The Viscosity Of Macromolecules In Relation To Molecular Conformation

The Viscosity Of Macromolecules In Relation To Molecular Conformation


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THE VISCOSITY OF MACROMOLECULES IN RELATION TO MOLECULAR CONFORMATION By JEN TSI YANG Cardiovascular Research Institute and Department of Biochemistry, University o f California Medical Center, San Francisco, California'

I. Introduction.. ................ ...................... 323 11. Definitions and Equations, .... ...................... 326 A. Definitions . . . . . . . . . . . . . . . . . ............................ 326 B. Determination of Intrinsic Viscosity. . . . . . . . . . . . . . . . . . .......... 328 C. Intrinsic Viscosity-Molecular Weight Relationshi . . . . . . . . . . . . . . . . . . 329 .................. 331 111. Newtonian Viscosities.. ............................ A. Theories for Rigid Particles .................... B. Determination of Particle Shape from Intrinsic Vi C. Other Hydrodynamic Properties. . . . . . . . . . . . . . . . . . D. Concept of Equivalent Hydrodynamic Ellipsoid.. E. Estimation of the Length of Equivalent Ellipsoid. F. Theories for Flexible Coils.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 G. Polyelectrolytes and Electroviscous Effect . . . . . . . . ............. 349 H. Examples of Several Common Proteins and Polype IV. Non-Newtonian Viscosities.. ............................................ 363 A. Theories for Rigid Particles. ........................................ 363 B. Determination of Particle Length (or Diameter). . . .............. 364 C. Experimental Confirmation, ......................................... 365 D. Comparison with Flow Birefringence Method.. ....................... 368 E . Theories for Flexible Coils.. ... ...................... F. Power Law of Viscosity.. . . . . . . ................... G. Complex Viscosity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Experimental Methods ........................................ 375 A. Types of Viscometers.. . . . . . . . . . . . . . . . . . . B. Corrections for a Capillary Viscometer . . C. Extrapolation to Zero Rate of Shear.. . . . . . . . . . . . . . . . . . . . . . . . . . 3% D. Design of a Capillary Viscometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 E . Viscosities of Extremely Dilute Solutions .................... 387 VI. Conclusions., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Appendix (Tables VIII-XIII) . . . . . . . . . . . . . . . References. .................................

I. INTRODUCTION Of the quantities which characterize a polymer solution the viscosity is one of the simplest and cheapest to measure; for this reason it is also among 1 This work was begun at and initially supported by the Department of Biochemistry, Dartmouth Medical School, Hanover, New Hampshire.




the most intensively studied, theoretically as well as experimentally. Thc importance of viscometry in industrial applications is reflected by the rapid progress of the science of flow and deformation-rheology. To protein chemists the determination of intrinsic viscosity (Section 11, A ), has become so fundamental that rarely is the size and shape of a protein in solutions characterized without a t least including a set of viscosity data. The specific excuses for the present review are threefold: first, the interpretations of intrinsic viscosity of protein solutions has recently undergone a more careful reevaluation. There have arisen doubts concerning aspects of viscosity treatments routinely applied in characterizing proteins, particularly the asymmetry of the protein molecules. These doubts must be examined and, if possible, resolved; the proper applications and limitations of viscometry in the studies of proteins will be emphasized. Second, the recent development of the theories of non-Newtonian viscosity has made the gradient dependence of viscosity less of a nuisance and more a source of additional information concerning the shape of polymers. It also offers a new means for studying the conformations and conformational changes of proteins. It is therefore opportune to put this new technique in the hands of biochemists, although little experimental work has as yet been done. Third, it appears desirable to summarize the basic equations employed in viscosity measurements, and the complications that may arise and precautions that are required under normal experimental conditions. No attempt will be made in t,his review to describe all the theoretical t,reat,ments even in a cursory manner. Only the physical significance of several well-known equations will be quoted without derivations. This it is hoped should be adequate for our purpose. A good “feel” for the relation between viscosity and macromolecular structure may be gained by perusing Table I. Globular proteins generally have lower intrinsic viscosities than flexible polymers (e.g., polystyrene) of comparable molecular weight; it is to be expected that compact, rigid proteins will move through solvent easier than “swollen” chains. There is however, no correlation between intrinsic viscosity and molecular weight when proteins of different types are considered; the intrinsic viscosity of rigid particles is dominated by particle asymmetry, not size. Thus, for example, in spite of its high molecular weight tobacco mosaic virus sohtions have a much smaller intrinsic viscosity than collagen solutions. On the other hand, in homologous series of one molecular type, rigid (e.g., a-helical poly-7-benzyl-L-glutamates) or flexible (e.g. , polystyrenes), intrinsic viscosities do increase with molecular weights. This is attributed to an increase in asymmetry (for rigid particles) or in effective volume (for flexible polymers) with increasing molecular weights. (Idealized hard spheres are an exception; see Section 111, A . ) The intrinsic viscosity of a



flexible polymer is strikingly dependent on the medium in which it dissolves; it is always greater in a "good" solvent (high polymer-solvent affinity) than in a "poor" one (low polymer-solvent affinity), again due to the difference in effective volumes. Among tlhe flexible polymers the TABLEI Intrinsic Viscosities of Several Proteins and Polymers i n Solutionsa Substance

Molecular weight

Proteins in aqueous solutions a t isoelectric points Ribonucleaseb Myoglobinc Ovalbumind Serum albumin6 Hemoglobin' Fibrinogen# Collagenh Myosin i Tobacco mosaic virusi Poly -r-beneyl-L-glutamatesb (1) I n dimethyl formamide (2) In dichloroacetic acid at 25°C

Polystyrenes (1) In benzene at 20"CE (2) In 0.869:0.131 (v/v) cyclohexane/carbon tetrachloride at 15"Cm Cellulose trinitrate in acetone a t 25"Cn a For the definition of intrinsic viscosity and related terms, see Table 11. Buzzell and Tanford (1956). c Wyman and Ingalls (1943). d Polson (1939). "Loeb and Seheraga (1956); Champagne (1957). f Tanford (1957b). g Scheraga and Laskowski (1957).

13,700 17,000 44 000 66 ,000 67,000 335,000 345,000 493,000 39,000,000

[s] dl gm-1

0.033 0.031 0.043 0.039 0.036 0.25 11.5 2.17 0.37

66 ,500 340 ,000 66,500 340,000

0.45 7.20 0.45 1.84

65,500 262,000 2,550,000 3 220,000

0.36 1.07 5.54 1.35

77 000 400,000

1.23 6.50

Boedtker and Doty (1956). Holtzer and Lowey (1956, 1959). j Boedtker and Simmons (1958). k Doty et aE. (1956). (1) a-helicnl form; (2) randomly coiled form. 2 Fox and Flory (1951). I'assaglia et al. (1960). n Holtzer et al. (1954). h


intrinsic viscosity also depends, as expected, on structure, being greater the more extended the polymer chains (cf., cellulose trinitrate and polystyrene). The foregoing observations merely suggest, the many complexities of the viscosity of macromolecules in solutions and emphasize the need for co-



ordinating viscometry with other physicochemical methods. For instance, viscosity measurements alone cannot distinguish the contributions due to the asymmetry of the protein molecule from the effective volume of the molecule in solution. Indeed there are many assumptions involved in the interpretation of intrinsic viscosities of protein solutions which will be discussed in more detail in Section 111. Intelligently used, however, viscometry can be a surprisingly versatile method. One of the most fruitful approaches has been the study of conformational changes of the protein molecules in solutions. The viscosity measurements have been used to detect the ill-defined process “denaturation,” to follow its kinetics, and sometimes even to establish its reversal. But even here success must be tempered with caution, for had such studies been made with only one poly-y-benzyl-L-glutamate having a molecular weight of about 66,500 (Table I) they would have failed ignominously. Likewise the increase of intrinsic viscosity on denaturation of a globular protein and its decrease on reversal of denaturation would not be sufficient enough to demonstrate a direct reversible process. But viscometry can provide powerful evidence when taken in conjunction with other measurements such as optical rotatory power and deuterium-hydrogen exchange. Today protein chemists are no longer preoccupied with the elucidation of the bare outlines (size and shape) of the protein molecules and are beginning to probe deeply into the internal structures of these molecules. It is in this realm that viscometry along with other methods will find new importance in the studies of conformations and conformational changes of the proteins.


A . Definitions By Newton’s definition the viscosity or, more appropriately, the viscosity coefficient, q , of a fluid in a laminar steady-state flow is expressed as the tangential force, F , per unit area, A , required to maintain a unit rate of shear (or velocity gradient), G, in the liquid. If the liquid fills the space between two parallel planes of area, A , one of which moves at a constant distance, r, from the other with a relative velocity, u, then we have Shearing stress: r = F / A Rate of shear: Viscosity:

in dynes cm-2

G = du/dr in sec-‘ 11 = r / G

in dynes sec cm-’


(1b) (1c)

The viscosity is said to be Newtonian, if it is independent of the applied shearing stress; and non-Newtonian, when it does not conform to New-



ton’s original definition. Some workers, however, have also suggested the term “generalized Newtonian viscosity” instead of using the prefix “non.” The viscosity as defined in Eq. ( l c ) may also be considered as a measure of the energy dissipation in the fluid required to maintain a viscous flow. The rate of work, dW/dt, done in overcoming the frictional resistance in a volume, V , of the liquid is equal to G2q per unit volume (Robinson, TABLEI1 Definitions and Symbols of Several Viscosity Terms Definition



Conventional Symbol

Name Solution viscosity Solvent viscosity Kinematic viscosity Relative viscosity Specific viscosity Reduced viscosity Inherent viscosity Intrinsic0 viscosity

Proposed Symbol


Viscosity of the solution lo Viscosity of the pure solvent q/p Viscosity/density ratio )I/)I~Solution/solvent ratio or viscosity ratio 9


Viscosity number


Logarithmic viscosity number Limiting viscosity numberb

First suggested by Kraemer (1938) (see also, Kraemer and Lansing, 1935). Concentration unit: grams per 100 ml (or grams per deciliter). More recently, also designated as the “Staudinger Index” by the International Union of Pure and Applied Chemistry (1957). Concentration unit: grams per milliliter.

1939). The introduction of macromolecules into the fluid disturbs the streamlines, thus causing an additional energy dissipation, the amount of which depends upon the size and shape of the particles. As a result the viscosity of a polymer solution is always greater than that of the pure solvent. The conventional names and symbols of viscosity of solutions and also the proposed terminology as approved by the International Union of Pure and Applied Chemistry (1952) are listed in Table 11. Logically speaking, the designation of [v], vre1 , and vsp as “viscosities” is indeed incorrect since these quantities do not have the dimensions of



viscosity. The proposed new terminology, however, appears not to have been accepted universally, perhaps partly because of tradition and partly because some of the terms are rather clumsy in practice. It is noted that the concentration units in the two systems differ by a factor of 100. For example, the intrinsic viscosity of serum albumin is 0.040 dl gm-' or 4.0 ml gm-'. Unfortunately this has caused some confusion in the recent literature because many publications, both experimental and theoretical, still prefer the old expressions. Without an explicit statement of the units employed, misunderstanding often occurs. In this review the old conventions are used, mainly because the majority of publications on proteins appear not to have adopted the new symbols.

B. Determination of Intrinsic Viscosity Xumerous empirical equations have been proposed to express the viscosities as a functioii of concentrations for the determination of intrinsic viscosity. These equations have been tabulated in several textbooks (see, for example, Philippoff, 1942), but many of them have only limited applications and are already obsolete. Here we will only mention three commonly used ones. For dilute solutions the well-known Huggins equation (1942) can be written as


+ k"rl12C

(2a) where the coefficient k' is believed to be some function of solute-solvent interaction. Another form which is related to Eq. (2a) has also been used extensively (Kraemer, 1938) : %P/C


(loge B r e l ) / C

= [B]



By expanding (log, qr,l)/C into a power series of ?lSp/C",as the conccntration approaches zero, and combining them with Eq. (2a) we have


+ k"

= 0.5

'l'hus by plotting both qs,/C and (log, vre~)/Cagainst the concentration according to Eys. (2a) and (2b) one should obtain the same intercept, [?I, and the limiting slopes at C = 0 should satisfy the above condition for k' and Id'. This double check would enable us to determine the intrinsic viscosity with more confidence. Another useful equation for moderately concentrated solutions (up to 5 % or more) was proposed by Martin ( 1942) (see also Spurlin et al., 1946) ; log10



log10 [a1

+ I'C[rllC

which at low concentrations is related to the Huggins equation with lc = lc'/2.303




Martin's equation may find applications in the cases where proteins denature readily at interfaces, especially when flowing through a capillary a t high dilutions. It has been used also in the study of non-Newtonian viscosities (see Section IV). However, in a so-called good solvent, i.e., where strong solute-solvent interaction exists, the Martin plot [log,, ( T*,/C) versus C ] frequently shows a downward curvature as the concentration decreases below a certain range (Weissberg et al., 1951). As a consequence one may obtain a false intrinsic viscosity if the experimental data are limited only to the range of high concentrations. For solutions of high concentrations, Raker ( 1913) has suggested the following equation : ?re1


which can he rewritten as orel

+ ac)"


+ [olC/n)"



= (1

where a and n are two constants. It reduces to the Huggins equation as the concentration approaches zero. On the other hand, it converts into the Arrhenius equation when n becomes infinite, i.e., qrel = e"'" (or log, vrel = [qJC). Equation (4) may be of limited use for routine viscosity measurements of protein solutions, but Oncley et al. (1947) have tried the Arrhenius equation for very concentrated protein solutions in an effort to avoid the complication of surface denaturation. It is pertinent to note that all these empirical equations are useful only in certain limited ranges of concentrations and therefore one should be careful in applying them to experimental results, It is always a good practice to design the experiments so that Eq. (2a) can be applied, unless other considerations prevent its use.

C. Intrinsic Viscosity-MolecularWeight Relationship The dependence of the intrinsic viscosity on the molecular weight for a homologous series has customarily been expressed by the modified Staudinger equation, often called the Mark-Houwink equation (Mark, 19%; Houwink, 1940) :





where K and a are two constants for a given solute-solvent system, the latter being characteristic of the molecular conformation. For rigid spheres a = 0 and for rigid ellipsoids and rods a = 1.7-2, whereas for flexible polymers it varies from 0.5 to about 0.8. It has further been observed experimentally that stiff polymers such as cellulose derivatives and nucleic acids have values for a ranging from 0.8 to 1.2. Staudinger was among the first to emphasize the importance of viscosity measurements for molecular



weight determinations of high polymers and to arrive empirically at a value of unity for the exponent for certain systems studied (Staudinger, 1932). If the polymer is polydisperse, the intrinsic viscosity is obviously related to some kind of average molecular weight, appropriately called the viscosity average by Flory (1943), which depends on the magnitude of a. It is a common practice to define the different averages of molecular weight by the general relationship:

A ? = ( Zr M, ,F / Z c,MP-l) = ( CN , M q + ' / C N , M f ) a


where c , is the weight concentration, N i the number of moles, and M , the molecular weight, of species i. Thus, for number average, M , , for weight average, M , , for z-average, M , , for ( z 1 ) = average, M , + l ,


a = o

a = l a = 2 a = 3

and so on. The viscosity average, M , , however, obeys a different relationship. At infinite dilution ?JSP



and Then we have according to Eq. (5)



(cc i M i " / C c;)''"


( CN i M ; + " / C NiMi)""


Thus, for a = 1, M , is identical with M , . For a values other than unity M , has usually been found to be closer to M , than to M , , although quantitatively it depends upon the molecular weight distribution in each particular case. For example, for polymers having the "most probable distribution" (Flory, 1953)

+ a > r ( i + ~)]"":2 where r ( 1 + a) is the gamma function of (1 + a ) . Thus the ratios beM,:M,:M,



come 1:1.76:2 for a = 0.5, 1:2:2 for a = 1, and 1:2.31:2 for a = 1.7. As another illustration consider a protein containing 10 % end-to-end dimer, which gives the ratios M,: M,: M , = 1.05: 1.13: 1.10. (The exponent of 1.7 for rigid ellipsoids assumes that the components have the same minor axis and their molecular weights are proportional to the axial ratios. ItJ can be shown that the intrinsic viscosity will actually decrease if the particles form side-by-side aggregates.)


