Dissipative stresses in dilute polymer solutions

E L S E V I E R J. Non-Newtonian Fluid Mech., 68 (1997) 61-83

Jemulet"

Dissipative stresses in dilute polymer solutions

J . M . R a l l i s o n

Department of Applied Mathematics and Theoretical Physics, Cambridge University, Cambridge CB3 9EW, UK

Received 19 April 1996

Abstract

Planar extensional flows of a dilute polymer solution are investigated using a free-draining bead-rod model. For steady flows, an analytic expression for the probability density of the polymer configuration is available. It is found that part of the associated steady polymer stress is unambiguously viscous at all time scales, in the sense that on cessation of flow it disappears instantaneously, but, except at very high flow rates, the elastic component is larger.

A Brownian dynamics simulation of the chain is constructed for start-up flows for which no analytic expression is known. A stress that is apparently viscous is found to develop alongside the elastic stress, having comparable magnitude at moderate flow rates. An interpretation of this result for a system having a wide spectrum of relaxation times is given. This feature is not captured by conventional FENE constitutive equations, and a novel model is developed. The consequences for calculations of complex flows are briefly discussed. © 1997 Elsevier Science B.V. All rights reserved.

Keywords: Dilute polymer solutions; Dissipative stresses; Constitutive equations

I. Introduction

Dumbbel l models o f po lymer solut ions suppose tha t the de fo rma t ion of an individual molecule m a y be described in terms of a vector R, or equivalently by a m o m e n t o f inert ia tensor A, the ensemble average o f RR. It is convenient to scale the molecular size so tha t in the absence o f flow A = I. The de fo rma t ion o f the molecule in a flow field u(x, t) then evolves in t ime by an equa t ion of the fo rm [1]

= A . Vu + Vu T. A - ~lwf(tr A) (A - I). (1) A

Times here have been scaled by the inverse fluid strain rate, W is a Weissenberg number , and f is an elastic force generated by entropic f luctuations. A t h e r mo d y n a mi c a rgument [2] for a tens ioned chain o f inextensible links at equi l ibr ium gives the F E N E spring f as an inverse Langevin funct ion, bu t for ma themat ica l convenience a c o m m o n choice [3,4] having the same physical character is

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62 J.M. Rallison / J. Non-Newtonian Fluid Mech. 68 (1997) 61-83

f = ( 1 ---L-~-]tr/i'~ '

This limits the maximum value of (R 2) to L 2, and hence the maximum linear distortion of the chain to L. Since a random-walk coiled polymer containing N monomer subunits has a radius of gyration proportional to N 1/2, whereas the fully stretched length of the same chain is proportional to N, a value of L scaling as N m should be appropriate.

For weak flows, with W<< 1, (1) has the steady solution

,4 = I+ 2 W E + O(W 2) (2)

where E is the rate of strain tensor. Furthermore, this equilibrium is attained in a (short) dimensionless time of order W. On the contrary, for strong flows with W>> 1, the steady state has stretched molecules for which trAi= L 2. The coil-stretch transition between these two behaviours takes place at W= ½.

The corresponding stress in the solution is usually taken (to within isotropic terms) to be of the form

¢

a = 2E + -~ f(tr A)A, (3)

the first term arising from the Newtonian solvent with unit viscosity, and the second proportional to the equilibrium volume concentration c < 1 of the polymers. For weak flows the zero-shear viscosity of the solution is given by (2) as 1 + c. In fact the model given by (1) and (3) predicts a steady shear viscosity of 1 + c for all values of W. The FENE-P variant [5] exhibits shear thinning by an amount c as W increases, but its behaviour in extensional flows is similar to that of the model discussed here.

In examining viscous and elastic contributions to the stress, it is of interest to consider the energy budget for such a fluid. For an inertialess incompressible flow the rate of working by external forces on a region V is given by the integral of the stress power a:E and is thus

f a:EdV=2f E:EdV+wff(trA)E:AdV. At high flow rates the polymer deformation is affine until full extension is reached, and Eq. (1) gives

tr A[ = A : V u

and so, writing U(tr A) as the internal energy in the spring defined as

f3 tr A

U = f ( x ) dx,

we have

f f ~ : E d V = 2 E:E d V + - ~ U ( t r A ) d V . (4)

For strong enough flows, then, Eqs. (3) and (4) imply that viscous dissipation arises from the solvent, and at a rate given by the solvent viscosity (there is no viscosity enhancement) and,

J.M. Rallison /J. Non-Newtonian Fluid Mech. 68 (1997) 61-83 63

provided that full extension is not reached, the polymer contribution to the energy budget is generated solely by storage within the elastic springs.

For rapid flows through constrictions in channels, all the elastic energy stored as the molecules stretch upstream is recovered as they relax downstream, and hence the overall pressure drop arises from the solvent viscosity. This value may nevertheless be larger than the pressure drop that would arise from the flow of the solvent alone, because the elastic stresses can modify the flow field. Since the elastic stresses do not directly cause the increased flow dissipation seen in some contraction flow experiments for which the overall strain experienced by a molecule is insufficient to extend it fully [6,7], the suggestion has been made [8,9] that an additional polymeric stress proportional to E should be included in Eq. (3) for only moderately extended dumbbells. Hinch [9], on the basis of computer simulations of stretching chains, specifically suggests a contribution proportional to A:EA.

Similar considerations apply for rapid flows past obstacles, but the associated flow fields are more complex, and hence the interpretation of the drag results is more difficult. The presence of a stagnation point at the rear of the obstacle implies that, no matter how rapid the flow, molecules that form the birefringent strand in the wake become fully stretched, the corresponding stress is dissipative, and this increases the drag. This contribution is additional to the recoverable stress that arises near the sides of the obstacle, and the balance between the elastic and viscous effects depends in a complex way on both W and L [1].

