3.B Cascade theory of gelation
4.2 Affine network theory
4.2.1 Local structure of cross-linked rubbers
Rubbers have complex structures made up of mutually entangled polymer chains. Let us focus on a cross-link point P in the sample, and study the chain paths and spatial distribution of the cross-links around it (Figure4.5) [4, 5].
The next cross-link along the path of the selected subchain starting fromthe cross-link P is a topological neighbor of the cross-link (• in Figure4.5). In contrast, the cross-links that lie far fromP along the path of the subchains, but very close to P in spatial distance, are spatial neighbors (× in Figure4.5). Let
Q ≡
4π 3 s2 30/2
µ
V (4.12)
be the number of spatial neighbors in the spherical region (the broken line in Figure 4.5) with the radius s2 10/2, the mean radius of gyration of the subchains, whereµ is the total number of cross-links in the sample andV its volume. For common rubbers, Q is as large as Q = 25–100. There are many cross-links in the neighborhood which are not directly connected to the particular one in focus. For randomcross-linking of prepolymers, s2 0µ−1, so thatQ is proportinal to 1/√µ; it decreases with the degree of cross-linking.
< r2 >01/2 P
Fig. 4.5 Spatial (×) and topological (•) neighborhoods around a cross-link in a rubber sample.
Due to such densely packed molecularly interpenetrated structures, rubbers are incom-pressible under deformation. Each chain takes a Gaussian conformation following the Flory theoremfor screened excluded-volume interaction. On the basis of these char-acteristics, we can derive the elastic properties of rubbers froma microscopic point of view.
4.2.2 Affine network theory
Consider a cubic rubber sample with side lengthL in equilibriumstate to be elongated along itsx-axis by a tension f to λx times its initial length (Figure4.6). The sides in they and z directions are deformed λy= λztimes. At the initial equilibrium state before the force is applied, each subchain in the sample has its end-to-end vector r0with the probability with Gaussian distribution
P0(r0) =
3
2π r2 0
3/2
exp
− 3r02
2 r2 0
, (4.13)
with the mean square value r2 0=na2(n is the number of repeat units on the subchain).
This is called Gaussian assumption. The chain vector r0connects the neighboring two junctions.
Kuhn [7] and Wall [8] assumed further that the chain vector r0deforms in proportion to the macroscopic deformation(λx,λy,λz) in spite of the molecular interaction and topological entanglements in the network. On the basis of this affine deformation, they found the relationship between the applied force and the deformation of the sample.
This assumption of affine deformation is written as
r0−→ r = ˆλ·r0, (4.14)
r0 r
f f
equilibrium
deformed deformation tensor
λ
Fig. 4.6 Relation between macroscopic and microscopic deformation of a rubber sample.
where ˆλ is the deformation tensor, and r0and r are chain vectors before and after the deformation. The deformation tensor takes the form
ˆλ≡
λx 0 0
0 λy 0
0 0 λz
(4.15)
for a uniaxial elongation. In more general deformation, the tensor has finite off-diagonal elements.
The physical base of affine deformation lies in the intricate structure of rubber described above; polymer chains are randomly coiled, highly interpenetrated and entan-gled, but may follow the deformation freely by adjusting the positions of the chain segments.
Letν be the total number of subchains in the sample. The number of subchains whose chain vector falls in the region r0and r0+ dr0is given byνP0(r0)dr0. They take the vector between r and r+dr after deformation, and hence the following relation holds:
νP0(r0)dr0= νP (r)dr, (4.16)
whereP (r) is the chain distribution after deformation. The assumption of affine deforma-tion connects r to r0by the relation (4.14). Because the free energy stored in a subchain whose chain vector is r is given by (1.36)
φ(r) = 3kBT
2 r2 0r2, (4.17)
in Gaussian approximation, the total free energy of the deformed sample is F (ˆλ) =
φ(r)νP (r)dr. (4.18)
Substituting (4.17) into this equation, and by using the affine deformation (4.14), we find In the usual deformation of rubber, the volume change is negligibly small, so that we can setλx= λ, λy= λz= 1/√
λ.
