Frequency Dependence of Intrinsic Stress and Birefringence Tensor of Bead/Spring Model of Polymer Solutions

The recently obtained complete solution of the simultaneous diagonalization of matrices H A and H in the hydrodynamic diffusion equation has basically changed the diagonal values vp of the symmetric matrix H of hydrodynamic interaction between all the beads of the elastic random coil model of the isolated macromolecule in solution. Since these values enter explicitly the expressions for the intrinsic stress and refractive index tensor in an alternating flow field if based on the concept of internal viscosity of the model one had to recalculate all values obtained formerly by using the then generally accepted erroneous set of vp data. The new vp equal unity independent of p while the old values were larger than 1 for small p and smaller for large p. Hence their too large contributions in the former range are partially compensated by their too small contributions in the latter region. As a consequence in the whole range investigated, between 3 and 300 chain links, the differences in rheological and rheooptical effects are relatively small, up to a factor of 2, although at higher link number the differences tend to grow with the logarithm of this number.


Introduction
difference is tn th e diago nal elements vp of th e te nsor (2) The correct simultaneous diagonalization [lJl of H A and H matrices in Zimm's hyd rodynam ic equati on [2] for the ideally flexibl e nec klace mod el of rand omly coiled isolated linear macromolec ule in laminar fl ow ma kes possible a more adequate calc ulation of intrinsic stress te nsor in all those cases where the coil is not yet noticeably deformed by th e flow. Such a zero gradient case includes the freq ue ncy depend ence of viscosity [2], s hear modulus, shear birefringe nce, normal stress difference, and acousti c birefrin ge nce but not the gradi en t de penden ce of these effec ts.
which turn s out to be th e unity te nsor with all vp mak es th e diagonal e lements of

Thi s
The main c ha nge introd uced by th e new soluti on as compared with th e old er incomplete soluti ons [3][4][5][6][7][8] is not in the eigenvalues Ap of H A Q -J H A Q = A (1) which were already calcul ated correctly in rece nt papers [6][7][8] and even tabulated for Z between 1 and 15 and h* = (3/7r) 112 ah / bO = .01, .1, .2, .3, [6] and forZ = 250, h* = .3 and Z = 300, h* = 0.4 [1]. Here ah is the hydrodynamic radius of the bead, Z + 1 is the number of beads , Z is the number and bo the root mean square length of the links. The ag ree wit h th ose of A , i.e. /-tp = Ap. Here Q is the transformation matrix of the original 3(Z + 1) dim ensional vector r of bead coordinates to that of dimensionless normal cOOl·d inates u r = bo Q u (4) and Q1" its transpose. Equation 3 completely differs from the ori gin a l e stim ate [3] that in first approximation the diagonal ele ments of the matrix M equal the eigenvalues ApR of th e Rouse model with va n ishing hydrodynamic interac tion , i. e. /-t~~ = ApR = 4 sin 2 [p7T / 2(Z + 1)] (5) and hence (6)  The subscripts Z and R refer to Zimm and Rouse [9] mod el, respec tively. Such a n estimate was based [3] on the s upposition that the tran sformation matrix Q c hanges so little by the introduc tion of hydrod ynamic interac tion that M remains practically th e same as in th e free draining case, i. e. M = A R · The diago nal e leme nts Vp for Z = 100 a nd h * = 0 (Rouse), . 1, .2 , .3, and .4 (Zimm) a re plotted in figure 1 a nd the eigenv alues Ap coll ected in ta bl e 1. The old values of v~l) = ApZ / ApR are connec ted with a broke n lin e. They are partially s itua ted below and partiall y above Vp = 1. This reduces the differences be tw een the old and new values in quantities dependent on Vp. They show up in the excess s tress te nsor as soon as internal viscosity, defin ed by the frictional parameter cp, is expli c itly cons ide re d . The very large old values V p for s mall p do not matter very much because in all formulae they a re multiplied by very s mall values CPP = pcp / Z . Since the differences be tween old and new vp increase with h * , in that whi c h follows, the comparison of calc ulated effec ts will be mainly done for h * = .4-, i.e., for a very large hyd rod ynamic interac tion. The differences are smalle r for smaller h * and of course disappear for the free draining coil with h * = 0 where ApZ = ApR' The new values J.Lp a nd Vp do not e nter the conven tional expression s for the intrin sic s tress or birefringence tensor of pe rfectly flexibl e nec kl ace mod e l so that no c hanges occur in these quantities. The s itua ti on is com ple tely different if one consid ers the effects of internal viscosit y whi c h depend on v p [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] . They will be most co ns picuous in th e values of viscosity corresponding to hi gh frequen cy fl ow fi eld a nd in the phase angle between the stress a nd strain ra te or between the birefringe nce and strain rate.
The inte rac tion tensor H depe nds on the inve rse intrabead dista nces I / rjk whic h ma kes the hyd rodynamic diffus ion eq uatio n intrins icall y non-linear. By replacing 1/ rjk with its average value (I/rjk) the te nsor H becomes a constan t whi c h ma kes the diffus ion equation lin ear and hence allows the introduc tion of normal coordinates according to eqs (1) to (4).
Such a procedure eliminates th e possibility of any realis ti c conside ration of the gradient dependence of a ny rheological or rheooptical effect because it does not tak e into account the c ha nge of shape of the random coil in flow which expands the molecule and he nce increases th e interbead distances rjk' Note also th a t by preaveraging over all angles be tween th e velocity and th e interbead vec tor the formulation of H as fun ction of I/rjk co mpletely evades the cons ideration of anisotropy of hydrod ynamic inte rac ti on whi c h b y itself yields a gradie nt dep ende nce of intrins ic viscosity [27] la rge e nough for explaining experime ntal data.
The general toleration of s uc h a profound modifi cation of hyd rodynamic interac tion by the replace ment of I / r) •. with its average (I / rjk) makes hard to und ers tand the almost ge neral objecti ons to the introduction of interna l viscosity as a res istance of the nec klace model against the defo rmationa l componen t of the normal e ige nmodes [28][29][30]. If one accepts th e rathe r questionable linearization of the hydrodynam ic diffus ion eq ua tion one has to accept also th e next ste p , i.e. , the co ncept of internal viscosity based on this linea rit y a nd its introduc tion in s uc h a ma nne r that th e ma th emati cal treatment re main s as s impl e as possible.
In th a t which foll ows the results of the ne w theory will be compared with those of the old o ne for h* = .4 a nd Z = 100 in th e whole freque ncy ran ge and th e depende nce of the limiting values for w = 00 on Z in the range be twee n Z = 3 a nd 300 a nd on h * in the ra nge be tw een . 1 and .4. In all cases the ra tio between th e inte rna l viscosity coeffi cie nt cp a nd th e fri ctiona l coeffi cie nt of th e bead f = 67TahYJs will be assumed constant, cpl! = 2. H e re YJ s is viscosit y of th e solve nt. The subscript s applies to the properties of the solve nt. The correspondin g non-subscripted qua ntiti es relate to solution .

