The theory of anomalous angles in the flow birefringence of polymer solutions

THE THEORY OF ANOMALOUS ANGLES IN THE FLOW BIREFRINGENCE OF POLYMER SOLUTIONS* YU. YA. GOTLIEB a n d YU. YE. SVETLOV Institute of Macromolecular Compounds, U.S.S.I~. Academy of Sciences (Received 18 March 1963)

INVESTIGATION of the g r a d i e n t d e p e n d e n c e of the e x t i n c t i o n angle for solutions of p o l y m e r s w i t h negative i n h e r e n t anisotropy, carried out b y F r i s m a n , T s v e t k o v , Syui Mao a n d S h t e n n i k o v a [1-4] h a v e led to the discovery of two t y p e s o f " a n o m a l o u s " angles (see Figures a a n d b).

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G,~O-3,~c -'

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Dependence of orientation angle on rate gradient: a --for fractions of polystyrene in dioxan 1 - - M = 3 . 4 × 1 0 s, c=0.2 g/100 ml; 2-_~I=6.9 ×10 s, c=0.185 g/100 ml; 3 - - M = l O ×10 s, c ~-0.197 g/100 ml. [ I-3]; b -- for solutions in dioxan of a polystyrene fraction of M = 69 × 10s: 1 --c =0.455; 2--c =0.375; 3--c =0.281; 4 - c =0.185 (c is the concentration in g/100 ml) [3]. I n reference [5] it was shown t h a t a n a n o m a l o u s g r a d i e n t d e p e n d e n c e of t h e o r i e n t a t i o n angle, ~, can be associated w i t h polydispersity o f b o t h t h e shape o f the maeromoleeules a n d o f molecular weight. I n t h e theoretical work o f ~opic [6] it was established t h a t for t h e dumb-bell chain model [7] a n o m a l o u s angles can be f o u n d for kinetically rigid, Gaussian chains b u t n o t for absolutely flexible chains. I t should be n o t e d t h a t the results in reference [6] for rigid chains relate o n l y to small gradients. T h e r e f o r e in reference [6] t h e possibility was discussed o f t h e a p p e a r a n c e o f a n o m a l o u s angles as a result o f change in t h e difference b e t w e e n t h e r e f r a c t i v e indices o f the solvent a n d o f the dissolved macromolecules, n2--n~. * Vysokomol. soyed. 6: No. 5, 771-776, 1964. 845

846

Yu. YA. GOTLIEBand Yu. Y~. SVETLOV

The difference between the angles of orientation of the positive shape effect, Ayf, and the negative effect of the inherent anisotropy, A Ys, leading to an anomalous dependence of a on G or (n2--n~) in fact arises as a result of differences in the degree of orientation of short and long chains, possessing different ratios of A~f to Ays . It should be remembered that the angle of orientation for rigid chains decreases with increase in T, the time of rotational diffusion, ~ being greater for longer chains. As was shown in references [6], [7] and [13], in the case under consideration the chain-length distribution functions do not have planar symmetry, whereas for absolutely flexible dumb-bells the distribution function is symmetrical with respect to the plane of the most probable orientation of the particles under flow. In other words the absence of symmetry in the distribution function and the above-mentioned difference in the dependence of Ay, and Ayf on chain length are among the conditions giving rise to difference in the orientation of the effects of shape and inherent anisotropy. More recently Kojama [8] has shown that an anomalous dependence on (n~--n~) for low values of G can be obtained for absolutely flexible chains also, if account is taken of the segmental structure of the macromolecule. The calculation was made on the basis of the molecular model proposed b y Kargin and Slonimskii [9] (see also [10]). This communication discusses both the gradient dependence of a for an absolutely flexible, free-draining multisegmented chain, and also the dependence of a on molecular weight and (n*--n~) making a comparison over a fairly wide range of values of G. ANISOTROPY OF SHAPE

It is well known that a macromolecule exhibits anisotropy of shape if the mean refractive index differs from that of the solvent, even if it consists of optically isotropie segments. This anisotropy is associated with deviation of the shape of the molecule from the spherical. The shape anisotropy of a macromolecule is due to interaction of dipoles induced b y the field of the light wave. The averaged tensor of the polarizability of a macromolecule takes the form:

A=AI÷A 2

(1)

where A 1 is the tensor of inherent anisotropy [11] and A 2 the averaged tensor of shape anisotropy

A2= ~ 0~>-I- ~ ak> <~k ~t> + " " i~k

(2)

itk¢:l

where ai is the polarizability tensor of ith segment and is the mean tensor of dipole-dipole interaction of the ith and kth dipoles (see [6,8]). In what follows we shall consider only the first term in equation (2) because our estimate of the second term, made b y means of Kuhn's dumb-bell model, shows that for not

