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The Relation Between Dynamic and Equilibrium Moduli, with Consideration of Entanglements B. E . EICHINGER University of Washington, Department of Chemistry, Seattle, WA 98195
The relation between the e l a s t i c modulus and the minimum nonzero eigenvalue of the force constant matrix for an elastomer is derived by means of a coarse-graining approximation. The treatment entails i d e n t i f i c a t i o n of the macroscopic fundamental mode with the longest wavelength mode that might be calculated from molecular theory. The coarse-grained view suggests a method by which the problem of entanglements may be approached. The quantitative analysis of some aspects of t h i s problem is developed, and the long range part of the entanglement coupling is shown to be of dipolar strength. Introduction Elastomers are s o l i d s , even if they are soft. Their atoms have d i s t i n c t mean positions, which enables one to use the well-established theory of solids to make some statements about t h e i r properties i n the l i n e a r portion of the s t r e s s - s t r a i n r e l a t i o n . For example, in the theory of solids the Debye or macroscopic theory is made compatible with l a t t i c e dynamics by equating the spectral density of states calculated from either theory i n the long wavelength limit. The relation between the two macroscopic parameters, Young's modulus and Poisson's r a t i o , and the microscopic parameters, atomic mass and force constant, is established by t h i s procedure. The only differences between t h i s theory and the one which may be applied to elastomers is that ( i ) the elastomer does not have crystallographic symmetry, and ( i i ) dissipation terms must be included in the equations of motion. The absence of a space group makes the spectral problem a difficult one. Our work i n t h i s area is f a r from complete, 1-3
0097-6156/82/0193-0243$06.00/0 © 1982 American Chemical Society Mark and Lal; Elastomers and Rubber Elasticity ACS Symposium Series; American Chemical Society: Washington, DC, 1982.
244
ELASTOMERS
A N D RUBBER
ELASTICITY
but we have been gaining a general idea of what the spectral densities of amorphous materials are l i k e , and i t now seems that some statements can be made with s u f f i c i e n t conviction so as to develop theory a l i t t l e further. In the f i r s t part to follow, the equations of motion of a soft s o l i d are written in the harmonic approximation. The matrices that describe the p o t e n t i a l , and hence the structure, of the material are then considered in a general way, and t h e i r properties under a normal mode transformation are discussed. The same treatment i s given to the dissipation terms. The long wavelength end of the spectral density i s of i n t e r e s t , and here i t seems that detailed matrix calculations can be replaced by simple scaling arguments. This shows how the i n e r t i a l term, usually absent in molecular problems, i s magnified to become important in the continuum l i m i t . These observations are equivalent to a coarse-grained view of the system, which i s tantamount to a description in terms of continuum mechanics. [It i s clear that "points" of the continuum may not refer to such small collections of atoms that thermal fluctuations of the coordinates of t h e i r centers of mass become substantial fractions of t h e i r strain displacements.] The elastomer i s thus considered to consist of a large number of q u a s i - f i n i t e elements, which interact with one another through dividing surfaces. A dividing surface provides a convenient means to think about the problem of entanglements. A chain from one element may meander across a dividing surface into a neighboring element, and then return. The number of chains from the neighboring element with which i t i s entangled w i l l certainly be proportional to the square of the number of steps that i t has made in the adjoining neighborhood. This approach to the entanglement problem i s only given a preliminary treatment here, but what i s done has the advantage of being well defined. Molecular Equation of Motion The potential energy V of the elastomer i s presumed to be given as a function of the atomic coordinates χ . ( l l o K 3 , K i < n ) , where η i s the number of atoms in the system. Since an elastomer has a well-defined equilibrium shape, there must be equilibrium positions x? for a l l atoms that are part of the continuous net work. Expand the potential in a Taylor series about the equilibrium positions, and set the potential to zero at equilibrium, to obtain V=(l/2) Σ α,3
Σ 1.j
kf. J
(x° - χ « ) ( χ ξ - χ ξ Η ΐ / 2 ) ( χ - χ ) Κ ( χ - χ Τ J
J
Mark and Lal; Elastomers and Rubber Elasticity ACS Symposium Series; American Chemical Society: Washington, DC, 1982.
(D
12.
