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Intrinsic viscosity of rigid macromolecules in dilute solution in steady shear flow
0014-305717810901-0623502.0010
E u r o p ean P o l y m e r Journal, Vol. 14, 623 to 624 © P e r g a m o n Press Ltd 1978. Printed in Gre a t Britain
INTRINSIC VISCOSITY OF RIGID MACROMOLECULES IN DILUTE SOLUTION IN STEADY SHEAR FLOW D. MELE Departamento de IngenierfaQuimica, Universidad Polit6cnica de Barcelona, Spain (Received 27 July 1976, in revised form 14 February 1978) Abstraet--A rigid linear array of beads in a Newtonian fluid is used to model a rod-like macromolecule in a dilute solution. Following the work of Kotaka, an expression is obtained relating the intrinsic viscosity to the velocity gradient. Computed results are compared with the experimental results of Str6mberg.
INTRODUCTION
It is w e l l - k n o w n t h a t t h e intrinsic v i s c o s i t y of a p o l y m e r s o l u t i o n m e a s u r e s t h e relative v a l u e of e x c e s s s h e a r s t r e s s e s in l a m i n a r flow d u e to t h e p r e s e n c e of i s o l a t e d m a c r o m o l e c u l e s . T h e g r a d u a l o r i e n t a t i o n of a n y n o n s p h e r i c a l particle in t h e flow d i r e c t i o n a l w a y s r e d u c e s this e x c e s s s t r e s s a n d h e n c e leads to a d e c r e a s e in intrinsic viscosity. S u c h a n effect is well e s t a b l i s h e d w i t h rigid rodlike particles like t o b a c c o m o s a i c virus. So, stiff chains and polyelectrolytes show a much more m a r k e d influence o f rate of s h e a r o n intrinsic v i s c o s i t y at relatively low m o l e c u l a r w e i g h t s t h a n d o flexible m a c r o m o l e c u l e s as s h o w n , for i n s t a n c e , b y A k k e r m a n et al. [1], F u j i t a et al. [2] a n d L o h m a n d e r [3-4] f o r s o d i u m c a r b o x y m e t h y l c e l l u l o s e , Eisenberg [5] and Strfmberg [6] for p o l y ( m e t h a c r y l i c acid), E i s e n b e r g [7] f o r D N A , S t a u d i n g e r [8] a n d L o h m a n d e r [9] f o r nitrocellulose, V i n k [10] f o r c a r b o x y m e t h y l a t e d cellulose e t h e r s a n d Zaragozfi [11] f o r p o l y ( s t y r e n e s u l p h o n i c acid). S e v e r a l m a t h e m a t i c a l m o d e l s h a v e b e e n u s e d to p r e d i c t t h e r h e o l o g i c a l b e h a v i o u r of dilute p o l y m e r s o l u t i o n s b u t t h e rigidity effect of the p o l y m e r c h a i n is n o t yet well e x p l a i n e d , as s h o w n b y S u b i r a n a [12] a n d N o d a et al. [13]. In this p a p e r w e d e v e l o p a n e x p r e s s i o n f o r the intrinsic v i s c o s i t y u n d e r s t e a d y s h e a r flow of stiff m a c r o m o l e c u l e s in dilute s o l u t i o n in a N e w t o n i a n fluid.
where q is the velocity gradient
av°
q ~m.
OXi
Kotaka [16] has calculated the increment of stress caused by the presence of macromolecules in a dilute solution in steady shear flow without any explicit knowledge of the distribution function by the use of Prager's method [17] and taking into account the hydrodynamic interactions. From Eqn 9 of Kotaka, we have for the difference of xy components of normal stress between the solution and solvent, 0 _ 2 . nokTq zxr - zxy - 1V D ~
where no is the number of molecules per cm 3, k is the Boltzmann constant, T the absolute temperature, D ~ the rotatory diffusional constant and a a parameter which is given by a = ql(6D °e)
(2)
Zxy - roy = q(~ - no)
(3)
But
n being the viscosity of solution, On the other hand, D '6 = kTIf,
(4)
where fr is the rotational friction coefficient. Therefore Eqn 3 becomes n - n0 = 4~:;,(1 _ 0.7285a2 2 + 1.0106a .... )
~g'r
THEORY
We use the rod-like molecule introduced by Kuhn [14] and investigated by Kirkwood and Auer [15] for viscoelastic properties of rod-like molecules under the action of periodic shear waves. We suppose that each molecule is composed of (2n + 1) beads of radius a, joined by a rigid connector. The distance between two consecutive centres of the beads is constant. Following Stokes's law, it is assumed that each bead has a friction coefficient 6 = 6wn0a,n0 being the viscosity of solvent. The centre of the polymer molecule is fixed at the origin of the coordinate system. The unperturbed solvent velocity is
(l_0.7285a2+l.0106a4 .... ) (1)
(5)
IJ
The generalized parameter/30 is given by [n]0n0Mq
[30
~
.