Internal Viscosity
The concept of inte rnal viscosit y was introduced in order to ex press the ina bilit y of the ra ndomly coiled p ol yme r molec ul e to c hange ra pidly its s hape [3 1, 32, 10, ll]. Suc h c hanges occur during the rotation of the macromolecule in a fl ow with a rotational component, e.g., the lam in ar fl ow with tra nsverse gradient , when the individual segmen ts a re alte rn ati vely passing from th e direction of compress ion to the di rec tio n of extension a nd vice versa. The direc tions of maximum compression and extension of the volume e le me nt are in th e flow plane perp endic ular to eac h other. The rapidit y of c ha nge is give n by the transverse gradie nt whic h equ als twice th e a ngular velocity of the id eally fl exible coil whi c h rotates with th e volume element. The other case is th e oscillating flow fi eld where the oscillation frequ ency determines th e rapidity of c hange from co mpression to extension and vice versa .
In the limiting cases of zero gradient and zero frequency the deformation ina bility of th e macromolecule does not play a ny rol e. All the changes occur so slowly that the effects are identical for completely rigid a nd id eally fl exi ble coils if onl y the ir conformational distribu tions agree with each other. With inc reasing grad ie nt a nd/o r freq ue ncy, howeve r, th e time effec ts a re playing a graduall y increasing role. Th ey are maximum in the second New toni a n regime cOITesponding to it ~ 00 or w ~ 00.
The rigidity of th e mac romol ec ule can be ass igned to different properti es of the c hain. It can be caused by the e nergy b arrie r se parating th e ga uche a nd tra ns conform atio ns which makes a ny co nform ati onal c hange more time consuming than in the case of perfectl y soft model [10). Since th e height of the e nergy barrier is inde pendent of the viscosity of the solvent its relative effec t as measured by the ratio of internal fri ctional resistance cp caused by the barrier and the fri ctional resista nce f of the segment decreases with the viscosity of the solvent. The macromolecule acts as very rigid in a low viscosity liquid, e .g. , acetone with TIs = .322cPoise = 32 .2 mNs/m 3 , and as very fl exibl e in a high viscosity sol ve nt, e. g. , Arodor with TIs up to 100 Poise (= 1 kNs/m 3 ) a nd higher. Another cause of slow molec ular response to the rapid ly c hangi ng flow fi eld res ides with the conform ati onal restra ints of the chain which permit only an inte rc hange of ga uc he and trans conformations [20, 2 3, 24). With a lmost ri gid le ngth of vale ncy bonds this mean s that most cha nges of length and position of a ny chain segment re quire a mu ch la rge r segme nt displacement than formul a ted in th e id eall y ll exible nec kl ace model which does not cons ider a ny inhe rent I imitati on of bead moti on. Generall y an ax ial di splace me nt of th e segme nt requires also some lateral di splace ment and vice versa . As a consequence the res istance of beads to pos ition c hange is larger than assumed on the bas is of hydrod yna mi c radius a h.
The ratio of the so obtained coefficient cp to f is indepe ndent of solvent viscosity because cp andf are both proportional to 'Y/s. Their ratio just measures the ratio of true displacement to the minimum displacement explicitely considered in the diffusion equation. It seems to be close to 2 for vinyl polymers The effect of internal viscosity is formulated in the system of normal modes [10]. For the pth normal mode of deformation one has a res istan ce coefficie nt cp p/Z. Such a choice is reasonable for both origins of internal viscosity as just disc ussed. In the first case one can argue that the changes to comply with any deformational mode are linearly increasing with the number of c hain atoms between s ubsequent nodes, i.e., with Zip , because a conformational c hange takes place with equal probability at any of these chain atoms. This makes the resistance increase with p/Z. In the seco nd case th e displacement at lower modes ca n be achieved in many ways so that the ac tual le ngthe ning of displacement path is much less noticeable than at higher mod es where the conformational restrictions are soon beco ming of ovenvhelming importance .
One may argue that the whole concept of inte mal viscosity can be discarded because it is not based on so me s tri ctly fundam ental analysis of c hain dynamics . It was indeed introduced in a rather pragmatic ma nn e r which also permitted an easy mathe n:Iati cal treatme nt [10]. But it turn s out that all more detailed treatments of Brownian motion of beads or of correlation between th e motion of two or more beads [33][34][35] lead to some, often hidde n , statement of mol ec ular rigidity whic h is needed for the results of s uc h a study to reproduce the characteristic rheological features of polymer systems, e.g. , the non-vanishing limiting intrinsic vi scosit y at very high frequency [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. Such a s tate of affair seems more to support than to refute the concept of internal viscosity in spite of its more pragmatic than fundame ntal way of introduction.

