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Shear and elongation in a theory for elastic fluids
69
Journal of Non-Newton&m Fluid Mechanws, 8 (1981) 69-79 Elsevler Sclentlfic Pubhshmg Company, Amsterdam - Prmted m The Netherlands
SHEAR AND ELONGATION IN A THEORY FOR ELASTIC FLUIDS
P A DASHNER
and W E VANARSDALE
Department of Theoretrcal 14853 (US A ) (Received
April 7.1980,
and Applred
Mechanrcs,
Cornell Unrversrty,
Ithaca, NY
In revised form August 18, 1980)
Summary A model for elastic fluids is assessed m the context of shear and elongational flows. This model provides a unified quahtative description of optical and rheological properties and, in its present form, seems to model polymer solutions with notable accuracy. This is achieved using only four material constants which can be related to the asymptotic values of the viscosity and primary normal stress functions m steady shear Urnaxial elongation data for more concentrated solutions suggests that modifications will be required m order to describe these materials
Introduction A phenomenological model for polymenc fluids, based on the notion of stored elastic strain, has been proposed [ 11. Here we would like to mvestigate the predi&ions of this model for shear and extensional flows. The particular form of the model to be mvestigated is 2D=L+LT,
tr(D) = 0 , det(c) = 1 ,
(1 1) (1.2)
where s is the devlatonc part (dev) of the stress tensor, L 1s the velocity gradient (I,,, = au,/ax,), c is a measure of stored elastic strain, and $ is the free energy per unit mass, a function of the mvanants I = tr(c) and II = tr(c’). The 0377-0257/81/0000-0000/$2
50 @ 1981 Elsevler SclenMic
Pubhshmg Company
70
constants pl and p2 must be non-negative to satisfy thermodynanuc considera tions. Usmg (1.3) and the Cayley-Ham&on theorem, the evolution equation (1.2h can be wntten as : =qQI+f#Jsc+##,
(1.4)
where
In this mvestlgatlon, we wfl use the energy function 2p$ = h[tr(l)]2 + 2M tr(12); = (h/4)(3 -I)2
2l=Z-c,
+ (M/2)(3 - 2I+ II) .
(1.5)
when specific results are needed. The matenal constants X, M are analogous to the Lame constants m lmear elasticity and satisfy the restnctlons
M>
3h+2M>0,
0,
which msure that 9 ISposltlve for all c # I. 2. Shear flow We consider the matron of a fluid between two parallel plates of mfimte extent. The components of the velocity field are assumed to have the form [u(y, t), 0, 0] with respect to a smtably oriented Cartesran coordinate system. In the absence of a pressure gradient m the direction of flow, the momentum equations reduce to
as,, ay
-=pjJ,
au
-P(Yv &bYY
(2-l) Ol=
0,
sublect m the no&p condrtlon on the plates and a smtable uutral condltlon. The evolution equation (1.4), with the mitral condltlon c = I, reduces to a system of differential equations for the nonvanrshmg components cXX, cXjrr cyy, cz,, SubJect to the constramt (1.2),. Usmg the fact that se ls an isotropic function of c, the dlfferentlal equations can be written m the alternate form
71
(1 .2)1 in terms of these components a at c,, = (c,,sL
9
+ c&,)/cc0
se )/j&s XX9 ac XY= ~~x,4y+cXY YY at aY au % -cxy 9 at YY= (c,&y + cyys;y)lc10 - 2 ay L * at 2%= Gzs~,lPo
(2.2)
We seek the solution of these equations for steady flow as well as their predlctlons for small oscillation and impulsive movement of one plate. A Sample shear For steady flow, we focus attention on the steady state equations obtamed from (2.2). Ehminatlon of the components of Se from the system of equations leads to the identity Cc:, +c,2y)c,, = 1. This result and the constramt
det(c) = cz,(cX,cyy -c:,)
(1.2)s
=1 ,
(2.3)
suggest that c can be described m terms of two parameters r, X such that c,, = r sin(2x), czz In
-2
=r terms
CXY= 7
cos(2x),
cyy = r[l + coss(2x)]/sin(2x)
,
.