(n-n0]
= O a k nor, /
= 12a
15
(6)
where M is molecular weight, R is the gas constant and [n] is the intrinsic viscosity at zero shear rate expressed in cm3/g.
The rotational friction coefficient .f, of rod-like molecules with sufficient large n has been determined recently by Kamakawa-Kamaki [18], ~T
3
f, - ~- noL [In n - 0.4228 + (2h)-I] -t
V ° = {qy,, 0, 0} 623
(7)
D. MELE
624 where L is the length of the chain, L = b(2n + 1), and h=
~ 3a 8~r'00b = 4-b"
(8)
Operating with Eqns 5, 6 and 7 we find [rl]q = [rl]o(l -1.1382/3o2+ 1.263/3o4.... )
(9)
2NArI°L3 [In n - 0.4228 + (2h)-1] -l 45
(10)
where [7]0 =
NA is Avogadro's number and [~]q is the intrinsic viscosity at shear rate q, expressed in cm3/g. DISCUSSION The static value of the intrinsic viscosity is essentially the same as that of Hearst [19] for a weakly bending rod w h e n the flexibility goes to zero. For h = 3/8 (spheres touching), E q n 10 is also in agreement with that of Y a m a k a w a [20] for straight cylindrical molecules of infinite length. In the limit of very large n, E q n !0 b e c o m e s identical to the result of K i r k w o o d and Auer-[15]. It should be noted, however, that the rod-like macromolecules c o m m o n l y e n c o u n t e r e d do not have values of n sufficiently large for the c o n s t a n t
--;C--:z_. . . . 09
08
I Ol
02
03
04
3° Fig. 1. Relative intrinsic viscosity [~]q/[~]o plotted against the generalized parameter /30. Curves 1, 1', 1"--+60% NaPMA at various ionic strengths, respectively 0.649 x 10-3 M, 2.264 × 10-3 M and 4.490 x 10-3 M NaCI[6]. Curve 2--Theoretical curve according to Eqn 9. Curve 3-Ellipsoid of revolution from ref. [21].
correction terms to the logarithm in E q n 10 to be neglected. With E q n 9, we have represented in Fig. 1 the reduced intrinsic viscosity vs the generalized parameter /3o. The results are only valid for low values of /30 ( < 1) because of the assumptions in Prager's method. On this figure we have represented a curve for the ellipsoid of revolution from Sheraga [21] with a ratio of semi-axes of 300. Its position is near to the rod-like molecule as we could assume. Some experimental results have been plotted in Fig. 1 for 60% N a P M A at various ionic strengths from Str6mberg[6]. It is clear that the effect on the reduced intrinsic viscosity increases considerably w h e n the ionic strength is lowered. The computed curve is near to the real values for moderate ionic strength; for lower ionic strengths, the theory must be improved. Acknowledgement--The author is indebted to Dr Juan A. Subirana for suggesting this problem and for valuable comments during the work.
REFERENCES
1. K. Akkermans, D. T. L. Pals and J. J. Hermans, Rec. Tray. Chim. 71, 56 (1952). 2. H. Fujita and T. Homma, J. Polym. Sci. I5, 277 (1955). 3. N. Lohmander and R. Str6mberg, Makromolek. Chem. 72, 143 (1964). 4. N. Lohmander, ibid. 72, 159 (1964). 5. H. Eisenberg, J. Polym. Sci. 25, 579 (1957). 6. R. Str6mberg, Ark. Kemi 25, 579 (1957). 7. H. Eisenberg, J. Polym. Sci. 25, 257 (1957). 8. H. Staudinger, Sorkin. Ber. 70, 199 (1937). 9. N. Lohmander, Makromolek. Chem. 72, 159 (1964). 10. H. Vink, Die Makromolek Chem. 131, 133 (1970). 11. J. Zaragoza, Thesis Doctoral. Universidad Polit6cnica de Barcelona, Spain. 12. J. A. Subirana, J. chem. Phys. 41, 3852 (1964). 13. I. Noda, Y. Yamaka and M. Nagasawa, J. phys. Chem., Ithaca 72, 2890 (1968). 14. W. Kuhn, Z. phys. Chem. AI61, 1 (1932). 15. J. G. Kirkwood and P. L. Auer, J. chem. Phys. 19, 281 (1951). 16. T. K0taka, J. chem. Phys. 10, 1566 (1959). 17. S. Prager, Trans. Soc. Rheol. 1, 53 (1957). 18. H. Yamakawa and J. Yamaki, J. chem. Phys. 58, 2049 (1973). 19. J. E. Hearst, J. chem. Phys. 40, 1506 (1964). 20. H. Yamakawa, Macromolecules 8, 339 (1975). 21. H. A. Sheraga, J. chem. Phys. 23, 1526 (1955).