Anomalous angles in flow birefringcnce of polymer solutions

847

v e r y low molecular weights ( M > 0 . 5 × 106) a n d for all gradients attainable experimentally, its contribution is negligibly small. For averaging Tik-~ik) w e obtained the conjugate distribution function (r~k): 2

o

¢)

f (rit) =cik exp {--/~ik [rik - - 25ikXikYik-- ~ikXik " - YikYik-] }

(3)

The m a g n i t u d e s cik, I % etc. which are complex functions of the gradient, can be determined from the conditions of normalization, using m e a n values of (r~.), (:c~k) etc. We calculated these means accurately for the case of a freedraining chain. (In e q u a t i o n (3) the direction of flow is alol~g the x axis and the gradient along the y axis). The function given in reference [8] is an expansion of equation (3) in the region of small gradients. O m i t t i n g the i n t e r m e d i a t e , cumbersome calculations we present the found results for the m e a n tensor of shape anisotropy 8 = - - BN1/2 ~1 (fl)

(4)

3

8

= 3 -

B2¥1/2~2(]~)

B = ~ - (2u-J/2n0 - \ 4 u p N A N ]

n --n5

(5) _12

L (n2+ 2) (no+ 2)

(6)

p=G10 lV /36 kT. Here n o is t h e refractive index of the solvent, n 2 the refractive index of the d r y polymer, 1o the m e a n square length of the segment, ~ the coefficient of friction, p the density of the d r y polymer, N~ the Avogadro n u m b e r a n d N the n u m b e r of segments in the chain. Using the results for the intrinsic viscosity of a free-draining molecule [11], it can be shown t h a t fl=.M[q] t1 G I R T where q is the viscosity of the solvent. The functions ~l(fl) a n d ~'2(fl) have been t a b u l a t e d a n d sented below fl 0 0.5 1 3 5 7 ~//1(/~) 0 0"221 ' 0"810 4 " 1 2 3 7 " 3 6 4 1 1 " 0 6 1 ~'~(fl) 0 0'243 0"915 5"134 9"702 14'794 a (fl)/O 1.584 1.474 1'350 0.763 0.492 0'377 b (fl)/O 1.361 1 . 2 9 6 1 . 2 2 0 0 - 7 6 0 0-517 0.403 a (fl)/b (fl) 1 . 1 6 3 1 - 1 3 6 1 . 1 0 6 1 . 0 0 4 0.949 0.935

(7) the results are pre10 16"453 22"607 0.273 0'301 0-910

20 34"572 48'839 0.143 0.163 0.885

EXTINCTION ANGLE

The extinction angle, ~, is determined b y means of the well k n o w n relationship [11]: tan 2~=

2



. (8)

848

YU. YA. GOTLIEB and Yu. YE. SVETLOV

Using equations (4) and (5) and the result for the mean tensor of inherent anisotropy [11] we obtain tan 2g---- 2.51-4-b(ri)

,8 l+a(ri) 5

4

a(ri)= -~- 8~q(ri) ri-~, 15"38/z__ O : 2~/2 u

b(ri) = -3- 8~2 (,8) ri-"

(9)

10)

no-2/. __9M

~2 F n2--n~ ]~ IV1/2 ~4upNaN. ] Lin2+ ~ + 2) ~ (6¢1--g2)-1

(11)

where (%--~2) is the anisotropy of the statistical segment. From (9) it is seen that a----45° (i.e. tan 2~----± c~) not only for the trivial case fl-->0 but also for certain values of ri=rio where

a(flo): -- 1

[a(flo)/b(rio)]:/:l .

(12)

The numerical data presented above show the dependence of the functions

a(ri)/O and b(ri)/O on ft. It is easy to see (when 8 ~ M 1/2 that there is a minimal molecular weight, Mmin, and correspondingly a 0min, for which the anomalous angle condition (12) is satisfied even at zero gradient, rio=0. I t is obvious t h a t w h e n M < M m l ~ the first equation (12) has no solution, i.e. there are no anomalous angles. With increase in M, rio(M) increases and the ratio a(rio)/b(rio) decreases, as is seen from the numerical data. At a certain value, M----Me, this ratio becomes equal to u n i t y and with further increase in M the ratio a(rio)/(b(rio) decreases monotonically. The value of a(rio)/b(rio) determines the type of anomaly. When the ratio is greater than unity an anomaly of type A (curve 1, Figure a) must occur, if however it. is less than unity the anomaly must be of type B (curves 2 a n d 3, Figure a). Thus when M m i n < M < M ~ an anomaly of type A occurs, and when M > Me, an anomaly of type B. When rio is large ~l(ri0) flo2=l'Trio 1. Substituting this in (12)and bearing in mind equation (7) we obtain: Go---- 1 . 7 x

