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Structure dependent hydrodynamic properties andbrownian motion of polymeric molecules

R. Cerf

To cite this version:R. Cerf. Structure dependent hydrodynamic properties and brownian motion of polymeric molecules.Journal de Physique, 1977, 38 (4), pp.357-376. �10.1051/jphys:01977003804035700�. �jpa-00208595�

LE JOURNAL DE PHYSIQUE

STRUCTURE DEPENDENT HYDRODYNAMICPROPERTIES AND BROWNIAN MOTION OF POLYMERIC MOLECULES

R. CERF

Laboratoire d’Acoustique Moléculaire (*)Université Louis-Pasteur, 4, rue Blaise-Pascal, Strasbourg, France

(Reçu le 20 mai 1976, révisé le 20 dicembre 1976, accepté le 22 décembre 1976)

Résumé. 2014 Cet article traite de problèmes en suspens en dynamique des polymères, notammentde la validité du concept de viscosité interne pour des chaînes de grande flexibilité cinétique. La dis-cussion est fondée sur l’hypothèse que des réarrangements locaux non diffusionnels (qui pourraientêtre coopératifs) coexistent avec les mouvements diffusionnels les plus rapides ; elle utilise le modèlelimite de l’auteur qui permet de tenir compte d’une catégorie de mouvements locaux, soit diffusion-nels, soit non diffusionnels.

L’association de coefficients de viscosité interne aux modes normaux est justifiée par l’approchestochastique en dynamique des polymères. On peut exprimer la viscosité interne indépendante de ~0en fonction de caractéristiques dynamiques locales. La symétrie du tenseur des efforts résulte de laloi de variation des coefficients de viscosité interne avec le mode, lorsqu’on calcule la vitesse derotation de la chaîne comme si elle était rigidifiée. La comparaison avec des résultats expérimentauxfournis par des techniques diverses confirme que la viscosité interne indépendante de ~0, donc ladynamique locale, peut se manifester dans les temps de relaxation les plus longs. I1 est ainsi confirméque les processus locaux non diffusionnels peuvent contribuer au renouvellement des conformations.Le modèle est compatible avec l’existence simultanée d’une viscosité interne indépendante de ~0

et d’une viscosité interne proportionnelle à ~0. On suggère toutefois que cette dernière résulte essen-tiellement de mouvements diffusionnels qu’il est impossible d’expliquer en considérant les seulsmodes de Rouse.On propose à titre de conjecture une description du mouvement brownien de la chaîne, et de sa

déformation dans un gradient de cisaillement haute fréquence, qui explique qualitativement à lafois que la viscosité interne indépendante de ~0 doit s’annuler à haute fréquence, et que la valeurlimite [~’]~ de la viscosité intrinsèque est sensiblement indépendante de la masse moléculaire.

Abstract. 2014 Pending problems in polymer dynamics are treated, particularly regarding the rele-vance of the concept of internal viscosity for kinetically flexible chains. The discussion is based on theassumption that local non-diffusional (possibly cooperative) segmental rearrangements coexistwith the fastest diffusional ones; it makes use of the author’s limiting model that permits one toaccount for one type of local movement, either diffusional or non-diffusional.

Association of internal viscosity coefficients with the normal modes is shown to be justified by thestochastic approach to polymer dynamics. The ~0-independent internal viscosity parameter can beexpressed in terms of local dynamical characteristics. Stress-tensor symmetry follows from the mode-dependence of the internal viscosity coefficients when the rotational velocity of the chain is calculatedas if it were rigidified. Comparison with experimental results obtained by various techniques confirmsthat ~0-independent internal viscosity, hence local dynamics, may be manifested in the longest relaxa-tion times. Thus, it is also confirmed that local non-diffusional processes may contribute to therenewal of conformations.The model further allows for ~0-proportional internal viscosity. It is however suggested that the

latter one arises for the most part from diffusional movements which cannot be accounted for bythe sole consideration of the Rouse-modes.A tentative description of the Brownian motion of the chain and of its deformation in a high

frequency shearing gradient is proposed that would qualitatively explain both that the ~0-inde-pendent internal viscosity must vanish at high frequency and that the limiting value [~’]~ of theintrinsic viscosity is nearly molecular weight independent.

Tome 38 No 4 AVRIL 1977

Classification

Physics Abstracts5.660 - 6.310 - 6.339 - 7.270

(*) (E.R.A. au C.N.R.S.).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003804035700

358

1. Introduction. - It is known that the zero gra-dient intrinsic viscosity [17]0 of macromolecular solu-tions yields structural information about the dissolvedmolecules, e.g., about their volume and shape if theparticles are rigid, and about the effective bond lengthif flexible chains are considered.We shall be mainly concerned here with local

structural characteristics of a different nature, i.e.,kinetic parameters such as the height of potentialbarriers and probabilities of transitions betweenrotational isomers. It is often considered [1] thatthese structural parameters do not in fact, enter intothe description of the normal (Rouse) modes ofdeformation of chain molecules, and therefore notinto most of their hydrodynamic properties.For example, in the dynamical theory based on

the gaussian bead-and-spring model [2], the relaxationtimes are merely proportional to the volume occupiedby one molecule. This is seen by using Zimm’s expres-sion of the relaxation times [3] :

in which L 2 is the mean-square end-to-end distanceof the chain, 170 the solvent viscosity, k Boltzmann’sconstant, T the absolute temperature and the £l’snumerical coefficients which are functions of theindex p of the mode. An expression equivalent to (1)is :

in which M is the molecular weight, [17]0 the zero

gradient intrinsic viscosity and R the gas constant.As a result of shielding of the interior of the mole-

cule by hydrodynamic interactions the friction coeffi-cient f of the beads has disappeared in the end result.It is true that by neglecting hydrodynamic interactionsRouse obtains relaxation times that are proportionalto f and Orwoll and Stockmayer [4] have shown howthe stochastic approach to polymer dynamic permitsone to express these times in terms of local jumpprobabilities. However, it follows from the abovethat in the limit of strong hydrodynamic interactionsZimm’s relaxation times cannot depend on local jumpprobabilities and this is a drawback of the theorybased on the classical bead-and-spring model. Indeed,the relaxation times should, at least in principle,depend on hindrance of internal rotation and theyshould be lengthened when hindrance is large.The influence of such hindrances had been investi-

gated before the advent of stochastic models. Kuhnand Kuhn [5], in particular, added in the diffusionequation for the dumbbell model an internal viscosityforce which arises from the barriers that must beovercome when the particle is deformed. The corres-ponding internal viscosity coefficient does not dependon the viscosity of the solvent. Most of the questions,

however, could be answered only with the aid of moreelaborate models.One would like first to be able to determine for a

given polymer whether internal viscosity effects aremanifested in hydrodynamic measurements, suchas the flow birefringence or the non-Newtonian flow.A study based on the elastic sphere model led tothe following qualitative result [6] (which thereafterappeared to be general, regardless of whether oneuses simple models or the more detailed bead-and-spring model with internal viscosity [7]-[10]) : wheninternal viscosity effects are taken into account,macroscopically measurable relaxation times are

encountered which are not simply proportional tothe solvent viscosity 170’ but are the sum of an qo-proportional and of an ?10-independent term. Thisoffers the possibility of answering the question raisedabove by carrying out measurements in solvents ofvarious viscosities [11].Most subsequent studies dealing with kinetic stiff-

ness of chains were based on the gaussian bead-and-spring model with internal viscosity. This model,the generalized diffusion equation to which it leads

(eq. (2) below) and the new set of relaxation times thataccount for hindrance of internal rotation (eq. (6)below) were proposed by us in 1958 [8].The model associates an internal viscosity coeffi-

cient (pp with each purely deformational normal modep. The gp’s are the elements of the internal viscositymatrix (D. Pure deformations are considered in orderthat no internal viscosity effect be associated withthe overall rotation of the molecule in a shearinggradient G.The diffusion equation then reads :

The transposed differential operator V’ is the diver-gence of the column-vector on which it operates. Thechain units, of effective friction coefficient C, havebeen grouped in N sub-chains, each of g units andof mean square end-to-end distance 1’, T is the dis-tribution function of the junction-points of the sub-chains, s the normal coordinate specifying these

points; go denotes the unperturbed velocity field andsn the rotational velocity of the molecule. The threeN-dimensional components of s will be denoted u, v, w.