The Distribution Function of the Beads
The continuity eq uation of the ideally flexible necklace in laminar flow which determines the distribution function t/J(ro, rl ' • • r z) reads -at/J/at. (7) Here Do = kT /f is the translational diffusion coeffi cient of the bead. By introduction of normal coordinates , eq (4), one transforms eq (7) into a system of Z + 1 partial differential Note that Vp in normal coordinates has th e dime nsion S-I. The distribution function of the coil is the product of all t/Jp (9) The functions t/Jp depend on the kind of flow fi eld v . The Oth mode does not show up in t/J because it represents a uniform translation of the whole necklace which does not affect t/J. The introduc tion of internal viscosity adds a viscous type resistance coefficient CPP = pcp/Z opposing the pth e igenmode of the true deformatio n rate of the coil. This rate is obtained by subraction of pure rotational velocity D X Up from the total deformation rate au p/ at. This yields an internal viscosity force [10,11] If one introduces this force in the pth diffusio n equation (eq (8)) one obtains after some rearrange men ts The distribution fun c tion t/Jp depends on th e kind of flow and on the angular velocity vertor !l.
for relatively soft mol ecules which rotate in phase with the volume element. This is the case with practicall y all conventional macromolecules if the degree of polymerization is so high tha t a truly random coil is formed. Very s hort chains, c hains with a great man y double bond s, ladder type and multiple s trand molecules, however, are more rigid and tend to rotate with a non-uniform angular velocity whic h de pe nds on orientation of the molecule. It is different from that of he volume ele me nt. Assymptotically, at very high rigidity and full y extended s hape of the macromolecule, it approaches that of rigid bodies, e.g., rods or ellipsoides. In that which follows only the case of practically und eform ed relatively soft coils with f! =y/2 will be considered .
In the stead y s tate flow with transverse gradient the pth eigenmode distribution function of the soft necklace reads The index p run s from 1 to Z. The value ° is excluded.
W ithout internal viscosity , CPP = 0, one has {3' p = (3p and one obtains the conventional distribution function. Note that eq (13) and the distribution function eq (6.1) in Ref. [1] refer to different flow fields: y(y,o,o) in the former and y (z,o,o) in the latter case.
In the oscillating flow field the gradient is a function of time (IS) The amplitu de Yo is so s mall that th e molec ul e re mains practically unde form ed so that th e zero gradi en t eigenvalues Ap and d iagonal ele ments lip, calc ulated for the coil a t rest, are applicable.

