1102
J. Phys. Chem. lS82, 86, 1102-1106
to A. C. Wahl for providing us with copies of BISON and used in the binding energy computations, and to M. L. Olson for use of some of his datahandling routines. As a graduate student, D.D.K. had the pleasure of being infected with Joe Hirschfelder's enthusiasm for understanding long-range interactions betweeen atoms and molecules. It is inspiring to see that Joe is still as enthusiastic as ever and is contemplating new solutions for unsolved problems in molecular theory.
in which only excitations from the valence shell of electrons is considered. That we agree moderately well with C3 values deduced from both calculated and experimental transition moments (when such agreement depends so sensitively on so many aspects of the calculation) suggests that the OVC premise is basically sound.
BISON-MC programs
Acknowledgment. Thanks are due to G . Radlauer for help with the computations of the long-range parameters,
A Klnetlc Theory for Polymer Melts. 3. Elongational Flows R. Byron Bhd,' H. H. Saab,+ and C. F. Curtisst Department of Chemicel E n g M n g and Rheokgy Research Center, and Department of Chembby and Theoretical Chembtry InstiMe, Unlversky of Wlsconsln-Mdson, Madison, Wlsconsln 53706 (Received: July 2, 1981; In Final Form: September 15, 1981)
The Curtiss-Bird constitutive equation, obtained from a reptation model for polymer melts, is applied to elongational flows. The steady-state elongational viscosity and the elongational growth function are calculated; it is found that these properties are very sensitive to the value of the link tension coefficient. The theoretical calculations are compared with the limited experimental data available.
Introduction In two recent publications Curtiss and Bird's presented a kinetic theory for undiluted polymers, based on the earlier Curtiss-Bird-Hassager phase-space In this polymer melt kinetic theory the macromolecules are modeled as freely jointed bead-rod chains (the Kramers "pearl-necklace model"), and the "reptation" concept used by Doi and Edwards5is introduced to account for the constraints imposed on a macromolecule by its neighbors. The Curtiss-Bird theory contains several parameters describing the macromolecule: N , the number of beads in the chain; a, the length of a rod connecting two successive beads; {, a bead friction coefficient; E , the link tension coefficient; and p, the chain constraint exponent. Some of these parameters occur together in the time constant, which arises in the derivation of the equation for the orientational distribution function: X = N3+s{a2/2kT. The Curtiss-Bird theory gives an expression for the stress tensor in terms of several kinematic tensors (i.e., a "rheological equation of state" or "constitutive equation"). For e = 0 and = 0 the constitutive equation is very similar to that of Doi and E d ~ a r d s .These ~ authors computed a number of the rheological functions for shear and elongational flows, and it is generally accepted that their results are qualitatively in agreement with experimental observations; they did not, however, present any data comparisons. In this paper we develop expressions for the elongational properties from the Curtiss-Bird theory and compare the calculated results with available experimental data. In the next paper in the series we do the same for shear properties. The Constitutive Equation The constitutive equation from the Curtiss-Bird theory is an expression for the stress tensor ?r = p6 + T in terms t Department of Chemical Engineering and Rheology Research Center. *Department of Chemistry and Theoretical Chemistry Institute.