I is the unit matrix, M the diagonal matrix whoseelements are Rouse’s eigenvalues, N the diagonalmatrix whose elements are Zimm’s vp’s ;

is the diagonal matrix whose elements are the ratiosof the internal viscosity coefficients qJp to the frictioncoefficient f of the beads :

359

and iinally :

For particles of fairly high kinetic rigidity therotational terms to be subtracted from the overallvelocities of the beads clearly are those which wouldhold for the supposedly rigidified particles [7]-[10] :

in which up and wp are normal coordinates.To obtain the rotational term which must be

subtracted from the velocity of each bead when

kinetically flexible chains are considered, we use

the same procedure of considering the overall velo-city Q of the molecule [10]. In the present case, thisstep obviously amounts to neglecting fluctuationsthat might be of importance in particular instances(see subsection 2.4, below).

In addition to the family of relaxation times of theRouse-Zimm model, of which the subset top entersinto the hydrodynamic quantities, eq. (2) leads to asecond family :

Each r) is associated with one of Rouse’s modes andcontains an additional internal viscosity term ip pp vp,independent of qo.

It is worth wondering why eq. (2) associates twosets of N relaxation times with the N modes and to

enquire about the physical significance of these times.The zp’s are characteristic of a system of frictionpoints and connecting springs. Although these timesdo not depend on local dynamic properties, theypermit the calculation of the intrinsic viscosity [q]oat zero gradient, and [q]o turns out to be proportionalto the sum of the ’tp’s. That the ’tp’s furnish the valueof [q]o is understandable, since Mo can be calculatedconsidering a rigid cloud of friction points (Debyeand Bueche [12]) and is therefore insensitive to thelocal dynamics. Thus, the Zimm times could be calledthe viscosity relaxation times.The zp’s, on the contrary, hopefully contain phy-

sically relevant features of local dynamics and assuch may be considered the true relaxation times ofthe chain. Moreover, it is seen from eq. (2) that in theabsence of a velocity field the experimentally measuredquantities depend on the T"S only. Thus, for example,the study of the scattering of light in solvents ofvarious viscosities may lead to new possibilities ofcomparing the theory with the measurements.Of course, it is by no means evident that the trick

of introducing an internal viscosity in the bead-and-spring model is sufficient for treating all dynamicalproblems and limitations may be expected.

The relaxation times Tp exhibit just the behaviourof the inverse rate of escape of a particle embeddedin a viscous medium over a potential barrier. In thisconnection, we may recall from the theoretical studyof Kramers [13] that when the potential barrier is

large compared with kT, the inverse rate of escapeis proportional to the solvent viscosity rio in the diffu-sion limit (large 17o’S) and contains an rio-independentterm in the non-diffusional case (small 17o’S). This

similarity in behaviour between the characteristictimes of chain models and of a Brownian particleis of course by no means fortuitous, as can be shown[14] using the stochastic approach to polymer dyna-mics in Iwata’s version [15].We start with Iwata’s coarse-grained diffusion

equation, which contains the probability of an

average local conformational change, and expressthis probability with the aid of the above-mentionedresult of Kramers. When the average local confor-mational change is assumed non-diffusional, this

procedure leads to associating an rio-independentinternal viscosity coefficient with each of Rouse’smodes, as was done in eq. (2).

For qo-independent internal viscosity effects to

show up, it is therefore necessary that non-diffusionallocal processes exist and when no rio-independentinternal viscosity effect is manifested, we may considerthat only diffusional processes are of importance.However, non-diffusional local processes do not

necessarily produce an qo-independent internal vis-cosity effect and there may be an no-proportional inter-nal viscosity effect associated with both diffusionaland non-diffusional processes.The fastest local movements (as detected in fluo-

rescence depolarization or spin lattice NMR measure-ments), the time constants of which are of a fewnanoseconds, are found to be diffusional [16]. Directevidence of rotational isomerism can also be obtained

by ultrasonic measurements. It is of interest thatultrasonic absorption [17-19] and dispersion [20]measurements show that there is a strong absorptionband around 1 MHz (i.e., far above the usual domainof viscoelastic measurements). For various vinylpoly-mers this high frequency band starts rather abruptly.It is therefore tempting to assume [21, 22] that theultrasonic absorption in the MHz range is not produc-ed by a cascade of fast movements of minimal seg-ments, but by local cooperative movements of longersegments of the chain. A two-state picture showsthat for segmental rearrangements to occur in theabove frequency range free energy barriers of severalkT’s are required, a necessary condition for non-diffusional behaviour to show up. Moreover, ultra-sonic measurements for polystyrene in solvents ofvarious viscosities are in favor of the non-diffusionalnature of the preceding movements [23].Even if the non-diffusional local movements are

slower than the diffusional ones, they may wellcontribute to the renewal of conformations by produc-

360

ing reorganizations of the chain over larger distances.Under these conditions such movements would bemanifested in the slowest deformational modes and

17o-independent internal viscosity effects would arise.We thus proposed to describe the Brownian motion

of chain molecules as a competition between localdiffusional and non-diffusional movements [21, 22].A complete theory would then require a full descrip-tion of both these types of processes as well as oftheir competing effects. However, it still seems diffi-cult to assign an exact probability to each of themany possible movements that involve, say, 3 to

10 bonds, especially when considering such factorsas inhibition of double occupancy, nearest neighbourcontact, obstructing effects of neighbouring parts ofthe chain, and distortions of bond angles and bondlengths in the crossing of potential barriers.

Therefore, we shall use our model, assuming thatthere is only one type of local movement present,either diffusional or non-diffusional, depending onthe problem which we consider. We feel justified inproceeding in this manner, as new insights are obtainedwhen expressing local friction coefficients in termsof characteristics of the hindering potential, alongthe line mentioned above and described in the pre-liminary notes [14] and [21]. Of course, the competi-tion between the two kinds of local processes, diffu-sional and non-diffusional, will be kept in mind whencomparing the theoretical results with the measure-ments.

We shall first present in section 2 the hydrodynamicsof the bead-and-spring model with internal viscositycoefficients satisfying the requirements of the sto-

chastic theory. Three different problems will thenbe considered, according to the experimental condi-tions : 1) low gradient and low frequency; 2) highgradient ; 3) high frequency. The latter term heredenotes conditions which are encountered in vis-coelastic measurements in a frequency range whichis still much below those frequencies at which localconformational changes are manifested.

1) LOW GRADIENT AND LOW FREQUENCY. - Asan example of a quantity that characterizes the lowgradient and low frequency behaviour we take theinitial slope, tan a, of the extinction angle curve of theflow birefringence [8, 9] :

Taking into account eq. (23) of section 2, the summa-tion furnishes :

where :

When the local processes are non-diffusional, weconsider in particular the situation [8, 9] in which

n170 is independent of 170 (see section 2.1). Further-more, in the limit of very high internal viscosity,eq. (2) describes the orientational effect of the particlein a shearing gradient. The corresponding macro-scopically observable relaxation time is [9] :

where the index OR means orientation. The followingvalues have been found for the numerical coefficient :a = 3.5 [8] or a = 2.5 [24], according to whether thedistribution function of the molecular axes is takenin two or in three dimensions, respectively. Using (9)we have :

For the chain molecules to behave as if they wererigidified with regard to the observable tan ex, i.e.,for eq. (10) to hold, it is required that (tan (x)op tan a

given by eq. (8). Thus, it is required that :

