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TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGE-SCHOOL DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN, HOOGLERAAR IN DE AFDE-LING DER TECHNISCHE NATUURKUNDE, VOOR EEN COM-MISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 20
DECEMBER 1967, TE 16 UUR
DOOR
JACOB VAN DAM Sclieikundig Ingenieur
geboren te Klundert
\Z^ö P
Druk: V.R.B.—Offsetdruklcerij - Groningen58
1967DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR. W. PRINS
Gaarne wil ik de lieren J . G . K e n n e d y en H. C. Nieuwpoort danken voor hun medewerking bij het ontwerpen en c o n s t r u e -r e n van de e l a s t o - o s m o m e t e -r .
1. I N T R O D U C T I O N AND SURVEY OF N U M B E R A V -E R A G -E M O L -E C U L A R W-EIGHT M -E T H O D S 7 2 . T H E O R Y 11 2 . 1 . E q u i l i b r i u m t h e o r y f o r n o n - p e r m e a t i n g s o l u t e s H 2. 1. 1. T h e c h e m i c a l p o t e n t i a l of the s o l v e n t in a s w o l l e n gel H 2 . 1 . 2 . E f f e c t of the d e s w e l l i n g by p o l y m e r s o l u t i o n s on the r e t r a c t i v e f o r c e 14 2 . 2 , E q u i l i b r i u m t h e o r y f o r p e r m e a t i n g s o l u t e s 15 2 . 2 . 1 . D i s t r i b u t i o n of t h e c o m p o n e n t s o v e r the two p h a s e s 15 2 . 2 . 2 . I n f l u e n c e of p e r m e a t i o n on the r e t r a c -tive f o r c e i n c r e m e n t 18 2. 3. Diffusion p h e n o m e n a in e l a s t o - o s m o m e t r y 22 3. D E S C R I P T I O N AND O P E R A T I O N O F T H E E L A S T O -O S M -O M E T E R 27 3. 1. D e s i g n of the e l a s t o - o s m o m e t e r 27 3. 2. O p e r a t i o n p r o c e d u r e 30 3. 3. S e n s i t i v i t y of the m e t h o d and s o u r c e s of e r r o r 30 3 . 3 . 1 . T h e b a l a n c e t r a n s d u c e r r e c o r d e r c o m -b i n a t i o n 31 3 . 3 . 2 . E r r o r s due to v a r i a t i o n s in b u o y a n c y 34 3 . 3 . 3 . R e l i a b i l i t y of the r e p l a c e m e n t p r o c e -d u r e 34 3 . 4 . T h e gel s t r i p s 35 38 38 40 4 . P R E P A R A T I O N AND C H A R A C T E R I Z A T I O N OF THE P O L Y M E R S 4 . 1. P r e p a r a t i o n 4 . 2. C h a r a c t e r i z a t i o n m e t h o d s 39 4. 2. 1. T h e r m o e l e c t r i c v a p o u r p h a s e o s m o m e -t r y 4 . 2 . 2 . M e m b r a n e o s m o m e t r y 41 4 . 2 . 3 . G e l p e r m e a t i o n c h r o m a t o g r a p h y 42 4, 3. M o l e c u l a r w e i g h t d a t a of the p o l y m e r s 43 4 . 3 . 1 . P o l y s t y r e n e s 43 4 . 3 . 2 . P o l y ( o x y p r o p y l e n e ) g l y c o l s 44 4. 3, 3, P o l y ( v i n y l a c e t a t e s ) 46 5, D E S C R I P T I O N AND I N T E R P R E T A T I O N OF T H E E L A S T O - O S M O T I C E X P E R I M E N T S 49 5, 1, E l a s t o - o s m o m e t r y by c a l i b r a t i o n 49
6 -5 . 1 . 1 . P o l y ( d i m e t h y l s i l o x a n e ) s t r i p s 49 5 . 1 . 2 . P o l y ( c i s - i s o p r e n e ) s t r i p s 53 5 . 1 . 3 . P o l y ( b u t a d i e n e - c o - s t y r e n e ) s t r i p s 57 5 . 1 . 4 . P o l y ( b u t a d i e n e - c o - a c r y l o n i t r i l e ) s t r i p s 58 5 . 1 . 5 . P o l y ( c i s - b u t a d i e n e ) s t r i p s 59 5 . 1 . 6 . C o m p a r i s o n of the e f f i c i e n c i e s of the s t r i p s 62 5. 2 . A b s o l u t e e l a s t o - o s m o m e t r y 63 5, 3, T h e effect of p e r m e a t i o n 68 5. 3. 1. E q u i l i b r i u m d e s w e l l i n g f o r p e r m e a t i n g s o l u t e s 68 5. 3, 2. T i m e - d e p e n d e n c e of the r e t r a c t i v e f o r c e f o r the c a s e of p e r m e a t i n g s o l u t e s 72 5, 4, C o n c l u s i o n s 73 A P P E N D I X I, Solution of the diffusion e q u a t i o n s f o r
a s t r i p in an infinite m e d i u m 75 A P P E N D I X II. L i s t of i m p o r t a n t s y m b o l s 79
R E F E R E N C E S 81 SUMMARY 83 S A M E N V A T T I N G 85
INTRODUCTION AND SURVEY OF NUMBER AVERAGE MOLECULAR WEIGHT METHODS
The task of the investigator who is concerned with the study of polymers is hampered by the fact that his mate-rials are not well defined. This holds in particular with regard to the molecular weight which for synthetic polymers may vary between a few hundreds and a few millions. In view of the fact that many physical and mechanical proper-ties are strongly molecular weight dependent, it is easily understood that research on rapid and sensitive methods fot determining molecular weights and molecular weight distri-butions has become very intensive during the last decennia.
Now, even for a particular polymer sample, it is impos-sible to speak about the molecular weight because all actual samples are mixtures with a molecular weight distribution, As a consequence we may define several molecular weight averages which are, in a mathematical sense, moments of the distribution. In this thesis we are only concerned with the methods for determining the number average molecular weight, defined as
M„ = E n^MjLn. ( i . i ) where Wi is the number of molecules with molecular weight
Mj. If the distribution can be considered to be continuous, we may write
CO OJ
M^ = j f(M) M AMI ( f(M) dM (1.2)
o o
where f(M) dM is the number of molecules having a molec-ular weight between M and M + dM.
In this average each molecule, regardless of its size, gives the same contribution. Therefore many methods for determining the number average are based on the coUigative properties of solutions, that is, one "counts" the number of solute molecules and calculates the number average mol-ecular weight from the known concentration^. To the col-ligative methods belong ebuUioscopy, that is measuring of the boiling point elevation, cryoscopy, based on the measure-ment of the freezing point depression, vapour phase osmome-try, in which the vapour pressure lowering as a result of
8
-addition of a solute is m e a s u r e d , and membrane or r e g u l a r o s m o m e t r y , where the hydrostatic head built up between a solution and the pure solvent, separated by a semipermeable m e m b r a n e , is m e a s u r e d .
The three first mentioned methods a r e no longer suitable if the molecular weight of the substance under investigation is l a r g e r than, say, 30,000 because in that case the effects to be m e a s u r e d a r e too s m a l l . Below this value they give excellent r e s u l t s and this is p a r t i c u l a r l y true for the vapour phase o s m o m e t r y . The ebullioscopic and cryoscopic methods p o s s e s s some drawbacks due to the extreme t e m p e r a t u r e s one is forced to work at. T h e r m o e l e c t r i c vapour phase o s m o m e t r y ^'"^, in which the vapour p r e s s u r e difference b e -tween a solution and the pure solvent is converted into a t e m p e r a t u r e difference, can be used at each t e m p e r a t u r e between melting and boiling point of the solvent,
The osmotic p r e s s u r e of solutions of polymers with m o l ecular weight above 20, 000 can be m e a s u r e d quite a c c u -rately, c o n t r a r y to the effects mentioned above, and this holds up to molecular weights of the o r d e r of 500, 000, Thus, the three methods mentioned before and membrane o s m o m e -try a r e complementary methods because the l a s t one gives r i s e to erroneous r e s u l t s for molecular weights below about 20,000 due to diffusion of the solute through the m e m b r a n e .
