effects of hampered draining of solvent on the translatory and rotatory motion of statistically...

JOURNAL OF POLYMER SCIENCE VOL. IX, NO. 1, PAGES 1-33

Effects ox Hampered Draining of Solvent on the Translatory and Rotatory Motion of Statistically

Coiled Long-Chain Molecules in Solution. Part 11. Rotatory Motion, Viscosity,

and Flow Birefringence*

HANS KUHN and WERNER KUHN, Institute of Physical Chemistry, University of Basel, Switzerland

V. ROTATION OF STATISTICALLY COILED CHAIN MOLECULESt

a. The Freely Draining Chain Molecule. Definition of

In his 1934 paper W. Kuhn had already pointed out that the viscosity and flow birefringence of solutions of macromolecules were essentially determined by the rate at which, due to Brownian motion, the long axis of an- isotropically shaped solute particles changes its orientation in the static, or streaming, solution. As this rate, in its turn, depends on the hydrodynamic forces resisting the rotation of the long axis HI of the coil through the solvent, the problem reduces to finding the functional dependence of this resistance on the molecular dimensions. For this purpose the procedure which served in Part I to determine the resistance to translatory motion of the freely, partially and nondraining molecule, will now again be employed and it will be seen that, in some respects, the rotatory resistance is related to the translatory resistance.

As before we can expect that in their hydrodynamic behavior the molecules will be essentially freely draining coils when their degree of polymerization is low, and essentially nondraining coils when it is high. At intermediate degrees of polymerization the resistance characteristics will also be intermediate, and to describe the transition we can, exactly as in the case of translatory motion, introduce a resistance parameter By virtue of the theorem of hydrodynamic similarity discussed in Part I, this parameter can be completely determined as a function of certain molecular dimensions by means of experiments on macroscopic models. Certain special cases

* Translated by Alexander Silberberg. The numbering of the sections, formulas and references follows on from Part IZ6 and

the symbols used retain their previous significance with the exception of the viscosity of the solvent.

1 In Part I we designated it by 7 and in the present paper by 70.

2 H. KUHN AND W. KUHN

can, however, be treated theoretically and one of these, namely, the ideal case of the freely draining chain molecule, will now briefly be discussed.

As has been demonstrated in a paper by W. Kuhn and H. Kuhn7 we can obtain a reasonably good approximation to the resistance experienced by a freely draining chain molecule during a rotation of its long a x i s HI (i.e., during a rotation of the vector h between the end points of the chain) if we replace the coil by a dumbbell model of such dimensions that the distance between the two spheres equals the distance h between the end points of the corresponding chain and that each sphere offers the same resistance to translation as a quarter of the chain molecule. If therefore L is the hydrodynamic length of the chain molecule, as defined in Part 1, then a sphere equivalent in resistance to a chain of length I/& is attached to each of these end points (or to each of two other characteristic points such as PI and Pz of Figure 5).

Fig. 5. Mode1 by which a chain molecule may be replaced in order to describe its rotational reaietance. A quarter of a chain is thought concentrated in each of the points PI and PO. These two points lie on the long axia HI of the mil, each being at a distance HI from the midpoint 0 of the axis.

In fact, it is found to be of advantage, in the interpretation of the experimental results obtained with large-scale models (see below), to choose another, equivalent, formulation of the above approximation. Instead of attaching the two quarter molecules to the two end points of the chain, two points such as PI and Pz in Figure 5 are chosen. These are situated on the longest dimension H1 of the coil and lie a distance H1 apart, that is, they lie each at a distance H1/3 from the center of rotation at 0. On the average, therefore, the distance between the points Pl and Pz is z/arl;, and this, as was shown in Part I (see equation 6), is equal to i.

The two models are quite equivalent, but, as the particular hydrodynamic characteristics of an individual codguration are better described by thegarameter Hl than by h, the model (Fig. 5) will be preferred in the present instance.

Let us now consider the rotation of the dumbbell model (Fig. 5 ) about an axis passing, perpendicularly to HI, through the point 0. If the angular velocity of the rotation is i, then clearly the velocity with which the points

HAMPERED DRAINING AND MOTION OF COILED MOLECULES. I1 3

PI and Pz move relative to the solvent is 5 = 1/3Hl& In the event, then, that the chain molecules can be regarded as freely drainingeoils, we see that in the model each of the two quarter molecules (of length L/4), situated at the points Pl and Pz, respectively, will have acting on it a force F given by the relation:

(42) where X is the friction factor dehed by equation (2) (if h is written instead of A,,,,,), and where according to equation (3) X approximately equals 3u/2.

The torque M which must be acting on the rotating molecule in order to maintain an angular velocity will thus be given by a couple formed by the forces given in equation (42). Remembering that the arm of this couple is z/3Hl we have:

F = (L/4)705h = (L/~)~O(HI/~)?X

As soon as the chain molecule can no longer be regarded as freely draining, that is, at higher degrees of polymerization, the factor X in equation (42a) ceases to be constant and the limits of applicability of equation (42a) have been reached. We are, however, free to use this equation by way of definition. That is, we may write:

M = ( L / ~ ~ ) ~ I o H ~ ~ X , , ~ (43)

where now so depends on the degree of polymerization that it charac- terizes the rotational resistance of the molecule at any degree of draining. Equation (43) thus defines A,,, in the same way as equation (2) defined A,,,, i in equation (43) is therefore the mean angular velocity just as 4 in equation (2) represented the mean linear velocity.

The actual instantaneous angular velocity + of a coil rotating under an applied torque M about an axis perpendicular to HI depends upon the coil's incidental shape and the inclination of Hz relative to the rotational axis.* In analogy to @a), therefore, we can write the following expression for the rotational resistance couple of a given chance configuration:

M = (L/WrloH?&, rot (434

where X , lol is the shape factor of a given molecule at a certain instance and for a particular orientation.

Some ideas about the general resistance factor now introduced can immediately be formed. Thus, in the limiting case of the nondraining coil (very high degrees of polymerization) we can simply use the equations of R. Gans for the hydrodynamic rotational resistance of ellipsoids (see Sec- tion d below). Similarly it is possible to treat the other limiting case (freely draining coil) by the method employed in Part I Section IIc, p. 525

* HZ is the maximum extension of the coil perpendicular to HI (see Part I, p. 523).


et seq., obtaining in this manner an at least semiquantitative value for A,, (see section f below). Even in the case of intermediate draining conditions an approximate value of A,, can be obtained on the basis of the empirical equation (29) for Atlans (see Section c, below).

b. Determination of A,, by Means of Experiments on Large-Scale Models

We are, however, more certain to obtain the right result if instead of such estimates we carry out actual experiments using large-scale wire models and interpreting the result by the theorem of hydrodynamic similarity discussed fuUy in Part I, section IId. In this way we shall arrive at an expression for A,, which is generally valid and incorporates all draining conditions from the freely draining coil to the nondraining case.

Fig. 6. Apparatus used to determine the rotational resistance of macroscopic models of chain molecules.

In its essentials our approach will be to determine the value of A,, ,ol in the case of a chain molecule of submicroscopic dimensions directly by measuring the hydrodynamic resistance to rotation (i.e., A,, of a geometrically similar but enormously enlarged model of the molecule.

The apparatus employed to measure the rotational resistance of such a large-scale model of a chain molecule is illustrated in Figure 6. The model, C, is dipping into a viscous liquid (a concentrated invert sugar solution) and has a torque applied to it by the pull of a weight G which is attached to a thread running over pulleys R1 and Rz. If r is the radius of pulley Rz and G is the mass of the weight G the torque M applied is given by:

M = Ggr

where g is the acceleration due to gravity. Furthermore the velocity of descent &/a? of the weight G is given by :


&/dt = rQ

where tj is the angular velocity with which the model is rotated in the liquid. As the other quantities, qo, L, and HI, entering equation (43a) are easily

determined, it is obvious that Xi, can be calculated from the velocity of descent &/dt of weight G.

For a more exact determination of X i , rot it becomes necessary to take the rather small frictional resistance of the bearings 11 and 12 into account. It is seen that the torque M applied to the axis 1112 has to overcome this frictional resistance besides the rotatory resistance of the model itself, and instead of equation (43a) the more exact relation ob- tains:

M = (L/18)~oEG+i,rot f QT (4-W

where T is a bearing friction coefficient. We can determine the magnitude of the quantity 7 by rotating spheres of known diameter in place of wire models in the apparatus in Figure 6. It is found, however, that when roller bearings are used at points l1 and 12

the bearing friction coefficient T becomes of such small relative magnitude that this cor- rection can in practice be neglected.

