Polymer Characterization - American Chemical Society

41 Recent Advances in the Use of Scattering for the Study of Solid Polymers Downloaded via UNIV OF MINNESOTA on July 10, 2018 at 04:24:28 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

R. S. STEIN and G. P. HADZIIOANNOU

1

University of Massachusetts, Polymer Research Institute and Materials Research Laboratory, Amherst, MA 01003 Polymers in the solid state may be characterized by scattering of light, x-rays, or neutrons. These three techniques may give information about the size, the shape, and the distribution in the space of the domains (or structural elements). Light scattering is useful for studying larger species having dimensions comparable with the wavelength of the radiation used. Rapid data acquisition techniques were developed that permit following the time dependence of scattering accompanying phase transformation or sample deformation.

General Principles

of Scattering

Origin of Scattering. W h e n a b e a m of radiation impinges on a sample, a portion of the radiation is scattered. T h i s scattered fraction depends on both the nature of the radiation and the composition of the sample. T h e scattering of v i s i b l e light is related to the fluctuation of the refractive index or p o l a r i z a b i l i t y of the sample and its anisotropy. T h i s polarizability is related to the m o b i l i t y of the electrons, w h i c h is affected by the molecular structure of the sample. T h e scattering of x-rays, however, is related only to the fluctuation of the electron d e n sity and is not affected by the electron strength of b i n d i n g to the n u c l e i . N e u t r o n scattering power, on the other hand, is a nuclear property, that varies with the composition of the nucleus and is independent of the c h e m i c a l nature of the species w i t h i n w h i c h these n u c l e i reside. T h u s , it may be affected by isotopic substitution, w h i c h proves to be a useful l a b e l i n g technique as w i l l be discussed. 1

Current address: I B M Research Laboratories, San Jose, C A 95193.

0065-2393/83/0203-0721$10.00/0 © 1983 American Chemical Society

Craver; Polymer Characterization Advances in Chemistry; American Chemical Society: Washington, DC, 1983.

722

P O L Y M E R

Scattering T h e o r y .

C H A R A C T E R I Z A T I O N

A M P L I T U D E OF T H E SCATTERING POWER.

In

a l l cases, the amplitude of the scattering may be described by an equation of the type:

E«(q) = ^ p

5

exp

[t(q · τ,)]

(1)

3

where p is the scattering p o w e r of the j scattering element of the system located at position η, q is the scattering vector of the system defined by th

5

q

= Q i ~ Qo

(2)

where qx = (2π/λ)§ι and q = (27r/X)s where s and Si are u n i t vectors along the i n c i d e n t and scattered rays, and λ is the wavelength of r a d i ation w i t h i n the scattering m e d i u m (Figure 1). T h e magnitude of q is 0

0

0

q = |q| = (4ττ/λ) s i n (β/2)

(3)

where θ is the scattering angle b e t w e e n the i n c i d e n t and scattered rays measured w i t h i n the m e d i u m . (The parameter q often is desig nated elsewhere by Q, h, K, or μ. F o r x-ray scattering, the angle normally is designated b y the Bragg angle, Θ , e q u a l to h a l f of 0.) F o r light (J), x-ray (2), and neutron (3) scattering, p is propor tional to the p o l a r i z a b i l i t y (a ), the electron density (pj), and to the neutron scattering length (bj) at position j . F o r light scattering, the light interacts w i t h the valence electrons, so the scattering p o w e r may be dependent on the molecular orienta tion. I n such cases, the p w i l l be a tensor quantity g i v e n by: Β

5

5

ô

P i

= K (Mj · O) L

(4)

where K is a constant, M , is the i n d u c e d dipole at position j g i v e n by L

Μ, = |«,|Ε

(5)

where a is the polarizability tensor at position^, and Ε is the electrical f i e l d of the i n c i d e n t l i g h t wave w i t h i n the m e d i u m acting o n the scattering element. F o r uniaxially symmetrical scattering elements w i t h p r i n c i p a l polarizabilities α and CL%, M J proves to be 5

υ

Mj = δ,· (Ε · a ) α 3

ά

+ a%E


(6)

41.