33 1

111. NEWTONIAN VISCOSITIES A . Theoriesfor Rigid Particles The additional average rate of energy of dissipation per unit volume as caused by the presence of a solute or suspended particle over that by solvent in a fluid can be expressed as

(dW/dt)/V = G 2 ( q -

qo) =



which in turn becomes (q




nuv/V =



Here n is the number of the noninteracting, identical particles, v the volume of each particle, V the volume of the solution or suspension, @ the volume fraction of the solution occupied by the particles, and v may be regarded as the viscosity increment which depends on the shape of the rigid particles. Einstein (1906a, 1911) was the first to treat thoroughly the viscosity of a solution or suspension of spherical particles which are rigid and large relative to the size of the solvent molecules, the latter being considered in effect as a continuous medium. In other words, the particles are regarded as small enough to exhibit Brownian motion, but large enough to obey the laws of macroscopic hydrodynamics. By straightforward hydrodynamic reasoning he obtained a value of 2.5 for v in Eq. (9b) in the limiting case where @ approaches zero. It is noted that the absolute size of the spheres does not enter into the viscosity equation. For solutions of nonspherical particles the situation is more complicated and the physical picture can be described qualitatively as follows: for a system of particles in a fluid one can define a distribution function, F (Peterlin, 1938), which specifies the relative number of particles with their axes pointed in a particular direction. Under the influence of an applied shearing stress a gradient of the distribution function, aF/at, is set up and the particles tend to rotate at rates which depend upon their orientation, so that they remain longer with their major axes in position parallel to the flow than perpendicular to it. This preferred orientation is however opposed by the rotary Brownian motion of the particles which tends to level out the distribution or orientations and lead the particles back toward a more random distribution. The intensity of the Brownian motion can be characterized by a rotary diffusion coefficient 8. Mathematically one can write for a laminar, steady-state flow: aF/dt


8V2F - V . ( F w ) = 0


where V 2is the Laplacian operator, the operator V . denotes the divergence w is the angular velocity of the rotating par-

of the function ( F a ) , and



ticles. The function F is further characterized by two parameters, a and R. Here a is the ratio of the rate of shear to the rotary diffusion coefficient, i.e., G / 0 . According to Jeffery (1922-1923) R = ( p z - l ) / ( p 2 I ) for ellipsoids of revolution having an axial ratio of p . Thus the fundamental problem becomes to find a general solution for the distribution function I" in Eq. (10), (in terms of a and p ) which determines the distortion of the streaming lines that contributes to the increase in viscosity of the solution over the solvent. Simha (1940) employing Eq. ( 10) solved the equation for the viscosities of solutions of ellipsoids of revolution for the limiting case a -+ 0. Under this condition the distribution of the particles can be regarded as almost completely random. Simha also found that for large axial ratios the viscosity increment a t a -+ 0 for very dilute solutions or suspensions (i.e., Q, 4 0) can be approximately represented by



= p2/15(10g, 2 p - 1.5)

+ p2/5(10ge2 p - 0.5) + 14/15


for prolate ellipsoids and Y =

16q/15 arctan (in radians) y

( 1 lb )

for oblate ellipsoids. Here p = a / b arid q = l / p , a and b being the semimajor and -minor axes of the ellipsoids. Mehl et al. (1940) have tabulated the numerical values of Y covering a wide range of p and q (Table VIII of the Appendix). It is noted that the viscosity increment is a function of the axial ratios only and does not depend upon the absolute molecular size of the particles. Each Y vtllue however corresponds to two axial ratios, p and q. A choice between the prolate and oblate models cannot be made from the viscosity measurements alone. For a homologous series of prolate ellipsoids having the same minor axis Hq. ( l l a ) can be approximated as (Simha, 1945) Y =

0.233 pl'"*,


5p 5



(llc) Y =

0.207 p'




p I 300

Thus, Y should increase with the molecular weight or length to the 1.7 power. For cylindrical rods, the suggestion has also been made to substitute 1.8 and 0.8 for 1.5 and 0.5 in the denominators of the two terms in Eq. ( I la) (Sadron, 1953a), presumably on the basis of Burgers' theory ( 1938), for cylindrical rods. For very large p values such modification is insignificant, although Burgers' treatment has now been found inadequate (Haltner and Zimm, 1959; Broersma, 1960). Independently Kuhn and Kuhn (1945) obtained the following viscosity



equations for rigid rods and discs: v = 0.4075(p

- 1)'

v = p2/15(10g, 2p


+ 2.5,

- 1.5)

+ p2/5(10ge2p - 0.5)

< 15 + 1.6, p > 15

and v = ( 3 2 / 1 5 ~ )( ~1)

- 0.628(~- l ) / (-~ 0.075)

+ 2.5,



(12a) (12b) (12~)

It is noted that Eqs. ( l l a ) and (12b) only differ slightly in the constant

term which is negligible for large axial ratios. By using as a model a linear array of spheres for the rodlike particles Kirkwood and Auer ( 19511 have calculated the intrinsic viscosity of these rods as a function of the axial ratio. (The relationship between intrinsic viscosity and viscosity increment is to be mentioned in the next section.) The asymptotic form of their equation is: [q] = ~NL'b/2250Mo loge ( L / b )


where N is t,he Avogadro number, L the length and 0 the radius of the rodlike particle, and M o the monomer molecular weight. It can be shown that for large axial ratios Eq. (13) is essentially identical with Eqs. ( l l a ) and (12b). Simha's equation [Eq. ( l l a ) ] , however, has been used most extensively although it is only recently that its experimental confirmation seems to have been definitely achieved.

B. Determination of Particle Shape from Intrinsic Viscosity Since the volume fraction, @, of the solute in Eq. (9b) cannot in general be determined experimentally, it has been a common practice to express the viscosity of Eq. (11) in terms of the intrinsic viscosity by the relationship [q] =

lim ( q -


qO)/voC' =



noting that is equal to V,,C/lOO. Here Vspis the specific volume of the solute in solution, the over-all volume occupied by 1 gm of the (dry) solute plus that due to bound solvent. The concentration C in grams dry solute per 100 ml solution can be calculated from straightforward analytical procedure. However, since VBPis not measurable experimentally, the solution of Eq. (14) with two unknowns, V,, and v, thus becomes indeterminate. To circumvent this difficulty it is customary to replace Vapby the experimentally measurable partial specific volume, of the solute. Equation (14) then becomes





(with no solvation)




This still does not solve the problem if the molecules are known to be hydrated (solvated) mainly because the volume occupied by the hydrated (solvated) particle will obviously differ from that of the dry particle. Mathematically, consider the volume of the solution as a function of the concentrations of dry solute, m, and solvent, mfl, the latter being the sum of the free solvent, m’o, and bound solvent, my . Thus I/‘ = V(m, mo) = ~ ( mmi ,

+ my)

Let dmb’ = wdm where w is the grams of solvent bound to 1 gm of solute. We then have by partial differentiation




+ w(aV/amo),]dm + (aV/amo),dmk

and at constant free solvent,





( a v / a m ) , ;=



P + wVfl = 7 + w / p


where is the partial specific volume of the solvent which can be considered as the reciprocal of its density, p. The term on the left side of Eq. (16) turns out to be identical with the partial specific volume of the hydrated (solvated) protein on the basis of dry protein mass, P h . The reason is as follows: imagine that one can first prepare a hydrated protein and then dissolve it in a large quantity of free solvent. Under these conditions no additional free solvent will be bound to the already hydrated (solvated) protein molecules, i.e., the free solvent, m i , remains constant. Thus the over-all increase in volume of the solution per unit (dry) protein mass represents PJ,. Accordingly Eq. (16) can be rewritten as 8 h



+ w/p



+ w/Bp)


Following Oncley (1941) one can rewrite Eq. (14) as [q] =

7h;av~,/lOO= 8 ( 1

+ w/vp)vh/lOO

(with solvation)


In the above derivation the bound solvent is assumed to have the same composition as the free solvent. This may not be exactly true for proteins in salt solutions, although the amount of salt involved in the bound solvent is so small that the resultant errors are quite insignificant (Tanford and Buzzell, 1956). Despite popular misunderstanding, Eq. (16) is derived without first assuming that V,, represents the sum of two volumes, one corresponding to the dry protein and the other the bound solvent. To substitute for V,, in Eq. (15a) obviously P must be positive and a nega-



tive value has never been reported for proteins. On the other hand, to use Eq. (15b) for hydrated proteins it is immaterial whether P in Eq. ( l e a ) is positive or negative as long as the sum of 7 and w/p is positive. It is generally recognized that proteins are hydrated in aqueous solutions, but the precise ranges of hydration are still little understood and not well characterized. To quote Edsall (1954), the lower and upper limits of the degree of hydration probably lie between 0.04-0.1 and 0.6-1.1 gm of water per gram of protein. In the hydrodynamic sense this water of hydration should not only include that tightly bound to the molecules but also that loosely dragged along with them. In any event the estimate of the axial ratio from Eq. (15b) is made possible only if the quantity 20 is known in advance. Oncley (1941) employed the assumption that:


= (1

+ W/PP)VA(P)


and designed a contour diagram in which the axial ratio, p, is plotted against the degree of hydration, w, with v as a parameter. Thus with known [ q ] , P, and thereby v [Eq. (Ma)], one can calculate the V h values for various w values and the contour line of this particular v describes the range of possible p values. The usefulness of this treatment lies in the fact that the 7is probably very close to the specific volume of the dry protein and w is assumed to vary over a not too large range. Therefore Eq. (15b) provides a rough estimate of the shape of the proteins. Let us now examine the assumptions underlying the Oncley treatment’. First, thermodynamically P is defined as the increment of the volume of the solution per unit mass of the solute added and therefore is not identical with V,, of the solute. These two quantities may be equal in magnitude if and only if the system is an ideal solution, that is, there is no solute-solvent interaction whatsoever present. To eliminate one unknown V,, in Eq. (14) by introducing we have at the same time added another uncertain term w into the equation. Thus this treatment offers at most a rough estimate of the shape of proteins for a chosen model, a prolate or an oblate ellipsoid. Furthermore the estimated p value corresponds only to the hydrated particle, which is slightly different from that of the unhydrated particle unless the bound water is so distributed throughout the protein molecule that it does not change its axial ratio because of hydration. Secondly, the use of Simha’s viscosity increments for protein molecules implies automatically that the shape of the molecule can be approximated by an equivalent hydrodynamic ellipsoid having the same volume as that of the protein. The uncertainty involved in this second assumption is not just a problem of hydration. Even if the degree of hydration were precisely known and even if 7and Vsp were identical the axial ratio as determined from the intrinsic viscosity may still not represent the real molecule.



Mathematical treatments such as Simha’s are exact, but the simple fact is that the actual shape of a protein molecule could be far from any fictitious model. The particle might be very irregular in shape and it might not even be perfectly rigid. The mathematical theories simply cannot handle completely the real physical situation, and the models are so chosen that they can deal with it in not too complicated a fashion. Thus all one can hope for is to deduce from the hydrodynamic measurements the dimensions of an equivalent model which behaves hydrodynamically as if i t were the actual molecule. It seems rather unfortunate to find that some workers have a tendency to accept Eq. (15b) without reservations as if it can give axial ratios even accurate to the first decimal. The truth is just the opposite. If there is good agreement between the hydrodynamic properties and other physical measurements, it may imply that the chosen model is probably not too far from the actual picture. On the other hand inconsistent results may actually shed some light on the real shape of the molecule as distinguished from a simple mathematical model.

C. Other Hydrodynamic Properties For the sake of discussions in a later section we will summarize here the theoretical equations of two other important hydrodynamic properties, the translational and rotary frictional coefficients. The former, designated as J can be determined from either sedimentation (Svedberg and Pedersen, 1940) or diffusion (Einstein, 1905, 190Ab; Smoluchowski, 1906) measuremrnts: fa


M ( l - Pp)/Ns

( 1 7%)





and HereM is the molecular weight and P the partial specific volume of the solute, N the Avogadro number, k the Boltzmann constant, and T the absolute temperature; s and D are the sedimentation and translational diffusion coefficients (after extrapolation to infinite dilution). The translational frictional coefficients from both measurements are regarded as identical, i.e., fa = fd . The rotary frictional coefficient, designated as {, can be determined from either flow birefringence or non-Newtonian viscosity measurements. For ellipsoids of revolution Perrin (1936) has shown that the translational frictional coefficient is given by the equation:




(1 - q2)1’2/42’3 log,([l

for a prolate model ( q = b / a

< 1), and

+ ( 1 - 42)1’2]/9)

Do/D = f/fo = (q2 - l)”2/q2’3arctan (in radians) (q2 - 1)”’

(18%) (18b)



for an oblate model ( q = b/a > 1). [Very similar equations have also been derived by Herzog et al. (1934).] Here f/fo is known as the frictional ratio and its numerical values as a function of the axial ratio are given by Svedberg and Pedersen (1940), Cohn and Edsall (1943), and Sadron (1953b). (Table I X of the Appendix.) Symbols fo and Dorefer to the frictional and diffusion coefficients of a sphere of the same volume as the ellipsoid and can be calculated from Stokes’ law:


6 ~ 7 0=~6 ~ 7 0 ( 3 V / 4 ~ ) ” ~

(18c) where r is the radius of the equivalent sphere, and V the volume of each particle. Thus the frictional coefficient,f, depends on both the volume, V , and axial ratio, ( p = l/q), of the particles. According to Perrin (1934), the rotary frictional coefficient, {, and thereby the rotary diffusion coefficient, 8, for ellipsoids of revolution can be given as = fo =

for a prolate model, and ~

0= ,


(3/2)([qz(q2- 2 ) / ( q z - 1)1/2~ arctan (in radians) ($’ - 1)‘”

+ $}/ti




for an oblate model. [Gans (1928) has also derived a set of formulas which are not exactly identical with Perrin’s, but such that the numerical values of 0 calculated from both equations are nearly the same.] The numerical values of {/{a for various axial ratios have been listed by Scheraga and Mandelkern (1953). (Table I X of the Appendix.) Here {,, and 0, again refer to the coefficientsof a sphere of the same volume as the ellipsoid and can be calculated from Stokes’ law: kT/&

= {O

= 8?rq0r3 =



Thus, like f the rotary diffusion coefficient, {, also depends volume and the axial ratio of the particles. For large axial ratios Eq. (19) reduces approximately to


both the

qoeb/T = (3k/16za3)[2 log, ( 2 ~ / b ) 11

and @Ob/1’

= (3k/16nb3) [arctan @ / a )

+ (a/b)]= 3k/32b3

(19d) (19e)

Equations (19d and e) have been extensively used in flow birefringence measurements and, more recently, in non-Newtonian viscosity studies. (For an ellipsoid there are three translational and rotary frictional coefficients,



fi ,f2

,f3 and {I, C2 , , each characterizing the resistance to motion of the ellipsoid parallel to one of its principal axes. For an ellipsoid of revolution, two of the f’s and {’s are identical respectively. The experimentally determined diffusion coefficient and frictional coefficient are given by the relationship :

+ + D3)/3

D = (01 Dz or


= kT(l/ji



+ l/fi + l/f3)/3




I n the case of rotary diffusion and frictional coefficients, only the rotation of the major a-axis about the minor b-axis is in general experimentally measurable. Accordingly only the equations for 6,and { b will be considered here. ) Following Oncley (1941) the frictional ratio for hydrated particles can be separated into two factors




w - 0 )


where the second term on the right side of the equation, designated as the hydration factor, is given by

j*/fo =

(Bh/P)”3 =


+ w/Pp)1’3


(2Oc) Again Oncley has constructed a contour diagram in which the axial ratio is plotted against the degree of hydration on a double logarithmic scale with j / j o as a parameter. By assuming a w value one can immediately read off the p value corresponding to the calculated f / j h . By the same reasoning one may also describe the rotary diffusion coefficient as

and {o =

8~70(3PM/4sN)= 67aPM/N


D. Concept of Equiualent Hydrodynamic Ellipsoid In view of the uncertainties raised in the Onclcy treatment a different



approach to the problem was proposed by Sadron as early as in 1942 although it did not then receive wide attention, probably owing to wartime conditions. More recently this analysis has been discussed in detail by Scheraga and Mandelkern ( 1953). These authors suggest the use of a rigid equivalent ellipsoid of revolution characterized by its effective volume, V , , and axial ratio, p , in the interpretation of the hydrodynamic properties of protein solutions. The shape of this fictitious model may not necessarily resemble the dimensions of the real particles at all. They have also correctly pointed out the necessity of a combination of two hydrodynamic measurements for the determination of both Y e and p . The interpretation of any single hydrodynamic measurement could sometimes lead to erroneous conclusions. For example, the increase in intrinsic viscosity upon denaturation could arise from either increasing asymmetry or increase in V , or both, although in the past such a change was regarded frequently in terms of increasing asymmetry as a result of unfolding of the polypeptide chains. Scheraga and Mandelkern specifically reject the implication of Eq. (16) as representing the specific hydrodynamic volume, because it ('neglects possible flow of solvent through the domain by the hydrodynamic forces, selected adsorption from mixed solvent, electrostriction, and similar effects," Instead they rewrite Eqs. (14), (18c), and (19c) as [q]




fo = 6 ~ q o ( 3 V e / 4 ~ ) ~ ' ~


To = 6qoVe


and By combining [q] [Eq. (22)] with either s (or 0)[Eqs. (17a, b, Ma, b, 23)] or 0 [Eqs. (19a, b, 24)] and eliminating V , they have evaluated two functions designated as 0 and 6.

and = Jv

Here y = N"3/(162007r2)1/3, F = fo/f, and J = ~oO/C.Both the P- and 6functions depend on the axial ratio only and their numerical values are listed in Table X of the Appendix. Once the axial ratio is determined the



effective volume, V , , can be calculated from one of the equations, Eqs. (22), (2X), and (24), which in turn gives the major and minor axes of the ellipsoid of revolution. This iiew analytical approach has since become a controversial subject (see, for example, Tanford and Buzzell, 1954, 1956; Loeb and Scheraga, 1956; Tanford, 1957a; Scheraga and Mandelkern, 1958; also Lauffer and Bendet, 1954). One argument is concerned with the use of an effective hydrodynamic volume. For years we have been accustomed to the Oncley treatment as a convenient method for estimating the axial ratios of the proteins. Now this new analysis raises serious doubts about the use of partial specific volumes and therefore seemingly uproots the very foundation of the conventional treatment. Naturally one may wonder what can be gained by characterizing this equivalent ellipsoid the shape of which might be quite different from the actual particle, and therefore the calculated values of V , and p might lack any physical significance. This argument actually points up the major contribution of this new concept which recognizes the uncertainties involved in t,he conventional oversimplified treatment and warns against its indiscriminate use that could lead to illusory conclusionx (see Section 111, B ) . (The word “effective” is introduced to emphasize the nature of this equivalent ellipsoid which effectively represents the hydrodynamic behavior of the real molecule. It would be redundant to call the exact mathematical models “effective,” but by the same token it is questionable to regard any actual molecule as corresponding to an exact model.) Another oft-repeated argument involves the expression of Eq. (16). Scheraga and Mandelkern reject it on the grounds that it incorrectly interprets (1) Q as the specific volume of the dry protein and (2) V, as the sum of two volunies, one for the dry protein and the other for the bound water. The second reason actually has nothing to do with the argument, since Eq. (16) can be derived without such an assumption. In this respect Lauffer and Bendet (1954) have argued for the use of v h instead of V s p (for hydrated proteins) by considering the viscosity as a function of the total amount of liquid displaced by the particles. But the real problem is and V,, have the same numerical values which could be not whether almost true a t least for the protein solutions. Rather, the question is: can we equate the particle volume with its hydrodynamic volume? It is the acceptance of this equivalence without reservations that has been disputed convincingly by Scheraga and Mandelkern. It is somewhat unfortunate that the current controversy has overshadowed the real test of the new analysis which at present is still lacking. Both the conventional and new analyses assume an equivalent ellipsoid and attempt t o solve for two unknowns, the volume and the axial ratio. Actually Oncley has also pro-



34 1

posed to combine tJwo contour graphs from viscosity and sedimentation (or diffusion) and determine t,he overlapping portions of the two sets of curves (after taking experimental errors into consideration) which define the possible range of the axial ratios (see, for example, Mehl et al., 1940). It caii be shown that by so doing the conventional treatment has implicitly eliminated the volume term, v(1 w / v p ) , just in the same manner as the p-function which, however, removes the volume restriction as defined in Eq. (15b). The shape of these curves is such that they never cross sharply, a fact merely reflecting the insensitiveness of the axial ratio determinat.ion from this procedure. The same difficulties are encountered in the








10 10' n/b -+ RATIO


FIG.1. Viscosity increments, Y, translational frictional coefficients, f/fO , arid @-functionsof the ellipsoids of revolution at various axial ratios. Left: oblate; right: prolate.

