Sedimentation of naturally occurring polyelectrolytes

SEDIMENTATION OF NATURALLY OCCURRING POLYELECTROLYTES

D. A. I. Goring and Carol Chepeswick Maritime Reqional Laboratory, National Research Council, Halifax, Nova Scotia Received May 2, 1955 ABSTRACT A n invariance of sedimentation rate with molecular weight was observed between certain concentrations for the polyelectrolytes sodium carrageenate and sodium alginate. This resultwas explained by assuming a network model which, at low concentrations, breaks up to permit sedimentation of the molecules individually. B y applying the Kozeny equation to the flow through the network the effectivespecificvolume and surface of the molecules were determined.

INTRODUCTION

It is known that the sedimentation constant is insensitive to the chain length of high molecular weight polymers (1, 2). Results obtained recently with polysaccharides from seaweeds have clearly demonstrated this point for naturally occurring polyelectrolytes (3, 4). The substances studied were sodium carrageenate, a sulfated polygalactose prepared from Chondrus crispus, and sodium alginate from a variety of brown algae. For wide changes in molecular weight, the graphs of the sedimentation constant, s, vs. the concentration, c, were concurrent above a certain concentration. Below this concentration the curves separated to give appropriately different values of (s)c.0. Similar behavior has been reported for two samples of deoxypentose nucleic acid of different molecular weights (2). In this paper an attempt will be made to explain these results on the basis of an analogy proposed by Signer and Egli (5) and later by Fessler and Ogston (6). Above the critical concentration, the sedimenting system is assumed to be an interconnecting fibrous network moving through the solvent. It may then be treated as a permeable bed of fibrous material by the semiempirical Kozeny equation as modified by Sullivan (7) and Robertson and Mason (8). If the pore size is smaller than the length of a molecule, changes in molecular weight, M, will have little effect on the flow characteristics of the network. This would produce the observed invariance of s with molecular weight. Below the critical concentration the network breaks up. The molecules sediment singly and variations of s with M are observed. The present work will be concerned only with the region of concentration ill which the network may be assumed to be intact. 440

SEDIMENTATION OF NATURALLY OCCURRING POLYELECTROLYTES

441

THEORETICAL

The model assumed is a random network of long cylindrical fibers of constant radius. The charge and associated counter ions are assumed to have no direct effect on the flow. However, the double layer will make the network rigid as well as binding a layer of solvent. This model is now considered a bed of fibrous material for which the permeability, K, may be written (7), K -

Q~L AP '

[1]

where Q is the volume of solvent flow per unit time, ~ the viscosity of the solvent, L the length of the bed, A the area of cross section of the bed, and P the fall in pressure across the bed. For sedimentation, K is replaced by a function of s, f ( s ) . To obtain f ( s ) we consider a boundary at a distance x from the center of the rotor. Let the bottom of the cell be r from the center of the rotor. In Eq. [1], ~ remains the viscosity of the solvent; L and A become the length of the sedimenting column (r - x) and the area of cross section of the cell. The pressure, P, may be replaced by the total sedimenting force on the molecules per unit area of cross section. Thus, if p is the density of the solvent and p the partial specific volume of the unhydrated solute, P = o~2xc(r -- x)(1 -- pp),

[2]

where c is the concentration of the solute and ¢0 the angular velocity. The rate of flow will be given by dx p

Q = - ~ ~A,

[3]

where dx~/dt is the true velocity of the boundary with respect to the solvent and e is the void fraction or the porosity. The void fraction is the fraction available for flow of any given plane perpendicular to the direction of flow. It is also the fraction of any given volume available for fluid flow. Enoksson's correction will be necessary to allow for the back flow of solvent owing to the piling up of material at the bottom of the ceil (9). If the hydrated solute is assumed incompressible, dx r dt

1 dx -

~ dr'

[4]

where d x / d t is the observed velocity of the boundary. From Eqs. [3] and [4] dx

Q = 3-[ A.

[5]

442

D. A. I . G O R I N G A N D C A R O L C H E P E S W I C K

Substituting in Eq. [1] K -

~ dx/dl c(1 - p~) ~ x

_

[6]

~s - f(s), c(1 - p~)

which is identical with the relationship proposed by Signer and Egli (5). The Kozeny equation can be written (7, 8), $

K -

~ ~ koS02 (1 -- ~)~'

[7]

where ~o and k0 are factors determined, respectively, by the orientation and shape of the channels and So is the surface area per effective unit volume of the solid. If a is the effective volume of the hydrated molecules for unit weight of solute, then =

1

--

[8]

ac.

Also, the surface area, ~, per unit weight of solute will be given by -- aS0.

