Ind. Eng. Chem. Res. 1995,34, 3355-3358
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Polymer Solutions and Melts: Viscosity, Diffusion, and Elasticity Arthur S. Lodge The Bannatek Co., Znc., P.O. Box 5472, Madison, Wisconsin 53705
In an empirical attempt to extend the “rubberlike liquid” temporary-junction network theory to polymeric liquids whose viscosity 17 depends on shear rate p, it is assumed that, in steady shear flow, the sample-average strand lifetime (t)may depend on p, and that, for linear polymers of a given molecular weight M , (t)is inversely proportional to the self-diffision coefficient D in a liquid at rest. It is assumed that the plateau modulus Go may be used to determine the strand concentration Y. Literature data for molten polystyrene and for a 15% solution of polystyrene in toluene give approximate superposition of the curves for u/Go vs p/D, where u denotes the shear stress; curves for u vs p differ by a factor of about 10 000. Curves for Go(N1 - Nz)/u2vs u/Go are about 100 times closer together than the corresponding curves for (NI - Nz)/u2vs u, but do not superpose; this suggests that (t2)/(z)2depends on the polymer concentration. N I and NZdenote the first and second normal stress differences.
Before Bob Bird joined Lightfoot and Stewart, Engineers found math hard-couldn’t do it. With Transport Phenomena, Advances are commoner. Even undergrad students get through it. Network Theory for Liquids
only; it is deduced for the ensemble-average positions of network points. Because recoil (following a homogeneous strain history) involves global affine motion, no valid molecular explanation can invoke any global affine motion assumption; otherwise one would be assuming that which ought to be deduced. For slower strains where changes in network connectivity are significant, theory A does not enable one to calculate the magnitude of recoil, or of other important quantities. “Theory B”, a descendant of theory A, invoked additional assumptions (constant strand creation and loss rates; neglect of effects of strand creation and loss on ensemble average network points’ affine motion) and led to the “rubberlike liquid” constitutive equations (Green and Tobolsky, 1946; Lodge, 1954, 1956):
Concentrated solutions and melts, containing linear, flexible, polymer molecules of degree of polymerization above a few hundred, exhibit the following “fast strain” properties: (a) large recoil; (b) coaxiality of strain, stress, and refractive index tensors following a single step shear from rest; (c) a plateau modulus Go whose magnitude (for melts) is close to the range of shear (d) moduli for lightly cross-linked elastomers above Tg; P,(t) = kT At-t’) B,(t’,t) dt’ (i,j = 1, 2, 3) (1) GOalmost independent of polymer molecular weight M. These properties strongly support the view that, for ny = constant 6, CPy (2) fast strains at least, the behavior is governed by a temporary-junction network. We know of no other Here, Py, Bv, and nu denote rectangular Cartesian analytic molecular theory capable of accounting for all components of the Cartesian space extra stress, Finger, of these properties. No single-molecule-in-an-unper- and refractive index tensors, respectively; At-t’) dt’ turbed-mean-field model can be regarded as giving an denotes the concentration at time t of network strands acceptable molecular explanation for a cooperative formed in the interval t‘, t’ dt‘; K denotes Boltzmann’s phenomemon (Domb, 1962) such as recoil (Lodge, 1988, constant; T denotes the absolute temperature. Theory 19891, which is, accordingly, a crucial property in the A also yields (2). context of the search for a deeper understanding of the For a low-densitypolyethylene melt at 150 “C, (1)and molecular mechanism underlying polymeric liquid be(2) give a constant stress-optical coefficient C and havior. coaxiality of stress and refractive index tensors in shear It is more helpful to consider the different stages in flow consistent with measurements (Wales, 1976);with the development of temporary-junction network theory a suitable choice of values for five constants, the in relation to the assumptions invoked rather than to equations also give a quantitative correlation of elonthe order of publication. Properties a and b can be gational stress growth and free recoil data (Chang, understood on the basis of “theory A” (Lodge, 19601, 1973), shear flow stress growth (Lodge and Meissner, which assumes the existence of a Gaussian network 1973), and constrained recoil (Huang, 19761, but only whose connectivity changes are very