STRUCTURAL CHARACTERIZATION OF NON-CRYSTALLINE HYDROGELS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HO-GESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFI-CUS DR. IR. C.J.D.M. VERHAGEN, HOOGLERAAR IN DE AF-DELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COM-MISSIE UIT DE SENAAT TE VERDEDIGEN OP DONDERDAG
18 DECEMBER 1969 TE 16.00 UUR
DOOR
JOHANNES HENDRIKUS GOUDA Scheikundig ingenieur
Geboren te Waalwijk
3\
V.R.B.-Offsetdrukkerij Kleine der A 3-4, GroningenDIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR
Gaarne wil ik mijn dank betuigen aan allen die bij het in dit proefschrift beschreven onderzoek behulpzaam zijn g e -weest, Prof. Dr, W, P r i n s dank ik voor zijn leiding en voort-durende hulp en belangstelling gevoort-durende dit onderzoek. Dr. I r . J . J . van Aartsen en D r . I r . A . E . M.Keijzers ben ik z e e r erkentelijk voor hun voortdurende interesse en vele s t i m u -lerende d i s c u s s i e s . De Heren Y . T i m m e r m a n en B . J . S p i t
dank ik voor het uitvoeren van een gedeelte der e x p e r i -menten. De Heren R. van Donselaar, J.G.Kennedy, H . C . Nieuwpoort en K. Sjoer dank ik voor hun steun bij het tot-standkomen van de meetapparatuur. Tot slot dank ik de Heer A . J . Dekker voor het vervaardigen van de tekeningen.
The author wishes to thank M e s s r s . T . C . W a r r e n , J . Schechter, and K. Provodator for carrying out parts of the experiments. Finally, I thank Mr. J . M. Lipton, who made it possible to write this thesis in the American language.
T A B L E O F C O N T E N T S C h a p t e r I INTRODUCTION C h a p t e r II I I - 1 I I - 2 I I - 3 L I G H T SCATTERING T H E O R Y 11 I n t r o d u c t i o n l l T h e s c a t t e r i n g of a r a n d o m c o l l e c t i o n of a n i s o t r o p i c r e g i o n s with c y l i n d r i c a l s y m m e t r y 14 T h e r a n d o m o r i e n t a t i o n c o r r e l a t i o n m o -d e l 29 C h a p t e r III I I I - l III-2 P H O T O E L A S T I C I T Y 32 I n t r o d u c t i o n 32 T h e s t r e s s - s t r a i n b e h a v i o r of a m e d i u m c o n t a i n i n g r i g i d a n i s o t r o p i c r e g i o n s 34 I I I - 3 C a l c u l a t i o n of t h e b i r e f r i n g e n c e 36 I I I - 4 T h e effect of s w e l l i n g 33 C h a p t e r IV E X P E R I M E N T A L A R R A N G E M E N T S 39 I V - 1 L i g h t s c a t t e r i n g 39 IV-2 P h o t o e l a s t i c i t y 43 I V - 3 E v a l u a t i o n of the m e a s u r e m e n t s 44 C h a p t e r V S A M P L E P R E P A R A T I O N AND SOME G E N E R A L OBSERVATIONS 49 V - 1 T h e polyvinyl a l c o h o l g e l s 49 V - 2 P o l y g l y c o l m e t h a c r y l a t e g e l s 50 V - 3 E l e c t r o n m i c r o s c o p y 51 V - 4 S p e c t r o m e t r i c m e t h o d s 52 C h a p t e r VI R E S U L T S V I - 1 I n t r o d u c t i o n V I - 2 P o l y v i n y l a l c o h o l g e l s V I - 3 P o l y g l y c o l m e t h a c r y l a t e g e l s V I - 4 E r r o r a n a l y s i s 53 53 53 59 61 5
C h a p t e r VII CONCLUSIONS 69 V I I - 1 I n t r o d u c t i o n 69 VII-2 L i g h t s c a t t e r i n g 69 V I I - 3 P h o t o e l a s t i c i t y 72 V I I - 4 Finsul c o n c l u s i o n s 74 S u m m a r y 76 S a m e n v a t t i n g 78 Appendix I T H E V E C T O R o EN T H E O R Y AND P R A C T I C E 80 A I - 1 I n t r o d u c t i o n 80 A I - 2 T h e t h e o r e t i c a l definition of o 80 A I - 3 T h e definition of o for a c t u a l e m a l y s e r s ( p o l a r o i d s ) 81 Appendix II F O R T R A N P R O G R A M 90 G l o s s a r y of S y m b o l s 92 R e f e r e n c e s 95 6
C H A P T E R I
INTRODUCTION
The classical picture of polymer networks a s a collection of randomly crosslinked, randomly kinked polymer chains has increasingly come under attack in recent y e a r s . Du§ek and Prins^ have summed up the arguments and available evidence in favor of some structuring in polymer networks. One may divide the possibilities for structuring into t h r e e p a r t s : before, during, and after the formation of the n e t -work. Networks may be formed by crosslinking existing polymer chains - in solution or in the solid state - or by crosslinking polymerization (e. g. copolymerization ofavinyl and divinyl compound).
Before the network formation t h e r e may be a non-equilib-rium ordering that depends upon the pretreatment of the polymer. Also, equilibrium ordering may exist, analogous to, for example, the packing of chains in block copolymers^ or diluent induced structured solutions ( e . g . lamellar or cylindrical). Kargin and coworkers^ have found globular and fibrillar s t r u c t u r e s in noncrystalline polymers, using m a i n -ly electron microscopy. If such pre-existing o r d e r is foxind in the non-crosslinked state, it may very well be trapped upon crosslinking the chains into a network.
During the formation of the network one may distinguish between two main factors which a r e , however, not independ-ent of each other: inhomogeneous crosslinking and incom-patibility of the polymer with the diluent. Inhomogeneous crosslinking may occur for several r e a s o n s . During the copolymerization of a vinyl and divinyl compound the m o r e reactive compound becomes m o r e rapidly exhausted than the l e s s reactive compound. Either short chains b e -tween the crosslinks a r e formed in the beginning and long chains at the end of the reaction or vice v e r s a . Also, pref-erential crosslinking next to an already formed crosslink may take place. It is conceivable that the crosslinked regions in the initially formed network a r e not compatible with the
diluent. This can lead to further inhomogeneous c r o s s l i n k -ing because the existence of segregated regions means that t h e r e is an inhomogeneous reaction mixture in which random crosslinking is unlikely.
T h e r e may be several r e a s o n s for the incompatibility of the local network s t r u c t u r e with the diluent. Excessive local crosslinking (an elastic free energy effect), may r a i s e the chemical potential of the diluent above that of the pure liq-uid. This leads to the exudation of diluent throughout the network (microsyneresis). Another possibility is that the polymer-diluent interaction is such that only limited swell-ing, r e g a r d l e s s of crosslink density, occurs (a free energy of mixing effect). Excessive dilution during network forma-tion will, for the same reason a s menforma-tioned above, then lead to s y n e r e s i s . Due to the low mobility in the polymer m a t r i x the diluent droplets may remain trapped in the net -work. This m i c r o s y n e r e s i s will lead to a heterogeneous network phase.
After the network has been formed the addition of a diluent may induce ordering. It is well known that this may occur in biopolymer solutions'*. In some c a s e s the same phenom-ena a r e found in synthetic polymer solutions^'^ and g e l s ' . If the diluent is water, aggregation phenomena a r e often ascribed to hydrophobic bonding^, i . e . the water "squeezes" the hydrophobic p a r t s of the polymer chains together.
It should be s t r e s s e d that the above paragraphs represent only a schematic outline of the arguments and available evidence for structuring in non-crystalline polymer networks.
An investigation on hydrogels is of interest for obtaining m o r e information about structuring in polymer networks and the causes of it. At least some of the factors causing
structuring in these gels may be analogous to the factors leading to gel s t r u c t u r e s occurring in biopolymer a s s e m -blies in the plant and animal world. It is worth mentioning that one of the gels (polyglycol methacrylate) investigated in this study, was originally conceived and developed for various biomedical applications such as flexible contact lens material**. Lenses of this type a r e now commercially avaible.
The investigation described in this thesis was c a r r i e d out along the following line. F i r s t an attempt was made to find direct evidence for supermolecular structures in the n e t
work, using light scattering, electron microscopy, and s p e c -t r o m e -t r i c me-thods. Secondly, in order -to find suppor-t for the picture of the gel s t r u c t u r e s that emerged from these m e a s u r e m e n t s , their photoelastic behavior was m e a s u r e d .
Light scattering is the predominant technique which was used. Some light scattering r e s u l t s on gels have been r e -ported in the l i t e r a t u r e previously *'^^, but these were ob-tained by means of unpolarized light or by measuring only the parallel component of the scattered light. If one u s e s the perpendicular component of the scattered light, one only " s e e s " the light scattered by anisotropic units. This makes the interpretation of the experimental r e s u l t s e a s i e r . (For the definition of the parallel and perpendicular component of the scattered light see Chapter 11).