use of P- arid &functions. The reason is simple. Any hydrodynamic eyuatioii contains a product of (volume X shape factor) and by eliminating the volume from a combination of two hydrodynamic measurements only the ratios of the t>woshape functions remain in the final equations. The insensitiveness of the latter is therefore not unexpected even though each shape function varies significantly with the axial ratio. This can best be illustrated graphically in Figs. 1 and 2, where the viscosity increment, Y, and the frictional coefficient,s,f/joand { / T o , are plotted as a function of the axial ratio, p , and compared with the corresponding p- and &functions. Thus the intrinsic viscosity increases and the sedimentation coefficient decreases (i.e., the translational diffusion coefficient increases) with increasing axial ratio, but the product ~ [ 7 7 ] ~ in ' ~ the @-function does not increase much over the same range of axial ratios. For example, if the experi-



mental p is 2.15 f 0.06 X lo6 it is not possible to ascertain whether the particle is a sphere, or an oblate ellipsoid having an axial ratio of almost any value, or a prolate ellipsoid having an axial ratio less than five. The situation is somewhat better for the &function, but unfortunately the experimental rotary diffusion coefficient is also more uncertain than the sedimentation coefficient. Thus both functions can only give a rough estimate of the axial ratio even if the experimental data are of the highest possible precision. With axial ratio determination uncertain it becomes meaningless to calculate the effective volume and the major and minor axes. The p-func-







, a/b





FIG.2. Viscosity increments, Y, rotary frictional coefficients, , and b-functions of the ellipsoids of revolution at various axial ratios. Left: oblate; right: prolate. Notice the difference in scales between Figs. 1 and 2.

tion however has found an unexpected use in molecular weight determinations. With known 1771 and s and by assuming a reasonable p value the molecular weight as calculated from Eq. (25) usually agrees very well with that obtained from other methods. One fundamental question concerning the new approach seems to have passed unnoticed despite the many arguments that have been raised aginst it. In the derivation of the p- and b-functions, Scheraga and Mandelkern have assumed that the three hydrodynamic properties can be represented by an identical equivalent ellipsoid. If either V , or p of this fictitious ellipsoid is arbitrarily fixed the other is automatically defined. But there is no a priori reason tjo assume t,hat the particles under shearing stress and sedimentation should fit the same hydrodynamic model. If V , is kept identical the appropriate p value for describing one property might differ from the



appropriate value for the other and vice versa. If the particles are very rigid and resemble some simple model, say, a cylindrical rod, one might expect a certain simple relationship between the actual particle and its equivalent ellipsoid. On the other hand if the particles have very irregular shapes and if they are not perfectly rigid and solvent-impenetrable one cannot accept the equivalence of the two ellipsoids under shearing stress and sedimentation without serious reservations. This is based on the fact that the motions of the particles could be quite different in the two measurements. Suppose, for example, that the particles have holes which let the solvent molecules penetrate freely. Conceivably there will be a constant backflow of the solvent through the interior of the particle under sedimentation, whereas the particles which rotate under the influence of velocity gradient will merely drag the solvent along. The same argument has been well discussed by Flory in his treatment of flexible coils (see below). The differencebetween the two equivalent ellipsoids might be very small. Nevertheless it seems just as dogmatic to disregard this point as to accept the arbitrary restriction of P = V,, in the conventional treatment. I n particular since the ,&function is so insensitive to the variation of p any small deviation from equivalence of the two ellipsoids could manifest itself significantly in the final determination of the axial ratio and the effective volume. In this respect the &function should in principle be more reliable since both [q] and 8 can be determined under the same applied shearing stress. If the particles possess a certain degree of flexibility Scheraga and Mandelkern (1953) have also noted another complication owing to the possible deformation of the particles. Under these circumstances the equivalent ellipsoid as determined at any finite velocity gradient may not necessarily be the same as that at zero gradient. This discussion does not cast doubt on the concept of the equivalent ellipsoid. It merely emphasizes the necessity of extensive tests of the /3- and &functions. The new approach has sharpened our understanding of the hydrodynamic properties and compelled us to take a critical look at the interpretations of the experimental data, which in itself is a significant contribution. The foregoing discussion seems to give a rather pessimistic point of view. On the one hand the conventional treatment is now in serious doubt and on the other the new analysis still leaves much to be desired. This however does not mean that one can obtain no information at all about the shapes of proteins in solutions. It merely implies that one can never be too critical about the interpretations of the experimental results. Viscometry is, and will always be, a powerful tool in the studies of proteins and other macromolecular systems. It is most useful in the studies of conformational changes such as protein denaturation, degradation, and aggregation. It will continue to be valuable in providing us with a rough picture of the general



shape of the protein molecules. The Oncley treatment has played an important role in the past in spite of its many shortcomings. It seems certain that this procedure will continue to be used by many workers because of its simplicity. But it will be most unfortunate if one takes its conclusions too literally and ignores completely the arguments that have been raised against it. This is particularly true in the studies of protein denaturation where the application of the Scheraga-Mandelkern analysis will often allow us to deduce whether the change in intrinsic viscosity upon denaturation is primarily due to a change of the shape or the effective volume or both, even though it may be only of semiquantitative nature. However, this new approach, sound in concept, has yet to be subjected to extensive tests. Until this is done it seems rather drastic to condemn the use of the conventional treatment as completely out of date, especially since the 0- and &functions do riot guarantee any better and more precise determination of the shape of proteins. There is a tendency to put too much emphasis on the differences rather than the similarities of the two treatments, thus causing a rather nnneressary controversy. 13. Estimation of the Length of Equivalent Ellipsoid

Although both the p- and &functions are too insensitive and uncertain for the determination of axial ratios, the length, L, of the equivalent ellipsoid (prolate) can be calculated with confidence from either the intrinsic viscosity a t zero gradient or the more commonly known flow birefringence. This can be shown as follows: Eq. (22), (19a and c ), arid (18a and c) can be rewritten as

[VIM = (4rN/300) (a3)bh’)


V O ~= / T(k/’%)( 1 b 3 )( J p ’ )





( A ~( a~)( 1~ / ) ~ ~ ~ / ~ )


which in turn become

L = 2a


I: = 2a

= 3.53

L = 2a


G.82 X 10-S([~]M)1’a(p2/y)1’3



1 0 - 6 ( ~ ~ ? t a 0 ) 1 1 3 ( ~ ~ p 2 ) 1 ’ 3 ( 2%


and 1.76 X 10-26[M(1- ~ p ) / 9 0 s ] ( P p ~ ~ ~ )(20b)

( L is expressed in centimeters.) In Table XI of the Appendix are listed the values of ( p ’ / ~ ) ~ ’(Jp’)113, ~, and (8‘~’’~)which for large axial ratios, p , can be approximated as



( p 2 / v ) ” 3 = constant


( JP’)’’~ = ( 3 log, 2p


- 1.5)1’3,

> 20










Fp2’3 = log, 2p,

Thus at least for intrinsic viscosity at zero gradient and flow birefringence appreciable errors in the calculated lengths are unlikely to result if a somewhat incorrect p value is assumed in Eqs. (27b) and (28b). In fact it has been shown that the viscosity method (at zero gradient) [(Eq. 27b)I is as good as the more common flow birefringence technique and both methods given entirely comparable errors for the IIRC: of incorrect p values (Yang. 1961b). Flow birefringence is known to be extremely sensitive to the degree of polydispersity of the rigid particles as reflected in the variations of the calculated lengths with velocity gradient. Precisely because of this it is very difficult to define a mean length for a polydisperse system. This difficulty is not encountered in the viscosity method (at zero gradient) and the viscosity-average length is usually regarded as close to the weight-average value. Unlike flow birefringence the viscosity measurements are simple, precise, and usually less time consuming. Another difference is that flow birefringence becomes impractical if the partihes are not very elongated whereas the viscosity method (at zero gradient) as discussed here is subjected to no such limitation so long as the axial ratio of the particles is fairly large. The requirement of a known molecular weight in Eq. (27b) seems to be a disadvantage of this viscosity method, although this quantity is usually available in the characterization of any macromolecule. As an illustration the calculated lengths of several proteins and polypeptides using Eq. (27b) are listed in Table 111. The agreement between viscosity and other methods is self explanatory. According to Eqs. (27b), (28b), and (29b) and Table XI of the Appendix, the use of an assumed axial ratio larger than the true value always results in a positive deviation in the estimated length. This in turn will reduce both the minor axis (or diameter), 2b, and the effective volume, V , . An upper limit of the p value can however be estimated from the Oncley treatment, since it is very unlikely that V ecan be much less than M v / N . Similar calculations can be given for oblate ellipsoids, but the errors will be larger than those for the prolate ellipsoids if an incorrect p value is used. In any case the flattened disc model rarely finds use in biological colloids. The same uncertainty has also been found in the use of Eq. (29b) and it will therefore not be discussed further.



F . Theories for Flexible Coils Theoretical treatments on the viscosity of solutions of polymer chains are too numerous to give even a brief summary. Originally their principal objective was to explain the intrinsic viscosity-molecular weight relationship as described in Eq. ( 5 ) . Now the major interest goes far beyond that and toward a better understanding of the solution properties of polymers. Our brief discussion will be confined only to general terms. The approach TABLEI11 The Calculated Lengths of Several Proteins and Polypeptides from Intrinsic Viscosity at Zero Shearing Stress Proteins and polypeptides Fibrinogen Tobacco mosaic virusb Myosin Collagen Poly-7-benzyl-L-glutamates M , = 130,000 M, = 208,000

From light From flow birefringence scattering (A) (A)

Range of axial ratio assumed

From [do (A)

10-20 15-25 30-70 100-200

580420 3450-3710 1600-1740 2760-2890

62O-68Oa 3340-3530'

50 80

920 1510





3000" 1620d 3000' 910, 1410f

Scheraga and Laskowski 4957). The calculated lengths for this virus, as obtained from [ q ] and ~ from flow birefringence, require revision to take account of the fact that the virus is more accurately represented by a rod of uniform thickness than by an ellipsoid of revolution. When due allowance is made for this, the calculated lengths are very close to those determined by electron microscopy. See Haltner and Zimm (1959) and the discussion in Section 111, H , 4 below. 0 Boedtker and Simmons (1958). Holtzer and Lowey (1956,1959). Boedtker and Doty (1956). f Doty et al. (1956). a

adopted by Brinkman (1947) and Debye and Bueche (1948) is to replace the actual polymer coils by a porous sphere having an effective radius, R. The effective porosity of this equivalent sphere is expressed by a shielding ratio, RIL, where L represents the depth to which the solvent can almost freely penetrate into the sphere. This model is so general that it does not depend on the particular assumed molecular structure, such as Gaussian or non-Gaussian chains, branched chains, etc. A more realistic and rigorous treatment was first developed by Kirkwood and Riseman (1948) using a necklace model and taking into consideration all the intramolecular interactions. I n both approaches tlhe solvent is regarded as a n "inert" continuous medium and its only influence on the intrinsic viscosity manifests itself



through certain parameters such as the frictional coefficient which in turn is related to the permeability of the solvent. An alternative approach has been formulated by Flory and Fox (1951) (see also Flory, 1953) who take into consideration the excluded volume effect and introduce an “equivalent hydrodynamic sphere,” the radius of which is postulated to be proportional to the root-mean-square end-to-end distance, (r2)l/’,or the radius of gyration, (s2”/’ [Note that (r’) = 6(s2) (for a Gaussian chain).] I n all these treatments the intrinsic viscosity can be expressed as [TI n: (T’)~/’/M= ( S ~ ” ~ ’ / M


In Flory’s treatment the term (r2)3’2/Mis rearranged into ( ( T : ) / M ) ~ ‘ ’ M ~ ’ ~ ~ ~ , where (r:) refers to the ideal, unperturbed state. The expansion factor a (not to be confused with the same symbol used in Section 111, A ) is defined as ( T ~ ) ~ ’ ~ / ( T which ; ) ~ / ~ reduces to unity in an ideal solvent or, in Flory’s terminology, at the &temperature.’ Equation (30) is very similar to that for rigid particles [Eq. (22)], since both (T’”’~ and (s’”’’ are proportional to the hydrodynamic volume of the polymer chain. Unlike the equations for rigid, impermeable particles those for the viscosity of flexible polymers do not contain a shape factor such as Y, but the hydrodynamic volume is dependent on the solute-solvent interaction, In the limiting case of complete solvent immobilization inside the molecule (according to Debye and Kirkwood) or at 0-temperature (according to Flory ) the above-mentioned treatments give essentially identical results (see also Peterlin, 1959). Debye-Bueche:

[T] =


2 312

13.57 X 10 ( s ) / M


4.17 X 1022(~2)3/2/M(30a)

Flory :




2 The osmotic pressure, H , of dilute polymer solutions may be expressed as a function of the concentration, C , in the form

T/C = RT(B1

+ BzC +

. a * )

(R = gas constant and T = absolute temperature). The so-called first virial coefficient, BI , is simply equal to the reciprocal of the molecular weight, M. The evaluation of the second virial coefficient, B Z, has become one of the principal theoretical developments in recent years. In his theory of polymer solutions Flory (1953) has shown that B I is proportional to a term (1 - e/T), which vanishes a t T = 8. Accordingly e may be considered as the “ideal” temperature a t which the above equation is reduced to the well-known van’t Hoff’s law, i.e., H / C = RT/M To avoid possible confusion with the symbol for rotary diffusion coefficient here we have used the symbol e for the theta temperature, rather than 0 which was originally used by Flory.




The numerical constant, of the Kirkwood-Riseman equation was originally and later reduced to 4.8 X 10” ( Kirkwood et al., 1955). given as 5.3 X A further revision by Auer and Gardner (1955) gives a valw of 1.25 X loz2 which agrees well with Zinim’s 4.17 X 10’” (1956). The experimental value by I’lory and his co-workers has risen from the original average 3.1 X 10“ to a more recent value of about 3.7 X loz2.Krigbaum and Carpenter (1955) reported that the Flory constant actually increases with decrease in the second virial coefficient (see footnote 2). A theoretical explanation for this variation for systems close to the 0-temperature has been given by Kurats and Ysmakawa (1958) and Yamakawa and Kurata (1958). The apparently too-large-value of the Debye-Bueche theory probably reflects its unrealistic assumption of a uniform segment distribution inside the porous sphere which underestimates the hydrodynamic interac tion in the coils. Nevertheless it is indeed surprising to find the closeness of various values despite the use of different models. Altogether both theories and experiments seem to converge to a good agreement at least in this limiting (we. Similarly, the theoretical treatments of the sedirne~it,ationfor. romplete solvent immobilization or at 0-temperature ran be written as: Debye-Bueche :



(1/24.3) M ( 1 - P ~ ) / N T o ( s ~ ) ” ~

Kirkwood-Riseman :

(1/12.73) M ( l - P ~ ) / N ~ O ( & *(81) ’*

Flory :

(1/12.56) M ( 1 - B ~ ) / N ~ I o ( s ~ ) ” ~

By combining Eq. (30a) with Eq. (31) and eliminating the term s2 we have : Debye-Bueche : Kirkwood-Riseman : 1i‘ior.y:

~~[~]*/~~~ /Pp) n i ~= /2.12 ~(x 1 10‘ 2.72 2.10 (theor.)


2.5-2.7 (expt,l.)

It is not,ed that the left side of Eq. (32) is exactly identical in form with the &function for rigid particles (Section 111, 0). I n lglory’s treatment the numerical coefficient on the right side of the equation should be a universal constant, designated by him as $1’3P-1. This constancy has been confirmed experimentally for many polymers studied, although the experimental values are always higher than the theoretical one. This has been attributed by Flory to the possible difference in the equivalent spheres under shearing stress and under sedimentation. It is also interesting to note that this constancy applies to those highly extended polymer rhsiris such as celliilose



derivatives even though the coefficients for viscosity and sedimentation in Eqs. (30a) and (31) are no longer constant separately. With increasing solute-solvent interaction or a t temperatures other than the &temperature Flory’s theory still predicts a constant coefficient in Eq. ( 3 2 ) ’ whereas according to the Debye-Bueche treatment the increase in solvent permeability results in an increase in the coefficient in the same manner as the ,&function increases with the axial ratio. This raises a n interesting quest,ion concerning the use of the p-function for rigid particles (Section 111, D ) . A constant value of 2.5 X lo6 according to Flory will correspond to an axial ratio of 15 from the p-function for a prolate ellipsoid, not to mention the unavoidable experimental errors. Since the theories for linear polymer chains are not directly applicable to most proteins it is not possible a t present to predict whether the coiled form of, say, a denatured protein having many cross-linkages would give a p-value closer to 2.1 X lo6 for a hard sphere or 2.5 X 10’ for flexible coils or even to some other values. This problem can only be clarified by extensive experimental analyses. Thus in the absence of any information concerning the rigidity of the molecules under study there is reason for caution in the quantitative interpretation of the &function. Branching. So far our discussion has been limited to linear polymer chains. The effect of branching on viscosity is still not well understood. The statistics of certain simple types of branched chains has been studied by Zimm and Stockmayer (1949) and Stockmayer and Fixman (1953). Since branching produces a less extended hydrodynamic volume than would be expected for a linear chain of the same molecular weight, conceivably the intrinsic viscosity for a branched polymer would be smaller and the sedimentation coefficient larger than those for a linear polymer. At present quantitative treatments are still scarce.

G. Polyelectrolytes and Electroviscous E$ect By definition polyelectrolytes are a class of macromolecules having ionizable groups distributed at intervals along the polymer chains. I n the absence of any other restriction such as cross-linkages, a flexible coil possessing ionic charges of like sign expands into a more extended chain conformation as a result of electrostatic repulsion. This polyelectrolyte effect is therefore dependent, on the degree of ionization. It can be partially neutralieed by the distribution of ions of opposite charge (counter ions) and thus repressed by an increase in ionic strength. If ionic charges of both signs are present along the chain in approximately equal number the molecule tends to coil more tightly than if it were uncharged, mainly because of the strong electrostatic attraction among the oppositely charged groups. A polyelectrolyte such as denatured protein a t its isoelectric point is an example of this class.