From the result of experiments with a number of models corresponding to different chance configurations of the molecule we can then determine

0.4

0. :

0.2

0. I

T - \IN*

I 0 1 2 3 4 . 5 6 7 8 9

Fig. 7. Plot of 1/X,,c against XrOi is the shape factor defined by equation (43) and determined by rotation experiments with macroscopic models of chain molecules. N,,, is the number of preferential statistical chain elements in tbe model. Dots: reciprocals of the values obtained in the case of the individual models by rotation about an axis perpendicular to the longest dimension of the coil. Each dot is related to one individual model and represents the mean obtained when rotating the model at different inclinations of the cross axis HZ to the axis of rotation. Circles: reciprocals of the XTo, values found by averaging over the values obtained with the various individual models with common N,. The individual dots display an even more pronounced scatter than was the case in the translation experiments (Fig. 1, Part I).

in a case where A,,,/& = 3.2.

The mean values, however, lie as before on a straight line.

the form factor A,,, defined by equation (43) by averaging over the individual At , values found. To establish X,,, as a function of the number, N,,,, of statistical chain elements and the length A, and thickness d , of the


statistical chain element, models were constructed corresponding to different values of Nm and to different values of the ratio Am/dh. The results in the m e where &/dh = 3.2 are recorded in Figure 7. In this figure we have plotted l /Arof (circles) and l / A i , rot (dots) as ordinates against dK as abscissa. As in Part I each dot represents the result for l/hr, obtained in the case of a particular model, while the small circles give the mean value l /Arof taken over all results corresponding to a particular value of M,,,. We note that the scatter of individual points about the mean is in the present case both absolutely and relatively larger than in the case of A,,,,,. Nevertheless the mean values again come to lie on a straight line in a plot of l/Arot against flm. Besides such experiments with models where Am/dh = 3.2, experiments were also performed with models in whose cases the Am/dh ratio was 1.6, 6.4, and 12.8, respectively. With the accuracy which these results permit we can summarize the findings in the following expression:

l /Arof = -0.05 + 0.12 log (A,/dh) + 0 . 0 3 w N , (44)

This is a formula whose structure is analogous to that of equation (29), the two formulae being distinguished only by Merent numerical coefficients.

c. Approximate Relationship between A,, and A,,,,

If the approximate treatment of section Va were strictly correct, it should be possible to determine Are, from the corresponding value of A,,, whatever the degree of polymerization. In this approximate treatment we obtained the rotational resistance of a chain molecule in solution in the following manner: we imagined that a quarter of the total chain was located in each of the two points PI and Pa in Figure 5, and assumed that the hydrodynamic effects of the rest of the molecule could be neglected. Insofar as this model is applicable one should be able to obtain the correct force F from (42) by introducing for A the value of A,,,, in the case of a chain molecule of hydrodynamic length 1/4L. From equation (29) we End:

A(Nm/49 Am, dh) = 1/[0-02 + 0.16 log (Am/dh) + O.1-I (45)

which on introduction into (42a) and comparing with the definition (43) yields the result :

(454 (approximation on the basis of X,,,,)

We see that this is only a rough approximation to the experimentally determined expression (PP), and the importance of carrying out the described experiments and making ourselqes independent of this approximation will thus be appreciated.

Comparing equations (29) and (44) we see that the coefficient multiply- ing dK in (29) is 0.1 while the corresponding coefficient in (44) is consider-

1/A,ot = 0.02 + 0.16 log (Amldh) + 0 . 0 5 f l m


ably smaller, 0.037. This fundamental difference between the values of A,,, and A,,, is a consequence of the differing hydrodynamic behavior of the random coil in translation and rotation. In the rotation, the more freely draining outer periphery of the molecule, say the two outer quarters, offers all the effective resistance and the rest could be neglected. This is the signiiicance of equation (45a) and the measure of agreement between (45a) and (44) is substantiation for this point of view. If we thus consider the model in which the two quarter molecules have been attached to the points PI and P2 in Figure 5, we note immediately that these two short molecules of only N,/4 statistical chain elements are much closer to the limiting case of free draining than the whole molecule of N , chain elements in translation. We see therefore that the same coil which in a rotation n a y still be largely draining may in a translation be practically nondraining.

d. in the Cases of High Degrees of Polymerization (Considera- tion of the Nopdraining Coil)

If we are considering molecules with very large values of N,, we shall find that despite its small coefficient the term containing N,,, in equation (44) will become so large that the other two can be neglected in comparison. Thus:

(from equation (44) for large values of N,)

We now compare this with the resistance factor which would be expected for nondraining ellipsoids whose external dimensions are related to the molecule as discussed in Section Ira.

According to Gansn the resisting moment of an oblate ellipsoid of revolution, whose semiaxes are c and a, is given by the following expression:

4 r p 2 + 1 M = 470 - ca2G 3

p2 - 1 P" - d p 2 - 1J

where : p = c/a (474

is the axial ratio of the ellipsoid. In the case of statistically coiled chain molecules the m h value of the axial ratio p is approximately 2.0. (An exact discussion of this mean with the aid of the distribution function has shown that all practically occurring values of p lie between 1.2 and 3.6.) For such values of p the fraction on the right-hand side of equation (47) reduces to about 1 . 2 ~ = 1.2 (c/a) so that the whole expression can more simply be written as:

M = (16r/3)qoc2aG - 1.2 = 207roac2U; (47W


Remembering that c = H1/2 and that (see equation (7)) a = H2/2 N

0.35flAm we can rewrite equation (47b) and obtain:

M = 20q,6[0.35dxAm(H12/4) ] = 1.75$%H12$dxA,

which in view of equation (4) gives:

M = (1.75/fim)L70Hi2G (474

X,,/18 = 1 . 7 5 / d x ; ArOr = 31/dx (48)

(484

By comparison with equation (43) we now find that:

or :

l/A,,, = d N , / 3 1 = 0.0321/K (nondraining coil, the coil regarded a s ellipsoid)

On comparing with equation (46) we conclude that here, as in the case of X,,,,, (equations (31) and (31a)), the experimentally found resistance factor XrOt = 27Ni1la is in the region of large values of N, nearly equal to the value obtained from equation (48), namely, X,,, = 31N;'/'.

e. Dependence of A,, on the Special Shape of the Molecule at a Given N,,,

The empirical relationship (44) gives A,, as a function of the dimensions A,,, and dn and the number N , of preferential statistical chain elements occurring in the molecule. In this equation X,, represents a mean which a t given N,, i.e., given degree of polymerization 2, is taken ouer very many differently shaped molecules which have this N , value in common but whose other parameters, particularly HI, are subject to wide variations. In the case of each large-scale model, however, the value of Xi, is determined from the definition (43a) and applies to the model individually..

It is interesting to investigate whether, or not, the Xi, values of molecules which agree in A,, d,,, and N,, but differ in their Hl values, are for practical purposes in agreement with each other. As will be seen from the following discussion, an independence of the parameter, A,, of the value of H1 can be expected in the limiting case of free draining and nondraining:

(1) I n the case of a freely draining chain molecule, X in equation (42a) can be regarded as the force required to shift a piece of chain, 1 cm. long, with a velocity of 1 cm./sec. relative to a solvent of viscosity of 1 poise. This force, under free draining conditions, must be independent of the degree of polymerization 2. It must, however, also be independent of Hl, as it is independent of whether the fraction of the chain is far from or close to the point about which the molecule is being rotated.

(2) In the case of a completely nondraining coil we know that the conditions are primarily as described by equation (47b). From this we obtain equation (47c) by remembering that the large semiaxis of the ellipsoid c = '/ZHI, where HI represents maximum linear external dimension of the


particular coiled chain molecule (Fig. 5 ) . For the small semiaxis, on the other hand, we can substitute the mean value H2/2 = 0 . 3 5 d R A m , it having been shown by W. KuhnZs that the mean value H2 of the lateral extension of the coil is practically independent of the incidental value of Hl.

Introducing equation (47c) into the definition for A,,, equation (43), we find that eliminates itself so that the resulting expression (equation (48)), i.e., the shape factor determining the rotational resistance of the investigated chain molecule, is independent of the longitudinal extension HI.

We conclude that the rotational resistance factor A,,, of nondraining ellipsoids with different long axes Hl will be the same, if the different values of Hl do not materially affect the lateral extension H2. Provided, therefore, that the lateral extension of nondraining coils remains the same we can state that the value of A,,, at a given degree of polymerization will be independent of the incidental values of the longitudinal extension HI.