slow in comparison for moderate (e.g., up to about 2 shear units in shear with the thermal motions needed to generate the usual flow) strains from rest. For larger strains, measured strand tensions. For a liquid initially at rest, the stresses were much smaller than those given by the application and removal of a homogeneous stress during theory. a time interval z*, long enough to avoid a “glassy” We define the sample-averaged nth power of the response yet short enough for changes in network strand lifetime z = t - t’: connectivity (due to loss and creation of strands) to be negligible, results in 100% elastic recovery, for the liquid’s behavior is indistinguishable from that of a lightly cross-linked elastomer described by the Gaussian where Y = f l z ) dz denotes the total concentration of network theory (James and Guth, 1946; Lodge, 1960, strands (of all ages). For a step shear strain and for 1976). Affine motion is assumed for the boundary pointa
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0888-5885/95/2634-3355$09.00/00 1995 American Chemical Society
3356 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
steady shear flow, (1) gives results which may be expressed as follows:
G(0)= vkT
(4)
G(O)N,/c? = (?)/(T)~
(6)
Here, G(0) denotes the value of the relaxation modulus at time 0; p denotes shear rate; u denotes the polymer contribution to shear stress (negligibly different from the total shear stress a t the polymer concentrations of interest); N I denotes the first normal stress difference. N2, the second normal stress difference, is zero, because of the Gaussian network assumption.
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Viscosity and Diffusion Theory B gives a viscosity 7;1= u/p that is independent of 9, because of the assumption that strand creation and loss rates are constant. We believe that it is unrealistic to hope to extend the theory analytically to take account of possible variations of strand creation and loss rates with p: the idealizations required would probably be so extreme that it would be difficult t o assess the significance of any results obtained. Can anything simpler be attempted, instead? The object of the present paper is t o follow up an early suggestion (Bueche et al., 1952) that viscosity and diffusion might be simply related in concentrated polymeric liquids. The self-diffusion coefficient D has dimensions (length)2/(time)and is a measure of motion of the mass center of a given polymer molecule (see, e.g., Lodge et al. (1990), eq 8). Processes leading t o strand creation and loss and to the return to an isotropic strand vector distribution during stress relaxation would be expected t o involve segmental motions similar to but continued for a shorter period than those required for significant motion of the mass center. It seems plausible, therefore, t o consider, as one simple possibility, that the average strand lifetime may be expressed in the form
(4 = {L(M)12/D(c,T,~sfl,i,)
(7)
where L(M) is an unknown function of dimension length, c denotes the polymer concentration (or density, in the case of a melt), and denotes the solvent viscosity. This would be assigning the y dependence of 7;1 entirely to that of D. Unfortunately, D has not yet been measured in flowing polymeric liquids, so we cannot yet explore fully the feasibility of (7). Instead, we will use published values of D which have been measured for polymeric liquids at rest, and see whether, in particular, the c dependence of (z) can be entirely accounted for by that ofD, as suggested in (7). In order to postpone consideration of the unknown function L(M) in (71, we will here consider data for one and only one M . If (7) proves to be of interest, one could choose L(M) empirically so as to give the observed dependence of 70 on M . Theory B does not include a “glassy response”, but one could simply add to the relaxation modulus G(t)an extra exponential term with a small enough time constant in order to take a first step t o include a glassy response. If the difference in value between that time constant and the next larger one (representing a strand loss rate) is large enough, then the graph of log G(t) vs log t would show what normally passes for a “plateau
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log y/D Figure 2. Data of Figure 1 replotted using measured values of the plateau modulus GOand self-diffusion coefficient D as scale factors. The observed superposition suggests that the concentration dependence of the mean strand lifetime (t)is determined by that of D alone.