This, in turn, means that n must be larger than a fewunits, say, n > 5. The condition (11) means that theinternal viscosity is quite high, as will be seen later. If :

n170 is of the order of a few cP

condition (11) is fulfilled for 170 1 cP. For thesevalues of qo the quantity (2 a - 1) 170 plays the roleof n170 (compare eqs. (8) and (10)).The corresponding tan a versus 170 curve showing

the passage from orientational to deformationaleffect when 170 is increased has been published pre-viously (Fig. 9 of reference [9]). The result is in accordwith observations of Schwander and Cerf [25] and ofLeray [26] for DNA. In figure 1 we use a slightlydifferent representation in which we plot

against qo. Curve I shows the passage from orientationto deformation. Its initial slope is equal to 2 a - 1and at high 17o-values the ordinate reaches the limit-ing value n17 o.Curve II represents the Zimm-limit in which there

is no internal viscosity present at all, i.e., the casen=0.When both diffusional and non-diffusional processes

are acting, the former ones become fastest at low17o-values [21, 22] and they may control the renewalof conformations in this range. Passage from then170 = const. situation at high 17o-values to the Zimmlimit at low qo-values must then be observed, as shownby curve III. In the present context, this curve givesan example of how we attempt to account for thecompeting effects of diffusional and non-diffusionalprocesses in the frame of a theory that furnishes thelimiting curves ni7o = const. and n = 0. In the repre-

361

FIG. 1. - Theoretical results for various situations wherein therelative importance of diffusional and non-diffusional processesdiffers. Curve I : non-diffusional processes dominate for all valuesof tjo. Curve II : diffusional processes dominate for all values of re-Curve III : non-diffusional processes are present, but diffusionalones dominate at low qo-values. The theory furnishes only thestraight lines. The curves I and III show conjectured theoreticalbehaviour for intermediate ?10-values. For definition of the ordinate,

see text.

sentation used here, curve III corresponds to

figure 14b of reference [22]. In an actual experiment,it may of course be difficult to decide whether a curveof type I or of type III is obtained.

Statements by Stockmayer [1] and Yamakawa [27]amount to assuming that one cannot have n # 0for kinetically flexible molecules at high molecularweight. According to these authors practically allconformational changes are diffusional, because theyare cascades of basic processes that are all diffusional,contrary to what has been assumed here. To us it

seems that cases intermediate between those of curves, Iand II should exist. One may hope that comparisonwith the experimental results will eventually showwhich assumption is correct.

2) HIGH GRADIENT. f - To determine experimen-tally tan a a double extrapolation to zero gradientand to zero concentration is required. This representsa major experimental difficulty and it would be

helpful to extend the theory to finite gradients.However, approximations introduced in the calcula-tion become less reliable as the gradient is increased.Furthermore, factors that are neglected in the model,such as the change of hydrodynamic interactionswith chain extension and effect of chain expansionin good solvents, are likely to become more importantas the gradient is increased. Useful conclusions arenevertheless obtained when the predictions of themodel for the non-newtonian flow (section 3) and theflow birefringence (section 5) are confronted with

measurements at finite gradient.

3) HIGH FREQUENCY. - It is a matter of trivialcalculation to deduce from eqs. (2) and (6) properties

in an oscillatory field such as the viscoelasticity,oscillatory flow birefringence and acoustic birefrin-

gence. However, the experimental results showedthat the ’1o-independent internal viscosity effect cannotcontribute much to the real part [17’] of the intrinsicviscosity at high frequency and we proposed that thiseffect must decrease with increasing frequency [9].On the other hand, eqs. (2) and (6) together with

the special form which we had chosen for the gp’sin connection with flow birefringence measurementsof Leray [26] :

were applied by Peterlin [28] (with frequency-inde-pendent jf), first to the study of the frequency-depen-dent intrinsic viscosity M = [17’] - i[q"]. In eq. (12)the internal viscosity coefficient Y is 17o-independentand N is the number of sub-chains. Several authorsuse eq. (12) writing T instead of Y (e.g., [28-30]).Contrary to a current opinion [28] the inverse pro-portionality of Tp on N has little to do with thesimilar N-dependence of Kuhn and Kuhn’s internalviscosity coefficient B [5]. Indeed, these authorsconsider relative movements of two rigidified moietiesof the chain, which is not a realistic picture of itsthermal motion. It can also be shown that for Kuhnand Kuhn’s model the internal viscosity term intan a is independent of molecular weight (see sec-

tion 6.1), instead of proportional to it as requiredby eq. (12). Furthermore, the proportionality to pof the Tp’s given by eq. (12) is not directly connectedwith Kuhn and Kuhn’s model of internal viscosity,contrary to a statement by Stockmayer [56].

It proved that Peterlin’s results cannot be reconciledwith the measurements, in accordance with our

expectation [9]. If we take the example quoted inreference [28] of polystyrene with M = 239 000, forwhich at high frequency [r¡’]00/[17]0 ,...., 0.2 in aroclorof viscosity no = 2.5 P, following Peterlin one wouldpredict that in a low viscosity solvent like toluene[ti’l. - [nlo and this was known not to be so [31].

Studies of the viscoelastic properties of solutionsof DNA by Meyer, Pfeiffer and Ferry [32] are consis-tent with a decrease of the 17o-independent internalviscosity with increasing frequency.

Their measurements in highly viscous solventslater led Massa, Schrag and Ferry [30] to introduce theconcept of 17o-proportional internal viscosity and topropose that Y must be considered proportional to no,instead of independent of it. They reported considera-ble success in fitting experimental results of viscoelas-tic measurements and oscillatory flow biref-ringencefor polystyrene with those values obtained [28, 33]from eqs. (2), (6) and (12).The concept of an ilo-proportional internal vis-

cosity is entirely compatible with the stochastic

approach ([21, 22] and section 2 below). Contraryto stands by Stockmayer [1] we consider, however,

362

that the need of introducing an qo-proportionalinternal viscosity does not preclude the existence ofan 17o-independent internal viscosity that depends onfrequency. Reasons for this have already been men-tioned and others will be discussed below.We also disagree with Peterlin’s revision of the

concept of internal viscosity [34], for his new approachis still based on all three eqs. (2), (6) and (12). Thusthe result that [17’]00 is nearly independent of molecularweight [28, 35], in accord with the measurements onpolystyrene [30], must still be traced back to the

presence of N in the denominator of eq. (12).The form [12] of the qJp’s has an intuitively simple

meaning, i.e., the pth eigenmode is resisted by aviscosity coefficient inversely proportional to thenumber of chain joints between two consecutivemodes. However, it turns out that when the qJp’sare chosen in accord with the requirements of thestochastic theory, another expression of the qJ p’ sis obtained ([21, 22] and eq. (23) below), with theresult [17;]00/[170] instead of [r¡’]oo is now molecular-

weight independent (see eq. (A, 6’) below).One may therefore wonder whether a mechanism

can be found that would both explain that the qo-independent internal viscosity must vanish at highfrequency and that the 17o-proportional internal

viscosity leads to a value of [17’]00 which is nearlyindependent of molecular weight. It may be thatthese effects could be explained by a detailed considera-tion of the competition between diffusional and non-diffusional movements and of the way in which they -generate medium and long range movements of thechain.

In section 7 we shall tentatively propose a pictureof the Brownian movement of the chain molecule andof its deformation in a high frequency shearing gra-dient that may well explain both these effects, howevernot strictly speaking in the existing frame of thebead-and-spring model.

2. Hydrodynamic theory. - In this section weconsider the limiting model in which all local move-ments are assumed to be of one type, generally non-diffusional ; in particular instances we consider onlydiffusional movements.