Summarizing, we conclude that, in the case of coiled chain molecules which are in one of the limiting conditions, i.e., either freely draining or completely nondraining, the values of A t , r o t will be nearly the same in cases where individual molecules possess the same degree of polymerization, but differ with respect to their longitudinal extension Hl. It is sug- gestive to suppose that this applies also in the case of partially draining coils, and that the quantity Xi, ,Of will, a t any given degree of polymerization, bear no systematic relationship to the incidental dimensions of the coil. This supposition has actually been confirmed by the indicated experiments with macroscopic models.2 It is, however, impossible to discuss this aspect in greater detail in the present paper.

f. A,, in Cases of Very Low Degrees of Polymerization (Consideration of the Freely Draining Coil)

It was pointed out in Part I (when discussing the case of A,,,,,) that it is actually hopeless to expect that expression (44) will apply in cases where N, < 1. Even when N , = 1 it is doubtful whether equation (44) is valid, as it has been verified experimentally only for values of N , 2 5. If we, nevertheless, try and introduce the value of N , = 1 we obtainfrom equation (44) the resiilt:

l / A r o , = -0.013 f 0.12 log (Am/dh) (49)

(from equation (44) for N , = 1)

As was done in the case of the translatory resistance factor we will compare this result with the resistance factor in the case of an ellipsoid with semiaxes c = '/2A,, a = 1/2dh (see equations (47) and (47a)). We equate this expression for the torque M in the case of an ellipsoid with the torque obtained from'equation (43). Putting L = Hl = A, and solving for A,,, then gives:


p 2 + 1

- A,, = 12n

p2 - 1

where p now has to be put equal to A,/dh. For large values of p, i.e., when A , 3 dh, this expression reduces to:

12n -1 + 1.386 + 4.6 IOg (Am/dh)

- - - - 12r -1 + 2 In 2p Xro, =

0.386 + 4.6 log (A,/&)

or : l / A f o t = 0.010 -k 0.12 log (A,/dh) (494 (for N,,, = 1; the individual statistical chain element regarded as ellipsoid)

We note that the difference between the expressions (49) and (49a) is remarkably small particularly when we take into account that in putting N , = 1 we have used the expression (44) in the extreme limit of its applicability. The difference between the results obtained from equations (49) and (49a) is approximately 20% of the value of A,, when Am/dh, for example, equals 10.

In the case of oery s d l values of the degiee of polymerization we must replace the quantity A,,, in equation (49a) by Zb (where b represents the hydrodynamic length of the monomer residue). We then obtain:

l /Afog = 0.010 + 0.12 log Zb/dh (49b)

(from equation (49a) for N , < 1; 2% >>d,,)

In general we may remark that it is possible, by means of equation (44, to express Afar as a function of the degree of polymerization Z, provided the parameters A , and dh of the substance under investigation have been de- tennined. This could, for example, be done by measuring the rate of diffusion or sedimentation. In this manner, using data reported in Part I, Section 4, we obtain the following equation for tpe case of nitrocellulose in acetone:

l /Arot = 0.03 + 0.0132/2

g. Rotational Mobility and Rotational Diffusion Constant

As rotatio+ mobility prOt we define the angular velocity which the coil would acquire in a fluid of viscosity 7 0 when acted upon by a torque of 1 dyne cm., and when rotated about an axis perpendicular to HI. From equation (43) we then obtain:

pros = hLZbos1 (50)


If we now substitute for A,, from equation (44) we obtain the following expression for the rotational mobility of a coil about an axis perpendicularly through the midpoint of the vector HI:

mi = (~~/[BoL%I)[-O.O~ 0.12 log (Am/&) 0-037flmI (51) In the place of Nm we can introduce the degree of polymerization Z or the

molecular weight M of the chain. Thus using equation (34a) or (34c), as the case may be, we find, respectively:

pFOt = (18/[qoLH;]) [ -0.05 + 0.12 log (A,/d,) + O.O372/blA,d5] (51a)

= (18/[qoLH;]) [-0.05 + 0.12 log (A,/&) + 0.037db/(AmM,)dKf] (51b)

The rotational mobility is related to the rotational diffusion constant Mo being the molecular weight of the monomer.

D,, by an equation which is analogous to expression (33) :

O706 = .&OfkT (52) where k is the Boltzmann constant, and T is the absolute temperature. The rotational diffusion constant can thus be expressed as:

Dro, = (18kT/[~oL~])[-O-05 + 0.12 log ( A m / & ) + 0.037flml (53) In this expression we can, as was done in equations (5la) and (51b), re-

place N , by the degree of polymerization 2, or the molecular weight M. Furthermore, it is also possible by the use of equations (4) and (6) to replace L and HI (in the expressions (51a) and (51b) as well as in (53)) by N,, etc. For instance, instead of equation (53) we obtain:

D,, = (9kT/[qoA$Ni]) [-0.05 + 0.12log (Am/&) + 0.037flI (53a)

Once D,, is known one can determine the time 1 it will take the molecule to rotate ita axis through an angle Aq. For this one uses the relationship

- A@ = 4D,,t (54)

The macrotime for a revolution, &,, Le., the average time in which the Brownian motion rotates the axis of the chain molecule through 360°, is then obtained when Ap in equation (54) is put equal to 2s, and the equation is solved for the time. For the macrotime of revolution e,, we thus obtain from equations (50), (52), (54), and (6) the general expression:


Consequently 8,0, has the general form:

eroi = arOIZ2/(1 + LdZ) (554

The quantities aror and b, , which are constants in the case of a polymer homologous series, can be determined from equation (55) once the viscosity qo of the solvent, the hydrodynamic length b of the monomer residue, the length A,, and the thickness dh of the preferential statistical chain element are known. We have shown in Section I1 that the last two of the above quantities are experimentally determinable by measuring the sedimentation constant sL or the translatory diffusion constant D. For example, using the data given in Section IV we find that, in the case of nitrocellulose in acetone, this results in:

8,, = 2.9 - 104Z2/(1 + 0.432/2) (seconds) (55b)

VI. DEPENDENCE OF VISCOSITY AND OF MAGNITUDE AND ORIENTATION OF FLOW BIREFRINGENCE ON DEGREE OF

POLYMER1 ZATION

a. General Relationships

In the beginning of Section V we saw that the viscmity and streaming birefringence of solutions of chain molecules were intimately connected with the hydrodynamic effects associated with a rotation of such a molecule in the solvenl. This is of course a natural consequence of the fact that, in a solution subjected to simple shear, the most significant motion performed by a solute molecule is its rotation (taking place with nonuniform angular velocity) about its center of gravity.

On the basis of this fact we were, therefore, in our earlier papers enabled to express the viscosity and streaming birefringence of such solutions in terms of a hydrodynamic resistance factor X which was specific to the type of molecule considered, and whose exact value could be left open until the final expressions were being discussed. This has proved of advantage in two ways. First, it was possible to derive a series of relationships from which the resistance factor could be simply eliminated. This applies in particular to combinations of the expressions for the viscosity and the orientation and magnitude of streaming birefringence. Such relationships from which X has eliminated itself are then in large measure independent of whether the chain molecules are freely draining, partially draining, or completely nondraining coils. Second, in those instances where it was not possible to eliminate X this approach enabled us to arrive a t generally valid final formulas in which the dependence on the degree of draining was completely contained in A. If then, as is now the case, the value of X has been determined, it only remains for us to introduce it into the final equation without having to repeat the entire calculation in detail.

In view of the above it is to be expected that the effects of partial draining


can almost entirely be accounted for if in our previously published expressions for the viscosity and streaming birefringence we introduce A,, from equation (44) in place of A. Thus, according to W. Kuhn and H. K ~ h n , ~ ~ the spec& viscosity of a solution containing C chain molecules per cubic centimeter is, in the case of small velocity gradients q in the fluid, given by:

~ s p = (X/48) [LNrnAHCl

or : [TI = (Tsp/C)limqqq = o = @/48) (N4/Mo)Amb2Z (56) c = o

where c is concentration in grams per cubic centimeter, * NA is Avogadro's number per mole, and M, is the molecular weight of the monomer. Re- placing X by A,,, from equation (44) in the above expression we find:

= (%%) = A,b21Yk ___ 2 C l irnmq=o 48Mg -0.05 + 0.1210g (Arn/dJ + 0.037db/AmdZ

c = o

(57) A rnb21\$ 0.562 -~ -

Mg -1.3 + 3.2 log (Arn/dh) + db/A,dZ I n analogy to this we obtain the following expression for the streaming birefringence index:%

where nl - n2 is the birefringence, no is the refractive index of the solvent, a1 - a2 is the anisotropy of the polarizability of the preferential statistical chain eIement, and q is the velocity gradient. Similarly the orientation index in the case of large and small shape resistance3' can be expressed as follows:32

3 X Amb2 2 4 8 kT - 2* (large shape resistance)

1 X A,b2 2 48 kT - - Z2 (small shape resistance)

In these equations w is 45 O direction.

the deviation of the angle of orientation from the

If we now replace A by from equation (44) in all above relationships we obtain expressions which will apply in all cases of partially draining chains:

* The symbol c has also been used to represent the small semiaxis of the elongated ellipsoid of revolution, but this duplication in the notation should not lead to confusion.


