Intrinsic Viscosities and
Molecular Weights of Cellulose and Cellulose Derivatives The determination of molecular weights from viscosity data by means of calibration suffers from differences in molecular weight averages and distribution, and from variations of molecular structure which express themselves in the [17]-M relationship. For accurate work calibration and unknown samples ratio. The theoretical interpretation o f visought to have the same cosity data depends on the applicability of the models of molecular shape and rigidity, the state in solution, concentration effects, and the influence of the rate of shear. Progress has been made recently in all these analyses, notably by Debye, Flory, Kirkwood, I h h n , Peterlin, and Simha. Although questions regarding stiffness and extension of the cellulose molecules and their derivatives have been brought nearer a solution, they are still not decisively answered. Experimentally, there are new data concerning the variation of the exponent in the Mark-Houw-ink relation with molecular weight and solvent. Staudinger’s equation is not a limiting law but fits intermediate cases. There is further evidence of a shear dependence o f the intrinsic viscosity at high rates of shear or high molecular weights.
E. H. IRIhIERGLT AND F. R. EIRICH Institute of Polymer Research, Polytechnic Institute of Brooklyn, Brooklyn, .Y.1..
M
ETHODS for determining molecular weight with the aid of
viscosity measurements shoiv a number of recent developments which are not yet definitely settled or accepted. The following survey is a brief progress report. The relation betn-een viscosities and average molecular T? eights, or degrees of polymerization, DP, is neither direct nor unambiguous. Viscosity as a frictional process is not a measure of mass but of the molecular resistance to the flow of a given solvent. Frictional resistances of suspended particles are in general functions of the type of flow or drag, of the dimensions of the particles, and of the interaction, dynamic and thermodynamic, betxeen them and the medium. Compact spheres, being geometrically defined by a single parameter, exhibit frictional resistances which can be described by their radii, such as by Stokes’ equation: ftransiatlon = 6m7r, or frfrotatlou= 8 x 7 9 . Ellipsoids of rotation require two parameters, and three-axial bodies require three to define their resistances. .4s there is no a priori relation between molecular dimensions and molecular weights, two model relations are needed, one describing the changes of dimensions v i t h molecular weights, and the other formulating hydrodynamic resistances according to type of flon and dimensions. Because increases in viscosity of a solution relative t o the pure solvent are due to the resistance to flow of dissolved or suspended particles, a knowledge of molecular dimensions becomes an essential part of the relation between intrinsic viscosity and molecular weights. Any such relation has t o be further qualified concerning the assumptions that have been made about solvent, concentration, and rate of shear. The nature of the solvent enters by means of the coefficient of solvent viscosity and on account of the therniodynamics of the solvent-solute system which determine the molecular state of solution. The finite concentrations employed introduce solute-solute interactions, which are not considered in
a,
2500
the derivations of the intrinsic viscosities. The rate of shear influences the configurations of flexible particles and the aveiage state of orientation of anisodiametric particles toward the streamlines and thus their resistance and contribution toward solution viscosity. Of these three factors, the part played by the solvent is the most fundamental one and is discussed further below. Recent advances in understanding of concentration and shear effects are summarized, which are of help in extrapolating to zero concentration and gradient. EXTRAPOLATION TO ZERO COVCENTRATIOS
The initial concentration effects in the case of spheres beyond Einstein’s equation have been investigated by Simha, Gold, and Guth (21, 22). The theory confirmed that in dilute solutions of nonelectrolytes the viscosity increase can be described by a series of integer powers in e. The coefficient of the square term was found t o be 14. Reinvestigations indicate that this value may be too high. A factor of 10 has been calculated in a simplified manner by Riseman (Id). The factors proposed by T’and ( 7 0 ) and by hIooney (4.5) are derived on the basis of special assumptions Tvhich are open to question. Recently, substantial progress has been made in the understanding of more concentrated solutions and of the concentration effectsof nonsphericalparticles. In the former case, Simha(63)has pointed out that the total hydrodynamic interaction will undergo a relative decrease if the particles are on the average so close that the nearer neighbors cut o f f interaction with the more distant ones. Simha obtains:
where ‘p is the volume concentration, f = ?’la/( a/E), ais the radius
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 11
Cellulose
*
.a
of the spheres, and b is the radius of a concentric outer “substitute” sphere which exerts the same amount of interaction as all the individual interacting particles together. Based on the pearl necklace model and using the methods of calculation so successfully employed for intramolecular interaction, Simha (65) and Riseman and Ullman (69) have developed relations which potentially cover a wide concentration range for coiled macromolecules. General numerical factors cannot be derived with accuracy because of the obvious deviations of the monomer units from being true compact spheres, because of solvent and volume effects, etc. Simha’s discussion of these factors deals in particular with the possibilities of pair formation and of osmotic configurational changes. Accordingly, a certain amount of aggregation occurs even in dilute solution and in good solvents, consisting largely of accidental pair formation as a consequence of ordinary concentration fluctuations, This is distinct from association due to impaired solubility, but still has a significant bearing on the concentration dependence. There is further a thermodynamic influence on the particle configuration due t o concentration when the environment of the solute particles changes from pure solvent more and more to one of pure solute. It amounts to an “osmotic” change, with the result that ih good solvents the molecular chains become relatively compressed and in poor solvent extended, when the concentration is increased (71 ). These theories help to define the usefulness of the well-known empirical equations of Eilers ( l l ) ,Baker (S), or Martin (44) and of partly empirical equations such as Mooney’s (46). Baker
qre1
= (I
+ F)”
n is a, number
(3)
where a is a constant-Le., for uniform spheres is equal t o 0.74, the maximum volume fraction Martin
qsp/c =
Mooney
qrel =
[ a ] exp (kl[q]c);
ki
=
k’ constant = - (4) 2.3
exp. [2.5?/(1 - k ? ) ] , k = constant
(5)
Such “closed” expressions can fit limited Concentration ranges only and should be considered as approximations of the infinite series of powers in c, derived by the hydrodynamic theory, in which subsequent coefficients can take care of different interaction mechanisms. The equation qsp/c = [ q ] k’ [q12c, or e
%
+
= [q]
- (0.50 - k’) I?]%,
introduced by Kraemer (Sf?),
Huggins (%), and others, is of such wide applicability because i t consists of only the first two terms of the general power series by which most concentration laws can be expressed, and because its plot usually stays linear within a n easily measurable concentration range- see, for instance, Table I (89).
