SEDIJfCST.iTIOS LQL-ILIBRIA O F POLYDIbPCR>O SOLUTE:,
235
SEDIXESTh'L'IOX E:QIXIIBRIA OF l'OI~~~I>ISl'E;KSE SOX-IDEAT, SOLLTEK
I.
THEORI-~~'
hIIC".U:L WALES Dcpartment
uj'
C h e m i s t r y , l - n i c e r s i l y oj' lf.iscorL.si)i, J l u d i s o n , It'isconsin
ReceitNed :liigust 65, 19.$7 ISTRODCCTIOS
The usdulness of sedimentation methods in p h y s i d chemistry has h e n amply tlcmonstratetl. There are tn-o distinct niethotls of approach, one based on the sediment ation-equilihriiunin and the other 011 sedimrntation-~-elocityobservations. The possibility of niakiiig a molecular-weight analysis of a mixture is one of the great advantages of the ultracentrifugal techniques. Much progress in this tlirtvtion has been made in the case of the globular protein and polysactmis, but the interpretation of the data in the case of solutions containing long-rhain solute molecules has had its difficiilties. I n an ;u?icle by Svedberg ( l o ) ,having to don-ith the cellulose inolecule~detailetf consiclerution is given to thc application of the combined sedimentation-relocity and cllffusion data to the study of solutc size and size distribution in solutions containing cellulose and its derivatives. However, there are here Some serious experimental problems to be overcome before success can be complete. IT-e shall prefw at thi? time t o investigate the possibility of the application of the sedimentation-eciuilibrium method to systems containing long-chain molecules. The interpretation of equilibiium ultracentrifuge data for systems of this kind has been rrndered difficult by pronounced deviations from Henry's law at \-cry low concentmtionP. How\.er, progress can be made 11ytaking account of thc tleriations frum ideality, if 1r.e can assume that at L: girrn tcmperat'ure these tleviations are the same ;is in the other themiotlynnmic properties, such :is osniotic prc'swrcb i 12 ' . 1 l'reseiit~(I a t the Tn-erity-first Sat,iorial Colloid S y n i p x i u n l , \\-Iiicli \\-as Leld iiiitler t h e [Luspices of rlic Division of C'olloitl (:lieinistry of tlic .Inicrican C'hemical Society at Palo .Uta, C'aljfolnia, June 18-20> 1947. This p:Lper is based 0 1 1 a thesis subriiitted by Micliael !Vales t o t h e F:iculty of the [Tniversity of Wisconsin i n partial fulfillmerit of t h e requireiiieiits for t h e degree of Doctor of Philosopliy, October, 194T. 2 This ~ o : . hrvas supportcx(l i n p a r t Iiy a grant froiii t h e Wiscorisin .llumni Rcsc:trcll F ~ U I I tlaticin.
236 THEORETIC.1L
Consider a column of solution of unit cross-section and length, dx, in a centrifugal field. -kt equilibrium, the change in potential energy in moving cI grams of materials through the distance c1.r to a point \\-here the concentration is cL clc,
-+
\vi11 be balanced by the ~ v o i clone k against the osmotic-pressure difference Let the osmotic pressure he given by
at any point at a cliqtance .I‘from the center of rotation in the column of .elution of unit cross-iection. Then the condition for equilibrium in the column i, c,(l - f p ) w 2 s c l . l :
=
(a)
ds
here w is the angular velocity and f the partial specific volume of solute. This is equivalent to stating that the partial molal iree energy of the solute is the same at all points, at equilibrium. Ilifferentiating equation 1 and remeinhering t h a t
IT-herethe suliscript refer.: t o species i, I\ e ol)tain
If this equation is broken up into i equations, using the separation function, h ,
--
c,w2x.(1 - T p l =
I2T dc - -> If L d.l:
+
tic c, d.t‘
/I,--”
where
I’
=
C
1 -a___’ bici
~~
2 c c i
(6)
S o asminiption is made as yet as to the behavim of the sZpnrati3n functions b , . Suppose, however, that for the system under consideration
(from experimental measurements of osmotic pressure)
SEDIMESTITIOS EQU1LIBRI.k O F POLYDISPERSE SOLUTES
237
(for measurements on successive fractions). Furthermore, suppose that b’ does not vary with molecular-weight distribution for polydiaperse material.3 It must be also emphasized that equation 8 and the last condition would not be expected t o hold for material containing molecules of different composition and/or very different degrees of branching. Employing equation G and reformulating the conditions
where y z is the weight fraction of species i in the mixture. Since db’/ac is zero for any distribution of yZ’s,
and from equation 8
then it f o l l o n . ~that
6, = f ( y l . .
