72
STEPHEN PRAGER
Variational EIounds on the Intrinsic Viscosity1
by Stephen Prager Department of Chemistry, University of Minnesota, Minneapolis, Minnesota
(Receiaed J u n e 5, 1970)
66466
Publication costs assisted by the National Science Foundation
The principle of minimum energy dissipation is applied to obtain lower bounds on the intrinsic viscosity of a polymer molecule consisting of A T spherical beads. The forces which the beads exert on one another need not be restricted to Rooke's law interactions-the main result, inequality 29, holds for any internal potential energy function. Brownian motion and hydrodynamic interaction effects are taken into account, and the preaveraging of the diffusion tensor used by most authors working in this area is avoided. For ai2 isolated bead ( N = 1) the present treatment gives the correct value of the inixinsic viscosity as obtained by Einstein, whereas previous theories, which are designed for use at high molecular weights only, give [ q ] = 0 in this limit. Explicit results are obtained for dumbells ( N = 2 ) and for looped chains; the effect of preaveraging oq calculated values of [ q ] is discussed for the case of a Gaussian loop in the limit of large N.
Introduction Two types of a,pproaches have been developed for the treatment of intrinsic viscosity in polymer solutions : the porous sphere model of Debye2 and the hydrodynamic interaction theory of Kirkwood and Riseman.a In the former, the actual random coil is replaced by a uniform porous sphere, which permits an exact solution of the hydrodynamic problem; the latter approach uses a much more reali,stic model, the Gaussian chain, but introduces approximations in the hydrodynamic equations, whose effect is difficult to assess. An improved theory of intrinsic viscosity should seek to (1) treat hydrodynamic interactions between different parts of the polymer chain at, the level of rigor achieved by Debye, and ( 2 ) take proper account of the forces between both adjacent and nonadjacent chain segments. I11 a recent paper4 we developed a variational treatment of the friction coefficient in polymer solut:ons, a treatment which we feel meets both criteria. I n the present artiole we extend our method to obtain rigorous lower bounds on the intrinsic viscosity. The a4~sumption,srunder which we claim rigor are fairly broad. (1) We represent the polymer molecule as a set of splrreried beads connected by bonds. The bonds need not be R ookean springs, and the chain which they $arm ma$ have branches and loops; van der Waals type interactions between unconnected beads are included (in particular no two beads may approach to within less than one diameter of one another), as are restrictions on bond angles. Only the beads offer hydrodynamic resistance-the bonds in between have no dimension. Each bead may rotate freely about its center, but such a rotation is in no way coupled to any change in bond angles. ( 2 ) The solvent is represented as a continuum h i d , whose motion is governed by the creeping flow Navier-Stokes equations. Inertia effects in the motions of the beads are also neglected. (3) Brownian molion of the beads is assumed to be The Jozirnul of Physical Chemistry, Vol. 76, N O . 1 , 1971
governed by a many-particle diffusion equation for the configurational distribution function of the polymer molecule. Following Kirk~vood,~ we introduce hydrodynamic interaction between beads by replacing the usual scalar diffusion coefficient with a diffusion tensor; however, our tensor differs from that used by Kirkwood.5 Our variational inequalities are restricted to steady, small shear rates. Oscillating shears and nonNewtonian behavior lie outside the scope of this paper.