alytical expressions for the linear extrapolation of cubic equations:
- qap/cz = Ul(C1 - cz) + az(c: - c2) 11SP/C1 - l a p c2 = a1 + adc1 + c2) c1 - c2
Vap/C1
Cellulose Nitrated Cotton Linters Wood pulp Rayon a Data of Lindsley (99).
h1 31 15
8.6 3.3
Maximum Value Conon., Relative g./100 ml. viscosity 0.1 20 0.3 60 0.5
1.0
50
15
For the very high molecular weights recently coming under investigation, the relative viscosity increases of very dilute solutions may require description b y power series including cubic terms. I n these cases, Bawn el al. (6)have applied the corresponding anNovember 1953
(7)
(For the fittings of curves requiring higher terms see Weissberg, Simha, and Rothman, 71.) A graphical help used by Mdnster (48)for the extrapolation of curved concentration functions is t o plot qsp/c once versus c, and then versus qsp, on the same plot. Both curves ought t o converge a t zero concentration. This fact, as a rule, greatly facilitates the determination of the intercept. EXTRAPOLATION TO ZERO SHEAR GRADIENT
The influence of rate of shear on the intrinsic viscosity was pointed out on theoretical grounds in the early thirties (6, 13, E!,$) and was supported by a few scattered observations. At t h a t time it was also shown (67) t h a t the rotatory diffusion constant of a molecule provides a measure for the gradients a t which orientation effects begin t o influence intrinsic viscosities beyond the experimental error. With the preparation and measurement of higher and higher molecular weights, especially of fairly stretched molecules such as linear polycarbohydrates or polyelectrolytes, the effect has become more pronounced and several authors have recently dealt with it The phenomenon consists in a decrease of intrinsic viscosities as determined by extrapolation a t constant rate of shear. Figures 1 and 2 show this dependence for nitrocellulose in ethyl acetate (51)and in butyl acetate (55)over a range of gradients up t o 5000 set.-* It is immediately apparent t h a t the well-known shear dependence for finite concentrations persists here t o the highest dilutions. As all theories are based on the absence of appreciable gradients, failure to extrapolate to intrinsic viscosities a t zero gradient could result in values which cannot be used for the calculation of molecular weights, just as the constants of the empirical equations vary with the gradient from sample t o sample. Although expedient from the experimental point of view, it is therefore not advisable to compute viscosities of different samples at a given rate of shear, Such values progressively tend to become relatively smaller with improved solvent or higher molecular weight, and would thus reduce actual differences. Several recently proposed extrapolation equations should b e distinguished from similar equations for absolute or relative viscosities. lnVsp = inqapo - OG
Fox, Fox, and Flory ($0)
1
Hall and Fuoss ( I S ) D e Wind and Hermans (10)
h1 =
A[?] = eonst
log (?S,/C)
(8)
+ = ~ I ~ I ~ / M@a) (l
- kp)
Newman, Loeb, and Conrad (61) log qap = log qapo Schurz (61,69)
TABLEI. VALUESOF CONCENTRATION AND VISCOSITY BELOW WHICHLoa (qaP/c) Is LINEARWITH CONCENTRATION”
’
(6)
- P log G
4- ~ [ V ] ~ G
= log
hl
+ Khlc
(9) (10)
(11) (12)
I n these equations qSPo is the specific viscosity (qrei - 1) at gradient G 4 , [ q ]D and [q]we the intrinsic viscosities at lim G = 0 and at a given G, respectively, Aq = ( [ q ] - f q ] ~ ) / [ q ] j~3,and the k’s are constants, and p is the actuating pressure. Fuoss’s equation is the limiting form of Flory’s equation for small p (since p Q G), but the other equations express a different behavior. All the above dependencies show G as entering t o the first power. They are in marked disagreement with the theory according t o which the angle e of orientation of particle axes toward the streamlines, as well as the number of particles oriented, are functions of the gradient. The reduction in viscosity should
INDUSTRIAL AND ENGINEERING CHEMISTRY
2502
thus be a function of G*. Kuhn and Kuhn ( S 4 ) in fact deduced that [q la =
[?lo (1 - E2/228
+ E4/2200 .
.)
(13)
where 5 = GRo2/Dt= 6 GqoM [qlo/RTis a measure of the particle orientation, RCis the average end-to-end distance, Dt is the translational diffusion constant, 70 is the viscosity of the solvent, and R is the gas constant.
mit, at least in principle, the calculation of parameters which belong to well defined models and give a better picture of the molecular structure or behavior. Recent developments follow the earlier equations by Einstein ( l a ) ,Simha ( S d ) , and Kiihn (56, 97, 38) for: Rigid, compact spheres
[ q ] = 2.5-
Rigid, compact ellipsoids f_”
(15(1n2f
9000
4000
G, SEC. - 1
Figure 1. Dependence of [q] on Shear Gradient Data of Newman, Loeb, and Conrad (52)
Peterlin (55) has recently simplified and extended these calculations and taken more explicit account of very soft, medium soft, and stiff molecules. H e concludes that very soft coils, because of compensation of stretching and orientation effects, will not shov any shear dependence if they are freely draining; if they are partially or wholly impermeable, jq] might even increase. The stiffer the chain, the more pronounced the shear effect and the less depending on the solvent. However, shear effects due to individual molecules and those due to concentration have to be held apart. The magnitude of the formcr is proportional to the concentration, c; that of the latter, to c2. Unless measurements can be extended to dilution where the shear effects depend on c, extrapolations to [ q ]will ~ be erroneous. Basing their calculations on the statistical theory of irreversible processes, Kirkwood and Auer (30) also investigated the frequency dependence of the intrinsic viscosity of rodlike molecules in dynamic measurements. They found that, in the limit of zero frequency, three fourths of the intrinsic viscosity is due t o energy dissipation by the rotatory diffusion torque of the rodlets. As the frequency of a periodic shearing motion is increased, the particle orientation falls increasingly out of step and [ q ] drops eventually t o one fourth of the zero shear rate value. Kirkwood and Auer find agreement of their theory with the data of Baker et al. ( 4 ) ; comparison with steady flow data cannot be carried out a t this state of the theory but, as above, an inverse dependence of [q] on G2 would be obtained, since the Kirkwood-Auer diffusion equation is essentially the one used by the other authors. The few experimental results available do not agree with the theory, b u t Peterlin quotes in support unpublished work by Singer. In most ca6es the extrapolations from higher concentration are presumably not good enough. Pending clarification, experiments which cannot be plotted according to the theory should be plotted by any empirical method that permits a linear extrapolation t o zero gradient. HYDRODYNAMIC THEORIES
FLEXIBLE MOLECULES.After [n] a t zero concentration and shear has been obtained, the recent hydrodynamic theories per2502
’
[T] =
- 3/2) -I- 5(ln 2f
(14)
”>
- 1/2) + 15
Va (15)
rh’A ERo [s] = 2 ___ LCO ill
Flexible, free-draining coils
0
4e3Ni 3M
The latter is a form of Kuhn’s equation which permits better comparison. I n these equations r is the radius of the suspended spheres or of the monomer unit (bead) in the molecular chain, M is the molecular weight, h’a is Avogadro’s number, 1 and d are the long and short axes of the ellipsoid, p is the polymer density, Z is the degree of polymerization, 7n is the monomer molecular weight, and E the mean square end-to-end distance; ( [ q ]in ml./gm).