In light scattering theory t h e r e a r e two approaches to describe the experimental r e s u l t s . The first, more general one, makes use of correlation functions. F o r example, a c -cording to the original Debye-Bueche treatment ^^ one can write the following proportionality for the light scattering from an optically isotropic but inhomogeneous solid:
CD
Is (e>)oc<n2> J r(r) sin(hr)/(hr) r^dr (1-1)
o
where 7(r) is a correlation function; the meaning of the other symbols can be found in the Glossary of Symbols. With the aid of a F o u r i e r inversion one can in principle calculate the correlation function. However, one then needs the scattered intensity until such am angle that the intensity becomes zero and remains zero at l a r g e r angles. If this is not the c a s e ,
- a s indeed it was for our samples - then one has to rely on the second approach which is based on models. One calculates the theoretical light scattering curves for these models and compares these with the experimental c u r v e s . It should however be realized that one never can prove by this procedure that a certain model is unique. Therefore it is n e c e s s a r y to find additional support for a certain model by using other techniques.
In Chapter II the calculation of the light scattering for a model consisting of a random collection of uniaxial a n i s o
-tropic regions (rods or disks of finite c r o s s section and thinckness respectively) is given. F u r t h e r a short review of the random orientation correlation model is presented. Chapter III gives the derivation for the s t r e s s - s t r a i n and birefringence-strain behavior for a combination model of a network consisting of randomly coiling chains in which anisotropic regions a r e embedded.
Chapter IV describes the various experimental a r r a n g e -m e n t s , a s well a s the conversion of the raw data.
In Chapter V the sample preparations and an account of some attempts to use electron microscopy and s p e c t r o m e t r i c methods on the gels a r e given.
In Chapter VI the experinaental r e s u l t s of the light s c a t -tering and the photoelasticity a r e presented.
In Chapter VII a discussion of the r e s u l t s i s given and the final conclusions a r e drawn regarding the applicability
of the models to the gel s t r u c t u r e s .
In Appendix I a discussion on the vector o, which is needed in light scattering calculations is presented.
Appendix II shows the computer program needed to t r a n s -form the raw light scattering data into Rayleigh r a t i o s .
C H A P T E R I I
LIGHT SCATTERING THEORY
II-1 Introduction
In order to calculate light scattering patterns in polymer science from models one often makes use of the Rayleigh-Gans approximation. K e r k e r 14 has recently proposed that a better name would be the Rayleigh Debye approximation, because the theoretical contributions of Debye a r e both e a r -lier and more significant than those of Gans. There a r e two conditions'^ which must be fulfilled for this approxi-mation to hold. F i r s t , the difference in refractive index between the particle and its surrounding must be small or
| m - l | « 1 (2-1) if one denotes by m the relative refractive index of the
particle with respect to. the surrounding medium. Secondly, the "phase shift" should be small:
^ | m - l | « l (2-2) In this expression a is a c h a r a c t e r i s t i c dimension of the
particle. One is then able to apply the same electrical field to all volume elements of the particle. Since the systems under investigation a r e swollen polymer networks, the use of the approximation i s , in general, justified. Heller'^ and Kerker et al. have given a quantitative survey of the a p -plicability of the Rayleigh-Debye approximation.
Before considering the light scattering pattern of a specific model it will be useful to derive the equation for the a m -plitude of the scattered light for one particle of volume V.
The XYZ-axes a r e chosen in the usual way (see Fig. II-1) . s^ and g' a r e the directions of the primary and scattered beam respectively. The scattering angle 9, is the angle between s^ and s'; n is the angle between the XZ-plane and the plane through s and s ' . The amplitude of the lin-early polarized p r i m a r y beam can be written as:
Figure II-l. The coordinate systems 1. i, k and a, b, c.
E = EoEp exp [i(tüt-kx)] ( 2 - 3 )
with EQ as the maximum amplitude; u is the circular fre-quency, t the time; k(=2jr/x) is the wave number, with X the wave length in the medium; x is the distance along the Xaxis, and Ep is a unit vector in the direction of the e l e c t r i c a l field of the p r i m a r y b e a m . It follows from e l e c t r o -magnetic light theory^^ that the amplitude of the scattered light of volume element i is given by:
1 K- - ( l / c ' R i ) j — ( P i - o )
.dt'
'•exp(-ikR.) ( 2 - 4 )
where c is the velocity of light in vacuum; Rj is the d i s -tance of the volume element to the observer; pj is the in-duced dipole moment, which equals g^E with a the po-larizability tensor; exp (-ikRj) is the phase factor, which takes into account the phase difference between the volume element and the place of the observer; the vector o d e t e r -mines, which conaponent of the induced dipole moment is
"seen" by the o b s e r v e r . Differentiation gives:
E ' = Eoto^Mi.o)/(c^Rj) exp[i(ut-kx-kRi)] (2-5) with Ml = g E p.
In order to obtain the total scattered amplitude, it is n e c e s -s a r y to -sum over the whole volume of the particle. If one writes for R, = RQ + ARj, with RQ the distance between the center of the particle and the observer and ARj equals (r. .§') (see F i g . I I - 2 ) , with r. the vector, which connects the center of the particle with volume element i and if one also a s s u m e s that R is l a r g e , so that it is possible to write in the denominator R • RQ then
Etot = ( w^/c^Ro) ^^P [ i ( w t - k R j ] E (M.. o) exp [-i(kx + kAR.)] (2-6) where E^^j is the component of the total scattered amplitude, which is seen by the o b s e r v e r . F r o m Fig.II-2 one has
AR
i — T
Figure II-2. Diagram defining s.
-(kx + kARj) = k(r., s), with s equals SQ - S ' . If the particle has a center of symmetry then for each volume element with Tj there exists one with r^. It is then possible to a r -range them in p a i r s , so that:
E,„, = ( U ) 2 / C 2 R J exp [i(ut-kR^)] L (M . o) cos [k(rj . s)]
j
(2-7) Replacing the summation by an integration, one has:
Etot = exp [ i ( u t - kR„)] E3 (2-8)
with,
E , = ( t ü V c ^ R J J (M^ . o) cos [k(r. . s ) ] dV (2-9) V
If one uses polarized light a distinction can be made b e -tween the perpendicular component of the scattered light and the parallel component. The definition of the vector o can be found in the book by Cabannes^^ . In both c a s e s (perpendicular and parallel) g is a unit vector perpendicular to the scattered ray. In the case of the perpendicular com-ponent g is perpendicular to the plane through s' and Ep. For the parallel component g lies in the plane through s' and Ep. In practice it is usually only possible to measure the light scattering in one plane, normally the horizontal plane. In order to measure in such an instrument the whole spatial light scattering envelope, several experimental a r rangements a r e possible. The definitions of o for two a r -rangements will be treated in Appendix I . In this chapter only the definition of g given by Cabannes will be used.
In connection with the experimental results (Chapter VI) two models will be considered: (1) anisotropic regions with cylindrical symmetry, e . g . r o d s , and (2) the random orien-tation correlation ( r . o . c . ) model.
II-2 The scattering of a random collection of anisotropic
regions with cylindrical symmetry
Benoit^"'^'' and coworkers have calculated the scattering of a random collection of rods but their approach was limited to infinitely thin r o d s . Recently Stein and Rhodes ^"^ have given a calculation for a collection of oriented r o d s . How-ever, in this case the calculation was limited to two dimen-sions and again the rods were assumed to be infinitely thin. A calculation for the other extreme of this model, infinitely thin disks, has recently been performed by Picot ^* . A more general theory then the one given here, has very recently been presented by van Aartsen^^.
The XYZ-axes a r e chosen in the same way as in Sec. I I - l . In order to facilitate the calculations (especially of
the phase factor) a coordinate system a, b, c (see Fig. II-l) is chosen in the following way. The axis a lies in the d i -rection of the bisectrix of 0, c lies in the plane through s ' d s' and is perpendicular to a; b completes a right-handed Coordinate system. The following equations for a, b, and c thus apply:
a = COSÖ/2 i + sinö/2 sin^u j + sinö/2 cosy k (2-10)
b = - coBU i + siniuk (2-11) c = sin0/2 i - cos0/2 siny j - cos0/2 cos/u k (2-12)
c lies thus in the direction of g {=s^ - s'). (see also Eq. r2-18)).
Figure I I - 3 . The coordinate systems a, b , c and 1, £.• S.'
The coordinate system of the anisotropic region r e p -resented by 1, p, and g (see Fig. II-3) is chosen as fol-lows:
1 lies in the direction of the axis of the anisotropic region, c[ is perpendicular to 1 making an angle <p with the plane through c and 1; g is perpendicular to both g and 1 a is the angle between 1 and c and X is the angle between the
a-axis and the projection of 1 on the (a,b) plane. One can now e x p r e s s 1, p , and g in a, b, and c.
