rans-zigzag (Fig. 1). The relative position of the molecules in the transverse plane is shown in Fig. 1 [3]. The different relative position of the projection of the planes of the trans-zigzags for the different polymers may be seen. In addition, a difference exists in the mutual binding of the molecules. While in PE the molecules are joined by only weak van dcr Waals interaction, for PCA stronger hydrogen bonds pass along the zigzag planes and the whole PVA lattice is densely "pierced" with hydrogen bonds. I n the work we measured the contours of the e q u a t o r i a l and meridional reflexions. The equatorial reflexions correspond to the reflexion from the planes passing paralle] to the axes of the molecules. The interplane distances correspond to the distances between t h e axes of t h e a d j a c e n t molecules in the t r a n s v e r s e direction. l~igure 1 shows the corresponding planes a n d gives the interplane distances: for P E t h e reflexion 110, for P C A the reflexion 002 a n d for P V A the reflexion 101.

a

b

f

]

L

A

L

._$ z _: c

A

I3.7 T

d C

,2J-I.27A I

C

FzQ. 1. Schemes of the elementary cells of crystalline PE (a), PCA (b) and PVA (c) lattices. a, b, c--Projection of the fiat skeletal tran,s-zigzags on plane perpendicular to the axes of the molecules (values of interplane distances arc indicated for room temperature); d--flat trans-zigzag of the skeleton of the molecules. * These samples were prepared in the Kavpov Physical Chemistry Research Institute. The authors are deeply grateful to Yu. A. Zubov for kindly making them available.

Expansion of lattices of some flexible chain polymers

1839'

The meridional reflex:.ons from the planes passing perpendicular to the a x e s of the molecules through the adjacent (along the axes) carbon atoms (Fig. 1): for P E and PVS the reflex~on 002 and for PCA the refl(,xion 0140. As an example FJg. 2 gives the equatorial and melidional reflexioi;s at two temperatures: 293 (room) and ~ 100 K. It wi!] bo seen th :t temperature influences both the angular position of the centre of the refl;.xions ~0m and their inte~.)sity. The problem of change in the late :sity (assvciate:l with the temperature ,hange in the amplitude of the vib"atio; s of the m'>lecules) w l ! be e:>~!sidereg is nat~lrally conspicuous for a large diffraction angle ( + m - 75°). For the equatorial reitexions with their angles (+m ~-20-25 °) splitting is practically not observed. a

b

d

C

i --~ ZI °

7~°501

Zl°JDI.,

75 °

Z2 °

75°J0 '

Zy °

74°50 r

,Zf°30 z 2Z °

75"

75"J0 t

/ 8 ° 19 ° 2 0 ° 2y °

75°75"30t76 °

~ - ~" ZJ °

~ "~r 2~ °

"-

25 ~

714° 75" 77"

Fie. 2. X-ray reflexions ofPE (a, b, e,]), PVA (c, g) and PCA (d, h) at 293 (1) and 100 (2) Kr a--macro; b--microcrystalline PE sample. Equatorial reflexions 110 (a, b), 101 (c) and 002 (d); meridional reflexions 002 (e-g) and 0140 (h). As these conditions are absent for micro-crystalline PE, PCA and PVA, the separation of the components of X-radiation for these is not observed. As Fig. 2 shows, all the equatorial reflexions with rise in temperature shift to lower angles and all the meridional to larger ones. The absolute A ~m and especially the relative zt~0m[~0mchanges in the positions of the equatorial refloxions are much stronger than the moridiona]. Measurements similar to those presented in Fig. 2 wore made over the whole temperature range (from 5 to 3 5 0 4 0 0 K) with steps from 5 to 50 K.