2.1 MOLECULAR MEANING OF INTERNAL VISCO-

SITY. - Association of an internal viscosity coefficientwith each normal mode is validated when local tran-sition probabilities are introduced [14]. For zero

external field the diffusion eq. (2) of the bead-and-spring model with internal viscosity reduces to :

On the other hand, Iwata’s coarse-grained diffusionequation [15] reads, for the case when the hydrodyna-mic interactions are accounted for :

When performing the coarse-graining procedure, letus take groups that are composed of the same number,g, of chain-units as are the sub-chains in eq. (13).The normal coordinate r then has as many elementsas s. Furthermore, let us disregard particular aspectsassociated with the ring-shaped character of Iwata’smodel.The elements of Iwata’s diagonal diffusion matrix D

are given by [ 15] :

where P, denotes the probability that a conformationalchange of a segment comprising p + 1 units occurswithin unit time; 9 denotes the average value of Jl.Two circumstances now make it possible to consider

eqs. (13) and (14) as being identical.

i) To first approximation, Iwata’s bp’s are mode-independent. From figure 2 of reference [ 15] we canindeed set, with sufficient accuracy :

provided p be smaller than, say, N/10. On the otherhand, all the series considered here converge fast

enough for the results to show no significant differencewhether we take 800 or 100 terms in the summations

(see below). Thus, we may indeed retain only modesfor which p N/ 10 and assume that the 6 p s are

mode-independent at high molecular weight. Weshall call Do the diagonal matrix whose elements areall equal to 60 p¡¡/g. Note that 60 = A 2, where d 2is the average of the mean squares of the displacementvectors of the atoms that take part in the local confi-gurational change.

ii) Due to the nearly linear dependence of P,,on 170’ as follows from Kramers’ general expressionof an inverse transition probability in the non-

difi’usional case [13], we may decompose Do 1 intothe sum of an qo-proportional part Do-1 and of an10-independent part D"-’

Thus :

where (D’ is qo-proportional and 0" is qo-independent.Using eq. (3), R is equal to :

where R’ is qo-independent and R" is proportionalto 1701. We may now write :

provided R’ and R" are chosen such that :

363

Retaining only qo-independent terms in 0, i.e.,putting :

eqs. (18’) and (21) furnish [14], [22] :

with :

where k and h are Boltzmann’s and Planck’s cons-tants and T the absolute temperature. In expressingP -1 use has been made of the one-particle schemeof Kramers ; m and z are the mass and friction coeffi-cient of the particle that crosses a potential welland m§ is the curvature at the top of the potentialcurve.

By eqs. (24) and (24’) the dimensionless internalviscosity parameter n is expressed in terms of mole-cular local characteristics; the higher the potentialbarrier W, the larger is n. Let a be the effective hydro-dynamic radius of the monomeric unit (( = 6 1t170 a) ;illustratively, if the conformational change of impor-tance relaxes in the ultrasonic range (say, around1 Megahertz), the parameter n as given by eq. (24’)is of the order of unity for 170 = 1 cP provided60 a - 102 A3. On the other hand, through its z-

dependence, n is inversely proportional to the solventviscosity 17 o.

For not too small p-values vp is proportional top- 1/2 N 1/2 (where N is the number of sub-chains);thus pp must be proportional to p1/2 N- 1/2 . Eq. (23)would also hold for the case of negligible hydrodyna-mic interactions since vp is then equal to unity.When R’ # 0, i.e., when 17o-proportional internal

viscosity is considered, eqs. (23), (24) and (24’)remain valid for the elements of the matrix R",and R’ can be obtained from eq. (20) in terms ofa),,,. Moreover, local diffusional processes not consi-dered in this section may also produce 17o-propor-tional internal viscosity; when only these processesare acting, the corresponding R’ is obtained by anequation similar to (20) with the right hand membermultiplied by 2 [22]. It is, however, likely that diffu-sional motions not accounted for by the model mustbe invoked for understanding the origin of 170--propor-tional internal viscosity effects (see section 7). In therange where these latter motions are negligible andif only diffusional processes are present, the equationsfor the observables (see Appendix) hold with an qo-proportional n.We shall nevertheless limit our consideration to no-

independent internal viscosity effects, i.e., use condi-tion (22) and eq. (24’). In principle, we should require[22] that both :

and We > nO-)A, where wà is the curvature at the bottomof the potential curve. However, for a chain moleculewith 17o-independent internal viscosity, the linear

dependence of the longest r"s on n is likely to holdin a large range of n-values, because the 17o-propor-tional term in top is the same in the diffusional and non-diffusional limits. Consequently, no lower limit willbe set to the value of n in the hydrodynamic theory,although eqs. (24) and (24’) may lose their validityat low values of n.

There remains to write the diffusion equation in thepresence of an external field, here a shearing gradient.The stochastic theory has not yet been developed sofar that the overall rotation of the molecule can be

separated from deformational modes, as requiredin particular for describing the behaviour of fairlyrigid and/or medium sized molecules. We thus feeljustified in adding in the force that governs the relaxa-tion of the normal modes the term [8, 9] :

instead of - Ol which we use when G = 0 and

0161n = 0. One is thus directly led to eq. (2).We should like to emphasize once more that, when

we consider ilo-independent internal viscosity, eq. (23)holds only for the model that neglects local diffusionalmovements. The effect of these movements will,however, be taken in consideration when discussingthe hydrodynamic properties, in particular their

dependence on the solvent viscosity ilo, the molecularweight M and the frequency OJ.

2.2 HYDRODYNAMIC PROPERTIES. - Stress tensor

symmetry follows from eqs. (5) and (23). The condi-tion of stress tensor symmetry indeed reads :

Using eq. (23), this reduces to :

Using

and interverting the order of the operations E and( > we see that condition (27) is rigorously obeyedif expression (5) is substituted for Q. The proof wouldhold even if R’ =1= 0, but is restricted to the limitingmodel considered here which accounts for only onekind of local process (i.e., either diffusional or non-diffusional).Now, at low gradient, the procedure of using a

distribution function ’P of the beads and considering

364

a mean Brownian current in itself leads [3] to an

overall rotational movement of the particle with :

We may note in this connection that, once the consi-deration of the overall velocity Q is accepted at lowbut non-zero internal viscosity, the eq. (7) for tan ais a rigorous consequence of (29).

This follows from the fact that at not too high inter-nal viscosity, when the polarizability tensor of thesub-chain obeys some simple conditions, tan a is

given rigorously by [8] :

Only the G proportional term ins is required tocalculate (30) ; thus we may use (29) and therefore :

which is identical to (7).For kinetically flexible chains at higher gradients

the averaging procedure involved in the considerationof an overall rotational movement of the particle maybe more drastic, as already pointed out in section 1.An attempt by Booij and Van Wiechen [36] to avoidsuch averaging procedures led to inconsistencies (e.g.,their eqs. (A. 9) and (A. 10) apparently do not fulfillthe condition that the right hand members are unevenfunctions of the velocity gradient G).At those higher gradients we shall retain the

approximations of reference [10]. In particular, weset :

Using eqs. (33), (34) and (28), eq. (5) reduces to :

On the other hand, we neglect the change of hydro-dynamic interactions between the beads with increas-ing values of the shearing gradient G and assume the

mean chain extension to remain small enough forthe elastic forces in the sub-chains to depend linearlyon their extension. These approximations result inthe various measurable quantities depending, at

high molecular weight, only on the two dimensionlessparameters P = GM[17]o 17o/RT and n. However, theyprevent the theory from being valid at very highvalues of the reduced gradient P. Even more stringentlimitations will be outlined later in this section.The previously derived [10] implicit equation for

c = G -’ 0 ), as well as the equations for the intrin-sic viscosity, mean chain extension, extinction angle,and birefringence as a function of the reduced gra-dient # are still valid when expressed in terms ofthe pp vp’s. However, eq. (23) must now replace theformerly used expression of the pP vp’s :

in which

where Ap is the pth eigenvalue of Rouse or of Zimmand M is the internal viscosity parameter used before.In reference [10] the reduced internal viscosity para-meter XRT/(M[17]o 170) was set equal to n. Here wecall this quantity n’ (see eq. (23’)), to make the distinc-tion with the parameter n used in the present paper.Note that :

Aside from the fact that the theory is now consideredbased on a limiting model, as stated at the beginningof section 2, the main difference between its presentand previous versions is that n’ is molecular-weightdependent whereas n does not depend on M. On theother hand, n’ is a function of the index p of the modeand n does not depend on p.

However, since the various expressions used in thetheory are unchanged in terms of the p p v p’s somesimilarity between the results is to be expected when,say, the values of pi vl given by eqs. (23) and (23’)are equal to each other, i.e., for :

Several of the values of n which have been used in thenumerical calculations correspond approximately tovalues of n’ considered previously. Thus, for exam-ple, n=0,222 ; 0,444; 0,667; 0,889 ; 1,j33 corres-

pond approximately to n’ = 5 ; 10 ; 15 ; 20 ; 30, etc...The curves of figures 2, 3 and 5 of the followingsections indeed show similarities with the corres-

ponding curves of figures 2, 4 and 6 of reference [10].