1 = sina cosX a + s i n a sinX b + cosa c (2-13) 2 = (-cos^) sinX + sirvp cosor cosX)a + (cosp cosX +
(2-14) + sincp cosa sinX) b - (sinip sino')c ^ '
q = (-sin(p sinX - cos*> coso cosX)a + (2-15) + (-cos(p costt sinX + sincp cosX)b + (cos^ sinff)c
The origins of all three coordinate systems coincide with the center of the anisotropic region.
F i r s t the phase factor is calculated.
s ' = COS© i + sine siim j + sin9 cos^ k (2-16)
§ o = i (2-17) thus:
s = s^ - s' = 2sine/2 [sin0/2 i - cos0/2sin;uj +
- COS 9/2 c o s / u k l =
--• (2-18) = 2 sin9/2c
r . = 1 1 + q q (2-19) cos [k(r. . s ) ] = cos (hi cosQr+ hq sinacoscp) (2-20)
with h equal to 2ksine/2.
Secondly (M.. g) can be calculated. It is assumed that the anisotropic region has two principal polarizabilities: a^ along the axis of the region and «g perpendicular to this axis. Since s lies along the X-axis, Ep is given, in general by:
Ep = y2 j + yak (2-21) The general equation for g is:
o = Xji + Xgj + Xgk ( 2 - 2 2 ) i, j , a n d k w i l l e a c h now b e e x p r e s s e d in t e r m s of a, b , and c . i = (i . a ) a + (i . b ) b + ( i . c ) c (2-23) j = ( j . a ) a + ( j . b ) b + ( i . c ) c (2-24) k = (k. a ) a + (k. b ) b + (k. c ) c (2-25) T h u s : o r , and o r fi = ( x i ( a . i ) + X2 ( a . j ) + Xg ( a . k ) } a + + { x i ( b . i ) + Xg ( b . j ) + X3 ( b . k ) } b + (2-26) + {x^ ( c . i ) + Xg ( c . j ) + Xg ( c . k ) } c o = A a + B b + C c (2-27) ^p = {y2 ( § 1 ) + y 3 ( 5 Ï ? ) } ^ + + ( 7 2 ( ^ J ) •*• ya ( ^ 1 ? ) } ^ "^ (2-28) + {yg (e-J) + ys ( c k ) } c E p = D a + E b + F c (2-29) In the c a s e of c y l i n d r i c a l s y m m e t r y of the p o l a r i z a b i l i t y
the "induced d i p o l e m o m e n t " , j ^ j , i s given b y :
M. = ( a i - «2) (Ep . 1) 1 + «2 E p (2-30) By u s i n g E q s . ( 2 9 ) , ( 2 2 0 ) , and (230) the following r e
-l a t i o n s h i p for E i s o b t a i n e d :
Es
2* +L/2 Q
(u Vc^Ro) J S J (Mj.o) COS [k(rj. s)] q dq dl d^
o -L/2 o ^ 2 - 3 1 )
( M . o ) depends only on the direction of 1 in space and is independent of 1, q, and <p, the integral extends thus just over cos Ec(r. .§)] . Subsequently all these integrations a r e performed.
2» L/2 Q
{w^/c R ) f J ƒ cos (hi cosa + hq slna c o s ^ q dq dl dip =
o -L/2 o 2ir L/2 Q
( u / c RQ) \ J J ƒ cos (hi cosa) cos (hq sina cos«^) q dq dl d^ <«• o -L/2 o
2ir L/2 Q
~ f J ƒ sin (hi cos a ) sin (hq sina cos9)q dq dl d> ^
° -^^^^ L / 2 Q ( 2 - 3 2 ) B (w^/c^R^ J ƒ 2» cos (hi cosa) J (hq sina) q dq dl =
-L/2 o ° / 2 / 2„ . f o , ,, , , L sin rh(L/2) cosa ]
- sin [h(L/2) cosa ] J^ (hQ sino) = (w / c Rp) 2ir LQ^
[h(L/2) cosa ] hQ sina
L is the length of the anisotropic region and Q is the r a d i u s . Jo(x) and Ji(x) a r e Bessel functions of the first kind and the zeroth and first o r d e r respectively.
The ratio of the intensity scattered by one anisotropic region and the p r i m a r y intensity (I^) is obtained by squaring the amplitudes (Eqs. (2-8) and (2-3) respectively) and av-eraging over one period. The average contribution to this ratio of one anisotropic region can be calculated by av-eraging this ratio over all possible orientations of the region, in space. Since interference between different particles is neglected the total scattered intensity, IjC^), is obtained by multiplying the average r e s u l t for one particle with the number of anisotropic regions
(v^^)-Is(^)/Io = (''an/2'r) J* ] E] sin« da dx (2-33)
( M j ö ) ^ follows f r o m E q . ( 2 - 3 0 ) .
(M..Q)2 - (a^-af ( E p . l ) ^ ( l . o ) + ^2-34) + 2^2(«1-02) < E p - l ) ( l - o ) ^ p . o ) + a 2 ( E p . o ) ^
with:
( E p . i ) = D sinof cosX + E s i n a sinX + F cosor
(1. o) = A sinof cosX + B s i n o sinZ + C cosor
( E p . a ) = AD + B E + C F
F i r s t t h e i n t e g r a t i o n o v e r X i s p e r f o r m e d ; s i n c e the p h a s e f a c t o r i s not a function of X , it i s only n e c e s s a r y to c o n -s i d e r M . . 2 ) , y i e l d i n g :
IT
(l/2it) J (M.. 0)2 dX = (a^ - 03)^ [(1/16) sin'*a {3(AD -f BE)^ -f o
+ (AE - BD)^} -f 1/4 cos^a sin^a {(AF + CD)^ + (BF + CE)^ +
+ 2CF(AD * BE)} + 1/2 (CF)^ cos'*o ] + ( 2 - 3 5 ) + 2oi2(a^ - ag) {(1/4) sin^a (AD ••• BE) + 1/2 (CF) cos^a}(AD •^ BE + CF) +
••• 1/2 a^(AD + BE + CF)^
I n t e g r a t i o n o v e r a a n d r e a r r a n g e m e n t of t h e v a r i o u s t e r m s g i v e s :
l3(e)/Io = 2(c^'*/cV) I? >)^^V^ [9ó2{{3(AD + BE)^ + + (AE - BD)2}l/8 Pg +
•f{(AF + CD)2 .f (BF -i- CE)2 -l- 2CF(AD + BE)} l/2(Pi - P^) + ( 2 - 3 6 ) + (CF)2 (P^ - 2P^ .f P^)} •••
+ 6Ö(1-A) (AD •^ BE -f CF) {(AD -t BE) 1/2 P^ + CF (P^j - P^)} ••• •f (1 - 6)2 (AD + BE -K CF)2 P ]
where 6 = (a^ - 02) I (o^ + 202), is a measure of the anisotropy; a = (a^ + 2a2)/3. is the average polarizability; V = T L Q ^ , is the volume of an anisotropic region, and
IT sin^ [h(L/2) cosa ] J^(hQ sina)
P (i - o, 1. or 2) = ƒ (sina) ^"^ * ^^ — d a o [h(L/2) cosa]'' (hQ sina)**
If the anisotropic regions a r e embedded in an isotropic medium with polarizability a^, then a-^ ^"^^ 02 should be replaced by {a-^ - a^ and (ja^ - a^) respectively.
Usually one e x p r e s s e s the scattering in the so-called Rayleigh ratio defined a s :
,2
1^9
1,(9) R (2-37) \ \
with Vj as the scattering volume. Using Eq. (2-36) one ob-tains for the Rayleigh ratio in this case:
Rg = KV^ S)*^a^ [9Ö^|Ï3(AD + BE)^ -f (AE - BD)^} 1/8 P^ + + {(AF + CD)2 + (BF -f CE) ^ + 2CF (AD + BE)) 1/2 (P - Pg) +
+ (CF)2 (P^ - 2P •»• P2)}+ ( 2 - 3 8 ) + 66(1 - 6) (AD -f BE + CF) {(AD + BE) 1/2 P, + CF (P - P,)> +
1 0 1 ^
•f (1 - 6)2 (AD -f BE -f CF)2 p 1 0^
K equals (32TT^) / X^ , with X^, a s the wave length of light in vacuum and V^g^^i is the number of anisotropic regions per unit of volume.