of this parametenxation,
the nonzero components
&X = PO+sin(%) cos(2x) = -sfy,
of se are
sIy = r-lo+sln2(2x) ,
where r = du/dy rs the rate of shear. The angle X 18 the onentation pnncipal elastic stresses s”l = /Lo+ sln(2X) = -s;
(2.4) of the (2.5)
,
with respect to the x and y axes. Usmg (2.4) and (l.l), the following expressions are obtained for the shear stress and normal stress differences 8 xy
=
1111+ PO~~*czx)l~= 71(;1) r ,
Q-6)
N1=sX, -s yy = 2Po;l sm(2x) cos(2x) = *I(+) f”,
(2.7)
N2 = sy y - szz = -+I/2
(2.8)
where the vlscometnc
=9,(f)
i.2 ,
functions I,
S,(f)
and q,(f)
are called the vrscoslty,
72
the pnmary and secondary normal stress coefficients, respectively. These results are mdependent of the choice of energy function. To complete the analysis, we mtroduce a free energy fun&on and solve the remammg mdependent equations to relate the parameters r, x to the rate of shear f. For the energy function (1.5) we obtam sm(2~) = (r3/2) {-[(3x -2(3x
+ 2) r2 + x] + [16(x
+ 2) r6 + (3x + 2)2 r4
+ 2)(3x + 8) r2 + 8(x + 2)2 + R]Y2}/[2r6
-(3x
+ 2) r2 + (x + 2)] , (2.9)
po”//M = cos(2x){ [X -(3X
+ 2) r2] sm(2X) + 2(X + 2) r3}/[2r
sm3(2X)3,
where x = h/M and 0 G r < 1,O Q x < n/4. Usmg these results, the matinal constants can be mterpreted m terms of the low and high shear-rate lumts
-3
-2
-I
0 LOG 7 (s ‘1
I
2
3
-2
-I
0
I
2
3
LOG y(s’)
Fig 1 ExperImental viscosity data, q(r), for three solutions at 298 K 0.1 5% polyacrylamide (Separan AP 30) m a SO/50 mixture by weight of water and glycerm, A, 2 0% polylsobutylene m Prlmol, n , 7% aluminum laurate m a mixture of decahn and m-cresol Solid @es are predictions using the constants (0) ~0 = 3 0 x lo2 N s/m2, +, = 6 0 X 10” N slm2, 91 = 2 0 x lo4 N s2/m2, x = 1 5 N/m2, (A) ~0 = 1 1 x lo3 N s/m2, vrn = 2.9 N s/m2, Jil = 4 0 X lo5 N s2/m2, h = 6 5 N/m2, (m) TJO= 9 0 X lo1 N s/m2, q-= 1 0 x 10” N s/m2, $1 = 3 0 X lo2 N s2/m2, i = 8 5 N/m2 The values of ~0, Q~, \kl were approximated when not exphcltly obtainable from the expenmental data The value of h was then chosen to obtain qualitative agreement with the viscosity and primary normal stress data Fig 2 Primary normal stress coefflclent data, 91(y) for the materials described m Fig 1 The solid lines are predictions using the constants given m Fig 1
73
of the vlscometnc functions: %=~$$)=110+111. rim=
(2.10)
lJ=?(^1)=bh,
Y-+_
With this identification of the material constants, the viscosity function (2.6) is somewhat similar to the sucessful, empirical model proposed by Carreau [3] (also see Bud et al. [4, p. 2101) (11-cl1MJo=
v+
a 2 (n-l)/2 , (~1~) I
where Xi 1sa characteristic tune and (n - 1) 1sthe slope of the viscosity curve m the power law region. A comparison of theory with data taken by Huppler et al. [ 51 for three polymer solutions 1sshown m Figs. 1 and 2. Comparisons were also made using the Mooney-Rmlm form of the energy function given m [ 11. Although the theoretical predictions were quahtatlvely identical, there are significant quantitative differences. For example, the pnmary normal stress coefficient plotted m Fig. 2 can be shown to approach an asymptote havmg slope -10/7, while the curve generated urlth the Mooney-Rivhn form has an asymptotic slope of -2. Although these asymptotic slopes are independent of the material constants, the elastic constants h (Hookean) and p (Mooney-Rlvhn) directly influence the lateral shift of the asymptotes. For either energy function, the four material constants can be determmed from the position of the low and high shear-rate asymptotes of the pnmary normal stress coefficient and viscosity curves. B Brefrrngence rn steady flow Buefrmgence, which is the difference m the principal values of the optical or refractive mdex tensor, indicates a material’s optical anisotropy. In fluids, this amsotropy occurs durmg flow as mduced stresses distort and onent molecules from their random position m the isotropic eqmhbnum state. This so-called streammg blrefnngence is reviewed by Janeschitz-Knegl [6] and Peterlm [7]. In the present theory, the tensor c characterizes the deformation of the polymer network from its natural state and se(c) is the elastic stress associated with this deformation. Here we assume that the elastic stress, and not the viscous part 2~.clD,is important m determmmg buefrmgence. Smce water is not blrefrmgent under normal circumstances, this assumption is consistent with the Newtoman limit of (1.1) m which se = 0. As we shall see, it also allows for materials which behave as Newtoman fluids m steady shear but are buefrmgent. In particular, we assume the optical tensor is proportional to the elastic
74
Fig 3 TheoretIcal predwtlons of blrefrmgence An and extmctlon angle X m steady shear flow for polylsobut lene solution described In Fig 1 The blrefrrmgence ISnormalwed by its value at y = lo- z s-l