[9]

Substituting for e and So in Eq. [7] we have K-

~o (1 -- ac) 3 ko a 2 c2

[i0]

If f ( s ) from Eq. [6] is now substituted for K in Eq. [10], we obtain s

p(l--p~){(1--~c) ,la2 k¢

3)

"

[11]

Equation [11] is the same as that proposed by Fessler and Ogston (6), except that the term (1 - ac) 3, which is e~, becomes 2 in their relationship. The difference arises because Fessler and Ogston apply the Enoksson factor directly to the Kozeny equation. In the present derivation it is shown that the Enoksson factor is canceled by the factor e necessary to relate the rate of flow through a plug to the true velocity of the boundary (see Eqs. [3], [4], and [5]). Before Eq. [11] can be used it is necessary to evaluate the shape factor, k0. The results of Robertson and Mason (8) show that at porosities lower than 0.8, k0 is constant. Fessler and Ogston (6) assumed k0 constant in their treatment. However, in the case of a sedimenting solute at a concentration of less than 1% it is likely that the effective void fraction is greater than 0.8. For such high porosities (0.8 --~ 0.98) Sullivan (7) has shown that k0 changes with e in a manner similar to that predicted theoretically by Emersleben (10) for cylinders with axes parallel to flow. The transition to random orientation is achieved by the orientation factor

SEDIMENTATION

OF NATURALLY

OCCURRING POLYELECTROLYTES

443

= cos2 0, where Ois the angle between the direction of flow in a pore and the axis of the bed. The value of ~ will be taken as 0.5 (11, 12). The experimental use of Eq. [11] involves the arbitrary choice of a value of the effective specific volume, a. The quantity (1 - ac)a/ko c is then evaluated for a series of concentrations, k0 values being obtained from the data of Sullivan (7). The concentration is assumed constant during a run. The graph of s vs. (1 -- ac)3/ko c is drawn. Several such curves are constructed for various values of a. The graph giving the best straight line through the origin is then assumed to correspond to the correct value of a. From its slope the effective specific surface, a, can be calculated. The equivalent pore radius, ~, can be computed for any value of the concentration (13) from _ 2(I

-

~c)

[121

qC

If the total drag is assumed to arise from a cylindrical fiber, the diameter, D, and length per gram of solid, L,, of the fiber m a y readily be computed from OL

--

7rD~L, 4

and a = ~DLs.

[13]

From L, it would be possible to calculate the length of the individual molecules making up the network if the molecular weight were known. Clearly, molecular weights derived from the Mandelkern-Flory equation (14) cannot be used for this purpose since they are based on a random coil model. However, by a treatment similar to that of Scheraga and Mandelkern (15), a molecular weight value, M,, can be obtained by assuming a prolate ellipsoid of large axial ratio, p. The intrinsic viscosity is given by NvV,

[7] - lOOM,'

[14]

where N is the Avogadro number, r is a shape factor which depends on the axial ratio, and V~ is the volume of the effective hydrodynamic ellipsoid. If there is no change in a when the network breaks up Ve - M, a N '

[15]

whence from Eq. [14] m [,7] -

100"

[16]

444

I). A. I. G O R I N G AND CAROL C H E P E S W I C K

Thus ~ can be found, and a corresponding value of p for the prolate ellipsoid can be obtained from the tables of Mehl, Oncley, and Simha (16). The Svedberg equation is s -

M,(1 - ~p) Nf

[17] '

where f is the frictional coefficient at infinite dilution. The frictional coefficient, fo, of a sphere having the same effective volume as the molecule is given by fo = (162 ~'2V~)1/3~,

[18]

which from Eq. [15] may be written f

=

( 1 6 2 ~ M ) 1/3 ~ ) '

rl

.

[19]

Substituting for f in Eq. [17]

M2~/3 = (162r2N2a) lf3~s(f /f°) (1 -

[20]

~p)

Svedberg and Pedersen (17) have compiled tables giving fifo in terms of the axial ratio p. Thus Ms may be evaluated and the length of the molecule, l, is given by

l = M'L" N

[21]

RESULTS The sedimentation rate was measured with a Spinco ultracentrifuge at 40 ° C. by the method outlined previously (3). The preparation and fractionation of the polysaccharides have also been described (3, 4). An acetate buffer of pH 5.5 was used throughout at an ionic strength of 0.2 for sodium carrageenate and 0.05 for sodium alginate. The values of (s°0)c~0 and the intrinsic viscosity of the various fractions are given in Table I. TABLE I

Physical Constants of the Fractions of Sodium Carrageenate and Alginate Fraction

(g.-1 dt.)