1840

G. DADOBAYEVand A. I. SLU'rS~r~.R

Figm'e 3a, b presents t h e d a t a on t h e t e m p e r a t u r e shift of t h e reflexions convorte(1 from expression (1) to the values of t h e r e l a t i v e d e f o r m a t i o n in t h e transverse a n d longi'tudina] directions. I t will be seen t h a t the d e f o r m a t i o n s of the lattice in the transverse a n d longitudinal directions have different signs. I n the t r a n s v e r s e direction t h e t e m p e r a t u r e d e f o r m a t i o n at 350-400 K reached high

'01 /

o

6O

3

e'I°'3

a

3O

b

l 6

oc , 10", 6O -

dej -I C

-#0

-6 -

I/Z

/

--4

-

20 -

1

-

d

2

-

-2 -

1

3 200

40O

3 20O

qO0 T,K

:FIG. 3. Relative change in the interplane distances (a, b) and temperature dependence of the linear expansion coefficient (c, d). PE (1}, PCA (2) and PVA (3); a, c--transverse expansion; b, d--longitudinal compression.

Expansion of lattices of some itoxible chain polymers

1841

v~.iues: from 2 in PVS to 10% in PCA. The longitudinal compression amounts to fractions of a percent. Thus, sharp anisotroI~y of thermal expansion in sign and value characteristic ~)f the crystals of chain structures exists [4-6]. Figure 3a, b shows that the temperature dependence as(T) is non-linear--the t~,mperature dependence becomes steeper. Differentiation of these temperature functions gives the linear thermal ~xt)ans'_on coefficients a(T)-~ds/dt (F.g. 3c, d) where the non-linearity noted is <~xl)ressed in inconstancy of the thermal expansion coefficients. The value a in different temperature ranges behaves differently. The region of fast rise (up to -~ 100 K) gives way to the region of slow r:se ( ,~ 100-200 K) and then the rate , f rise in a again increases. Since the functions a(T) are obtained by differentiating the functions a(T) plotted from the experimental points with their inescapable scatter, we compared the function a(T) with the temperature fanction of thermal ea,pacity for constant volume Or(T) for crystalline P E [4] (Fig. 4). It m a y be seen t h a t the function Cv(T) has the same characteristic inflexions and in much the same temperature region as for a(T). Since approximate proi~ortionality exists between a(T) and Cr(T) [7] then the form of the function Cv(T) m a y serve as confirmation of the functions a(T). Let us consider the general course of the functions a(T). We shall at first ~h-al with the functions g(T) in the transverse direction (Fig. 3c). Close to T = 0 the accuracy of measurement of c(T) is not high enough to plot tile function ~(T) here with confidence. Evidently in the region from zero to several degrees Kelvin this flmction must have the form ~ T a [4, 6]. Then (Fig. 3c) close ~o linear rise in a(T) to T~, 100 K occurs. According to the theoretical calculations [4] in chain molecules with the form o f a fiat zigzag the first to be excited are the torsional vibrations (rotatory vibrations of the zigzag plane about the axis of the molecule). The characteristic temperature of these vibrations is put at from 100 to 200 K [4]. It m a y be assumed t h a t the near linear rise in a(T) in the temperature region from ~ 10 to ,~ 100 K is connected with the gradual "unfreezing" of the torsional vibrations, i.e. with manifestation of the quantization of the vibratory states of the lattice. Since in all the three polymers the carbon skeleton is almost identical, the characteristic (Debaye) temperatures of the torsiona] vibrations must be practically identical. From the graphs of a(T) one m a y approximately evaluate this temperature as equal to ~150 K. Thus, in the region ~150 K in the polymer J~lttice torsional vibrations of the molecules are gradually excited. Then from -~200 to 350400 K there is further rise in ~ due to two factors: the influence of anharmonism of high orders for the already excited torsional vibrations and the gradual excitation of the transverse bending vibrations of the molecules. The characteristic temperature of these vibrations is 450-500 K [4]. The conclusions drawn on the gradual unfreezing of the torsional and bending vibrations are confirmed by evaluation of the "classical" expansion coefficients