stress. Usmg (2.5), this gives the blrefrmgence An = IQ& sm(2~) ,
(2.11)
m terms of the proportionality constant K. Here the angle x 1s called the extmctlon angle., It decreases monotonically from 7r/4 to zero as the shear rate increases. This agrees with experimental data for polymer solutions given by Janeschltz-Knegl [6] and Peterhn [7]. In the lmut, as M + 00 m (2.9), polymer chams become rigid, and the extmctlon angle remains a/4 for all shear rates. In this case, the blrefnngence (2.11) is proportional to the shear rate and the viscometnc functions (2.6)-(2.8) reduce to those for a Newtoman fluid. Tlus lmut seems to descnbe the rheo-optlcal behavior of low-molecular weight liquids like benzene [ 71 m steady shear. For polymer solutions, the blrefnngence (2.11) can be rewritten as An = K[r)(K) -+I
K/sm(2x) ,
(2.12)
usmg (2.6) and (2.10). At low shear rates (where r) >> T)_), this expression is consistent with the so-called rheo-optlcal law m which the entire stress 1s proportional to the optical tensor. The phenomenologrcal theory of Coleman et al. [2] is also consrstent with this empirical law for slow flows. Expenmental data presented by Janeschitz-Kriegl [6] and Peterlm [ 71 tend to confirm tlus relation for polymenc hquids. Predlctrons of buefrmgence and extmction angle mung (2.9) and (2.12) are shown m Fig. 3 for one of the polymer solutions described m Fig. 1. C Start-up and relaxatron The unsteady shear flows assocrated with the sudden start-up or cessation of plate movement were simulated, lgnormg mertlal effects. For stress growth after an imposed shear rate to, the stress growth functions are defined as
v’(t; fo) = %yU)/~O ,
l(S) Fig 4 TheoretIcal predrctrons of stress growth functrons for the alummum soap solutron descrrbed In Frg 1 plotted agamst time for varrous shear rates These functrons are normallzed by the correspondrng vrscometrrc functrons Experunental data are indrcated by a sohd symbol Frg 5 Theoretlcal predrctlons of stress relaxatron functions for the alummum soap solutron descrrbed m Frg 1 plotted agamst time for various shear rates These functrons are normalized by the correspondmg wscometrrc functions Experrmental data are mdrcated by a solid symbol
Smularly, the stress relaxation after cessation of simple flow with shear rate +O1s described by the stress relaxation functions rl-It* To) = sx,(t)/Yo , *i(t,
.;/o) E NI(~)/% ,
The stress growth and relaxation functions were obtamed through the numencal mtegratlon of (2.2) with the shear rate and mtlal condltlons appropnate to to the given test. Typical results assummg (1.5) are shown m Figs. 4 and 5 where all functions have been normahzed by the correspondmg vlscometnc
76
functmn (2.6)-(2.8) evaluated at +,. These curves are quahtatlvely the same as data obtamed by Huppler et al [8], but they typically overestimate the expenmental results. However, recall that we have neglected mertla and posable spatial mhomogeneltles m obtammg our results. D Small amplrtude
oscrllatory
shear flow
This type of flow lends Itself to a perturbation analyas. Consider the followmg nondlmenslonal vmables: t= tw,
Y = ylh,
u = ulwh;
B
q
s/wp,
,
where h 1s the gap width and o 1sthe frequency of the nnposed osclllatlons The momentum equation (2.1) becomes
as *=cYdt, au SUbpA
to
a = pwh*/po
(2.13)
,
the boundary condltlons
u(0, t) = 0,
E>O,
u(1, t) = f sm(t),
where u 1smltlally zero and E 1sthe nondunenslonal amphtude associated with the unposed oscfiatlons. Assummg f 1ssmall compared to one, we attempt a solution of (2.13) m terms of the expansions u = EUI(Y,t) + e2u*(y, t) + O(e3) , s = E&(Y, t) + E2s*(y, t) + O(2) c - z = ec,(y,
t) + ?c*(y,
,
t) + O(E3) ,
where cl and c2 satisfy the constramts tr(cI) = 0,
tr(c2) =D(c?)/2
,
which follow from (2.3). For the energy function (1.5), we obtam Se = -&fcl
- E*M[c: + c2 -$tr(cq)
I] + 0(e3) ,
where fi = M/~.c~w.The only nonzero component of cl 1s (cxy)l = -exp(-fit)
j %exp(&) 0
dr .
ay
Consequently, the normal stress differences are O(E*). From (2.13), the result mg equation for u1 is exp(&)
au1
d7 = a at,
77
which satlsfles the condltlons Ul(l, t) = sm(t) .
u1(0, t) = 0,
MY, 0) = 0,
The equation for u2 IS the same, except both boundary geneous. If (Y= 0, we can easily obtam the solutions u1= y sm(t),
condltlons
u2 = 0 .
are homo(2.14)
For 01# 0, a more complicated solution for u1 can be obtamed using Laplace transforms. Usmg (2.14), the shear stress IS given by the expresslon SXY= EW[$(0)
sm(ot)
where the dynamics V’(W) = [r)o +
q”(W) =
-q”(W)
vlscoslty
cos(wt)] functions
+ O(E3) , q’(w) and T)“(W) are
f\Illcd/[l+(h~)21,
XI = /JO/M
These results are the same as approxunate predlctlons Jefferys model described by Bird et al. [4, p. 3321.