Energy Dissipation Consider an isolated polymer molecule of N beads suspended between t x o parallel plates normal to the x axis, the force per unit area acting on the upper and lower plate being, respectively, y and -y in the x direction. Let the beads be momentarily located at positions rl, . . , rN, and let the forces acting on them be f,, . f N . The rate of energy dissipat,ion resulting from this system of forces should have the bilinear form 1 E = -- CDf,:fif, -I- 2 CAc:fiy B:YY (1) kT 2 , J 1. where the tensors D,, At, 2nd B will, in general, be functions of rl, . . , rN. An alternative expression for E may be obtained in terms of the stress distribution in the solvent. If d(r) represents the viscous stress a t some point r, and TO is the viscosity of this solvent, then ,
+
1
E
1
= -
2ro
"
J- d(r):d(r)d3r Clt
the integral extending over the multiply connected (1) Work supported by a grant from the National Science Foundation. (2) P. Debye, Phgs. Rev.,71, 488 (1947). ( 3 ) J. G. Kirkwood and J. Riseman, J . Chem. Phys., 16, 585 (1948). (4) J. Rotne and 8. Prager, ibid., SO, 483 (1969). ( 5 ) J. G. Kirkwood, Rec. Trav. Chim., 68, 649 (1949).
VARI~ZTXOINAL Berm13sON
THE
INTRINSIC VISCOSITY
region i l l comkting of all points located between the platee, but not in Ibc interior of a bead. The key l,o our variational method is the principle of minixruin energy dissipation, which states6 that if, in place of ithe t r u e viscous stress distribution d*(r), we uw a trriaX function a*@) satisfying the conditions t r d‘k -- 0
(34
d” = (d*)T
(3b)
73 is the symmetrized dyadic product of the unit vectors in the x and x directions (we may think of the T o r term as the contribution made by the parallel plates). For dt we shall use, following Gutlr and S i ~ x h a 6he ,~ viscous stress distribution around an isolated sphere of radius a, subject to a force ft, and suspended in a large body of solvent which is being acted upon by a shearing stress pyor d*(eO = d P ( e 0
+PdP(gr)
(7)
where a t all points r in 32’ as well as t,he further requirements and J;,[(d*
a_
Ci
s
4- p*l).C] X id2s = 0 (Id*@)
or^-
+ p*l).%d2r= -73
(4b) (5)
the resulting estiniaite E” of the energy dissipation rate will be too large. I n these equations p*(r) is a scalar function representing the trial pressure distribution, the superscript, T designates the transpose of a tensor, and B is the unit lensor; Si, S+, and 8-are, respectively, the suirfaces of sphere z, the upper plate, and the lower plate, $(r) is t)lieunit inward pointing normal to the surface or ilfat r. and P is the unit vector in the positive x direc tlon; a is the plate area. Condition 3a is nothing but the definition of a viscous stress as the nonhydrostatic part of a total stress tensor. The remaining conditions ensure that an ,zclmissible trial stress will a t least not give &e to un1)alanced forces or torques on any sphere or elcmeiit of” fluid. Minimization of E* subject to these conditions leads to an optimal d(r) equal to 770 times the sgmmetria,ecl gradient of a divergence-free velocity field which is compatible with rigid-body translation and rotation ai the spheres and the motion of the plates in the -5 and --2 diivections. Complete minimi zation is thus equivalent to solving the Navier-S tolccs equations subject to appropriate boundary conditions in a’. I n view of the complex geometry of a’,this is clearly not possible, but it is not ab all hard to construct trial functions d* which will give upper bounds on E. These bounds can then in turn be convwted into lower bounds on the intrinsic visco3ity, as we sliall show.
Trial Stress ~ ~ s ~ r ~ b i u t ~ ~ ~ s As in ref 4,we shall construct d* by superposition of contributions from individual spheres d*(r’) = yo^where
yo is
+
dZ(pl.)
(6)
are the familiar Stokes and Einstein velocity perturbations. That this choice for d* satisfies conditions 3 and 4 is easily verified; condition 5 can always be fulfilled by an appropriate choice of yo. The dimensionless parameter P may be adjusted to minimize E*. The energy dissipation rate corresponding to (6) is
E*
= -270 1 [yo2r:r
2 ~ o rE : 1.
s
n’
(v - 4su3N) 3 4..