The investigations described in this t h e s i s , a r e concerned with a relatively new method, the e l a s t o - o s m o m e t r y , which has a close resemblance with r e g u l a r o s m o m e t r y . The m e t h -od is based on the phenomenon that the degree of swelling of a swollen gel, normally a crosslinked polymer network, is dependent on the activity of the swelling agent around the gel, A gel submerged in a pure solvent r e a c h e s a swelling equilibrium by absorption of solvent, in which the chemical potential of the solvent inside and outside the gel is equal, Replacement of the pure solvent by a polymer solution d e -c r e a s e s the a-ctivity of the solvent around the gel a-cid in course of time a new equilibrium is attained in which the chemical potentials a r e again equal, but the degree of s w e l l -ing is lower. Thus, the swollen gel acts like the membrane in an o s m o m e t e r , allowing solvent molecules to diffuse freely through while retarding the polymer molecules,
As a r e s u l t of the. d e c r e a s e of the degree of swelling the volume of the gel also d e c r e a s e s . As e a r l y as 1945, Boyer^ used this effect to determine number average molecular weights of polystyrenes up to 300,000 by weighing the amount of solvent in polystyrene-divinylbenzene gels, swollen in solutions of different concentrations. However, the v a r i a -tions of the dimensions of the gel turn out to be so small
be v e r y insensitive o r experimentally unfavourable. P r e l i m i n a r y theoretical and experimental work by H e r -m a n s ^ ' ^ , Ya-mada^, P r i n s ^-^ and M i e r a s ^ h a s de-monstrated that if one keeps one of the dimensions, preferably the length, constant and c o ns i de r s the r e t r a c t i v e force caused by the deswelling, this force is of such an o r d e r of m a g nitude that it can be m e a s u r e d quite accurately by the s e n -sitive force indicators available at p r e s e n t .
The p r e s e n t study is concerned with an e l a s t o - o s m o m e t e r based on this principle, in which the sensitivity is c o m -parable with or even better than the sensitivity of the best m e m b r a n e o s m o m e t e r s . The theory of this type of e l a s t o -o s m -o m e t r y is presented in Chapter 2, Chapter 3 gives a description of the apparatus and its operation. In Chapter 5 the experimental r e s u l t s a r e given and compared with the-ory,
Naturally, the determination of one of the a v e r a g e s of the molecular weight distribution of a polymer is not sufficient for a complete characterization in. general, because the poly-m e r behaviour will often be dependent on the shape of the distribution, especially in the case of wide distributions, Information about the distribution can be obtained by s e p -arating the sample into s e v e r a l fractions and by determining the molecular weight of each of these fractions. This method i s , however, very time-consuming. F o r this reason one has tried to develop m o r e rapid methods. One method is based on the permeability of the membrane in a conventional o s m o m e t e r for low molecular weight substances ^^ , If permeation o c -c u r s , the osmoti-c p r e s s u r e d e -c r e a s e s with time at a rate dependent on the diffusivity of the permeating molecules and thus on their size, A c o r r e c t interpretation of this time effect provides the molecular weight distribution of the low molecular weight tail of the s a m p l e . Owing to the specific p r o p e r t i e s of the m e m b r a n e s , commonlyused in o s m o m e t r y , this method is r e s t r i c t e d to relatively low molecular weights, roughly below 10,000.
In view of the analogy between m e m b r a n e and e l a s t o o s m o m e t r y one could expect that information about the m o l -ecular weight distribution could also be obtained from a detailed examination of the diffusion p r o c e s s e s connected with permeation of low molecular weight substances into the gel of the e l a s t o - o s m o m e t e r . Owing to complications of both mathematical and practical nature this expectation has not been fulfilled for the g r e a t e r p a r t . It is possible, however, to determine the number average m o l e c u l a r weight in the case of penetration if the shape of the molecular
1 0
-weight distribution is known. The theoretical aspects of the penetration effect a r e presented in Chapter 2 and Appendix I.
Finally, it m u s t be mentioned that during the l a s t few y e a r s a new technique has been developed which shows e x t r e m e l y favourable p r o s p e c t s for a rapid and accurate d e termination of molecular weight distributions, viz. gel p e r -meation chromatography (GPC). This technique has been used in our investigation for the characterization of our polymer s a m p l e s . It is described in some detail in Chapt e r 4. A disadvanChaptage of GPC is Chapthe necessiChapty of a c a l i b -ration with samples of the same polymer as the polymer under investigation, but with a known distribution.
Chapter 4 also deals with the preparation of the various polymers used in our m e a s u r e m e n t s . Together with GPC, vapour phase o s m o m e t r y and membrane o s m o m e t r y were used for their c h a r a c t e r i z a t i o n .
THEORY
2 . 1 . EQUILIBRIUM THEORY FOR NON-PERMEATING SOLUTES.
2 , 1 , 1 , The chemical potential of the solvent in a swollen gel.
A rubber o r e l a s t o m e r is commonly considered to consist of long, randomly coiling polymer chains connected to one another by covalent c r o s s l i n k s . The whole network can be considered as one gigantic m a c r o m o l e c u l e . Such a m a t e r i a l , brought into contact with a solvent''', will not dissolve but will form a gel by absorption of solvent,
The solvent content of the gel in equilibrium swelling is determined by the condition that the chemical potential of the solvent inside the gel m u s t be equal to that outside the gel. The chemical potential inside the gel can be considered to contain two volume-dependent contributions. One is a con-sequence of the change in free energy due to mixing of the polymer network chains with solvent molecules, AF^, , The other contribution a r i s e s from the elastic o r conformational free energy change, A F ^ , which is a r e s u l t of the shift in population of the chain conformations upon swelling. Thus
^ aAF^ aAF,
^ i g = A^i + + ( 2 . 1 )
in which n-^ is the standard chemical potential of the solvent and Ni is the number of moles of solvent in the gel.
An expression for the free energy of mixing which has demonstrated its value in p r a c t i c e , is due to Flory^'' and Huggins ^^. In their derivation they make use of a liquid l a t -tice model in which a polymer solution is considered as a homogeneous distribution of polymer segments and solvent m o l e c u l e s . This will only be the case in r a t h e r c o n c e n t r a -ted solutions, viz, a few p e r c e n t s if the molecular weight is not too low. Most swollen gels fulfil this condition b e -cause the volume fraction of polymeric m a t e r i a l is seldom
• Diluent would be a more appropriate word. In elasto-osmometry, however, the polymer containing solvent phase around a swollen gel and not the gel itself is of chief in-terest. We will therefore use the word solvent ttiroughout.
1 2
-l e s s than 5 percent. In the F-lory-Huggins derivation the entropy and energy of mixing a r e treated separately. The e n -tropy of mixing between the solution and its pure compo-nents is calculated under the assumption that it is indepen-dent of the interaction energy between polymer segments and solvt,nt molecules. In o r d e r to calculate the energy of mixing, only first neighbour contacts are taken into account. This r e s t r i c t i o n is justified, because the forces between uncharged molecules d e c r e a s e rapidly with the distance of separation. F o r a solution of ATg moles of solute in Ni moles of solvent the r e s u l t a r r i v e d at, r e a d s :
A^n. =RT [^1 In 0, + N^ In 02 + xi2 (N, +p2 A^2 ) f^i 0^ (2. 2) In this equation R is the gas constant, T is the absolute t e m p e r a t u r e , ^^ and ^g ^^^ the volume fractions of solvent and solute, respectively and p2 is the m o l a r volume of the solute divided by the m o l a r volume of the solvent. In the original theory the Huggins interaction p a r a m e t e r X12 t which ranges between 0 (very good solvent) and 0, 5 (phase s e p a ration), is m e r e l y a function of the interaction energy b e -tween the two s p e c i e s , A m o r e extensive study of the mixing p r o c e s s shows that Xi2 should be regarded as a free energy p a r a m e t e r , X12 = a + b/T, because there may also be a contribution to the entropy of mixing from the interaction between the components of the solution. Besides, X12 often turns out to be concentration dependent. However, if X12 1^ considered as a s e m i e m p i r i c a l p a r a m e t e r , the F l o r y H u g -gins expression provides a satisfactory description of many polymer solutions.