E u r o p ean P o l y m e r Journal, Vol. 14, 623 to 624 © P e r g a m o n Press Ltd 1978. Printed in Gre a t Britain
INTRINSIC VISCOSITY OF RIGID MACROMOLECULES IN DILUTE SOLUTION IN STEADY SHEAR FLOW D. MELE Departamento de IngenierfaQuimica, Universidad Polit6cnica de Barcelona, Spain (Received 27 July 1976, in revised form 14 February 1978) Abstraet--A rigid linear array of beads in a Newtonian fluid is used to model a rod-like macromolecule in a dilute solution. Following the work of Kotaka, an expression is obtained relating the intrinsic viscosity to the velocity gradient. Computed results are compared with the experimental results of Str6mberg.
INTRODUCTION
It is w e l l - k n o w n t h a t t h e intrinsic v i s c o s i t y of a p o l y m e r s o l u t i o n m e a s u r e s t h e relative v a l u e of e x c e s s s h e a r s t r e s s e s in l a m i n a r flow d u e to t h e p r e s e n c e of i s o l a t e d m a c r o m o l e c u l e s . T h e g r a d u a l o r i e n t a t i o n of a n y n o n s p h e r i c a l particle in t h e flow d i r e c t i o n a l w a y s r e d u c e s this e x c e s s s t r e s s a n d h e n c e leads to a d e c r e a s e in intrinsic viscosity. S u c h a n effect is well e s t a b l i s h e d w i t h rigid rodlike particles like t o b a c c o m o s a i c virus. So, stiff chains and polyelectrolytes show a much more m a r k e d influence o f rate of s h e a r o n intrinsic v i s c o s i t y at relatively low m o l e c u l a r w e i g h t s t h a n d o flexible m a c r o m o l e c u l e s as s h o w n , for i n s t a n c e , b y A k k e r m a n et al. [1], F u j i t a et al. [2] a n d L o h m a n d e r [3-4] f o r s o d i u m c a r b o x y m e t h y l c e l l u l o s e , Eisenberg [5] and Strfmberg [6] for p o l y ( m e t h a c r y l i c acid), E i s e n b e r g [7] f o r D N A , S t a u d i n g e r [8] a n d L o h m a n d e r [9] f o r nitrocellulose, V i n k [10] f o r c a r b o x y m e t h y l a t e d cellulose e t h e r s a n d Zaragozfi [11] f o r p o l y ( s t y r e n e s u l p h o n i c acid). S e v e r a l m a t h e m a t i c a l m o d e l s h a v e b e e n u s e d to p r e d i c t t h e r h e o l o g i c a l b e h a v i o u r of dilute p o l y m e r s o l u t i o n s b u t t h e rigidity effect of the p o l y m e r c h a i n is n o t yet well e x p l a i n e d , as s h o w n b y S u b i r a n a [12] a n d N o d a et al. [13]. In this p a p e r w e d e v e l o p a n e x p r e s s i o n f o r the intrinsic v i s c o s i t y u n d e r s t e a d y s h e a r flow of stiff m a c r o m o l e c u l e s in dilute s o l u t i o n in a N e w t o n i a n fluid.
where q is the velocity gradient
av°
q ~m.
OXi
Kotaka [16] has calculated the increment of stress caused by the presence of macromolecules in a dilute solution in steady shear flow without any explicit knowledge of the distribution function by the use of Prager's method [17] and taking into account the hydrodynamic interactions. From Eqn 9 of Kotaka, we have for the difference of xy components of normal stress between the solution and solvent, 0 _ 2 . nokTq zxr - zxy - 1V D ~
where no is the number of molecules per cm 3, k is the Boltzmann constant, T the absolute temperature, D ~ the rotatory diffusional constant and a a parameter which is given by a = ql(6D °e)
(2)
Zxy - roy = q(~ - no)
(3)
But
n being the viscosity of solution, On the other hand, D '6 = kTIf,
(4)
where fr is the rotational friction coefficient. Therefore Eqn 3 becomes n - n0 = 4~:;,(1 _ 0.7285a2 2 + 1.0106a .... )
~g'r
THEORY
We use the rod-like molecule introduced by Kuhn [14] and investigated by Kirkwood and Auer [15] for viscoelastic properties of rod-like molecules under the action of periodic shear waves. We suppose that each molecule is composed of (2n + 1) beads of radius a, joined by a rigid connector. The distance between two consecutive centres of the beads is constant. Following Stokes's law, it is assumed that each bead has a friction coefficient 6 = 6wn0a,n0 being the viscosity of solvent. The centre of the polymer molecule is fixed at the origin of the coordinate system. The unperturbed solvent velocity is
(l_0.7285a2+l.0106a4 .... ) (1)
(5)
IJ
The generalized parameter/30 is given by [n]0n0Mq
[30
~
.