yn),
3
b,
=
26’ when y z = I ,
(12)
y3 = yn = . ” y n = 0
Formulating equation G in terms of the y 1 and performing an arbitrary variation in the yzwe hare
+ b,6y,] = 0
C{y16b, i
=
266‘,
Cay, = 0
(13)
where
Using the method of undetermined multiplier3 and combining the terms of equations 13 and 14 we obtain
and since the 67,are arbitrary
where n is an arbitrary constant. A n obvious solution to this set of equations is found by applying the Iioundary conditions of equation 12. T h e system polystyl.eiie-butanone is tieiicvcd t o exhibit t h i s behavior, on thr tiasis of 21 large number of osmotic measurements on solutions of 0.1-2 per cent by weight. (It. 1 1 . E w a r t and H. C. Tingey: Paper presented at, the 111th Xlceting of the hmerican (‘hcniicsi Society, which was held a t .Itlantic City, S e w Jersey, April 14-18, 1947.)
238
VICHAEL n’.lLES
Khile there is some doubt aq to n-hether thi- is the only solution of equation IG, it is certainly the simplest solution and hence t o be preferred on physical grounds, in the absence of any other information. SOK returning to equation 5 :
It can be seen from this equation that the more non-ideal the p / c 2’s. c curve i.e., the larger the h ’ s , the greater will be the deviation from ideality. The higher the concentration of the solution and the higher the molecular weight of the solute, the greater n ill he the deviation from ideality. For very low molecular weights the deviation becomes negligible (in practice, below about JP, = 60,000 for 0.1 per cent solutions in carbon tetrachloride, and for somewhat higher molecularv-eights in methyl ethyl ketone). It is also obvious that if b, is some involved function of the JI,’s or the yi’s or lioth. a theoretically exact calculation of a true 111, from the dc/dr and c, obtained experimentally would be extremely difficult, to say the least. If the DL’bare all the same function of total concentration, the calculations are simple. Hon ever, assuming that all the bt’s are equal and constant, as previously justified. in the caw of a suitable solvent tlC, ~
(19)
An alternative and niore compact formulation, proposed by D r . R. 11. E w a r t , General Laboratories, C . S.Rubber ConipaIiy, Passaic, S c x Jersey, is here reproduced. Making use of equation 1 it follows by t h e application of the Duhem equation t h a t
or if 2.1
dc, dX
=
= (2.4,
(1- r-p)w’ -
~
R2‘
- B
This represents t h e important correction for non-ideal solutions in a monodisperse system. With polydisperse systems it is not possible t o solve t h e Duhem equation i n order t o get
SEDII\IESTATIOS EQCILIBRIA O F POLYDISPERSE SOLUTES
239
where
B = -20gb’ RT
mm. solvent grams polymer 100 g . solution I n any event the maximum uncertainty in an experimental determination of B is about 10 per cent. Variations with molecular weight of this order of magniresulting from an uncertude could not be readily detected. The error in tainty in B is:
’
)
For polystyrene-butanone, an especially favorable case, this amounts to an error of about 1 per cent for a molecular weight of 100,000 a t 0.3 per cent by weight, assuming 10 per cent error in B. This indicates an error of 5 per cent due to this cause a t *Ifz = 500,000. This error could, of course, be decreased by going to lower concentrations. X concentration of 0.2 per cent by weight is still high enough t o obtain fair results. h considerable improvement would result from the use of the osmotic balance in determining B (4). In an actual ultracentrifugal experiment an excess pressure of 1/2 to 18 atm. may be developed in the sedimentation cell by the centrifugal field. The constant B is evaluated from measurements a t atmospheric pressure. An expression for its rate of change n-ith pressure should prove useful.’ It is estimated that ~~~~
the above for each solute component. However, since for such systems i t has been found that
i t is consistent with the Duhem equation if