The theory described above has originally been set up for solutions of polymers of finite chain length. Extension to swollen, crosslinked networks r e q u i r e s no essential a l t e r a -tions, however. We only have to take into account that in this case N2 = 1, because the network can be regarded as one gigantic m a c r o m o l e c u l e ,
Several theories have been proposed for the elastic free energy of e l a s t o m e r s ^2,13,14 _ j^^ g^n theories the e l a s t o m e r is considered to be mainly an entropy spring, the i n t e r m o l -ecular forces usually being considered s t r a i n independent. T r e a t i n g the network as an a s s e m b l y of independent Gaussian chains leads to the following result:
AF = RTv^\jA{Xl+X^^^ + xl - 3) - BlnX^X^x2 ^^^ ^^
where v^ is the number of moles of elastically effective network chains between c r o s s l i n k s ; X^, X and X^ a r e therelative deformations with r e s p e c t to the so-called reference state in the x, y and z directions, respectively. The r e f e r -ence state is defined as that state in which the chains a r e u n r e s t r a i n e d , that is in which the mean square end-to-end distance of the chains between crosslinks is the s a m e a s they would have in a solution of the same concentration. The factors A and B a r e here introduced to account for the fact that there a r e s e v e r a l v e r s i o n s of Equation (2. 3) in the l i t e r a t u r e . H e r m a n s ^^ finds A = B = 1, Flory^^ A = 1, B = I , J a m e s and Guth^"* A = ^, B = 0. The most recent experimental work-^^ regarding this question, offers strong support for A = \, B = | , which is really H e r m a n s ' r e s u l t , but with only half the number of effective network chains, a s also predicted by Duiser and Staverman-^^ .
When the e l a s t o m e r is allowed to swell in a solvent then the relative volume change is given by X^X X^ = qlq^; q, the inverse volume fraction of gel m a t e r i a l in the swollen s t a t e , is normally called the degree of swelling; q^ is the degree of swelling in the reference s t a t e .
In e l a s t o - o s m o m e t r y we consider the case of a swollen gel s t r i p stretched in one, say x, direction from a length L^ in the reference state to a length L. Then X = L/L^ and
Xy = X^ = qL^/q^L. Substitution of these A.-values into
Equation (2, 3) yields: A F = RTu
(I
\ o 2qL - 3 - B l n - ^ ^L (2,4)The chemical potential of the solvent inside the gel can be readily obtained now by differentiating Equations (2,2) and (2.4) with r e s p e c t to N-^ . Taking into account that the volume fraction of network m a t e r i a l , 0^, is equal to l/q and that 0-^ + ^g ~ 1» the r e s u l t i s :
*ig In {l-q~^) + q~^ +
x,^r
where Xig is the interaction p a r a m e t e r for the solvent-net-work interaction.
In deriving Equation (2,5) it must be noted that q = {V^ +
+ N^^Vi)/ Vj in which V^ is the volume of network m a t e r i a l
and Vi the m o l a r volume of the swelling agent. F o r conve-nience the quantity p = V, fi' v, has been introduced. It
1 4
-s t a n d -s f o r the effective c h a i n l e n g t h b e t w e e n c r o -s -s l i n k -s , a -s m e a s u r e d i n n u m b e r of s o l v e n t l a t t i c e s i t e s and h a s the a d v a n -t a g e of b e i n g i n d e p e n d e n -t of -the d i m e n s i o n s of -the gel s -t r i p . 2 . 1 . 2 . Effect of the deswelling by polymer solutions on the
retractive force.
T h e s t a t e of a gel s t r e t c h e d in one d i r e c t i o n in the p r e s -e n c -e of a s w -e l l i n g a g -e n t i s d -e t -e r m i n -e d by th-e v a r i a b l -e s T, V, L and N^a • An i n f i n i t e s i m a l c h a n g e of t h e f r e e e n e r g y of s u c h a gel i s given by dF = -S dT - p dV + fdL + ^^ dN^ ( 2 . 6 ) F r o m t h i s e q u a t i o n it i s a p p a r e n t that the r e t r a c t i v e f o r c e , ƒ, i s e q u a l to / 8 A F \ /L qL \
ƒ= - — -ARTvJ—-^] ( 2 . 7 )
^ 9 ^ ^ , v , N ^ g \ ^ o ^ o ^ y if we e m p l o y E q u a t i o n ( 2 . 4 ) , AFjj, b e i n g i n d e p e n d e n t of L. We a r e i n t e r e s t e d in the c h a n g e of r e t r a c t i v e f o r c e a s a function of the c o n c e n t r a t i o n of the p o l y m e r s o l u t i o n o u t s i d e the g e l , c, a t c o n s t a n t l e n g t h . T h i s c h a n g e in f o r c e a r i s e s f r o m the c h a n g e in q due to a r e d i s t r i b u t i o n of s o l v e n t a s the gel i s b r o u g h t f r o m the p u r e s o l v e n t into the p o l y m e r s o l u t i o n . F r o m E q u a t i o n (2. 7) it follows t h a t T h e v a r i a t i o n of q with c i s g o v e r n e d by the c o n d i t i o n t h a t in e q u i l i b r i u m the c h e m i c a l p o t e n t i a l s of the s o l v e n t i n s i d e and o u t s i d e the g e l a r e e q u a l ; lu = lu o r ig ^is l a c ; dc '9^ig\ _ ^ s o t h a t 'a^igX fdq\ da / g \ dM,3 / d c ir^] . ^ = 3 - ^ and t h u s [^] = ( 2 . 9 ) If we r e s t r i c t o u r s e l v e s to d i l u t e s o l u t i o n s , the c h e m i c a lpotential of the solvent in the solution outside the gel can be written as a virial s e r i e s in the concentration:
/u,3 = ^l-RT[{MJM^)c + A^c'' + . . . ] (2.10) Here the concentration c is e x p r e s s e d in t e r m s of the weight of polymer p e r weight of solvent. It should be noted that if the solution is sufficiently concentrated for the F l o r y -Huggins approach (Equation 2.2) to be valid, the second
v i r i a l coefiicient A2 is given by A2 - (i-Zi2 )(Pi/p2)^» 1"^ which Py and Pg a r e the densities of the solvent and the dry polymer, respectively,
Using Equations (2, 5) and (2. 10) for calculating {dq/dc)i ^'^^ introducing this r e s u l t into Equation (2,8) finally leads to
1 - ^ "
L J\qJ B{l-q-yp^q-'-2p^x,^q''(l-q-')
= KRT[^:r- + -c + . . . \ (2,11)
M„ Ml
J
where CTQ is the c r o s s s e c t i o n of the gel s t r i p in the r e f e r -ence s t a t e . F o r p r a c t i c a l purposes it is convenient to also have the integral form of Equation (2.11):
ƒ - ƒ = KRT [-^ c +-^ c^ + ...] (2.12) where f^ stands for the r e t r a c t i v e force when the gel s t r i p
is swollen in pure solvent.
2 . 2 . EQUILIBRIUM THEORY FOR PERMEATING SOLUTES
2 , 2 , 1 , Distribution of the components over the two phases, The equilibrium between a swollen gel and a polymer s o -lution, the solute molecules of which a r e able to penetrate into the gel, r e q u i r e s that the chemical potentials of all s o -lution components a r e equal inside and outside the gel. In
1 6 -o r d e r t-o c a l c u l a t e t h e s e c h e m i c a l p -o t e n t i a l s we a p p l y the F l o r y - H u g g i n s a p p r o x i m a t i o n to b o t h p h a s e s . T h e f r e e e n e r g y of m i x i n g of a s o l u t i o n of n c o m p o n e n t s i s g e n e r a l l y given b y ' ' AFn ^^m RT E N, n .1=1 ' 1—J- 1 » J ( 2 , 1 3 ) w h e r e the s u m m a t i o n o v e r i and j i s to be t a k e n o v e r a l l p a i r s j > i. F r o m o u r p o i n t of view the m o s t i n t e r e s t i n g c a s e i s t h a t of a m i x t u r e of p o l y m e r s of the s a m e h o m o -l o g o u s s e r i e s , c h e m i c a -l -l y i d e n t i c a -l and d i f f e r i n g in c h a i n l e n g t h o n l y . F o r s u c h a m i x t u r e the i n t e r a c t i o n p a r a m e t e r f o r a l l s o l v e n t - p o l y m e r p a i r s i s e q u a l ( x n = Xip) and the i n t e r a c t i o n p a r a m e t e r f o r a l l p o l y m e r - p o l y m e r p a i r s i s e q u a l to z e r o . T h e n the c h e m i c a l p o t e n t i a l s of the s o l v e n t (1) and a p o l y m e r c o m p o n e n t (k) in the s o l u t i o n a r e given by A* is M? + RT In (1 - ) E (1 -—)(Z(. + -'^lp(f ^ i s ) ' A', ks /u^ + RT In ^w, - ^u -1) + P. + A<Xip(l - E ^ . J ' (1 -—)0. ( 2 , 1 4 ) ( 2 , 1 5 ) in w h i c h the s u m m a t i o n s a r e to be t a k e n o v e r a l l p o l y m e r c o m p o n e n t s i{i > 1),
T h e f r e e e n e r g y of m i x i n g of the gel i s a function of the a m o u n t s of s o l v e n t , n e t w o r k m a t e r i a l and a l l d i s s o l v e d p o l y -m e r c o -m p o n e n t s . A s p o i n t e d out in S e c t i o n 2, 1, 1 the t o t a l f r e e e n e r g y , and t h u s the c h e m i c a l p o t e n t i a l , a l s o c o n t a i n s a c o n f o r m a t i o n a l f r e e e n e r g y c o n t r i b u t i o n . By d i f f e r e n t i a t i o n w i t h r e s p e c t to the n u m b e r of m o l e s of the c o m p o n e n t in q u e s t i o n we o b t a i n f o r the c h e m i c a l p o t e n t i a l s i n s i d e the g e l : ^^ = < + RT ig 1 ln{l-q'^ - L ^. ) + q'^ + E (1 -^)0 + i ig i Pi ig + X, q''^ + X, (E 0. f + (y + X, Ig IP i ig Ig IP V, ZdAF RTV^ \ dq X )q'^ L ^. + pg i ig ( 2 , 1 6 ) ^ g = ^ + « r , - 2 In ^,g - iP.-l) ^ P,q-' + / > , f (1 -j-)0,^^ ,2 P k X i g < 7 - ^ + / > , X i p ( l - E 0ig) ^
py Vy aAFg
^ ^ ^ g ' ^ ^ - i p - ^ g ^ ^ ^ ' - f ^ i g ) " : R r i r ^
(2,17) Xpg is the interaction p a r a m e t e r of the interaction between the dissolved polymer molecules and the network m a t e r i a l , In the derivation of Equations (2, 16) and (2. 17) we have used the definitionq = (V^ +,Lp^NiV^)/V^ (2,18)
The equilibrium conditions ^il„ - A<1S and /U^g = A'ks offer the possibility of e x p r e s s i n g ^^ in t e r m s of ^^^ and E ^jj . To this end we expand the logarithmic t e r m s of A^ig and
Hx% in powers of E 0jg and E ^jj , Then we s u b s t r a c t Pi^l^ia and p^^y^ from u^^ and ^^l•.