The polyelectrolyte effect manifests itself in many of the solution properties of the macromolecule (for a brief review, see Doty and Ehrlich, 1952). The titration behavior of polyelectrolytes has been studied extensively, both theoretically and experimentally, and its discussion is beyond the scope of this review. Hydrodynamically, this effect also influences the measurements of sedimentation, diffusion, flow birefringence, and viscosity of the polymers. At sufficiently high ionic strength where the ion atmosphere is clustered closely around the macromolecule, so that the charges on the molecule are well shielded, the polymer behaves as if it were uncharged. The sedimentation coefficient however begins to fall with decreasing ionic strength as a result of the dragging along of the counter ions and also the expansion of the polyelectrolyte. The diffusion coefficient however will increase with decreasing ionic strength owing to the fact that the counter ions diffusing ahead of the polyelectrolytes induce an accelerating effect which more than compensates for the increase in the translational frictional coefficient of the expanded polyelectrolyte. For flow birefringence both the extinction angle and the birefringence increment are very sensitive to the molecular asymmetry of the polyelectrolyte. I ts “abnormal” behavior in salt-free solution is believed to be due to the same cause as that of the viscosity discussed below (Yang, 1958b). I n the absence of added electrolytes the reduced viscosity, rlSp/C,of a polyelectrolyte rises upon dilution in a striking manner as a result of the expansion of the polymer chains (Fuoss and Strauss, 1948; Hermaris and Overbeek, 1948; Kuhn et al., 1948). Empirically, Fuoss and Strauss have found that the viscosity data can be represented by the equations qsp/C = a

+ b/(l + c d C I




in the absence of salt and [tll = A(1

in the presence of salt, where a, b, c, A and B are constants. Here A is the intrinsic viscosity in the limit of infinite ionic strength, p. The term a is usually very small as compared with the last term in Eq. (33) and thus can be neglected. The numerator b is sometimes considered as the intrinsic viscosity of the polyelectrolyte in its most swollen state, although, strictly speaking, this is not true since the ss,/C - C plot passes through a maximum and curves downward rapidly as the concentration approaches zero. Most proteins do not behave like polyelect


molecular weights, presumably because of the presence of cross-linkages, etc. in the proteins which greatly restricts their expanded volumes. Electroviscous Efect [for a recent review, see Conway and Dobry-Duclaux (1960)l. I n an ionic solution or suspension each charged particle is surrounded by an atmosphere of counter ions having a net charge of opposite sign to that of the particle. This ionic atmosphere undergoes deformation when subjected to an applied shearing stress, thus giving rise to a n additional dissipation of energy. This electrostatic contribution to the viscosity of the solution or suspension is commonly called the electroviscous effect. Smoluchowski (1916) was among the first to put forward a n equation, but without derivation, to explain this effect on charged colloidal particles. Twenty years later Krasny-Ergen (1936) derived an equation similar to Smoluchowski’s, except for a different numerical factor. Experimentally Bull (1940) studied this effect on the change in viscosity of ovalbumin with degree of ionization, but found that the experimental value was about ten orders of magnitude less than that predicted by the theory. This prompted Booth (1950, 1953) to derive a new equation for a spherical model which can be rearranged as (see, Tanford and Buzzell, 1956) [?]el


322[g][ 2z2$’ ( Ka )

( Cizz/xi)] / [ D kTgoU2


(35 )

where c = protonic charge, Z = charge on the particle, D = dielectric constant of the solvent, K = Debye constant (proportional to the square root of the ionic strength), a = radius of the sphere, $‘ = a decreasing function of KU (see Booth’s papers for its numerical values), ci = concentration of the ith component of the inorganic ions, Zi = charge of the corresponding ith component, and X i = ionic conductance in practical units. Comparison of Bull’s data with Booth’s theoretical prediction (under certain assumptions) appeared to be very favorable. Recently Tanford and Buzzell (1956) and Buzzell and Tanford (1956) further tested Eq. (35) with the viscosity data of bovine serum albumin and ribonuclease and found that the agreement between theory and experimental results was reasonably good. Some of Tanford’s calculations are listed in Table IV to illustrate the order of magnitude of this electroviscous effect. It is noted that the observed values were uniformly higher than the theoretical predictions, although the discrepancies were barely outside the experimental errors. On the other hand, these could be attributed partly to the nonspherical shape of the proteins which does not exactly fit with Booth’s simple model. Nevertheless the general opinion that the electroviscous effect is small for charged colloidal particles (Hermans and Overbeek, 1948) seems to have received strong support a t least in the case of globular proteins. Furthermore, this effect can be eliminated in the presence of high ionic strength. I n relatively high concentrations of polyelectrolytes where the double



layers and the charged particles overlap one another, a so-called second electroviscous effect may arise. These particles tend to move away from one another in the direction perpendicular to the flow. This mutual interaction (electrical repulsion) causes another extra dissipation of energy and therefore increases the viscosity of the solution or suspension. This contribution increases with the square and higher powers of the concentration a t constant ionic strength or with decrease in ionic strength at constant concentration. Frequently these particles also exhibit a non-Newtonian behavior mainly because the displacement of the encountering particles will be less the more rapidly t,hey pass each other. As a consequence the enhanced viscosity due TABLE1V The Electroviscous Effect Ionic strength

PH 5-5.6 5.0 7.3 7.3 5.9-11 2.8 2.8 9.6 b

0.014.05 0 0.01 0.05 0.01 0.05 0.15 0



Bovine serum albumin0.037(mean) 0.0406 Ob -11 0.0412 -16 0.0383 Ribonucleasec 0.0330(mean) 0.0358 +16 0.0350 +16.5 O* 0.0336

Tanford and Buzzell (1956). At the isoianic paint is zero but Buzzell and Tanford (1956).



(dl gm-l)




(Booth Eq.)

0.004 0.004 0.001

0.0009 0.0008 0.0005

0.0028 0.0020 0.0006

O.OOO9 0.0004 0.0003

z*as used in the Booth equation is not zero.

to this electroviscous effect will be smaller at higher shearing stress. In general, however, this effect is not considered to be particularly strong with macromolecules (see, for example, Dobry, 1955).

H . Examples of Several Common Proteins and Polypeptides Viscosit,y data for proteins are indeed too numerous to give even a cursory summary. Rather we will consider only a few examples to illustrate the information that can be obtained from viscometry and the limitations in its interpretations in the light of the previous discussions. 1. Synthetic Polypeptides

Until recently the validity of Simha’s equation appeared not to have been subjected to unambiguous experimental confirmation, although it has been used extensively by the protein chemists. The recent investigations of



synthetic polypeptides provide an ideal model for such a test. Doty et al. ( 1956) (see also Doty et at., 1954) found that poly-y-beneyl-L-glutamates exist in the form of Pauliiig and Corey’s a-helix (1950) (see also Yauling and co-workers, 1951) in solvents having low hydrogen-bonding properties.

.A P









I I I100d,000




200;000 Molecular weight




Fro. 3. Intrinsic viscosity-molecular weight relationship of poly-7-benzyl-Lglutamates. The open circles represent the randomly coiled form and other symbols the a-helical form. The line of steeper slope is plot of Simha’s equation. Abbreviations: dichloroacetic acid, DCA; chloroform saturated with formamide, C-F; dimethy1 formamide, DMF; light scattering, L.S.; weight-average molecular weight, M , . Reproduced from Doty ef al. (1956).

These authors have also shown that for a homologous series of polypeptides, the intrinsic viscosity varies with the molecular weight (from light scattering) to the power of 1.7,which is in perfect accord with Simha’s theoretical value (see Fig. 3 ) . By adopting the conventional treatment with P = 1/1.31 and w = 0 the [7]-M relation could be fitted to Eqs. ( l l a ) and (15a) for prolate ellipsoids

3 54


if the major axis of the hydrodynamic ellipsoid was equated with the length of the rod (with a pitch of 1.5 A per amino acid residue) and its minor axis was taken as 18.2 A (varying experimentally between 17.75 and 18.0 A in different solvents). Since the volumes of the rod and the equivalent hydrodynamic ellipsoid were assumed to be identical, the diameter of the former would become (2/3)''' of the minor axis of the ellipsoid. It turned out to be 14.9 f 0.3 A which agreed well with 15.0 A found from the X-ray study of Arndt and Riley (1955) for the average distance separating neighboring helices. Thus the conventional treatment seems to explain well the viscosities of the polypeptides. Such consistent results however did not prove the shape of the hydrodynamic ellipsoids and neither did they rule out other possibilities. It can easily be shown that if the axial ratios in the above calculations are changed by a constant factor the same [q]-M relation still holds, except that the hydrodynamic volume will differ from that calculated from V , = v M / N . Doty et al. (1956) also pointed out that the arbitrary equating of the rod and its hydrodynamic ellipsoid having the same length and volume can be avoided by using Eq. (13) for rods instead of Simha's equation. This however involves a similar arbitrary assumption in the calculations since the Kirkwood-Auer model is a rodlike array of spheres, thus requiring an adjusted radius to fit the experimental data. It, will also be interesting to find out what axial ratios the p-functions will yield for this simple model system, if the sedimentation coefficients were available. In Fig. 3 the intrinsic viscosity begins to deviate downward from the theoretical curve for rigid rods when measurements are extended to the molecular weight range of about one million. This was attributed by Doty et al. (1956) to the onset of a very small amount of flexibility for the long, thin a-helices. I n solvents having unusually strong hydrogen-bonding ability the intramolecular hydrogen bonds of the helices are no longer stable and the conformatlion of the polypeptides is that of a random coil. Doty et al. (1956) found that in dichloroacetic acid [77] = 2.78 X 10-6N0.87,the exponent falling in the range expected for flexible polymers. It should be pointed out that the curve for the coils in Fig. 3 could shift upward or downward depending on the solvents used. The two curves in the figure, however, cross and the intrinsic viscosity a t the intersection remains unchanged even if the helix is transformed into the coil or vice versa (see also Table I ) . On the right side of the intersection a shift from the helix to coil form is accompanied by a decrease in the intrinsic viscosity and the reverse is true on the left side of the figure. As a result of these studies Doty and Yang ( 1956) have also demonstrated a first-order helix-coil transition of polypeptides by measuring their viscosities or optical rotations in mixed solvents



of ethylene dichloride and dichloroacetic acid. Similar reversible transition was also observed for poly-L-glutamic acid in 0.2 A4 NaC1-dioxane (2: 1) in the pH range of 5.4-6.4 (Doty et al., 1957). For moredetailsonthis subject seea comprehensive review in this volume by Drs. Urnes and Doty. 2. Serum Albumins

a. Native Conformation. Both human and bovine serum albumins have been the subjects of numerous physical and chemical studies. They are TABLE V Physical Properties of Serum A l b u m i n , Fibrinogen, and Tobacco Mosaic V i r u s Physical property Intrinsic viscosity, [Q] (dl gm-l) Partial specific volume, P(m1 gm-1) Sedimentation coefficient, ~ 2 ~ . , , , (Svedberg units) Translational diffusion coefficient, D z ~ ,(Fick , units) Rotary diffusion coefficient, €h rUI (sec-l) Molecular weight, M (gm mole-’) Molecular length, L(A) Light scattering Hydrodynamic a

Serum albumina

Fibrinogens Tobacco mosaic


0.037-0.041 0.73 4.4

0.25 0.71-0.72 7.7-7.9

0.37 0.73 188









3000-3400 3600

640,000d 65,50067,500




4 O O ,0 O m ,

Loeb and Scheraga (1956) and Champagne (1957).

a Quoted from Scheraga and Laskowski (1957).

Boedtker and Simmons (1958). Corrected from 036,w = 830,000 (O’Konski and Haltner, 1957).

known to be indistinguishable by most physical criteria. Some of the physical data are summarized in Table V. The shape of the albumins however is still a matter of controversy at the present time. These molecules were considered approximately as prolate ellipsoids ( p E 4) on the basis of the early work of Oncley et al. (1947). The choice of a prolate rather than a n oblate model was based largely on the conclusions drawn from the dielectric dispersion measurements (Oncley, 1942; Oncley et al., 1952; Dintzis, 1952), which gave two relaxat,ion times and could not, fit the data of any chosen oblate ellipsoid. Oncley’s analysis however ignored the contribution to the dielectric dispersion from polarization of the ion atmosphere about polyelectrolytes (see, for example, O’Konski, 1955; O’Konski and Haltner, 1957). There is another possible uncertainty in the dielectric measurement.



Kirkwood and Shumaker ( 1952) have recently given a new interpretation of the dielectric const,ants of protein solutions in terms of moments which arise from charge fluctuation due to proton migration among the various basic sites of the protein molecules. Such fluctuations can apparently contribute to the moments of the order observed experimentally for many proteins even in the absence of permanent, dipole moments. Whether this new theory offers a better account of the experimental results is stJillundecided. Nevertheless it casts some doubts on Oncley's conclusions. An entirely different conclusion seems to have come from the low angle X-ray scattering studies by Anderegg et al. (1955) and Luzzati (1960). At very low angles the scattering curves gave a radius of gyration which appeared to be much too great for a sphere. At somewhat higher angles the scattering curves seemed to be in better accord with an oblate model having an axial ratio of 3.5. X-ray data do not prove the structure and only suggest the compatibility of such a model with the experimental data. The discovery by Hughes (1947, 1950) of human mercaptalbumin mercury dimer has led Oncley to reconsider his earlier model in order to account for the steric relations for the dimer formation. This revised model consisted of a prism of approximately 144 X 45 X 22 A3, which fits well into the unit cell of the crystal as employed by Low (1952) in her X-ray diffraction study. As Edsall (1954) has pointed out t>heX-ray scattering function for the prismatic model might be quite different, from that of a prolate ellipsoid. Thus one still cannot rule out other mode!s which might agree equally well with the X-ray scattering result,s. Furthermore there also exists the possibility that a model which fits with one physical method may not be identical with the equivalent hydrodynamic ellipsoid. Loeb and Scheraga (1956) and Champagne ( 1957) have independently made careful studies of the hydrodynamic properties of bovine serum albumin, each from a single lot, which eliminated the possible complications that may arise from the use of different preparations. [The only value not, measured by these authors was the partial specific volume of the protein which was taken from Dayhoff et al. (1952) and corrected from 25°C to 20°C by using the general temperature coefficient for P for proteins (Svedberg and Pedersen, 1940).] Loeb and Scheraga have taken into consideration a small (3%) faster component in their sample whereas Champagne has found 8 % of dimer in her preparation. The physical data and the corresponding p-functions of the two groups are listed in Table VI for comparison. Independently Tanford and Buzzell (1956) found that [q] = 0.037 and, together with ~ 2 0 = , ~ 4.42 X from Creeth (1952), and M = 67,000, obtained a @-valueof 2.04 f 0.06 X lo6. Harrington et al. (1956) used [q] = 0.038, S ~ O =, ~4.45 X lo-", and M = 67,500 which gave a p-value



of 2.05 x lo6. All four groups reached essentially identical results which are lower than the minimal theoretical value of 2.12 X lo6for rigid spheres. Loeb and Scheraga (1956) have claimed that the standard deviation cited by them was a minimum value which would be augmented by determinate errors in any of the measured values and they therefore do not regard this low @-valueas a failure of the theory. Champagne (1957) has also assumed a spherical shape for bovine serum albumin and attributed the lower than TABLEVI Calculations of the p-Functions of Bovine Serum Albumin Physical property PH Ionic strength Partial specific volume, P(m1 gm-1) Intrinsic viscosity, [ 7 ] (dl gm-1) Sedimentation coefficient, ~ 2 0 (Svedberg . ~ units) Translational diffusion coefficient, D z o , ~ (Fick units) Molecular weight, M (gm mole-') 8-Function X 10-6

Loeb and Scheragan

5.13 0.5 0.730b 0.041 (1) 4.41" (2) 4.48 (1) 5.90d (2) 5.81 (1) 67,000 (2) 69,200 (1) 2.05 f 0.06 (2) 2.04 f 0.06


5 0.15 0.730b

(1) 0.0388 (2) 0.0607 (1) 4.42c (2) (1) 6.07 (2)3.89 (1) 66,400 (2) 161,000 (1) 2.04. (2) 2.04

a In Loeb and Scheraga's paper (1956) (1) refers to the major component and (2) the whole protein, whereas in Champagne's paper (1957) (2) refers to the minor component. Taken from Dayhoff el al. (1952). c Corrected for the adiabatic cooling effect (Waugh and Yphantis, 1952;Biancheria and Kegeles, 1954). d Taken from Wagner and Scheraga (1956). 0 Champagne has expressed her data in terms of a function v / F 3 (in our terminology), which differs from the p-function by a constant only.

theoretical value to experimental errors. But one also cannot rule out the possibility of an oblate ellipsoid as a model for the protein, since such an ellipsoid has a @-valuewhich is essentially independent of the axial ratio and is very close to that for a sphere. Following the same reasoning one may even suspect the possibility of a prolate ellipsoid of low axial ratio if the experimental errors were slightly higher than the standard deviation. Harrington el al. (1956) suspected that the anomaly was caused by approximations in the Simha equation for axial ratios less than 20. This however was refuted by Loeb and Scheraga (1956) on grounds that the viscosity increments, Y, in the p-function calculations are taken from the



exact solutions of the Simha equation rather than from its approximate form. 1,auffer and Beiidet (1954) suspected that ambiguity might also arise from the lavk of precise knowledge of the theoretical equations such as Simha’s and l’errin’s, since any small deviations could lead to very erroneous p-values. On the other hand Tanford and Buzzell (1956) were of the opinion that serum albumin cannot be represented successfully by a solid equivalent ellipsoid, a suggestion which certainly should not be taken lightly. Indeed it would be less than critical to claim that the positive deviations of the p-values were more reliable than the negative ones in this case merely because the latter gave a lower than the minimum theoretical value and were therefore not acceptable. Since the serum albumins are known to have conformational adaptability and are perhaps not perfectly rigid even at the isoelectric point it is conceivable that the equivalent ellipsoid under shearing stress may differ from that under sedimentation. Such a discrepaiwy could result in an erroneous p-value. All one can say a t present is that the serum albumins are probably nonspherical, but that they :ire not, far from spherical. 6. Conformations in Urea Solutions and at Low p H . The denaturation of serum albumins by urea provides an example of the kind of information which one can draw from viscosity measurements and also the limitations of its interpretation. The increase in intrinsic viscosity and decrease in diffusioii roeffic.ient of horse serum albumin (Neurath and Saum, 1939) with increasiiig urea concentrations clearly indicated a marked change in the protein cmformation in such solutions. It had been commonly believed that protein denaturation involved the unfolding of the peptide chains, thus resulting in a molecule having a highly elongated rod shape. In terms of increasing asymmetry in urea solutions the axial ratio of the horse serum albumin was said to increase from about five in the native protein to about twenty in the denatured protein. Such interpretation based 011 any single hydrodynamic measurement is now known to be questionable, since it does not rule out the possibility of an increase in the effective volume rather than the change in shape. Scheraga and Mandelkern (1’353) found that the p-values of the above-mentioned results could be equally well interpreted in terms of a sphere and that the denaturation process involved an isotropic expansion of the molecule. True, the application of the p-function to nonrigid particles may still be open to question, but together with other information the inferences as drawn from the hydrodynamic meaburements seem to be very plausible in this case, although they may he only of seniiquantitative nature. Indeed Doty and Katz (1950) (see also Doty and Edsall, 1951; Kay and Edsall, 1956) concluded from their light scattering studies of bovine serum albumin in concentrated urea solutions that the principal change undergone by the albumin molecules is that