-1
0
(2.15)
r)4h~)211[1 + (h421 ,
i LOG w (s-‘1
2
of the corrotational
3 LOG w (s-‘1
Fig 6 TheoretIcal predIctIon of the dynamic wscoslty function q’(w) for the aluminum soap solution described m Fig 1 Experimental data are mdlcated by a solrd symbol The dashed hne ISthe correspondmg wscoslty function ~(7) replotted from Fig 1 for comparlson Fig 7 Theoretwal predlctlon of the dynamic vlscoslty function $(0)/w for the alummum soap solution described m Fig 1 ExperImental data are mdlcated b_* a sohd symbol
78
A typical compenson of the leadmg order predlctlons (2.15) with data taken by Huppler et al. [5] 18shown in Figs. 6 and 7. These results agree quantltatlvely only for a lmuted range of frequency. However, the approximate results (2.15) are basically those of a lmeanzed constltutlve theory and cannot be expected to represent the full theory accurately over the entire frequency range. These results do suggest that q’(w) lies below the vrscoslty curve, mdlcated by the dashed lme m Fig. 6, m accord with expelvnental observations. If mertla 1s mcluded, the dynamic viscosity functions comclde mth those shown m Figs. 6 and 7 for w < ~O’S-~ and he above these functlons at higher frequencies. 3. Elongational flow Usmg a method of classlflcatlon suggested by Stevenson et al. [9], the spateal gradient of velocity for this flow with respect to a suitably onented Cartesian coordmate system ISof the form L=[[i
f
_:_,1,
-lGrnG-+.
where g IScalled the extension rate. The parameter m denotes the type of flow with the extreme values correspondmg to planar (m = -1) and uniaxlal (m = - i) elongation for posltlve t. For c = Z mltlally, the evolution equation reduces to ~c=cls’/ro-2D), where c and se are diagonal. In the steady state, the elastic stress 1s se = 2Z.l& . resultmg normal stress differences
The
NI = 2(1are
m)(Cc0
+ kh)
t.
N2
= WJm
+ 1)(10
+ PI)
t ,
independent of the choice of energy function. For umaxlal elongation, these expressions reduce to
N1= 377&,
N2 = 0
.
such predictions have not been checked for polymer solutions, they have been experunentally venfled for various polymer melts at small extenslon rates (see Stevenson [lo]). However, experunental results by Melssner [ll] (also see Bud et al. [4, p. 1881) suggest that this steady state does not exist at larger rates. Although this 1s not a charactenstlc of our present model, It IS possible to mclude this effect wrthm the general theoretical framework. For mstance we could easily generahze these forms wlthm the confmes of the general theory by replacmg the material constant cl0 with a
While
79
positive scalar functron of the tensor mvanants of c. If this fun&on depends lmearly on 12, the pnmary normal stress difference vnll mcrease exponentrally above some cntlcal extension rate slmrlar to Lodge’s rubber-like hquld [4, p. 4491. From (2.6), we can see that tlus change will yield a more slowly decreasmg shear-vrscosity function m accord with experunental results for melts [4, p. 2101. Conclusron We have been able to accurately simulate the behavror of three polymer solutions m a number of basic flow situations usmg a particularly snnple form of the general theory mvolving only four constants. We have also mdicated how sunple modificattlons of this model make It possible to qualitatively descnbe the behavior of polymer melts m elongation. The compllatlon of these results summarizes our mltlal efforts at demonstratmg the utility and flexrbrhty of the general theory presented m [ 11. We believe that this theory, which 1s based on the concept of mtemal elastic stram, provides a unified descnptlon of both optical and rheological properties mthm a theoretical framework which is both highly mtuitlve and of sufficient flexlblllty to mcorporate a wide vanety of unportant mechamcal effects.