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A N D

H A D Z I I O A N N O U

Solid Scattering Studies

723

Ζ

Y

χ

Figure 1. Geometrical configuration of a scattering experiment. where δ, = a — a a n d is the anisotropy at position a n d a$ is a unit vector along the principal polarizability axis at position j . T h e vector Ο is a unit vector along the polarization direction of an analyzer i n the scattered light beam. A consequence of anisotropy is that the light scattering from an anisotropic system is dependent on the polarization of the i n c i d e n t and scattered rays. F o r neutron scattering, the neutrons interact w i t h the matter through nuclear or magnetic interaction (3, 70—72). T h e neutron scattering cross-section, w h i c h is the ratio b e t w e e n the scattered n e u tron flux to the i n c i d e n t one, is g i v e n b y the relation σ = Airb a n d contains two components, one coherent (a ) and one incoherent (o"inc) cross-section. O n l y part of the scattered neutrons take part i n the interference process, w h i c h is described b y the coherent scattering cross-section (a = 4nb ). T h e incoherent cross-section is related to the neutron or nuclear s p i n . T h i s incoherent cross-section does not exist for zero-spin nucleus, such as C or 0 but is very important for hydrogen (101). tj

2j

2

coh

coh

2

coh

1 2

1 6

S C A T T E R I N G INTENSITY IN A D I S C R E T E A P P R O A C H .

The

i n t e n s i t y of

scattering, energy per u n i t area per unit t i m e , is obtained from I.(q) = K [ E ( q ) · E,*(q)] s

(7)

where the constant Κ depends on the nature of the radiation, a n d E *(q) is the complex conjugate o f E ( q ) . T h e calculation of the scattering from any system involves the summation of E q u a t i o n 1 over a l l scattering elements. T h i s step re quires a knowledge of p for s

s

5


724

P O L Y M E R


= / p ( r ) e x p [ i ( q · r)]dh

EM

(8)

T h e integral is three-dimensional over a l l orientations and lengths of the vector r. F o r spherically symmetrical systems where p(r) = p(r), the integral becomes one-dimensional as 00

/

p(r)

E,(q) =4π(

£ î 5

\

-^rVr

(9)

and depends only on the variation of p(r) w i t h r. T h i s expression may be evaluated readily for particular m o d e l systems, for example, for isolated spheres of radius R and scattering p o w e r p i m b e d d e d i n a m e d i u m of scattering p o w e r p , l e a d i n g to (2, 5) s

s

0

E (q) s

= V (p s

s

- p ) (3/L7 ) (sin U - U cos U)

(10)

3

0

where V = (4J3)nR is the v o l u m e of the sphere and U = qR . T h i s treatment results i n a scattering intensity that oscillates w i t h U (and as a consequence w i t h angle Θ) but generally decreases w i t h θ at a rate that depends on (R /X). T h e scattering falls off more rapidly w i t h large (R /X). [This result i m p l i e s the R a y l e i g h - G a n s - D e b y e approximation, that is, that the direction of the i n c i d e n t f i e l d Ε is u n m o d i f i e d i n crossing the boundary of the scattering particle. T h i s approximation is good for x-rays and neutrons and applies for light p r o v i d e d that too great a polarizability difference does not exist b e t w e e n the particle and its surroundings and t h a t R / X is not too large. M o r e exact theories have b e e n advanced by M i e and others (1, 5).] E q u i v a l e n t expressions have b e e n found for other shape particles ( 6 - 9 ) . I n general, this observation of the rate of fall off of scattered intensity w i t h scattering angle provides a measure of the particle size. F o r isotropic spherically symmetrical particles, the scattered i n tensity is circularly symmetrical around the incident beam and is i n dependent of the azimuthal angle μ (Figure 1). T h i s observation also w i l l be true for anisotropically shaped particles such as rods that are randomly oriented. H o w e v e r , for oriented anisotropically shaped par ticles, the variation of the intensity w i t h θ depends on μ. T h i s relation ship provides a means for the measure of the dimensions of particles i n different directions. F o r polarized light scattering from isotropic particles, the scat tered light is polarized i n the same direction as the i n c i d e n t light. T h u s , intensity is observed for V polarization (vertically p o l a r i z e d i n c i d e n t and scattered light where vertical is the direction p e r p e n 3

s

s

s

s

s

s

v


41.

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A N D H A D Z I I O A N N O U

725


dicular to the plane i n w h i c h θ is measured), but intensity is zero for H polarization (scattered l i g h t v i e w e d through a horizontal analyzer). v

SCATTERING FROM SPHERULITIC-TYPE MORPHOLOGY.

Crystalline poly

mers often e x h i b i t spherulitic morphology (10). T h e s e spherulite s are spherically symmetrical aggregates of crystalline a n d amorphous regions (Figure 2). T h e y are anisotropic a n d exhibit different polarizabilities i n the radial, c ^ , a n d tangential, ct directions. T h e i r scattering can be approximated quite w e l l b y that from an anisotropic sphere l e a d i n g to the equations (4, 11) u

Figure 2. Spherulites of polyethylene.