There are three additional reasons for favouring a dissipative stress term. First, several, otherwise successful, calculations of non-Newtonian flow using FENE equations (e.g. [1,10,11]), find that the computed flow best matches the experimental data when values for L of 5 or less are used, and these are much smaller than reasonable estimates of N t/2. One effect of using a small value for L is that, after only a small degree of stretch, further stretching is impossible, and so (1) gives

f ( t r A) ( A - I) ~ W(A.Vu+ VuT.A)

and thus from (3)

= 2E+ c(A.Vu+ Vu T'A).

It follows that the polymer stress, although elastic, has a value proportional to the instantaneous flow rate, and thus appears to be anisotropically viscous (just as for a suspension of rigid rods). The choice of a small value for L may thereby mimic a dissipative stress.

Second, in sink flows where substantial polymer stretch occurs in the radial f direction, the elastic stress ( c / W ) f ( t r A)A has the form g ( r ) ~ . The corresponding body force V. a is conserva- tive, and so does not modify the flow in a confined region. The inclusion of an anisotropic viscous stress may therefore have a disproportionately large effect in creating vorticity in the flow.

Third, it is well known that a polymer chain exhibits a spectrum of relaxation modes and times, and only the mode having the longest relaxation time is captured by a dumbbell model. Faster-decaying modes, although elastic when viewed on a short enough time scale (even water is elastic on a time scale 10-14 S), will appear to be viscous on a longer time scale, and so, to account for these short-lived modes within a pragmatic dumbbell framework that suppresses the internal degrees of freedom, an explicitly viscous stress should appear. The alternative is to

64 J.M. Rallison /J. Non-Newtonian Fluid Mech. 68 (1997) 61-83

include a spectrum of relaxation times in a model to represent the internal degrees of freedom; this may indeed be necessary for flows that are strong enough, or which vary sufficiently rapidly in time. The corresponding stresses will then be elastic, but any computational procedure must then resolve the shortest times included.

In this paper we reconsider the inclusion of a dissipative contribution to the polymer stress. For reasons of simplicity and computational convenience we examine in particular a planar extensional flow given in cartesian coordinates by u = E(x, -y). The only measurable stress is then 0-1~- 022, and we define a planar extensional viscosity [4] as

/'/e = (0-11 - - 0"22)/E. ( 5 )

The planar extensional viscosity of the solvent is then 4, and as W---,0, that of the solution is 4(1 + c). In this limit we may therefore put

c = ( ~ e - 1 ) / 4 . (6)

In the following sections we calculate dissipative and elastic contributions to /~e by means of analytic results for steady flow (Section 2), and numerical results for start-up flow (Section 3). We find that although when viewed on an arbitrarily short time scale the viscous stress is relatively small, the effective viscous stress that accounts for the internal modes can be comparable with the elastic stress when viewed on the longest relaxation time scale. A model showing these effects is presented in Section 4 and compared with some experimental data. Conclusions are given in Section 5.

b2

f I

f Ro

"k

- J

u "-- E ( x , - y )

Fig. 1. Definition sketch of a planar bead-rod chain.


2. Steady extensional viscosity for a Kramers chain

We first examine elastic and dissipative stress contributions for a planar bead- rod chain, sketched in Fig. 1, subject to Brownian motion, and placed in a steady planar extensional flow. Some analytic progress can be made in this case.

Consider a free-draining chain having N inextensible connectors each of length b and orientations given by the unit vectors hi, i = 1,..., N. The rods are freely hinged. The N + 1 beads have positions R;, i = 0,..., N and each has a constant friction coefficient (. For a chain lying in the (x, y) plane we may put bi = (cos 0i, sin 0~). As noted by Hassager [4,12] an exact solution due to Kramers is available for the equilibrium probability density p for the vectors R~ which takes the form

N

p oc exp((E: Y' R ~ / 2 k T ) i = 0

N

-- 1 C O S ( 0 i AF Oj)], ocexp[P ~ As/ i , j= 1

the constant of proportionality being determined by the normalization of p. The flow strength is defined by a dimensionless monomer P6clet number P = (b2E/2kT. The quantity A,.j is the Kramers matrix [4] with

f 2 i = j Ai~= - 1 j = i + l 0 otherwise

and A~ 1 is its inverse, given by

Si(N + I - j ) / ( N + I) i <_j Ai] 1 [f iN+ l - i ) / ( N + l) j<_i.

The eigenvalues of A are 4 sin2[n~/2(N + 1)] for n = 1 .... , N. It follows that for large N the eigenvalues are widely spaced, the largest being a factor N 2 bigger than the smallest. This spread is of course reflected in the wide spectrum of Rouse relaxation times for the undeformed chain; the longest of these is ~ = ~b2N2/2~ZkT= N2P/rc2E. the coil-stretch transition should therefore take place when the strain rate exceeds 1/~, and, since this corresponds to a chain P6clet number w, of ½,

W= N2P/2r~ 2 = O.051NzP. (7)

For a steady state, Pe is conveniently given by the Giesekus expression [4] for the polymer stress as

N

/ ~ e - 4 = n ( b 2 ~ (A,)-' cos (0 ; -0 j ) ) i , j= 1

where n is the number density of polymers, and the angle brackets denote an average over the 0~ with probability density p. For P--,0, p = constant, and the sum on the right is easily evaluated as ~N(N + 2). We therefore have from (6) that the equilibrium concentration of polymers is given as


n(b 2 c = - - N(N + 2).