5

RT

- 8. 3 M [t/] t/

(13)

Since 0 ~ M 1/2, and for a free draining macromolecule [q] ~ M, the Go ~ M a/~ whereas rio(M) increases proportionally with M a/~. I t is possible to consider in a similar way the behaviour of the extinction angle with change in (n2--n2o)=A(n ~) for a given value of M. Here the change in A(n ~) can result from a change in the refractive index of the solvent, no2, or from a change of the concentration of polymer in the solution. I n the latter case no~ corresponds to the refractive index of the solution (el. reference [2]). From equation (11) it is seen t h a t the dependence of O on A(n~) at a given value of M is similar to the dependence of 0 on M 1/~at a fixed value of A(n2). Since when M is fixed ri is independent of A(n2) (see (7)), the dependence of rio on A(n9") determined from (12)

Anomalous angles in flow birefringence of polymer solutions

849

coincides with the dependence of Go oll A(n2). Consequently there is a minimal refractive index difference, Amin(n2)~g(M), such that when A(n~)~ Amtn(n2) there will be no anomalous angles, i.e. for all values of G the shape effect is less than the effect of inherent anisotropy. When A(n2) increases to a certain value A~(ne), an anomaly of type A must occur (curves 1 and 2, Figure b), and finally when A(n-)~/Ic(n 2 ), an anomaly of type B (curves 3 and 4, Figure b). From the above results it follows that the angles of orientation of the shape effect, ~j, and of the effect of inherent anisotropy, ~s, are not equal to one another and do not have the same dependence on the gradient. (tan 2 a J t a n 2 ~f)-~a(fl)/b(fl)t

(14)

Analysis of the distribution function f(r ~) for a multisegmented model (N ~ 1) (cf. equation (3)) shows t h a t it is symmetrical with respect to the plane of the most probable orientation of the particles in flow. I n contrast to particles possessing internal viscosity the mean angle of orientation is independent of the chain length r. It can be shown t h a t when one of the main axes of the polarizability tensor of each of the effects coincides with the vector-r and the anisotropies of the two effects in the molecular axes are derivatives of the (r) functions, then for a symmetrical functionf(~ the angles of orientation of the two effects coincide for all values of G and

tan

5

(15)

For the single-segment, dumb-bell model of K u h n and Oopi5 where the ellipsoid of the polarizability tensor is co-axial with r tan 2 ~ t a n

2~j=

1/fl

(16)

Thus for the multisegmente4 model the main axes of the polarizability tensors of the two effects do not coincide with the molecular axes, but form a certain angle with r . The mean angle of orientation, and consequently the effective relaxation time, of each of the effects differs from the angle of orientation of the v e c t o r r and generally speaking is characterized by a different gradient dependence. The gradient dependence of the effect of inherent anisotropy coincides with the gradient dependence of the mean angle of orientation of r, i.e. their effective relaxation times differ by a numerical factor. For the shape effect the gradient dependence of the angle of orientation is different and close to that for a chain with internal rigidity, i.e. the deformability of the chain with respect to the shape effect is less t h a n with respect to the effect of inherent anisotropy (or to r). This is understandable when it is remembered that the effect of inherent anisotropy is determined by the sum of the squares of the deformations of short

850

Yu. YA. GOTLIEBand YU. Y~. SV~TLOV

chain segments, whereas the shape effect also includes change in the distance between pairs of remote segments, which lead~ to an increase in the effective relaxation time. COMPARISON WITH EXPERIMENT

The proposed theory relates to the case of low concentrations of polymer, where hydrodynamic interaction and collision between the macromolecules can be neglected, and the "optical" interaction can be reduced to substitution of the refractive index of the solvent b y the refractive index of the solution, which is dependent on the concentration of polymer. Therefore the gradient dependence of the extinction angle at different molecular weights should be compared with experimental results extrapolated to zero concentration. However the existing experimental material does not provide the possibility of performing this extrapolation for a wide range of molecular weights and we are limited to the examination of experimental data on ~(G) at the same, low concentration. The experimental data are shown in Figures a and b [1-3]. The theory accurately describes the transition from one t y p e of anomaly to the other with change in molecular weight and in concentration. However it cannot be claimed that the theory gives quantitative agreement with experiment because it does not take account of deviation from the free-draining requirement for the macromolecules, of the effect of internal viscosity and of the: gradient dependence of Green's tensor. The dependence of//0 an M at A(n~)~-const was in our case (/30 ~ M 1 / 2 ) closer to the experimental relationship (rio ~ M [3, 4]) than that required b y the theory of reference [12] (~0"M1/4) • In conclusion we wish to express our sincere gratitude to V. N. Tsvetkov for presenting the problem and for valuable discussion and to E. V. Trisman for constant interest in the work and assistance in discussion of the experimental data. CONCLUSIONS