The Intrinsic Stress Tensor
In dyadic formulation the intrinsic stress tensor reads Here N is Avogadro numbe r, M is molec ular weight, and F is the vec tor of forces exerted by the beads of th e necklace model on the flowing liquid. One has in the space of normal coordinates The" type of laminar flow s hows up in v and n.
The b ilin ear coordinate averages in eq (16) can be derived from th e d iffusion eq uation, eq (11), by multipl icati on by 1;1'2 , I;pYJp, ... a nd integration ove r the whol e s pace.

Flow With Transverse Gradient
In The averages are c ut off beyond the lowest power it which is needed later in zero gradient expressions for intrinsic viscosity, normal stress difference and birefringence. These averages have to be inserted in the expression for the intrinsic stress tensor.
yielding the frequency dependence of intrinsic viscosity

MYl8
A rather similar but not ide ntical expression applies to intrinsic streaming birefringence Ll n* -Lln8 Lln* -Llns  The meaning of th e sy mbols ]/1 eqs (17) to (21) IS as follows: The s ums III and IV go to zero fo r va nis hin g inte rn al viscos it y cp ~ O. [n this limiting case th e intrin sic viscosity and s tream in g birefrin ge nce are propOl-tional to eac h oth e r.
He re R is th e gas constant, K is the rheoo pti cal coe ffi c ie nt , n is refra cti ve ind ex, 8 is the phase angle betw een th e fl ow gradi e nt and viscos ity or birefringe nce, a l and a2 are th e opti cal polariza bilities of th e lin k in th e directions para llel a nd perpe ndic ular to th e link respec tiv ely.
Both definitions of intrin s ic viscosity and birefrin ge nce in eqs (20) and (21)  It is important to note that as a consequ ence of the new terms III and IV, i.e. , (YJ1'2), caused by th e introduction of internal viscosity the ex press ion s for intrin sic viscosity and streaming birefringe nce, eq. (20) and (21)   The ofte n used real and imaginary part of intrinsic shear modulus y c->o cy (24) are plotted in figure 5. The difference betwee n the old a nd new theory is relatively small for Z = 100 but would be larger for Z = 300.
The consequence of the non-vanis hing second Newtonian viscosity, [7]] 00 f. constitute the main support of the theory. The intrins ic first normal s tress differe nce turn s out to be (25) [1 + C = A*e 2iwt + B* while the second normal s tress difference vanishes in all isolated necklace mod els with ideal elastic links inde pe ndent of internal viscosity. In app lying eq (25) one must not forget that the phase angle 01 of A * is dependent only on the factor of exp (2iwt) and that the angle 02 only red uces the constant vertical displace ment to B cos 02 but does not yield a ny phase shift.
The term 1 in th e parenthesis, i.e., B*, keeps th e firs t normal stress diffe rence positive during the whole period up to very high freque ncies. The oscillation is taking place with twice the frequency of the flow field. All these effects are very much the same as in the case of no internal viscosity. The difference is mainly in the replacement of T p by T' p and the term proportional to T' p -Tp. The amplitude A of the oscillating term These averages still co ntain th e transi e nt whi c h, in th e general case, cannot be easily separated from the s tatio na ry solution l·eached after the transien t has tapered off (fi g. 7).
The separation ca n be pe lformed if YOT p/WT' p is so s mall th a t one can re place the ex pone ntial fun c ti on by its linea r ex pan s ion. In such a case one obtain s for th e stationa ry solu ti o n