726

P O L Y M E R

l„

v


= KV {(3/U )(o — Or) cos (0/2) sin μ cos μ • (4 sin 17 - U cos U - 3 Sri/)} 2

3

2

k

(11)

2

Zr, = K V {(3/L7 ) (a, - a.) (2 sin 17 - U cos 17 - SiU) + (or - a ) (SiU - s i n 17) - (a* - a ) cos (0/2) • cos //, (4 sin L7 - U cos 17 - 3 SiC7)} 2

3

(12)

2

s

r

2

2

where V is the volume of the spherulite, 17 = qR, R is the radius of the spherulite, SiC7 = J (sin xlx)dx, and μ is the azirriuthal scattering angle. I n this case, the p o l a r i z e d scattering patterns d e p e n d on μ b e i n g fourfold symmetric for H (Figure 3) and twofold for V . T h e H intensity is finite a n d depends on the anisotropy (ctr — a ) of the spherulite. It exhibits a m a x i m u m w i t h 0 at a value of U = 4.08 such that u

0

v

v

v

t

Figure 3. A theoretically calculated H scattering pattern from an iso lated ideal spherulite (100). v


41.

STEIN

727



4.08 = (4nRA)

s i n (flU/2)

(13)

T h i s equation provides a means for the determination o f spherulite size, as discussed later. T h e intensity of scattering varies w i t h (c^ a*) , w h i c h depends o n the degree of crystallinity of the spherulite a n d may be used for f o l l o w i n g its change w i t h temperature or time. T w o of the terms i n the V expression ( w h i c h are i n d e p e n d e n t o f μ) d e p e n d on a , the p o l a r i z a b i l i t y of the surrounding o f the spheru lite. T h e s e surroundings may be amorphous phase w i t h a p o l a r i z a b i l ity da or other spherulites w i t h a n average p o l a r i z a b i l i t y (ov + 2a )/3. Results i n d i c a t e d (12) that 2

v

0

t

= Φ (α 8

ν

+ 2c%)/3 + (1 - Φ*) ota

(14)

where φ is the v o l u m e fraction of spherulites. T h u s , the shape of the V scattering pattern w i l l change w i t h φ as the spherulites grow, as was observed (13). W h e n s p h e r u l i t i c samples are deformed, the spherulites change from spherical to e l l i p s o i d a l objects. T h i s change may be m o d e l e d to y i e l d deformed scattering patterns as s h o w n i n F i g u r e 4 (14). T h e i r study d u r i n g deformation permits the analysis of deformation m e c h a nisms. 8

v

8

SCATTERING INTENSITY IN A D E B Y E - B U E C H E C O N T I N U U M A P P R O A C H

FOR ISOTROPIC SYSTEMS. Samples w i t h r a n d o m l y arranged structures can best b e treated i n terms o f a statistical description i n v o l v i n g the fluctuation of their scattering power from its average value ρ (15)

η* =Pi ~ Ρ D e b y e and B u e c h e (15) showed that for isotropic systems I.(q) = K i r \ f y(r) exp [i(q • r)]d r 3

(16)

where y(r) is a spacial correlation function d e f i n e d b y

y(r) = — ψ —

() 17

where the symbol ( ) designates an average over a l l pairs of scatter i n g elements separated b y r. T h e quantity r varies from u n i t y at τ = 0 to zero at r = o° i n a manner dependent o n the structure of the system. F o r spherically symmetrical systems, E q u a t i o n 16 reduces to r


728

P O L Y M E R


Unoriented

50% E l o n g a t i o n

100% E l o n g a t i o n Figure 4. Theoretically predicted and experimentally determined changes in the H small angle light scattering pattern of isotactic polypropylene film with elongation. The film stretch direction and the polarization of the incident beam are vertical while the scattered beam is viewed with a horizontal analyzer (14). v


41.

STEIN

729



I (q) = 4TT Κη Γ

(18)

y(r) ^~^r dr

2

s

2

In this case, y(r) may be obtained b y F o u r i e r inversion to give K' r °° sin (qr) y(r) == f I (q) — q*dq

, (19) v

s

F o r randomly dispersed two-phase systems, y(r) usually can be ap proximated b y an exponential function (15, 16) y(r) = e x p ( - r / a )

(20)

c

where a is a correlation distance characterizing the scale of the heter ogeneity. Kratky a n d P o r o d (17) showed that for a two-phase system con taining volume fractions φ a n d φ of the two phases, the average chord lengths through the phases are g i v e n b y c

χ

2

£ = α /φ and f λ

€

2

2

= α Ιφ α

(21)

1

T h e substitution of E q u a t i o n 20 i n E q u a t i o n 18 leads to the result (16)

'· " (

)=κν

Polymer Characterization - American Chemical Society

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