(ni + 2)' 4l1r ( ~ 1 - CYZ Amb'NA 0.232 -______ 6% 3 kT Mo -1.3 + 3.2 log ( A m / & ) + -42

Amb' 0.84Z2

(61)

-1.3 + 3.2 log ( A m / d h ) + d b / A , F Z c=o

(62) (small velocity gradient, large shape resistance)

1 Amb2 0.84Z2 [+(9 = - - VOQ limwq=O kT -1.3 -I- 3.210g ( A m / d n ) f m m d Z (63)

c = o

(small velocity gradient, small shape resistance)

It is seen from (57), (61), (62), and (63) that [TI, [n], and [w] have the general form:

(large shape resistance)

[wl = ' / ~ * a 3 / ( 1 + brotdZZ) (634 (small shape resistance)

l'he quantities a,, an, a,, and brOr are constants in a polymer homologous series being given by:

(57W

(6W

Amb2NA 0.56 a, = ~

Mu -1.3 + 3.2 log ( A m / & )

(71: + 2)' 4r (YI - (YZ Amb'NA 0.23 -______ a, = 6no 3 kT Mu -1.3 + 3.2 log (Am/&)

3 2 Q, = - a, M,/(NAkT)

l/Wm -1.3 + 3.2 log (Am/d,) brot =

In exactly the same way one can also obtain expressions which apply in the case of large velocity gradients by simply replacing X by A,, in the relevant equations +en from our earlier papers for the viscosity and the magnitude and orientation of the streaming birefringence.38


It must be remembered, though, that all these expressions are based on rotation experiments on models and that the relative motion between the parts of the chain and the surrounding fluid is in the case of a motion in- duced by a velocity gradient somewhat dii€erent from the case of a pure rotation of the molecule. Despite this, the new formulas, in which the degree of draining has been taken into account, represent an obvious im- provement over the previous ones in which X was simply regarded a8 a constant. So much, in fact, that before proceeding further with a discussion of the expressions now obtained we must investigate anew the other approxima- tions incorporated in our earlier hypotheses, and consider improvements with respect to these. We will then see, however, that the coefficients in expressions (56) or (57) are thereby left unaltered.

b. Taking into Account the Spatial Distribution of Vectors HI and the Lateral Extension of the Coiled Chain

It has been assumed, in our earlier papers quoted, that in the motions of the coil in the streaming liquid the end points of the chain always remain in the plane of flow (two-dimensional distribution). In reality there is a spatial distribution of vectors h and HI (Fig. 5 ) and as a further basic simplification in these derivations we ignored the lateral extension of the coiled molecule. We shall investigate these two points in turn, and it wi l l suffice if we determine the effects that any refinements of the theory have in the case of a small velocity gradient.

Our procedure shall be to evaluate the effects of these refinements (spatial distribution of the directions of the axes and lateral extension of the molecule) in the case of a rigid dumbbell model which allows the problem to be solved mathematically. In applying this to the behavior of chain molecules in the limit of small velocity gradients we shall avail ourselves of the fact repeatedly demonstrated in our previous papers that for small velocity gradients the viscometric and flow optical behavior of chain molecules with large and small shape resistance becomes identical.

( 1 ) Dumbbell model in the Case of a Planar and Spatial Distribution of the Orientalions.

Let us consider a dumbbell model consisting of two spheres of radius r which are held rigidly apart at a distance S. S is so large compared with r that the two spheres do not influence one another hydrodynamically. The force required to drive a sphere with a velocity u through a fluid of viscosity qo is then given by Stokes' law, equation (15), so that the torque which has to be applied to rotate the dumbbell with an angular velocity + about an axis passing perpendicularly to 5' through its midpoint becomes:

M = 2(S/2)(S/2)+6q0r = 37rmS2r+ (64)

(dumbbell model, rotation in plane)


Let us now apply this to a solution containing G such.dumbbell models per cubic centimeter and in which a velocity gradient q is being maintained. If we assume that the rotation of the axes of the dumbbells occurs exclu- sively in the plane defined by the direction of flow and the velocity gradient we can give an expression for the mechanical energy per cubic centimeter per secondZB which, due to the presence of the dumbbells, is transformed into heat. This is:

E = (3r/4)Gqoq2S2r (65)

The total energy transformed into heat per cubic centimeter per second will then be :

where q designates the viscosity of the suspension. we then obtain:

From these expressions

(simple dumbbell model, twodimensional distribution)

In the case of a suspension of ellipsoids, corresponding expressions for the heat generated when there is a spatial distribution of the axes have been derived by W. Kuhn and H. K~hn.~*J5 Applying a similar treatment now to the case of a spatial distribution of the axes of dumbbell models we obtain:

E = (i/2)Gqoq2S2r (654

(simple dumbbell model, three-dimensional distribution)

so that in the place of equation (66) :

(7 d / q o = ( ~ / 2 ) GS2r (66a)

(simple dumbbell model, spatial distribution of the direction of the axes in the solution)

(2) Application of Formula for Viscosity of Suspensions of Dumbbell to the Case of Solutions of Chain Molecules

According to Figure 5 we approximate a chain molecule of hydrodynamic length L by a dumbbell with spheres a t distance:

s I= 2/3Hl (67)

apart, where HI is the distance between the two points in the coiled chain which are furthest apart in space. If we introduce this into equation (64), and compare with equation (43), we shall be able to determine of what radius r the spheres of the dumbbells will have to be made such that, when the distance S between the spheres is given by equation (67), the dumbbell


models will have the same resistance to rotation as the chain molecules. And we obtain:

from which:

It is now reasonable to assume that a chain molecule in solution produces the same energy dissipation E as a dumbbell whose parameters S and r are matched to the molecule by equations (67) and (68), respectively. By introducing these equations into expressions (66) and (66a), therefore, we can obtain expressions for the specific viscosity of a solution containing G chain molecules (of hydrodynamic length L) per cubic centimeter. This gives:

37r L 4 AT, ,

f s p = - G ~ A,, - H,” = - GLH; 4 247r 9 72

(twodimensional distribution)

t l s 9 = ( b t / 7 2 ) (2/3)GLfC

(qpatial distribution)

Using equation (4) as well as the facts that, if c is the concentration in grams per cubic centimeter:

c = G(M,/NA)Z (69)

= 2.1NJ; = 2.1ZbAm (70)

where NA is Avogadro’s number, and that : -

we obtain as our final expressions:

(coil regarded as simple dumbbell model, two-dimensional distribution)

and :

(coil regarded as simple dumbbell model, spatial distribution of directions of the axes in the solution)

Comparing expressions (66) and (66a) we note that the specific viscosity of a suspension of dumbbell models is in the case of a spatial distribution of the direction of their axes only 2/a of that produced when the distribution is two-dimensional. Obviously the same factor must result for solutions of statistically coiled chain molecules when, as has just been done, the molecules are approximated by dumbbell models (equations (71) and (72)).


This difference which results when considering a planar instead of a spatial distribution was actually compensated for in our earlier papers.'.* While it is true that we assumed there that the axes remain in the plane of flow, we at the same time used as end-to-end distance of the planar model, not the spatial vector h, but its projection onto the plane of flow. The m a n square value of this projection (in the case of a random spatial distribution of the vector h) is given by

while the mean square value of the spatial vector h itself, according to equation (5), equals N,,,Ak. If then, by using a model whose axis remains in the plane of flow the viscosity is made to come out 3/2 too large, we com- pensate for this by reducing the square length of the axis HI (which is proportional to h) to 2/3 the value in the case of a spatial model. I n our previous formuhe we therefore obtained the same numerical factor as is found for the case of a spatial distribution of the axes of the mlcules.

The small Werence between the numerical factors 1/48 and 1/51 in the case of equations (56) and (72), respectively, remains to be discussed. As will be remembered, the mean value and the square value of the end-to- end distance h of the chain were used in the derivation of equation (56). In the case of equation (72), however, Hl (see Fig. 5) and its square were employed. This difference in the choice of parameter accounts completely for the dhcrepancy.