0022-365418212086-1102$01.25/0
of two kinematic tensors: i. = Vv + ( V V ) and ~ ~[O](t,t?. Here T is the total stress tensor, 6 is the unit tensor, 7 is that part of the stress tensor that vanishes at equilibrium, and ( V V ) ~is the transpose of Vv. The tensor y[OI(t,t') is a finite strain tensor defined else where;'^^ it contains information about the complete history of the deformation of a fluid element. The Curtiss-Bird constitutive equation is 7
= NnkT[y36 -
in which p and
u
st -m
~ (- tt?A(t,t? dt'-
are given by
and the tensors A and B are
A = uu[l
+ y[ol:uu]-3/2
B = '/,Xi.(t):uuuu[l+ ~ [ O ] : U U ] - ~ / ~
(4)
(5)
Here the overbars indicate averages over a unit sphere (...) = (1/4a)So2"Jo"( ...) sin 1!9 d0 d@,and u is a unit vector. The above constitutive equation differs from that of Doi and (1)C. F. Curtiss and R. B. Bird. J. Chern. Phvs.. 74.2016-25 (1981): Errata: In (5.6)change [uj]to [uj];in (6.2)change gvt; to x v t u ; in the first line of (A51 change - to +. (2)C.F. Cur& an8 R. B. Bird, J. Chern. Phys., 74,2026-33 (1981); Erratum: In (4.2)uuuu should be uuuu. (3)C.F.Curtiss, R. B. Bird, and 0. Hassager, Adu. Chern. Phys., 35, 31-117 (1976). (4)R. B. Bird, 0. Hassager, R. C. Armstrong, and C. F. Curtiss, "Dynamia of Polymeric Liquids", Vol. 2, 'Kinetic Theory", Wiley, New York, 1977,Chapter 14. (5)M. Doi and S. F. Edwards, J. Chern. SOC.,Faraday Trans.2, 74, 1789-1801, 1802-17,1818-32 (1978);75,38-54 (1979).
0 1982 American Chemical Society
Kinetic Theory for Polymer Melts
The Journal of Physical Chemktty, Vol. 86, No. 7, 1982 1109
r = (;
2
0
-1
0 1 P);(t)
(15)
The flow field may also be described by giving the displacement functions, which relate the location x', y', z'of a fluid element a t some past time t'to its location x , y, z at the present time t:
hv I
x ' = A1/2x
(16)
y' = A1/2y
(17)
Flgure 1. The functions p ( t ) and v(t) from (21, (3), (lo), and (11).
A(t,t? = expJtt' r:(t'?dt"
(19)
Edwards in the factor NnkT, the definition of A, and in the inclusion of the c term. Hassager6 has derived a variational principle for creeping flows for the limiting case of € = 0. We note for later use that
, t ? ,components 7:; = The finite strain tensor ~ [ ~ ] ( twith C k ( d x ~ / a x , ) ( a x k ' / a x , ) - ",6 is then
0 10-4
10-1
10-2
10-3
100
t/h
0
p q t , t ' )=
(;-l 0
;-1
A-'0
1
)
(20)
The tensors A(t,t?and B(t,t? are now found by using (20)
and therefore
AmA s )
x'
ds =
8 C
1
-=1
Then we let u, = sin 8 cos 6,uy = sin 8 sin 6,and u, = cos 8 and perform the integrations over 8 and 4. This gives
T 2 o , d d (Y2
v(s) ds =
16 r4a , d d
1 1 =CY^ 6
3 A,, - A , , = 2
The expressions in (2) and (3) are not numerically convenient for small values of the argument. Use of the Poisson summation formula' allows us to write p ( s ) thus
(for A
-(
A3
1-
~ 3 - 1
> 1)
3 A,, - A , , = 2
-(
A3
1 4 3
arctan (A3 - 1)lJ2 (A3 - 1)1/2
arctanh (1 - A3)lJ2 (1 - A3)lI2 (24)
(for A
and then ..
=---
x
< 1)
4, - B,, --
xr:
.
(E,,[ x
1
+2 3
8(A3 - 1)'
C (-1)" x ...
8A6 + 17A3 + 2 - (15A6
+
arctan (A3 - 1)'l2 12~3) (A3 - 1)1/2
n=1,2,
(for A > 1) (11) The functions p ( s ) and v(s) are plotted in Figure 1.
Bz, - B,, -
8A6 + 17A3
A6
]
The Kinematic Tensors for Elongational Flows We consider here flow fields of the type u, = -Y2r:(t)x
(12)
uy = -y..e(t)y
(13)
(14) in which i(t) is the elongation rate. For thisflow the tensor +(t)= Vv + (Vv)+is u, = e(t)z
(6) 0. Haasager, 'Variational Principle for a Restricted KBKZ Rheological Equation of State", DTH Report, Jan 22,1981. (7) E. Madelung, 'Die Mathematiachen Hilfsmittel des Physikera", 7th ed,Springer, Berlin, 1964,p 69.