% - ^ k ' ^ l g = '^ks - ' ^ k ' ^ i s • ( 2 - 1 9 )
and after r e a r r a n g e m e n t we find:
ln(^kg/^ks) = /'k l n ( l - ^ " ' ) + P k ^ i g + X,p - 5(pg)9-' +
+ higher terms in E 0. (2.20)
i ^^
It should be s t r e s s e d that an explicit knowledge of the con-formational part of the chemical potential i s not required for the derivation of Equation (2, 20),
Even at concentrations of s e v e r a l p e r c e n t s the higher t e r m s in the volume fraction of dissolved polymer a r e neg-ligible, so that in the concentration range normally employed the ratio between the volume fractions of a polymer c o m -ponent of a c e r t a i n chain length, inside and outside the gel is a constant for a given gel. Although the derivation is based on the Flory-Huggins expression for the free energy of mixing part, it is conceivable and even probable that this result has a much g r e a t e r validity. This i s , however, not easy to prove for the general c a s e ,
The ratio between the volume fractions of solvent is given in first approximation by
01 1 - ^-1 - E 0i_ ,
-18-2 . -18-2 . -18-2 . Influence of permeation on the retractive force
in-crement.
F o r permeating solutes the change in r e t r a c t i v e force of a gel s t r i p as a function of the concentration of the s u r -rounding polymer solution is again given by Equation (2.8). In the p r e s e n t calculation of {dq/dc)^ from the condition A'lg = /"is we have to take into consideration that now fji„ is not only a function of q but also of the volume fractions of the polymer components in the gel. Henceforward, for convenience, we shall denote E 0= by $ and L 0^^ b y $ s ' Hence i 8 8 i d^ Is d $ s and thus M.\ - ! i ('9^\ - ! i ( d ^ l s / d $ s ) - ( 9 ^ l g / 9 i g ) L . q ( % / 9 ^ s ) ( 2 . 2 2 )
^ck= p,[^^J,-Pi (a.,g/a,),.$ ('-''^
owhere Pp is the density of the solute and c the total weight of solute p e r unit weight of solvent.
In e l a s t o o s m o m e t r y we a r e chiefly interested in dilute s o -lutions, i, e, $s~*"0 and thus $g—"O, Introduction of these limiting conditions simplifies the mathematical t r e a t m e n t of the problem considerably. F o r a dilute solution the chemical potential is in first approximation linearly dependent on the concentration (see Equation (2. 10)) and thus on the volume fraction of polymer: M. . ^is= ^1 -^^W' - < -^^h ^s ( 2 . 2 4 ) JWn p^ SO t h a t
(Sr)
--f
C O P npjj is the number average chain length of the solute as e x -p r e s s e d in number of solvent molecules.
As a consequence of the diffusion into the gel each polymer component i will r e a c h there a certain volume fraction 0^^. This affects the chemical potential of the solvent, again in first approximation, by an amount - RT 0^„ /p^ . Summation over all species i yields for the chemical potential A<ig (com-pare also Equation (2,5)):
^ ig ^l + RT ln{l-q-^) + q-'' + x,g ? " ' +
%M-<V-«^-)-?i^'«
F r o m t h i s e q u a t i o n we o b t a i n : /dl^ ig ,a$oRT
J - L ^ 0,
and L.q a$g i Pi 'g IS dq = RT L . $ „--'"-^\'"*t^«"
(2,26) (2,27) (2.28) By i n t r o d u c i n g E q u a t i o n s ( 2 . 25), ( 2 . 27) and ( 2 . 28) i n t o E q u a -t i o n s ( 2 . 2 3 ) and ( 2 . 8 ) w e find: dc = KRTL,c-»o PpMi 1 ^ as, i p.
E ^ ^ J (2.29)
w h e r e K h a s the s a m e m e a n i n g a s in E q u a t i o n ( 2 . 1 1 ) . In S e c t i o n 2. 2, 1 we d e r i v e d an e x p r e s s i o n f o r the r e l a t i o n b e t w e e n the v o l u m e f r a c t i o n s of p o l y m e r i n s i d e and o u t s i d e the gel s t r i p . A c c o r d i n g to E q u a t i o n ( 2 . 20) the r a t i o b e t w e e n t h e s e two i s a c o n s t a n t w h i c h i s e q u a l t o : is ( 1 - ^ - ^ ) e x p | ( x i g + Xip - Xpg
)'i
Pi (2.30) T h i s , t o g e t h e r w i t h the c i r c u m s t a n c e t h a t f o r a given p o l y -m e r s a -m p l e d $ s / $ s = d0is/0is , -m e a n s t h a t i Pi -^ig T h u s we finally a r r i v e at: 9 ? ^ 0i,/a$3 = ? ^ ^ i g / ^ s dc KRT L,c-»o V^ 1 Pi - ^ \ Ip: ^ 0i> E 0. ( 2 . 3 1 ) (2.32)2 0 -T h i s e x p r e s s i o n h a s l i t t l e p r a c t i c a l u t i l i t y , h o w e v e r , e v e n if the s a m p l e c o n s i s t s of p o l y m e r m o l e c u l e s of the s a m e c h a i n l e n g t h . Of c o u r s e , in t h i s l i m i t i n g c a s e , e c a n a l -w a y s be o b t a i n e d by a c a l i b r a t i o n -with s a m p l e s of kno-wn m o l e c u l a r w e i g h t . T h e c h a i n l e n g t h of an unknown s a m p l e c a n t h e n be found by m e a n s of a n u m e r i c a l c a l c u l a t i o n . H o w e v e r , t h i s p r o c e d u r e i s not a p p l i c a b l e if the s a m p l e p o s s e s s e s a m o l e c u l a r w e i g h t d i s t r i b u t i o n , a s i s n o r m a l l y the c a s e . T h e only p o s s i b i l i t y f o r m a k i n g E q u a t i o n ( 2 . 32) a c c e s s i b l e f o r t h i s g e n e r a l c a s e l i e s in a s e r i e s e x p a n s i o n of ePi. T h i s i s p e r m i t t e d o n l y if pj In e i s s m a l l c o m p a r e d to one and t h i s will be the c a s e if the d e g r e e of s w e l l i n g i s s u f f i c i e n t l y high a n d / o r (Xig + Xin - Xpg) i s l a r g e . T h e l a t t e r c o n d i t i o n will be fulfilled if the s w e l l i n g a g e n t i s a r e l a t i v e l y p o o r s o l v e n t f o r both the n e t w o r k p o l y m e r and the d i s s o l v e d p o l y m e r . At f i r s t s i g h t t h i s s e e m s c o n t r a d i c t o r y to the r e q u i r e m e n t of a high d e g r e e of s w e l l i n g , but it i s i n d e e d p o s s i b l e to h a v e h i g h d e g r e e s of s w e l l i n g e v e n in p o o r s o l v e n t s , v i z . if the c r o s s l i n k i n g h a s t a k e n p l a c e in s o l u t i o n . I n t r o d u c t i o n of the s e r i e s e x p a n s i o n i n t o E q u a t i o n ( 2 . 32) finally l e a d s to a ac L,c-»o = KRT KRT Pi Pi PpM, -In e -In e - 1
(In ef p^
-P i ( l n ef _ 2 p p M , ^ w -. -. -. -. ; — (2.33) in w h i c h M^ i s the w e i g h t a v e r a g e m o l e c u l a r w e i g h t of the d i s s o l v e d p o l y m e r , defined a s : E iViMf ) f(M) M^ d M ( 2 . 3 4 ) M , E N M i i I i i{M) M dM It f r e q u e n t l y o c c u r s t h a t only a p a r t of the p o l y m e r s a m -p l e , i. e. the low m o l e c u l a r w e i g h t t a i l , i s a b l e to -p e n e t r a t e i n t o the g e l . L e t u s s u p p o s e t h a t the c r i t i c a l c h a i n l e n g t h f o r p e n e t r a t i o n i s p ^ . T h e n the s u m m a t i o n in E q u a t i o n ( 2 . 32) only c o n t a i n s c o n t r i b u t i o n s of the c o m p o n e n t s of c h a i n l e n g t h s h o r t e r t h a n 6 : ^ c 'Bf\ ac L,c-»o KRT P ^ 1 ^ P 1 1=2 Pi n E 0. i=2 "^isC
/ (2.35)where n is the total number of components.