In the next section we shall see that the numerical factor remains almost unaffected if we approximate the chain molecule by a crossed, instead of a simple, dumbbell model.

(3) Crossed Dumbbell Model in the Case of Planar and Spatial Distributions of Orientations

A feasible objection to the use of a simple dumbbell as model for the statistically shaped chain molecule would be the implied neglect of the lateral extensions of the coil. In order to perfect our theory also in this respect we shall therefore consider a crossed dumbbell model such as is shown in Figure 8. As in the case of Figure 5, let us consider that in each of the points P1 and P2 in Figure 8, which are a distance Sl apart, there is placed a sphere of radius r'. In addition, however, let us place two similar such spheres in the points P3 and P4 whose distance apart is S2. S, and S2 are perpendicular and cross each other centrally between PI and P2, and PJ and P4, respectively. In order to match the rotational resistance of this model with that of a statistically coiled chain molecule consisting of N,,, statistical chain elements of length A,, we let, in analogy to equation (67) :


id put:

8 2 = SJ2 = 2/3(H1/2) (734 ?cause we know from equation (7) that in the mean H2 = 1/2H1. Let us now consider the rotation of the crossed dumbbell model in a ,edhn of viscosity lo about an axis passing perpendicularly to Hl through s midpoint. Let the angular velocity in this rotation be + and, as we do it intend to assume that the cross axis H2 will always lie in the plane in hich Hl is being rotated, we define an angle 8 between H2 and the axis lout which Hl is being rotated. The velocity of points Pl and P2 in Figure relative to the surrounding solvent will therefore be given by l/pSl+, while Le relative velocity of points P3 and Pk will be ‘/2& (sin a)+.

P 2

Fig. 8. G o d dumbbell model. Spheres of radius r’ are attached to the The points PI and PZ lie on the long axis HI at a distaqx

The points PI and P, are on the cross axk H2

points PI to P,. &/2 = &/3 from the midpoint 0. at a distance S2/2 = H1/6 from 0.

The torque required to maintain this angular velocity will then, in anal- y to equation (64), be given by:

(74) M = 3rqort+<$ + S: sin2 S>

It will be remembered that, while in the experiments with macroscopic odels (described in Section Vb) the models were always rotated about axes hich were a t right angles to HI, the orientation of the cross axis H2 was ;ried such that Merent angles were in turn included between the axis ‘i and the axis of rotation. The average torque obtained from such varia- 3118 was substituted in equation (43) and is thus fundamentally included 1 the expression for x , ~ ~ . ]In analogy to this we shall regard, as measure for the rotational resistance the model (Fig. 8), a mean value of M which is obtained from equation %) by averaging over all angles 6 which H2 can form with the axis of rota- I


tion. As sin2 29 = ‘/z we can immediately write an expression for this average torque to be applied in order to rotate the model in Figure 8 with an angular velocity i about an axis perpendicular to HI. In this rotation the axis H2, though remaining a t right angles to HI, is otherwise free to assume statistical orientation. We find that:

M = 37~7)~r’$[S: + (S,2/2)] (75) The behavior in streaming solution of such models as shown in Figure 8

has already been investigated by W. K ~ h n . ~ ~ Using these results, as well as the theories evolved in a later paper,34 we can give an elipression for the heat generated per cubic centimeter by the presence of G dumbbell particles when the orientations of their axes have a spatial distribution. In good approximation w-e find that:

(crossed dumbbell model, spatial distribution of the longitudinal axes)

In the same way as we obtained equation (66) from (65). we now can write for the specific viscosity:

7) - 70 n- s: + (9/5,s:s; + s; - - Gr’ To 2 8:: + s,”

(crossed dumbbell model, spatial distribution of the longitudinal axes)

If according to equation (73a), S2 = ‘/zSI, equation (76a) still further reduces to:

(7) - 7 ) o ) / q o = ( ~ / 2 ) G r ’ S i . 1.21 (76b) (crossed dumb-bell model, spatial distribution of the longitudinal axes S, = I/&,)

(4 ) Application of These Equations to the Case of Solutions of Chain Mole- cules

Exactly as was done above under (2), we shall now apply the results obtained for suspensions of crossed dumbbell models to solutions of chain molecules. As the first step we must fi?d what value is to be assigned to the radius r‘ of the spheres which (see Fig. 8) are placed in the points PI t o Pq. This radius must be so chosen that the model in Figure 8 has the same resistance to rotation about an axis perpendicular to S, as a statistically coiled chain molecule with long axis HI. Substituting equations (73) and (73a) in (75) we obtain:

M = 3~7)~r’?(9/8)(4/9)H: ( 7 5 ~ )

Comparing this with the definition (43) then gives:

(L/18)7)aH;&,,, = (3n- /2 )qOH~r’~


or :

r’ = X,,,L/(27a) (77)

We see that the dumbbell model (Fig. 8) will offer the same resistance to rotation as a chain molecule of hydrodynamic length L. and of coil length Hl when its dimensions S1, S2, and r’ are given by equations (73), (73a), and. (77), respectively.

If as before we again assume that a solution of chain molecules will have the same viscosity as a suspension of the above dumbbell models, which are equivalent to the chain molecules in their resistance to rotation, then the specific viscosity of the chain molecule solution is obtained by substituting equations (77) and (73) in the expression (76b) :

If we further put for G, HI, and L their values given by equations (69), (70), and (4), respectively, we arrive a t the following expression:

(coil regarded as crossed dumbbell model, spatial distribution of the coil’s longitudinal axes)

If we change the above conditions and postulate that HI remains in the plane of flow and that H2 while remaining a t right angles to HI is otherwise statistically oriented, we obtain in analogous fashion :

(coil regarded as crossed dumbbell model, two-dimensional distribution of the coil’s longitudinal axes)

On comparing expression (78) with (72) and expression (78a) with (71) we note that these are practically identical. We therefore conclude that almost the same specific viscosity will be derived whether we replace the chain molecule by a simple or by a crossed dumbbell model. This applies, of course, only on condition that the dumbbells are in either case so chosen that their external dimensions and those of the chain molecules agree in the respects relevant to the theory and that, second, their resistance to rotation about an axis perpendicular to their long axis is the same as that of the chain molecules.

It will be noted that the agreement found applies equally to the case where the‘long axis of particles remains in the plane of flow (equations (71) and (78a)) as to the case where these axes are statistically oriented in space (equations (72) and (78)). Therefore, because the expressions derived for a two-dimensional and three-dimensional distribution of the long axes


are not affected by the lateral extension of the coil, we can conclude that they will in all likelihood also remain unchanged if the coil's third dimension were taken into consideration. (This could be done by replacing the molecules by three-dimensional models, which would more closely resemble the spatial coils.)

It is of particular 'significance that the numerical factor 1/48 in equation (78) agrees so well with the numerical factor in equation (56), which bas been taken from our previous papers. We are therefore led to the conclusion that the earlier results retain their validity in spite of the rejinements now introduced. Consequently we were entitled, under retention of the remaining numerical factors, to replace X by (equation (44)) in our earlier expressions for the viscosity and the streaming birefringence.

We may, in particular, regard expression (57) as a good approximation to the specific viscosity of solutions of chain molecules in cases of small velocity gradients.

In Part I of this paper we paid particular attention to what happened to translatory resistance of a coiled chain molecule in the extremes of low and high degrees of polymerization, using as basis for comparison the translatory resistance of ellipsoids. We shall now do the same in the case of expression (57) and discuss what viscosity is to be expected in the various limiting cases which can occur.

c. Behavior in the Extremes of High and Low Degrees of Polymerization

( I ) Equation (57) in the Limit of Large Values of Z

When the degree of polymerization is high, the first two terms in the denominator of equation (57) can be neglected in comparison with the last, so that the expression reduces to:

(equation (57) in the case of high degree of polymerization)

In the limit of very high degrees of polymerization the viscosity will thus increase in proportion to the square root of the degree of polymerization Z, i.e., show the dependence on Z already previously described by W. Kuhnl for the case of a nondraining coil.

(2) Comparison with the Expression for Viscosity of Suspensions of Non- draining Spheres or Ellipsoids

As we have seen in Section IIa, the coiled chain molecule can in rough approximation be considered as a sphere of volume V given by equation (lo), or in somewhat better approximation as an oblate ellipsoid of revolution of axial ratio p = 2.0.


