5682 Mergell and Everaers Macromolecules,Vol.34.No.16.2001 (69) 品=风 6 =GA (60) =-+-6 (70) a- 61) nd 70 the phantom m0=2-G (62 relauions tor the Money-Rvin r 2C]Gph +G 2C=GM (63) (72) 2C2=0 64) comp aube/Rubins ein-Par and Rubi io a=-a1 (73) the tube diameter representing the ne A)and there n四=零 ent ngler ents (model B).Thus (65) rich uating eq 25 one obtain and 36,one obtains (74) ances c》=i2x-1+ 2R2 (2-1)x-Yle-cp ur-rondinP() (raa2》 2R2 =a21x-x1十 r rite this result in t 影，2-4-eg R,2 dcl4a+号4u例 ra2》-a2(1》」 (75) d420 60=1+0+-1 (67) deforr a ndent.Introducing )=drca) Eq tion 67 show ro31-33() is incorrect.However.the two functions are dr(),eq 75 can be rewritten as For the shear modulus and the stress-strain relation '》-ay1x-WΦ= 2-10ra21》 d) 0+中(60-0(76 (68) the elastic propere of the double tube mode weFurthermore, one obtains for the shear modulus and the stress-strain relation so that the Mooney-Rivlin parameters are simply given by B. The Heinrich-Straube/Rubinstein-Panyuk￾ov Model. Heinrich and Straube25 and Rubinstein and Panyukov43 have carried out analogous considerations for model B, i.e. an affinely deforming tube. The relation between the strength of the springs lB and the tube diameter in the unstrained state is identical to the previous case. However, the tube diameter now becomes deformation dependent: Thus, the typical width of the fluctuations changes only with the square root of the width of the confining potential. Using equations eqs 54 and 36, one obtains for the mean square internal distances: Again, we can rewrite this result in terms of a universal scaling function for the degree of affineness of the polymer deformation: Equation 67 shows that Straube’s conjecture31-33 fA(y) ) fB(y) is incorrect. However, the two functions are qualitatively very similar. For the shear modulus and the stress-strain relation, we find in agreement with Rubinstein and Panyukov.43 To account for the network contribution to the shear modulus, these authors add the phantom network results to eqs 69 and 70. This leads to the following relations for the Mooney-Rivlin parameters:43 Note that eq 70 holds only for λ ≈ 1. For large compression or extension the approximation k . l (λ) breaks down and one regains the result of Heinrich and Straube:25 C. The “Double Tube” Model. In the following, we discuss a combination of two different constraints, one representing the network (model A) and therefore deformation independent and the other representing the entanglements (model B). Thus, we use model C to combine the Warner-Edwards model with the Hein￾rich-Straube/Rubinstein-Panyukov model. Evaluating eq 25 one obtains for the tube diameter The deformation dependent internal distances are given by In this case, it is not possible to rewrite the result in terms of a universal scaling function, because the relative importance of the two types of confinement is deformation dependent. Introducing Φ(λ) ) dTCR 4(λ)/ dTBR 4(λ), eq 75 can be rewritten as For the elastic properties of the double tube model we find gA(λ) ) Fb2 xklA 6 ) GA (60) GA ) 1 4 Fb2 kBT dTA 2 (61) σT(λ) ) (λ2 - 1 λ)GA (62) 2C1 ) GA (63) 2C2 ) 0 (64) dTBR 2 (λ) ) λR dTB 2 3 (65) 〈rxx′R 2 (λ)〉 2Rg 2 ) λR 2 |x - x′| + 1 2 (λR 2 - 1)|x - x′|e-(Rg 2|x-x′|)/(dTBR 2(λ)) - 3 2 (λR 2 - 1) dTBR 2 (λ) Rg 2 (1 -e-(Rg 2|x-x′|)/(dTBR 2(λ))) (66) 〈rxx′R 2 (λ)〉 - 〈rxx′R 2 (1)〉 (λR 2 - 1)〈rxx′R 2 (1)〉 ) fB( Rg 2 |x - x′| dTBR 2 (λ) ) fB(y) ) 1 + 1 2 e -y + 3 2 e -y - 1 y (67) gB(λ) ) 1 8 Fb2 kBT λdTB 2 ) GB λ (68) GB ) 1 8 Fb2 kBT dTB 2 (69) σT(λ) ) (xλ - 1 xλ + λ - 1 λ)GB (70) 2C1 ) Gph + 1 2 GB (71) 2C2 ) 1 2 GB (72) σT(λ) ) (λ - 1 xλ)GB (λ , 1, λ . 1) (73) 1 dTCR 4 (λ) ) 1 dTAR 4 + 1 dTBR 4 (λ) (74) 〈rxx′R 2 (λ)〉 2Rg 2 ) λR 2 |x - x′| + 1 2 (λR 2 - 1)|x - x′| dTCR 4 (λ) dTBR 4 (λ) e-(Rg 2|x-x′|)/(dTCR 2(λ)) - 3 2 (λR 2 - 1) dTCR 2 (λ) Rg 2 (1 - e-(Rg 2|x-x′|)/(dTCR 2(λ))) × dTCR 4 (λ) (dTAR 4 + 2 3 dTBR 4 (λ) ) dTBR 4 (λ)dTAR 4 (75) 〈rxx′R 2 (λ)〉 - 〈rxx′R 2 (1)〉 (λR 2 - 1)〈rxx′R 2 (1)〉 ( Rg 2 |x - x′| dTCR 2 (λ) , Φ(λ) ) ) fA(y) + Φ(λ)(fB(y) - fA(y)) (76) 5682 Mergell and Everaers Macromolecules, Vol. 34, No. 16, 2001