By making different choices for o and E it is possible to obtain equations for any geometry desired, from the general r e s u l t (Eq. 2-38), because E is connected with the transmittance direction of the polarizer and g with that of the analyser. In this chapter the emphasis will be on the theoretical light scattering patterns, so the definition of o,
m e n t i o n e d in S e c . I I - l , will be u s e d . U il> is the a n g l e in the Y Z - p l a n e b e t w e e n the Z - a x i s and the d i r e c t i o n of E p , t h e n E = (- s i n ^ j + c o s ^ k) t h u s y = - siml/ yg = coB(// (2-39) F o r the p e r p e n d i c u l a r c o m p o n e n t , o i s g i v e n by ( s e e A p -pendix I) g = ( l / N ) r - s i n 9 s i n e i - l - c o s 9 cosii» j + sin(i/ c o s In t h i s c a s e : x^ = - (1/N) s i n 9 s i n e Xg = (1/N) c o s 9 cos^^ Xg = (1/N) COS0 sin(i'
F o r the p a r a l l e l c o m p o n e n t , one h a s (see Appendix g^^ = (1/N) - s i n 9 c o s 9 c o s e i - {sinip + sin29 sin/u
+ { c o s ^ - s i n 2 9 coSiU c o s e } li T h u s :
x^ = - ( 1 / N ) s i n 9 c o s 9 c o s e
X = - ( 1 / N ) ( s i n ( ^ + sin29 s i n / u c o s e } 2
X = (1/N) {cos(^ - sin29 cos/u c o s e }
9 k] (2-40) I) c o s e } j (2-41) r~ 2 2 2 ~l ^ In t h e s e e x p r e s s i o n s N e q u a l s s i n 9 s i n e + c o s 9 ; £ {=iu+<l/) i s the a n g l e b e t w e e n the p l a n e s t h r o u g h E and s^ and Sjj and s ' r e s p e c t i v e l y (see F i g . I I - l ) .
With the aid of E q s . (2-26) t h r o u g h ( 2 - 2 9 ) , A, B , C . D , E , and F c a n be c a l c u l a t e d for e a c h c a s e and by i n s e r t i n g t h o s e v a l u e s into the g e n e r a l E q . (2-38) the R a y l e i g h r a t i o s a r e o b t a i n e d , for the p e r p e n d i c u l a r (/) and p a r a l l e l ( / / ) c o m p o n e n t s of the s c a t t e r e d light:
L . 2 X ; 0 - 0 0 \ ; & - O . I
Figure 11-4. Small-angle light scattering patterns for rods as a function of the radius.
L.O.OX;Q.1JOX.8,0.1
Figure II-5. Stnall-angleli^t scattering patterns for disks as a function of the thickness.
R-' = 95 ^•**3„KV^ N"^5^ [{3 sin^« cos^e cosS/2 + •fsin^e/2 (cos e cose - sin^e)^} 1/8 P2 +
+{cos^9/2 (cos^« cos8 - sin^e) ^ - 2 sin^t cos^e cos^e/2} 1/2 (P - P ) + 1 ^
•I- sin^e cos^« cos'*e/2 (P - 2?^^ •»• P ) ] ( 2 - 4 2 )
and
R"= ^* KV^ N"^ a"^ [952 {{3(sin^« - cos^t cos6 sin^e/2)^ + 2 2 2 4
•f 4 sin e cos e sin 9/2 cos 8/2} 1/8 P„ + ( 2 - 4 3 ) + {4 sin^t cos^t cos^e/2 +
+ 2 cos^e cos^e/2 cose (sin^e - cos^c cosB sin^e/2)} 1/2 (P - P ) + * cos^c cos^e/2 cos'^e (P - 2P, + P )} -f
o 1 2
+ 66(1 - 6)(sin^c •» cos^t cos^e) {(sin'^e - cos^» cosö sinV2) 1/2 P^ ••• + cos^« cos^e/2 cose (P^ - P^)} + (1 - 6)^(sin^« -f cos^e cos^e)^ P^
These equations were programmed in F o r t r a n IV. The calculations were performed on an IBM 360/50 computer for various combinations of L, Q, and 6. The r e s u l t s were plotted as contour plots; some r e s u l t s are shown in F i g s . IIr4 and H-S.
F r o m these E q s . (2-42) and (2-43) it is easy to calculate some special c a s e s , which a r e useful because of their easy experimental accessibility. Four components of the scattered light in the XY-plane or horizontal plane are calculated: H , H. , V , and Vj^ (the capital letter r e f e r s to the s c a t -tered light, the small one to the incident light). For H, and Vj, the angle e equals 180°, and for Vy and Hy this angle equals 90°; thus:
Vj, = Hy = 96^ v'^ KY^'a^ {1/8 sin29/2 Pg +
+ 1/2 008^9/2 (P, - P 2 ) ) ^^'*^^
o » 030 075 10
SIN e/z
to
Vy =^rnKV^ö^[(27/8) 6^ Pa -1-36(1-6) Pi + ( l - 6 ) ^ P j (2-45)
" h " *ln*^^^"^ [962{(3/8) sin'*e/2 P2 - sin^e/2 cos^e/2 (Pg - P^) +
+ cosV2 (Pg - 2P^ •^ P )} * (2-46) + 66(1 - 6) {(-1/2) cose sin2e/2 P + cose cos^e/2 (P^ - P^)} *
•f (1 - 6)^ cose P^]
T h e s e e q u a t i o n s w e r e a l s o p r o g r a m m e d in F o r t r a n IV. T h e r e s u l t s a r e shown in F i g s . I I - 6 and 11-7 for r o d s and in F i g . I I - 8 for d i s k s .
F o r s m a l l v a l u e s of 9, one c a n expand the p h a s e f a c t o r in a s e r i e s . T h e i n t e g r a t i o n c a n t h e n be p e r f o r m e d by hand and y i e l d s : Vfa = "v = (9/2) b^S>\^KV^a^ {(1/15) - (1/105) [(L/2)^ + 0 ^ ] h^ -f . . } (2-47)
^^r-^^w^]w ] '""'
, / o ,* T^-,r2-2r46^ -I-5 (l3 /35 + 566 + 446^\ ^ Hh = 1 / 2 V ^ K V a [ — - [— ( 35 ) + ^ Q^ ƒ 35 - 286 + 206^^ 1 , 2 ^ l " " e " ! ï~5 li^ " " • • • • ] (2-49) T h e s e e q u a t i o n s show t h a t it i s , in p r i n c i p l e , p o s s i b l e to obtain the length L , the r a d i u s Q a s well a s the a n i s o t r o p y 6 f r o m m e a s u r e m e n t s , of H , V , and Hu a t s m a l l a n g l e s , by taking the s l o p e of the line for e a c h r a t i o v e r s u s h^ devided by the i n t e r c e p t for h e q u a l s 0. A n o t h e r way^^ to obtain the a b s o l u t e v a l u e of 6 is e x t r a p o l a t i n g the r a t i o of H and V to 9 e q u a l s 0;( H y / V , ) e = o = 36^/(46^ + 5) (2-50)
I\5 CO
As can be seen from F i g s . I I - 4 and II-5 the scattering envelopes of the perpendicular a s well as the parallel com-ponent become more and more c i r c u l a r at increasing thick-n e s s . The shape of the patterthick-ns for rods (Fig. II-4) athick-nd disks (Fig.II5) a r e about the s a m e . The shape of the p e r pendicular component does not depend on 6, because 6 a p -p e a r s as a factor in front of the equation for R'' . The -plots of the various components in the horizontal plane show that especially the wide-angle scattering is strongly influenced by increasing thickness (Figs.II-6 through 11-8). Fig. II-7 shows that the location of the minimum, with r e s p e c t to 9 = 90°, in the H^ curve yields the sign of the anisotropy, provided that the radius of the rod is not too large and the anisotropy not too small. If the radius is too large, there a r e s e v e r a l minima, if the anisotropy is too small the minimum appears so close to 90°, that it is impossible to say if it appears at s m a l l e r or at l a r g e r angles than 90°. II-3 The random orientation correlation model
This model was introduced by Stein and Wilson ^^. The scattering medium is assumed to consists of small a n i -sotropic volume elements, which have cylindrical s y m m e t r y . F u r t h e r m o r e the following assumptions are made. There is no correlation between the polarizability and anisotropy fluctuations and all fluctuations a r e spherically s y m m e t r i c . The medium does not have an overall macroscopic orien-tation. Finally it is assumed that the ratio of the two polarizabilities for each volume element is constant. Keijzers 27,28 has shown that the scattering is given by:
Re = (xi+Xg +X3) Q H- (xgyg + x^y^f (O-Q) (2-51) Since in the theoretical case o is a unit vector, one has ^ 1 ' + ^ 2 ' -" ^ 3 ' = 1-Q = 27rK (1/15) ^2 f f(r) ^{r) EÏILS^ r ^ d r (2-52) / hr o = 2.K [<,2> ƒ ,(0 £ilgH) ,2,^ ^ 4_ ^2 J'f(^) ^^^j s i n ^ ^2,^-] o o 29
In t h e s e r e s u l t s <r]^> is the m e a n s q u a r e p o l a r i z a b i l i t y f l u c t u a t i o n . jS i s the m e a n a n i s o t r o p y of the s a m p l e , y (r) i s the p o l a r i z a b i l i t y c o r r e l a t i o n function defined a s y (r) = < r j . 1 . > r / < ' ) ^ > , w h e r e rj. and r\. a r e the p o l a r i z a b i l i t y f l u c t u a t i o n s of the ith and j t h v o l u m e e l e m e n t r e s p e c t i v e l y , s e p -a r -a t e d by -a d i s t -a n c e r . f(r) i s t h e o r i e n t -a t i o n c o r r e l -a t i o n function defined a s f (r) = ( 3 < c o s ' ' 9 i j > , - l ) / 2 , w h e r e 9.j i s the a n g l e b e t w e e n the m a i n o p t i c a l a x e s of the ith and jth v o l u m e e l e m e n t , s e p a r a t e d by a d i s t a n c e r ; ;u (r) = 1 + <n >v(r)
with a the m e a n p o l a r i z a b i l i t y . " Using E q s . (2-40) and (2-41) for Oy and g* r e s p e c t i v e l y ,
one o b t a i n s
R-' = Q (2-54) R#= O - sin^9 c o s ^ e (O-Q) (2-55)
F r o m t h e s e e q u a t i o n s it follows t h a t the s c a t t e r i n g e n v e l o p for the c r o s s e d c o m p o n e n t i s c i r c u l a r , and for the p a r a l l e l c o m p o n e n t e l l i p t i c a l .