b
o 0.5 0
10
90
40
30
-*c
Figure 2.
Stiff RIolecules, a, and Completely Soft Ones, b (55) [ ~ ] G / [ ? ] Ofor
Experimental data for cellulose nitrate i n butyl acetate (68). Equation 13
See
Einstein’s formula for spheres may be taken as the prototype, shoviing the intrinsic viscosity to be proportional to the ratio between volume of the dispersed phase and molecular weight, but independent of the latter because of the proportionality between volume and M . The molecular weight dependence in the other two equations enters through the geometry or shape factor and requires finding of the correct functional relationship between molecular weight and dimension. Kuhn’s equation, R ith E and ,If, can be seen to bear out Staudinger’s equation [VI = KmM (174 and d slowly increasing, Simha’s equation leads IVith 1 a to a proportionality with Ai‘ with an exponent smaller than 0.85, approximating the exponents found in Mark’s ( 4 1 ) more general equation
111 = KMM4
INDUSTRIAL AND ENGINEERING CHEMISTRY
07b) Vol. 45, No. 1
-Cellulose
I*
The relation between M and the shape of solventrpermeable molecules is complicated by the fact t h a t with increasing molecular weight the flow through the molecule must become increasingly hampered. This results in increasing the effective volume and reducing the geometry factor and the average frictional contribution of the chain elements. The recent theories dealing with hampered flow show that the decreasing penetration b y the solvent leads t o a decrease of a. Kuhn and Kuhn have elaborated their theory of the viscosity increase due to statistically coiled strings of beads by deducing t h e intramolecular hydrodynamic interaction from model experiments. Instead of Equation 16, they find:
where b, is the hydrodynamic length of the monomer unit,
dh
is
its thickness, and A , is the length of the so-called statistical chain element-Le., the length of the average number of monomer units after which a kink in the molecular chain occurs. I n view of @ = A k N , = b;Z, A,,, becomes equal to b, and N , equal to Z , for chains t h a t possess unhindered rotation. For very high molecular weights the first two terms in the denominator of Equation 18 can be neglected and [?] becomes proportional t o 4%: [ q ] = 0.60 NA(Ambm)*/2Z1/g/m
(18a)
A , follows if a reasonable assumption can be made about b,. If this impenetrable coil were to be treated like a full sphere, the only change would be in the numerical factor, which would become 1.03. Using instead a short ellipsoid of axis ratio 2, the factor would become 1.05. For rigid chains of low molecular weight, for which A , = bmZ, [ q ] becomes approximately proportional t o 22. I n a recent publication (36), Kuhn, Moning, and Kuhn have recalculated the values of the constants in Equation 18 and their new equation is
which for very high molecular weights, like Equation H a , reduces to: [ q ] = 0.43N~(A,b,)’/:Z’/a/m
.
-
(18a’)
I n the Debye-Bueche theory ( 9 ) , the model adopted is that of beads of radius T and frictional coefficient f, forming a spherical swarm of radius R, with uniform bead density. If the swarm is very open, the aggregate friction is proportional to the number of beads and leads to Staudinger’s law as one limit. As the swarm becomes denser and the solvent penetration falls off, the viscosity exponent drops in accordance with the function (p), which depends on bead friction and bead concentration in the swarm.
hl
=
4rR.S N A
If, in the other limit, the swarm becomes impenetrable, the viscosity is determined by Einstein’s law-Le., (p) approaches the value 2.5. However, since the radius of the swarm, R,, is still t o depend on the mean square end-to-end distance of the chaint h a t is, since R, = ( 6 / 1 8 %)‘Iz 0: M ‘ I L t h e volume of the swarm becomes proportional to M*Iz, and the viscosity t o Practically the same results were obtained independently by Brinkman (7). The discussion of Debye’s theory shows the operation of the relevant factors most clearly, but in general his model may not be expected t o serve for many polymer solutions. The theory of Kirkwood and Riseman ( 3 1 ) employs two more applicable models-namely, t h a t of hampered flow through beads arranged as in a random coil, and flow interference between beads
November 1953
strung out in a straight line. Applying a different method of calculation, a similar result as by Debye and Bueche is obtained for the coil.