24

So by writing

6 N

Vl¢- N(N + 2) Z i,j= 1

we have

(A,; cos(0,- 0j)), (8)

/re = 4(1 + cv/¢);

thus v/o is the contribution to/z~ of just one chain. Now We contains both elastic and dissipative parts. We may calculate how much of the stress

is unambiguously dissipative even on the shortest time scales by imagining that the Brownian forces on the chain were suddenly to cease, but that the flow continued. Then, at that instant of time, the elastic recoil of the chain would also cease, but dissipative processes would continue, because the rigid parts of the chain cannot deform affinely. In the linear viscoelastic regime, this is the part of the stress that contributes to the high frequency plateau in the complex viscosity (see the discussion in Larson [13], Section 8.2.3).

If the tension in the ith link of the chain is T,, then, because inertia is negligible, the force balance on the beads gives

7",.+ l b ; + l - - 2 T,.b, + T~ _ , b , _ , = (b(hg - E ' b i )

and, using the inextensibility constraints, b,-b~---0, the tensions are determined by the matrix equation

N

E A~jTj =~l~i '~' ' l~i j = l

where

f 2 i = j A*= - b i ' b j j= i+_l 0 otherwise.

The polymer stress is nb Z N (T,b,bi), and thus the viscous part V/v of the planar extensional viscosity, analogous to V/e, is given by

6 N qV- -N(N+ 2) ~ ( c ° s 2 0 ; A * - I c°s20i)" (9)

i,j= 1

In sufficiently strong flows for which the monomer P6clet number P-~ 0% all the rods become aligned in the x direction, each 0; ~ 0, and hence .4 * ~ A, giving qe = V/v. Thus, ultimately, all the stress is viscous.

On the contrary, the condition in a chain of N links that a coil-stetch transition should occur is that the chain P6clet number, proportional to N2p, should exceed a critical value, and for large N this will occur when P << 1. Indeed, if P itself becomes comparable with unity, in a


qe

qv f

I I I I I i I I I I I I I

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 P

0.55 1.10 1.65 2.20 W

Fig. 2. Steady planar extensional viscosities V/e and V/v for a b e a d - r o d chain with N = 6 and varying P. W = 0.18P. - - , results obtained from Eqs. (8) and (9); marked points and error bars show the steady state values of, and fluctuations in, qe obtained from simulations in Section 3; - - -, asymptotic value as P ~ ~ .

stretched state the tensions in the interior of the chain, which scale with N2p, will be so large as to exceed the strength of a ca rbon-ca rbon bond, and the chain will break [14]. Of practical interest, then, are flows for which W is comparable with unity.

Shown in Fig. 2 is a numerical computat ion of r/e and t/v for N = 6 and varying values of P. Similar computat ions have been performed by Doyle et al. [15]. Calculation of the six-dimensional integrals over 0~,..., 06 is, in principle, straightforward, using the exponential accuracy for periodic functions of the trapezoidal rule, but as N is increased the computat ional resources required for such an evaluation, which scale as K u for some K (and, in practice, K > 16), become impractical. For P ~ 0 we find that r/e ~ 1 and V/v ~ 0.29. For P--, ~ both have an asymptotic value of 3.5. It is apparent from Fig. 2 that a coi l -s t re tch transition for/~e takes place over a range of flow rates between P = 0.2 and P = 0.4, for which Eq. (7) gives W = 0.37 and W = 0.73 respectively. These values lie on either side of the critical value W = ½ as expected.

The corresponding rise in r/v is postponed to larger values of P. There is, however, an early increase in qv to a level comparable with unity when coi l -s t retch occurs. Similar trends are seen in Figs. 3 and 4 for the cases N = 4 and N = 8 respectively.

Direct evaluation of the integrals for larger values of N is difficult. A Monte Carlo technique proves adequate to evaluate v/e for values of N up to about 10, but, as explained below, the


integral for qv is sharply peaked, and, even with 107 evaluations, only poor accuracy can be obtained for N--- 10 by this method.

Of greatest concern, of course, is the behaviour of r/c and r/v for large N. Hassager [12] has noted that for large enough flow strengths the approach to the fully extended state can be examined by asymptot ic methods. Suppose that P is sufficiently large that all the 0; are small. Then as P ~ 0% it suffices to keep only quadrat ic terms in 0;, and

p o c e x p - P y' Lo.Ofl j i , j = 1

where

L,j= f ½(N+3)A I i= j

[ i¢ j .

It follows that for large N the fluctuations in the 0; are of size (NzP) - 1/2 for the central links with i close to ½N, and (NP)-1/2 for the end links with i close to 1 or N.

We may similarly simplify the expression for r/o to write N

r / c=½(U+ 1 ) - Z M~(O,Oj), i , j = 1

2 11e

1

I I I I I I I I I I I I

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 P

0.49 0.98 1.47 W

Fig. 3. Steady planar extensional viscosities t/~ and t/V for a bead-rod chain with N = 4 and varying P. W= 0.082P. , results obtained from Eqs. (8) and (9); ---, asympotic value as P--* oo.


3

t

| I I I I

0.1 0.2 0.3 0.4 0.5

0.65 1.30

I

0.6 P

1.96 W

Fig. 4. Steady planar extensional viscosities qe and r/V for a b e a d - r o d chain with N = 8 and varying P. W = 3.26P. - - , results obtained from Eqs. (8) and (9); marked points and error bars shown steady state values of, and fluctuations in, qe obtained from simulations in Section 3; - - - , asymptotic value as P ~ ~ .

where

3(NN__~6~ 1 ) M~= ~ A:~' i=j

~ - ~ Air K i ~j

To evaluate t/v we need the corresponding simplification of A* for small values of 0;. Noting that

A*. = A~ + 0 U

with


0~= f l Oi -- Oi) 2 j = i __+ 1

otherwise

we have

A*-1 = A~ ' - A~ lOk~Aif l + O(OA -z).