+ 2 - (15A6 +
arctanh (1 - A3)lI2 12~3) (1 - A3)'/' (for A
(26)
< 1)
Equations 23 and 24 are in agreement with the results of Doi and Edwards5 (see their In,4.11). For values of A near unity, we can use the Taylor series for arctan, arctanh, and exp to derive useful expressions 3 9 A,, - A,, = - In A + -((In A)2 ... (27) 5 70
+
Bzz-Bxx
xr:
1 6 = - + - In A 5 35
+ ...
(28)
1104
The Journal of Physical Chemistry, Vol. 86, No. 7, 1982
Bird et ai.
_.
- Ae
+op
10-2
10-1
IO0
h h - A,)
Figure 2. The functions (A, flow from (23)-(28).
and (B,
- &)/Xi
for elongational
which are valid for both positive and negative values of In A. The functions (Azz- A,,) and (Bzz- B,,)/Ai are shown in Figure 2. Steady Elongational Flow We first consider the steady-state flow for which i: is a constant, and we obtain the elongational viscosity q = (T,, - ~ ~ ~ )For / ismall . values of Ai-, both positive and negative 1400
+ -)X2i2 78400 58800
+ ... (29)
as reported earlier.* This shows that dqldi at i- = 0 is positive. For large positive values of A i , it may be shown that lim q = lim NnkT( 1 hl-m
A&--
€
+ :EA&)
= itNnkTX (30)
-
To obtain this we use (8) and (9) and the fact that both (Azz- A,,) and (Bzz- B,,)/Xi approach unity as In A m. From the last two equations we obtain
-
Similarly in the limit as In A -00, (Azz- A,,) approaches -1/2, and (Bzz- B,,)/Xi approaches 1/4, so that
IO'
Flgure 3. Elongatlonalviscosity as a function of elongation rate i: for various values of the link-tension coefficient t. The time constant X is N3+@{a22/2kT.For small values of IXil, (29) can be used. TABLE I : The Functions A€
1 2 5 10 20 50 100 200 500 1000
102{*
105a
5.021 5.040 5.090 5.144 5.153 4.750 3.876 2.718 1.415 0.7929 0.4268 0.1815 0.009345
3.363 3.394 3.498 3.704 4.211 5.956 8.317 10.99 13.77 15.06 15.81 16.30 16.48
sa and sRfor Use in (33)
-
-A;
loz;B
0.1 4.978 0.2 4.955 0.5 4.878 1 4.732 2 4.406 5 3.497 10 2.523 20 1.604 50 0.7726 1 0 0 0.4196 200 0.2216 500 0.09280 1000 0.04746
3.306 3.280 3.212 3.128 3.048 3.098 3.317 3.595 3.880 4.009 4.083 4.131 4.149
Note that for a given value of 6 , the steady elongational viscosity q(i-) for any value of E can be obtained from q(i) = q A ( k )
+ tqB(i)
(33)
Values of q A and qB, arising from the integrals containing the tensors A and B, respectively, are given in Table I for several values of A&. Elongational Stress Growth Next we consider flows for which k(t) = 0
These expressions are probably not useful for estimating t, since q ( m ) and q(-m) are too difficult to measure. The complete set of curves of q(i) for various values of t is shown for both positive and negative values of i in Figure 3. These curves were obtained from the elongational growth viscosity as t m; this is discussed in the following section.