1842

G. DADOBAYEV and A. I. SLUTSKER

(i.e. at temperatures above the characteristic). According to [8] there exist~ the approximate relation

a=k/DaE, where k is the Boltzman constant; E is the modulus of elasticity; D the linear dimensions of the vibrating "particle". From [9] the transverse elasticity moduli of the lattices amount for P E ~_4×10 a 1YIPa, P V A _ 9 × 1 0 a. Taking D z ( 3 - 4 ) × l O -1° m we obtain for P E ~ 1 0 × 10 -5 and ibr PVS ~ 5 × 10 -5. Figure 3c shows that the ~ values in the region T ~ 2 0 0 K for the given polymers approach those evaluated. Let us turn to the data on the temperature dependence of the longitudinal contraction of the lattice. The longitudinal shortwave vibrations in the carbon skeleton are the most rigid. Their characteristic temperature is put at 1500-2000 K [4]. At room and lower temperatures such vibrations are practically not excited. I n fact, longitudinal expansion of the molecules is not observed. Moreover, as Fig. 3b shows, there is longitudinal compression of the molecules. The cause of the observed longitudinal temperature contraction, as noted in theoretical [6] and experimental [4, 5] studies, lies in the above considered vibrations of the transverse type: torsional and bending. Such vibrations for virtual unstretchability of the molecules lead to reduction in the projection of the length of the C--C bonds onto the direction of the axis of the molecule. The uniformity of the form of the functions 8(T) and a(T} in the transverse and longitudinal directions (Fig. 3) confirms t h a t in the longitudinal direction we see not the independent longitudinal vibrations but the result of the transverse vibrations. Let us now look at the 8 and ~ values for different polymers. From Fig. 3 it is clear that with the uniformity of form of the temperature functions e(T) and g(T) considerable difference is observed in the value of thermal expansion (4-5 times on passing from PVA to PCA), P E in this respect occupies an intermediate position. With what factor is such a difference connected? With the difference in the amplitude of the vibrations or the index of anharmonism (the anharmonism of the vibration is the cause of thermal expansion in general)? With a two-term representation of the elasto-force interaction F(Ad)=fAd--gAd 2 [8] the mean thermal expansion is

Ad

g A dS

d

fd

where f and g are the harmonic and anharmonic coefficients characterizing the non-linear elasticity of the intermolecular bonds; Ad, is the mean square of the deviation of the "particle" from the position of equilibrium. Thus, the differences in the values 8 m a y be due both to the factor g/f and Ad 2. l~Ieasurement of only the transverse expansion for different polymers would

Expansion of lattices of some flexible chain polymers

1843

not allow one to isolate the dominant factor. However data on longitudinal contractions makes this possible. In fact, as noted, the longitudinal contraction of the molecules is a direct consequence of the vibrations of the transverse t y p e a,.ld does not depend on the anharmonism of the interaction cf the molecules sb we it is determined by the degree of coiling or bending of the molecules. Figure 3 shf)ws that the longitudinal contraction for various polymers differs and changes in value in much the same w a y as transverse expansion. This suggests that in different polymers the amplitudes of the transverse vibrations of the molecules (lifter; this factor plays a dominant role in determination of transverse expansion of the !attices. Variation in the degree of anharmonicity of the intermolecular iuteraction if it occurs has a much weaker influeuce. Cg, J/mole.K 40-

20

0

200

7;,K

qO0

Fro. 4. Temper~tm'e dependence of the specific thermal capacity of crystalline PE [4]. The difference in the amplitudes of the transverse vibrations well agrees with the special nature of the relative position and interaction of the molecules in the lattice. Thus, from the scheme in Fig. 1 it will be seen that the PCA molecules are most free: the planes of the zigzags are parallel to one another while the hydrogen bonds follow along the length of the molecule with division in seven units [3]. In such a quasi-layered structure the strongest interlayer expansion is also observed. For P E thougb having no hydrogen bonds, the planes of the skeleton of the molecules are arranged cross-like (Fig. 1). Evidently increase in the intorplane distance is most actively influenced b y those molecules the zigzag planes of which arc close to the plane 110. This is half of all the molecules and therefore the expansion of the P E lattice is ~ 2 times less than that of the PCA lattice (Fig. 3a, c). The difference observed in expansion for large and small P E crystallites is evidently connected with the enhanced influence of the amorphous regions ;surrounding the crystallites (for small crystallites).