3 59

of approximately isotropic swelling, although the details of their work are not yet published. The rapid reversibility of the urea denaturation of serum albumins as revealed from both optical rotation and viscosity measurements also manifest a lack of internal rigidity of the protein molecule (Kauzmann and Simpson, 1953; Frensdorff et al., 1953; Kauzmann, 1954). The concept of “configurational adaptibility” as proposed by Karush (1950) to account for the adsorptive power of serum albumins essentially leads to the same conclusion. Much attention has recently been focused on the behavior of serum albumins in acid media. Unlike most globular proteins serum albumins undergo marked structural changes at pH below 4. This is reflected in many physical measurements such as the abnormal sharpening of the titration curve (Tanford, 1952; Tanford et a,?., 1955a), increase in depolarization of fluorescence (Weber, 1952; Harrington et al., 1956), increase in intrinsic viscosities (Bjornholm et al., 1952; Yang and Foster, 1954; Tanford et al., 1955b), decrease in sedimentation coefficients (Harrington et al., 1956; Reichmann and Charlwood, 1954; Charlwood and Ens, 1957), and increase in levorotation (Yang and Foster, 1954). Bjornholm et al. suggested that the molecules aggregated end-to-end at low pH, thus implying an increasing asymmetry to account for the high viscosities, whereas Weber originally suspected a dissociation of the protein molecule into subunits to explain the depolarization behavior. Both interpretations seem now to be ruled out, since the light scattering studies (Doty and Steiner, 1949) indicate no change in molecular weight a t low pH. Both optical rotations and intrinsic viscosities of serum albumins were found to be extremely sensitive to pH and ionic strength in a manner analogous to the behavior of synthetic polyelectrolytes. Thus the overwhelming evidence seems to indicate an essentially isotropic swelling of the serum albumins in acid media. Here is just another illustration of the power of viscometry in detecting the conformational changes of proteins; but any single hydrodynamic measurement simply cannot give an unequivocal answer to the nature of such changes. (For more details on the subject of serum albumins see a recent>review by Foster, 1960.) 3. Fibrinogens

The human and bovine fibrinogens are also known to be remarkably siniilar in both size and shape. Their high intrinsic viscosity, strong birefringence, and significant angular dissymmetry of light scattering all indicate that the shape of the molecule is far from spherical. The interpretation of their physical properties has been complicated by the impure preparations employed in many iiivestigations except perhaps the most recqent ones. This subject has recently been reviewed by Scheraga and Laskowski (1957)



and the physical properties of fibrinogen in aqueous solutions are included in Table V. From electron microscope studies fibrinogen appears to resemble a string of beads and the lengths of the rodlike particles are in the range of 400 and 600 A (Hall, 1949, 1956; Siegel et aE.,1953). Mitchell’s (1952) preparation however seemed to consist primarily of spheres of 50 A in diameter and Porter and Hawn (1949) suggested a disk model for the protein. The diameter of the protein was found to be 3 0 4 0 A by Hall and 60-80 A by Siegel et al. The latter results would give a molecular volume two or three times that calculated from the specific volume of the protein. The electron microscope observations revealed considerable polydispersity for fibrinogen in the dried state, in direct contrast with the studies of solution properties which indicated that fibrinogen is fairly homogeneous. It is not yet clear whether this discrepancy could be attributed to artifacts which arose during the process of drying. The evaluation of the 8-function for fibrinogen in solution led to a very puzzling conclusion, i.e., that the axial ratio was only about five ! Edsall (1954) has considered the possible experimental errors and found that the lower and upper limits of 8 would be 2.05 X lo6and 2.28 X lo6.The former was below the minimum theoretical value for a sphere, whereas the latter corresponded to an axial ratio of six. This is in direct contrast t,o the weight of evidence available a t the present time and the explanation must lie elsewhere other than in experimental errors. As Edsall pointed out such an equivalent ellipsoid would indicate that the molecule is highly swollen by the embedded solvent and the effective volume would have been about five times that of the unhydrated molecule (assuming = V,,). A better explanation for such a large discrepancy between hydrodynamic and other physical properties is still lacking. Could it be that fibrinogen is represented by two different equivalent ellipsoids under shearing stress and under sedimentation? From the definition of the &function [Eq. (25)] one cLtn write



Y( V , l V a ) [ v ( p , ) ” 3 F ( p ~ ) l


where the subscripts refer to viscosity and sedimentation measurement s. If either Ti, < V,(p, = p . ) or p , < p8(V,,= V , ) the experimental /3 could easily give a false axial ratio much lower than either p , or p , . For example, if the volumes are kept identical, but p , = 14 and p , = 18, we have then



8i4[F(pie.)/ F ( pi41 1


= 818 4p14) l Y ( P 1 8 ) i1/3 =

2.28 X lo6

which corresponds to an axial ratio of 6. Likewise, if p is kept constant but


36 1

V , is smaller than V 8, a low &value will also result. This illustration by no means implies that fibrinogen must require two equivalent ellipsoids for viscosity and sedimentation. It is not clear why the p-function for fibrinogen behaves LLabnormally,”if the other weight of evidence is accepted as a correct description of the molecule. I t is even possible that the conclusion of Siege1 et al. (1953) may eventually turn out to be the correct one. There is need for more extensive tests for the @-function. As mentioned earlier, ideally the best answer should come from the determination of the &function. Unfortunately at present the rotary diffusion coefficient is usually the least reliable quant,ity in all hydrodynamic measurements because of errors inherent in the physical methods of flow birefringence and perhaps also non-Newtonian viscosity (see Section IV) . (Electric birefringence also may not give the same rotary diffusion coefficient as the other two methods, since the equivalent ellipsoids can be different under shearing stress and under electrical field.) Edsall (1954) has also illustrated the impossibility of evaluating the axial ratio from the 6function. The latter was about 0.80 for fibrinogen which corresponded to a prolate ellipsoid with an axial ratio of more than 300. If the rotary diffusion coefficient were only about 15% greater than that listed in Table V the calculated axial ratio would decrease to between ten and twenty. 4. Tobacco Mosaic Virus

The size and shape of tobacco mosaic virus has been the subject of numerous investigations both in solid state and in solution. From X-ray studies (Bernal and Fankuchen, 1941; Franklin and Klug, 1956; Caspar, 1956; Crick and Kendrew, 1957) and electron microscopy (Wilkins et al., 1950; Williams and Steere, 1951; Williams, 1954; Steere, 1957; Hall, 1958), it seems well established that the rodlike virus has a length of about 3000 A and a diameter close to 150 A. There is disagreement on the solution properties of the protein however, perhaps because of the difficulty in obt8ainingnearly monodisperse preparations. For example, the values of intrinsic viscosity from different laboratories were reported to vary from 0.25 to more than 0.6. In order to make any meaningful interpretations of the physical properties it is desirable to use the same preparation for all the measurements. Recently, Boedtker and Simmons ( 1958) have carefully reinvestigated this protein with a combination of light scattering, flow birefringence, viscosity, and sedimentation methods. Their results are summarized in Table V. The same preparation was also studied by Hall (1958) whose electron microscope measurements indicated that the number-average length was 3020 A and that more than 85 % of the 201 particles counted had lengths between 2800 and 3200 A. Thus the preparation was regarded as essentially uniform.



The evaluation of the equivalent ellipsoid after Scheraga and Mandel-

kerii ( 1 9 S ) led to @ = 2.61 x lo6 and 6 = 1.06. The former corresponded to L L ~ I:mid ratio of 18.5. If the experimental errors were considered to be

f 2 % , the axial ratios varied from 16 to 21, which fell into the rmge observed from the X-ray and electron micrograph studies. The &value gave an aria1 ratio of only 8. By assuming a f15 % error for the &value (about three times that for the calculated length) the axial ratio would vary from 5 to 20. On the other hand the length of the equivalent ellipsoid can be estimated with less ambiguity. As has been shown in Table 111 it varied from :3340 to 3550 A from flow birefringence to 3450 to 3710 A from intrinsic viscosity at zero gradient if the axial ratio was assumed to be 15 and 25. If one now turns to the conventional treatment by assigning an arbitrary water of hydration of 0,0.2, and 0.4 gm per grams of protein the corresponding lciigths of the hydrated protein are found to be 3680,3590, and 3540.4, respectively. These results compare very favorably with those mentioiied earlier, despite the fact that the use of hydration involves many debatable sssumpt ions. As Boedtker and Simmons (1958) have pointed out the equivalent length as obtaiiicd from the hydrodynamic measurements, about 3600 A, was appreciably larger than that from light scattering and electron microscope studies. The reliability of the Perrin equation to represent accurately a cylindrical particle has thus been questioned. Haltner and Zimm (1959) have measured the rotary frictional coefficient (kT/0) of carefully machined brass rods and prolate ellipsoids. The axial ratio of these models was so chosen that it approximated the actual ratio found in tobacco mosaic virus ( p = 20). According to these authors the reciprocal of the rotary diffusion coefficient for the rod (square ends) was 40% greater than that cdculated from the Burgers’ approximation for rods (1938) and the ratio of the coefficients between the ellipsoid and the rod was &/Or = 1.56. interesting discussion on the Burgers’ equation has recently been offered hy Broersma (1960) in his new theory on the same subject.] The latter finding seemed also to indicate the inadequacy of I’errin’s equation for rodlike particles. Yang (1961a) however has pointed out that the two models used in these experiments difkrent in volume, although they had the samc length and axial ratio. If one considers a hypothetical equivalent ellipsoid having the same volume and axial ratio as the rod, the ratio of the coefficients between the previous smaller ellipsoid and this equivalent ellipsoid is 1.50, which indeed agrees very well with Haltner and Zimm’s experimental value of 1.56. Thus the l’errin equation appears to be equally applicable for these “fat” rods, although the equivalent major and minor axes in this case were 14.5% greater than those of the rods. This fictitious equivalent ellipsoid is certainly by no means unique. Nevertheless it a t



least gives us a self-consistent approximate relationship between the rod and its equivalent ellipsoid. If the same argument can be applied to tobacco mosaic virus the length of the rodlike particle as deduced from the hydrodynamic measurements would have been 3600/1.145 or nkout :
A . Theories for Rigid Particles The theory of non-Newtonian viscosity for ellipsoidal particles was first explicitly stated by Kuhn and Kuhn (1945), using Peterlin’s distribution function (Peterlin, 1938) and Jeffery’s hydrodynamic treatment (.Jeffery, 1922-1923) [Eq. (lo)]. More elegant treatments have recently been developed by Saito (1951), using the same ellipsoidal model, and also by Kirkwood and his co-workers (Kirkwood, 1949; Kirkwood and Auer, 1951 ; Kirkwood and Plock, 1956; Riseman and Kirkwood, 1956) for rodlikc particles. The equivalence of the three theories has also been demonstrated by Saito and Sugita (1952). The general solution of Eq. (10) for the viscosity increment, v, can be expressed in the form v=a-bba2+ca4...


with the axial ratio, p , as a parameter. Here LY = G / @ and a, b, c, etc. are constants. The physical picture is also very simple. With increasing shearing stress the particles tend to orient more toward the streamlines and cause less energy dissipation than a t zern shearing stress; consequently, the viscosity drops. Precise numerical solutions of Saito’s equations ( 1951) have been computed by Scheraga (1955) in the manner employed previously for flow birefringence (Scheraga et al., 1951), using the Mark I computer. The values of v are tabulated as a function of p and LY (Table XII). The v values a t



a = 0 are identical with those listed in Table VIII according to the calou-

lations of Mehl et al. (1940). The ratios of v,/v,,o can also be converted to the experimental quantities, [q],/[q] ,=o , by the simple relationship

(37) (Table X I I I ) . Because of the limitations of the internal storage ettpacity of theMark I calculator these theoretical values are limited to a = 60. This range should be more than adequate for most of the experimental measurements. Empirically a plot of v or [ q ] , / [ q ] o E o versus a a t constant p on a double logarithmic scale was found to yield a straight line at a above five for both prolate and oblate ellipsoidal models (Yang, 1959). Thus the upper limit of a in the tables can be further extended by another order of magnitude or two through t)his simple extrapolation. Good agreement between the experimental results and the extrapolated theoretical values was recently reported. This procedure is arbitrary and without theoretical j ustification and therefore should be used with reservations.

B. Determination of Particle Length (or Diameter) The gradient dependence of viscosities of rigid particles leads to a new method for determining the particle shape. Like the flow birefringence technique (for recent reviews, see Cerf and Scheraga, 1952; Scheraga and Signer, 1960) it gives a direct measure of the rotary diffusion coefficient or the relaxation t>imeof the ellipsoids of revolution. For highly asymmetric particles of, say, p > 10 the ratio of [ q ] , / [ ~ ] ~ = oas a function of a( = G / e ) is very insensitive to the chosen axial ratio, p . Thus the calculations of particle length or diameter are essentially identical with those employed in the flow birefringence measurements. First, one has to choose either a prolate or an oblate model from other available information and then make a rough estimate of the axial ratio, for example, from the intrinsic viscosity a t zero gradient using the conventional treatment (with 8 ) .Next, from the experimental [q]o/[q]a=o or [q]r/[q]r=o values the corresponding a-values can be read from the Table XI11 in the Appendix, noting that a = G / O = (T/T)/(qoe/T)


(For detailed tabulations, see the original reference.) For axial ratios other than those listed the a-values can be obtained through interpolation or extrapolatlion, although this step is usually unnecessary for very large axial ratios. Thirdly, with a known for each chosen rate of shear or shearing stress, the quantity qo8/T can immediately be calculated according to Eq. (38). Finally, the particle length or diameter can be determined from Perrin's equations [Eqs. (19)]. If the calculated lengths or diameters vary with the gradient it is a clear indication of the degree of polydispersity. The



dimensions as determined from the non-Newtonian viscosities again represent those of the equivalent hydrodynamic ellipsoids. The arguments raised in Section I11 can equally well be applied here. According to theory a rigid sphere does not have non-Newtonian viscosity and for axial ratios very close to unity the decrease in the intrinsic viscosities with increasing shearing stress is also very small. If in addition the dimensions of the particles are small (i.e., large q o B / T ) it is necessary to employ very high shearing stress so that a becomes large enough to be measurable. Thus for practical purpose this method is most useful only for highly asymmetric particles having very elongated or flattened dimensions which usually exhibit significant gradient dependence of their viscosities. To illustrate this point for protein solutions, consider the commonly used Ubbelohde or Ostwald type viscometers having a shearing stress of about 10 dynes cm-2 (or a rate of shear of about 1,000sec-’ for aqueous solutions). No significant non-Newtonian viscosity will be detected within experimental errors (a 1) at, say, 25°C for particles having a length (for prolate) less than 1900 to 2900 A or a diameter (for oblate) of about 1500 A or less (for axial ratios from 5 to 300). If the applied shearing stress is increased by tenfold the corresponding limits would become 9001300 A and about 700 A, respectively. Thus for most solutions of globular proteins the gradient dependence of viscosity would hardly be detected in ordinary capillary viscometers. On the other hand for highly elongated particles such as tobacco mosaic virus or collagen the gradient dependence could be quite appreciable when the shearing stress is in the order of 10 dynes cm-2.

C . Experimental Conjirmation The theory of Saito has been confirmed experimentally by measuring the gradient dependence of the viscosities of poly-7-benzyl-L-glutamates (in a-helical form) (Yang, 1958a, 1959). Following the rheologists’ practice, one can present the experimental data in a series of flow curves as shown in Fig. 4, where the rate of shear is plot,ted against the shearing stress on a double logarithmic scale. The viscosities appear to approach Newtonian (where the slope is unity) a t low shearing stress, followed by a marked drop above a critical shearing stress, 7 , , and approaching another Newtonian region a t very high shearing stress. The S-shape curve shows a more striking curvature with increasing concentration. From these flow curves the intrinsic viscosities at either constant shearing stress or constant rate of shear can be calculated. Yang has chosen the former on the basis of Eq. (38) which in turn is related to Eq. (37) hy




[email protected]/T = constant

( 378 1



for any particular polymer. Thus by plotting [&/[&o against r / T the polymer under study will yield a composite curve independent of the solvent viscosity and/or temperature employed, provided that no conformational change occurs under these conditions. In this respect it is noted that measurements in an Ostwald or Ubbelohde viscometer are made under virtually constant shearing stress (see Section V ) . The corresponding rates of shear are no longer constant under these circumstances and vary inversely with




1 0

3 4 5 log f . FIG.4. Typical flow curves of an a-helical polypeptide in m-cresol at 25°C. The experimental points at T > lo4dynes cm-2 were omitted in the plot. Concentrations: 0 , 0.379%; A, 0.502%; X, 0.621%; A , 0.776v0; 0 , 0.918%. Reproduced from Yang (1959).



the concentrations of the solutions, although by tradition most>workers still prefer to calculate the rate of shear for each solution and express the viscosities as a function of the latter. The agreement between the theory and experiments can test be illustrated in Fig. 5 , where the solid and broken lines were calculated on the basis of the known physical properties of the polymers. The obvious feature in the figure is the sharp drop in [q] wit,h increasing shearing stress which is in striking contrast with thc rather mild non-Newtonian behavior of the same polymer in its randomly coiled form (see below). This marked difference between rods and coils clearly provides a new means for the study of



conformational changes in proteins, for example, protein denaturation. Also from the figure the non-Newtonian viscosity becomes significant a t a much higher shearing stress for the shorter particles than the more elongated ones, mainly because the former have a larger 7108/T [Eq. ( 3 8 ) ] .Furthermore, since each polymer has its characteristic viscosity curve the composite curve of many components in a polydisperse system would conceivably result in a broadening of the non-Newtonian region, as is indeed the case in Fig. 5 . Since the gradient dependence of the intrinsic viscosity of rigid particles gives a direct measure of the rotary diffusion coefficient it is possible in 1.0










4 0


z 7 0.4 Y


0.2 0



2 Log



FIG.5. Shearing stress dependence of the intrinsic viscosities of three a-helical polypeptides. The lines are theoretical curves (broken lines t o the right of the arrows being extrapolated theoretical curves): 1, [email protected]/T = 0.76; 3, q&/T = 0.054; 2, calculated on the basis of three parts curve 1 and one part curve 3. Reproduced from Yang (1959).

principle to calculate the Scheraga-Mandelkern &function [Eq. (26)] from viscosity measurements alone, provided that the molecular weight is known from other physical methods. Here again the same uncertainty as discussed in Section I11 applies particularly when the particles are heterogeneous. Cerf (195813) has suggested that since the shape of the theoretical curve for [v],/[o],,o as a function of a is dependent on the model and also to some extent on the axial ratios, one can in principle determine both simply by superimposing the experimental [ ~ ] a / [ v ] o = versus o log G curve on the theoretical [~]u/[q],Eoversus log a curve for prolate or oblate ellipsoids. From the logarithmic scale one then calculates the rotary diffusion coefficient, since log a = log G - log 8.With p and 8 known the Perrin equation gives the value of the major and minor axes and thereby the volume. Furthermore with p (and thereby the viscosity increment, v) and V , deter-



mined, Eq. ( 2 2 ) in turn gives the molecular weight,. Thus a single intrinsic viscosity curve can provide data on both the size and shape of the particle. Cerf’s suggestion has not been tested but some formidable difficulties in practice can be foreseen. First the particles must be monodisperse and the experimental data of the highest precision. From the theoretical tables it is clear that the effect of axial rat,io on the gradient dependence of the viscosity becomes significant only when p is rather small, say, less than 10. Yet precisely in this range the intrinsic viscosity is usually small and its non-Newtonian behavior is difficult to detect. On tjhe other hand it is almost impossible to distinguish the theoretical curve for one axial ratio from t,he other if they are fairly large. To further complicate the problem any degree of polydispersity will distort the experimental curve so that no comparison can be attempted. Nevertheless it will be of interest to test this method if the particles are monodisperse and have low axial ratio arid also small rotary diffusion coefficient. The last requirement makes the measurements possible without resort to the use of high shearing stress.