zone”, i.e., a region of slowly-varying ordinate log G(t), whose height, near the short time end, can be identified with the logarithm of the “plateau modulus” Go (sometimes denoted by the symbol W). Accordingly, it is reasonable to use the equation
v = GdkT
(8)
to determine the unknown strand concentration v from measured values of GO. To test these ideas, we need measured values of dp), D, and Go for a given polymer of narrow molecular weight distribution in systems of different concentration. Figure 1 shows values of log u vs log p for an undiluted polystyrene (M,= 8.6 x lo5 = 1.17Mn)at 214 “C(Wales, 1976) and for a 15 w t % solution of polystyrene (M,= 11.1 x lo5)in toluene at 30 “C(Kotaka et al., 1962). It is seen that values of 4.3)are about lo4 times higher for the melt than for the solution. Figure 2 shows the data of Figure 1 replotted in the form of log(u/Go) vs log(p/D). It is seen that the solution and melt data fall close to a common curve. Because of the curvature, the scaling of each of the two axes is significant. In Figure 2, a value D = 1.96 x cm2/s
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3367
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log 0 Figure 3. Normal stress difference data in steady shear flow for the solution and melt of Figure 1. N I and N2 denote the first and second normal stress differences.
was used for the toluene solution; this was interpolated (with respect to c ) from the D data of Kim et al. (1986, Figure 1)for toluene solutions at 20 "C of a polystyrene having M, = 9.29 x lo5 = l.lOM,,with small corrections for the different T and M, values being made on the basis of the empirical result D a T/(qs(T)Mw2}, where l;ls is the solvent viscosity. For the melt, a value D = 1.32x cmz/swas used; this was obtained from the 174 "C D values of Lodge et al. (1990,Figure 3.11, using the empirical result D = T / ( ~ o ( T ) Mvo(T) ~ ~ ;values a t 214 and 174 "Cwere related by using the usual shift factor UT, given by Laven (1985,p 113). 70denotes the melt viscosity at low 9. For the melt, a value GO= 2.6 x lo5 Pa was used (Isono et al., 1978,Figure 13;Lin, 1984,Figure 3). Values of Go for polystyrene in toluene could not be found, but there is some evidence that the variation with choice of solvent is slight (Isono et al., 1978,Figure 13;Os& et al., 1985,Figure 4),so a value GO= 3.2 x lo3 Pa was used from data for solutions in benzyl n-butyl phthalate (Isono et al., 1978,Figure 13). Support for above use of empirical results for D has been given by Bueche et al. (1956),Kim et al. (19861,and Lodge et al. (1990).
Elasticity Equations 5, 6,and 8 suggest looking at the dimensionless plot of Gal/u2 vs u/Go to see whether superposition of solution and melt data is obtained. We could not find published data for N1 for the toluene solution, so we have used values for N1 - N Zgiven by Kotaka et al. (1962)instead. Wales' melt data for N1 and NZwere used to calculate N I - Nz.Figures 3 and 4 show that values of Gf11/u2 for the solution and melt are about 100 times closer together than are the corresponding values of N1/u2but do not superpose. This suggests that r ) ~ , is a measure of the dimensionless ratio ( ~ ~ ) / (which width of the strand lifetime distribution, depends significantly on polymer concentration. The present results suggest that it would be useful to measure D(p) in steady shear flow t o see whether there are any systems for which q(P)D(p)is independent of y .
Acknowledgment For helpful comments, data, and references to published data, I am indebted to J. M. Dealy, W. W.
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log GIGo Figure 4. Use of the scale factor Go to construct dimensionless plots of the data of Figure 3. This does not give superposition; it suggests that (t2)/(t)2 is concentration dependent.
Graessley, T. P. Lodge, P. Manjeshwar, Faith Morrison, H. H. Winter, and Hyuk Yu.