1844

G. DADOBAYEV and A. I. SLlrrS~ma

In the P V A lattice although all the zigzag planes are also parallel to one another (Fig. 1) the mesh of hydrogen bonds is so dense (on each oxygen atom)~ that the amplitudes of the vibrations of the transverse t y p e (both tersJonal and bending) are comparatively low. We would again note that the values of the transverse expansion (for examplc~ the expansion coefficients at room temperature) are in sufficient measure proportional to the reciprocal moduli of the transverse elasticity of the polymer ClTSt~lilites (since the function ¢z~-I~/DSE is justified, see above), which is in good agreement with the influence discussed of the value and "density" of the intermoIecular interaction on the value of thermal expansion. The present work provides experimental proof of the dominant role of thevibrations of the transverse t y p e (torsional and bending) in the dynamics of th(~ straightened molecules of a number of polymers (PE, PCA, PVA) for the temperature region 0-400 K, observation of gradual excitation of these vibrations and evaluation of their characteristic temperatures. The presence of transverse vibrations in virtual absence of longitudinal in the region 0-400 K raises the question of the further specification of t h e fluctuation rupture of stressed (and thereby straightened) polymer molecules which until now has been c~ns'de:ed on the basis s~lely of longitudinal vibrations [10]. Since the results of the present work show that in the temperature rcgio~L 0 4 0 0 K the dynamics of the polymer molecules is largely non-classical, the problem of estimating the quantum effects in constructing the fluctuation theory of the~ disruption of chain molecules raised in [I1] acquires real meaning. REFERENCES

1. G. DADOBAYEV and A. I. SLUTSI~ER, Vysokomol. soyed. A24: 30, 1982 (TranslatodJ in Polymer Sci. U.S.S.R. A24: 1, 33, 1982) 2. Yu. A. ZUBOV, V. L SELIKHOV, M. B. KONSTANTINOPOL'SKAYA,A. P. KOROBKO. and G. P. BELOV~ 1bid. B12: 570, 1970 (Not translated in Polymor Sci. U.S.S,R.) 3. F. H. JELL, Polimernye monokristally (Polymer Monocrystals). p. 552, Khimiya, Leningrad, 1968 4. B. WUNI}ERLICH, and G. BAUER, Teploomkost' lineinylda polimerov (Thermal C~pac-ity of Linear Polymers). p. 238, Mir, Moscow, 1972 5. S. KAVESH and J. M. SHULTZ, J. Polymer Sci. AS, 8: 243, 1970 6. I. M. LIFSHITS, Zh. eksper, i teor. fiz. 22: 475, 1952 7. L. D. LANDAU and Yo. M. LIFSHITS Statisticheskaya fizika (Statistical Physics), Part 1, p. 584, Nauka, Moscow, 1976 8. Ya. I. FRENKEL', Vvedeniyo v tooriyu metallov (Introduction to the Theory of Met,ale). p. 291, GITTL, Leningrad, Moscow, 1948 9. I. SAKURADA, T. ITO and K. NAKAMAYE, Khiln. i tekhnol, polimerov, 10, 1.9~ 1964: 10. V. R. REGEL', A. I. SLUTSKER and E. Yo. TOMASHEVSKII, Kineticheskay~ priroda. prochnosti tverdykh tel (Kinetic Nature of the Strength of Solids). p. 560, Nauka, Moscow, 1974 ll. R. L. SALGANIK, Fiz. tverd, tela, 5, 1336, 1970