I n these equations B is a constant. However, it should be emphasized t h a t this is merely an assumption which happens t o be consistent with the Duhem equation. This is the important equation for treating polydisperse non-ideal solutions of polymers, and equation 19 follows directly from i t by summation. It can also be shown t h a t equation 25 can be obtained from i t with complete generality, provided B is a constant. 6
I n general, and using Lewis and Randall’s notation
and
240
JIICH.1EL \\'.LLES
this effect is negligible compared irith the uncertainty in B , on the basis of the data of Gee and Treloar (3) on the partial specific volume of rubber in benzene. I n any event the extremely high speeds are used only for samples of very Ion molecular Jveight n here the non-ideality correction is unimportant anyway. For cases n here this is appreciable, P < 2 atm. over atmospheric. ?;OK Jll has been obtained. If it is calculated correctly, J12,M3, JI, , . . etc. may be obtained from it and its change n-ith height in the sedimenting column in principle. This is demonstrated in the following way: P
I r
dc,
Then from equation 19
Substituting in equation 5 and remembering that
, we obtain
SEDIJIE?;T.iTIOS E Q T I L I B R I I O F POLYDISPERSE SOLUTES
24 1
It can be shon-n in a similar manner that
for all values of q , positive or negative, where M q is defined as:
Jlfc, z
It is thus possible to obtain “point” values of any moment (molecular weight) by knowing the moment below it as a function of concentration. In theory, the initial moment may be any moment and need not have been calculated from the change in concentration with distance. However, it can be seen that to do the reverse calculation involves an arbitrary constant of integration. Nevertheless in certain cases where at some region of the sedimentation cell M, and J12 are close together, the integration constant for Jl0 may be evaluated with a fair degree of precision.6 For Jlns= AInzwe have, from equation 24:
6 I t can be proven t h a t there is no point in the cell, except for the monodisperse case, at which t h e material has the same average molecular weights as the whole polymer. For t h e concentrations at which the zeroth, first, and second moments have the same values as t h e over-all values for the whole polymer, we h a v e :
I*
xc, { .1f32
c: =
.VzZ - Mi, d r
b
[
X
{ 113, .lfzz
-
I
dZ
tierr
CX = the conrentration tvhere M o , = J I o = 31, c t = the concentration where MI, = .111 = c r = the concentration where 112, = .112 = .lI, t i i d x I S t h c distance from the center of rotation t o any point It ( ~ H I Itw swri that all eiit I ~ T I O I I S are uiicqua~.e\c.c>pt when .Illz= .Jfzz = .113, c t c this niorrotiispcrsc cas?
(YJII-
242
MICH.1EL TVSLES
The solution of this equation is
where K , is a constant of integration. This form v a s used by Lansing and Kraemer (6). Criteria for estimating IC, can be developed. By combining two moment recursion formulae:
It is now possible t o make two approximations. The derivative in the brackets may be taken as zero or as equal to
Then one obtains a value for JIo2at some point which can be used to evaluate the integration constant in equation 28. This should be done a t a point in the sedimentation cell where all moments are as close together as possible, excluding the ends of the column. For positively skeIved molecular-weight distributions the derivative in equation 29 should be positive. Hence if it is taken to be zero, the result n-ill be too low. Conversely, if it is taken t o be equal t o
d In c2
the result will be too high. For fractionated material this choice is frequently immaterial, since there is a region in the cell \There
d In c2 is very close to zero. However, in other cases this procedure may not be very satisfactory. The best that can be done is to take the mean of the tivo approximations. Having derived relationships for calculating “point values” of various average molecular weights, or moments, it remains to convert them into the corresponding values for the whole polymer. This may be readily done by means of the following expressions:
SEDIMEXTATIOS EQUILIBRIA O F POLYDISPERSE SOLUTES
213
q= 2,3,4.*.