If we proceed in the same way as in the derivation of Equa-tion (2.25), we obtain the following result:
M.\ = KRT 4
-aC/L,c->o M Pi(ln ef(K).
Pi In e + 7',— ++ 2 Z M r ^^«^^ ^
}
PpMi (2.36)In this equation ip is the weight fraction polymer which is able to penetrate: M_ c E i=2 n E 1=2 ^is
0.
ff(M)
M dMfuM)
1 * (2.37) M dM(Mjj)j, and (M^)^, a r e the number and weight average m o l e c -u l a r weights, respectively, of the penetrating part of the solute: .^c M^
J f ( M ) M dM jf(M) M^ dM
( ^ n l = °-M: and (MJ^ = °-^ (2.38)
Mc |f(M) dM O' (f(M) o J M dMIt is apparent from Equation (2. 36) that a correction for the penetration of the low molecular weight tail of a given polymer sample is only possible if the type of molecular weight distribution and e a r e both known. In o r d e r to ob-tain e a calibration with a fully penetrating solute of known molecular weight and a known, narrow molecular weight distribution will be n e c e s s a r y .
In one r e s p e c t the elasto-osmotic method for evaluation of the molecular weight of a partially penetrating solute, as described in this Section, differs considerably from the analogous method in membrane o s m o m e t r y . In e l a s t o o s m o m e t r y the equilibrium state is considered, whereas in m e m -brane o s m o m e t r y the equilibrium state offers no information, because then the penetrating part of the solute does not contribute to the osmotic p r e s s u r e . In the l a t t e r case the p r o b
2 2
-lem is solved by a suitable extrapolation to zero t i m e . The extrapolated osmotic p r e s s u r e is proportional to the theoret-ical osmotic p r e s s u r e , i. e. the p r e s s u r e if the membrane were ideally s e m i p e r m e a b l e . The proportionality constant, which is normally called the reflection coefficient-^^, is a very complex function of among other things the m o l e c -u l a r weight. As a r e s -u l t a direct association with a simple average molecular weight is virtually impossible in most c a s e s 13.
2 . 3 . DIFFUSION PHENOMENA IN ELASTO-OSMOMETRY.
In Section 2. 2 we studied the influence of solute p e n e t r a -tion on the equilibrium degree of swelling. The establishment of equilibrium takes a c e r t a i n amount of time, which is d e -termined by the diffusion r a t e s and the concentrations of the components of the solute. In view of the fact that the diffusion coefficients of the various solute components a r e directly related to their molecular weights, we might expect a detailed study of the deswelling as a function of time to furnish some information about the m o l e c u l a r weight distribution. Such a t r e a t m e n t would at first sight appear to be analogous to the t r e a t m e n t of penetration data in membrane o s m o m e t r y ^'^. The problem of i n t e r e s t h e r e is a special case of the problem of diffusion out of a plate into an infinite medium o r vice v e r s a . This is so because the volume of the gel s t r i p is very small compared to the amount of solution around the gel and because the diffusion flows through the sides of the s t r i p can be neglected since the width is much l a r g e r than the thickness.
In the solution a s well a s the gel phase the diffusion p r o c e s s is governed by F i c k ' s second law, which can be written in our case a s :
a^^ig aVi„ B0. aVi,
We s t a r t from a gel, swollen in the pure solvent. This means that the initial volume fraction of solvent is equal ^o (0^ )^ = = 1 - q^^, in which q-^ is the degree of swelling of the gel in pure solvent. The initial volume fractions of the polymer components in the gel, (^ig)o. are equal to z e r o . At z e r o time the gel is brought into contact with a polymer solution
n
the initial volume fractions of solvent and solute components n
a r e ( ^ i J ^ = 1 - .E^ (0is)o and (0is)o, respectively. After the deswelling equilibrium has been reached, the volume f r a c -tions a r e fixed by the equilibrium condi-tions described in Section 2 . 2 . 1 :
(^ig)t««.= 0-1(^13)0 and (0ig)t=-= a'i(0is)o, in which a 1 = 1 - q'^ and a 1 = e Pi.
In o r d e r to solve the diffusion equations we also need the conditions for the boundary between the gel and the solution. One boundary condition is that the diffusion flow out of the gel must be equal to the diffusion flow into the solution. In the second place the relation between the volume fractions at the boundary m u s t be known. It s e e m s logical to a s s u m e that in this region the equilibrium conditions, described in Section 2 . 2 . 1 . a r e maintained from the first beginning of the diffusion p r o c e s s onward. As a r e s u l t the ratio between the volume fractions at the boundary is a constant throughout:
(«>ig/<^is)x=td = "'I and (</>ig/«>is)x=±d = 0-1
if the thickness of the gel s t r i p is equal to 2d.
The diffusion equations (2. 39) can be solved by using Laplace t r a n s f o r m a t i o n s . A full t r e a t m e n t of the procedure is given in Appendix I. One obtains the volume fractions of solvent and polymer inside the gel a s a function of the d i s tance to the centre of the s t r i p , x, and t i m e , t. The d i s -tribution of the solvent and the polymer components at a given time is not uniform over the thickness of the s t r i p . As a consequence the degree of swelling and thus the r e -t r a c -t i v e force v a r y wi-th -the dis-tance -to -the c e n -t r e . Thus, the s t r i p should be considered a s a continuous a s s e m b l y of .parallel springs of varying modulus all held at the same extension r a t e . Such an a s s e m b l y is equivalent to a spring with a c r o s s s e c t i o n which is equal to the sum of the c r o s s -sections of the component springs and with a modulus which is the average of all moduli. F o r the calculation of the a v erage degree of swelling we need the average volume f r a c -tions of the solvent and the polymer components. These fol-low directly from Equations (A. 17) and (A. 18) in Appendix I:
+d
<^^> -h
i ^g ^^ =
2 4 -(^ig )o ^ 2 V ^ 7 P ; { l - ^ l ( 0 l s ) o / ( ^ g ) o } ^ 1 - Vjf ^ 1 Ig Is . /7-> \ "^ r ayo -V£)j \ wrf . . (n+l)d ' .^ lerfc —^^ - lerfc ^ '—
VD;7
^ ^ ^
« / ^ I g + ^ ^ l s ( 2 . 4 0 ) ig 1 r*^ ^. dx = ig a V D . D. ig IS 1 Ig ISa?s~„ v5r\"
Ig ' ' Ig IS Vf ierfc nd ierfc {n+l)d (2.41) where Dig and Dis a r e the diffusion coefficients of the s o l vent molecules in the gel phase and the solution phase, r e s -pectively, and Dig and Dis those of a polymer component i. The complete solutions provide no possibility of obtaining the diffusion coefficients from the experimental c u r v e s . P o s -sibly, in some c a s e s , a way out is afforded by retainingonly the f i r s t t e r m of the ierfc-summation, which is equal
to ll'sfW. F o r not too large diffusion coefficients this may be allowed at small l v a l u e s .