The tension can be found by differentiationf = (∂defF /∂(λL))T to be f =νkBT
Dividing (4.21) by the initial areaL2of the cross section gives the elongational stressσ (tension by a unit area). The stress–elongation relation turns out to be
σ =νkBT
Because the cross section under deformation isL2/λ, the stress τ per unit area of the cross section under deformation is
τ =νkBT of the densityρ, and the molecular weight M of the subchain. Hence we have
τ =ρRT
The molecular theory of rubber elasticity on the basis of affine deformation assumption is the affine network theory, or the classical theory of rubber elasticity.
Young’s modulus
The Young’s modulusE of a rubber is found by further differentiation of the stress E = λ
The linear Young’s modulusE is defined by the Young’s modulus for an infinitesimal deformation. Fixing atλ = 1 in the above equation, we find
E =3ρRT
M (4.26)
For example, a sample of M = 104, ρ = 1gcm−3 at T = 300K gives ν/L3= 10−4mol cm−3andE = 7.4 × 106dyne cm−2. It is 10−5times smaller than the Young modulus of iron 9×1011dyne cm−3.
The Young’s modulus increases in proportion to the absolute temperature because the thermal agitation grows stronger with temperature. The result is opposite to metals, which become softer with temperature. In a metal, atoms are regularly positioned to minimize interaction energy in the form of crystal. Thermal motion intensifies on heating, so that the force for restoring the original positions is weakened with temperature, which results in a decrease in the elasticity.
In rubbers, however, the modulus increases in proportion to the temperature due to the reduction of the entropy by the constraint brought by deformation. Thus, entropic elas-ticity and energetic elaselas-ticity have opposite tendencies as a function of the temperature.
Tension–elongation curve
The main characteristics of rubber elasticity is well described by affine network theory, but the profile of the tension–elongation curve deviates in the high-elongation region from the experimental observation. Figure4.7compares the experimental data (circles) of the tensionf with the theoretical calculation (broken line) as functions of the elongation λ [9]. Data show the shape of the letter S. There is a sharp increase in the high-elongation region. They largely deviate fromthe theory because the chains are stretched beyond the linear regime, and the Gaussian assumption of the affine network theory breaks down at high elongation.
Toreloar improved this deficiency of the Gaussian assumption by introducing the Langevin chain (1.26) instead of the Gaussian chain to incorporate chain nonlinearity [9] (see Section4.6). The effect of nonlinear stretching can thus be studied by refining the single-chain properties of the subchains.
Chain entanglements
There are also some discrepancies between theory and experiments regarding the nonlin-ear stretching effect. Figure4.8plots the ratioσ/(λ−1/λ2) against the reciprocal defor-mationλ−1for a cross-linked natural rubber (Mooney–Rivlin plot) [10–13].Because the ratio increases in proportion toλ−1, the experiments can be fitted by the linear curve
σ = 2C1
with two constantsC1andC2. Equation (4.27) is called the Mooney–Rivlin empirical formula.
Elongation λ 5
4
3
2
1
0
1 2 3 4 5 6 7 8
Tension [mN m–2]
Fig. 4.7 Tension–elongation curve of a cross-linked rubber. Experimental data (circles), affine network theory by Gaussian chain (broken line), affine network theory (4.107) by Langevin chain (solid line).
4.0
3.0
2.0
0.4 0.6 0.8 1.0
B A C D E F G
1/λ σ/(λ–1/λ2) [kg cm–2]
Fig. 4.8 Mooney–Rivlin plot of a cross-linked natural rubber. The curves A–G have different degrees of cross-linking with sulfur content covering from3% to 4%. (Reprinted with permission from Gumbrell, S. M.; Mullins, L.; Rivlin, R. S., Trans. Faraday Soc. 49, 1495 (1953).)
Because the constant C2 decreases when the rubber is swollen by solvents, this extra termis deduced to be caused by the topological entanglements of the subchains.
The entangled parts serve as the delocalized cross-links which increase the elasticity.
Networks are disentangled on swelling, and the Mooney constantC2decreases.
