Rubber Elasticity J. E. Mark Department of Chemistry and The Polymer Research Center, The University of Cincinnati, Cincinnati, OH 45221 Rubber elasticity may be operationally defined as very large deformability with essentially complete recoverahility. In order for a material to exhibit this type of elasticity, three molecular conditions must he met: (1) the material must consist of polymeric chains, (2) the chains must be joined into a network structure, and (3) the chains must have a high degree of flexibility (1-3). The first requirement arises from the fact that the molecules in a rubber or elastomeric material must be able to alter their arrangements and extensions in space dramatically in response to an imposed stress, and only a long-chain molecule has the required very large number of spatial arrangements of very different extensions. This versatility is illustrated in Figure 1, which depicts a random spatial arrangement of a relatively short polymer chain. In this random arrangement the chain extension (as measured by the end-to-end separation) is quite small. For even such a short chain, the extension could be increased approximately fourfold by simple rotations about skeletal bonds, without any need for distortions of bond angles or bond lengths. The second characteristic cited is required in order to obtain the elastomeric recoverahility. I t is obtained by joining together or "cross-linking" pairs of segments, approximately one out of a hundred, thereby preventing the extended polymer chains from irreversibly sliding by one another. The resulting net-
Figure 1. A two-dimensional projection of the backbone of an m l k a n e chain (or sequence from a longer polyethylenechain) which consists of 200 skeletal bands (4).misrepresentative arrangementor spatial wnfigvatimn was computer generated using known values of the bond lengms, band angles, mtational angles about skeletal bands, and preferences between the corresponding rotational states (9.
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work structure is illustrated in Fieure 2. in which the crosslinks are represented by dots. ~ h e l cross-links e may be either chemical bonds las would occur in sulfur-vulcanized natural rubber ( I ) or p'hysical aggregates [for example the small crystallites in a partially crystalline polymer (1,6) or the glassy domains in a multi-phase block copolymer (7,811.The last characteristic specifies that the different spatial arangements be accessible, i.e., changes in thrse arrangements should not be hindered bv ronstralnts as might result trom inherent rlgidity of the c h s , extensive chain crystallization,or the very high viscosity characteristic of the glassy state ( 1 , 3 ) . The Orlgln of the Elastic Retractlve Force
The molecular origin of the elastic force f exhibited by a deformed elastomeric network can he elucidated through thermoelastic experiments, which involve the temperature dependence of either the force at constant length L or the length at constant force (I). Consider first a thin metal strip stretched with a weight W t o a point short of that giving permanent deformation, as is shown in Figure 3. Increase in temperature (at constant force) would increase the length of the stretched strip in what would he considered the "usual" behavior. Exactly the opposite, a shrinkage, is observed in the case of a stretched elastomer! For puruoses of comparison, the . . result observed for a gas at constant pressure is inrludcd in the Firure. Raisine its temperature would uf cuurw cause an increase in volume V. The explanation for these observations is given in Figure 4. The primary effect of stretching the metalis the increase AE in energy caused by changing the distance d of separation between the metal atoms. The stretched strip retracts to its original dimension upon removal of the force since this is associated with a decrease in energy. Similarly, heating the strip a t constant force causes the usual expansion arising from increased oscillations about the minimum in the asymmetric potential energy curve. In the case of the elastomer, however, the major effect of the deformation is the stretching out of the network chains, which substantially reduces their entropy (I, 2). Thus, the retractive force arises primarily from the tendency of the system to increase its entropy toward the (maximum) value it had in the undeformed state. Increase in temperature increases the chaotic molecular motions of the
Figure 2. Schematic sketch of a typical elastomeric network
ELASTIC BEHAVIQR
w
u
d
deform '
.
2. Rubber
sas
Figure 3. Results of thermoelastic experiments carried out on a typical metal. rubber, and gas.
chains and thus increases the tendency toward the more random state. As a result there is a decrease in leneth at const;lnt forcc, or an incrrase in furre at ronstant length. This is strikinelv similar to the behavior of a com~ressedem. in which the extkkt of deformation is given by the recip&l volume 1/V. The pressure of the gas is largely entropically derived, with increase in deformation (increase in 1/V) also corresponding to a decrease in entropy. Heating the gas increases the driving force toward the state of maximum entropy (infinite volume or zero deformation). Thus, increasing the temperature increases the volume at constant pressure, or increases the pressure at constant volume. This surprising analogy between a gas and an elastomer (which is a condensed phase) carries over into theexpressions for the work dw of deformation. In the case of a gas dw is of course -pdV. For an elastomer, however, this term is essentially negligible since network elongation takes place at very nearly constant volume (1,2).The corresponding work term where the difference in sign is due to the now becomes +fa, fact that positive w corresponds to a decrease in volume hut an increase in length. Similarly, adiabatically stretching an elastomer increases its temperature in the same way that adiabatically compressing a gas (for example in a diesel engine) will increase its temperature. The basic point here is the fact that the retractive force of an elastomer and the pressure of a gas are both primarily entropically derived and, &a result, the thermodynamic and molecular descriptions of these otherwise dissimilar systems are very closely related.