It follows that N

~/v=½(U+ 11- Z M~.(Oflj) i , j = 1

where

(lO)

f 3 ( N - 3)/(N + 2) 3i(N + 1 - i)[3 + (N + 1)i - i2]/N(N + 2)

M~. = - 3q(N + 1 - q)(q + 1)(N - q)/N(N + 2)

0

i = j = l , N i = j ~ l , N i = q , j = q + l or i = q + l , j = q otherwise.

For P---, ~ it may be shown [12,16] that

q~ ~½(N+ 1) + P -1L , 7 ~M~. + O(P -2)

~/v---,½(N + 1) + P - ' I. - l lt/tv + O(P -2) ~ i j " ' ~ i j - -

and on performing the matrix inversions for large N we have

,N(, 1) r/v = ~

It follows that ~/e approaches its asymptotic level when the chain P6clet number N2p exceeds a value of order unity, whereas the viscous part qv does not increase to the same level until the monomer P6clet number P is of order unity.

The physical reason for the latter result can be seen from the chain configuration sketched in Fig. 5. When Brownian motion is suppressed, the tensions T in the central segments are given for small 0 as

N 2 T ~ 1 + N202

and thus vary rapidly in a region of size N - 1 near 0 = 0. In other words the viscous tensions are much lower for a chain having a small amount of "give" than for a fully stretched chain. As noted above, the central links of the chain fluctuate in the steady state by amounts of order (N2p) - 1/2 about 0 = 0 and so these tensions, and similarly ~/v, reach their asymptotic values only for P larger than 1.


We may also note in passing that because the difference (of N / 2 P w. N3/W) between qc and V/v is accounted for by elastic effects, the elastic stress must saturate as P---, ~ for a single straight planar chain at a level proportional (in dimensional terms) to k T N 3, in agreement with the result in [17].

We are therefore led to the conclusion that for large N, and at physically realistic P6clet numbers W, the steady dissipative contribution to r/c is relatively small. The results of Figs. 2 and 3 and 4 suggest (for small N) that for such flow rates, the ratio of V/v to qe is perhaps of size N - or N-1/2. We have not been able to determine this ratio analytically when N is large. The mathematical difficulty is the inversion of A * for small 0i needed in Eq. (9). Eq. (10) is inadequate for this purpose because neglect of the error term needs A - 10 to be small. The Kramers matrix has some eigenvalues that are as small as N 2, however, and so the error term is negligible only if every 0i << N ~, and, as noted in the physical discussion above, this requires that P >> 1.

From the analytic solution we have nevertheless shown that there is a dissipative polymer stress and this stress is of dominant importance for steady, very strong flows. For weaker flows, however, the size of the steady dissipative contribution appears to be small when N is large. Its level in unsteady flows remains an open question.

An important but more subtle issue is not addressed by our analysis. We have noted that modes having very short time scales are present. In consequence, on cessation of flow these modes will relax very rapidly. Although the stress associated with such a mode is elastic, on the chain relaxation time scale such stresses will appear to be viscous. In other words, our calculation of V/v underestimates the effective viscous stress measured in an experiment that is unable to probe the shortest Rouse scales.

One way of examining these different contributions would be to perform a modal decomposition of r/~. Unfortunately, this requires the calculation of the eigenvalues and eigenfunctions of

5 ° p = ~-~A *-1 ~,j = l abj p

the operator and, for the ill-conditioned matrix A*, these eigenfunctions are not known.

0 0 -- 0 v~'-~--- -'~-'~0 0 -- "~ v 0

Fig. 5. Instantaneous tensions in a long straight chain having a small kink near its centre and subject to planar extension.


In one extreme case, however, the short-lived stress response has been determined. Grassia and Hinch [17] have examined the collapse under Brownian motion of an initially straight chain. The associated stress per molecule falls from a level k T N 3 to k T N via a cascade of time scales. In an initial phase, the stress drops to the level k T N 2 over a time r/N4; then, exploring the entire spectrum of Rouse relaxation times from r / N 2 to r, a further fall takes place to the level kTN. The stress finally relaxes to equilibrium by an exponential decay at a rate (close to that) given by the longest relaxation time r. An experiment that can measure stress only on the time scale r will see only this final exponential decay, and will thus conclude that the plateau elastic stress level for a straight chain is proportional to k T N (corresponding to r/e oc N~ 140, and the unseen fast initial decay from the true level k T N 3 would therefore be attributed to a viscous response.

In the next section we construct a Brownian dynamics simulation of a long chain in a transient flow, and seek to correlate the stress levels with the largest structures in the flow.

3. Numerical simulations

We next consider a numerical simulation of a bead-(stiff-) spring chain, immersed in a planar extensional flow, and calculate the steady and transient planar extensional viscosity defined by (5). The method is described in [17], is easier to code than a simulation involving inextensible connectors, and has already been used by Hinch [9] to examine the start-up of uniaxial extensional flow. We do not repeat the details here.

In [9] it is found that large viscosities can arise well before the chain is fully extended, that the associated stresses scale with the applied flow rate (i.e. they are viscous and not elastic in origin), and that they are proportional t o / I 2 rather than A. It appears that the origin of these transient viscous stresses is the appearance of substantially straight segments within the stretching chain (first seen in simulations of Acierno et al. [18]) which evolve by a relatively slow unfolding of kinks (see also [19]). The simulations in [9] suffer however, from the drawback that the monomer P~clet number P is (typically) 0.2, and thus the chain P6clet number N a p is unphysically large. As noted in Section 2, large viscous effects are to be expected in the steady state when P is of order unity, but may be greatly reduced for P comparable with N-2, so the conclusion may be misleading.