365

FIG. 2. - Rotational velocity of the molecule. Value of 8 = G - 1 0 > as a function of reduced velocity gradient for various valuesof the internal viscosity parameter n.

"

In practice, to derive the new results, we havesimply replaced the coefficient :

by :

in the equations of reference [10]. These equationsare rewritten in the appendix to account for condi-tion (23), i.e., using eq. (39). We also choose here thepositive sense of rotation such that at low gradient 0 > = G/2 (instead of 0 > G/2) ; thus s

is positive.In the various summations that must be carried

out to solve the implicit equation for s and to calculatethe measurable quantities, we have put N = 800.Similar calculations have been carried out choosingN = 100. No significant difference appeared providedn did not exceed 10. All results are for strong hydro-dynamic interactions between the beads.

In sections 3 to 5 we shall compare the theory withthe measurements, using a set of theoretical curveswhich summarize the results.

2.3 ROTATIONAL VELOCITY AND MEAN CHAIN

EXTENSION. THE C-DISCONTINUITY. - The parameters = G - ’ 0 > shows a discontinuity already des-cribed [10] (see also Figs. 2 to 5). Indeed, whenn nl, where nl is a limiting value

s - 0 when fl - 00. On the contrary, for n > nl,s tends to a finite limit when fl - oo, the limit

approaching 1 (i.e., ( Sl ) = G/2) when the internalviscosity is large. Furthermore, for n nl the product

sfl’ remains finite; thus 0 > -+ 0 when oo,as pointed out in the above reference.As already discussed [10], these results are conse-

quences of the approximations made : at small n’sthere is apparent extension of the chains to infinity,due to the linear restoring forces being used, whereasat large n’s the particles behave as rigid spheres dueto an average rotational velocity 0 > being used.Fluctuations must obviously produce departures fromthe behaviour thus described by the theory in theabove two limiting cases.

It is also interesting to note that the s-discontinuityis exhibited at a value n, for n which is much smallerthan the value n, at which the transition betweenthe orientational and the deformational effect occursat very small G’s. An order of magnitude of nt is

given by the condition that the same value of tan a(the initial slope of the extinction angle curve) beobtained whether one uses equations that are validat small or at large internal viscosity, respectively.From results recalled in section 1 we obtain nt - 6and therefore :

The effect of the 8-discontinuity has already beenillustrated [10] for the mean chain-extension

where ( L 2 > and ( L 0 I > are the-mean square end-to-end distances for an arbitrary P and for P = 0,respectively. The plot of the mean chain extensionas a function of P for various n’s with condition (23)being fulfilled is shown in figure 3.

366

FIG. 3. - Mean chain extension produced by the velocity held asa function of reduced velocity gradient j8 for various values of

the internal viscosity parameter n.

2.4 LIMITS OF VALIDITY OF THE THEORY. DIS-

CUSSION OF THE 9-DISCONTINUITY. - In a steadyflow, the molecules are stretched and compressedat a frequency coo = 2 0 > = 2 EG. Using Zimm’slongest relaxation time il (Eq. (9)), one obtains :

. Figure 4 is a plot of the chain extension

against coo il for various n’s. The B-discontinuity ishere exhibited in a dramatic manner. In particular,we see that for low n’s the frequency roo tends to zerowhen the chain extension increases indefinitely,whereas for high n’s the extension remains finitewhen coo increases indefinitely.The results shown in figure 4 now permit one to

consider in a more precise way the limitations of thetheory for chains of not too high internal viscosity.

i) If the chain extension is large, the distributionfunction T of the beads, the relaxation times them-selves, and of course Q and Sl z, are likely to losetheir significance. Fluctuations not accounted for

by the theory may then become so important that

the shape and the fate of individual molecules in theflow become relevant. For example, the situatioÏtdescribed above, in which a molecule of low internalviscosity is extended in the direction of flow at veryhigh shear rates is obviously an instable one.

ii) On the other hand, according to figure 4, thereexists the possibility that for chains of intermediatevalues of internal viscosity (n - 1) the chain extensionremains small whereas the product Wo ’t 1 becomesmuch larger than unity. Also in this case the use of( Q > is less straightforward.Thus, for the theory to remain valid, it seems safe

to require both that the chain extension remainssmaller than unity, and that mo T, does not exceeda few units. These conditions are seen from figure 4to require that fl remains smaller than a limit of afew units. Consider for instance the dotted curves D1and D2 which correspond to fl = 2.5 and fl = 7respectively. These special values of # will play a rolein the following sections. At p = 2.5, both the chainextension and WO’t1 remain smaller than unity.From the results depicted in figure 4, we can even statethat, for this to be so, fl should not exceed by muchthe value 2.5. Thus, the theory is likely to be somewhatless quantitative at fl = 7. Furthermore, it is clearlyimpossible to assert the 8-discontinuity before over-simplifications introduced by the theory at fl-valueslarger than a few units are removed.

3. Non-Newtonian flow. - Figure 5 shows that atconstant value of the reduced gradient # the non-Newtonian effect as measured by E = 1 - [17]/[17]0is equal to zero both for zero internal viscosity(Zimm-limit) and for infinite internal viscosity (Eins-tein-limit). We may recall in this instance that neitherEinstein’s rigid sphere [37] nor the Rouse-Zimm bead-and-spring model yield non-Newtonian behaviour.However, for intermediate values of n, E goes herethrough a maximum at given fl (although the positionof this maximum in the n-scale depends on the valuechosen for fl, see Fig. 5). This was the basis of our pre-vious interpretation [38] of wolff s measurements forPS in solvents of various viscosities [39].

Since diffusional processes can become predominantat low 17o’S departure from the behaviour depicted infigure 5 may result and Zimm-behaviour may beobserved at both ends of the rio-scale. Internal vis-cosity would then be manifested only at intermediatevalues of ilo.On the other hand, the parameter n is not a function

of the product [17]0 rio as in the previous derivation [8]and is now merely proportional to 1701 (see eqs. (3)and (23)). This, however, is a minor difference since [I]ois constant for a given value of the chain-expansiona = ([?1]01[q]0,0)’I’, where [q]o,o is the value of [17]0 in a0-solvent.

z

The results of figure 5 will be compared with themeasurements of Wolff [39] for the PS fraction S2 ofmolecular weight M = 10’. Figure 6 represents the

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FIG. 4. - Mean chain extension produced by the velocity field in function of roo 11 for various values of the internal viscosity parameter n ;mo is the frequency at which the molecule is streched and compressed in the flow; Tj is Zimm’s longest relaxation time. Note that the

curve is multivalued at small n’s. The curves D1 and D2 correspond to p = 2.5 and P = 7, respectively.

FIG. 5. - Variation of relative intrinsic viscosity as a function of reduced velocity gradient for various values of the internal viscosityparameter n.

values of E7 (i.e., the values of E for P = 7) plottedagainst the chain expansion a. The signs 0, All

0 correspond to solvents for which no 1 cP,1 cP qo 10 cP, no > 10 cP, respectively, andshow reasonable alignment. The value of no foreach solvent is indicated near the representativepoint. In the mixed-solvents used by Wolff neither a

nor qo seems to be a determining factor and thecorresponding values of E have been discarded.The figure discloses the known increase [39] of the

non-Newtonian effect with increasing value of a.