In the case of a suspension containing G nondraining spheres of volume

(q - ~ O ) / T O = 2.5VG (80) IGtroducing equations (10) and (69) in this we obtain an expression roughly applicable to solutions of chain molecules:

(81)

( ~ J c ) = 1.03 (N’/M,)Azb”/’<Z (82)

V per cubic. centimeter we can, according to Einstein, write that:

(7 - tlo)/(70c) = 2.5 * O.~~N?A:(NA/M,)(~/Z) This now transforms to the following by the use of equation (34a) :

(chain molecule of high degree of polymerization, regarded as nondraining sphere)

If instead we assume that the external shape of the coiled chain molecule is an ellipsoid whose axial ratio p, on the average, equals 2.0 (equation (12a)), we find that the viscosity of a suspension containing G such ellipsoids per cubic centimeter is given by:

(7 - t o ) / q o = 2.91vG (83)

qsp/c = 1.05(N~/M,)A2b”’l/Z (84)

Substituting in this equation then gives:

(chain molecule of high degree of polymerization, regarded as nondraining ellipsoid)

We note that the numerical factor is substantially the same as in equation

On comparing equations (82) and (84) with (79) we see that in the case of high degrees of polymerization the viscosity calculated on the basis of macroscopic model experiments (equation (57)) agrees reasonably well with the value predicted on the basis of the external dimensions of the coil, i.e., with the value obtained if the molecule is regarded as completely nondraining at these high degrees of polymerization. Moreover, we find an exactly similar result as when we were comparing equation (46) with (48), or equation (31) with (31a), namely, that the viscosity derived from equation (79) turns out to be smaller than the value estimated from equation (84). The lagging behind in the numerical factor in equation (79) is more marked than in the corresponding cases of equations (46) and (31). This is most likely due to the fact that the viscosity of a solution though mainly determined by the rotational diffusion constant is influenced also by other relative motions between the particle and the streaming solvent consisting in the main of a flow of liquid parallel to the axis of the particle.

(3) Behavior in the Limit of Low Degrees of Polymerization

When the degree of polymerization is low, i.e., when N,, the number of statistical chain elements in the molecule, is about equal to or less than 1, the relationship (57) will cease to apply.

(82).


In the case of such short molecules, however, we can try and replace the chains by ellipsoids of revolution whose long axis corresponds to the hydrodynamic length L and whose cross axis is the hydrodynamic thickness dh of the chain. In a paper already quoted,34 the viscosity of such a suspension was derived as a function of the axial ratio p of the ellipsoids, where in this case:

p = L/dh = Zb/dh (85)

This expression for the viscosity was then presented graphically in Figure 7 of the paper

If we take into account that the volume of an ellipsoid, whose length is L and whose thickness is dh, is given by:

we can find an expression for the volume G V which G such particles will occupy. If G, the number of particles per cubic centimeter, is expressed in terms of c, the concentration in grams per cubic centimeter, we find that:

Consequently, using equation (77) in -the paper p >> 1, we have:

and assuming

(elongated ellipsoid Zb/& >> 1) The behavior in the less important case where the values of p are small

can be seen from Figure 7 of the earlier paper34 to which reference is being made. It should be remarked that the values for the viscosity calculated from expression (86) must turn out to be somewhat too large. This is particularly true a t relatively large values of Z as the curvature of the chain, which was neglected in the derivation of equation (86), makes itself more apparent. Equation (86) thus correctly reproduces the behavior in cases of small values of Z and must be replaced by equation (57) when 2 becomes larger.

We should like to point out, however, that the expressions here given may only with great caution be applied in cases of small values of Z. As soon as the dissolved molecules are no longer large as compared with the molecules or molecular aggregates of the solvent, a hydrodynamic treatment of the problem becomes inadmissible. Under such circumstances, the actual viscosity may be markedly smaller than the viscosity that would be expected on the basis of hydrodynamics.37

This can, for example, be illustrated by the behavior of paraffin hydro-


carbons whose solutions in the region of small molecular weights ( G o , etc.) have an unexpectedly small viscosity. A 20-membered normal paraflin hydrocarbon in solution in CClk has a viscosity index, [ v ] = 2, a result even smaller than the value [ v ] = 3 calculated on the basis of the Einstein formula, which postulates a spherical particle shape.

d. Comparison with Some Experimental Results

As a specific example of an experimentally investigated polymer-solvent system we can consider the case of nitrocellulose (M, = 268) in acetone. Here we have measurements by M o ~ i m a n n , ~ ~ whose results for the viscosity index [ v ] we have plotted (as circles) against the degree of polymerization 2 in Figure 9. The curves in that figure, on the other hand, are obtained from equation (57) (curve 1) and equation (86) (curve 2) by putting A, = 135 A., dh = 9.2 A., and b = 5.15 A.* As expected, these measure-

200 300F - -

0 loo 200 300 400 Fig. 9. Viscosity index [v] 21s. degree of polymerization 2 (nitrocellulose in

acetone). C m e I : Calcu- lated from equation (57); the curve represents the relationship [v] = 2/(0.54 f 0.04342). Curne 2: Calculated from equation (86) (freely draining chain molecule regarded as extended ellipsoid of length 2%). Curve 3: Expected dependence of intrinsic viscosity on the degree of polymerization. Small circles: Actual viscosity measurements made by M ~ s i m a n n . ~ ~ [q] is, as the exact treatment shows, not directly proportional to the degree of polymerization 2 (as it would have to be according to Staudinger’s law). A t low degrees of polymerization, [q] is approximately proportional to 2 2 ; over an intermediate range, it is proportional to 2; and, at very high degrees of polymerization, it is roughly proportional to 4%.

Plotted for A, = 135 A., dh = 9.2 A,, and b = 5.15 A.

* It must be remembered that A, values between 120 and 150 A. and dh values between 7 and ll A. could represent the experimental data almost as well as the values chosen.


ments show agreement with curve 2 (equation (86)) at low degrees of polymerization, and with curve 1 (equation (57)) at moderately high and high degrees of polymerization. Consequently the unbroken curve (curve 3) may be considered to give the actual dependence of the viscosity on the degree of polymerization.

As the above has demonstrated, the viscosity index [ q ] in a polymer homologous series does not show that linear dependence on the degree of polymerization which the Staudinger law would lead us to expect. The actual behavior seems to be such [hat at lm degrees of polymerization the increase is proportional to Z2, in an intermediate region approacimately proportional to Z and at very high degrees of polymerization proportional to <Z. That this type of dependence on Z applies not only in the case of nitrocellulose as just demonstrated, but also in the case of a large number of further high polymeric substances, has been demonstrated by Wo. Ostwaldan on the basis of an abundant amount of experimental data.

A remark must be added with reference to the values of the parameters used to describe the nitrocellulose molecule in acetone. In Section IV we found that on the basis of sedimentation measurements the values of these parameters were A, = 44 A. and dh = 9.2 A. while the values found above from the viscosity relationships are A, = 135 A. and d,, = 9.2 A. This discrepancy between the A, value obtained from sedimentation and that from diffusion data might be due to a partial association of the chains in solution but, even more likely, to the limited accuracy of the experiments (lack of really homogeneous fractions, ambiguity in the extrapolation of the q8Jc to zero concentration).

e. Application to Other Phenomena

The experiments of partially draining statistically shaped chain molecules a l b w us also to perfect the expressions for the amount and orientation of the streaming birefringence. In the case of small velocity gradients this has already been done in equations (61), (62), and (63) in Section a above. The same applies also to the expressions for the viscosity and the amount and orientation of the streaming birefringence in case of high velocity gradients inclusive of the dependence of these phenomena on the shape resistance of the chains. The procedure in all these cases is the same, to replace X by A,,$ from equation (44) in the relevant equations of our previous papers. In this manner a better approximation to the actual behavior is obtained every time.

NOTE ADDED IN PROOF

Since this paper was submitted for publication a number of theoretical and experimental articles have appeared which are concerned with the same subject :


Theoretical Treatments

(1) Kirkwood and Riseman,14 and Debye and Bueche12 have derived formulas expressing q,,/c as a function of 2. The form of their equations is considerably more involved than the expressions derived earlier by us (equation (57)), but in their general behavior these formulas agree with ours. Thus, in the case of large values of 2, Kirkwood and Riseman’s expression reduces to:

( q s p / C ) = (0.60/M0) NA~’”A

an equation which M e r s from our corresponding equation (79) only slightly in the numerical factor.