If this approxinaation is valid. Equations (2.40) and (2.41) become: <>^Xg> = ( ^ l g ) o 2 ^ ^ V ^ {1 - ^l(^ls)o/(^lg)o} rf^i « 1 ^ + ^ and <^ig> = (^is)o ^ 7 F « i ^ i g ^ i s a VD. + VD. 1 IE i:
^fi
( 2 . 4 2 ) ( 2 . 4 3 )The average r e c i p r o c a l degree of swelling depends on these average volume fractions through:
<q
- ' - <^^i
E <0. >i=2 ig
The r e t r a c t i v e force at constant length (Equation (2.7)), however, contains <q>. Owing to the fact that the variations of q over the thickness of the s t r i p a r e l e s s than 10 percent, the e r r o r introduced by writing <q> = <q'^>'^ is negligible. Inserting Equations (2.42), (2.43) and (2.44) into Equation (2. 7) we get: ƒ = ARTu J e r 2 ^ > ^ 1 - )c + dy/iT i0i. )o [(^ig )o - Q'l(^ls)o] ' I g ^o A g -Dis
r^
a^/D. D.
( 2 . 4 5 )Because we r e s t r i c t ourselves to small ?-values and dilute solutions this equation, by means of a s e r i e s expansion of the denominator, can be approximated by
ƒ = / o -^ 2ARTv^L,q^ q^ L^ d-^ (0i. )o a-^D. D 1 I g IS i=2 o v 5 ~ + y/D' 1 Ig IS
{(^lg)o
vf
J a.-JD7 +VD, ^ i - ^ i g ( 2 , 4 6 )Unfortunately, even this simple, approximate expression offers no possibility of obtaining the diffusion coefficients of the p e rme a tin g components. This is in c o n t r a s t to a c o r responding expression in m e m b r a n e o s m o m e t r y for the o s -motic p r e s s u r e as a function of time, as derived by Hoff-mann and Unbehend^. In m e m b r a n e osmometry the osmotic p r e s s u r e depends on the concentration difference between the solutions on both sides of the m e m b r a n e . In e l a s t o o s m o m e t r y , however, the volume fractions inside the gel d e -termine the r e t r a c t i v e force. As a result the slope of the curve, obtained by plotting (f-fo) v e r s u s VT, contains both the diffusion coefficients and the volume fractions of the components. T h e r e is no means of separating these two contributions. Another, m o r e p r a c t i c a l , difficulty has already
been mentioned before, i. e. Equation (2.46) is valid only at small t. Anticipating the experimental r e s u l t s described
2 6
-in Chapter 5, it turns out that for the case of moderately swollen gels this condition is by no means fulfilled.
DESCRIPTION AND OPERATION OF THE ELASTO-OSMO-METER
3 , 1 , DESIGN OF THE ELASTO-OSMOMETER.
In designing an e l a s t o - o s m o m e t e r there a r e two main points to consider. In the first place we d e s i r e the highest possible sensitivity and in the second place the construction should be such that the length of the e l a s t o m e r s t r i p is kept a s constant a s possible. The l a t t e r condition is im-portant because the equations describing the elasto-osmotic effect (Chapter 2) are derived under the assumption of con-stant length. All force m e a s u r i n g devices, however, in-herently need a small displacement for measuring the exerted force. When this displacement is too large the length of the s t r i p and thus its r e t r a c t i v e force diminishes to a con-siderable extent (see Section 3 , 3 , 1 ) . F o r s t r a i n gauges, a s used in the instruments of Yamada^ and M i e r a s ^ , this ef-fect can lead to reductions in signal up to 20%.
A much b e t t e r instrument for measuring forces at constant length is an inductive t r a n s d u c e r . Such a force indicator displays a much s m a l l e r displacement than a s t r a i n gauge, F o r our instrument (Type Q l / l O - 5 0 , Hottinger, Darmstadt, Germany) this displacement amounts to only 30 m i c r o n s at a maximum load of 10 g r a m s .
Another disadvantage of the e l a s t o - o s m o m e t e r s of Yamada and M i e r a s is that the gel s t r i p is directly connected to the s t r a i n gauge. This a rr a ng e m e n t prohibits the use of very sensitive m e a s u r i n g devices because these have a small maximum load. The absolute force level is s o m e t i m e s fairly high because, in o r d e r to avoid slack and other i r r e v e r s i b l e non-elastic behaviour, the s t r i p has to be given a degree of elongation of 5 to 10 percent. F o r moderately swollen s t r i p s this can lead to forces of 10 to 40 g r a m s .
F o r this r e a s o n an e l a s t o - o s m o m e t e r was developed in which the gel s t r i p is connected to one end of a balance, whereas the inductive force pick up is attached to the other a r m of the balance.
Figure 1 shows a photograph of the instrument, a sche-matic diagram can be found in Figure 2. In this diagram the balance a r m is marked with the symbol A, the t r a n s d u c e r and the gel s t r i p with B and E, respectively,
I IN}
0 3 I
Figure 2. Schematic diagram of the elasto-osmometer. A = balance; B = inductive force pick up; C = thermostated measuring cell; D = thermostated solution vessel; E " swollen gel strip; F = micrometer, connected to lower clamp of strip; G = adjustment device for constant immersion level; K l - 6 " metal stopcocks for introduction, replacing and draining of polymer solutions.
In o r d e r to keep friction as low as possible the knife-edges of the balance and their bearings are made of agate. By placing a counterweight at the same side as the transducer the initial force due to the stretching can be compensated. A second advantage of a balance-type design is that an ad-ditional amplification can be obtained by moving the point of attachment of the force pick up c l o s e r to the centre of the balance.
The output of the t r a n s d u c e r is supplied to an amplifier (Type KWS/lI-50, Hottinger), connected to a 4 Volt r e c o r d e r (Goerz Servogor, type RE 511, Vienna, Austria).
A condition for attaining a high sensitivity is that the in-strument should be well isolated from mechanical vibrations. For example, in our case m e a s u r i n g at night-times, r a t h e r than day-time, reduced the noise level about tenfold.
The gel s t r i p is held between two clamps of which the upper one is connected to the balance a r m by m e a n s of an Invar rod. The lower clamp is attached to the base plate of the balance, also by means of an Invar rod. A m i c r o m e t e r
3 0
-(F in Figure 2) moves this lower clamp in o r d e r to adjust the length of the s t r i p . The s t r i p , together with its clamps, is i m m e r s e d in a swelling agent or a polymer solution in the m e a s u r i n g cell C. This v e s s e l has a volume of about 60 ml and is thermostated by a water jacket to exclude t e m p e r a t u r e influences.
The whole instrument is enclosed in a perspex box to avoid disturbance by air draughts.
3 , 2 , OPERATIC^ PROCEDURE.
At the beginning of an experiment the m e a s u r i n g cell is usually filled with pure solvent. After establishment of the swelling equilibrium the solvent is replaced by a polymer solution of known concentration from the storage vessel D (see Figure 2) by applying a small nitrogen p r e s s u r e and operating the metal stopcocks Kl, K3, K5 and K4, This replacing p r o c e s s must not be c a r r i e d out too fast in o r d e r to avoid too much mixing of the original and the new solution, A reasonable .time for this p r o c e s s is 10 minutes. During the replacing p r o c e s s the new swelling^'ëqiltttbTttmi begins to establish itself. In most c a s e s it is attained in 15 to 20 minutes in the case of non-permeating solutes and in about the same time or a little longer for permeating solutes. If the solution in the m e a s u r i n g cell has to be replaced by a solution of lower density, this latter solution is introduced via stopcocks Kl and K2, whilst the original solution is drained off by way of stopcocks K4 and K6.
In view of the buoyancy effect on the Invar rod of the upper clamp it is important to adjust the liquid level al-ways at p r e c i s e l y the same height. This is achieved by first lowering the liquid level, after the replacement, to below the end of a narrow steel tube G (Figure 2) and after that making it r i s e again slowly until the liquid surface takes hold of the end of the tube.
3 . 3 , SENSITIVITY OF THE METHOD AND SOURCES OE ERROR.