In near 0-conditions the non-Newtonian effect istoo small and the values of 170 are too close to eachother for the results to be conclusive. On the other

368

FIG. 6. - Experimental values of E7, i.e., E = 1 - [t’/]/[t’/]o at

p = 7, for Wolff’s [39] polystyrene sample S2 of molecular weightM =10’ as a function of the expansion parameter a in 14 solvents.The viscosity tjo of each solvent measured in cP is indicated nearthe corresponding experimental point. The signs 0, D, 0 cor-respond to the solvents for which qo 1 cP; 1 cP ilo 10 cP ;ilo > 10 cP, respectively. For the values of the expansion para-meter 1.15 a 1.35, a dependence of E7 on 10 is disclosed.The dotted arrows are corrections for the chain expansion effect.

hand, an internal viscosity effect is exhibited at inter-mediate values of the chain expansion. Take for

example the range 1.15 a 1.35 in which thereare eight representative points and the followingviscosities in cP : 170 = 0.4 ; 0.56 ; 0.74 ; 1.83 ; 1.97 ;11 ; 20 ; 106. The E7 versus 170 curve can be smoothedby correcting the E-values for the chain-expansioneffect (dotted arrows in the figure). When 170 isdecreased from 106 cP to 0.4 cP an increase of E7is first observed, followed by a decrease. The theo-retical and experimental E7 versus p curves are com-pared in figure 7 in a semilogarithmic plot.The experimental points are those for which1.15 a 1.35. It has been assumed that the non-

vanishing contribution at very high values of 170’(E7)00, simply adds to the lo-dependent contribution.The n-scale and the 17o-scale correspond to the theo-retical and to the experimental curve, respectively ;they can be shifted with respect to each other inorder to achieve the best fit.

The results are compatible with the expected effectof difi’usional processes that would come into playat low 17o’S and would result in a decrease of E fasterthan described by the theoretical curve of figure 7.Superimposing the curves in figure 7, as indicated

FIG. 7. - Comparison of the results of figure 6 with the theorythat accounts for non-d;.ffl.-s;.on-.1 local processes. The dotted curve

E7(?10) is obtained from the experimental points of figure 6 forwhich 1.15 a 1.35. The full line E7(n) is theoretical and isobtained from the results of figure 5. The non-vanishing value ofE7(tlo) at very high ilo’s has been added to the E7(n) values. Bothplots are semi-logarithmic and the two curves have been shiftedhorizontally with respect to each other, in order to determine thevalue of nt’/o. The discrepancy between theory and experimentswhich is observed at small ilo’s is attributed to the effect of local

diffusional processes not accounted for by the theory.

above, yields an estimate of the nilo value that cha-racterizes the non-diffusional processes. Since n = 1when yyo - 0.3 cP, we conclude that for Wolff’s PSsample S2 :

This value must be considered as a rough estimate,due to arbitrariness involved in substracting (E7),,,,,to the numerous approximations already pointed outin the previous sections and to the fact that the chainsare not gaussian for 1.15 a 1.35.The Peterlin effect [40], i.e., the increase of viscosity

due to reduced shielding by hydrodynamic interactionsas the chain extension increases may be larger inhigh-viscosity solvents. Hence the possibility mustbe examined that this effect is responsible for thedecrease of E7 at high 17o-values. It seems, however,that this can be discarded, since from WolfTs resultsthere appears no direct relationship between the

viscosity-upturn effect and the value of 170.No influence of 170 on E7 is apparent for Wolff’s

PMM sample M1 of molecular weight

However, significant qo-variation has only been

achieved, in this case, at quite high a-values ’and thesample is described by Wolff as being highly poly-disperse. Thus, no final conclusion can be drawnfrom WolfTs measurements on PMM.

369

Remark. - We should like to point out that somany factors are involved that an tjo-dependence ofthe observed effects can be neither assessed nor dis-

proved for a given sample unless measurements in alarge number of solvents of comparable a-values andwidely differing r¡o’s are available.

Concerning chains of intermediate kinetic flexi-

bility, we should like to suggest that non-Newtonianflow measurements be carried out for cellulose in a

large number of solvents, since from the available

data this polymer might exhibit a typical internal

viscosity effect [41].

4. Viscoelastic properties. - From remarks pre-sented in the Introduction, it follows that a study ofthe frequency dependence of the intrinsic viscosityM = M - i[n"], including the effect of the internalviscosity, requires that the frequency dependence ofthe latter parameter be taken into account. This,in turn, would require a knowledge of how diffusional

FIG. 8. - Real part of relative intrinsic viscosity as a function of reduced frequency for various values of the internal viscosity parameter n.

FIG. 9. - Imaginary part of relative intrinsic viscosity as a function of reduced frequency for various values of the internal viscosityparameter n.

370

and non-diffusional local processes are competingand of how their competition depends on frequency.In figures 8 and 9 the quantities [?I’]/[?I]o and M/Moare plotted against coM[?I]o qo/RT for various n’s,as deduced from eqs. (A. 6) and (A. 7) of the appendix,assuming that n does not depend on frequency, i.e.,neglecting diffusional processes..A way to explain high frequency behaviour that

seems appealing (see sections 7.5 to 7.7) consists inassuming that the response of the chain to the rapidestablishment of a small-amplitude velocity gradientfirst involves relative displacement of fairly largegroups of monomers. From this picture, ?10-indepen-dent internal viscosity effects must decrease with

increasing wand become negligible for :

with m1- - Tp1and p - MIMe, where MG is the

average molecular weight of these groups. Accord-ing to the estimate of section 7.6, from measurementsin aroclor M1- - 5 000 for polystyrene. Thus, for amolecular weight M = 106, we would expect :

i.e., WG M[tl]o r¡o/RT ’" 104. At lower molecular

weights, the vanishing of the 170- independent internalviscosity would of course occur at lower values of thereduced frequency.

Figure 8 shows that even at n = 0.3 it would behard to detect an effect of the qo-independent internalviscosity on [?I’]. From the estimate of n17 0 *for poly-styrene (section 3), n = 0.3 would require qo = 1 cP,an ilo-value for which diffusional processes wouldcome into play, further reducing the effect of qo-independent internal viscosity. For values of 170larger than one Poise, as used by Massa, Schragand Ferry [30], no ?10-independent internal viscosityeffect can be expected in polystyrene.

It would be of interest to know whether an 170-independent internal viscosity is disclosed in visco-elastic measurements by polymer-solvent systemsfor which there is indication of higher internal vis-cosity from other methods.

5. Flow birefringence. - The discussion will mainlybe devoted to the behaviour of the extinction angle x.The birefringence is considered briefly at the end ofthe section.

The x versus P curves, as deduced from eq. (49)of reference [10] (also, appendix, eq. (A.4)), are

shown in figure 10 as well as in figure 11, using anenlarged scale in the latter one. The curves correspond-ing to different values of n nearly cross each other ata #-value close to 2.5. The theory is, however, quanti-tatively less reliable for fl-values larger than 2.5 dueto the effects of the B-discontinuity (see section 2).On the other hand, Q cannot be averaged over theconfigurations at high internal viscosity. Thus eq. (A. 4)becomes invalid at high values of n. This can also beseen in figure 11 according to which the initial slopetan a of the x versus G curve increases indefinitelywhen n is increased, whereas it must reach an upperlimit of the order of 5 times its value for n = 0 (seesection 2).

Useful conclusions may be drawn when tan a

is measured in solvents of various viscosities. Thedouble gradient (G) and concentration (c) extrapola-tion is particularly difficult when the birefringence. isof the order of a few 10-8. The first experiments [6]carried out in solvents of various viscosities for

polystyrene, and the results obtained later [26, 42]for the same polymer are therefore not entirelyreliable.

Less flexible molecules usually exhibit a largerbirefringence. Although such molecules are not themain concern of the present study, we shall brieflyconsider the results for desoxyribonucleic acid (DNA),

FIG. 10. - Extinction angle of the flow birefringence as a function of reduced gradient fJ for various values of the internal viscosityparameter n.

371

FIG. 11. - Extinction angle of the flow birefringence on enlarged #-scale.

because measurements by various techniques are

available and their discussion will bring up a pointof interest.

5.1 DNA. EXTINCTION ANGLE. - It has beenmentioned in section 1 that the flow birefringencemeasurements of Schwander and Cerf (M = 6 x 106)and of Leray (M = 6 x 106) show passage fromorientational to deformational effect when 170 isincreased.On the other hand, from oscillatory flow birefrin-

gence measurements, Wilkinson and Thurston [43]reported rigid particle behaviour for a DNA sampleof molecular weight M = 6 x 105 and flexible chainbehaviour (zero internal viscosity) for a sample ofmolecular weight M = 5 x 106 dissolved in water.