References 1 P A Dashner and W E Van Arsdale, J Non-Newtonran Fluld Mech , 8 (1981) 59-67 2 B D Coleman, E H D111and R A Toupm, Arch Rat Mech Anal , 39 (1970) 358399 3 P J Carreau, Ph D Theme, Umverslty of Wlsconan, Madison, 1968 4 R B Bud, R C Armstrong and 0 Hassager, Dynamics of Polymeric Llqulds, Vol I FluId Mechamcs, Wdey, New York, 1977 5 J D Huppler, E Ashare and L A Holmes, Trans Sot Rheol , 11 (1967) 159-179 6 H Janeschttz-Krlegl, Adv Polymer SIX ,6 (1969) 170-318 7 A Peterhn, Ann Rev Fluld Mech ,8 (1976) 35-65 8 J D Huppler, I F MacDonald, E Ashare, T W Spnggs, R B Bird and L A Holmes, Trans Sot Rheol ,11(1967) 181-204 9 J F Stevenson, C K Chang and J T Jenkms, Trans Sot Rheol ,19 (1975) 397-405 10 J F Stevenson, AIChE J , 18 (1972) 540-647 11 J Melssner, Rheol Acta, 10 (1971) 230-242
Journal of Non-Newton&m Fluid Mechanws, 8 (1981) 69-79 Elsevler Sclentlfic Pubhshmg Company, Amsterdam - Prmted m The Netherlands
SHEAR AND ELONGATION IN A THEORY FOR ELASTIC FLUIDS
P A DASHNER
and W E VANARSDALE
Department of Theoretrcal 14853 (US A ) (Received
April 7.1980,
and Applred
Mechanrcs,
Cornell Unrversrty,
Ithaca, NY
In revised form August 18, 1980)
Summary A model for elastic fluids is assessed m the context of shear and elongational flows. This model provides a unified quahtative description of optical and rheological properties and, in its present form, seems to model polymer solutions with notable accuracy. This is achieved using only four material constants which can be related to the asymptotic values of the viscosity and primary normal stress functions m steady shear Urnaxial elongation data for more concentrated solutions suggests that modifications will be required m order to describe these materials
Introduction A phenomenological model for polymenc fluids, based on the notion of stored elastic strain, has been proposed [ 11. Here we would like to mvestigate the predi&ions of this model for shear and extensional flows. The particular form of the model to be mvestigated is 2D=L+LT,
tr(D) = 0 , det(c) = 1 ,
(1 1) (1.2)
where s is the devlatonc part (dev) of the stress tensor, L 1s the velocity gradient (I,,, = au,/ax,), c is a measure of stored elastic strain, and $ is the free energy per unit mass, a function of the mvanants I = tr(c) and II = tr(c’). The 0377-0257/81/0000-0000/$2
50 @ 1981 Elsevler SclenMic
Pubhshmg Company
70
constants pl and p2 must be non-negative to satisfy thermodynanuc considera tions. Usmg (1.3) and the Cayley-Ham&on theorem, the evolution equation (1.2h can be wntten as : =qQI+f#Jsc+##,
(1.4)
where
In this mvestlgatlon, we wfl use the energy function 2p$ = h[tr(l)]2 + 2M tr(12); = (h/4)(3 -I)2
2l=Z-c,
+ (M/2)(3 - 2I+ II) .
(1.5)
when specific results are needed. The matenal constants X, M are analogous to the Lame constants m lmear elasticity and satisfy the restnctlons
M>
3h+2M>0,
0,
which msure that 9 ISposltlve for all c # I. 2. Shear flow We consider the matron of a fluid between two parallel plates of mfimte extent. The components of the velocity field are assumed to have the form [u(y, t), 0, 0] with respect to a smtably oriented Cartesran coordinate system. In the absence of a pressure gradient m the direction of flow, the momentum equations reduce to
as,, ay
-=pjJ,
au
-P(Yv &bYY
(2-l) Ol=
0,
sublect m the no&p condrtlon on the plates and a smtable uutral condltlon. The evolution equation (1.4), with the mitral condltlon c = I, reduces to a system of differential equations for the nonvanrshmg components cXX, cXjrr cyy, cz,, SubJect to the constramt (1.2),. Usmg the fact that se ls an isotropic function of c, the dlfferentlal equations can be written m the alternate form
71
(1 .2)1 in terms of these components a at c,, = (c,,sL
9
+ c&,)/cc0
se )/j&s XX9 ac XY= ~~x,4y+cXY YY at aY au % -cxy 9 at YY= (c,&y + cyys;y)lc10 - 2 ay L * at 2%= Gzs~,lPo
(2.2)
We seek the solution of these equations for steady flow as well as their predlctlons for small oscillation and impulsive movement of one plate. A Sample shear For steady flow, we focus attention on the steady state equations obtamed from (2.2). Ehminatlon of the components of Se from the system of equations leads to the identity Cc:, +c,2y)c,, = 1. This result and the constramt
det(c) = cz,(cX,cyy -c:,)
(1.2)s
=1 ,
(2.3)
suggest that c can be described m terms of two parameters r, X such that c,, = r sin(2x), czz In
-2
=r terms
CXY= 7
cos(2x),
cyy = r[l + coss(2x)]/sin(2x)
,
.