103
xi
0.1 0.2 0.5
1050
too
10-1
= iofor
for t C 0 t20
(34)
so that from (19)
A(t,t') = exp[io(t - t')] = exp[iot] for
for 0 I t ' l t < t'5 0
--co
(35)
We use this result in (20) through (26); the elongational
The Journal of Physlcal Chemlstty, Vol. 86, No. 7, 1982 1105
Kinetic Theory for Polymer Melts
TABLE 11: Algorithm for Romberg Integration with Integrand f ( s ' )
-"
101
7 (e) -
IHDPE I,T
select value of A;, select set of values for t / h ( e . g . , 0,
continue until
C
'
IT^ - T ~ : ; I
G
3
i
Io-' l o 0tl
\
io-;o-l'
10-8
4
x €- = 1000
150°C X = 30 sec 3.T 150°C
v HDPE
2x 5 x 10-4,10-3,2 x 10-3,...I for i = 1, 2, .... d o set h =' (t'/h)it1- ( t / h ) i calculate T: = h { f [ ( t / h ) i ]+ f [ ( t / h ) i t , ] } / 2 f o r k = 1. 2. ..., do set h = h / 2 2k-1 calculate T i = 1/2T&-, +h f [ ( t / h ) i + (3l)h] j= 1 for m = 1, ..., k d o - T7-;1)/(4m- 1) Lcalculate TI: = T7-l + (T-I
;
' " ' ' ' ' ~
'
I
" "
'"",'I
100
'
'"',,"
J
1
'
IO 1
3
E = 0.05
'"",,I
A i
IO
IO
Flgure 5. Comparison of experimental steady elongational viscosity with calculated curves from the Curtiss-Bird theory; e = 0 corresponds to the Dol-Edwards theory: (H)data of Laun;' (V)data of Munstedt and Laun." Note that in plotting the experimental values of Q(i)/ij(O), the quantity HO) was taken to be 3q0, where qo is the zero-shear-rate viscosity, determined experlmentaliy.e~'O lot
1 10-5
10-4
10-3
t/ x
10-2
10-
'
4
10-1
1 1 . 1
3
IO0
Figure 4. Elongationai growth vlscosity as a function of time for different values of the elongation rate io. Each family of curves corresponds to one value of the link-tension coefficlent, e.
I E 10-2
=0.00 ,
,
,
,
,,I
v +€o ,
,
, , ,
I , ,
I
I
=0.3 =0.1
, , , , ,,
,I
I
,
,
,
growth viscosity, defined by q+(t;io) = [r,,(t) - r z 2 ( t ) ] / i O , is now q+(t;io) --
NnkTX
(36) The subscripts s = t - t'and t indicate that in these expressions A is exp(@) and exp(kot), respectively. By changing the variable of integration in (36) from s to s' = S I X , we can rewrite q+ in the following way:
1
r.m
which is a finite quantity independent of i,,. The experimental data do not go to sufficiently small values of t to test this prediction.
Data Comparisons Here we compare the elongational viscosity curves calculated in the two preceding sections with experimental data. Figure 5 shows steady-state elongational viscosity data for two HDPE meltsgJOat 150 "C. The values of X and e were chosen by finding the combinations that would give the best fit with the data at intermediate and high elongation rates. These values are Laun's HDPE 1 Munstedt and Laun's HDPE 3
The subscripts s' and t / X indicate that now A is given by exp(Xe0)(s?and exp(Xko)(t/X),respectively. To evaluate the second and fourth integral in this expression we make use of (6) and (7). The two remaining integrals were evaluated numerically by Romberg integration.8 The
h=30s h = 18 s
~=0.08 E = 0.08
(8) S.D.Conte and C. de Boor, "Elementary Numerical Analysis: An Algorithmic Approach", 2nd ed, McGraw-Hill, New York, 1972, p 315. (9) H. M.Laun in "Rheology",Vol. 2, G. Astarita, G.Marrucci, and L. Nicolais, Ed., Plenum Press, New York, 1980, pp 419-24. (10) H. Mhstedt and H. M. Laun, Rheol. Acta, 20, 211-21 (1981).
The Journal of Physical Chemistry, Vol. 86, No. 7, 1982
1106 IO)
'
'
" " ' I
""',
'
'
Bird et al.