It is not surprising that samples of DNA of mole-cular weight around half a million exhibit nearlyrigid particle behaviour. On the other hand, it ishard to follow Wilkinson and Thurston in consideringit meaningful that a model consisting of 10 freelyjointed gaussian sub-chains [43] will be quantitativelysuccessful for the same substance when M is ten times

larger. Precisely because for M = 6 x 105 rigid-particle behaviour is observed, one would expectthat some internal viscosity effect is still present forthe high molecular weight sample studied by Wilkinsonand Thurston, even though DNA is not an examplewhich is likely to be well fitted by a bead-and-springmodel.

In our work, we admitted that theories based ona sub-chain model are reliable only when it is not

necessary to fix the length of the sub-chain, i.e.,the number N of sub-chains per molecule. (Thetheory is therefore limited to high molecular weight.)

It is of course of interest to investigate the effect offixing the value of N, especially with regard to apply-ing the theory at lower molecular weights. The

results, however, should be manipulated with care,the more so because diminishing N at zero internalviscosity is known to result in qualitatively the sameeffect as increasing the internal viscosity parameter

at constant N [44] (except, of course, as far as theeffect of varying qo is concerned).

Thus, we consider Wilkinson and Thurston’s ana-lysis as a clear illustration of the risks involved inattributing quantitative validity to theories in whichthe number N of sub-chains is an adjustable parameter.For DNA it is also necessary to keep in mind that

the properties of the sample studied may depend onthe methods of preparation and purification.

5 . 2 POLYSTYRENE. EXTINCTION ANGLE. - Return-

ing to molecules that exhibit a small birefringence ina shear gradient we shall restrict the discussion tothose results which are presented in effecting a reduc-tion with respect to concentration, using the variable

as proposed by Peterlin and Signer [45] ; ’IN is theNewtonian zero shear viscosity and c the concentra-tion. The parameter fiN is a reduced shear rate in thesame sense as flo = (lim #N)c-o. The considerationof the variable /3N carries its share of arbitrariness,as does any method of extrapolating the results toc = 0, or of avoiding such extrapolation.For polystyrene, we shall consider the results of

Janeschitz-Kriegl [46], who studied fractions of mole-cular weight ranging from 0.388 x 106 to 1.59 x 101.All the measurements of this author furnish :tan a N 0.2/3N rather than the theoretical value0.1 /3N’ which is obtained for zero internal viscosity.Measurements for solvents of comparable a-values(see section 3) and widely different viscosities are,

however, not available. The results show that theinternal viscosity is not large for polystyrene, butthey are not incompatible with the estimate of n170made in section 3.

5.3 POLYMETHYLMETHACRYLATE. EXTINCTIONANGLE. - According to Tsvetkov and Budtov [47], [48],PMM shows a typical internal viscosity effect whenstudied in solvents of various viscosities. The effectis large enough for the results in the solvents of lowest

372

viscosity to resemble the behaviour expected whenorientation of the particles prevails. Thus n17 0 mustbe in this case of the order of several centipoise.

These measurements have been criticized by Jane-schitz-Kriegl [46] on the ground that Budtov [49]experienced considerable difficulties in extrapolating Xto zero concentration and made some assumptionsabout the shape of the x-curve. However, in refe-rence [47] and in a review by Tsvetkov [50] data fora PMM fraction obtained by bulk radical polymeriza-tion (M = 2.3 x 106) are given in sufficient detailin four solvents to permit using a PN plot. The resultsare compared with the theoretical cot 2 x versus fcurves, plotted on a double logarithmic scale (seeFig. 12). In the linear part of these curves, at lowfl-values, cot 2 x does not differ from 2 tan a and is,therefore, correctly determined using 0 = G/2 (seesection 2.2). In this range no ad hoc character ofthe theory can be invoked, although at higher#-values the use of the overall rotational velocity ofthe molecule involves approximations.

It is seen that tan a is increased by a factor ofabout 6 when the solvent viscosity is lowered by afactor of 50. If there were no internal viscosity effectno change in the value of tan a should be observed.

In the most viscous solvent (tetrabromethane at7°C), the Peterlin-Signer concentration-reductiondoes not work and only the results for the most dilute

solution have been plotted in figure 12. The discre-

pancy observed in this solvent with the theoreticaln = 0 behaviour may be due to :

i) the imperfect concentration-reduction,ii) non-gaussian chain statistics,

iii) a small 17o-proportional internal viscosity effect.

On the other hand, it is quite unlikely that thedependence of tan a on the nature of the solvent canbe attributed to a change in solvent power. Indeed,the value of Mo decreases regularly by no more thana factor 2 when qo is decreased from 15.3 cP to 0.3 cP.If the solvent power had an effect, one would ratherexpect that tan a would increase in better solvents.

In order to show the influence of polydispersity,a cot 2 x versus P plot of results obtained by Savvonand Grigor’eva [51] for polydisperse PS samples is

given in figure 13. The known increase of cot 2 xat low PN’S is exhibited. However, if the shift of thecurves of figure 12 were produced by polydispersityit would not depend on qo.

In conclusion, even if orientation is questionablefor PMM at low ilo’s and low PN’S, the results ofTsvetkov and Budtov strongly suggest that their

samples exhibit a large 17o-independent internal vis-cosity effect. Of course, it would be helpful to have thestereoregularity specified in any such experiment,the result of which should also be compared with

FIG. 12. - Extinction angle of the flow birefringence : comparison of theoretical cot 2 x versus P curves for various n’s with measurementsof Tsvetkov and Budtov [47], [50] for a polymethylmethacrylate fraction obtained by bulk radical polymerization (M = 2.3 x 106). Resultsin four solvents of various viscosities are shown. The following concentrations given in g/100 cm’ were used : in tetrabromo-ethane at7 °C (ilo = 15.3 cP); (*) c = 0.03 : in a mixture of bromoform and tetrabromo-ethane at 21 °C (ilo = 3 cP); (6) 0.23, (0) 0.18, (Q) 0.12;in methyl-ethyl-ketone at 21 °C (qo = 0.4 cP) : (Ic) 0.495. (f!) 0.405, (4) 0.325, (,A) 0.275; in acetone at 21 °C (ilo = 0.3 cP) : (A) 0.98,

0.81, (§) 0.67, ( zl) 0.55, (A) 0.28.

373

FIG. 13. - Extinction angle of the flow birefringence. The effect of polydispersity is shown using the same representation as in figure 12.The measurements are by Savvon and Grigor’eva [51] for polystyrene samples in benzene. The mean molecular weights as determinedby viscosity measurements are of the order of 3 x 106. Polydispersed samples : (0) and (0) ; mixtures of samples of different molecular

weights : (0) and (A).

those of other measurements, particularly the studyof the viscoelastic properties and of the scatteringof light.

5.4 BIREFRINGENCE. - The equation used here forthe G-dependence of the birefringence An (eq. (50)of reference [10] and appendix eq. (A. 5)) is valid whenthe optical internal field is neglected. This approxima-tion is acceptable only when the mean index ofrefraction of the solvated particle is close to that ofthe solvent and An then is very small. Thus, theresults of figure 14, in which #[An]/[An]o is plottedagainst fl for various values of n, have mainly illus-trative character; [An] is the intrinsic birefringencefor an arbitraI°y fl ; [An]o is its value at small fl (i.e.,at such fl’s that [An] is proportional to fl) :

All theoretical curves have the slope unity at small fl.A curvature would not necessarily be easy to detectin the range of f3-values for which the approximationsmade in the theory are not too drastic.

6. Comments. - 6.1 Bazua and Williams [52]have recently used our procedure of considering theoverall rotational velocity Q of the flexible chain.In their work, the internal viscosity terms that appearin the ip’s and in tan a are molecular weight indepen-dent, as in our first proposal [7]. This model leads toa negligible internal viscosity effect for kineticallyflexible chains at high molecular weight.The internal viscosity term is also molecular

weight independent in the expression of tan a as

calculated from Kuhn and Kuhn’s dumbbell model [5](see reference [53]).De Gennes [54] has recently presented an argument

intended to prove that internal viscosity is negligibleat high molecular weight. The basis, however, isthe model of reference [7], which we prefered to

abandon in favour of the normal mode approach ;(de Gennes’s results agree with those of [7]).On the contrary, from our second model [8, 10],

which is used in the present paper, internal viscosityterms that are proportional to M[illo are obtainedwhen condition (23) is accounted for (see eqs. (6), (8)and (9)).