In the h o r i z o n t a l plane one o b t a i n s in t h i s c a s e :
" h Hy = V, = Q Vy = 0 = O c o s ^ 9 + Q sirv^B (2-56) (2-57) (2-58) E q u a t i o n (2-50) g i v e s the r e l a t i o n b e t w e e n (H^V^) Q _ Q and 6 ; if one c a l c u l a t e s t h i s r a t i o a l s o for the r a n d o m o r i e n t a t i o n c o r r e l a t i o n m o d e l and s e t s the two r a t i o s e q u a l ,
2 2 t h e n one o b t a i n s the following r e l a t i o n b e t w e e n 6 and j3 :
2 1 ^ ' J f ( r ) A . ( r ) r ^ c i r ^^-59) * ~ 9 fP
<r}2> ƒ 7 ( r ) r ^ d r o
If one c o m p a r e s the p e r p e n d i c u l a r c o m p o n e n t for both m o d e l s then t h e r e is only a d i f f e r e n c e in s h a p e for thin r o d s . F o r t h i c k r o d s the s c a t t e r i n g envelope is c i r c u l a r a s it is
for the random orientation correlation m o d e l . The parallel component has in both c a s e s a s i m i l a r elliptical shape. It is thus only possible to distinguish between the two models
the case of thin r o d s .
C H A P T E R I I I
PHOTOELASTICITY
III-l Introduction
A vast amount of work^-^^ of both experimental and theoretical nature, has been published with respect to the elastic properties of non-crystalline polymer networks. F o r a Gaussian network of randomly coiling chains, the following equation for the s t r e s s - s t r a i n behavior in uni-directional extension applies^:
ag = Ai/kT ( < r ' > i / < r S „ ^ ) [ A ^ - A^'] (3-1) where a^ is the s t r e s s on the strained c r o s s section, P is the
number of chains per unit unstrained volume, k is the Boltz-mann constant, T is the absolute t e m p e r a t u r e , and A is a con-stant, which originates from an uncertainty in the effect of the crosslinks upon the chain s t a t i s t i c s , e. g, A equals 1 in the theories of Hermans^'' and Flory and Wall^^'^^ but equals
1/2 in the theory of James and Guth^s-as. (<r^>J<r^>^) is the so-called front factor of Tobolsky, with <r^> ^ and <r^>(jj the mean square end-to-end distance in the isotropic initial state and the dry reference state respectively, and Ajj is the extension ratio with reference to the initial s t a t e .
In general the experimenteil data, for moderate extensions, fit the empirical "so-called" Mooney-Rivlin equation (3-2) better than Eq. (3-1):
a = (C* + C * / A J ( A ' - A ; ' ) (3-2)
C* and Cg a r e empirical constants, which differ from substance to substsince. Blokland ^^ has given a survey of the attempts, that have been made to explain both constants in molecular t e r m s . One possibility is that the appearence of C2 is due to a certain structuring in the network. Ci is frequently identified with the constant, Ai'kT ^r^>^/<.r'^^
in the equation for a Gaussian network (Eq, (3-1)). T h e r e i s , however, no theoretical basis for this procedure ^•"^^.
F o r the case that the s t r u c t u r a l elements in the network consist of infinitely thin, stiff needlelike "anisotropic r e -gions", embedded in a m a t r i x of randomly coiling chains, a s t r e s s - s t r a i n relation has been derived by Kuhn and co-workers 37-40^ They a r r i v e at the following equation:
^an=(3^ankT/(2(A^ - 1)))(2A^ + 1 - 3 A 3 ( A 3 - 1)-*arctan(A3 - i)i) (3-3) a^n i s the s t r e s s on the strained c r o s s section due to the anisotropic regions and i/^^ is the number of anisotropic r e -gions per unit volume. According to Volkenstein, Gotlib, and Ptitsyn'*^. this equation can be approximated for the r e -gion A^ = 1.2 to A^ = 3, by:
«^an = 0-72 ^,„kT (A^ - A ; 2 ) (3.4)
By adding the s t r e s s e s for the randomly coiling chains and the anisotropic regions one gets:
°^ = ^G + «^an = [Aï'kT ( < r 2 > , / < r 2 > ^ ) +
(3-5) -H 0 . 7 2 ^ , „ k T / A j (Aj - A^-i)
This is a Mooney-Rivlin type equation. F r o m this r e s u l t it follows that i/^n has to be of the same o r d e r of magni-tude a s v if one wants to obtain a C* of about the s a m e o r d e r of magnitude a s C*, as has been experimentally ob-served. This s e e m s an unlikely result, if the anisotropic region is visualized as extending over an appreciable number of chain segments of different chains (see Chapter VII).
It is well known that a polymer network becomes b i r e -fringent upon stretching. This is attributed to the orientation of the chains a n d / o r the anisotropic regions in the network. Kuhn and Grün^^ have calculated the dependence of the birefringence on the s t r a i n for a Gaussiem network, as well a s for a collection of needles. They obtained the following relationships:
A n g =
= Ai/kT(<r2>i/cr2>od)(2jr/45)((n2 + 2f/n){al " « 2 ) ( ^ W ) (3-6) and
An,„ = i/3„(W9)((ri^ + 2)2/n)Oi - P2)((2A'^ + 1)I{A\ - 1) +
- 3A^^arctan(A^^ - 1)*/(A^, - 1)^^^) (3-7) where Ang is the birefringence of a Gaussian network, An^^
is the birefringence of a collection of needles, and ii is the average refractive index of the medium. {a\ - a%) is the anisotropyof a chain segment, and (P ^ - Pg) is the aniso-tropy of an anisotropic region (needle). It should be noted that the factor A(<r^>i/-cr^>Qj) was originally not introduced by Kuhn and GrUn.
If one uses the same approximation as used for the s t r e s s in Eq. (3-4), one obtains from Eq. (3-7):
An,„ = 0.17 !/,„ ((ii^ + 2)2/fl)(Pi - P2)(Ax - A^^) (3-8) In the same way, as for the s t r e s s , one can add the bi-refringence for chains and anisotropic regions to obtain an equation of the Mooney-Rivlin form:
An = (A* + A * / A J (A^ - A"^' ) (3-9) where A J equals (2?r/45) Av (cr >J<r >od) ((n + 2 ) /n)(aj-ar2)
and A* equals 0.17 v^^({n^ + 2 ) ^ / n ) 0 j - P^).
Ill-2 The stress-strain behavior of a medium containing
rigid' anisotropic regions
42
Volkenstein and Gotlib have extended the calculation of Kuhn to anisotropic regions with a finite c r o s s section. The calculation proceeds in a s i m i l a r way as for a Gaussicin net-work. F i r s t the new distribution function of the rods i s calculated after the sample is stretched. With the aid of this distribution function, the entropy is calculated.
ondly, from the entropy difference between the strained and the unstrained network the s t r e s s i s calculated.