The independent, molecular, parameters of the model are: Z, the number of C-C links in the chain and thus usually twice the degree of polymerization, b, the effective bond length, and 5, the frictional coefficient of the single beads; 90 is the viscosity of the solvent and (F).is a slowly varying function of b, and As .Z’/2appears in the denominator of function ( F ) which varies from unity at z + o t o a limiting value ( F ) a I/.z+ at z + m, the dependence of [ q ] on M varies again from M 1 t o MI’*. When these theories are compared with experiments, the few suitable data yield frictional coefficients (or the corresponding values, f, in the Debye-Bueche theory), which are very small compared with values obtained by inserting radii of monomers into Stokes’ equation. This is in itself not surprising, as there is no reason t o expect the validity of Stokes’ macroscopic coefficient of 6n, except t h a t this coefficient has rather surprisingly often been found t o hold in other molecular applications. It is therefore possible that the small values of t reflect deviations of the model from the actual behavior, such as suggested below. The values for the parameter, b, defined by the relation
r,
r
+
b z = ( 1 cos y 1 - 005 Y
) (1 + -cos )bz p
1where bo is the actual length of the glucose unit, y is the skeletal bond angle, and p is the spatial angle between successive bond planes, however, are of the right magnitude, since the R j , calculated according t o statistics as = b2Z, agree with similar values from light-scattering or sedimentation-diffusion measurements. This is of particular importance in the following context. The Kirkwood-Riseman theory, like Debye’s, predicts practically impenetrable molecules at high molecular weights, or otherwise for sufficiently densely clustered coils. The frictional contributions of the individual beads then vanish and, apart from constants calculated by the theory, the viscosity depends on b and Z only. It should therefore be possible, from the knowledge of [?I and M , or E, of a single high molecular fraction, to obtain b for the whole polymer homologous series in a given solvent and temperature. Knowledge of M of a lower fraction would yield 5. With the help of these two values and [TI, one could calculate M of every polymer of a given series in a given solvent. Using the tabulated function ( F ) , b and r can, of course, be determined from any two pairs of [s] and M , but the above method is more accurate and convenient. All the above theories were developed for sharp fractions only. If applied t o nonfractionated materials, an appreciable error may be made, which can be approximately corrected b y selecting the viscosity-molecular weight average applying t o the exponent a in Equation 17b. Fortunately, the light-scattering molecular weight average is also a weight average, and the sedimentation-diffusion value an approximate weight average, but in comparing K2’s those derived from light scattering are Z averages. The constants KMand a of the empirical exponential law follow equally from two values for [?] and M ; not being derived from a well defined hydrodynamic model, K.w and a do not give nearly so meaningful a picture of the polymer chain in solution as b and or A,, b,, and dh, which describe dimension and resistance of successive monomer units and the average angle between them. However, a remains valuable as a guide for the permeability of the molecular coil and, in the KirkwoodRiseman theory is connected with the function ( F ) . FLEXIBLE MOLECULES, FOX-FLORY MODIFICATION. Discussing the limiting formula of Kirkwood and Riseman for high molecular weights
zz
r
INDUSTRIAL AND ENGINEERING CHEMISTRY
2503
plot of [q]2/3/~1f versus M / [ q ] according to the theoreticah equations: one realizes that apart from the molecular parameters Ro, or b and Z , this equation contains only constants. Rewriting it as ['j@ ( R " ) 3 / 2 / M , @ = constant Equation 20b is seen t o be of the form of Einstein's equation. The same was previously found true for Equations 18a and 19, and all three equations can be rewritten as ['] = number R i / 3 f , where the numbers are (gram per nil.): 0.0051 for Kuhn and Kuhn, 0.015 for Debye and Bueche, and 0.0069 for Kirkwood and Riseman (43). This in itself means no more than that impenetrable ellipsoids can be replaced by effective spheres, the radius of which is proportionalto acharacteristic dimension of the coil. At the same time, as was mentioned before (34j, the enveloping ellipsoids of high molecular weight coils have axis ratios not far from unity. Conperning Equation 20b, Flory (17) noted that the equation covered a wide molecular weight range instead of being limited to the highest values. This was particularly true for poor solvents where it is more readily understood on account of dense coiling. However, based on the following considerations, Fox and Flor? ( 1 8 , 1 9 ) adapted the equation also for general use. The original statistical theories deriving the average end-to-end distance of randomly kinked molecules did not take into account either "short range" (steric hindrance t o rotation) or "long range" (attraction or repulsion between distant members of the chain during back-coiling) forces. The former can be taken care of by suitable alteration of the statistic elements, unless there is a systematic trend. The extent of the attractive forces depends on the thermodynamics of the solute-solvent system. I n an athermal solvent-Le., no heat of solution-solute-solute attraction is just balanced by the attraction from the solvent molecules. The repulsive forces cause the "volume effect" and prevent segments from taking up positions 1% here others are already located. h s compared with the unbiased random dimension, Ro, the diameter of the coils thus becomes enlarged by a factor 01, unless attraction and repulsion balance:
- -
R2 = R: mand
['I
=
[q]2/3/iTf1/3
= K2/3
+K5/3C~Jf/[q]
C T = 2$, C M ( ~ e/T) =
(a5
(22)
- CY~)/~TI'/~
where $, is a parameter characterizing a given polymer-solvent pair and C.V is a constant determined by M ,@, and V S . Sadron (60) and his group have also demonstrated that t h e available experimental data admit interpretation equally well on the basis of porous and impenetrabIe molecules. They point out very clearlv that the question of the function between real and hydrodynamic dimension, which is the same as the question of the degree of permeability, is tied up with the question whether the dimensions follow true random or biased statistics, and thus are proportional to M 1I2 or not. The theory of Fox and Flory is criticized because it seems unlikely that K should reach its limiting value a t such low molecuIar weights, and because it is doubtful whether a rigorous calculation of m, on which the constancy of K rests, would lead t o a dependence of the volume effect on the molecular weight. However, Fox and Flory have accumulated impressive experimental material in their favor. Unless this evidence can be explained on a different basis, and until a generally accepted theory of the volume effect is evolved, the Fox-Flory theor>-provides a very useful relation for viscosity evaluations. RIGIDMOLECULES. Although the discussion so far has referred to coiled molecules, it serves to illustrate the present status of understanding as well as to provide a basis for conclusions regarding size and shape of cellulose molecules. If the latter were coiled, the viscosity data should fit the above deductions. If they were stiff, the data should conform with the theories f o r elongated ellipsoids or rods. Using the above approach and a stiff straight chain as model, Kirkwood and Riseman (58) calculated the frictional and viscosity coefficients, and Kirkwood and Auer (30) improved the theory by taking into account the effects of rotatory Brownian motion. Kirkwood and huer find:
@ ( z ) 3 / 2 0 1 3 / M=
+
-4ccording to the Flory-Fo.; theorj-, thevalue of cy in everjr case can be determined from viscometric and thermodynamic data and is approximately proportional to MO.'. The intrinsic visFlory and cosity thus depends on 'Vf to a higher power than '/2. Fox could show that, with CY properly determined, K is in fact very nearly constant. I n view of all the approximations involved. theexperimental value of @ = 2.1 to 2.6 x 102'is in fair agreement n-ith 3.6 X loz1from the Kirkaood-Riseman theory (51j. Thus, while in the Debye-Bueche and Kirkwood-Riseman theories the emphasis was on the degree of coil permeability as a function of M ,determining a,Fox and Flory concentrate on the change of the coil dimensions with M. According t o Fox and Flory, coiled molecules of molecular R eights from a few 10,000 upward would be practically impermeable for the solvent, as shown by the constancy of K. The experimental values of a larger than would be due to volume effects, and the small frictional coefficients, 5; could be explained as apparent values which in the absence of individual bead friction express the friction per bead of the molecule as entity. In increasingly poor solvents, the volume effect becomes more and more compensated by the molecular attraction; with Af -+ m, the factor a: becomes unity when the second osmotic virial coeficient A2 = 0, and follows directly from [q] and M,or vice versa. This is in good accord with the observation t h a t close to the precipitation point the intrinsic viscosities of a given polymer are practically the same in all solvent systems. For the experimental determination, the critical miscibility temperature, 0, is either found from a series of fractions, extrapolating to &I = m, or K and 01 are obtained from a 2504
where Z = (2% 1j, 2n = 2 r P is the number oi C-C bonds in a vinyl polymer, or twice the number of glucose residues in cellulosic molecules, and the other symbols are as before. ( F ' j is again a someTvhat different function of c, b. and Z . For large values of 1) b-Le., for high molecular weightsthe length L = (272 (F'j approaches its asymptotic value:
+
and 7TlVALZ
[TI = 9000m log L / b (1 +
3
iTm>
Thus, the intrinsic viscosity is a function of ,TP/log M . For short or medium molecular weights, the influence of the frequency depends on the molecular parameters, according t o ( F ' ) , but for very large molecules the effect of the frequency is seen to be the same, percentagewise, for all intrinsic viscosities. In steady flow, o = 0, the equation becomes the same as the limiting form of Simha's equation ( 1 5 ) ; i t differs from the Kirkwood-Riseman equation by a factor of and from Kuhn's equation (36,57, 38) for rods (24)
r
by the explicit numerical factor and the factor between and f, and by the absence of the logarithmic term in the denomina+or. The lat,ter, i t is interesting to note, enters into the Kirkwood-
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 11.