[71

(s%)~0

F60 FC100 [FC120-A ~Protonal H [23 B

13.1 11.7 7.0 14.1 3.6

13.2 11.8 8.0 7.4 4.3

•

Polyelectrolyte

Sodium carrageenate Sodium alginate

I

S

SEDIMENTATION OF NATURALLY OCCURRING POLYELECTROLYrrES

12 i S4Ol 0o ~( (S)8

. . . . F60 ...FC I00 F 120 A

6 4 2 I

I

2

I

4C x 103 (~m.c~.~1

I

I

8

tO

o

F r o . 1. s~0 vs. c for f r a c t i o n s of s o d i u m c a r r a g e e n a t e s h o w n i n T a b l e I .

S:o° (S)

2

I

I

I

I

2

4

6

8

C X 103 (gm.cn3a)

FzG. 2. s~0 vs. c for f r a c t i o n s of s o d i u m a l g i n a t e s h o w n in T a b l e I.

"I

S

~=20

E u

to_ X

I

s~o(s)

2

FIG. 3. (1 -- ~c)~/koc vs. s°o for c a r r a g e e n a t e n e t w o r k .

~:45

446

D,

A.

I.

GORING

AND

CAROL

CHEPESWICK

:20

3 v

2 X

,g I

I

I

t

I

2

3

4

5:0 (s) FIG. 4. (I -- ~c)3/k0c vs. s~0 for alginate network. T h e graphs of s°0 vs. c are given in Figs. 1 and 2. F o r sodium carrageenate (Fig. 1) s°0 was independent of molecular weight f r o m concentrations of 0.01 g. c m 9 d o w n to 0.0025 g. cm. -L, below this concentration the graphs of s40 o vs. c separated, and it was assumed t h a t the n e t w o r k model no longer applied. As shown in Fig. 2 this range of concentration for sodium alginate was 0.01 g. cm. -3 to 0.002 g. c m 9 0

The graphs for (1 - ac)3/ko c vs. s40 for a values from 5 to 20 are shown in Figs. 3 and 4. The best straight line through each set of points was established by the method of least squares. B y trial and error the value corresponding to a line through the origin was obtained. For a below and above this critical value, the lines passed, respectively, above and below TABLE I I Dimensions of the Network for Sodium Carrageenate and Alginate a

Polyelectrolyte

Sodium carrageenate Sodium alginate

(era) g.-])

11 12

~r X 10-~

D

(era) g.-O

(A.)

49 77

89 62

L~ X 10 -12

L ~ X 10"t2

18 39

99 153

(era. g.-O

(cm. g.-t)

TABLE i I I Molecular Weights, Lengths, and Equivalent Pore Radii for Fractions of Sodium Carrageenate and Alginate

CA.) Polyelectrolyte

Sodium carrageenate Sodium alginate

Fraction

M~

~F60 ~FC100 ~FC120-A ~Protonal H (23 B

790,000 630,000 290,000 290,000 74,000

l (A.)

2400 1900 900 1900 500

c = 0.01 g. cm.-a

¢ = 0.0023 g . c m . -a

360

1580

230

1270 (c = 0.002 g. cm. -8)

SEDIMENTATION OF NATURALLY OCCURRING POLYELECTROLYTES

447

the origin. The line through the origin represented a value of a which satisfied Eq. [11]. From the slope of the line, (1 - p~)/~ 2 was determined, from which ~ was computed. Values obtained for c~, a, D, and L, are given in Table II. Also included is the chain length per gram, L=, of the extended molecule based for carrageenate on the dimensions derived by Bayley (18) and for alginate on a Fisher-Taylor-Hirschfelder model. In Table III values of Ms and 1 are given for each of the fractions. Also included are values of the equivalent pore radius for the highest and lowest concentrations in the range for which the network model was applied. DISCUSSION