We have therefore performed simulations at much lower values of P for which the monomer dynamics is dominated by Brownian motion. Our initial condition for the polymer configuration is a planar random walk having N links of unit length. This free-draining chain is subjected to a planar extensional flow, and Brownian motion is introduced using the algorithm described in [17]: within each time step of size 5t a constant force (6/St)l/2ni acts on the ith bead, with the (two) components of ni independently and randomly distributed on [ - 1, 1]. As shown in [17] the expression for the entropic polymer stress may be written so as to reduce the size of fluctuations to a magnitude of order unity, rather than (6t)-1/2, and this device improves the quality of the data over the algorithm used in [9].

The simulations are expensive to run for large values of N (greater than 100, say) because 8t is restricted in size by the magnitude of fluctuations permitted in the length of a single monomer. A steady state configuration cannot in general be reached until the chain has experienced a total strain of at least one. In consequence the total number of time steps for a chain P6clet number


of unity scales as N 2, and the number of arithmetic operations involved for each run is thus proportional to N 3. In addition, to obtain acceptably accurate ensemble averages, it was found necessary to perform an arithmetic mean over at least 30 runs; a substantially higher figure would be desirable.

3. I. Steady results

In Figs. 2 and 4 we show, for some short chains (N = 6, N = 8), the long-time average values of ~/e for varying P. The error bars indicate (for the arithmetic mean of 30 runs, and further time-averaged over a strain of 0.2) the size of fluctuations in the apparent steady state. The errors are larger, in both absolute and relative terms, for smaller values of P. Notwithstanding this difficulty, the results from the simulations show agreement with the analytic solutions of Section 2, lending confidence to the accuracy of our computational algorithm, and showing that for this purpose a stiff-spring system is equivalent to a bead - rod chain. Similar results have been produced by Doyle and Shaqfeh [15]. The interpretation of r/v for a bead-spr ing chain is more ambiguous, so we have not computed it here.

3.2. Start-up results

We next consider, for start-up flows, the time evolution of the polymer deformation, and of r/e.

For a random-walk polymer the moment of inertia tensor, defined as [1/(N + 1)] EN= 0 (RiRi ) , is equal to ~(N + 2)I. We therefore define

12 N

A = ( N + 1)(U+ 2) (,.=o ~ RiRi) , (11)

so that A = I at equilibrium. Of greatest interest is the behaviour of A and r/e for large values of N: we show here results for N = 2 5 , 50 and 100. We choose flow rates P so that the corresponding values of W given by (7) are comparable with unity.

In all the simulations we find that, apart from small random fluctuations, A12 remains close to zero, as required by symmetry. A convenient scalar description of the extent of the polymer distortion is therefore provided by

AA = All --A22

Plotted in Fig. 6 against fluid strain Et is a time evolution for N = 100, P = 0.0025, and thus W = 1.3, of both AA and r/e. It should be noted that since we no longer have a steady state here, a further error is introduced by time averaging over a strain of 0.2. For short times the data are rather noisy, but the stresses in this regime are small. For intermediate strains, the growth of both the distortion and the viscosity on this log-linear plot is approximately linear (thus both are in fact growing exponentially) at a rate that is roughly half that of an affine deformation. For larger strains the results start to show a departure from exponential behaviour as the finite length of the chain is approached.

A striking feature of the results is the apparent proportionality, for intermediate strains, of ~/e and AA. This is not the A 2 dependence seen by Hinch [9] at larger P. From a strain of 1-4, the


10

3

AA

qe

0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 Strain

Fig. 6. Start-up of planar extension. Log-linear plot of AA and t/e vs. strain for N= 100 and P = 0.0025. The corresponding value of W is 1.3. Note the proportionality of AA and r/e.

extensional viscosity grows by a factor of 10. The coefficient of propor t ional i ty between r/e and AA over this range is 0.35 _+ 0.03. For strains less than 1, the data are noisy and inconclusive, but polymer stresses are small. For strains larger than 4, full extension of a chain of 100 links is approached, and the ratio of t/e to AA falls. Some recent work by Doyle and Shaqfeh [20] suggests that for larger strains r/~ becomes propor t ional to IAA ]3/2. Near full extension, tensions within the chain are much higher, and thus the springs must be stiffer in order to mainta in their inextensibility. Smaller time steps are therefore needed, and it is computa t ional ly more expensive to obtain reliable results. We therefore show data only for strains less than 4.

A second example (N = 50, P = 0.02) is shown in Fig. 7. The coefficient of proport ional i ty for intermediate strains is found in this case to be 0.30. Table 1 gives the corresponding coefficients for a range of values of N (25, 50, 100) and P, corresponding to modes t values of W > 1.

J.M. Rallison / J. Non-Newtonian Fluid Mech. 68 (1997) 61-83 75

Provided that tr ,4 is not close to L 2 (i.e. the steady state has not been reached), the non-linearity f of the spring is not important, and the proportionali ty of r/e and AA is precisely as expected for an Oldroyd B fluid for which r/e = AA/4 W. For a purely viscous stress, on the contrary, qe/AA would be independent of W. In Table 1 we show the values of ~le/AA obtained in the computer experiments, and the residual value if the 'Oldroyd B' contribution of 1/4 W is removed. It appears that for each N there is an extra additive term, independent of W so that

qe 1 ~-/~. (12)

AA 4W

The data suggest that fl is a decreasing function of N, but it is unclear from the results (which involve errors of + 0.03) whether an asymptote is reached for large N, or whether a scaling (fl ~: N-1/2 for example) is appropriate.