Literature Cited Bueche, F.; Cashin, L. M.; Debye, P. Diffusion and Viscosity in Concentrated Polymer Solutions. J . Chem. Phys. 1952,20, 1956. Chang, Hui. Elongational Flow and Spinnability of Viscoelastic Fluids. Ph.D. Thesis, University of Wisconsin-Madison, 1983. Domb, C. Co-operative Phenomena. Encyclopaedic Dictionay of Physics; Thewlis, J., Ed.; Pergamon Press: Oxford and London, 1962;Vol. 2,p 92. Green, M. S.;Tobolsky, A. V. Stress relaxation in Polysulfide Rubbers. J. Chem. Phys. 1946,14, 80. Green, P. F.; Kramer, E. J. Tracer Diffusion Coefficients in Polystyrene. J . Muter. Res. 1986,1, 202-206. Huang, T.-A. Time-Dependent First Normal Stress Difference and Shear Stress Generated by Polymer Melts in Steady Shear Flow. Ph.D. Thesis, University of Wisconsin-Madison, 1976. Isono, Y.; Fujimoto, T.; Takeno, N.; Kjiura, H.; Nagasawa, M. Viscoelastic Properties of Linear Polymers with High Molecular Weights and Sharp Molecular Weight Distributions. Macromolecules 1978,11, 888. James, H. M.; Guth, E. Theory of the Elastic Properties of Rubber. J . Chem. Phys. 1943,11,455. Kim, H.; Chang, T.; Yohanan, J. M.; Wang, L.; Yu, Hyuk. Polymer Diffusion in Linear Matrices: Polystyrene in Toluene. Mucromolecules 1986,19, 2737. Kotaka, T.; Kurata, M.; Tamura, M. Non-Newtonian Flow and Normal Stress Phenomena in Solutions of Polystyrene in Toluene. Rheol. Acta 1962,2,179. Laven, J. Non-Isothermal Capillary Flow of Plastics Related to Their Thermal and Rheological Properties. Ph.D. Thesis, Delft University, 1985. Lin, Y.-H. Stress Relaxation Line Shape Analysis. Macromolecules 1984,17,2846. Lodge, A. S.Some Finite-Strain Generalizations of Boltzmann's Equations. Proc. 2nd Znt. Congr. Rheol.; Butterworths: London, 1954;p 229. Lodge, A. S. A Network Theory of Flow Birefringence and Stress in Concentrated Polymer Solutions. Trans. Faraday Soc. 1956, 52, 120. Lodge, A. S. The Isotropy of Gaussian Molecular Networks and the Stress-Birefringence Relations for Rubberlike Materials Cross-linked in Stressed States. Kolloid 2.1960,171,46. Lodge, A. S. On the Gaussian Network Theory for Cross-Linked Elastomers: Isotropy and Stress-free Shape Uniqueness Proofs for the Compressible Case. Proc. VZZth Znt. Congr. Rheol.; Kubat, J.,Ed.; Chalmers University of Technology: Gothenburg, Sweden, 1976;p 79. Lodge, A. S. Constitutive Equation or Stress Calculator? J . Rheol. 1988,32,93.
3358 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 Lodge, A. S. Elastic Recovery and Polymer-Polymer Interactions. Rheol. Acta 1989,28,351. Lodge, A. S.; Meissner, J. Comparison of Network Theory Predictions with Stress/Time Data in Shear and Elongation for a LowDensity Polyethylene Melt. Rheol. Acta 1973,12,41. Lodge, T. P.; Rotstein, N. A.; Prager, S. Dynamics of Entangled Polymeric Liquids: Do Linear Chains Reptate? Adv. Chem. Phys. 1990,79, 1. Osaki, K.; Nishimura, Y.; Kurata, M. Viscoelastic Properties of Semidilute Polystyrene Solutions. Macromolecules 1985,18, 1153.
Wales, J. L. S. The Application of Flow Birefringence to Rheological Studies of Polymer Melts. Ph.D. Thesis, Delft University, 1976.
Received for review January 3 , 1995 Accepted May 12,1995@ IE9500127 Abstract published in Advance ACS Abstracts, August 15, 1995. @