D . Comparison with Flow Birefringence Method With the development of the non-Newtonian viscosit,y theories it is now possible to compare the rotary diffusion coefficient and thereby the calculated length (or diameter) of the rigid particles as obtained from this technique with that from the commonly used flow birefringence method. Since both measurements depend upon the same molecular distribution funct,iori (Section 111) they should give an identical measure of the rotary diffusion coefficient. Differences, however, will arise if the system under study is heterogeneous. The mean intrinsic viscosity is calculated from Eq. (7) whereas the mean extinction angle, x , for flow birefringence is defined by the Sadron equation (1938) : tan 2x


Z A n i sin 2 x i / C A n ; cos 2xi


where the subscript i refers to the ith component which, if present alone in the solut,ion, would give an extinction angle, x i , and a birefringence, Ani. Consequently the mean length as calculated from both methods will not only vary with the applied shearing stress but also depend on the equation used for such calculations. This can best be illustrated by the recent results on tobacoo mosaic virus (Yang, 1961a) (Fig. 6 ) . The flow birefringence average length decreases, as expected, montonically with increasing shearing stress, since the orientation of the particles is heavily weighted by the longer ones a t lower shearing stress. The viscosity-average length however reveals a maximum for the following reason. At zero gradient the mean length can be calculated from Eq. (27b), which is usually closer to a weight average. As soon as the shearing stdress increases, however, the



longer particles are first oriented, resulting in a drop in viscosity and thereby a decrease in the mean rotary diffusion coefficient which in turn causes an increase in the calculated mean length. At still higher shearing stress even the shorter particles begin to be oriented and as a consequence the mean rotary diffusion coefficient gradually increases and the mean length decreases again. Once all the particles are oriented parallel to the streamlines (at infinite shearing stress), there will be no further drop in viscosity and the mean length probably approaches the same average as that a t zero gradient. On the other hand the flow birefringence averages are quite complicated and still not well known. Goldstein and Reichmann (1954) have shown that as shearing stress approaches zero the flow birefringence average 3 1/3 length, ( L ) = ( ( L 6 ) / ( L)) which is even more heavily weighted than the z-average. The same authors have also suggested a number average when





[email protected] *J





10 ao 30 40 so SHEARING STRESS, f dyner cm-'


FIG.6. The hydrodynamic lengths of a tobacco mosaic virus sample. Reproduced from Yang (1961a).

the shearing stress becomes infinite. This perhaps explains the crossover of the two curves in Fig. 6. It is clear that the flow birefringence technique is extremely sensitive to the degree of polydispersity, much more so than the viscosity method. At very high shearing stress both curves in Fig. 6 appear to level off gradually. This merely reconfirms the fact that any apparently constant length as calculated over a narrow range of shear can be quit,e misleading. On the other hand for flow birefringence the extrapolated length (to zero gradient,) may approach the upper limit of the longest particles but definitely does not represent the mean length a t zero gradient>.

E. Theories for Flexible Coils The gradient dependence of viscosity of flexible coils is a much more complicated problem than that of rigid particles. I n addition to orientation under shearing stress the permeable and deformable nature of these molecules leads to such theoretical and conceptual difficulties that no adequate theory has as yet been found to be of general applicability. Kirkwood and



his co-workers (Kirkwood and Riseman, 1948; Kirkwood, 1949, 1954; Riseman and Kirkwood, 1956) first developed a rigorous theory using a general statistical model with hydrodynamic interaction. Since the gradient dependence of molecular conformation is not known explicitly for random coils these authors have considered only the undeformed equilibrium conformation, thus leading to Newtonian viscosity only. Rouse (1953) and Zimm (1956) employed two somewhat different models, elastically connected beads without and with hydrodynamic interaction, respectively, but also found no gradient dependence a t all. This conclusion is not in accord with existing experimental evidence. It, is noted that the three theories are derived by the use of a perfectly flexible Gaussian coil model, which does not fit any real polymer chain. Earlier this idea of imperfect flexibility led Kuhn and Kuhn (1’346) to introduce the concept of “internal viscosity,” which has been further extended by Cerf in a series of papers (1951, 1955, 1957, 1958a, 1959). In either case the net effect is to stiffen the molecular chain and both treatments Iead to a gradient dependence. On the other hand, by considering the effect of anisotropy of the polymer conformation on hydrodynamic interaction rather than internal viscosity Peterlin and CopiE (1956) and Ikeda (1957) reach essentially the same conclusion as Kuhn and Kuhn and Cerf. The results of these four theories can all be expressed in a series expansion of the intrinsic viscosity in powers of the rate of shear: [ ~ ] ~ / [ & = o = 1 - constant




alt,hough the numerical constants are not identical with one another, At present there is no unanimity of opinion among those who hold various theories concerning the origin of the gradient dependence of viscosity. It may also be mentioned that Bueche ( 1954), using the same model as Rouse and Zimm, developed a theory which predicts a drastic drop in intrinsic viscosity with increasing shearing stress in the same manner as that for rigid particles. This surprising contradiction between Bueche’s and Zimm’s theories has been discussed by Peterlin and CopiE (1956) who suspect that Bueche’s conclusion probably arises from an incorrect treatment of hydrodynamic interaction. Very recently, Zimm (1960) has given a more exact treatment of the hydrodynamic interaction than in his earlier work. This refined theory also leads to a gradient dependence. Experimental evidences of the non-Newtonian viscosities of polymer chains are too numerous to list here. Most experimental and theoretical results can be described by a general equation: [qIG =

[qjc=o(l - constant G ” )

(41 )

It has the same inverse8 shape as that found for rigid particles (Fig. 5 )



although the extent of drop in viscosity is usually much milder than in the latter case. Many factors influence the constant in Eq. (41) and all experiments are consistent in demonstrating that the effect increases sharply with molecular weight [see Eq. (40)]. Of equal importance is the effect of solvent, the gradient dependence of viscosities of any polymer being much more pronounced in good solvents than in poor ones (Sharman et al., 1953). It does not disappear, however, even in an “ideal” solvent, i.e., at Flory’s 0-temperature (Flory, 1953), although the effect is much reduced under these conditions (Passaglia et al., 1960). At 0-temperature the root-meansquare end-to-end distance, (ri)1’2,of any real polymer chain is still consistently greater than that calculated on the assumption of completely free rotation about each chemical bond, (T~,,,)~”, and thus the molecular conformation is not quite equivalent to a perfect Gaussian coil. On the other hand, for polymers of different chemical compositions having different degrees of flexibility (or stiffness) the gradient dependence of viscosities is reported to be much more pronounced, the more extended the chain conformations (Passaglia et al., 1960). In the extreme case where the polymer chains are completely stiff and fully extended they resemble rigid particles, for which both theory and experiments are in complete agreement. These observations are interrelated and a general pattern emerges, that is, the non-Kewtonian viscosity of polymers is predominantly determined by their molecular conformations. If the polymer chains are coiled up through the use of a poor solvent they invariably exhibit little gradient dependence of viscosity. Conversely, for very stiff chains or for those which are highly extended because of strong solute-solvent interaction in a good solvent, a pronounced gradient dependence of viscosity usually results. Empirically this dependence has been found to disappear only when experimental data in various solvents are extrapolated to the ideal conditions under which the polymer chains have the conformation dictated by the hypothetical completely free rotations about each chemical bond (Passaglia et al., 1960). Viscosity theories for flexible coils are not directly applicable to most proteins. Qualitatively, the fact that flexible polymers have a less marked non-Newtonian viscosity than the rigid particles should be of interest in the studies of protein denaturation. This is further illustrated in Fig. 7, representing the viscosities of a poly-y-benzyl-L-glutamate in both the a-helical and randomly coiled forms (Yang, 1958a). The striking drop in viscosity for the rods is a direct contrast to the small gradient dependence for the coils. In this particular case the curves for the two forms at comparable concentrations cross each other. One is therefore led to believe that the a-helices remain stable even when subjected to very high shearing stress. Strong evidence against any conformational change of the helices under shearing stress comes also from the close agreement between theory



and experiments for these helices over the entire range of shearing shress studied (see Fig. 5 ) . From the findings in Fig. 7 one may also deduce that most denatured proteins, if present as very flexible coils, will exhibit little or no non-Newtonian viscosity under normal experimental conditions.

F . Power Law of Viscosity The value of the exponent, n in Eq. (41) has been the source of considerable controversy among the rheologists: t,he so-called power law of vis-




I-?\\ !-

\ ,.~m




2 Dichloroocrtic







Lop 7

FIG.7. Corn srison of the non-Newtonian viscosities of a polypeptide in helical (in m-cresol) and randomly coiled (in dichloroacetic acid) forms. Reproduced from Yang (195th).

cosity. By simple geometric analysis viscosity should be an even-function of shearing stress or rate of shear independent of the direction of flow; that is, only when the exponent n is an even number can the numerical value of viscosity remain unchanged by reversing the sign of T or G from positive to negative. Indeed most theoretical treatments predict a value of 2 for n. A notable exception is Bueche’s theory (1954) which reaches values of 2 and $5 for free-draining and nonfree-draining coils respectively. It is noted however that the agreement with experiment which Bueche demonstrates



could equally well be achieved with other functions and the curves he presents are by no means unambiguous on this point. Experimentally most published data for polymer coils suggest an odd-function, i.e., n = I, as T or G approaches zero (for a brief review, see Hermans, 1957). To obtain the value of this exponent involves extrapolation to zero gradient and this is often difficult. For rigid particles both theories and experiments are in accord with an even function (see Section IV, C) . If the macromolecular system is polydisperse i t is possible to obtain a pseudo-odd function even though each component obeys an even-power equation (Yang, 1959). Very recently Eisenberg (1957) has shown that even for monodisperse rigid particles a linear approximation can be obtained over certain limited ranges of rate of shear. By examining a theoretical curve for rigid particles having a particular rotary diffusion coefficient he has illustrated that the correct limiting quadratic law in this special case was [v]a/[q]o=o =

1 - 3.4 X 10PG2

for G < 140 sec-'. If a term in G' was added to the above equation the range of linear extrapolation could be extended to G < 280 sec-'. Unless [qlo=O was known from measurements a t G < 30 sec-' in this particular case several incorrect empirical equations could be found to fit the theoretical curve within different ranges of rate of shear. For example, [q]a/[q]o=o =

between G


1.065 - 9.1 X 10-4G

100-400 sec-I, and [~Ia/[q]u=o = 1/(0.2

+ 5.97 X 10-"G''2)

over a wide range of rate of shear provided that G > 300 sec-'. Although Eisenberg only discussed the intrinsic viscosity of rigid particles the same argument is expected to apply to flexible coils as well. Thus the power-law controversy appears to have been overemphasized. At least for polymers which are not well fractionated the effect of polydispersity can complicate the interpretation of the power law.

G. Complex Viscosity The viscosity of a macromolecular solution can undergo changes when subjected to a periodic shear wave of frequency, w , instead of a steadystate shearing stress. The response of the particles to such a sinusoidally oscillating shear can be expressed in terms of a complex viscosity, q*: ?I* = 7]R




where the subscripts R and I refer to the real and imaginary parts of a



complex number and i = 4-1.The real part is the viscosity and the imaginary part is the complex modulus of elasticity of the solution. I n steady-state flow where w = 0, the q1 term, together with the components of the complex modulus of elasticity, vanishes and the real part in Eq. (42) simply becomes the steady-state viscosity. By plotting 7~ as a function of the frequencies, w , on a logarithmic scale the values of 7~ fall from a low a-plateau to another plateau at very high frequencies, in the same manner as the non-Newtonian behavior of the solution varies as a function of the shearing stress. The physical picture is as follows: a t low frequency the particles are oriented (and also extended for flexible coils) by the applied shearing wave, but a t the same time they rotate in the flow gradient. The input energy is thus gradually dissipated and a true viscosity appears. At high frequencies, however, the particles are oriented (and extended for flexible coils) only slightly in one phase of the motion which immediately reverses itself. Thus the stored energy in the particles does not have time to dissipate but returns to the fluid. Accordingly there is very little loss in energy and the viscosit*ybecomes small. Since the complex viscosity is concentration-dependent just in the same manner as the steady-state viscosity, one can define a complex viscosity increment, v*, and a complex intrinsic viscosity, [7]*,similar to that for steady-state viscosity v* =


- ivI

( 42%)


( 4% )



= [7lR


For rigid particles the frequency dependence of the real part in Eq. (42a), V R , has been solved by Cerf (1952) and can be written as

For ellipsoids of revolution the numerical values of v A and V B have been tabulated by Scheraga (1955), and the sum of v A and V B (i.e., V R a t w = 0 ) is identical with the viscosity increment from Simha’s equation. Thus Eq. (43) provides an alternative method to that of the non-Newtonian viscosity for the determination of the rotary diffusion coefficient, 0. Cerf has also pointed out that 8 is determinable from the slope a t the inflection point (1.P.) of the vR versus w-curve, i.e., w(1.P.) = 2 d 3 0 . At present, however, no experimental test of Eq. (43) has as yet been reported. Several theoretical treatments of complex viscosities of flexible polymers have recently been developed among which Rouse’s theory (1953) has received wide attention. The general agreement between the theory and ex-



periments (Rouse and Sittel, 1953) is very good, although there appears to exist a systematic difference amounting to about 50% between the calculated relaxation time and the experimental one. This discrepancy is small in view of the wide range of molecular weights employed for various polymers. This subject is of great interest to the rheologists but as yet it has not been applied to most proteins.

V. EXPERIMENTAL METHODS Viscometry has become such a routine technique in many laboratories that it seems unnecessary to mention the usual precautions such as the clarification of sample, the cleanliness of the viscometer, the temperature control (to 0.01"C or better), the timing device, the vertical alignment of capillary viscometer, etc. In this section we will only describe the basic equations used in the viscosity measurements and various corrections and precautions which are necessary to insure reliable results. (F'or a monograph, see, for example, Barr, 1931.)

A. Types of Viscometers Viscometers include the capillary, coaxial cylindrical, cone-and-plate, falling-ball type, etc. The capillary and, to a lesser extent but of increasing importance, the coaxial cylindrical viscometers, in numerous modifications, are the two most commonly used in scientific laboratories. 1. Capillary Viscometers

The flow in a capillary is inhomogeneous in the sense that the shearing stress, r , and the rate of shear, G, vary with the position of the fluid inside the capillary. The velocity of the flow is maximum along the central axis but gradually drops to zero at the wall, whereas the reverse is true for the shearing stress and rate of shear. For a Newtonian flow the viscosity, q( = r / G ) , remains constant at any point inside the capillary even though both r and G vary considerably from one point to another. On the other hand, for a non-Newtonian flow the viscosity varies along the radial distance of the axis. a. With Essentially Constant Pressure Head. For a Newtonian flow the viscosity can be calculated by the following equations: T,,,






4Q/?rR3= 4V/lrR3t




q =

Here T~ is the maximum shearing stress at the wall, G the nominal rate of shear, AP the applied pressure, Q( = V / t ) the volume flow rate, and R



the radius and L the length of the capillary. Equation (44c) is the wellknown Hagen-Poiseuille equation. [For the derivation of Eq. (44c), consult any st,andard textbook on hydrodynamics.] In an Ostwald or Ubbelohde viscometer where the hydrostatic pressure can be represented by A h . p . g , Eq. (44c) becomes 7 =


( 44d )

Here Ah is the pressure head, p the density of the solution, and g the acceleration of gravity. Strictly speaking, Ah in this type of viscometer varies during the measurement. It is not correct merely to take the average Ah at the beginning and ending of the measurement since the flow is faster a t first than near the end. The average Ah can however be calculated by the Meissner equation Ah


(Ah1 - Ahz)/log, ( A h i / A b )


where Ah1 and Ah2 are the hydrostatic heads at the beginning and ending of the measurement, (see Barr, 1935). The Hagen-Poiseuille equation is no longer directly applicable if a flow is non-Newtonian. It can be easily shown that instead of Eq. (44b) t,he volume flow rate in this case becomes (Hermans, 1959)

By st,raightforward differentiation one obtains 77, =

aR37kd7rn/d(Q T ~ )

and the apparent, experimental rate of shear, G, (44b) should then be corrected by the relation G(corr.)



, as calculated

+ 3)/4


from Eq.


where n is defined as the slope of the flow curve, i.e., dlog G,/dlog rrna t any chosen T~ or G, . Alternately, the apparent viscosity, 7, , as calculated from Eq. (44c) can be corrected by the relation vT,




1 - @logsa/dlog7rn)


(see also, Krieger and Maron, 1952). Several other modified forms of Eq. Equation (47) has been derived previously by the rheologists from the so-called power law G = k7. which assumes a constant n a t any chosen applied pressure ( k being a constant). This derivation is not exact since n varies with the position of the fluid inside the capillary. It is unity a t the center of the capillary where the flow is Newtonian and becomes greater than one toward the wall when the flow is non-Newtonian. Equation (47) can now he derived without this unrealistic assumption,



(48) have been reported in the literature which are essentially identical to one another. Since we are more interested in the correction to intrinsic viscosity due to inhomogeneous flow in a capillary it can be further shown that [ql,,,, (con-.