Cellulose300
500
c
I00
0
100
Po0
300
400
Figure 3. Relation of Limiting Viscosity Number to Degree of Polymerization for Cellulose Nitrate i n Acetone (36) Plotted for A m = 135 A., d = 9.2 A., bm = 5.15 A. 1. Calculated from Equation 18; curve represents relationship [TI = Z/(0.54 0.043 4% 2. Calculated for freely draining chain molecule regarded as extended elli soid of length Zbln 3. Expected dependence of [qfonZ 0. Actual viscosity measurements made by Mosimann ( 4 7 )
+
[ 71 is as the exact treatment shows, not directly proportional to degree of polykerization, 2 (as it would have t o be according t o Staudinger’s law). At low D P ’ s [ q ] IS approximately proportional t o Zz; over a n intermediate range, it is proportional t o 2; a t very high DP’s it is roughly proportional to
42
Riseman-tluer theory by taking the hydrodynamic interaction along the chain into account, while in Simha’s theory, and before in the theories of Jeffery (29) and of Eisenschitz ( 1 6 ) , the logarithmic term arose from approximating the molecule by an ellipsoid. In steady flow the above equations are strictly valid only for negligible gradient. The changes in intrinsic viscosity due t o preferred particle orientation, as produced by larger gradients, have to be deduced from the theories of Boeder (6), Kuhn (35, 37, 38), Peterlin ( 5 7 ) ,and others as discussed above. Summarizing the above theories, Figure 3 shows diagrammatically the intrinsic viscosity as a function of the 400 molecular weight - for stiff chains:
,
[q] a
ever, and even the stiffest become progressively flexible as the weight increases in a homologous series, A transition from rodlike to coiled molecules will therefore be found as indicated by the broken line in Figure 3. Polymers then differ according to the molecular weights a t which the transition from stiffness to flexibility is reached. For an appreciable range of molecular weights in this region, Staudinger’s rule, [ q ] = KmM’, will represent the relation reasonably well. As there will always be some interaction between some parts of a molecule, the concept of completely freely draining chains is recognized as unrealistic. Staudinger’s rule results rather from a compensation of effects-Le., chain stiffness tending to give rise to an exponent a larger than unity versus intramolecular interaction reducing a below 1. A similar compensation-e.g., a permeability falling off more slowly with increasing encompassed volume than expectedmay also be the cause of the often observed constancy of a over a Ride range of molecular weight at exponents smaller than 1. I n the above hydrodynamic theories, the constancy would have t o he explained in terms of very slowly changing functions ( F ) , ( F ’ ) , or (p); it fits well into Flory’s theory which, on the othw hand, does not allow for any but thermodynamic changes in the exponent after complete impermeability has been reached a t rather low molecular weights, and K has obtained its limiting value. The transition from stiff to flexible chain character with increasing molecular weight has been very well illustrated by Iiuhn and Kuhn ( 3 4 ) and by Peterlin (56). Some of the sets of curves of [ q ]us. M calculated by him as a function of stiffness and intramolecular interaction, are shown by Figure 4 a, and b. Here the curves designated g = 1 (where g is the cosine of the angle of deviation of subsequent chain segments from the straight line) represent rigid rods, while g = 0 stands for ideally flexible chains. The ratio a / b ( a bead radius, b average bead distance) stands for the extent of intramolecular interaction. The set of curves in Figure 4,a, being drawn for a / b = 0.1, thus is valid for very weak interactions; a / b = 0.3, in Figure 4, 6 , shows the case for medium degree of interaction. Figure 5 finally shows the same data plotted logarithmically as N / [ q ]us. NIIa,and contains some data for cellulose acetate, plotted by Peterlin for comparison 400
,!w __ and
log 144’ for flexible chains and random coiling: [ q ] 0: M The latter 1 kM’/*‘ relation, pointed out by Kuhn and Kuhn ($4) a n d empirically b y Wilson (7%)and derived by Peterlin (16, 64), is the form t o which the K i r k woo d -R i s e m a n equation can be simplified if only the first term of the series represented by function ( F ) is considered. T R A N s I T I o N FR o 11
300
300
PO0
PO0
ioa
100
+
7
RIGIDITY TO FLEXIBILITY. None of the type of polymer molecules considered a r e perf e c t 1y r i g i d , h o w -
November 1953
a
PO
40
60
BO
Figure 4. Relative Intrinsic Viscosity, [qIre1 = [v]a/[q]o a s a Function of Number of Chain Elements (56) For very small interaction, ao/bo = 0.1 b. For medium interaction, ao/bo = 0.3 Parameter LI varies from 0 (most flexible thread) t o 1.0 (rigid rod) o. Radius of bead representing chain element bo. Distance between two consecutive beads a.