30

10

3

AA

I i I I I , I I I

0.4 0.8 1.2 1.6 2.0 2.4. 2.8 3.2 3.6 4.0 Strain

Fig. 7. Start-up of planar extension. Log-linear plot of AA and qe vs. strain for N= 50 and P= 0.02. The corresponding value of W is 2.6. Note the proportionality of AA and qe"


Table 1 Values for intermediate strains of the ratio qe/AA, compared with the (Oldroyd-B) value 1/4W

N P W = 0.051N2p ~Ie/AA qe/AA- 1/4W

25 0.02 0.64 0.66 0.27 0.04 1.3 0.46 0.26 0.08 2.6 0.42 0.32

50 0.005 0.64 0.55 0.16 0.01 1.3 0.37 0.17 0.02 2.6 0.30 0.20

100 0.00125 0.64 0.52 0.13 0.0025 1.3 0.35 0.15 0.005 2.6 0.26 0.16

The data are subject to statistical errors of _+ 0.03.

We have noted in Section 2 that for steady extensional flows the contribution of qv to qe is small (except at much higher flow rates than those examined here), and the question arises as to the origin of the fl AA term in r/e given by (12). One possibility is that the stretching chain explores configurations (e.g. the kinked configurations found for large W in the computations of Hinch [9] and Larson [16]) that are far from those at equilibrium, so that transient viscous stress contributions arise. This hypothesis seems unlikely, in that at a moderate W internal modes of the chain, which relax faster than the chain as a whole, should have time to equilibrate. More likely therefore is that the apparent viscous stress is a manifestation of the fast-relaxing elastic stresses generated by the internal modes. Because the effective Weissenberg numbers associated with these internal modes are low, the associated stresses must have a magnitude proportional to the instantaneous strain rate as noted in Section 1.

A partial explanation of the proportionality of these stresses to AA is provided by the Grassia and Hinch analysis [17] of the relaxation of an initially straight chain. After a fast initial relaxation, during which the reduction in chain size is small, so that AA = N, the residual elastic stress contributes an amount proportional to N / W = AA/W to q~. The same idea applied to a partially stretched chain suggests a residual elastic stress on flow cessation that is proportional to AA. If then, as in the simulations here (but not those of Hinch [9] at higher W) the total value of ~/~ is proportional to AA, then the viscous part of the stress must also scale with AA. Furthermore, the coefficient fl of the viscous term should be constant for large N. The data suggest that the constant is approximately 0.1-0.15, and hence that the contributions from viscous and elastic parts are equal for Weissenberg numbers as low as 3.

3.3. More complex strain histories

The results described above have been obtained only for the start-up of planar extensional flows. They suggest that the polymer extensional viscosity at time t is given for W > 0 as

qo=4 + f l AA


30

10 AA

3

1

Flow doubles

I ! I

0.4 1 2 3 4 Strain

Fig. 8. Double-step start-up of planar extension. Log- l inear plot o f AA and qe vs. strain for N = 100, wi th P = 0.0025 for strains less than 2, and P = 0.005 for strains greater than 2. The corresponding values of W are 1.3 and 2.6 respectively.

We may generalize this equation to the case where W < 0, by noting that the transformation W ~ - W switches the x and y directions. It follows that r/e is unchanged, but AA changes sign. The symmetry can be preserved, and the dissipative character of the final term guaranteed, by putting

= 4 f A A I)- re PlAA

The question arises as to whether this equation can describe the stress response if E and hence W are also functions of time. It is plainly impossible to investigate all possible strain histories, but we examine below two illustrative cases.


3.3.1. Double-step strain rate increase If the strain rate is suddenly doubled, then for a purely elastic stress r/~ will be halved. On the

contrary, if the stress is purely viscous, r/~ will be unaffected. Fig. 8 shows a double-step numerical experiment of this kind: for a total strain of 2 the imposed value of W is 1.3; for strains between 2 and 4 the value of W is 2.6. A chain of 100 monomers is considered. During the first phase, r/e is approximately proportional to AA, the coefficient of proportionality being 0.35 as in Table 1. For strains beyond 3 the proportionality is less close, and the constant has fallen to approximately 0.24, again in line with Table 1. For strains between 2 and 3 it appears that r/~ is no longer proportional to AA; just after the imposition of the higher value of W, r/e falls (though by less than a factor of 2), and thereafter, over a total strain of about 1, r/e recovers its proportionality to AA. This behaviour suggests a complicated rearrangement of the chain, and appears to depend in a subtle way on the internal degrees of freedom. It is thus not susceptible to measurement by a gross feature such as ,4. Equivalently, under a sudden change of flow rate, short time scales for chain relaxation are involved, and, for Weissenberg numbers of about 3, a unit strain takes place while this internal rearrangement occurs.

3.3.2. Flow reversal A more dramatic version of the same phenomenon is shown in Fig. 9 where a chain (having

50 monomers) is subjected to a strain of 2 with W = 1.3, and then the flow is suddenly reversed and continues for an equal time, so that the ultimate total strain is zero.

When the flow reverses, r/e changes sign, but its magnitude falls by a factor of about 0.5. It is difficult to obtain a precise value of r/e at flow reversal owing to our time-averaging process. This behaviour is again intermediate between a purely elastic response (where r/e would change sign but retain its magnitude) and a purely viscous one (where r/e would be unchanged). At flow reversal, the computed value of AA is 8.5. Eq. (12) with fl = 0.15 gives r/~ as the sum of - 1.6 (elastic part) and + 1.7 (viscous part).

After the flow reverses, the chain contracts faster than affinely (because both the entropic spring and the flow pull in the same direction) and AA falls. Correspondingly r/~ rises, and becomes positive again even though, for a short time, AA remains positive (this is again incompatible with a purely elastic response).