7
"'
54
E =
E 3 T=15O0C DPE
E -
3 k
Io-'
€ = 0.2'
0
10'2
t I0"l 10-4
,
, ,
,,,
, 1 .
10-3
, ,
1
1
1
,
1
10-2
,
,
t/X
.,,
,
,
10-1
,
, , ,
,
I
,
\ \
POLYSTYRENE T = 170°C X = 31 sec
I
I00
Flgure 7. Comparlson of experimental elongational growth viscosity with calculated curves from the Cwtiss-Bird theory. The combination of X and t (18 s, 0.08) that gives the best fit with the steady elongational viscoslty is used. Data of Mirnstedt and Laun.''
These two particular samples were chosen because the data covered a wide enough range to make a curve-fit possible. Theoretical curves of q+ obtained with these combinations of X and e are compared with the corresponding elongational growth data in Figures 6 and 7. The fit of the theoretical curves for both q and q+ is encouraging; however, it must be kept in mind that the theory was developed for monodisperse systems, whereas the data are for polydisperse samples (&,/&fn = 13.9 for Laun's HDPE 1, and 13 for Miinstedt and Laun's HPDE 3). Recently van Aken and Janeschitz-Kriegl"J2 have inferred fl for negative values of i from birefringence measurements in a biaxial extension experiment utilizing two impinging fluid streams guided by lubricated trumpets. They found that +jdecreases monotonically with increasing -$ in qualitative agreement with the left side of Figure 3. Their data, for polystyrene melts with Mw/iCln= 2.76, are shown in Figure 8.
Conclusions Inclusion of the link-tension coefficient t in the kinetic theory of melts leads to elongational viscosity curves with shapes rather different from that of the t = 0 curve. In particular it is seen that for high elongation rates, the limiting value of the elongational viscosity increases with increasing t. One possible interpretation of this is that the link-tension coefficient would be larger for chains with large side groups or with some branching; this would mean (11) J. A. van Aken and H.Janeschitz-Kriegl,Rheol. Acta, 19,744-52 (1980). (12)J. A. van Aken and H. Janeschitz-Kriegl, Rheol. Acta, in press.
IO"
IO0
IO'
102
loJ
-A2 Figure 8. Comparlson of the biaxial extension data for a polystyrene melt (Hostyren N 4000 V) of van Aken and JaneschitzXriegl" with the Curtiss-Bird theory. The data have been temperature shifted to 170 "C. In plottlng the experlmental values of Q(t)/Q(O), Q(0) is taken to be 31,, where v0 is the experimental zero-shear-rate viscosity.
that the high elongation-rate limiting values, q ( m ) would be expected to increase with increased size of the appended groups-that is, with the "roughness" of the chain. A correct experimental test of the theory has yet to be made because of the lack of elongational viscosity data for monodisperse samples. Alternatively, the theory needs to be generalized to take polydispersity into account. It has long been known that elongational viscosity comparisons provide a very good test for molecular theories, inasmuch as this property seems to be quite sensitive to the molecular model used. Figure 3 suggests that for IXil I1, it is a good approximation to take q = 377& This may be a useful estimate for determining whether or not stretching motions are important in viscoelastic fluid dynamical calculations. Comparison of Figure 3 with the corresponding theoretical results for polymer solutions suggests that melt elongational viscosity changes much less dramatically with elongation rate than does the solution elongational viscosity.
Acknowledgment. We are greatly indebted to the National Science Foundation for financial support provided for molecular theory studies, R.B.B. and H.H.S. for Grant ENG78-06789 and C.F.C. for Grant CHE79-05685. In addition R.B.B. acknowledges support provided by the Vilas Foundation of the University of Wisconsin. Also conversations with Professor H. Janeschitz-Kriegl (Linz), Professor J. Meissner (Zurich), and Professor W. W. Graessley (Northwestern) have been helpful.