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
2505
are flexible and thus less subject to orientation. Fitting the observed shear dependence into the theoretically calculated pattern (Figure 2), one might conclude that cellulosic molecules are never quite as soft as vinyl polymers (see also Table 11).
0.4
0.1
O F SHE.4R GRADIENT ON I N T R I N S I C VISCOSITY TABLE 11. EFFECT
0.1
[Calculations of K u h n a n d Kuhn ( 5 3 ) ]
0.4 0.3
L
/
,'
\
0.4
[?IO
Butylacetate
[?IO
= 30 0
Butylacetate
.
Cellulose nitlate D P = 7000
[?IO
0.1
= 11.7
Cellulose nitrate D P = 2300
0.5
0.6
Sample Cellulose nitrate D P = 640
,
= 60 0
G, Sec.-1
[?Ic
1000 2000 4000 8000 16000
11,7 11.7 11.4 11.1 10.5
500 1000 2000 4000 8000
29.3 28 8
500 1000 2000
58 0 66 5 53 5
29 8 28 0
28.6
[?IlG/ [?
10,
Obsd. 1 1 0.914 0.948 0.897
0.996 0 976 0 960 0.934 0,853 0 964 0 948 0 888
-
dhg
[?lG/[?1o
0.022 0.044 0.088 0,172 0.344
1 0.99 0.96 0.91
1
0.101 0 203 0.406 0.812 1.624
0.984 0.950 0.895 0.842 0.755
0 62 1 24 2 48
0 86 0 80 0 69
[?lo = intrinpio viscosity a t (f = 0. [?]G = intrinsic viscosity a t finite gradient. a'ha2 corresponds t o E in Equation 13 and in Figure 2, except for a nilmerical factor. Gtaudinger a n d Sorkin (68), Buchheim a n d Philiprioff (8).
I+
Figure 5. Logarithm of IV/[77lre1 0.3 (5.5)
+.
TS.
Log .Y'/2 for aolbo =
-
Experimental data for cellulose acetate in acetone ('V = D P ) (1,66)
COMPARISON WITH EXPERIMEhT
The following figures and data give examples FLEXIBILITY.
DIJIESSIONS.The actual dimensions of the cellulose nitrate molecules as calculated by the various theories permit the rounding out of the picture. Table I11 shows average molecular endb e n d distances as a function of M ,which may be compared with the fully extended lengths. Some [ 7 ] - M data were also interpreted according to the Fox-Flory theory, b y determining K graphically according t o Equation 22 and calculating O( therefrom (Table IV). Mandelkern and Flory (40) published data on the extension ratios (length in solution over fully extended length) for cellulose nitrate, cellulose butyrate, and cellulose caprylate. The longer side chains of these two derivatives seem t o effect impermeability at lower degree of polymerization than for cellulose nitrate, since K is here reasonably constant, and the dimensions are found to be smaller than anticipated. The greater stiffness of cellulosic molecules as compared with vinyl polymers, questioned earlier, is accepted in Flory's latest paper with Newman ( 5 1 ) on cellulose nitrate, as quoted above. Unfortunately, similar tables for other derivatives and for cellulose itself in a variety of solvents could not be prepared
of the viscosity-molecular weight relationship of cellulosic solutions. Figures 6 to 8 show the extent to which the rodlilte behavior is followed. Departure from the straight line of the plot of [ q ] 1 s. MZ/log M, in Figure 6 occurs a t about [VI = 5 , or = 400. Plotting as it were for coiled molecules-i.e., X7/17] us. M1/2 (Figure 7)-a straight line is obtained for the high molecular weights with a departure now a t the lower end, again corresponding to about Dp = 400. The same observations were made by Badger and Blaker (1). The authors' data scatter more widely, but the plot follows the same pattern (Figure 8). The results (27, 28) in cupriethylenediamine scatter still worse, reflecting the difficulties of the technique (Figure 8). The transition from rigid to flevible molecules should be paralleled by decreasing permeability. Since, according t o Fox and Flory (18, 19) Equation 21 is valid and the value of K becomes constant when the chains become impermeable, the transition should also be reflected in the values of K. Newman and Flory (51) have discussed such data and arrive a t the conclusion that cellulose nitrate becomes impermeable a t Jl S 400,000, very much higher than in the case of flexible molecules. Thus, cellulosic solutions contain fairly stiff molecules up to a degree of polymerization of approximately 400, above which the chains become increasingly flexible and eventually impermeable; from the slopes of the curves in Figures 7 and 8, the order of increasing stiffness is: cellulose in cupriethylenediamine (CED), cellulose acetate, cellulose nitrate. The flexibility does, of course, influence the shear dependence of [ q ] and explains why the effect is not much greater than actuallv observed: while the molecules are short and stiff, they are so Figure 6. Deviation of Experimental Data for Cellulose Nitrate in Acetone (27, 28) from Kirkw-ood-Riseman Equation for Rods small as t o require very large gradients for 0. Cellulose nitrate i n acetone marked orientation. When they are large, they
-
2506
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Vol. 45, No. 1 1
Cellulose 10
x
mentation-diffusion measurements. The figures are shown in Table VI11 and exhibit fair agreement between the two methods. Miinster (48, 49), by calculating b values of 32 to 35 A. from his viscosity data, confirmed Newman's values. He also found differences in b for cellulose nitrate from cotton and wood. Summing up what is known about dimensions, a good general understanding has been reached; the data relative to each other are fairly consistent and, for the higher molecular weights, certainly do not indicate fully stretched molecules. Many more data will be necessary, however, in order to establish absolute values of any accuracy.
102
0
6
4 lNFLUENCE OF THE SOLVENT
P
1
0
Figure 7.
P
, 5
4
3
-d 6 X 102
M/[77]us. MVB(56)
f.