The accuracy and sensitivity of the elasto-osmotic method is affected by three sources of e r r o r s . In the first place the construction of the balance and the inductive t r a n s d u c e r may be the origin of imperfections such a s irreproducebility and non-linearity. Secondly, the measuring cell, together with the gel strip, the clamps, e t c . can be a possible source of e r r o r . Finally, e r r o r s may originate from the procedure
for replacement of the solutions,
3 , 3 , 1 . The balance-transducer-recorder combination. In o r d e r to obtain information concerning the sensitivity of the mechanical and e l e c t r i c a l part, the instrument was calibrated by means of known weights. The a r r a n g e m e n t during this calibration was such that the distance between the point of attachment of the pick up and the centre of the balance was 0,6 times the distance between the point of attachment of the s t r i p and the c e n t r e . In section 3,1 the possibility of an amplification of the force exerted on the transducer, by moving the point of attachment c l o s e r to the c e n t r e , was mentioned. However, due to an increasing in-fluence of mechanical vibrations the overall sensitivity does not increase beyond a certain point. In our instrument the value 0,6 can be regarded a s an optimum value. Moreover, the d e c r e a s e in r e t r a c t i v e force originating from the fact that the length of the gel s t r i p d e c r e a s e s a s a r e s u l t of the displacement of the transducer, is the g r e a t e r the m o r e the point of attachment approaches the pivot, as pointed out at the end of this Section,
A calibration was c a r r i e d out on the 1 /J, 2 yu, 5 yu and 10 yu scale of the amplifier. These displacement values correspond with forces of 0,13, 0,26, 0,65 and 1.30 g r a m s , respectively, In all r a n g e s the deviations from non-linearity a r e l e s s than 0,4%. The reproducebilities for the two most sensitive r a n g e s , which are of most interest from an experimental point of view were both found to be about 1 mg. This sen-sitivity is l e s s than that of an analytical balance (0,1 mg), The r e a s o n is that, although the assembly r e s e m b l e s an analytical balance, its equilibrium position depends on the t r a n s d u c e r r a t h e r than on the weight of the beam and the location of the centre of gravity with r e s p e c t to the pivot,
The effect of the extensibility of the t r a n s d u c e r will now be subjected to a c l o s e r inspection (see Figure 3).
The equations describing the change in r e t r a c t i v e force as a function of the concentration of the polymer solution outside the gel a r e derived under the assumption that the length of the strip, L, is equal before and after the de-swelling, In that case the t r a n s d u c e r m e a s u r e s a force k which is related to the r e t r a c t i v e force of the s t r i p , ƒ, by
k = {a/b)f . (3.1)
3 2 -f ( -f ) V V gel strip /////////// transducer
Figure 3 . Effect of the extensibility of the transducer,
of the s t r i p in reality changes to L'. The corresponding r e t r a c t i v e force becomes ƒ ' and the transducer m e a s u r e s a force ^ ' :
k' = {a/b)f'
( 3 . 2 )Since we want to calculate the deviation resulting from the displacement effect in the case of a given solute concentration and thus a given chemical potential of the solvent, we a r e interested in
Ak= k - k > =a (d£
b \dL hj. 'ig A L ( 3 . 3 )
where AL = L - L'.
By differentiating/, as given by Equation (2. 7), with respect to L, we get:
(lt).ig = ^^^^^
2^1-0 ? o ^ ' ^ o ^ dq 2 \dL J yU Ig ( 3 . 4 )Noting that
I I ) ( ^ ] ( ^ ) = - l a n d thus ( f t ) --^-^^1^1^
dL}^.,^\d^,Jq\ dq JL
\BL)U,^{d^^Jdq)^
(3.5) we obtain, if we employ Equation (2,5) for the chemical potential of the solvent:
lr)Mig=^^^^e
2^io Kq^L^
o^PyL-( 3 , 6 )
where K has the same meaning as in Equation (2,11).
k^ is related to the displacement of the t r a n s d u c e r which
in our case is equal to {b/a)AL, through 6, the displacement of the t r a n s d u c e r per unit force:
a 6 ( 3 . 7 )
6 is a constant for a given t r a n s d u c e r ,
Combining Equations (3,3), (3,6) and (3.7) we finally a r r i v e at:
#=^ieT...mv' . ü i
-t'O K Qo ( 3 , 8 )Owing to the fact that the deviation is proportional to the force itself, the slope of the experimental f-c curve will be too small and consequently the molecular weight calculated from the slope at its limit of infinite dilution will be too high. F o r a poly(dimethyl siloxane) s t r i p an estimate has been obtained by substituting a i/g -value obtained from the swollen modulus and assuming A and q^ to be unity. It turns out that A^/^' = 2.48x10"^ (a/&)2 in this c a s e . This means that for ö / a = 0.6 the deviation is only 1.5%, but for
bla = 0.1 the deviation amounts to 25%.
It a p p e a r s from Equation (3.8) that the deviation i n c r e a s e s with increasing v^ and thus with increasing modulus. T h e r e -fore a c o r r e c t i o n for the effect will be n e c e s s a r y in general. Fortunately, it will be obvious from the foregoing that we can get round the difficulty by a calibration of the instrument
3 4
-with solutions of known activity.
3 . 3 . 2 , Errors due to variations in buoyancy.
The buoyancy of the gel strip, the upper clamp and the rod which connects this clamp to the balance can be affected by two effects. One is the deviation of the liquid level from its proper height, influencing the rod only, the other is found in the change in density of the solution around the gel upon its replacement by a solution of higher concen-tration.
The deviation brought about by variations in the liquid level can easily be calculated from the d i a m e t e r of the In-var rod which forms the connection between the upper clamp and the balance a r m . Because in our instrument the dianneter is 2 mm, a variation of the liquid level of 1 mm gives a difference in buoyancy of 3 mg, if the density of the solution in the measuring cell is 1 g/cm"^. By means of the adjust-ment device, described in Section 3 . 1 , the liquid level can be adjusted to within 0,2 m m . This reduces the e r r o r to about 0.6 mg which is well within the sensitivity of the in-s t r u m e n t .
The difference in density between a polymer solution and the solvent may exercise m o r e influence. The volume of the gel strip, the upper clamp and that part of the rod which is i m m e r s e d in the liquid amounts to 1.75 cm^, For a 1% solution of polystyrene in toluene of which the density dif-ference with regard to the solvent is 0.0017 g/cm^, the buoyancy difference is equal to 3 mg. This example shows that, generally speaking, this effect is not negligible. However, in molecular weight determinations one usually performs m e a s u r e m e n t s on a s e r i e s of solutions of different concen-trations in order to eliminate non-ideality by extrapolating to infinite dilution. This extrapolation also eliminates the density effect which is in first approximation linearly proportional to the concentration. On the other hand, if one is interested in the second v i r i a l coefficient of the polymer solution (A2 in Equation (2,10)), a correction will be n e c e s -s a r y ,
3.3.3. Reliability of the replacement procedure.
During the introduction of a solution from the storage v e s s e l into the measuring cell, concentration changes may occur by mixing with the original solution in the cell. In o r d e r to get an impression about the magnitude of the dif-ferences in concentration between the storage vessel and the
m e a s u r i n g cell, the replacement procedure was c a r r i e d out with aqueous solutions of ammonium thiocyanate. The ex-p e r i m e n t s were ex-performed using two solutions, differing 0,0018 g/cm^ in density. In one experiment the dilute solution was brought into the m e a s u r i n g cell and subsequently replaced by the more concentrated solution in about 10 m i n u t e s . F r o m the overflow of the cell 10 ml aliquots were col-lected, in which the concentrations were determined by volumetric titration with a standard silver nitrate solution. In a second experiment the m e a s u r i n g cell contained the m o r e concentrated solution, w h e r e a s the dilute solution was introduced at the top of the cell, as described in Section 3 , 2 ,
In both experiments a boundary layer of not m o r e than about 5 ml volume was observed. The concentrations in all aliquots, except the one containing the boundary layer, agreed with those of the original solutions to within 1%. In view of the fact that the density difference of these test solutions is comparable with that of the dilute polymer solutions normally employed in e l a s t o - o s m o m e t r y , we may assume that in the latter case the concentration r e m a i n s constant as well.
3 . 4 . THE GEL STRIPS.
F o r use in e l a s t o - o s m o m e t r y an overwhelming variety of possible e l a s t o m e r s is at our disposal. In principle, the only demand we make on the e l a s t o m e r s is that they exhibit a c e r t a i n degree of swelling in a suitable swelling agent, Of course, the gel s t r i p s must also be manageable in this respect that their consistency allows them to be mounted in-to the clamps and in-to be stretched in-to about 10%,
However, in o r d e r to obtain a high sensitivity it will be profitable to aim our r e s e a r c h at s t r i p s with a high K-valae (Equation (2,11)). An a p r i o r i prediction of K r e q u i r e s know-ledge of all network p a r a m e t e r s , some of which are not so easy to obtain. Besides, in the derivation of Equation (2,11) the e l a s t o m e r was solely regarded as an entropy spring. Energy effects may also play a role, however. This altogether makes the s e a r c h to a certain extent a question of trial and e r r o r .