As the deformation continues, as shown in Fig. 9, A22 overtakes All, SO that both the elastic and the viscous contributions to r/e are again positive, and as the overall strain falls from 1 to 0 Eq. (12) is again able to provide a good estimate for r/e.

4. Discussion

We have shown on the basis of molecular simulations in Section 3 the existence of a significant dissipative polymer stress or, at any rate, a polymer stress that appears to be dissipative, at moderate strain rates. Although our simulations are restricted to planar extensional flows, we seek to provide a hypothesis for the form of the full constitutive equation consistent with continuum mechanical principles, against which more complex flows can be tested. A natural extension of the FENE equations to incorporate this effect is


10

5

0

- 5

AA

1 2 , qe ~ 1 . 0

s S

Flow reverses

Fig. 9. P lanar extension with flow reversal. L i n e a r - l i n e a r plot of AA and ~/, vs. s t ra in for N = 50, with P = 0.01 for s trains less than 2, and P = - 0.01 for strains greater than 2. The cor responding values of W are 1.3 and - 1.3 respectively.

A = A . Vu + Vu T. A - l f ( t r A ) ( A - I ) (13)

with

c A:EA (14) ~r = 2 E + - ~ A + o~c tr-----A

The final term in (14) has been chosen to be explicitly dissipative (since tr A > 0) and to be proportional to the magnitude of A. For large enough deformations in extension (All >> A22 or vice versa) Eq. (14) agrees with Eq. (12), but, unlike (12), is also an analytic function when Al l = A22. The coefficient ~ = 4fl gives the ratio of dissipative to elastic polymer stress contributions. A value for a of about 0.5 is suggested by the simulations in this paper. If the coefficient is left as an additional disposable parameter, greater flexibility could be maintained, but more experimental data would be needed to fix its value. As noted by Hinch [9], since the


elastic stress tends to a finite limit as W o oo, it is no longer necessary to multiply A by f i n (14): the explicitly viscous term now provides the high strain plateau viscosity.

4.1. Properties of the model

Like other FENE equations, as L ~ oo the elastic stress associated with the Oldroyd-B fluid is recovered. In consequence, the normal stress behaviour, and those phenomena associated with normal stresses (e.g. shear flow instabilities), should not be greatly affected. However, the presence of the new viscous term will tend to increase the flow resistance or drag in a given geometry.

We discuss some homogeneous flows below. Since we seek to provide a model appropriate for L comparable with N 1/2, for illustrative purposes we take L = 50, c = 1, a = 0.5.

4.1.1. Steady flows In steady uniaxial extension, the Trouton ratio increases from 3 at W = 0 to a plateau value

of 3 +oceL 2~ 1250 as W ~ oo. The transition takes place near W = ½. the contribution of the dissipative term in strong steady flow dominates the elastic term, consistent with the estimate made in Section 2.

In steady simple shear the model exhibits a slight shear thinning. The zero-shear viscosity is still 1 + c, but for W ~ oo the shear viscosity decreases to 1 + 1 ~c, the decrease taking place when W--~ L. For the parameter values above, the viscosity decrease is from 2 to 1.25. The first normal stress difference is initially quadratic, but ultimately linear in shear rate; the second normal stress difference vanishes. (The FENE-P version has an asymptotic steady shear viscosity of 1 as W ~ oo and therefore shear thins to a slightly greater extent). Choosing a slightly larger value for a can reduce the extent of the shear thinning to produce a Boger-like behaviour.

4.1.2. Unsteady flows For large values of W in start-up of pure extension, Eq. (14) predicts a transient extensional

viscosity independent of W, whereas the conventional FENE model (3) gives no polymer contribution to the viscosity for large W until the polymers are fully stretched. The new model predictions are shown for several values of W in Fig. 10. The curve for W = 5 is very close to the asymptote for W ~ oo. An interesting feature of the model is that, because the elastic part of the stress saturates as W ~ oo, the plateau value of ~/o is a decreasing function of W.

Tirtaatmajda and Sridhar [21] have measured the uniaxial extensional viscosity of a variety of polymer solutions in start-up flows. A remarkable feature of their data is that the viscosity of each solution appears to be a unique function of the total strain suffered by the fluid, and this would appear to imply that the extensional stresses must scale with the applied flow rate, and hence be purely viscous in character. At first sight, this observation provides strong evidence for a viscous stress, and is certainly consistent with the existence of a high W asymptote for /re suggested by the model. In practice, however, the scatter (on a logarithmic plot) in the data in [21] is quite large, and, for suitable choices of c and L, may be consistent with an elastic stress over a decade of W from about 1 to 10. More important is that extensional viscosities in excess of 1000 are observed in [21], and these are inconsistent with the choice of a small value for L. The greater challenge for the conventional FENE models such as (3) is to reconcile for the same

1000

material (the M1 fluid, say) the use of a small value for L for complex flows, but large L to explain the extensional data. In agreement with the model, the data reveal a decrease in the plateau in r/o as W increases.

On sudden cessation of extension, the model predicts that the viscous contribution will instantaneously disappear, leaving only an elastic stress. For large A (so that the solvent stress is negligible) the ratio of the viscous contribution to this residual elastic part is simply a W independent of A. Some recent data from Sridhar [22] on 'fluid B' suggests a decomposition into viscous and elastic parts of this kind, with the viscous contribution to stress certainly proportional to W, but scaling perhaps as A 3/2 rather than A.