The invariance of s with molecular weight makes ultracentrifugation unsuitable for studying polydispersity of such molecules. For if the concentration is low enough to be in the region where molecules sediment separately, the resolution is poor. At higher concentrations the rate of sedimentation is independent of M. In addition, the shape of the peak is governed largely by the concentration dependence. This lack of sensitivity to polydispersity is advantageous when studying the naturally occurring polyelectrolytes. For although a given component may be polydisperse a sharp peak will be obtained; another component may then give rise to another peak, thereby permitting a differentiation of components in spite of each component's being present in a range of molecular weights. The values obtained for a are an order of magnitude higher than the specific volume measured pycnometrically. However, a represents an effective partial specific volume and therefore includes water of solvation around the chain as well as closed pores or occluded liquid within the network. Robertson and Mason (8) and Goring and Mason (13) have found values of from 1.5 to 5 from the permeability of cellulose pads. Fessler and Ogston (6) have reported values of 4.2 and 18 for polysarcesine and nucleic acid, respectively. It seems likely, therefore, that a relatively large volume of solvent is immobilized by the network. At the concentrations at which the network begins to disintegrate, the pore radius and the molecular length should be similar. This is the case as shown in Table III. However, the values of ~ at the low concentrations are greater than the lengths of the shorter molecules. This might be expected since molecular interaction would tend to retain the structure for pore sizes which were somewhat greater than the molecular length. The length per gram, L,, of the fibers in the network is considerably shorter than L~ for the fully extended chain (Table II). This implies some coiling. A possible mode] is shown in Fig. 5, based on a helical configuration of the molecule. Each fiber is made up of a long regular coil. Water is bound in and around the coil to give the effective diameters shown in Table II. Watson and Crick (19) and others (20, 21) have proposed helical struc-

448

D. A. I. GORING AND CAROL CHEPESWICK . ~o ° . . "

J. ............

o.**

*o

FIG. 5. A possible model of the polyelectrolyte network. The dotted lines are the effective hydrodynamic boundaries of the fibers. The full lines are the coiled polyelectrolyte chains.

tures for nucleic acid. It is realized that the present analysis presents no direct evidence for a helix. However, such a configuration would fit previous viscometric (22, 23) and light-scattering (24) observations on sodium carrageenate which indicated a rodlike shape in buffers of relatively high ionic strength. When the effective charge on the molecule increased at low ionic strength, the helix would be capable of considerable expansion, giving the observed large increase in viscosity (23). The treatment lacks a rigorous theoretical background. A more detailed derivation of the flow characteristics for random networks of low porosity might profitably be used in the above manner. ACKNOWLEDGMENT The authors wish to thank Dr. E. Gordon Young for his interest and advice throughout the work.

]:~EFERENCES MOSIMANN,H., Helv. Chim. Acta 26, 61 (1943). PEACOCKE,A. R., AND SOHACHMAN,H. K., Biochim. et Biophys. Acta 15, 198 (1954). GORING,D. A. I., AND YOUNG,E. G., Can. J. Chem. 33,480 (1955). VINOENT,D. L., GORING,D. A. I., AND YOUNG, E. G., in press, J. Appl. Chem. 5, (1955). 5. SIGNER, R.~ ANDEGLI, H., Proc. Intern. Colloq. Macromolecules, Amsterdam, 19~9, pp. 179-192. 6. FESSLER, J. H., AND OOSTON,A. G., Trans. Faraday Soc. 47,667 (1951). 7. SVLLIVAN,R. R., J. Appl. Phys. 12, 503 (1941). l. 2. 3. 4.

SEDIMENTATION OF NATURALLY OCCURRING POLYELECTROLYTES 449 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

ROBERTSON,A. A., AND MASON, S. G., Pulp & Paper Mag. Can. 50, 1 (1949). ENOKSSON,B., Nature 161,934 (1948). EM~.RSLEB~N,0., Physik. Z. 26, 601 (1925). GORING,D. A. I., AND MASON, S. G.,Can. J. Research B28, 307 (1950). SULLIVAN,R. R., ANDHERTEL,K. L., Advances in Colloid Sc/. 1, 37 (1942). GORING,D. A.I., ANDMASON, S. G., Can. J. Research B28, 323 (1950). MAND~LX~RN,L., AND FLORr, P. J., J. Chem. Phys. 20, 212 (1952). SCHERAGA,H. A., ANDMANDELKERN,L., J. Am. Chem. Soe. 75, 179 (1953). M~.HL, J. W., ONCLEV,J. L., AND SIMaA, R., ~cienee 92, 132 (1940). SVV.DB~.RO,T., AND PEDERSEN, K. 0., "The Ultracentrifuge." Clarendon Press, Oxford, 1940. BxvLSr, S. T., Bioeh~n. et Biophys. Acta 17, 194 (1955). WATSON,J. D., AND CRICK, F. H. C., Nature 171, 737 (1953). WILXINS, M. H. F., STOXV.S,A. R., ANDWILSON,H. R., Nature 171,738 (1953). FRANKLIN,R. E., AND GOSLINO,R. G., Nature 171, 740 (1953). MASSON,C. R., AND CAIN-.S, G. W., Can. J. Chem. 32, 51 (1954). MASSON,C. R., ~.NDGORINO,D. A. I., Can. J. Chem. 33,895 (1955). GORING,D. A. I., Can. J. Chem. 31, 1078 (1953).