of this parametenxation,
the nonzero components
&X = PO+sin(%) cos(2x) = -sfy,
of se are
sIy = r-lo+sln2(2x) ,
where r = du/dy rs the rate of shear. The angle X 18 the onentation pnncipal elastic stresses s”l = /Lo+ sln(2X) = -s;
(2.4) of the (2.5)
,
with respect to the x and y axes. Usmg (2.4) and (l.l), the following expressions are obtained for the shear stress and normal stress differences 8 xy
=
1111+ PO~~*czx)l~= 71(;1) r ,
Q-6)
N1=sX, -s yy = 2Po;l sm(2x) cos(2x) = *I(+) f”,
(2.7)
N2 = sy y - szz = -+I/2
(2.8)
where the vlscometnc
=9,(f)
i.2 ,
functions I,
S,(f)
and q,(f)
are called the vrscoslty,
72
the pnmary and secondary normal stress coefficients, respectively. These results are mdependent of the choice of energy function. To complete the analysis, we mtroduce a free energy fun&on and solve the remammg mdependent equations to relate the parameters r, x to the rate of shear f. For the energy function (1.5) we obtam sm(2~) = (r3/2) {-[(3x -2(3x
+ 2) r2 + x] + [16(x
+ 2) r6 + (3x + 2)2 r4
+ 2)(3x + 8) r2 + 8(x + 2)2 + R]Y2}/[2r6
-(3x
+ 2) r2 + (x + 2)] , (2.9)
po”//M = cos(2x){ [X -(3X
+ 2) r2] sm(2X) + 2(X + 2) r3}/[2r
sm3(2X)3,
where x = h/M and 0 G r < 1,O Q x < n/4. Usmg these results, the matinal constants can be mterpreted m terms of the low and high shear-rate lumts
-3
-2
-I
0 LOG 7 (s ‘1
I
2
3
-2
-I
0
I
2
3
LOG y(s’)
Fig 1 ExperImental viscosity data, q(r), for three solutions at 298 K 0.1 5% polyacrylamide (Separan AP 30) m a SO/50 mixture by weight of water and glycerm, A, 2 0% polylsobutylene m Prlmol, n , 7% aluminum laurate m a mixture of decahn and m-cresol Solid @es are predictions using the constants (0) ~0 = 3 0 x lo2 N s/m2, +, = 6 0 X 10” N slm2, 91 = 2 0 x lo4 N s2/m2, x = 1 5 N/m2, (A) ~0 = 1 1 x lo3 N s/m2, vrn = 2.9 N s/m2, Jil = 4 0 X lo5 N s2/m2, h = 6 5 N/m2, (m) TJO= 9 0 X lo1 N s/m2, q-= 1 0 x 10” N s/m2, $1 = 3 0 X lo2 N s2/m2, i = 8 5 N/m2 The values of ~0, Q~, \kl were approximated when not exphcltly obtainable from the expenmental data The value of h was then chosen to obtain qualitative agreement with the viscosity and primary normal stress data Fig 2 Primary normal stress coefflclent data, 91(y) for the materials described m Fig 1 The solid lines are predictions using the constants given m Fig 1
73
of the vlscometnc functions: %=~$$)=110+111. rim=
(2.10)
lJ=?(^1)=bh,
Y-+_
With this identification of the material constants, the viscosity function (2.6) is somewhat similar to the sucessful, empirical model proposed by Carreau [3] (also see Bud et al. [4, p. 2101) (11-cl1MJo=
v+
a 2 (n-l)/2 , (~1~) I
where Xi 1sa characteristic tune and (n - 1) 1sthe slope of the viscosity curve m the power law region. A comparison of theory with data taken by Huppler et al. [ 51 for three polymer solutions 1sshown m Figs. 1 and 2. Comparisons were also made using the Mooney-Rmlm form of the energy function given m [ 11. Although the theoretical predictions were quahtatlvely identical, there are significant quantitative differences. For example, the pnmary normal stress coefficient plotted m Fig. 2 can be shown to approach an asymptote havmg slope -10/7, while the curve generated urlth the Mooney-Rivhn form has an asymptotic slope of -2. Although these asymptotic slopes are independent of the material constants, the elastic constants h (Hookean) and p (Mooney-Rlvhn) directly influence the lateral shift of the asymptotes. For either energy function, the four material constants can be determmed from the position of the low and high shear-rate asymptotes of the pnmary normal stress coefficient and viscosity curves. B Brefrrngence rn steady flow Buefrmgence, which is the difference m the principal values of the optical or refractive mdex tensor, indicates a material’s optical anisotropy. In fluids, this amsotropy occurs durmg flow as mduced stresses distort and onent molecules from their random position m the isotropic eqmhbnum state. This so-called streammg blrefnngence is reviewed by Janeschitz-Knegl [6] and Peterlm [7]. In the present theory, the tensor c characterizes the deformation of the polymer network from its natural state and se(c) is the elastic stress associated with this deformation. Here we assume that the elastic stress, and not the viscous part 2~.clD,is important m determmmg buefrmgence. Smce water is not blrefrmgent under normal circumstances, this assumption is consistent with the Newtoman limit of (1.1) m which se = 0. As we shall see, it also allows for materials which behave as Newtoman fluids m steady shear but are buefrmgent. In particular, we assume the optical tensor is proportional to the elastic
74
Fig 3 TheoretIcal predwtlons of blrefrmgence An and extmctlon angle X m steady shear flow for polylsobut lene solution described In Fig 1 The blrefrrmgence ISnormalwed by its value at y = lo- z s-l