+ td[qlO/dloge.r, [91a(1 + tdlog[qla/dlogr,)




where [q], is the uncorrected, apparent intrinsic viscosity a t shearing stress, r m , as customarily calculated from the flow times of the solutions and solvent. For Newtonian flow where n = 1 no correction for the nominal rate of shear [Eq. (44b)l is necessary. This is also obvious from Eqs. (48) and (49) where q, and [q], are independent of the applied shearing stress when the flow is Newtonian and therefore their derivatives become zero. On the other hand the n values could be as high as 3 or 4 for concentrated solution in the non-Newtonian region (see Fig. 4). For most viscosity measurements where the solution viscosity may be only several times that of the solvent this correction to n appears to be insignificant. Nevertheless such precautions should by no means be overlooked since any small errors in the absolute viscosity could be enlarged several times in the specific viscosity. Indeed according to Eq. (49) the errors are by no means negligible even if all the viscosities are measured in very dilute solutions. Precise graphical determination of n, dlogq./dlogr, , d[&/dlog,r, , or dlog[q],/ dlogr, is rather difficult. An improved procedure which can be adapted to Eqs. (48) and (49) has recently been suggested by Maron and Belner (1955), which however still requires some guesswork. For very precise calculations a computer program can also be set up to resolve this difficulty. In the past the Kroepelin equations (1929) for the mean rate of shear




and the mean shearing stress 5 = AP*R/3L

( 50b 1

were widely used. These equations are derived oiily for a Newtonian flow and there seems no justification to apply them to a non-Newtonian flow (see also, Goldberg and FUOSS, 1954). If one is interested only in the extrapolation of the viscosities to zero gradient it is immaterial whether Eqs. (44a and b ) or the Kroepelin equations are employed. On the other hand with [q]a/[q]a-o as a function of a the calculated rotary diffusion coefficient, 8 , will differ by a factor of y$ from the two sets of equations. However Eqs. (44a and b ) and (47) are preferred to the Kroepelin equations, since the former are derived without any assumptions.



b. With Continuously Varying Pressure Head. Instead of multibulbs, a wide precision-bore tubing can be attached to the capillary of an Ubbelohde viscometer which constitutes a suspension type with a continuously varying pressure head when filled with the solution or solvent to be measured. Thus the volume flow rate in this case becomes Q = dV/dt instead of V / t . Again by straightforward differentiation the viscosities can be calculated from the following equation (Hermans and Hermans, 1958) : 1/vrm = - ( 2LS/[email protected])[4dlogeAh/dt


( dt/dlogeAh)d210geAh/dt2] (51)

where Ah is the pressure head a t time t, S is the cross-section area of the wide precision-bore t,ubing, and the other symbols have been defined previously. For a Newtonian liquid log,Ah is proportional to the flow time t and the second term in the bracket drops out. Thus a plot of log,Ah versus t yields a straight line and the viscosity can be calculated from the slope. For a non-Newtonian liquid the -dlog, Ah/dt decreases with decreasing Ah, since the viscosity is higher at lower pressure head. Thus the log,Ah versus t plot will show an upward curvature as the time t increases. I n most cases in particular with very dilute solutions, the second term in the bracket of Eq. (51) can usually be omitted without the introduction of significant errors. For the relative viscosity a t any chosen pressure head one can simply write as a first approximation vrel




Precise graphic determination of the differential dlog,Ah/dt is again rather difficult when the flow is non-Newtonian. Equation (51) can equally well be applied to a U-tube type viscometer, in which a capillary is connected to essentially a manometer (see, for example, Maron and Relner, 1955). The only modification is that the pressure head, Ah, is calculated from the difference in the decreasing meniscus of the capillary arm and that arising in the manometer 5rm. 2. Coaxial Cylindrical (Rotational) Viscometer I n a rotational viscometer the solution is filled in the annulus between two concentric cylinders of which either the external ( Couette-Hatschek type) or the internal (Searle type) cylinder rotates and the other, which is connected to a torsion-measuring device, is kept in position. Let Ri and Robe the radii of the inner and outer cylinders, h the height of the cylinder which is immersed in the solution or its equivalent height, if end effects are present, w the angular velocity of the rotating cylinder, and T the torque (or moment of force) required to keep the velocity constant against the viscous resistance of the solution. It can be shown that the shearing stress (see, for example, Iteiner, 1960): r m = T/2*R2h (53a)



the rate of shear4:

and the viscosity: 9 =

T(R%- R:>/4xhRTRfw


Here R refers to Ri or R, if the inner or outer cylinder rot,ates. Just as in a capillary flow the rate of shear in Eq. (53b) for a non-Newtonian flow can be further corrected by an equation (see, Reiner, 1960) G(corr.)


G,.n(l - a ) / ( l - a”)


where n is again equal to dlogG/dlog.r and a = (Ri/R,)‘. The rotational viscometer has an obvious advantage over the capillary one, due to the fact that the rate of shear across the annular gap is nearly constant and approaches constancy as the ratio of the gap to the radius of the cylinder becomes smaller. At the bottom of the inner cylinder the rate of shear is quite different from that across the gap, thus introducing some uncertainty into the measurements. This end-effect problem can be minimized or eliminated. An excellent discussion has been given by Mooney and Ewart (1934). The range of usage and accuracy of this type of viscometer is limited only by the response of the torsion-measuring device. I t can be so designed as to cover extremely low range of shearing stress. Unlike the capillary viscometer the construction of a rotational viscometer of high precision is a delicate problem.

B. Correctionsfor a Capillary Viscometer 1. Kinetic Energy

The derivation of Eq. (44c) assumes no acceleration along the axis of the capillary. This is not true at both ends of the capillary where the rate of flow is much faster inside the tube than outside. Thus the pressure drop, AP, is not caused entirely by the viscous resistance of the fluid and a portion of it is attributed to the velocity of the flow according to the well4The velocity of the rotating cylinder, u, equals the radius, r, times w . Thus, du/dr = rdw/dr O . The term on the left aide of the equation is the velocity gradient and the first term on the right is the rate of shear. Strictly speaking, the two terms in this case differ by a quantity, w . 6 Equation (54) is derived from the so-called power law of viscosity which, strictly speaking, is not exact (see footnote 3 on p. 376). According t o Mooney (1931) a general solution for a non-Newtonian flow in a rotational viscometer is not possible. Recently Krieger and Maron (1952) and Krieger and Elrod (1953) have proposed an approximate solution using Euler and Maclaurin’s method.




known Bernoulli theorem. From energy considerations (per unit time) one can write AP(effective) .Q = AP(expt1.) .Q -


Ql - p d Q d 0 2


The integral is the kinetic energy correction where p is the density, Q = V/t the volume flow rate of the fluid, and u the velocity of the flow a t point r from the axis of the capillary. For Newtonian liquids it can be shown that the integral term becomes pV3/n2R4t3, noting that dQ = 2nrdr.u, -du/dr = G T / T , = (4V/nR42)r, and u = (2V/rR4t)(R2 - r 2 ) .Usually an empirical constant m is introduced to correct the nonideality of the construction of the visrometer. Thus we have T(corr.) = AP(R/BL)

- mpV2/2r2R3Lt2


and q(corr.)


nR4.AP.t/8LV - mpV/8nLt


TCquation (55b) is frequently called the Hagenbach correction. The value of m is found to be of the order of unity; many workers prefer a value of 1.12, although there have been many arguments about its correct value. The best procedure seems to determine it experimentally. The above correction is derived for a Newtonian liquid and therefore becomes somewhat uncertain if the flow is non-Newtonian. According to Eq. (55b) the kinetic energy correction can be minimized by increasing the flow time per unit volume (i.e., t / V ) or the length, L, of the capillary or both. Practical consideration however limits the feasible range of the length. Increasing the flow time by using a larger volume of the solution also does not help a t all. The best method is to reduce the capillary radius since the flow time is inversely proportional to the fourth power of the radius. On the other hand if the capillary is too fine cleaning of the capillary will become a problem. The kinetic energy correction is also obviously smaller if the density of the solution is lower. 2. End Effect

A flow undergoes sudden contraction and expansion upon entering and leaving a capillary. Thus the velocity of the flow is less near the ends than that in the middle point of the capillary. This effect may be considered as equivalent to an increase in the effect,ive length of the capillary. It is frequently called the Couette correction. The Hagen-Poiseuille equation corrected for both the end and kinetic energy effects, can then be written as : t = nR4AP.t/8(L nR)V - mpV/8n(L nR)t (50)




38 1

Here nR is usually of the order of several times the radius, R. Thus this correction becomes insignificant if the ratio of L / R is very high, say, over 50. On the other hand in the non-Newtonian viscosity measurements where the length of the capillary may be reduced c.onsiderahly in order to cover a wide range of shearing stress this factor may become appreciable. For an entirely different reason the rheologists have studied the effect of L / R on the apparent viscosity by deliberately reducing it to close to unity. By so doing they are able to determine the so-called recoverable shear which is one of the important properties of the viscoelasticity (Philippoff and Gaskins, 1958). This aspect is beyond the scope of the present review. [For experimental determination of the constants m and n in Eq. (56), see an excellent paper by Swindells et aZ. (1952).] Equation (56) is more commonly written as 7] =

at - @ / t


To determine the constants (Y and @ one measures the flow time of two liquids, or one liquid at, two different temperatures, of kiiowri viscosities and d v e s the two simultaneous equations. Or, if one can vary the applied pressure then Eq. (5Ga) can be rewritten as g =

a'AP*t - a/t


By plotting AP.t against l / t or APet' against, t one can determine the viscosity from the intercept or the slope of the straight line respectively. The experimental a- and &values should be comparable t,o those calculated from the dimensions of the capillary.

3. Anomalous Flow Near a Wall The distribution of particle orientations near a capillary wall is restricted hy the presence of such a rigid body (the wall) which introduces a preferred direction in an otherwise isotropic medium. As a consequence the rate of shear near the wall may not be a function of the shearing stress alone and there may be an effective velocity of slip a t the wall. This complication however is usually ignored in routine work. 6 Strictly speaking, t,he (Y- and 8-values in an Ostwald or Ubbelohde viscometer remain constant only if the liquids to be measured have the same densities. If this is not the case Eqs. (56s) and (56b) can be rewritten as:


= ffPt


and q = a'AP+t

- pp/t



4 . Surface Tension and Drainage Errors

Correctlion to the pressure head due to surface tension can in principle be determined from the curvatures of the surfaces at both the upper and lower meniscuses inside the viscometer: AP8




where y is the surface tension, r the radius of curvature, and APa the pressure head due to capillary rise. Since this effect occurs a t both meniscuses it cancels out provided that the two radii of curvature are identical. The drainage error arises from the fact that a small amount of solution or solvent adheres to the wall of the reservoir of a capillary viscometer during measurements. According to a theory by Zweegman, Tuijnman, and Hermans as quoted by Tuijnman and Hermans (1957) the fractional loss of the volume, - AV/V, to the wall for a Ubbelohde viscometer having a radius R, for the reservoir (assumed cylindrical) can be written as

- AV/V


(4/3R,) ( &/aRipg)l’’

( 5% 1

which in combination with the Hagen-Poiseuille equation [Eq. (44c)l becomes - AV/V =

(4/3R,) (AP. R4/8LR?pg)’I2

( 5%1

0.471Ah”2*R2/L’‘2* Ri



- AV/V


noting that AP = Ah.p.g(Ah = pressure head, p = density of the liquid, and g = acceleration of gravity). Thus the drainage error increases with increasing pressure head and radius of the capillary, and with decreasing reservoir radius and length of the capillary. The drainage correction applies to both the solution and solvent and therefore becomes relatively insignificant when only the relative viscosities are considered. For practical purposes both surface tension and drainage corrections can be approximately incorporated into the constants in Eqs. (56a and b ) . 5 . Turbulent Flow

The viscosity as calculated according to Eq. ( 4 4 ) is meaningful only if the flow is laminar. For a capillary flow the rate of flow should not exceed a critical velocity which can be determined from its Reynolds number




where D is the diameter of the capillary, u the velocity of the flow, p the density, and the viscosity of the fluid. The critical Reynolds number for



most liquids has been found to lie in the range of 2,000 to 4,000. The situation is more complicated in a non-Newtonian flow where the viscosity varies with the shearing stress and thereby with the velocity of the flow. Various criteria for the onset of turbulence have been suggested but so far no common agreement seems to have been reached on this subject. One reasonable suggestion is to use the viscosity at the applied shearing stress rather than at zero shearing stress, although many other relations have also been reported in the literature. Equation (59) can be rearranged into

RN = 2Vp/rRqt

= 2Qp/rRt

( 59a 1

For aqueous solutions, say, at 25°C the lower limit of the critical Reynolds number will not be reached as long as V/Rt is well below 30. Thus for all practical purposes this complication will not arise in routine viscosity measurements. On the other hand if the non-Newtonian viscosity measurements are extended to a very high range of the shearing stress, it is desirable to check the possibility of this effect. 6. Density

The densities of a dilute solution and solvent are usually assumed to be almost identical and as such this correction was thought to be negligible in an Ostwald or Ubbelohde viscometer. According to Eq. (44c) the relative viscosity is ( 60a 1

d t o = Pt/POtO

Thus the ratio of flow times is actually a measure of the kinematic viscosity, v (see Table 11). tho = ( ? / P > / ( t l o / P o )

= v/vo

( “31

(The symbol v as used in this section should not be confused with that for the viscosity increment.) To obtain the intrinsic viscosity

one must in principle know the accurate values of densities a t different concentrations. This procedure is certainly too laborious for routine work. By defining an intrinsic kinematic viscosity

Tanford (1055), however, has shown that




where is the partial specific volume of the solute. This simple relation permits the determination of the intrinsic viscosity with ease. As an illustration Tanford has given the data on bovine serum albumin a t 25°C in 0.01 M KC1: [v] = 0.03436, P = 0.734, and po = 0.9975, which in turii yields [T] = 0.03704 [the last term in Eq. (62) being 0.00268], An accuracy of fO.O1 for 7 is sufficient for routine calculation since experimental data usually do not warrant the use of four significant figures for the intrinsic viscosity. This density correction may amount to several per cents for those globular proteins having low intrinsic viscosities. It becomes negligible if the intrinsic viscosities are very high.

C. Extrapolation to Zero Rate of Shear It has been a common practice to measure the viscosities in a multigradient viscometer. The relative viscosities a t each concentration are then plotted against the rates of shear, followed by extrapolation to zero rate of shear and the intrinsic viscosity is calculated from these intercepts for various concentrations in the customary Huggins plot. From both theoretical and experimental considerations, however, the shearing stress rather than the rate of shear should be preferred, although tradition has emphasized the latter (Section IV, C). I n view of the discussions on the power-law controversy (Section IV, F ) one may argue that extrapolation to zero rate of shear should be done by plotting the viscosities against the squares of the rates of shear. It can further be shown by simple differentiation that an equation such as Eq. (40) gives a zero slope at zero rate of shear in a q-G plot (i.e., the viscosities level off as G approaches zero). Thus a linear extrapolation will lead to a viscosity higher than its true value, although normal experimental errors usually make it difficult to detect such subtle differences. The situation is complicated further by the problem of polydispersity since the gradient dependence of the viscosities is less sharp for a polydisperse system than for monodisperse particles. It may not be difficult to fit the experimental data with a linear plot over almost any narrow range of the shear employed, thus giving a completely erroneous extrapolated value. The situation will be much worse if such a plot exhibits an upward curvature as the gradient approaches zero, since in this case the intercept on the viscosity axis depends entirely on how one draws a smooth curve through the experimental points and arbitrarily extrapolates them to zero gradient. The use of a semilogarithmic scale or some other relations may minimize the curvature, but this could be only illusory and the uncertainty of the extrapolated value still exists. All these considerations warn against any indiscriminate use of a linear approximation, unless one is certain that the range of the applied shearing stress is sufficiently low to warrant this pro-



cedure. For rigid particles some information about their shape would immediately reveal the critical shearing stress above which the non-Newtonian behavior becomes significant.

D. Design of a Capillary Viscometer Many types of capillary viscometers have been described. Here we will discuss only some general considerations. According to Eq. (44a) one can construct a multigradient Ubbelohde-type viscometer by adjusting the length, L , the radius, R, of the capillary, or the applied pressure head, A P , or a combination of the three. The shearing stress can be lowered by increasing the length or reducing the radius. The latker however is limited hy the practical range of the flow time, and also the problem of cleaning (to remove dust, etc.). Since according to Eq. (44b) the flow time varies inversely as the fourth power of the radius, a reduction of the radius to one-half results in a sixteenfold increase in the flow time. It is also impractJical to compensate for this vast variation by reducing the volume of the flow. If the liquid flows by gravity, g, then A P = Ah'pvg. Thus the applied shearing stress can also be lowered by using a shorter Ah. There is however a lower limit of Ah beyond which appreciable errors will be introduced by the effects of surface tension, etc. To construct a high gradient viscometer one reverses the procedure by using a shorter length of the capillary, or by using a higher pressure head by either increasing the Ah or applying an external pressure with compressed (dry) nitrogen, etc. The use of a larger radius, however, is undesirable in this case, since it usually makes the rate of flow too fast. For very short length one has to correct for the kinetic energy and end effects. As an illustration the dimensions of three hypothetical Ubbelohde-type viscometers are listed in Table VII including both the upper and lower limits. One of the procedures for designing such a viscometer can be briefly described as follows: first specify the ranges of shearing stresses to be used which in turn determine the ranges of rates of shear, provided that the viscosity of the solvent is known. For example, the viscosity of water is 1.005 centipoise at 20°C or 0.894 centipoise at 25°C. Thus the magnitude of G is roughly a hundred times that of T for pure water. Next estimate the desirable flow time, t, for each bulb and also its volume, V , say, between 1 to 5 ml. These in turn determine the radius, R, of the capillary according to Eq. (44b). Thirdly adjust the ratio of A P / L or A h / L so as to obtain the desired T according to Eq. (44a). With the radius fixed both T and G remain unchanged so long as the ratio A h / L and V / t are kept constant respectively. Thus for viscometer I1 in Table VII the length of the capillary and the pressure head can be reduced to, say, 80 cm and 5 cm (for the upper bulb) without affecting the value of T . Likewise the



flow time can be either increased or decreased by proportional increase or reduction of the volume of the bulb. [It is noted however that the kinetic energy correction remains the same in both cases since it is only related to the volume flow rate according to Eq. (55b) .] Viscometers I1 and 111in the table merely illustrate the different ranges of shearing stresses that can be reached. As a rule a smaller radius and shorter length of the capillary are used for high-gradient viscometers. With some simple calculations one can easily design a capillary viscometer which serves the particular purpose of any experimental work. The dimenTABLEV I I The Dimensions of Three Hypothetical Multibulb Ilbbelohde Viscometers Dimension Length, L (cm) Radius, R (cm) Pressure head, Ah (em) (1) Upper bulb (2) Lower bulb Volume of flow, V (cm3) (1) Upper bulb (2) Lower bulb Flow time, t (sec). (1) Upper bulb (2) Lower bulb Shearing stress, T (dynes cm-2) (1) Upper bulb (2) Lower bulb Rate of shear, G (sec-')a (1) Upper bulb (2) Lower bulb a



400 0.061

400 0.032

20 4 5 1 170 170 1.5 0.3 167 33

25 5 1 0.2 350 350 1 0.2 110 22


4 0.016 25 5 5 1 280 280 50 10 5,500 1,100

Based on water at 25°C.

sions of the precision-bore tubing and bulbs can be calibrated with pure mercury. The viscometer should further be tested with water or other liquids of known viscosities and the corresponding shearing stresses and rates of shear should compare favorably with those calculated from mercury calibration. This test is especially important when a very long capillary is used, which by necessity is very likely bent into loops or other compact shapes. By so doing the cross section of the capillary might be subjected to distortion and a mean radius should be determined and used in Eqs. (44a and b). A slightly different procedure is adopted for the design of viscometer I in which a rather unusually large radius of the capillary is first specified. The flow time for each bulb can be kept fairly high by using a large volume



of the bulb and also by lowering the shearing stress and rate of shear. Such a viscometer is desirable when the danger of clogging the capillary with solid particles (through surface denaturation, etc. ) becomes a problem.