dt(eJd3r
+ E [a’.,(e,) z
:d,(eJd3r
+
where V is the volume of solution being sheared, which, since we are ultimately calculating an intrinsic viscosity, we shall regard as containing just, a single polymer molecule. The integral in the last term of (9) is awkward, because of the complexity of the region 9’ and leads to terms involving three spheres simultaneously. T o avoid this we resort to a device employed in our earlier paper.4 The last two terms in (9) represent the integral of a perfect square over Q’. Their contribution will thus be increased if we replace Q’ by the full volume V ; since E* is already an upper bound on E, it Will retain this property after the replacement has been made. Indeed, we can obtain a somewhat improved result by applying Schwartz’s inequality to the integral over the region (V - Q’) which has been d d e d by this operation
‘I
a scalar constant, ea e r - si,and
r E 2%+ 23
(6) W. Prager in “Studies in Mathematics and Mechanics Presented to R. yon Mises,” Academic Press, New York, N. Y., 1954,p 208. (7) E. Guth and R. Simha, Kolloid Z., 74, 266 (1936).
The Journal of Physical Chemistry, Vol. 76, No. 1 , 1971
STEPHEN PRAGER
74
Before making this substitution, however, me should realize that we know neither the f, nor the chain configuration, nor the relation between them. This is the problem we shall take up in the following section. Configurational Averaging
I n order to perform the integrals in (lo), we must specify d, i n the interior of sphere i as well. The most convenient choice is &(et)
= 0
(Pi < a)
(11)
which. leads, in the ] h i t of large V , to the relations
, r:L,dz(p,)dar = - 2 m 3 y o p r : r 3. 2 ~ , r : ? f -
Clearly it does not make sense to treat, as we have done so far, a particular configuration of the chainsome sort of average is called for. Since the shear stress y is assumed to be small, it is appropriate to use the equilibrium, zero shear configurational distribution function iPe(rl, . . , , r,) for this purpose. Accordingly we seek to calculate the mean rate of energy dissipation
(E)
J . . .J~ ( r . ~. . ,, r,; V
fi . . .
V
t)x
qe(rl, . . . , rN)d3rl . . . d3rN (14) The forces f, t o be used in the expression 1 for E consist of the direct interaction between adjacent and nonadjacent beads in the chain, plus st thermodynamic contribution coming from gradients in the configurational distribution function. More usefully, we can say that the fi are produced by the deviation of the actual distribution function * ( T I , . . ., Y ~in)a steady shear from the no-shear equilibrium form \ke
where the following definitions have been introduced
The distribution function \k satisfies 5~ many-body diffusion equation of the type used by K i r k ~ o o d ;in ~ the steady state, we have
where vi is the velocity of the ith bead. Now let us consider the variational problem of minimizing ( E )with respect to fit subject to the condition
bW
with cij = rs - ri and x i the distance of the point rf from a plane midwaJy between the two parallel plates. Substitution of (1.2) into (10) will produce a quadratic expression m the ft which is of the same form as (l), and which, for any configuration of the chain, constitutes an upper bound on the true rate of energy dissipation produced by the forces f, in the presence of a shear stress
acting on the solvent a t large distances from the chain. The Journal of Ph,ysicuE Chemistry, Vol. 7 5 , No. 1, 1071
fi = -
br
It is readily found that the Euler-Lagrange equation for the optimum choice of W is
with boundary conditions
Equation 19 is identical with what would have been
VARIATIONAL Borr~nsON
THE
INTRINSIC VISCOSITY
15
obtaincd by connbi niiig (E),(16), and (17), and making the ~ ~ , ~ n t ~ ~ f i c a ~ ' i ~ ~
The boundary condition 20 merely confines each bead to the volume 8. What we h a w shown is that if, in place of the optimum set of fl, wl.ricb are rather inaccessible, we choose a trial set a,* satisfying (18) for some function W*, the resulting tislimate of the mean energy dissipation rate will always be larger than the true value. This inequality will be further reinforced if, in place of uiing (I) Lo1 calciilate E , we substitute E* as given by (ne)> @@I,
. . . , r,;
f ~ ., . I t fN))