stress. Usmg (2.5), this gives the blrefrmgence An = IQ& sm(2~) ,
(2.11)
m terms of the proportionality constant K. Here the angle x 1s called the extmctlon angle., It decreases monotonically from 7r/4 to zero as the shear rate increases. This agrees with experimental data for polymer solutions given by Janeschltz-Knegl [6] and Peterhn [7]. In the lmut, as M + 00 m (2.9), polymer chams become rigid, and the extmctlon angle remains a/4 for all shear rates. In this case, the blrefnngence (2.11) is proportional to the shear rate and the viscometnc functions (2.6)-(2.8) reduce to those for a Newtoman fluid. Tlus lmut seems to descnbe the rheo-optlcal behavior of low-molecular weight liquids like benzene [ 71 m steady shear. For polymer solutions, the blrefnngence (2.11) can be rewritten as An = K[r)(K) -+I
K/sm(2x) ,
(2.12)
usmg (2.6) and (2.10). At low shear rates (where r) >> T)_), this expression is consistent with the so-called rheo-optlcal law m which the entire stress 1s proportional to the optical tensor. The phenomenologrcal theory of Coleman et al. [2] is also consrstent with this empirical law for slow flows. Expenmental data presented by Janeschitz-Kriegl [6] and Peterlm [ 71 tend to confirm tlus relation for polymenc hquids. Predlctrons of buefrmgence and extmction angle mung (2.9) and (2.12) are shown m Fig. 3 for one of the polymer solutions described m Fig. 1. C Start-up and relaxatron The unsteady shear flows assocrated with the sudden start-up or cessation of plate movement were simulated, lgnormg mertlal effects. For stress growth after an imposed shear rate to, the stress growth functions are defined as
v’(t; fo) = %yU)/~O ,
l(S) Fig 4 TheoretIcal predrctrons of stress growth functrons for the alummum soap solutron descrrbed In Frg 1 plotted agamst time for varrous shear rates These functrons are normallzed by the correspondrng vrscometrrc functrons Experunental data are indrcated by a sohd symbol Frg 5 Theoretlcal predrctlons of stress relaxatron functions for the alummum soap solutron descrrbed m Frg 1 plotted agamst time for various shear rates These functrons are normalized by the correspondmg wscometrrc functions Experrmental data are mdrcated by a solid symbol
Smularly, the stress relaxation after cessation of simple flow with shear rate +O1s described by the stress relaxation functions rl-It* To) = sx,(t)/Yo , *i(t,
.;/o) E NI(~)/% ,
The stress growth and relaxation functions were obtamed through the numencal mtegratlon of (2.2) with the shear rate and mtlal condltlons appropnate to to the given test. Typical results assummg (1.5) are shown m Figs. 4 and 5 where all functions have been normahzed by the correspondmg vlscometnc
76
functmn (2.6)-(2.8) evaluated at +,. These curves are quahtatlvely the same as data obtamed by Huppler et al [8], but they typically overestimate the expenmental results. However, recall that we have neglected mertla and posable spatial mhomogeneltles m obtammg our results. D Small amplrtude
oscrllatory
shear flow
This type of flow lends Itself to a perturbation analyas. Consider the followmg nondlmenslonal vmables: t= tw,
Y = ylh,
u = ulwh;
B
q
s/wp,
,
where h 1s the gap width and o 1sthe frequency of the nnposed osclllatlons The momentum equation (2.1) becomes
as *=cYdt, au SUbpA
to
a = pwh*/po
(2.13)
,
the boundary condltlons
u(0, t) = 0,
E>O,
u(1, t) = f sm(t),
where u 1smltlally zero and E 1sthe nondunenslonal amphtude associated with the unposed oscfiatlons. Assummg f 1ssmall compared to one, we attempt a solution of (2.13) m terms of the expansions u = EUI(Y,t) + e2u*(y, t) + O(e3) , s = E&(Y, t) + E2s*(y, t) + O(2) c - z = ec,(y,
t) + ?c*(y,
,
t) + O(E3) ,
where cl and c2 satisfy the constramts tr(cI) = 0,
tr(c2) =D(c?)/2
,
which follow from (2.3). For the energy function (1.5), we obtam Se = -&fcl
- E*M[c: + c2 -$tr(cq)
I] + 0(e3) ,
where fi = M/~.c~w.The only nonzero component of cl 1s (cxy)l = -exp(-fit)
j %exp(&) 0
dr .
ay
Consequently, the normal stress differences are O(E*). From (2.13), the result mg equation for u1 is exp(&)
au1
d7 = a at,
77
which satlsfles the condltlons Ul(l, t) = sm(t) .
u1(0, t) = 0,
MY, 0) = 0,
The equation for u2 IS the same, except both boundary geneous. If (Y= 0, we can easily obtam the solutions u1= y sm(t),
condltlons
u2 = 0 .
are homo(2.14)
For 01# 0, a more complicated solution for u1 can be obtamed using Laplace transforms. Usmg (2.14), the shear stress IS given by the expresslon SXY= EW[$(0)
sm(ot)
where the dynamics V’(W) = [r)o +
q”(W) =
-q”(W)
vlscoslty
cos(wt)] functions
+ O(E3) , q’(w) and T)“(W) are
f\Illcd/[l+(h~)21,
XI = /JO/M
These results are the same as approxunate predlctlons Jefferys model described by Bird et al. [4, p. 3321.