EiCHiNGER
Dynamic and Equilibrium Moduli
245
to f i r s t approximation. The coefficient matrix ( k - p = Κ i s the matrix of force constants. For now the potential w i l l be taken to be isotropic in a l l atom p a i r s , so that k?^ = k..6 . It i s then convenient to rearrange the lx3n matrix into a 3xfrmatrix X, with transpose X', and write g
J
V = (l/2)Tr[(X-X)K(X-X)']
(2)
where the elements of Κ are now just the k... The trace operation Tr has the effect of computing the scalar product of the displacement vectors. The Lagrangian L for the system i s L=
(l/2)Tr(M') -
(l/2)Tr[(X-X)K(X-X)"]
where M i s a diagonal matrix of masses m.. motion i s simply
(3)
The equation of
ΧΜ + (X-X)K = 0
(4)
We now need to add the dissipation term. A Rayleigh dissipation function w i l l suffice for t h i s purpose, since the hydrodynamic interactions in the.elastomer should be well screened. Let F be a matrix such that XF has elements of the form f.[*« - ζ
-
1
Σ
x«]
j
J
where the sum i s over the ζ neighboring atoms which contact the i t h atom. This term represents the average velocity f i e l d against which the i t h atom moves. The f r i c t i o n depends upon the instantaneous configuration, but we w i l l assume that a pre-averaging approximation has been applied (similar to that used to pre-average the Oseen tensor in single chain dynamics) so that i t s structure i s determined by the mean positions of the atoms. The f r i c t i o n factor f. i s not necessarily the same for a l l atoms, and since the structure of the matrix F i s already complicated, no harm comes from the generalization. The equation of motion i s further generalized by imposition of a 3xn external force matrix a, so that we f i n a l l y have XM + XF + (X-X)K = σ
(5)
which requires a solution. This i s d i f f i c u l t in general because F and Κ do not necessarily commute, and they cannot then be simultaneously diagonalized. Nevertheless, there i s an argument that one can make for the long wavelength l i m i t that w i l l show what the structure of the solution to eq. (5) must be.
Mark and Lal; Elastomers and Rubber Elasticity ACS Symposium Series; American Chemical Society: Washington, DC, 1982.
246
ELASTOMERS
A N D RUBBER
ELASTICITY
Suppose that a s t r i p of rubber i s placed under modest tension, and i s then plucked. On performing t h i s experiment, one may observe the fundamental o s c i l l a t i o n of the form sin πχ/L, where L i s the length of the s t r i p . The o s c i l l a t i o n dies away on a time scale of about one second for a typical rubber band, for example. The motion i s clearly not overdamped, as are the motions of individual polymer molecules in a solution. How i s i t that the high frequency modes of eq. (5) have a small effective mass and a large f r i c t i o n constant, while the low frequency modes have large effective mass and small f r i c t i o n . The answer, of course, l i e s in the structure of the matrix F. For long wavelength modes the corresponding eigenvalues of F are very small, while the same i s not true for the short wavelength modes. Imagine now that the elastomer i s repesented by an array of cubes, or more generally by an array of polyhedra that are topologically isomorphic to cubes, and that these are packed together i n a simple cubic array. Each polyhedron might be several hundged or more Angstroms on an edge, and hence contain well over 10 atoms. The atoms in each c e l l w i l l be given labels U ) a r b i t r a r i l y assigned, but the c e l l s w i l l carry indices ( i , j , k ) corresponding to the usual crystallographic convention. The coordinates of the matrix X might then be written and F and Κ now carry eight indices. The advantage of t h i s supernumeration i s that F and Κ are now blocked into a form such that they w i l l be approximately diagonalized by a transformation whose long wavelength eigenvectors are of the same form as those that diagonalize the Κ matrix for a simple cubic l a t t i c e . Within each c e l l the coordinates are v i r t u a l l y constant with respect to an eigenvector whose wavelength i s very much greater than the average edge length a of a single c e l l . For these modes, the equation of motion must admit of a scaling argument. On making the transformation X — QT, where Q are the approximate normal mode coordinates, and Τ i s the matrix of approximate eigenvectors (T would diagonalize a regular cubic l a t t i c e ) , ΤΜΤ' scales r e l a t i v e l y as _a , TFT' scales as
1
Mark and Lal; Elastomers and Rubber Elasticity ACS Symposium Series; American Chemical Society: Washington, DC, 1982.
12.
EICHINGER
Dynamic and Equilibrium Moduli
247
On the basis of t h i s argument, the equation of motion for the longest wavelength mode in the elastomer i s m
eff^l
+
K
l l q
=
c
r
t
l
^
where m ^ i s an effective mass, κ- i s the minimum non-trivial eigenvalue of K, and t , i s the eigenvector corresponding to κ^. The f r i c t i o n term i s omitted since i t i s small, and because dropping i t at t h i s stage does no harm to the relations with continuum equations which w i l l follow. The solution of eq. (6) is q^j^ = atj/Kj + q!j* cosa^t
(7)
where