dw
=
zcLdz for a sector-shaped cell)
(33) dw = cz dx for a linear cell J Sometimes in the sector-shaped cell a factor (x i.6 ) must be substituted for x in case the point of the sector does not coincide n i t h the center of rotation. S o n - the problem of obtaining information about the molecular-weight distribution of the polymer presents itself. TI-e may proceed in tn-o different ways: first, to calculate as many moments of the distribution as possible and try t o fit them by some empirical distribution curve; or second, to solve an integral equation. ( 1 ) Since the yth moment depends essentially on the derivatives of the concentration up to and including d4c+/dxq, the uncertainty in the moments will be the larger, the higher the moment. This is due to the fact that dc,/dz is obtained experimentally using the Lanini scale method (6, l l ) , and as one goes to higher and higher derivatives the uncertainty in the derivatives becomes larger and larger. -11~0,the portion of solution very near the high-concentration end of the cell becomes the decisive factor in determining the value of the moments, as higher moments are calculated. I t has been found possible in practice t o go only as high as M3 = XzTl with any degree of reliability. The foregoing recursion formulae are used, slopes being computed graphically n-ith a tangent meter. The criteria for reliability are admittedly meager. T-alues of Tvhich increase regularly nith concentration without too much irregular variation are taken as being reliable. At that, JI, may be in error by as much as i 20 per cent, judging from experiments on mixtures of polymers (13). The recursion formula is also used in calculating ;If, in preference to the older methods n-hich gave average values of JlZover an interval instead of point values (6, 12). I n practice, four moments are usually available. Thus, these moments may be used to estimate the constants in the follou-ing distribution functions ( 2 , 6, 9):
244
MICHAEL W.ILES
where
n-here p ,
=
yth moment expressed as a degree of polymerization
=
b- + q + l 1-ff
and functions of the type
JIne-a.li2
(36) What we have here, then, is a curve which can be considered t o approach the true distribution on a basis of agreement with four moments. I t must be also remembered that there is an added uncertainty in JI,,due to the approximation procedure we have described. Unfortunately, equations 34, 35, and 36 frequently cannot be made t o fit the behavior exhibited by imperfectly fractionated polystyrenes at all (13). In this event the total distribution curve (or moments) may be resolved into t w o superimposable distributions by splitting the In c, 1's. JIuzcurve into two parts and proceeding with each part as before. This leads to a rough visualization of the shape of the true distribution curve. This method \\-as applied t o data for some polystyrene fractions, and the results will be discussed with other esperiniental data (13). (a)lf one attempts to fit a generalized curve, such as a sum of terms of the form of equation 34, 35, or 36, to a set of moments, the labor is so great that it is just as easy to solve the integral eyuat,ion of sedimentation equilibrium, n-hich ]vi11 now be developed. This gives the distribution function directly' and the solution offers no theoretical mathematical difficulties for a continuous distribution of molecular weights. This procedure is similar in principle to that of Rinde (8). From equation 3 it follo\\-s that ci = c l , exp M.liG(z)
(37)
where
G(z) = A(s' -
2 ) - B(c, - c,)
(38)
It has been shown t h a t . in theory, an infinite number of moments of any distribution may he determined from a sedimentation equilibrium experiment. This means t h a t one and only one real positive function, the differential molecular-weight distribution curve, is defined by t h e experiment anti in principle the solution of equation 41 is unique. The restriction of csontinuity of j ( M ) is not necessary for this conclusion. The only source of trouble is that an inordinately large number of terms may be required in equation 44 t o It r q m s m t - y ( l l ) , This is t o he espected, of course, with discontinuous distiihutiona. nay also o w u r \$it11 a distribution with t w o xidely separated peaks, as in associated polyvinyl c~hloritlc(1 ) . This difficulty may be removed i n sonic cases b y use of equation 45. .4 Fourier iiitcxgral rcprcwntation of j ( X ) o r y ( W ) has tieen t r i d but has pruven inipract ic~alilc~.