Cellulose acetate in acetone (2) X. Cellulose acetate i n acetone (67) 0. Cellulose nitrate in acetone ( I )
I
because of the scarcity and uncertainty of data. A comparison of the hypothetical dimensions in solution of different polymers of the same fully extended length is given in Table V from interpolated data. The solution data were calculated according to the Debye-Bueche theory and show the greater extension of cellulose nitrate and cellulose acetate, the latter again being found more kinked than cellulose nitrate. Unfortunately, establishing the essential parameters from viscosity and molecular weights is tedious and needs very good fractions and carefully extrapolated values. Further data regarding dimensions were taken from light-scattering or sedimentation and diffusion measurements, and where possible compared with viscosities. Peterlin (66) recalculated the data of Sookne and Harris (67) and of Singer (66) for cellulose acetate in acetone. Table VI reproduces his figures, referring to the points in Figure 7. The extension ratios illustrate the transition from rods to coils, and the values for R show the dimensions in solution t o be expected over this range of molecular weights. The Kirkwood-Riseman parameter b is given as 23.7 A. Badger and Blaker ( 1 ) have determined values of R for cellulose nitrate direct from light scattering, comparing the calculation procedures for spheres, rods, and coils as molecular models. Their values are given in Table VII. Whatever the model employed, the dimensions calculated are seen to lie much below the fully extended length for the samples of higher molecular weight. Newman (50) has calculated the Kirkwood-Riseman parameter, b, from the values of Table VI1 and from his own sedi-
9
Good solvents prefer solvent-solute contact; reacting with, or solvating, the polymer exothermally, they stiffen and stretch the molecules where possible, thus influencing the entropy of mixing and giving rise to a large second virial coefficient. Regular solvents are less exothermic and hardly influence the entropy of dilution. Thermodynamically ideal solutions are not possible because of the nonideal entropy due to the size of the cellulosic molecules, but athermal solvents may exist. In poor solvents solute-solute contacts are preferred, so that the polymer has a more compact structure than that due t o random coiling and volume effects. In very poor solvents intramolecular attraction compensates for volume effects and gives rise to statistically ideal
I
0
O
0
I 40
(pya
I 60
m/[77]
Figure 8. us. ( B ) ' / s for Cellulose Nitrate i n Acetone and Cellulose in Cupriethylenediamine (27, 28)
0
Cellulose nitrate in acetone Cellulose i n eupriethylenediamine
TABLEIV. NITROCELLULOSE DATA"CALCULATED ACCORDING TO FLORY A N D Fox Ra, calcd.C, Ratio aab A. Ra/ Rext. 1,84 123 0 264 1 45 1,88 0.234 138 2 32 2.69 0.214 199 2.85 2 95 24 1 0.180 3 68 3.21 0.175 271 3.42 4 70 347 0.148 3.73 6 35 385 0.136 84 650 4.33 7 10 440 0.131 4 58 38 865 8 65 0.116 518 9 4 4.75 a5 975 554 0.111 68 1000 9 78 4.80 564 0 109 12 7 5.47 89 1250 652 0.101 86 1745 16 3 6.07 803 0.088 16 1 6.00 90 1830 0.087 817 91 2330 21 0 6.77 964 0 . oao sa 2700 25 3 7.57 1080 0.078 Unpublished data of Immergut and Mark. b = KM1/2a.8, From plot of [ q ] 2 / 8 / M 1 / 8 ~ M 8 . / [ ? ] (see Equation 22), we ob"Ling, Knowing K ,we obtain amL = R i a (see Equation 2 0 , where Ra is the random ooil value, 2boZ'/a, bo = 5.15 A. Sample NO. A6
TABLE 111. CELLULOSE NITRATE IN ACETONE"
I 20
FPN
4n
[?I
1 12
~~
Q
45 46 54 63 83 84 68 89 86 90 91 Data of
90 1.12 465 100 2.32 142 930 iao 2.95 1340 170 260 3.58 1550 184 300 4.70 2350 230 460 7.10 3350 270 650 9.78 5150 334 1000 12.7 6480 370 1250 440 16.3 1740 9000 16.1 1820 9300 450 21.0 2350 11900 510 Immergut, Sohnrz, and Mark (.a?,
November 1953
.
158 255 311 349 439 566 738 855 1040 1050 1247
285 4 02 4 83 518 642 765 948 1062 1254 1280 1450
I
Q
INDUSTRIAL AND ENGINEERING CHEMISTRY
-
2507
TABLE kr.
COhlPARISON O F
Polymer Polyisobutylene GR-8 He\ ea rubber Pollstqrene Po15 methylmethacrylate Cellulose acetate Cellulose nitrate
DIFFEREXT POLYhlERS
Sol\ ent DiisobutLlene Toluene Toluene Toluene Benzene Acetone Acetone
Temp,
c
20
[?I 0 63
WITH SAVE
-
EXTENDED LENGTH (68)
-
M7l
DP
a
112,000 71,500 68,000
2000 1180 1000
0 64 0 66 0 67
L, A
Ea,
5000 5000 5000 5000 5000 5000
115 122 120 160 ZOO
I
A.
solvent should exhibit a higher intrinsic viscosity, but solvent effects should be smallel than for highly flexible vinyl polymers of
comparable length, the ‘Iutions should be distinguished 200,000 2000 0 70 25 o 94 200,000 2000 0 76 by constant osmotic intercepts 30 15 0 262 000 970 0 78 565 and L, factors from ap30 9 5 292 000 970 0 92 5000 713 parently high intrinsic viscosities in associating solvents. [For experimental values of [77] in a variety of solvents see ( 2 7 , 28, 40, 46. 6R).] In terms of the hydrodynamic theory, the quality of the solvent will express itself in the values of R,, or b; decrease in solvent power, for example, will increase the flexing of the chain and thus decrease the value of b. There will be another, minor, solvent effect, inasmuch as viscosit>- of the solvent enters into the equations by way of vu. In terms of the exponential equation, decreasing solvent poFer will, of course, reduce exponent a. For the experimental verification of the behavior just outlined, i t is necessary to rely largely on the data already quoted for molecular size a n d shape (Table VIII). For the most part recent data were employed, as the older ones have been discussed in previous publications, but some are again quoted for comparison. Apart from hydrodynamic parameters, Table VI11 contains Kx and a values from measurements intended for I 1 I 1 the calibration of viscosities only. Figure 9, finally, 1000 p zoo0 3000 shows graphically the course of the molecular diFigure 9. (iMI77]/0)‘/3 z’s. E for Cellulose and Cellulose Derivatives mensions with number-average molecular weight a5 in Different Solvents (27, 28, 40, 51, 52, 56) a function of the solvent. 