(n-n0]
= O a k nor, /
= 12a
15
(6)
where M is molecular weight, R is the gas constant and [n] is the intrinsic viscosity at zero shear rate expressed in cm3/g.
The rotational friction coefficient .f, of rod-like molecules with sufficient large n has been determined recently by Kamakawa-Kamaki [18], ~T
3
f, - ~- noL [In n - 0.4228 + (2h)-I] -t
V ° = {qy,, 0, 0} 623
(7)
D. MELE
624 where L is the length of the chain, L = b(2n + 1), and h=
~ 3a 8~r'00b = 4-b"
(8)
Operating with Eqns 5, 6 and 7 we find [rl]q = [rl]o(l -1.1382/3o2+ 1.263/3o4.... )
(9)
2NArI°L3 [In n - 0.4228 + (2h)-1] -l 45
(10)
where [7]0 =
NA is Avogadro's number and [~]q is the intrinsic viscosity at shear rate q, expressed in cm3/g. DISCUSSION The static value of the intrinsic viscosity is essentially the same as that of Hearst [19] for a weakly bending rod w h e n the flexibility goes to zero. For h = 3/8 (spheres touching), E q n 10 is also in agreement with that of Y a m a k a w a [20] for straight cylindrical molecules of infinite length. In the limit of very large n, E q n !0 b e c o m e s identical to the result of K i r k w o o d and Auer-[15]. It should be noted, however, that the rod-like macromolecules c o m m o n l y e n c o u n t e r e d do not have values of n sufficiently large for the c o n s t a n t
--;C--:z_. . . . 09
08
I Ol
02
03
04
3° Fig. 1. Relative intrinsic viscosity [~]q/[~]o plotted against the generalized parameter /30. Curves 1, 1', 1"--+60% NaPMA at various ionic strengths, respectively 0.649 x 10-3 M, 2.264 × 10-3 M and 4.490 x 10-3 M NaCI[6]. Curve 2--Theoretical curve according to Eqn 9. Curve 3-Ellipsoid of revolution from ref. [21].
correction terms to the logarithm in E q n 10 to be neglected. With E q n 9, we have represented in Fig. 1 the reduced intrinsic viscosity vs the generalized parameter /3o. The results are only valid for low values of /30 ( < 1) because of the assumptions in Prager's method. On this figure we have represented a curve for the ellipsoid of revolution from Sheraga [21] with a ratio of semi-axes of 300. Its position is near to the rod-like molecule as we could assume. Some experimental results have been plotted in Fig. 1 for 60% N a P M A at various ionic strengths from Str6mberg[6]. It is clear that the effect on the reduced intrinsic viscosity increases considerably w h e n the ionic strength is lowered. The computed curve is near to the real values for moderate ionic strength; for lower ionic strengths, the theory must be improved. Acknowledgement--The author is indebted to Dr Juan A. Subirana for suggesting this problem and for valuable comments during the work.
REFERENCES
1. K. Akkermans, D. T. L. Pals and J. J. Hermans, Rec. Tray. Chim. 71, 56 (1952). 2. H. Fujita and T. Homma, J. Polym. Sci. I5, 277 (1955). 3. N. Lohmander and R. Str6mberg, Makromolek. Chem. 72, 143 (1964). 4. N. Lohmander, ibid. 72, 159 (1964). 5. H. Eisenberg, J. Polym. Sci. 25, 579 (1957). 6. R. Str6mberg, Ark. Kemi 25, 579 (1957). 7. H. Eisenberg, J. Polym. Sci. 25, 257 (1957). 8. H. Staudinger, Sorkin. Ber. 70, 199 (1937). 9. N. Lohmander, Makromolek. Chem. 72, 159 (1964). 10. H. Vink, Die Makromolek Chem. 131, 133 (1970). 11. J. Zaragoza, Thesis Doctoral. Universidad Polit6cnica de Barcelona, Spain. 12. J. A. Subirana, J. chem. Phys. 41, 3852 (1964). 13. I. Noda, Y. Yamaka and M. Nagasawa, J. phys. Chem., Ithaca 72, 2890 (1968). 14. W. Kuhn, Z. phys. Chem. AI61, 1 (1932). 15. J. G. Kirkwood and P. L. Auer, J. chem. Phys. 19, 281 (1951). 16. T. K0taka, J. chem. Phys. 10, 1566 (1959). 17. S. Prager, Trans. Soc. Rheol. 1, 53 (1957). 18. H. Yamakawa and J. Yamaki, J. chem. Phys. 58, 2049 (1973). 19. J. E. Hearst, J. chem. Phys. 40, 1506 (1964). 20. H. Yamakawa, Macromolecules 8, 339 (1975). 21. H. A. Sheraga, J. chem. Phys. 23, 1526 (1955).