215
SI.;DI 1I E S T .? T 1O S 1.: Q Z-I I. I H RI .\ 0 F P 01,TD ISP E rt8 I.; 5 0 LUTES
using eome referenctl point r p , c,. species i be cn,. Then
Let the initial over-all concentration of
frum a material 1)alant.c. on species 2 . Here 6 is a correction for non-roincidence of the center oi t h e cell hcctor \ \ i t t i the cwter of rotation and is included since it \\a,+found necessary for the cell used in this investigation. The distances of thc end> of the column of wlrition from the center of rotation are denoted by n and 0. Hence
PaFQing t o
;I
“continuouq” distribution of moleciilar )\.eights,
io‘ - as
c, Cll
=
1 --2--
f(X)exp JIG(x) _ _d-M_+
(.I,
+ 6 ) esp X G ( . r ) d x
(11)
Here f ( J f 1 i* thc diffeiwitial \\eig.ht-tlihtrihution i~inrtionof the polymer, c,/c0, the i:itio of the cwnccntration at clibtance .r to the initial concentrtition, i n t i G ( i ) arc knon n funvtionh of .r. -411 cwnhtanti are knm\ n e.;perimmtally. ‘I’hcl arbitrary ronstant in G(.r) may no\\. he yet eclri:il t o zero? +inre it cancels out in equation 41. This c.quL\tion may he yolved by letting .m
A*bwning that y(J1) may Iir represented by a finite number oi terms of the form J ~ ’ < c ~ . ~ ~ =
e
(44)
xw p 1
Thia is justified in general. since -yi31) is zero at .lI 1h r expression r .
=
0 and at M
-+
30.
246
MICHAEL WALES
could also be used, although equation 44 is more general.
Here, the exponent
m is selected from a consideration of the moment ratios, a narrow distribution requiring higher values of m than a broad distribution (10). This solution
requires a great deal more numerical labor than equation 44. Substituting equation 44 into equation 42 and integrating,
where Q =
K - G(x)
(47)
The constant K is arbitrary but must be taken so that Let 1 P = -
Q
is always positive.
Q
t'hen
is known from the experimental data and equations 46 and
47. This is then fitted by a polynomial in p of the form $(PI
=
go
+ g1p + g2p2 + . .
'
(50)
Then, equating coefficients:
The use of equat'ion 45 in the same manner gives:
Here one selects experimental points (cn, z) to determine as many constants IK R A E h l E R , F:, 0 . : .J. Am. Cheni. SO('.54, 1369 (1935j. ( 7 ) M O S I M A N N , H . : IIclv. ('him. Acta 26, 369 (1942). (8) RINDE,H . : Ihsscrtation, I:psala, 1928. (9) SCHUI,~, ( i . V . : %. physik Chcm. B43, 25 (1939'1; for ~lisc~i~ssion wil I ~ O Y E R , R. I:.: Iiirl. Eng. C:hcni., Anal. E X . 18, 342 (1942). (10) SVEDBERG, T . : , J . Phys. ('olloiti Chem. 51, 1 (1!44i). (11) SVEDBERG^ 'r,,4N1) P m I . : K s i m , I i . 0 . : The I ' ( [ r r t c e n l r i j i ~ g c . Oxford University Press, I,o1~doI~ (1040). (12) WALES,AI., HEXI)ER,11,A I , , \ V ~ I , I , ~ , ~ ,M I . S\V.. , ANI) P:w.+R,r, I t . 1 1 . : J. Cheiii. l'hys. 14, 353 (1946). (13) \VAI,E;S. h I . , ' r H O S c i w i N , , J , 0.. \ ~ I I , I , I A M S ,. j . by.,ANI) 1';WARl. 11.: Part, 11 O i t h i s serics, i n prcparation.