For the start-up of simple shear, the model contains a weakness. Shown in Fig. 11 are transient shear viscosities, plotted for fixed W against time. Times here are normalized by the

5 f . / -

Extensional viscosity

100

10


I I I I I I I I I

0 1 2 3 4 5 6 7 8 9 Strain

Fig. 10. Start-up of uniaxial extension for the model (13), (14), with ~ = 0.5, c = 1 and L = 50. Plot of extensional viscosity against strain for W = 1, 2, 5. The curve for W = 5 is indistinguishable from that at W--* 0o.


polymer relaxation time, rather than the inverse shear rate. On this basis all the curves for a Oldroyd-B fluid would coincide. For finite L, however, curves for higher values of W lie above those for the linear viscoelastic limit, W<< 1. These features are not seen in experiment [4] where the linear viscoelastic curve appears to provide an upper bound to the transient shear viscosity at non-zero W. This undesirable feature is common, however, to all F E N E models, and occurs when W becomes comparable with L (hence its failure to appear for L ~ ~ ) . The large value for L proposed here at least postpones the difficulty to high shear rates.

5. Conclusions

Flexible b e a d - r o d polymer chains deformed by flow are commonly supposed to exert stresses that depend only on their distortion, and not on the instantaneous flow in which they are immersed. Our calculations show that additional dissipative stresses are present, whose apparent magnitude depends on the shortest time scale of an experiment. The presence of rigid, or near-rigid, elements within the chain undoubtedly generates dissipative stresses that can be seen on an arbitrarily short time scale, but except at very high extension rates the elastic part is significantly larger. More importantly, elastic stresses associated with fast internal modes (for which the effective Weissenberg number is small) appear to be viscous on the longest chain relaxation time. In principle these modes could be treated by using an elastic model that incorporates a spectrum of relaxation times [23], but, as far as complex flow calculations are

2

Shear v iscosi ty

10

I

30

50

0 I I I I

0 1 2 3 4 Time

Fig. 11. Start-up of simple shear for the model (13), (14), with ~ = 0.5, c= 1 and L = 50. Plot of shear viscosity against time normalized by the polymer relaxation time, for W = 1 (very close to the linear viscoelastic result), and for W= 10, 30, 50.


concerned, the shortest relaxation time would then need to be resolved, and the pragmatic alternative is instead to include a viscous stress in the model. The computations here suggest that this term should be proportional to the polymer distortion.

A new model that incorporates dissipative stresses of this kind is proposed in Eqs. (13) and (14). Further work is needed to evaluate its usefulness, or otherwise, in complex flows.

A potentially important omission from the simulations (and hence the model) here is of hydrodynamic interactions between different parts of the chain. The change from a Rouse to a Zimm spectrum will certainly affect the scaling of W with N (from N 2 to Nl5), and may necessitate the introduction of a conformation-dependent friction term in (14). Including this effect in simulations for large N is a formidable task, however ( N 4 rather than N 3 operations are needed for each run), and we have not yet attempted to do so.

Acknowledgements

I am grateful for the stimulating comments of Pat Doyle and others during a visit to Stanford supported by NATO Research Grant 941212, and also for helpful discussions with Tam Sridhar, David James and John Hinch at the Isaac Newton Institute in Cambridge. The support of the SERC Computer Science Initiative GR/H57585 is also acknowledged.

References

[1] M.D. Chilcott and J.M. Rallison, J. Non-Newtonian Fluid Mech., 29 (1988) 381-432. [2] P.J. Flory, Statistical Mechanics of Chain Polymers, Wiley, New York, NY, 1969. [3] H.R. Warner, Ind. Eng. Chem. Fundam., 11 (1972) 379-387. [4] R.B. Bird, C.F. Curtiss, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. 2, Kinetic

Theory, 2nd edn., 1987. Wiley. [5] R.B. Bird, P.J. Dotson and N.L. Johnston, J. Non-Newtonian Fluid Mech., 2 (1980) 213-235. [6] U. Cartalos and J.M. Piau, J. Non-Newtonian Fluid Mech., 45 (1992) 231 285. [7] A. Ambari, These de Doctorat d'Etat, Universit6 de Paris, June 1986. [8] J.M. Rallison and E.J. Hinch, J. Non-Newtonian Fluid Mech., 29 (1988) 37-55. [9] E.J. Hinch, J. Non-Newtonian Fluid Mech., 54 (1994) 209-230.

[10] J.V. Satrape and M.J. Crochet, J. Non-Newtonian Fluid Mech., 55 (1994) 91-111. [11] R.A. Keiller, J. Non-Newtonian Fluid Mech., 46 (1993) 143- 178. [12] O. Hassager, J. Chem. Phys., 60 (1974) 2111. [13] R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworth, Stoneham, MA, 1988. [14] J.A. Odell, A. Keller and A.J. Muller, Colloid Polym. Sci., 270 (1992) 307-324. [15] P. Doyle and E.S.G. Shaqfeh, J. Non-Newtonian fluid Mech., (1996). [16] J.M. Rallison, J. Fluid Mech., 93 (1979) 251-279. [17] P. Grassia and E.J. Hinch, J. Fluid Mech., 308 (1996) 255. [18] D. Acierno, G. Titomanlio and G. Marrucci, J. Polym. Sci., 12 (1974) 2177. [19] R.G. Larson, Rheol. Acta, 29 (1990) 371-384. [201 P. Doyle and E.S.G. Shaqfeh, J. Non-Newtonian Fluid Mech., (1996). [21] V. Tirtaatmajda and T. Sridhar, J. Rheol., 37 (1993) 1081-1102. [22] T. Sridhar, J. Non-Newtonian Fluid Mech., (1996) in press. [23] L.E. Wedgewood, D.N. Ostrov and R.B. Bird, J. Non-Newtonian Fluid Mech., 40 (1991) 119 and 48 (1993) 211.

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Dissipative stresses in dilute polymer solutions