MTJs 1'=1
The add itional term L (T~ -Tp)/3 is independ e nt of w and he nce re prese nts th e Trouto n viscosity in th e seco nd New tonian range wh e re I/w a nd JI/w di sappear. As in th e case of conve ntional intrins ic viscos ity, eq (20), th e finite va lue a t W -7 00 is a co nseque nce or inte rnal vi scos ity, i. e., of parti al co il ri g idity. The rreq ue ncy dependence or [1/]", and phase a ngle 011/ a re plotted in fi gure 8 for Z = 100, h * = .4, a nd cplf = 2.
The intrin s ic birefrin ge nce reads (33) whi c h ror s mall amplitud e red uces to Thi s ex press ion diffe rs from th e s trea min g bire frin ge nce in an osci lla tin g fl ow fi e ld eq (21), onl y in th e rac tor 3. The no npropo rti onality be twee n viscosit y a nd birefrin ge nce is aga in th e co nseq ue ncy of the addi ti onal inte rnal viscos it y term in eq (33) .  Acousti c birefring e nce is in many res pec ts closely related to birefrin ge nce in a n osc illating je t Oow. The main difference is the absence of lateral co ntrac tion as represe nted by -yx/2 and -yy/2. In c ontras t to je t Oow with cons tan t spec ific volume (incompress ible liquid ) the volume element s ubjecte d to a n acous ti c wave is pe riodicall y compressed and expande d. That means a cons ta nt (gp2) yieldin g in eq (34) th e re place me nt of th e facto r 3 by 2 .
One has N WT = -K ac ( He re Zo is the location of th e ce nt e r of hydrod yna mi c res ista nce of th e mac romolec ule, B is a mplitud e, l ac is intens it y a nd Cae is propagati on veloc it y of acous ti c wave, a nd p is th e dens it y of solution . The frequency de pe nd e nce of aco usti c bire frin ge nce a nd phase angle is exactly the same as in th e case of osc illatin g jet Oow.
The differencp be tween acoustic birefringence according to the correct new and the incorrec t old values Vp can be seen in figure 3 whe re [Jln lac is plotted vers us WT 1 for Z = 100, h * = .4 and <p/j = 2 . The phase angle Qac is identical with an and Oln.

Conclusions
The pape r presents th e calcul atio n of mos t of th e intrin sic rh eological a nd rheooptical effects of linear homopolymers in an oscillating Oow fi eld wh ic h may be explored ex perime ntally. In the case of rh eooptical e ffects only the intrinsic birefrin gence of the polymer is includ ed. The form-birefringence is comple tely neglected. The same appli es to th e inOue nce of la rge s ide groups whi c h may effec t inde pe nde ntl y the optical anisotropy of the segme nt and its freq ue ncy de pende nce .
The introdu cti o n of th e a ppropriate vp = 1 values in stead of the old values vp = ApZ/ ApR does not c hange dras ti cally th e effects de pe nding on internal viscosit y. As a rul e the ratio be twee n th e new a nd old values of intrins ic viscosit y, birefringe nce, and first normal s tress diffe re nce is less th a n 2 , at leas t in the ra nge of Z be tween 3 and 300. W ith hi ghe r Z the diffe re nces inc rease as a linear fun c tion of log Z.
As already me ntio ned , th e smallness of the diffe re nce is a co nseque nce of the pec uliar de pe nd e nce of old vP(I) o n p: muc h larger than 1 a t s mall p and s malle r than 1 a t hi gh p. H e nce the larger contributions in the form e r part of the s ums are partially compe nsated by th e smalle r contributions in th e la tt e r part.
The mo st importan t c hanges occ ur in the seco nd Ne wtonian intrinsic viscosi ty whi c h is the mos t co nspic uous co nsequence of inte rnal viscosity. H e re th e de pende nce of (1)l ", on Z is muc h less accord ing to the new th eory than it was in th e case of the old theo ry.
No calc ul ation of the grad ie nt effec ts was attemp ted because one knows that in a Oow with a finite ve loc ity grad ient the rando ml y coiled mac romolec ul e is deform ed with a conseque nt c hange of all inte rbead dis tances. Thi s yields a c hange of inte rac ti on te nsor H whic h leads to a mod ifi cati o n of "Ill e ige nvalues Ap. The rest valu es Ap used in thi s paper a re on ly applicable to effe c ts whe re an extrapola tion to zero grad ie nt is straightforward. This is th e case wit h d ynamic effects where one uses very small grad ie nts a nd co nce ntrates on the frequ ency dependence of th e effects measured . The s ituatio n, however, is basicall y diffe re nt in the non-linear ran ge of gradient de pe nd e nce of excess stress and optical te nsor.