The anisotropic regions a r e characterized by three o r -thogonal vectors 1, jg, and £[. It is assumed that the regions have cylindrical symmetry. _1 makes an angle rj with the X-axis of a Cartesian-coordinate system. The sample i s now stretched in the direction of the Xaxis, until the r e -gion makes an angle n' with the X-axis. With the aid of the equations of motion for the region and by making sev-e r a l simplifying assumptions Volksev-enstsev-ein and Gotlib a r r i v sev-e at the following relation between rj and r}':
3 Q^-L^
. 2 Q 2 + L 8 , ( 3 - 1 0 ) tan n = Ax tan n ' where Q is the radius of the anisotropic region, and L is
its length. F o r Q = 0 this relation reduces to the one given by Kuhn:
(3-11) Thus if one puts
tan Axeff
n'
= = A""'''' Q 2 - L 2 Ax tan =n
p 2 - l (3-12) with p = L / Q , one can use Kuhn's expression for the en-tropy by replacing A,; by \^ff The force is then given b y :f = - ( T / L i ) O A S / a A x ) =
- - (T/Li)OAS/aAx^ff)(aA^^fj/aAx)
(3-13)
where f i s the force, Lj the initial length, and AS the entropy difference between the strained state and the isotropic reference state. Carrying out this calculation, and t r a n s f o r m -ing the force to the s t r e s s on the strained c r o s s section, one obtains:
^an= (3-14) = y i^j,„kT/(2(Ax-l))(2AT;+l- 3A];(A^- 1)"^ arc tan (A^ - 1)*)
where y = 3(p^-l)/(p -i-l). F o r p = oo(infinitely thin "needles") this relation reduces to Eq. (3-3); for p = 0 (infinitely thin "plates") one obtains:
<^an = (3/2) l / a n k T / ( A ^ l ) ( 2 - ^ A ^ 3 ( l - A f ) * a r c t a n h ( l - A - 3 ) * ) (3-15) In Table HI-1 the constants a r e given to approximate Eq. (3-14), in the s a m e manner a s is done in Sec. III-l, for different values of p. F o r values of p < l the function cannot be approximated by an equation of the type represented by Eq. (3-4).
Ill-3 Calculation of the birefringence
It is assumed that the anisotropic region has a polariza-bility ^i in the direction of 1 and jSg in every direction perpendicular to 1. The contribution of one region to the polarizability of the stretched sample is given by:
6i = ^1 cos^ n' + ^2 sin^ n' (3-16) «2 = (Pi - |32) sin^ ij' cos2^ +32 (3-17) where 61 and 62 a r e the polarizabilities in the direction of
the stretch and perpendicular to it, respectively; ^ is the angle, which the projection of 1, on the YZ-plane, makes with the Y-axis. The average contribution of one region to the polarizability follows from:
< 6 i > = jj Vi6idn'd^/(//Vidrj'd(^) (3-18) < 6 2 > = ƒƒ Vió^dri'déUjJ Vjdn'd*) (3-19) where Vj is the probability that region i m£ikes an angle
between TJ! and TJ.' + drj'. with the X-axis, rj' follows from Eq. (3-10). Replacing again A^ by A^eff (see Eq. (3-12)), one gets after integration:
^6i> = ^2 + O i - P 2 ) P (3-20)
<62> = (1/2) (jSi -Hj32) - (1/2) (/3i - /32) P (3-21) with
P = A ' , , „ / ( A , \ „ - l ) - A L f a r c t a n ( A L f - l)*/(ALff- ^f' (3-22) With the aid of the Lorentz-Lorenz relation, one obtains for the birefringence An^^
(n^ - l)/(n^ + 2) - (n^ - l)/(n2 + 2) S 6;iAn^„/(n^ + 2-)^ =
= (4jr/3) Van(<*i> - <*2=^) (3-23) where An = n , - n„
an 1 /
An^„ = (7r/9)((R2 + 2)Vn) u^ifi^ - ^j) ( 2 A J + 1)/(A>; - 1) +
- 3A];arctan ( A J - 1)*/(A>; - 1)^^^) (3-24) F o r p = CO (infinitely thin "needles", this gives Eq. (3-7),
for p = 0 (infinitely thin "plates") one gets:
An,„ = (;r/9)((ii'+ 2)Vn) i/,„ O^ - ^^U^'^ ' l ) ' ' ( 2 A t +1 +
- 3A"jf a r c t a n h ( l - A;^ )*/(l - A;^ )*) (3-25) In Table III-l, the constants. A* a r e also given as found
by approximating Eq. (3-24) in the same manner as done in Sec. I I I - l . Tabl» I I I - l -X)0 40 20 4 2 0.72 0.72 0.72 0.72 O.Si 0.2t 0 . 1 7 0 . 1 7 0 . 1 7 0 . 1 7 0 . 1 5 0 . 1 0 37
Ill-4 The effect of swelling
If one uses for swollen networks as the reference state the isotropically swollen gel then it is possible to write for a swollen network, instead of Eq. (3-5):
CT = Af^kT q p q j ^ ^ ^ [A^^ - ^'x] + 0. 72 t/^* kT [ A ^ - A 2
r3-26)
with q. the degree of swelling of the isotropically swollen network, q^ the reference degree of swelling, and i^* and^an ^^^ number of chains and anisotropic regions per unit volume of the swollen network, respectively.
F o r the birefringence it is possible to write for relation (3-9), in the same manner as was done for the s t r e s s ,
An = Ai/*kT qf\f^{n^+ 2)^/n)(27r / 45)(al-a%){Al-A]^ ) + + 0. 17 ((n' + 2)^ln) «/*„ (^i - P2) kT (A^ - A^ ) (3-27) for p > 4.
It should be r e m e m b e r e d that the final Eqs. (3-26) and (3-27) are based upon a model, the correctness of which is by no means beyond doubt. F u r t h e r m o r e the s t r u c t u r e of the network may depend upon the nature and anaount of diluent, as well as upon the imposed deformation.
Finally, it should be noted that form birefringence is not included in the above considerations. The effect on the anisotropy of the statistical chain element will not be im-portant because our gels are concentrated systenas. In con-centrated systems form birefringence is absent due to the practically uniform segment density. It seems reasonable to assume that for our gels the refractive index difference between the anisotropic regions and the enviromnent will be small. If that is the case then the form birefringence of the anisotropic regions is also negligible.
C H A P T E R IV
EXPERIMENTAL ARRANGEMENTS
IV-1 Light scattering
I V - 1 - a WIDE-ANGLE LIGHT SCATTERING
The wide-angle light scattering of the gels'*^ was measured with a Cenco-TNO turbidimeter (Cenco Instruments, Breda, The Netherlands). The gel s t r i p s were mounted between glass slides, held in a specially constructed stainless steel frame, which was placed in a standard cylindrical light s c a t -tering cell (see Fig. IV-1). In o r d e r to m e a s u r e the scat-tering
—|-STAM£S5 STEEL FRAhC
CEU
Figure IV-1. Stainless steel frame for measuring gel strips in the wide-angle l i ^ t scattering photometer.
f r o m 10 t o 150 d e g r e e s , the v e r t i c a l p l a n e of the s a m p l e could be s e t a t a n a n g l e f with the i n c i d e n t b e a m (v = -45°, 90°, and +45°).
I V - l - b SMALUANGLE UGHT SCATTERING
T h e s m a l l a n g l e l i g h t s c a t t e r i n g m e a s u r e m e n t s w e r e p e r f o r n i e d on the s m a l l a n g l e l i g h t s c a t t e r i n g p h o t o m e t e r d e -v e l o p e d in t h i s laboratory2''.44, A d e t a i l e d d e s c r i p t i o n of
t^
the apparatus and the conversion of measured intensities to Rayleigh ratios has been given by Keijzers2''. Because the scattering levels of the gels were low the photometer had to be modified to improve the sensitivity*. A chopper d e -vice consisting of a perforated rotating disk was placed in the p r i m a r y beam. The chopper gives the incident beam a frequency of 750 c p s . This frequency was monitored with the aid of a small light bulb and a photodiode. The output from the diode was fed as a reference signal into a lock-in amplifier (HR - 8, PAR, Prlock-inceton, N . J . , U . S . A . ) . The signal of the detecting photomultiplier was also fed into the lock-in amplifier. The amplifier selectively utilizes only that part of the signal that has the same phase and frequency as the reference signal. Since noise is distributed over all frequencies, one is able to eliminate most of the noise. The amplifier has also a time averaging circuit, which averages the signal over a selected time. It is possible with this s e t -up to improve the sensitivity of the light detection by a fac-tor of 300 to 400.
F o r the m e a s u r e m e n t s on gel s t r i p s a special cell was constructed (see F i g . IV-2). This cell basically consists of
SCREW EYE INNER CYLINDER
Figure IV-2. Cell for measuring gel strips in the small-angle light scattering photometer,
two concentric cylinders, the inner cylinder being movable with a screw mechanism enabling an adjustment to the thick-•This instrument is now commercially available through: N. V. Nederlandsche
Optiek-en InstrumOptiek-entOptiek-enfabriek D r . C E . S l e e k e r , Zeist, The Netherlands.
ness of the gel s t r i p . The remaining part of the cell i s filled with the swelling agent. The polyvinyl alcohol gels were m e a s -ured in a small rectangular (5.0 x 2.50 x 0.2cm) glass cell in which the gel was made. In this case the cell was mounted in a sample holder s i m i l a r to the one used by Keijzers^''.