E. Viscosities of Extremely Dilute Solutions Streeter and Boyer in 1954 reported that the reduced viscosity, qsp/C, of polystyrene exhibited a marked upward curvature in the Huggins’ plot when the concentration diminished below a certain critical concentration but it passed through a maximum and dropped again at still lower concentrations. Two years earlier Takeda and Tsuruta (1952) had found a similar behavior for the viscosity of polyvinyl chloride, although their paper was unnoticed at that time. These reports have rekindled an old controversy on the concept of a so-called “critical concentration” (Staudinger, 1932). According to one school of thought the properties of polymer solutions should undergo a marked change at this critical concentration; above it the polymer chains cannot move independently of one another whereas below it this molecular interference disappears. As a consequence many physical measurements in the range of concentrations commonly employed have been suspected to be of doubtful value. The upturn in the reduced viscosity below the critical concentration can be attributed partly t)o the untangling and partly to the expansion of the polymer chains, whereas its downturn at still lower concentrations can be due to the loss of polymer molecules by adsorption on the capillary wall of the viscometer. A second school, however, argues that various interactions have already manifested themselves in the concentration-dependence expressions of these physical properties. The common practice of extrapolation to zero concentration of a power series, for example, in the Huggins plot, should remain valid both above and below the critical concentration. Thus there is no need for a drastic revision of the whole concept of dimensions of polymers in solutions as implied by the former school. The “abnormal” viscosity behavior of the polymers at extremely low concentrations could very well be explained by adsorption alone. Numerous experimental results support one school over the other. As is often true for a controversial subject many of the data quoted as evidence were so conflicting that they caused further confusion of an already confusing topic. True, the experimental difficulties are extremely great, since the very nature of extremely dilute solutions demands special instrumentation of highest precision. In the case of viscosity measurements, problems of temperature control, atmospheric pressure fluctuation, cleaning, etc. become far more serious than in routine work. For example if qsp/C = 1 at C = 0.001 % the flow time must be reproducible to the order of one part in 100,000 for the desired accuracy. At present there is still no agreement as to whether adsorption is the



only cause for the “abnormal” behavior of the viscosity as described above (see, for example, Umstatter, 1954, 1958; o h m , 1955, 1958; Claesson, 1960). The situation is indeed very complicated: adsorption leads to a loss in polymer concentration with a corresponding decrease in flow time, while the adsorption film on the wall narrows the radius of the capillary, which according to Eq. (44c) varies inversely with the fourth power of the flow time. At moderate concentrations the increase in flow time due to smaller radius, however, more than offsets the decrease from the first effect, which i n turn results in a sharper increase in the tls,/C. Assuming that the thickness of the film is r , then we have v,,I(obsd.)


qx(corr.) (1

+ 4 r/R)


where R is the radius of the capillary. The corrected qr,l can be obtained by measuring the viscosities in a series of viscometers of different radii and extrapolating them to I/r = 0 ( o h m , 1955; Takeda and Endo, 1956). There is much argument about the correct equation for adsorption and the actual picture of the adsorbed film. For example, it has been asserted that the amount of solute in an extremely dilute solution is insufficient for the formation of a thick, uniform adsorption film. Nevertheless one ran define an effective thickness of the adsorbed film, r, for Eq. (64). This adsorption also depends on the concentration employed in a manner compatible with the Larigmuir isot,herm. These discussions merely point, out the complications tlhat may arise from the viscosity measurements of extremely dilute solutions. So far as the author is aware no “abnormal” viscosity behavior seems to have been reported for proteins in extremely dilute solutions (((1 5%). Nevertheless it seems desirable t,o keep in mind this controversy and its possible complications in the interpretation of viscosity data.

VI. CONCLUSIONS In this review we have briefly discussed the theoretical and expcrimerital tLspectnsof both Newtonian and non-Newtonian viscosities of polymer solutions. To protein chemists one of the interesting developments is no doubt the re-examination of the (Newt,onian) viscosity treatments of protein solutions. There are many assumptions involved in the effective use of intrinsic viscosity measurements for evaluating the asymmetry of the protein molecules, however attractive the conventional treatment may have appeared for the past two decades. Carefully interpreted, the intrinsic viscosity (at zero gradient) can still provide a reasonable estimate of the axial ratios of the protein molecules. The concept of equivalent hydrodynamic volume, sound in principle, has put the viscometry of protein solutions in a proper perspective, although the quantitative aspects of this new approach still



await extensive tests and the insensitivity of the proposed p- and &functions to variations in the axial ratio of the equivalent ellipsoid makes this task formidable. Viscometry has been one of the most powerful tools for detecting the conformational changes or “denaturation” of proteins, but intimate details of the process cannot be deduced from the viscosity alone. The use of a combination of hydrodynamic measurements such as those involved in the evaluation of the @-functionmay enable us to deduce whether such changes are primarily due to the changes in shape or in effective volume of the protein molecules, even though the inferences are only of a semiquantitative nature. Of greater importance is the coordination of viscometry with other physicochemical methods such as the optical rotatory power and deuterium-hydrogen exchange ; together, they can shed a deeper insight on the internal structures of proteins. The theories of non-Newtonian viscosity for rigid particles are all in good agreement and are satisfactorily established by experimental data. This recent development offers a new measure of the rotary diffusion coefficient and also the degree of polydispersity of the macromolecules. Like flow birefringence, however, the technique is useful only for highly asymmetrical particles which show significant gradient dependence of the viscosities. For the Newtonian viscosities of flexible chain polymers various theories and experiments appear to converge to a good agreement, at least in the limiting case of ideal solutions. Little, however, has been exploited on the effect of branching on the viscosities of polymer solutions and it is not known whether the present forms of these theories can be applicable to solutions of certain proteins having a high degree of flexibility. For the non-Newtonian viscosities of polymer chains present theories are in a stage of rapid progress. Experimentally the gradient dependence of the viscosities is found to be closely related to the expansion of the polymer chains. Moreover, the striking difference in the non-Newtonian behavior between rigid particles and flexible chains appears to furnish a new means for studying the conformational changes of proteins in solutioiis.

ACKNOWLEDGMENTS The major part of this review was written at the Centre de Recherches sur les MacromolBcules, Strasbourg, France. The author is indebted t o Professors C. Sadron and H. Benoit and their colleagues for their valuable assistance and warm hospitality during his stay there. It is a pleasure to thank Professor J. T. Edsall for helpful criticism and numerous suggestions of the manuscript. Thanks are also due the management of the American Viscose Corporation for providing the author the opportunity to complete his work on viscosities (1956-1959). The financial support from a John Simon Guggenheim Memorial Foundation Fellowship, the National Science Foundation (G-10471), and also in part the United States Puhlic Health Service (SF-435, A-4008) is gratefully acknowledged.



VII. APPENDIX(Tables VIII-XIII) TABLE VIII Viscosity Increments, v , as a Function of the Axial Ratios of the Ellipsoids of Revolution", Axial ratio



Axial ratio



1 1.5 2 3 4 5 6

2.50 2.63 2.91 3.68 4.66 5.81 7.10 10.10 13.63 17.76 24.8 38.6 55.2 74.5 120.8

2.50 2.62 2.85 3.43 4.06 4.71 5.36 6.70 8.04 9.39 11.42 14.80 18.19 21.6 28.3

50 60 80 100 125 150 200 300 400 500 600 700 800 900 lo00

176.5 242.0 400.0 593 .o 882.0 1222 2051 4278 7247 10,921 15,216 20,250 25,880 32,200 39,117

35.0 41.7

8 10 12 15 20 25 30 40

-~ ~


MehI et al. (1940). Sadron (1953b).



102.3 136.2 204.1



TABLEIX Frictional Ratios, f/fo and , as a Function of the Axial Ratios of the Ellipsoids of Revolutiona*


Axial ratio

flfo Prolate

Zko Oblate

Prolate ~

1 1.2 1.4 1.6 1.8 2 3 4 5 6 7

8 9 10 12 14 15 16 20 25 30 35 40 50 60 70 80 90 100 200 300

1 1.003 1.010 1.020 1.031 1.044 1.112 1.182 1.255 1.314 1.375 1.433 1.490 1.543 1.645 1.739


1.829 1.996 2.183 2.356 2.518 2.668 2.946 3.201 3.438 3.658 3.867 4.067


1 1.003 1.010 1.019 1.030 1.042 1.105 1.165 1.224 1.277 1.326 1.374 1.416 1.458 1.534 1.604 1.667 1.782 1.908 2.020 2.119 2.212 2.375 2.518 2.648 2.765 2.873 2.974 -


Oblate ~

1 1.505 2.340 3.395 4.638 6.061 9.401

1 -



1.132 1.464 1.843 2.240 2.645 3.471 4.305 5.143 6.407

41.80 61.05 83.45

8.519 10.64 12.75


13.37 17.94



137.3 202.9 279.9 465.9 694.5 2428 5085

Svedberg and Pedersen (1940), Cohn and Edsall (1943). Scheraga and Mandelkern (1953).


16.99 21.24 25.48 33.96


42.44 84.86 127.3



TABLEX 6 - and &Functions for Ellipsoids of Revolution" Axial ratio

1 2 3 4 5 6 8 10 12 15 20 25 30 40 50 60 80 100

200 300










2.12 2.13 2.16 2.20 2.23 2.28 2.35 2.41 2.47 2.54 2.64 2.72 2.78 2.89 2.97 3.04 3.14 3.22 3.48 3.60

2.12 2.12 2.13 2.13 2.13 2.14 2.14 2.14 2.14 2.14 2.15 2.15 2.15 2.15 2.15 2.15 2.15 2.15 2.15 2.15

2.50 1.93 1.57 1.37 1.25 1.17 1.07 1.02 0.990 0.959 0.923 0.904 0.893 0.880 0.870 0.865 0.859 0.854 0.845 0.841

2.50 2.42 2.34 2.20 2.10 2.03 1.93 1.87 1.83 1.78 1.74 1.71 1.69 1.67 1.65 1.64 1.62 1.62 1.60 1.60

Scherttga and Mttndelkern (1953).



TABLE XI Shape Factors for the Calculations of Lengths of Prolate Ellipsoids Axial ratio


1 1.2 1.4 1.5 1.6 1.8 2 3


5 6 7 8 9 10 12 14 15 16 20 25 30 35 40 50

60 70 80 90 100 150 200 300

Viscosity‘ (P/v)l/a

0.7368 -


0.9493 -

1.112 1.347 1.509 1.626 1.718


1.850 1.943 2.009


2.086 2.180 2.245 2.295 2.366 2.420 2.459 2.520


2.564 2,641 2.692 2.761

Rotary diffusion coefficientb (JP)’‘3



1.385 1.567 1.676 1.753 1.811




1.956 2.002




2.123 2.171 2.209 2.267 2.292 2.343 2.395




2.544 2.606

Translational diffusion coefficientc (FPz/3)

1.Ooo 1.126 1.239 1.341 1.435 1.521 1.871 2.132 2.330 2.513 2.661 2.791 2.904 3.008 3.186 3.340


3.472 3.691 3.917 4.098 4.249 4.384 4.607 4.788 4.940 5.076 5.193 5.297


L(=2a) = [ (600[q]M/Nr)(pz/v))l/*= 6.82 X 10-S([q]M)1’3(p*/v)1/3. L(=2a) = [(k/r)(T/qoe)(~p2)ll’3= 3.53 X 1 0 - ~ ( T / ~ o e ) 1 ~ 3 ( J p z ) 1 ~ 3 . L(=2a) = [M(1 - ~ p ) / 3 ~ ~ ~ N s l ( F = p1.76 ~ / ~X)10-2s[M(1- ~ p ) / q ~ s ] ( F por ~ ’= ~) (k/3r) (T/qoD)(Fpz/3) = 1.46 x 10-17(T/qoD)(Fp2//8).



TABLE XI1 Viscosity Increments, Y, as a Function of a for Various Axial Ratios of the Ellipsoids of Revolutions a

0.00 0.25 0.50 0.75 1.oo 1.25 1.50 1.75 2.00 2.25 2.50 3.00 3.50 4.00 4.50 5.00 6.00 7.00 8.00 9.00 10.00 12.50 15.00 17.50 20.00 22.50 25.00 30.00 35.00 40.00 45.00 50.00 60.00 a




27.18 27.15 27.07 26.94 26.76 26.54 26.28 25.98 25.66 25.32 24.97 24,24 23.49 22.77 22.07 21.41 20.22 19.19 18.30 17.53 16.86 15.51 14.42 13.69 13.05 12.52 12.06 11.32 10.74 10.27 9 A47 9.477 8.845

55.19 55.13 54.96 54.68 54.29 53.81 53.25 52.62 51.93 51.20 50.44 48.86 47.26 45.70 44,21 42.80 40.24 38.04 36.14 34.50 33.07 30.21 27.89 26.34 24.98 23.86 22.90 21.35 20.12 19.12 18.24 17.46 16.12

Scheraga (1955).



176.8 176.6 176.0 175.1 173.8 172.1 170.3 168.1 165.8 163.4 160.8 155.5 150.1 144.9 139.9 135.2 126.6 119.2 112.9 107.5 102.7 93.19 85.54 80.43 75.93 72.28 69.12 64.05 60.39 56.80 53.94 51.40 47.04


593.7 593.0 591 .O 587.7 583.2 577.6 571.1 563.8 555.8 547.4 538.5 520.3 501.8 483.8 466.6 450.3 421 .O 395.7 374.0 355.3 339.1 306.7 280.7 263.3 248.0 235.7 225.0 207.9 194.5 183.6 174.0 165.5 151.O


18.19 18.18 18.15 18.09 18.01 17.91 17.80 17.67 17.52 17.37 17.22 16.89 16.55 16.22 15.90 15.60 15.04 14.54 14.11 13.72 13.38 12.66 12.07 11.64 11.26 10.93 10,64 10.15 9.753 9.411 9.114 8.844 8.391

100 69.10 69.06 68.91 68.68 68.36 67.97 67.51 66.99 66.42 65.82 65.18 63.87 62.53 61.20 59.92 58.70 56.46 54.48 52.73 51.19 49.82 46.96 44.57 42.86 41.33 40.03 38.86 36.91 35.32 33.94 32.75 31.66 29.82



hl./[vl.-o a

0.00 0.25 0.50 0.75 1.OO 1.25 1.50 1.75 2.00 2.25 2.50 3.00 3.50 4.00 4.50 5.00 6.00 7.00 8.00 9.00 10.00 12.50 15.00 17.50 20.00 22.50 25.00 30.00 35.00 40.00 45.00 50.00 60.00

TABLE XI11 as a Function of a for Various Axial Ratios of the Ellipsoids of Revolutiona Prolate








1.oooo 0.9989 0.9960 0.9912 0.9846 0.9765 0.9669 0.9559 0.9441 0.9316 0.9187 0.8918 0.8642 0.8378 0.8120 0.7877 0.7439 0.7060 0.6733 0.6450 0.6203 0.5706 0.5305 0.5037 0.4801 0.4606 0.4437 0.4165 0.3951 0.3779 0.3623 0.3487 0.3254

1.0000 0.9989 0.9958 0.9908 0.9837 0.9750 0.9649 0.9534 0.9409 0.9277 0.9139 0.8853 0.8563 0.8281 0.8011 0.7755 0.7291 0.6893 0.6548 0.6251 0.5992 0.5474 0.5053 0.4773 0.4526 0.4323 0.4149 0.3868 0.3646 0.3464 0.3305 0.3164 0.2921

1.oooO 0.9989 0.9955 0.9904 0.9830 0.9734 0.9632 0.9508 0.9378 0.9242 0.9095 0.8795 0.8490 0.8196 0.7913 0,7647 0.7161 0.6742 0.6386 0.6080 0.5809 0.5271 0.4838 0.4549 0.4295 0.4088 0.3910 0.3623 0.3416 0.3213 0.3051 0.2907 0.2661

0.9988 0.9955 0.9899 0.9823 0.9729 0.9619 0.9496 0.9362 0.9220 0.9070 0.8764 0.8452 0.8149 0.7859 0.7585 0.7091 0.6666 0.6300 0.5985 0.5712 0.5166 0.4728 0.4435 0.4177 0.3970 0.3790 0.3502 0.3276 0.3093 0.2931 0.2788 0.2543


1.0000 0.9995 0.9978 0.9945 0.9901 0.9846 0.9786 0.9714 0.9632 0.9549 0.9467 0.9285 0.9098 0.8917 0.8741 0.8576 0.8268 0.7993 0.7757 0.7544 0.7356 0.6960 0.6636 0.6399 0.6190 0.6009 0.5849 0.5580 0.5362 0.5174 0.5010 0.4862 0.4613

0.9994 0.9973 0.9939 0.9893 0.9837 0.9770 0.9695 0.9612 0.9525 0.9433 0.9243 0.9049 0.8857 0.8672 0.8495 0.8171 0.7884 0.7631 0.7408 0.7210 0.6796 0.6450 0.6203 0.5981 0.5793 0.5624 0.5342 0.5111 0.4912 0.4740 0.4582 0.4316


Calculated from Scheraga (1955). See also Yang (1958a). For axial ratios other than those listed here see t h e original papers.



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