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The Elastic Free Energy and Elastic Equations of State The simplest molecular theories of rubberlike elasticity (I, 2,9-11) are based on a Gaussian distribution function w(r) (1, 2, 12) for the end-to-end separations r of the network chains (chain sequences extending from one cross-link to another). Specifically, the function
where (r2)orepresents the dimensions of the free chains as unperturbed by excluded volume effects ( I ) , is applied to the network chains in hoth the stretched and unstretched states. The Helmholtz free energy of such a chain is given by a simple variant of the Boltzmann equation
Figure 4. Sketches explaining the obsewations described in Figure 3 in terms of the,mOleCUlar origin of the elastic force or pressure.
T. Consider the process of stretching it from its random undeformed state with r = ro components of x , y , z to the deformed state with r components of a , x , a Y y , a , z (where the a's are molecular deformation ratios). The free energy change for a single network chain is then simply
Since the elastic response is essentially entirely intramolecula~(1,4,9,IO, I3), the free energy change for v network hains with average components (x2), (y2), ( 2 2 ) is
It has generally been assumed that the strain-induced displacements of the cross-links or junction points are aftine (i.e. linear) in the macroscopic strain. In this case, the deformation ratios are obtained directly from the dimensions of the sample in the strained state and in the (initial) unstrained state: a, = LJL,;, a, = L,lL,i, a= = L,IL,i
(5)
The dimensions of the cross-linked chains in the undeformed state are and, in the simplest theories (1,2), (r2); is assumed to be identical to (r2)0;i.e., it is assumed that the cross-links donot significantly change the chain dimensions from their unperturbed values. Also, the isotropy of the undeformed state requires that and thus The expression for the elastic free energy of deformation is then simply
where C(T)is a constant a t a specified absolute temperature Volume 58 Number 11 November 1981
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Equation (9) is basic to the molecular theories of ruhherlike elasticity and can he used to ohtain the elastic equations of state for any type of deformation U,2, %11,13,14). Its application is best illustrated in the case of elongation, which is the type of deformation used in the great majority of experimental studies (1.2). This deformation ocurs at essentiallv constant volume and thus a network stretched by the amount a, = a > 1 would have its oeroendicular dimensions compressed by the amounts ~ l 'aa-= y a-lI2 < 1. Accordingly, for elongation,
Since the Helmholtz free energy is the "work function" and the work of deformation is fdL (where L = aLi) f
= ( ~ A F I ~ L ) T ,= V
(11)
The nominal stress f* I f/A*, where A* is the undeformed cross-sectional area, is then f* = vkT(a - a-2)
(12)
where u is now the density of network chains, i.e., their number per unit volume V = LjA*. Also frequently employed is the "reduced stress" or modulus
p]s f*l(a- a-2)= ukT
(13)
I
CONSTANT T
CATHETOMETER
Figure 5. Apparatus for carrying ouf sterstrain measurements on an elastomer in elongation.
O6
7
The elastic equation of state in the formgiven in eqn. (12) is strikinelv similar to the molecular form of the eouation of state for & ideal ga's, p = NkT(1lV)
(14)
where the stress has replaced the pressure and the number of network chains the number of eas molecules. Similarlv. .. since the stress was assumed to be entirely entropic in origin, f* is oredicwd to bedirectlv urooortional to T at constant u land i/), as is the thk ideal gas a t constant 1/V. The strain function ( a - a-2) is somewhat more complicated than is 1/V since the near incompressibility of the elastomeric network supetposes compressive effects (-ae2) on the simple elongation (a) being applied to the system. Recent work suggests that the elongation of a network can he represented as affine probably only in the range of small deformations (4, 10, 15, 19). At higher deformations, it is thought that junction fluctuations diminish the modulus by a factor Am< 1 in eqn. (13):
p]= &ukT
(15) In the limit of the very nonaffine deformation which would be exhibited by a "phantom" network (where the chains are portrayed as being able to transect one another) where 4 is the cross-link functionality (the number of chains emanating from a network cross-link) (4,16,19). Some Experimental Details and Results
The apparatus typically used to measure the force required to give a specified elongation in a ruhherlike material is very simple, as can be seen from its schematic description in Figure 5 (20, 21). The elastomeric strip is mounted between two clamps, the lower one fixed and the upper one attached to a movable force eauee. A recorder is used to monitor the outout of the gauge as a function of time in order to ohtain equilihrium values of the force suitable for comoarisons with tbeorv. The sample is generally protected with an inert atmosphere, such as nitrogen, to prevent network degradation, particularly in the case of measurements carried out at elevated temperatures. Both the sample cell and surrounding constant-temperature bath are glass, thus permitting use of a cathetometer or travelling microscope to ohtain values of the strain by
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Figure 6. Stress-eiongation curve tor nahlrai rubber in Me vicinity of r w m temperature (2.22).
measurements of the distance between two lines marked on the central portion of the test sample (20,21). A typical stress-strain isotherm thus obtained on a strip of cross-linked natural rubber is shown in Fieure 6 (2.22). . . . The units fur the force are generally Newcons, and the curves 01,~ainedare usuallv cherkrd for reversibilitv. In this tvoc of representation,
