NIPPON GOMU KYOKAISHI

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Eq. (1) is proposed for the relation between the tensile force and elongation of a cross-linked polymer which shows the rubbery region based upon the ideal Affine deformation.
f=ν(0)RT(α-α^-2) (1)
Eq. (2) or (3) which is a corrected eq. (1) holds, however, for a tightly cross-linked polymer because of its abnormal structure.
f=r²(network chain)/r²₀(free chain)•ν(0)RT(α-α^-2) (2)
or f=Φ•ν(0)RT(α-α^-2) (3)
Here, Φ=r²/r₀².
When the shear modulus G is used, eq. (3) becomes,
G=Φ•ν(0)RT (4)
Taking elongation into consideration,
G=Φ•ν(0)RTΓ(λ_m) (5)
Here, λ_m=r_m/r_i, r_m is the maximum chain length and r_i is the average chain length without elongation.
Moreover,
Γ(λ_m)=1+6/5λ_m²+297/175λ_m⁴+……(6)
Assuming polymerization degree of n, and segment length of l, most probable average square length r₀², and maximum chain length r_m are respectively expressed by eqs. (7) and (8).
r₀²=C_n•nl² (7)
r_m=qnl (8)
Here, C_n and q are constants regarding the Polymer structure, and eq. (9) is obtained.
λ_m=(q²•n/Φ_n•C_n)^1/2 (9)
Eq. (6) is approximated by eq. (10) in the range of 0<1/λ_m<0.88.
Γ_app(λ_m)=1/(1-6/5λ_m²)=5λ_m²/(5λ_m²-6) (10)
The next equation (11) is derived from eqs. (5), (6), (9) and (10).
ν(0)RT/G=1/Φ_n-6•C_n/5q²•n (11)
Assuming the real cross-linking density is ν′(0), and the apparent one is ν(0) for a tightly cross-linked polymer, eq. (12) is derived from eq. (11).
ν′(0)/ν(0)=1/Φ_n-6/5(0.83)²•n^0.43 (12)
The reasonable relation of ν_M(0)>ν_S(0) holds in the smaller range of ν(0), while the reciprocal relation of ν_M(0)<ν_S(0) which is not understandable does in the higher range of ν(0). The latter has been made understandable by using eqs. (12).

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