The scattering at a s e r i e s of 9 values was measured a s a function of ijj with the aid of two rotatable Polaroid filters
(Polaroid HN22, between glass slides) see F i g . I V - 3 . The angles 9, tp and n a r e defined in Fig. I I - l . ip could be
ad-«njwKNn
WORM
GCAR-OEABS
POLAROID
Figure IV-3. Schematic diagram of the rotatable Polaroid filter.
justed to an accuracy of + 0.1°, In this way it i s possible to m e a s u r e the whole light scattering envelop by varying ip, instead of n. The angle iJi is fixed at 90° in this photometer as in most other light scattering photometers. The Vy and Hy components were measured with the aid of fixed polaroids. F o r r e a s o n s that will be explained below (Sec,IV-3-b), the analyser was placed in the detector a r m . The angle w, i. e. the angle which the transmittance direction of the analyser makes with the vertical, was calculated from the relation:
tanu = ( l / c o s ö ) cotip (4-1) The derivation of this equation and the necessity of using it,
will be given in Appendix I.
l\9
Figure- IV-4. Schematic diagram of photoelasticity set-up,
1 - Mercury Arc; 2 = CoUimating Lens; 3 and 6 = Diaphragms; 4 = Interference Filter; 5 "= Neutral Density Filters; 7 • Rotatable Polarizer; 8 = Sample Chamber; 9 =(X/4) plate; 10 « Rotatable Analyser; 11 s Photomultiplier; 12= In-ductive Force Transducer; 13 = Screw Micrometer; 14 = Amplifier for 12; 15 = High Voltage for 11; 16 «OpticalBench;
IV-2 Photoelasticity
The equipment*^ used is shown in Fig. IV-4. The gel s t r i p was clamped in a glass cell, which was filled with the swelling agent. The upper clamp is movable by means of a screw m i c r o m e t e r . The thickness and the length of the s t r i p were measured with a low power traveling microscope and a cathetometer respectively. To m e a s u r e the birefring-ence, An, a de Sénarmont compensator was used. This compensator consists of a rotatable analyser and a X/.4 plate. The sample is situated between the X/4 plate and the p o l a r i z e r . The transmittance directions of the polarizer and analyser make an angle of +45° and -45° respectively with the stretch direction of the sample. The slow direction of the X/4 plate is parallel to the transmittance direction of the polarizer. The birefringence can then be calculated from the relation*^:
2K = * = (2;rAndJ/X„ (4-2) K. is the angle through which the analyser has to be turned
to obtain minimum intensity, * is the phase retardation of the sample and dj its thickness. The intensity was naeas-ured with the aid of a photomultiplier.
In o r d e r to obtain a X/4 plate for the wave length of the light used (XQ = 546lA), a device was constructed, which made it possible to tilt a mica sheet through a certain angle around its slow direction. The procedure is as follows. One puts this mica sheet in the position of the sample with its slow direction vertically and uses another mica sheet, which has a phase retardation of 90° t P as an approximate "X/4 plate". The value of p is unimportant. The relation for minimum intensity is^^:
tan2K = cosp t a n * (4-3) Since in this case * has to be 90°, tan 2K = co thus K = 45°.
One thus sets K at 45° and tilts the first mica sheet until minimum intensity is obtained. * is now 90° for that sheet and can be used as an exact X/4 plate.
To obtain the s t r e s s a calibrated inductive t r a n s d u c e r ( M o d e l Q l / 5 , 5Kcs, Hottinger Messtechnik, Darmstadt, G e r
many) was used. The signal of the transducer was amplified by means of a c a r r i e r wave amplifier (Model KWS-II, Hot-tinger). F r o m relaxation measurements it was deduced that three hours was a sufficient long time to approximate equi-librium, after the s t r i p was brought at a certain elongation. After the m e a s u r e m e n t s the width of the s t r i p was d e t e r -mined with a measuring microscope.
IV-3 Evaluation of the measurements
I V - 3 - a WIDE-ANGLE LIGHT SCATTERINGAs explained in Sec. IV1a, in o r d e r to obtain the s c a t -tering for the wide-angle range of 9 (10° to 150°), it was n e c e s s a r y to set the sample under different angles with the p r i m a r y beam. As a consequence of this a different volume of the sample is seen by the detector for each setting. It was thus n e c e s s a r y to convert the data to the same reference s t a t e . It turned out that a theoretically calculated conversion factor could not be used, probably because of the uncertain influences of reflections at the different interfaces. T h e r e -fore, use was made of an empirical conversion factor, which was estimated as follows. F i r s t an a r b i t r a r y AgJ-sol was measured in the cylindrical cell and the usual corrections were made. Secondly a small rectangular glass cell was filled with the same sol. The cell had approximately the same dimensions as the gel s t r i p s . The scattering was now measured in the same way as for the gels. Finally the small cell was filled with water and measured again. After the subtraction of the water plus glass scattering, for each angle the ratio of the scattered intensity of the AgJ-sol in the cylindrical cell and the intensity of the sol in the rectangular cell was calculated. This procedure was repeated several t i m e s . It was also performed with a Ludox solution (1.5 w%) (E.I. du Pont de Nemours, Wilmington, D e l . , U . S . A . ) . Unfortunately the reproducibility was r a t h e r poor (see F i g . IV-5). To minimize the e r r o r s , inherent to this procedure, a s much as possible an average of several s e r i e s was taken. Because of the poor reproducibility the data can only be considered to be semiquantitative. By
reference to the scattering of pure benzene, all data were converted to Rayleigh r a t i o s .
Figure IV-6. Conversion factors for the wide-angle light scattering measurements. The solid line is the average of a number of measurements, some of the measured points are also shown,
I V - 3 - b SMALUANGLE UGHT SCATTERING
All m e a s u r e m e n t s were performed with the sample perpen-dicular to the p r i m a r y beam. Due to the geometry of the cell and cell holder it i s , in practice, only possible to m e a s -ure from 9 = 40°down to the resolution limit of the instrument which is 30'. After the measurement of the scattering of each gel, the gel was replaced by the swelling agent and the scattering was measured again. Because of the low s c a t -tering levels of the gels the scat-tering of the blank (the cell filled with the swelling agent) is of the same o r d e r of m a g -nitude (10-30% of the gel scattering). In order to estimate which was the main cause of this, relatively high scattering of the blank s e v e r a l liquids, water, 0. 5M magnesium p e r -chlorate, benzene etc. and the empty cell were m e a s u r e d . F r o m the r e s u l t s it was concluded that the scattering was mainly caused by the a i r - g l a s s and glass-liquid interfaces. Since these cannot be eliminated at small angles, a c o r r e c
tion had to be applied. The value of the blank was subtracted from that of the gel at the same external scattering angle. Above 25° to 30° the scattering of the gel and the blank were about the s a m e . It was thus impossible to m e a s u r e the gel s t r i p s beyond these angles in this instrument.
Because of the interface scattering it was impossible to use the coupled polaroid set-up that Keijzers27 used for tjj variation. This would mean that the scattering of two more interfaces (polaroid-air) would be included in the signal. The gel scattering would then have been completely obscured. F o r the same reason it was impossible to make photographs of the light scattering pattern of the gels using a l a s e r s e t -up 27.
The product of the turbidity r and the thickness ds of the sample was determined by measuring the p r i m a r y intensity with (Ijo), and without (I^) the sample; rdj follows from: exp (-TdJ =
= dso/Io) [ l - (-g - l ) ' / ( - g + l ) ' ] ' ' [ l - ("s - n g ) ' / ( n 3 ^ g ) ' ] " ' (4-4) with n and x\^ the refractive indices of the supporting glass windows and the sample respectively. This product is needed for the correction of the raw data (see below). Moreover it s e r v e s as a guide line for the absence of multiple s c a t -tering. Roughlyis one can say that if Tds<0, 1, there is only single scattering; for 0. l < T d s < 0 . 3 multiple s c a t -tering may be present, and for Tds>0. 3 multiple scat-tering is usually present.
The thickness of the sample was obtained by the following procedure. F i r s t the sample was placed between two glass slides and the thickness of sample plus slides was measured with a m i c r o m e t e r . Secondly the thickness of the slides was m e a s u r e d . Subtraction of these two values gives the thick-ness of the sample. The refractive index of the sample was measured with an Abbe refractometer. The concentrations of the magnesium perchlorate solutions were determined by titration of magnesium with EDTA^^.
Owing to the geometrical arrangement of the photometer s e v e r a l corrections, have to be made. Stein and Keane^' have given a survey of these c o r r e c t i o n s . Keijzers^'' has given the corrections needed for this particular photometer.
His equations have been used with the exception of the r e -flection corrections for the scattered r ay . This was n e c e s s a r y because in the present experiments the analyser was placed in the detector a r m r a t h e r than in the p r i m a r y beam d i r e c tion. The equations for these corrections a r e given in A p pendix I. The final equation, which combines all the n e c e s -s a r y correction-s i -s :
1^9) n^Tds [l-(l-sin2e„/n2)i] [R2cos2a; + R2 sin2a]'^
lo cose^[exp(-rdj)-exp{-Tdj (l-sin^/n^^-»}] [l-(ng-l)2/(ng+l)2][l-(n,-ng)2/(ns+ng)^]
(4-5) 9u is the scattering angle outside the sample. The meanings of Rqand Rp a r e explained in Appendix I; 1,(9) is the m e a s -ured s c a t t e r e d intensity.