-1
0
(2.15)
r)4h~)211[1 + (h421 ,
i LOG w (s-‘1
2
of the corrotational
3 LOG w (s-‘1
Fig 6 TheoretIcal predIctIon of the dynamic wscoslty function q’(w) for the aluminum soap solution described m Fig 1 Experimental data are mdlcated by a solrd symbol The dashed hne ISthe correspondmg wscoslty function ~(7) replotted from Fig 1 for comparlson Fig 7 Theoretwal predlctlon of the dynamic vlscoslty function $(0)/w for the alummum soap solution described m Fig 1 ExperImental data are mdlcated b_* a sohd symbol
78
A typical compenson of the leadmg order predlctlons (2.15) with data taken by Huppler et al. [5] 18shown in Figs. 6 and 7. These results agree quantltatlvely only for a lmuted range of frequency. However, the approximate results (2.15) are basically those of a lmeanzed constltutlve theory and cannot be expected to represent the full theory accurately over the entire frequency range. These results do suggest that q’(w) lies below the vrscoslty curve, mdlcated by the dashed lme m Fig. 6, m accord with expelvnental observations. If mertla 1s mcluded, the dynamic viscosity functions comclde mth those shown m Figs. 6 and 7 for w < ~O’S-~ and he above these functlons at higher frequencies. 3. Elongational flow Usmg a method of classlflcatlon suggested by Stevenson et al. [9], the spateal gradient of velocity for this flow with respect to a suitably onented Cartesian coordmate system ISof the form L=[[i
f
_:_,1,
-lGrnG-+.
where g IScalled the extension rate. The parameter m denotes the type of flow with the extreme values correspondmg to planar (m = -1) and uniaxlal (m = - i) elongation for posltlve t. For c = Z mltlally, the evolution equation reduces to ~c=cls’/ro-2D), where c and se are diagonal. In the steady state, the elastic stress 1s se = 2Z.l& . resultmg normal stress differences
The
NI = 2(1are
m)(Cc0
+ kh)
t.
N2
= WJm
+ 1)(10
+ PI)
t ,
independent of the choice of energy function. For umaxlal elongation, these expressions reduce to
N1= 377&,
N2 = 0
.
such predictions have not been checked for polymer solutions, they have been experunentally venfled for various polymer melts at small extenslon rates (see Stevenson [lo]). However, experunental results by Melssner [ll] (also see Bud et al. [4, p. 1881) suggest that this steady state does not exist at larger rates. Although this 1s not a charactenstlc of our present model, It IS possible to mclude this effect wrthm the general theoretical framework. For mstance we could easily generahze these forms wlthm the confmes of the general theory by replacmg the material constant cl0 with a
While
79
positive scalar functron of the tensor mvanants of c. If this fun&on depends lmearly on 12, the pnmary normal stress difference vnll mcrease exponentrally above some cntlcal extension rate slmrlar to Lodge’s rubber-like hquld [4, p. 4491. From (2.6), we can see that tlus change will yield a more slowly decreasmg shear-vrscosity function m accord with experunental results for melts [4, p. 2101. Conclusron We have been able to accurately simulate the behavror of three polymer solutions m a number of basic flow situations usmg a particularly snnple form of the general theory mvolving only four constants. We have also mdicated how sunple modificattlons of this model make It possible to qualitatively descnbe the behavior of polymer melts m elongation. The compllatlon of these results summarizes our mltlal efforts at demonstratmg the utility and flexrbrhty of the general theory presented m [ 11. We believe that this theory, which 1s based on the concept of mtemal elastic stram, provides a unified descnptlon of both optical and rheological properties mthm a theoretical framework which is both highly mtuitlve and of sufficient flexlblllty to mcorporate a wide vanety of unportant mechamcal effects.
References 1 P A Dashner and W E Van Arsdale, J Non-Newtonran Fluld Mech , 8 (1981) 59-67 2 B D Coleman, E H D111and R A Toupm, Arch Rat Mech Anal , 39 (1970) 358399 3 P J Carreau, Ph D Theme, Umverslty of Wlsconan, Madison, 1968 4 R B Bud, R C Armstrong and 0 Hassager, Dynamics of Polymeric Llqulds, Vol I FluId Mechamcs, Wdey, New York, 1977 5 J D Huppler, E Ashare and L A Holmes, Trans Sot Rheol , 11 (1967) 159-179 6 H Janeschttz-Krlegl, Adv Polymer SIX ,6 (1969) 170-318 7 A Peterhn, Ann Rev Fluld Mech ,8 (1976) 35-65 8 J D Huppler, I F MacDonald, E Ashare, T W Spnggs, R B Bird and L A Holmes, Trans Sot Rheol ,11(1967) 181-204 9 J F Stevenson, C K Chang and J T Jenkms, Trans Sot Rheol ,19 (1975) 397-405 10 J F Stevenson, AIChE J , 18 (1972) 540-647 11 J Melssner, Rheol Acta, 10 (1971) 230-242