6.2 Internal viscosity laws presenting the latterfeature have already been proposed in the appendix

374

FIG. 14. - Magnitude of the flow birefringence; #[An]/[An]o is

plotted against the reduced gradient fl for various values of theinternal viscosity parameter fl. All curves are tangent to the dotted

straight line.

to reference [7] and by Budtov and Gotlieb [55].However, the result of reference [55] has surprisingfeatures, since the conclusion holds even if there isno friction associated with hinging at the sub-chainjunctions. One would perhaps expect that in thiscase Budtov and Gotlieb’s model should exhibit abehaviour close to that of a freely jointed chain andthat there should be no important internal viscosityeffect at high molecular weight.

In our model, hindered hinging at the junctions istaken care of by associating internal friction withcollective deformational modes.

6.3 The molecular weight dependence found herefor the internal viscosity terms holds only at highmolecular weights (when the results do not dependon the length chosen for the sub-chains) and in thelimiting case in which diB’usional local processes arenot accounted for. The question remains open whetherthe competition between diffusional and non-diffu-sional local processes would lead to molecular weightdependence of the internal viscosity parameter n.

The experiments tell us that for low molecular

weight (below a few 104) rigid particle behaviouris approached [1, 42]. Also, they offer examples,

which have been discussed herein, of qo-independentinternal viscosity effects at high molecular weight.The data are, however, too scarce for a final conclusionto be drawn on the effect of the molecular weightin the full range of M-values.

Accounting for the competition between diffusionaland non-diffusional local processes could also leadto a p-dependence of the Tp’s differing from that usedhere.

6.4 Stockmayer [56] proposes a simple calculationshowing in particular that a bead-and-spring modelby itself discloses no internal viscosity. This wasevident a priori. Also, the criticisms about internalviscosity contained in reference [56] are met to a largeextent by our preliminary note of 1971 [14] and bythe present paper.

7. Conclusions and conjectures. - 7.1 The sto-

chastic approach justifies our model in which internalviscosity coefficients are associated with the normalmodes. When only one type of local motions, diffu-sional or non-diffusional, is accounted for, the sto-chastic theory imposes a condition upon the internalviscosity coefficients. Stress tensor symmetry followsfrom this condition, when the rotational velocityis chosen equal to that of the rigidified chain. Therio-independent internal viscosity can furthermore beexpressed in terms of local dynamical characteristics(eqs. (24) and (24’)).

7.2 Reliable experimental results confirm that therip-independent internal viscosity, hence local dyna-mics, may be manifested in the longest relaxationtimes.

7.3 Application of the model to high frequencybehaviour using eq. (12) has no meaningful founda-tion, unless a justification for this equation can befound that considers the concept of an ?I 0-proportionalinternal viscosity. Thus, the experimental observa-tion that the high frequency limit M is nearlymolecular weight independent remains unexplainedin current theories based on the bead-and-springmodel.

7.4 When qo-independent internal viscosity is exhi-bited, quite precise conditions are imposed for theBrownian motion of the chain. It is required that localnon-diffusional (possibly cooperative) segmental rear-rangements contribute to the renewal of conformationsover a time scale of the order of the longest relaxationtime ’t 1 .However, the qo-independent internal viscosity

must disappear at high frequency and this requiresthat for large values of ev the behaviour be controlledby diffusional processes.

7. 5 A picture of the response of the chain to ahigh frequency shearing gradient, or to the rapidestablishment of a small amplitude velocity gradient G,can now be proposed that renders the two conditionsof 7.4 compatible.

375

Suppose the rapid establishment of a small cons-tant shearing gradient G first involves relative dis-placement of fairly large groups of monomers. In thispicture, local deformations would first occur mainlyin segments connecting these groups and in whichthe configuration is favourable to deformation. Thisnon-Rousian process would involve mainly diffusionalmovements because of the size of the moving groups,but would result in a non-steady state conformation.We may note that the concept of collective modes thatare generated by a cascade of basic local movementsis partially abandoned here, when considering veryfast deformations.

The establishment of the steady state would needmedium range rearrangements of the chain, involvingthe local diffusional and non-diffusional movements,and would require that the velocity field be main-tained over a time much larger than T p with

where MZ7 is the average molecular weight of themoving group. Thus, the ilo-independent internal

viscosity would become negligible at frequenciesmuch lower than the value :

7.6 According to the picture of 7.5, we nowassume that the high frequency limiting intrinsic vis-cosity [q’] is determined to first approximation bythe motion of (short lived) groups of monomers, eachgroup moving quasi-freely relative to other parts ofthe chain. With this assumption we arrive at a straight-forward interpretation of the independence of [tl’] .and the molecular weight M, since the limiting intrinsicviscosity would be roughly equal to the intrinsic

viscosity []c of the average mobile cluster :

From the results of Massa, Schrag and Ferry [30],the molecular weight of the average group wouldthen amount to approximately 5 000 for polystyrenein aroclor. Furthermore, according to this picture,the lower M, the lower the value of o-)-r, at which highfrequency behaviour should set in, in agreementwith the experimental results [30].

If this suggestion is supported by future work, thetheory of the 17o-proportional internal viscosity mayhave to be revised to account for it.

7.7 In summary, the proposed picture of the

dynamics of a linear array of Z carbon-carbon linkswith hindered rotation makes use of several of the

many possibilities offered by the - 3Z degrees offreedom of the system.At lower frequencies, diffusional and non-diffu-

sional local processes each play their part. The relativeimportance of both these contributions to Rouse-Zimm modes must depend on the nature of the poly-mer, the solvent viscosity and the molecular weight.

In a high frequency shearing field, intra-chain

mobility of frozen clusters of monomers may prove tobe of importance (cluster motion). Normal-modedescription would be recovered at intermediate fre-quencies, due to competition between cluster motionand the local motions. We have suggested collectivecluster motion in polymeric chains at high frequencyfor the first time in reference [22].

7.8 One may finally expect that internal viscositycan manifest itself in situations of practical impor-tance. Kohn [57] has already suggested that dragreduction by polymers might be related with internalviscosity and used high frequency viscoelastic datato support his view. This, however, amounts to

considering qo-proportional internal viscosity, forwhich current theories are not likely to be reliable.

In our opinion it would be of interest to investigatethe possible relation of drag reduction, especiallyof the corresponding onset gradient, with 110-inde-pendent internal viscosity. This question may seemrelevant since, from the data collected by Kohn,the onset of drag reduction appears to occur generallyat gradients wherein wo (as defined in section 2.4)is not much larger than T

Appendix. - Eqs. (A. 1) to (A. 5) are derived fromthe diffusion eq. (2) using the approximations ofsection 2. Eqs. (A.I) to (A. 5) agree with those ofreference [10] when eq. (39) is accounted for and sis changed into - e, in accordance with the positivesense of rotation chosen in section 2. A printing errorhas been corrected in eq. (49) of reference [10] (eq. A. 4below).The implicit equation for e reads :

The following expressions are obtained for themean chain extension :

for the intrinsic viscosity :

376

for the extinction angle (note that the angles 2 Xused in reference [10] and in the present article arecomplementary) :

and for the birefringence :

According to sections 1 to 4, the study of the visco-elastic behaviour requires that the frequency depen-dence of the internal viscosity parameter n be takeninto account. If, however, n is considered independentof frequency, as is permissible for WT 1 1, the

equations for [17’] and [17"] (the frequency-dependentreal and imaginary parts of [il]) are quite easy to derivefrom the diffusion equation (eq. (2)). Indeed, at lowgradient s = 0.5 and the implicit equation for e

(eq. A.1) need not be solved. Inserting eq. (6) intothe expressions thus derived by Peterlin [28], weobtain :

where

The expressions to be summed from 1 to N in

eqs. (A. 1) to (A. 7) may be transformed into functionsof therp I s and of n, using eq. (37).Although the use of eqs. (A. 6) and (A. 7) is not

permissible at high frequency (for reasons indicatedabove), it is of interest to note that, if n is consideredconstant, eq. (A. 6) yields :

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