In o r d e r to get an impression of the p r o p e r t i e s which govern the efficiency, s t r i p s of varying chemical composition and degree of crosslinking have been used in our investiga-tions. Measurements have been c a r r i e d out with s t r i p s of poly(dimethyl siloxane) (General E l e c t r i c C o . , Schenectady, N. Y., U,S, A.), poly(cis-isoprene), poly(cis-butadiene),
3 6
-poly(butadiene-co-styrene), containing 23% styrene and poly-(butadiene-co-acrylonitrile), containing 36% acrylonitrile (courtesy of the Shell P l a s t i c s Laboratory, Delft, The Netherlands),
All s t r i p s were crosslinked in the dry s t a t e . This has the advantage that q^ has a low value, because the chains in such a dry crosslinked polymer should have conformations which a r e close to those in the non-crosslinked, unrestrained s t a t e . As the volume fraction of network m a t e r i a l in the dry state is close to 1, q^tsa I. According to Equation (2.11)
K is l a r g e s t when q^^ has a minimal value,
Before crosslinking the s t r i p m a t e r i a l s were formed into thin sheets of about 0, 2 mm thickness. This was performed by p r e s s i n g the m a t e r i a l , after a p r e l i m i n a r y diminution of the thickness by means of a calender, in a steel p r e s s at a p r e s s u r e of about 10, 000 atm*. During the pressing the p r e s s was heated to about 100°C by s t e a m . Although K is linearly dependent on the c r o s s - s e c t i o n of the gel strip, it is not advisable to use much thicker s t r i p s , as the time of attainment of equilibrium i n c r e a s e s with increasing c r o s s section. So one h a s to find a compromise between the sen-sitivity and the speed of the instrument.
The crosslinking was accomplished by y- or|3-irradiation, The irradiation normally gives r i s e to s e v e r a l types of reactions like chain scission, gas evolution and c r o s s l i n k -ing^-^ . For our purpose the crosslinking effect must be predominant. All m a t e r i a l s used in our investigations could readily be crosslinked by irradiation without too much damaging,
F o r the irradiation of poly(dimethyl siloxane) and poly(cis-isoprene) use was made of a Co^° source with a dose rate of 0,1 m e g a r a d s per hour. During the irradiation the s t r i p s were kept under nitrogen in sealed glass tubes. In the case of poly(dimethyl siloxane) a dose of 10 m e g a r a d s turned out to be sufficient to produce an insoluble network of reasonable strength. F o r poly(cis-isoprene) the proportion between the reactions leading to crosslinking and the chain scission reactions is l e s s favourable and here doses of 30 to 40 m e g a r a d s are n e c e s s a r y .
The s t r i p s consisting of the other m a t e r i a l s mentioned above
were crosslinked by irradiation with electrons (energy
0,5 MeV) from a Van de Graaff generator**. This source h a s the advantage that the intensity and the effectiveness
• The assistance of Rubberinstituut TNO, Delft, is gratefully acknowledged.
*• The author is grateful to Dr. W.A.Cramer, Reactor Instituut, T . H . , Delft, for his assistance.
of the radiation is much g r e a t e r than that of a Co^° source, so that the radiation times a r e shortened considerably,
In Table I the doses and the p r o p e r t i e s of the resulting e l a s t o m e r s a r e listed. The degree of swelling q was d e t e r -mined by m e a s u r i n g under a m i c r o s c o p e the dimensions of a piece of crosslinked polymer before and after swelling in the swelling agent, which in aU c a s e s was toluene,
Table I Gel material poly(dimethyl siloxane) poly(cls-isoptene) poly(cis-butadiene) poly(butadiene-co-styrene) poly(butadiene-co-acrylonitrile> Dose (megarads) 10 30 42 15 15 7,5 Q 6,23 1,34 5,96 3.87 5.72 2.85 Young's modulus in the dry state
(dynes/cm^)
1,55 xlO^
1 . 1 5 „ l o ' '
Young's modulus in the swollen state
(dynes/cm 2) 5,2 xlO^ 1,62X10^ 1 , 8 5 x 1 0 ^ 7 . 5 2 x 1 0 ^ 3 . 3 4 x 1 0 ^ 8 . 5 3 x 1 0 ' '
In most c a s e s the original polymer m a t e r i a l contained a s m a l l e r or l a r g e r amount of additives like s t a b i l i z e r s and dyes. Also, some low molecular weight, soluble m a t e r i a l is formed through side reactions during the irradiation. F o r this r e a s o n all s t r i p s were subjected to an extraction with toluene for several d a y s .
All gel s t r i p s , prepared in this way, behaved well in this r e s p e c t that they could be easily mounted into the clamps and that after a short initial period the r e t r a c t i v e force at a c e r t a i n degree of elongation remained constant,
C H A P T E R 4
PREPARATION AND CHARACTERIZATION OF THE POLYMERS
In o r d e r to investigate the possibilities and limitations of the elasto-osmotic technique a calibration with well defined polymer m a t e r i a l s is of prime importance. To this end t h r e e different types of polymers were used, viz, polystyrenes, poly(vinyl acetates) and poly(oxypropylene)gly-cols,
4 , 1 , PREPARATION
A number of polystyrenes were prepared at Kunststoffen-instituut TNO, Delft, The Netherlands, by anionic polyme-rization of styrene using a technique described by Szwarc ^^. In this method the reactants and the conditions a r e chosen such that (i) the rate of initiation is much g r e a t e r than the propagation r a t e and (ii) termination does not occur during polymerization. Under these c i r c u m s t a n c e s the polymer chains, after a simultaneous initiation, grow at equal r a t e s . When all the monomer is consumed they will all have nearly the s a m e length. As a result the polymer, after termination, will have a very narrow molecular weight distribution.
Similar polystyrene samples were also obtained from ArRo Laboratories I n c , , Joliet (Illinois), U, S , A , , in a l a t e r stage of this study.
The poly(oxypropylene)glycols were obtained from Dow Chemical C o . , Rotterdam, The Netherlands (trade name: Voranol) and Shell Chemie Nederland N . V , , The Netherlands (trade name: Caradol). These polyglycols a r e prepared by base-catalyzedpolyadditionof propylene oxide to a bifunctional or trifunctional alcohol like propylene glycol or glycerol, The polymerization p r o g r e s s e s by way of a stepwise mechanism 2-^ without termination and thus it p r e s e n t s the same c h a r a c t e r i s t i c s as the anionic polymerization of vinyl polymers described before. T h e r e f o r e , the polyglycols also have a very narrow molecular weight distribution.
The poly(vinyl acetates) were p r e p a r e d by conventional radical polymerization of about 40% vinyl acetate solutions
in benzene at 70 C, using azobisisobutyronitrile as initiator, Triethyl amine was used as a chain t r a n s f e r agent to adjust the desired molecular weight. P r e c i s e data concerning the preparation a r e given in Table II, together with the limiting viscosity numbers in acetone which were determined by means of an Ubbelohde v i s c o m e t e r . The limiting viscosity number is defined as
n - rio
[ n ] - l i m - ^ ^ - ^ - (4,1)
c -» o ' o
in which rj and n^ a r e the viscosities of the solution (con-centration c) and the solvent, respectively,
Table II Polymer PVAc-1 PVAc-2 PVAc-3 PVAc-4 PVAc-5 PVAc-6 Concentration initiator (g/1) 0 . 4 1.2 1.8 2.0 2.0 4.0 Concentration chain transfer agent (ml/1) _ -2 10 20 40 [»)] (dl/g) 1,080 0,865 0,704 0.515 0.476 0.341
In all c a s e s the conversion was kept below 10%. This means that the monomer concentration during the p o l y m e r i -zation is practically constant. Polymeri-zation at constant monomer concentration in the presence of a chain t r a n s f e r agent yields polymers with a Flory-Schulz-type distribution^*:
/ ( M ) d M = C exp (-pM)dM (4,2) because termination by combination is supgressed. In this
equation C and /3 a r e constants. The ratio My,IM^ for such a distribution is equal to 2.
4 . 2 . CHARACTERIZATION METHODS.
In o r d e r to obtain data concerning the number average molecular weights as well as the molecular weight distributions of the polymers described in Section 4. 1, s e v e r a l techniques were used, A s u m m a r y of all r e s u l t s will be found in Section 4, 3,