(1) A theory developed for a segmented macromolecular m o d e l predicts the existence of two types of anomaly in the gradient dependence of the extinction angle.

(2) Transition from one type of anomaly to the other occurs with change in molecular weight or in t h e difference between the refractive indices of the dry polymer and the solution, due for example to a change in concentration. (3) The theory is found to be in good qualitative agreement with the existing experimental data. Translated by E, O. PtnLLII, S REFERENCES

1. E. V. FRISMAN, Dokl. Akad. Nauk SSSR 118: 72, 1958 2. E. V. FRISMAN and V. N. TSVETKOV, Zh. tekh. fiz. 29: 212, 1959 3. SYUI MAO, Dissertation, A. A. Zhdanov State University. Leningrad, 1962

ttydrodynamics of polymer solutions - V I I .

851

4. V. N. TSVETKOV and I. N. SHTENNIK()VA, Vysokomol. soyed. 2: 649, 1960 5. V. N. TSVETKOV and I. N. SHTENNIK()VA, Vysokomol. soyed. 5: 740, 1963 6. M. (~()PI(~, J. Chem. Phys. 26: 1382, 1957 7. W. KUHN and It. KUItN, Helv. chem. acta 28: 1533, 1945 8. P. KOYAMA, J. Phys. Soc. Japan 16: 1366, 1961 9. V. A. KARGIN and G. L. SLO)NIMSKII, Dokl. Akad. Nauk SSSR 62: 239, 1948 10. P. E. ROUSE, J. Chem. Phys. 21: 1272, 1953 l l . B . ZIMM, J. Chem. Phys. 24: 269, 1956 12. V. N. TSVETK()V, E. V. FRISMAN, (). B. PTITSYN and S. Ya. KOTLYAR, Zh. tekh. fiz. 28: 1428, 1958 13. W. KUHN H. KUHN and P. BUCHNER, Ergebn. exakt. Naturw. 25: l, 1951

THE HYDRODYNAMICS OF POLYMER SOLUTIONS--VII. THE EFFECT OF LONG RANGE INTERACTION ON THE INTRINSIC VISCOSITY OF LINEAR MACROMOLECULES NEAR THE ~-POINT* Yu. YE. EIZNER Institute of Macromolecular

and O. :B. PTITSYN

Compounds,

U.S.S.R. Academy

of Sciences

(Received 18 ~iarch 1963)

I N RECENT t i m e s a n u m b e r o f a u t h o r s h a v e discussed t h e effect of long r a n g e i n t e r m o l e c u l a r i n t e r a c t i o n on the intrinsic viscosity, of flexible, linear m a c r o molecules [1-9]. T h e general result of t h e s e discussions has b e e n the e s t a b l i s h m e n t o f t h e f a c t t h a t w i t h i m p r o v e m e n t of t h e solvent, i.e. w i t h increase in t h e coefficient of swelling, t h e F l o r y coefficient in t h e e q u a t i o n

D ] =(6R2) ~ ~s/2

(1)

decreases (R is the radius of inertia of the m a c r o m o ] e c u l e a n d M the m o l e c u l a r weight). T h e effect o f long r a n g e i n t e r m o l e c u l a r i n t e r a c t i o n was t a k e n into a c c o u n t differently in references [1-9], b u t in all cases t h e m e t h o d s were similar. T h e p r e s e n t c o m m u n i c a t i o n is d e v o t e d to calculation of this effect on the basis of t h e strict t h e o r y of long r a n g e i n t e r a c t i o n , applicable to t h e case w h e n these i n t e r a c t i o n s are small (see reviews [10, 11]). R e s u l t s o b t a i n e d b y m e a n s of the K i r k w o o d - R i s e m a n t h e o r y [12] should be t h e m o s t a c c u r a t e . H o w e v e r for this it is n e c e s s a r y to m a k e a series solution of t w o i n t e g r a l e q u a t i o n s [13], which w i t h the insufficient a c c u r a c y of existing * Vysokomo]. soyed. 6: No. 5, 777-871, 1964.