This equation was programmed in F o r t r a n IV. The c o m -puter program is given in Appendix H.
I V - 3 - C PHOTOELASTICITY
The swelling ratio, q, was calculated from the weights of the swollen, Ws, and the dry gel, wj, with the aid of the densities of the swelling agent, Ps, and the dry polymer,
Pd:
(Wd/Pd) + K - W d ) / P 3
q = (4-6)
K/^'d)
The initial length and the thickness were obtained by e x t r a -polating both s t r e s s and birefringence to z e r o . With the aid of the m e a s u r e d initial width the unstrained c r o s s section, Oi, can be calculated. The strained c r o s s section, 0, follows ironn the relation:
0 = Oi/A, (4-7) withAx the extension ratio. Eq. (4-7) holds if it is assumed
that there is no volume change upon stretching. This is an approximation, which is justified within the precision of our data.
T h e s t r e s s - o p t i c a l coefficient, defined a s :
C = An/CT (4-8) with a the s t r e s s p e r s t r a i n e d c r o s s s e c t i o n w a s o b t a i n e d
a s the s l o p e of the v i r t u a l l y l i n e a r plot of An v e r s u s the s t r e s s u p to e l o n g a t i o n of 50%. T h i s i s not c o n s i s t e n t w i t h the p r o p o s e d m o d e l , b e c a u s e a plot of An v e r s u s cr will not in g e n e r a l yield a s t r a i g h t l i n e ( E q s . (3-26) and ( 3 - 2 7 ) ) . At t h e s e s m a l l e x t e n s i o n r a t i o s (Ax-$1.5) i t i s , h o w e v e r , v e r y difficult to d i s t i n g u i s h b e t w e e n the p r o p o s e d m o d e l and a G a u s s i a n n e t w o r k .
C H A P T E R V
SAMPLE PREPARATION AND SOME GENERAL OBSERVATIONS
V-1 The polyvinyl alcohol gels
The polyvinyl alcohol (PVA) gels were prepared as follows. PVA (Elvanol 72/30; Dupont, DP^ = 1830, e s t e r residue
< 1%) was dissolved in doubly distilled water by refluxing a mixture of water and PVA for 24 h o u r s . The solution, approximately 5% by weight was filtered at elevated t e m p e r a -ture and p r e s s u r e , through a sintered glass filter into a cylindrical dust free glass cell. Different weights of the c r o s s -linking agent - thiodiacetaldehyde orterephthalaldehyde - w e r e added. The thiodiacetaldehyde was added as the acetal. Next a filtered, weighed amount of 4N sulfuric acid was added to increase the pH of the solution to approximately 0. 3. In about three hours gelation took place. The same procedure was followed to obtain PVA gel s t r i p s . The solution was poured out into a plexiglass mold, (7 x 2 x 0.2 cm), or into a small rectangular glass cell, ( 5 x 2 x 0 . 2 ) , just p r i o r to gelation. The concentration of PVA was calculated afterwards from the weights of the swollen éind dry gel. After the gelation had taken place, the sulfuric acid was removed by thoroughly washing with water. To make s u r e
Gel PVA-I PVA-II PVA-III PVA-IV •"HJO 28 28 28 28 Table V-n 1.34 1.34 1.34 1.34 1 Td s 0 0 0.04 0.04 crosslinks per monomer unit 0.02 0.01 0.006 0.004 49
that the gel was in an equilibrium state, a period of at least fourteen days was allowed before making measurements on the gel. Some data on the gels a r e collected in Table V - 1 . X - r a y diffraction did not reveal the presence of c r y s t a l s .
V-2 Polyglycol methacrylate gels
Polyglycol methacrylate (PGMA) gels were kindly supplied by the Institute of Macromolecular Chemistry at P r a g u e , Czechoslovakia. The gels were p r e p a r e d ^ by copolymeri-zation of ethylene glycol methacrylate (GMA) with a smeill amount of ethylene glycol dimethacrylate (DGMA). During the polymerization different amounts of water were present. Table V-2 s u m m a r i z e s the circumstances under which the gels were made.
M l P O U - l - o POIA-l-0 f O I » - 3 - o r » A - 4 - o Pa<A-5-o P O U - I - S ratt-2-a p c m - 3 - i i PCMA-i-n KHK-i-n P O t t - l - l PCHA-l-2 POUL-l-3 «Mtar present « t p o l y M T l s a c i o n wX 0 . 0 2 0 . 0 i l . O W . S » 7 . 0 0 . 0 2 0 . 0 t l . O U.i 4 7 . 0 0 . 0 0 . 0 0 . 0 DOU yl 0 . 1 6 0 . 1 6 0.16 0 . 1 6 0.16 0 . 1 0 0 . 1 0 0 . 1 0 0 . 1 0 0 . 1 0 0 . 1 0 0.41 1.04 ' « , 0 I.7B 1.81 1.82 2 . 0 0 2 . 0 0 1.78 1.78 1.82 -1.78 1.77 1.64 i d 0 . 0 3 0 . 0 6 0 . 0 4 . 0 . 1 4 -0 . -0 6 0 . 0 6 0 . 0 3 -0 . -0 6 0.07 0 . 0 4
The gel s t r i p s had a thickness of 2-3 mm and were cut from sheets in the. proper dimensions to fit the measuring cells. Other swelling agents were introduced by gradually replacing the water.
The PGMA gels have been extensively studied in the
stitute of Macromolecular Chemistry at P r a g u e . Mechan-ical " and optMechan-ical p r o p e r t i e s have been reported. Although no X - r a y diffraction r e s u l t s have been reported, it s e e m s unlikely, in view of the fact that the polymer is atactic, that c r y s t a l s will exist in the network. If one fol-lows the network formation, as a function of the diluent present at polymerization, three regions with respect to water content a r e found, 0-46 w%, 46-80 w%, and above 80 w% water. In the first region the gels look homogeneous and t r a n s p a r e n t . In the second region the gels become very turbid. The lowest concentration of water at which the gels become turbid is called the critical concentration. The t u r -bidity is believed to be caused by m i c r o s y n e r e s i s a n d / o r microphase separation; water droplets a r e exuded throughout the network due to the incompatability of the polymer and water at these concentrations. Above 80 w% the gels a r e spongy and very turbid. If the gels a r e prepared in the presence of ethylene glycol r a t h e r than water then they remain transparent for all concentrations of ethylene glycol. This is ascribed to the better compatibility of PGMA and ethylene glycol.
F r o m a study of the swelling behavior in different swell-ing agents for PGMA gels Refojo' has drawn the conclu-sion that the gels have a secondary s t r u c t u r e stabilized by hydrophobic bonding.
V-3 Electron microscopy
In o r d e r to get a m o r e direct proof of the possible presence of supermolecular s t r u c t u r e an attempt was made to make electron micrographs from the gels. Owing to the mechanical properties of the gels it was impossible to cut very thin (<1000A) slices on a microtome directly. After several methods were tried, the best way to circumvent this problem seemed to be, to freeze-dry the gels, to cleave them at room t e m p e r a t u r e and to make replicas from the freshly cleaved surfaces. After shadowing with platinum, photographs
' These experiments were carried out by B.J.Spit, Institute of Applied Physics TNO-TH, Delft, The Netherlands.
were made. However, the photographs taken from these samples showed too little contrast to draw any conclusion. It is possible that the freeze-etching technique would give better r e s u l t s , but this technique was unfortunately not a-vailable.
V-4 Spectrometric methods**
If the gels contain supermolecular units, then it is pos-sible that in a n d / o r around these units specially bonded water o c c u r s . If that is the case then it might be possible with IR and NMR to detect this.
IR-spectra between 4000-3000cm "•^, which is where the OHstretching frequencies occur, were measured on a P e r k i n -E l m e r Infrared Spectrophotometer, Model 521 ( P e r k i n - -E l m e r Corp. Palo Alto, Calif., U. S, A , ) , The samples were approxi-mately 20;u sections of the gels cut with a microtome. This could be achieved most conveniently after swelling the gel in ethylene glycol and evaporating most of it. The samples a r e then still somewhat pliable; in water they a r e too pli-able and in the dry state they a r e too brittle.
IR measurements were made on sections swollen in water, on dry sections and on pure water. It turned out, however, that there is no detectable difference between the hydrogen bonding in pure water and that in the water-swollen gels.
spectra were taken on a Varian Model A60 NMR-s p e c t r o m e t e r at room temperature (Varian A NMR-s NMR-s . , PEIIO Alto, Calif., U.S. A , ) , The samples were either long thin gel s t r i p s o r suspensions of gel p a r t i c l e s . No difference was found between the proton resonance of the sample and that of pure water, which indicates that if there is any special water s t r u c t u r e near the polymer chains, the exchange rate with regular water is fast at room t e m p e r a t u r e .
•• These experiments were carried out by T.C.Warren and J.Schechter, Department of Chemistry, Syracuse University, Syracuse, N.Y., U.S.A,