(2) Kirkwoodm,14 claims that the influence of Brownian motion has not been adequately taken into account in previous theories of intrinsic viscosity. It must be pointed out, however, that a complete treatment of this effect has been given in our 1945 paper.34 We discussed there the influence which a nonisotropic distribution of particle orientations in the streaming solution must have on the viscosity. Clearly the Brownian diffusion of the particle orientations will in the presence of a distribution gradient become directed, i.e., more particle axes will diffuse the one way than the other. Such a directed Brownian translation or rotation move- ment of the particles gives rise to an energy dissipation which has been neglected by all authors before 1945. As all our papers since 1945 have however been based on that treatment, the influence of Brownian motion has thus, in our opinion, been correctly allowed for in our formulas.

(3) A fundamental hypothesis, in this as well as in our earlier papers has been the postulate that the viscosity of a solution of chain molecules depends essentially on the rotational resistance, and thus on the rotational diffusion constant of the coils. This point of view has been adopted recently also by Riseman and Kirkwood,*l who derive a relationship between viscosity index and rotational diffusion constant which is essentially the same as that which follows from our treatments.

(4) In Part I a comparison of hydrodynamic thickness dh, found from sedimentation and dif€usion measurements, with the actual thickness d, of the chain has shown that it is admissible to use hydrodynamics to describe the behavior of chain molecules in solution. Similar conclusions can now also be drawn from available viscosity measurements, the dn values derived from such data again being in rough agreement with the d, values. Such large deviations from hydrodynamic behavior as Kirkwood, Riseman, and o t h e r ~ ‘ ~ , ~ ~ have assumed to occur are therefore not borne oh, by the experimental evidence.

(5) A treatment has recently been g i ~ e n 4 ~ in which influence of cross- sectional dimensions on the quantities [q] and [n] have been considered somewhat more extensively than in the present paper. We have thus obtained slightly Merent numerical factors in the comesponding eqres- sions, namely the factor 0.6 instead of 0.56 in equations (57) and (57b),


and 0.17 instead of 0.23 in equations (61) and (61b). Thus, for example, instead of (57):

Experimental Investigations

Sookne and Hamis,43 and Bartovics, Jelling, and Badgley44 have determined the Viscosity of solutions for a series of cellulose acetate fractions in acetone. The results of their measurements are recorded in Figure 10 as the points marked with circles. The curve shown in this figure is drawn from equation (57c) on the basis of the values A, = 92 A. and dh = 8 A.,

Z-

Fig. 10. Viscosity index [v] us. degree of polymerization 2 (cellulose acetate in acetone).42 Plotted for A, = 92 A., dh = 8 A. Unbroken cwve: Calculated from equation (57c). Dots: Measurements by Sookne, Harris, and Bartovics, Jelling, and Bedgley.

which, as shown in Part I, are the values found from sedimentation data for the same substance in the same solvent.* It is seen that there is, in this case, excellent agreement between theory and experiment. This might be partly fortuitous, but it points to the reliability of the theoretical method.

It should be mentioned, however, that light scattering dissymmetry measurements tend to give considerably higher A , values than the value 92 A. found above. Thus values between 570 and 1040 A.45 have been reported, but in view of the extreme sensitivity of the method to traces' of foreign matter the possibility that these high values are due to extremely fine dust cannot quite be excluded.

* The values given in Part I and in a paper by H. Kuhn (J . Colloid Sci., 5,331 (1950), see p. 340) are slightly different from the above due to a computational error which has now been corrected.


Recently, Badger and Blaker46 critically investigated several fractions of nitrocellulose (M, = 284) in acetone (vo = 0.0030). In Figure 11 their experimental results for [v] have been plotted against 2 (as circles). These results can best be interpreted by using for A, and day in equations (57c) and (86), the values A, = 156 A. and db = 10 A., giving the curves shown in the

8 00

700

600

500

400

300

200

100

- 100 200 300 400 500 600 700 800 900 1ooo 1100 1200

Fig. 11. Viseosity index [v] us. degree of polymerization 2 (nitrocellulose in acetone). Plotted for A, = 156 A., dh = 10 A. Curve 1: Calculated from equation (57c). Curve 2: Calculated from equation (86). Curve 3: Expected dependence of the intrinsic viscosity on the degree of polymerization. Small circles: Measurement by Badger and Blaker.46

figure. The value A, = 156 A. agrees reasonably well with the results A, = 120 to 150 A. with which, as shown above, Mosimann’s data can be fitted. As will be seen, this value also agrees reasonably well with the results of other measurements. Expressing the diffusion constant according to the data of Badger and Blaker by the approximate formula:

DZ = (4 + 0.6 ~’ /~)10-5

one finds by making a comparison with equation (35) that A, = 100 ang- stroms, dh = 15 A. This value for A, is thus in better agreement with the viscometrically determined value, than the value A, = 44 A. found from Mosimann’s sedimentation data. From light scattering dissymmetry


measurements on the same samples, Badger and Blaker find A, values of between 200 and 630 A. Similar conclusions can be drawn from small angle x-ray scattering experiments; thus, from measurements on nitrocellulose in acetone Kratky and Pored" have found that A, should be larger than 150 A., a value which at any rate does not contradict our results from the viscosity measurements.

SUMMARY OF PART IZ6 AND PART I1

When estimating the resistance of statistically shaped chain molecules t o translatory or rotatory motion in solution it is important to remember that the motions of the chain or parts of the chain, relative to the surrounding solvent, do not occur without some retention of solvent molecules in the folds of the coil. Moreover, in the qase of a polymer homologous series, the low polymers are near the limit of completely free draining while, with increasing degree of polymerization, a gradual transition to the case of the nondraining coil takes place (W. Kuhn and H. Kuhn, 1943).

In order to obtain more precise information about this transition from the free draining to the nondraining case, experiments were carried out with statistically shaped macroscopic models. Using a theorem of hydrodynamic similarity which is fully described, a form factor h is defined which applies equally to the macroscopic models and to the geometrically similar submicroscopic originals. The shape factor A,,,,, in the case of translatory motion Mers considerably from the factor A,, applying in the case of a rotation of the model. Both factors are determined experimentally with these models, whose mode of construction is described, and results obtained from the measurements are collected in a formula which expresses the shape factor as a function of the number of statistical chain elements in the molecule, of their length, and of their thickness. The translatory and rotatory resistances which these formulas predict in cases of low and high degrees of polymerization agree well with the resistances calculated theoretically in the limiting cases of free draining (low polymerization) and nondraining (high polymerization) which are capable of exact hydrodynamic treatment.

On the basis of the shape factor htrans, which determines the translatory resistance, quantitative expressions are developed for the sedimentation and diffusion constants of statistically shaped chain molecules. These expressions are functions of the degree of polymerization and the length and the thickness of the statistical chain element. From them one can immediately deduce that the sedimentation constant so = a, + b,(Z)’”, where 2 is the degree of polymerization and Q, and b, are constants in the case of each polymer homologous series. A similar dependence is found also for the product, DZ, of diffusion constant and degree of polymerization.

The validity of these expressions is demonstrated by comparison with experimentally found results. These are used to calculate the degree of


coiling, and the hydrodynamic thickness dh of nitrocellulose, methylcel- lulose, and cellulose acetate.

one can in similar fashion demonstrate the degree of draining on the viscosity, and on the magnitude and orientation of the streaming birefringence. It can be shown that, contrary to the Staudinger viscosity law, the viscosity index [77] = (v8Jc)~irn c = o in a polymer homologous series is not directly propor-

tional to the degree of polymerization over the entire range. At low degrees of polymerization 2, [q] is approximately proportional to Z2, and a t very high degrees to 4.2; only over an intermediate range of values is it approximately proportional to 2. These changes with degree of polymerization also correspond to a gradual transition from the case of the freely draining coil (low polymerization) to the other extreme of the nondraining coil (high polymerization).

Here too, in the case of the viscosity relationship, good agreement is oh- tained in the limits of low and high degree of polymerization between the foi-muIa resulting from the above method and the expressions which can in these cases be derived theoretically by means of exact hydrodynamic treatments.

The relationship between the viscosity and the degree of polymerization which has been described is confirmed by comparison with experimental data.

Characterizing the resistance to rotation by a shape factor

q - 0

References

26. H. Kuhn and W. Kuhn, J. PoZymer Sei., 5,519 (1950). 27. R. Gans, Ann. Physik, 86,628 (1928). 28. W. Kuhn, J. Polymer Sci., 1,380 (1946). 29. Compare reference 7 in Part I. See also W. Kuhn and H. Kuhn, HeZu. Chim.

