316 Jarry, Monnerie
Macromolecules
(19) K. L. Wun and F. D. Carlson, Macromolecules, 8, 190 (1975). (20) A. M. Hecht and E. Geissler, J. Phys. (Paris),39, 631 (1978). (21) E. Geissler and A. M. Hecht, J. Phys. (Paris),39, 955 (1978). (22) The variance has been determined from estimates of pipetting
errors, limitations of instrumental resolution, and uncertainty in fitting of data points. Note that our former suggestion, that a = 4, is in error because of an earlier misconception' concerning the identity of p .
Effects of a Nematic-Like Interaction in Rubber Elasticity Theory Jean-Pierre Jarry* and Lucien Monnerie Laboratoire P.C.S.M., Ecole Supdrieure de Physique et Chimie de la Ville de Paris, 75231 Paris Cede+ 05, France. Received November 7, 1978
ABSTRACT: A modified theory of rubber elasticity is proposed which includes anisotropic intermolecular interactions favoring the alignment of neighboring chain segments. These interactions are described in the mean field approximation by an intermolecular potential having the same form as that used for the study of nematic phases. It is shown that such interactions increase orientation of chain segments but do not significantly alter the stress-strain behavior of a network. Finally, experimental consequences of the theory are discussed. In recent years, the local structure of amorphous polymers has been a subject of considerable interest. In the rubbery state, the stress-induced orientation distribution of chain segments should be very sensitive to the presence of local order. The second moment S of this distribution has been calculated by Roe and Krigbaum' in the framework of the classical gas-like t h e ~ r yof~ rubber ,~ elasticity. Observation of deviations from this theory is a way to approach the problem of local order in rubbers. A few data are already available in this respect. Fukuda, Wilkes, and Stein4 reported values of the stress-optical coefficient which are too high in dry rubbers and which are reduced upon swelling. They suggested that this inconsistency could be due to some local order. Besides, the fluoresence polarization techniqueM has been recently extended to the study of rubbers and preliminary results' also show that S is anomalously higher in dry networks than in swollen ones. This emphasized the need for a modified theory of rubber elasticity including intermolecular interactions which could give rise to anisotropic packing of chain segments. Di Mario8 and Jacksonget al. developed lattice models with anisotropic interactions which enhance the alignment of neighboring segments. Their aim was not to study orientational properties of networks but rather to account for the additional C2 term of the phenomenological Mooney-Rivlin equation: u = (C, + C2/A)(A2 - A-1) where u is the true stress and A the extension ratio. Both models lead to a ratio C2/C1,proportional to 1/N ( N being the number of segments per chain), which is much smaller than experimental values (C2/C1 1). The Di Marzio theory was improved by Tanaka and Allenlo who showed that his C2 term should be doubled. Recently, de Gennesl' discussed the behavior of polymeric networks containing nematogenic segments near the isotropic-nematic transition. In the present work, the chain segments of usual rubbers are considered as cylindrically symmetric objects submitted to a weak nematic-like interaction. Segment-segment interactions are described in terms of a thermodynamic potential of the same form as that introduced in the study of nematic phases.12J3 The stress-strain and orientation-strain relations are derived for uniaxial stretching.
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0024-9297/79/2212-0316$01.00/0
It is concluded that such interactions principally increase the orientation and that the mechanical behavior is not significantly altered, according to the results of Di Marzio, Jackson, and Tanaka and Allen. Finally, experimental procedures are discussed which should allow one to distinguish between nematic-like interaction and entanglement effects.
General Considerations In a uniaxial medium, the intermolecular potential U123 between two molecules or chain segments, which behave as cylindrically symmetric objects, depends on the intermolecular distance r12 and on the angles el and e2 between the molecular axes and the symmetry axis of the medium. It can be expanded in the form:
U12 = ,E uh-12)Pl(01)Pd02) 1 even
(1)
where P i ( 0 )are Legendre polynomials. Restricting the development to the first anisotropic term (1 = 2), one obtains a contribution Fintto the free energy per molecule: where S is the second moment of the molecular distribution function and U is a positive parameter of interaction. As a preliminary step, let us consider a chain of N segments interacting with a uniaxial medium of axis 2, the end-to-end vector R lying in the 2 direction (Figure 1). The free energy per segment can be expanded in the form:
F = aS2 + bS3
+ . + cr2 + dr4 + . . . + er2S + , ,
fr2S2+ gr4S + . . . (3)
where S is the second moment of the surrounding medium and r2 = R2/NRo2, Ro2referring to the free chain. The first set of terms characterizes the nematic-like i n t e r a ~ t i o n ' ~ and the second one describes the usual entropic elasticity.2 cr2is the dominant Gaussian term c = 3/2kTand dr4 is the first non-Gaussian term corresponding to a possible increase in the tension when the chain approaches its limit of extension. The remaining coupling terms arise from the alteration of chain elasticity due to intermolecular in0 1979 American Chemical Society
Nematic-Like Interaction in Rubber Elasticity Theory 317
Vol. 12, No. 2, March:-April 1979
t'
lz (b)
(a)
/
Figure 1. Schematic representation of chain conformations: (a) isolated chain, (b) chain interacting with a uniaxial medium. Its conformation tends to an unidimensional folding along 2.
Figure 3. Representation of the three identical chains in a three-chain cell. The end-to-end vectors are mutually perpen-
dicular.
where (Y is a coefficient depending on A , B , E, F , and G. Placing eq 6 into eq 5, the free energy is obtained as a function of R only:
us
c
Figure 2. Schematic evolution of the Gaussian chain stiffness t / R = ( 2 / R o 2 ) ( c+ eS -k fS2) with increasing interaction.
teraction. As a matter of fact, they modify the tension t a t the chain ends: a 2 t = -(NF) = -$c + eS + fS2 + . . .)R t aR 8 0 4 -(d gS)R3 . . , (4) NRO4
+
+
The signs of the coulpling terms can be simply predicted from the symmetry properties of the interaction which privileges the *Z direction of the segments: An isolated tridimensional chain tends toward the unidimensional state when submitted to nematic-like interaction (Figure 1). The first Gaussian term of eq 4 must then decrease from 3kTIR; to kT/R; when interaction increases. It is thus obvious from Figure 2 that e is negative (as previously shown by de Gennes") and f is positive. Moreover, g is negative since the non-Gaussian term is smaller for unidimensional chains than for tridimensional ones.15 The free energy per segment is finally rewritten with explicit signs on the form: F = A S 2 - BS3 + . . . + Cr2 Dr4 . . , -Er2S + Fr2S2- Gr4S + . . . ( 5 )
+
+
Let us now consider an assembly of mutually interacting chains, all having the same end-to-end vector R. Such a situation is not a r'ealistic description of a polymeric network but it simp1.y reveals the main properties of an ensemble of flexible self-interacting chains. For a given value of R, the second moment of the orientation distribution of the chain segments is determined by:
(g) = o r
Hence: r; S = -r2 2A
+ ar4 + . . .
The broad conclusions to be drawn from the above results (eq 6 and 7 ) are the following: (i) A nematic-like interaction does not affect the first term of F (eq 7) which dominates the stress-strain behavior in the Gaussian range (r4