1. Fully extended length Being derived from so many different sources, 2, 3, 4, 8 , 9. From (&2):/ (light scattering) the values contained in Table VI11 must be con~ i n 5 , 6, 7, 10, 11. From M 2. Cellulose nitrate in methyl ethyl ketone sidered against their background. As it is 3. Cellulose nitrate in acetone 4. Cellulose nitrate i n ethyl acetate important whether or not the measurements were 8. Cellulose acetate in acetone carried out on fractions, this has been recorded 9. Cellulose tributl rate i n butanone where possible. Similarly, the type of molecular weight average, resulting from the method employed, is given. coil dimensions-Le., t o a 2nd virial coefficient zero for III +- m , Values for K , could obviously be given only where Staudinger’s and a factor of cy = 1. Physically and thermodynamically these solutions are not ideal, but owe their behavior to two law was deemed by the author t o apply. High figures occur balancing deviations from ideality; the same situation can be where the exponent is appreciably lower than unity. According to brought about, for example, in real gases by means of pressure, Husemann and Schulz (26), higher values also pertain for fractions than for unfractionated samples. and is characterized by rapid departure from ideal behavior by slight changes in conditions. I n the case of very poor solvents, The exponents a are unexpectedly low in view of the supposed stiffness of the molecules. Intramolecular hydrodynamic minor changes in solvent pow-er may cause precipitation, or. on interaction therefore appears high, which is not too surprising in the other hand, increases of cy. view of the bulkiness of the substituents. From the order of a, The danger of association, or partial precipitation, in even the most dilute solutions of poor solvents which can be measured ethyl acetate is the best solvent for cellulose nitrate. Acetone is about equal for cellulose nitrate and cellulose acetate. (thus affecting the extrapolation t o [q]),is greater with cellulosic niaterials than vinyl polymers. The latter possess the more The values for b and 6 quoted are those calculated originally flexible molecules; conditions enhancing solute-solute contact will. in dilution, primarily lead to tighter coiling. Cnder the same circumstances, all contacts of molecules unable to fold TABLE VI. DIMENSIONS FOR CELLULOSE ACETATEIN ACETOKE back on themselves will be intermolecular. Therefore. while, [Calculated b y Peterlin ( 5 4 , 56) I with flexible molecules, working in poor solvents may be justified Red, L, R, in order to obtain quasi-ideal solutions, this is not indicated for 121, A. -4. A. L/R cellulosic solutions of low or medium molecular weight, as such 10,350 148 190 190 163 1.2 molecules do not follow random statistics, nor are they free to 325 419 930 447 2.1 51,000 143,000 545 701 2600 770 3 .4 respond t o solvent changes in an as yet predictable manner. 3500 903 3.9 194,000 634 817 The above remarks, together with the earlier discussion, lead R calculated from viscosity data one t o expect rather different solvent effects for low and high R;d calculated from sedimentation data) (64) L = length of stretched molecule molecular weights. In the case of the former the small, or absent. R = effective radius ( 6 6 ) L / R = coiling ratio flexibility should render the intrinsic viscosity insensitive to the nature of the solvent. For large molecular weights the bettei 30 25 25
0 86 0 85 0 67
I
2
2508
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 45, No. 11
Cellulose DIMENSIONS FOR CELLULOSE NITRATEIN ACETONE lCalculated bv - Badaer - and Birtker (1)1
TABLE VII.
-
Sample S4a SM sa14
PSl
Pa B A
Sphere
0 30 1 30 2 22 2 98 6 86 14 90 21.00
TABLE lrIII. x
M , Light Scattering Rod Coil 9,400 9,400 9,400 35,000 35,000 35,000 50,000 49,000 49 ,000 87,000 89,000 93,000 298,000 312,000 319,000 356,000 370,000 400.000 394,000 518,000
[?I
Cellulose Derivative Cellulose nitrate
d (sphere),
A.
...
..
..
1.1
0.91
88-110
... ...
(0.98) (0 91)
..
..
*
30 30-35
10
..
CELLULOSE DERIVATIVES
b, A.
1010 G./Seo. 12
(=m/[?])([?]/Me)
Solvent Acetone
..
32 (-57)
8
46
98-122 77-100
... ... ...
Ethyl acetate
Cellulose dinitrate Cellulose acetate
Ethyl lactate Butyl acetate .4cetone Scetone
...
80 62 71
.,. 230
...
150 230
Cellulose acetate butyrate dcetone Cellulose butyrate MEK Cellulose caprylate Toluene Cellulose Cuprammonium CED
... .,. ... 260 230 170 124-156
..
0 80
0 67 0.48
0.99 0.99 1 03 0 92 0 71 0 79 0 82 0 80
2.57
.. 0.38 1 22 21 1.91 149 2.4
..
.. ,.
..
1.37 0.46 0.93
0 83 0 90 0 75
o:s5
0’81
..
*.
.. ..
1 33
0 90
80
11 o
..
.. .. *.
.. .. ..
.. .. ..
..
..
RO/(op)“2. A peculiar difficulty should here be pointed out. The statistical model employed by Kirkwood and Riseman sets the number of links in the chain, Z,equal t o the number of atomic bonds along the chain backbone. In vinyl polymers, this number is twice the average degree of polymerization, and has t o be used in calculating 6. I n cellulosic solutions, from the authors’ recalculations, most authors seem t o have used Z = This implies one link per monomer unit, an assumption open t o question, since the connecting glucosidic oxygens exert two symmetric bonds, and Z = 2 might be assumed for the same reasons as in vinyl polymers. This question is further cpmplicated by the fact that not one, but sequences of glucose rests are certain t o act a s statistical units, with a corresponding reduction of the number of links. I n principle the number of monomer units in a z c a l l e d statistical unit is determined by the factor containing cos Q in Equation 20a, which is the correction factor for lack of free rotation. The model bv Kuhn and Kuhn (36‘) is in this respect more explicit, introducing monomer sequences as statistical units. Calcula-
(m).
(m)
November 1953
33
-
(+)
References (80)
,M
+ M,
(87,181
’- % s) +
++
Light scattering B/(DP)1’2
Diffusion ( 1 )
M,
M,
35
(+) (+)
by the authors. They were all derived b y using the KirkwoodRiseman equations for coils. This is due t o the fact t h a t usually more and better data were available for the high molecular weight range and the evaluation procedure there is more convenient. The same is true where the b values were obtained from sedimentation or diffusion data. The equations for rods were used only t o test for the nature of the [VI-M function. Calculating from light scattering data where the choice is of lesser consequence, the coil model too was employed and b obtained as
P
34
Frac- Avertions age ? M,
(B6)
(47) Cotton (48) Wood pulp(48)
.. ++ nMl,, 3 5 52 (87) 4- x} .. .. .. .. .. .. ( f ) M, M , .. .. + M,
16
Lext,? A.
170 630 900 1670 5700 6550 9200
445
725 960 1250 (1120) 2008
890 1210 1690 (1450)
AND