30. W. Kuhn and H. Kuhn, Helv. Chim. Ada, 29, 71 (1946), in particular equation

31. The term “shape resistance,” as introduced in our earlier papers (see W. Kuhn and H. Kuhn, J. Colloid Sci., 3 , l l (1948), must be explained. It is clear that if a polymer molecule is subjected to a rapid constellation change in solution, such a change is to a large measure opposed by energy barriers inside the molecule, i.e., there is a certain tendency of the molecule to resist rapid changes in its shape. This tendency, formally identical with a viscosity, is what we have defined as shape resistance.

32. W. Kuhn and H. Kuhn, HeZv. Chim. Aclu, 29, 71 (1946), in particular equations (5,lO) and (5,14).

33. For the viscosity index, refer in particular to W. Kuhn and H. Kuhn, HeZu. Chim. Ada, 28, 1533 (1945); if the shape resistance is small use equation (4,191; if it is high, equation (4,23). For the magnitude of the streaming birefringence, see W. Kuhn and H. Kuhn, Helu. Chim. Ada, 29,71 (1946) ; if the shape resistance is small use equations ($4) and (3,40b); if it is high, equations (5,5) and (3,40b). See also Figure 9 in this paper. For the orientation of the streaming birefringence refer to the same paper, equations (5,9a) and (3,40b), if the shape resistance is small; equation (5,12), if it is high. See also Figure 10 in this paper.

Ada, 28,1533 (1945), in particular equations (4,15) and (3,41a).

(5,3) *

34. W. Kuhn and H. Kuhn, Helu. Chim. Ada, 28, 97 (1945). 35. R. Simha (J. Phys. Chem., 44,25 (1940)) has obtained expressions for the viscosity


index of suspensions of ellipsoids which are different from ours in the general case, hut which agree with ours in the special case of very elongated ellipsoids. Simha's equations are, however, derived on the basis of incorrect assumptions, a fact which was first pointed out by H. C. Brinkman, J. J. Hermans, L. J. Oosterhoff, J. T. Overbeek, D. Polder, A. J. Staverman, and E. H. Wiebenga (Proceedings of the International Congress on Rheology, Holland, 1948).

36. W. Kuhn, 2. physik. Chem.,A161,1 (1932). 37. A more extensive discussion of that point has been given by W. Kuhn. See refer-

38. H. Mosimann, Helu. Chim. Acta, 26,61,369 (1943). 39. Wo. Ostwald, Kolloid Z., 106,l (1944). 40. J. G. Kirkwood, Proceedings of the International Colloquium on Macromolecules,

41. J. Riseman and J. G. Kirkwood, J . Chem. Phys., 17,442 (1949). 42. W. Kuhn, H. Kuhn, and P. Buchner, Ergebnisse der exakten Naturwissenschaften,

43. A. Sookne and M. Harris, Ind. Eng. Chem., 37,475 (1945). 94. See H. Mark, T. Alfrey, and R. B. Mesrohian, h a t actwl de la Chimie et de la

45. R. S. Stein and P. M. Doty, J. Am. Chem. Soc., 68, 159 (1946). 46. R. M. Badger and R. H. Blaker, J. Phys. Chem., 53,1056 (1949). 47. 0. Kratky and G. Porod, Proceedings of the Znternational Colloquium on Macro-

ence 23.

Amsterdam, 1949. Centen, Amsterdam, 1950, p. 133.

25, l(1951).

Physique des Molf?cules Gkantes. Masson, Paris, 1950, p. 117.

molecules, Amsterdam, 1949, p. 250.

Synopsis

In the two limiting cases of the freely draining and the completely nondraining coil a theoretical treatment was given, while for intermediate draining conditions an experimental method, involving large-scale models of the molecule, was shown to be available. The same treatment is now extended to deal with the analogous case of the hydrodynamic rotational resistance of the random coil. It is on this characteristic resistance that the intrinsic viscosity and the flow birefringence of solutions of chain molecules principally depend.

In Part I the translational resistance of the random coil was discussed.

auZ2 1 u,z2 [o] = (-) =

~ O Q limc=O 1 + b r o t d Z ; q = o q = o

(large shape resistance of molecular (small shape resistance of molecular chain) chain)

Expressions are found for the viscosity index [ ? I , the orientation index [o], and the birefringence index [n ] respectively. In these formulas 2 is the degree of polymerization while the parameters aV, a,, a,, and b,t are independent of 2. These parameters are, however, functions of the length A, of the preferential statistical chain element and the hydrodynamic thickness dh of the chain. a, also includes the anisotropy of polarizability of the statistical chain element. It is seen moreover that [ w ] (in contrast to [?I and [ n ] ) also depends on the shape resistance of the chain (i.e., on the resistance which the chain inherently offers to a rapid change in configuration).

R6sumB

On a determine dans la premihre partie la rhistance au mouvement de translation Dans les deux cas limite de la pelote d'une pelote statistique par un medium visqueux.


compl&tement perm6able et de la pelote imperm6able pour le solvant une consid&ation theorique a b6 developpb et dam le cas g6n6ral de mol&ules partiellement perm6ablea cette r&sistance a b6 determinb experimentalement en employant des mod&les B grand khelle. La r&tance au mouvement de rotation d'une pelote statistique est deter- mini% d'une fawn analogue. Cette dernihe rbistance est easentielle pour la visoosit6, ainsi que pour la valeur et l'orientation de la bir6fringence d'ikoulement de suspensions de molecules en & h e . On trouve que l'index de viscosit6 [q], l'index de birefringence [n] et l'index d'orientation [w] sont donnb par les relations suivantes:

101 = (-) = a Z z , IW] = ( Y ) = - 1 &Z* V O ~ l imc=o 1 + b r o c d Z ' r l ~ p l imc-0 3 1 + b r o t K Z

q =o Q =.o

(grande viscosit6 interne) (petite viscosit6 interne)

Z indique le degr6 de polymerisation; a,, an, ow et brat sont dea wnstantes dam une s6ie homologue de polym6rimtion; elles dbpendent de la longueur de 1'616ment statistique A,,, et de I'6paisseur hydrodynamique dfi de la chaine. a, est, en outre, une fonction de l'anisotropie de la polarisation de l'blbment statistique de la chaine. [w] (contraire- ment au [q] et [n]) d6pend encore de la grandeur de la Vjscosit6 interne de la pelote, i.c. de la rbistance que la chaine oppose 1 un changement rapide de sa configuration.

Zusammenfassung

In Teil I wurde der Reibungswiderstand ermittelt, der bei einer Translatio&wegung eiues statistisch gekniiuelten Fadenmolekiils durch ein viskoses Medium auftritt. In den Fallen gleichmiissig bespiilter und undurchspiilter Kniiuel wurde dieaer Widerstand auf Grund theoretischer Betrachtungen ermittelt und im Falle teilweise durchspiilter Molekiile wurden Versuche an makroakopischen Molekiilmodellen herangemgen. In entsprechender Weise wird nun der Reibungswiderstand ermittelt, der bei der Rotation eines geknauelten Fadenmolekiils relativ zum umgebenden Liisungsmittel auftritt. Dieser Rotationsreibungswiderstand ist weaentlich fiir die Viskositilt, sowie fiir den Betrag und die Orientierung der StrSmungsdoppelbrechung von Fadenmolekiilsuspen- sionen. Es zeigt sich, dass die Viskositatszahl [q], die StrSmu~doppelbrechungszahl [n] und die Orientierungszahl [o] durch die Beziehungen gegeben sind:

= p) = a 2 [n] = (-) nr - n2 = a 3 c limc=O 1 + b r o r d g ' VoqC 1imc=o 1 + brer.\//Z

q = o q = o

1 a,Z2 a,Zn -. r,.,] = (w> = - tloq limc=O 3 1 + b r o r d 3

q = u P = o

Molekdfadens) . Molekiil fadens) (grosse innere Viskosjtat des

Darin ist Z der Polymerisationsgrad; a,,, a,,, a, und b,t sind Konstanten innerhalb einer polymerhomologen Reihe, die von der LEU@ A, des statistischen Vorzugselements und der hydrodynamischen Dicke d h der Fadenkette abhiingen. a, hiingt mdem von der Anisotropie der Polarisierbarkeit des statistischen Fadenelements ab. Ausserdem hiingt [o] im Gegensatz zu [q] und [n] noch von der GrSsse der inneren Viskositiit des Fadenkniiuels ab, d.h. vom Widerstand, mit welchem sich der Molekiilfaden einer raschen Aenderung seiner Gestalt entgegensetzt.

Received December 1, 1947

(kleine innere Viskositit des

effects of hampered draining of solvent on the translatory and rotatory motion of statistically...

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