4.2.3 Elastically effective chains
Apart fromthe problemof entanglements, there remains another difficult problemin affine network theory. It is how to count the numberν of subchains that contribute to the elasticity. Obviously, dangling chains and self-loops should not be counted. They are elastically inactive because they do not transmit the stress.
Flory clarified the activity of subchains by using the words elastically effective chain, or active chain [1]. An elastically effective chain is a chain that connects two neighboring cross-link junctions in the network.
Flory’s correction
When networks are formed by cross-linking of the prepolymers, the end parts of the pre-polymers remain as dangling ends in the network. Flory made a correction by subtracting the number of such trivial free ends from the numberν of chains which appeared in the stress [14].
LetM be the molecular weight of the prepolymers, and let Mcbe that of the subchains after cross-linking. The latter is the average value over the subchains whose distribution is assumed to be sufficiently narrow. The number of subchains in a unit volume is ν = ρ/Mc, whereρ is the density of the rubber sample. Because the number of ends of the prepolymers is given by 2ρ/M, the number of the effective chains should be
νeff= ρ
by subtraction (Figure4.9). The tension is then τ =ρRT
where the factor 1−2Mc/M has appeared to exclude the free ends.
Criterion for elastic activity
There remain many inactive subchains after such an end correction is made. For instance, a group of chains, such as shown in Figure4.10, may be dangling from one junction as a whole. They are inactive. Scanlan and Case applied the graph theory and introduced a criterion for judging the activity of a given subchain [15,16]. It is stated as follows.
Let(i,k) be the index to characterize the topological nature of a junction; the index i (path number) is the number of paths emerging from it and connected to the skeletal structure of the network, andk is the multiplicity of the junction (the number of subchains connected to the junction). Letµi,k be the number of junctions whose index is(i,k).
Mc
M
Fig. 4.9 Flory’s correction for the dangling chains.
dangling group
i =1, k =2
Fig. 4.10 An end group which is dangling froma junction by a single path.
Junctions whose path number is larger than or equal to 3 are called elastically effec-tive junctions. Junctions with path number 1 connect dangling chains (Figure4.10);
junctions with path number 2 are not active because they merely extend the already existing paths. Both types should not be counted as effective chains.
The Scanlan–Case (SC) criterion states that a subchain is elastically effective if both its ends are connected to the elastically active junctions, i.e., whose path numbersi and iare larger than or equal to 3 (Figure4.11). The criterion leads to
νeff=1 2
∞ k=2
2k
i=3
iµik (4.30)
for the number of elastically effective chains, where the factor 1/2 is necessary to avoid counting the same subchain twice.
( i , k )
( i' , k' )
Fig. 4.11 Scanlan–Case criterion for elastically effective chains. Chains withi ≥ 3 and i≥ 3 are effective.
The numberµi,k of junctions with specified type can be found as a function of the degree of reaction for polycondensation systems and the random cross-linking of pre-polymers. Also, there has been much research into the nature of the active chains and elastic moduli near the gelation point. Some results will be presented in Section8.2.
4.2.4 Simple description of thermoelastic inversion
As a simple application of the affine network theory, let us study thermoelastic inversion by taking the effect of thermal expansion into the theory [17]. Consider a sample of rubber to be elongated byλ0= 1+ 0 ( 0 1) at a reference temperature T0. The tension is σ ∼=3νkBT0 0. If the sample is heated to the temperatureT under the constant length, the sample increases its volume, so that the length in equilibrium atT0is elongated, which results in a reduction in the degree of elongation by
= 0−β is the thermal expansion coefficient at a constant pressure. The thermal expansion is caused by the motion of subchains and their molecular interaction.
The tension should be given by σ 3νkBT
If the temperature coefficient is calculated under the condition that the degree of elon-gation 0measured relative to the equilibrium length at the reference temperatureT0is constant, it is ∂σ
The critical degree of elongation at which the sign changes is found to be 0=β
3(2T −T0). (4.35)
The tension reduces on heating in the small elongation region below this critical value.
For instance, if we takeT0= 293 K, T = 343 K, and β = 6.6 × 10−4K−1, the critical elongation is 0= 0.086. It turns out that, for the deformation below 8.6%, thermal expansion dominates the entropic elasticity.