5678 Mergell and Everaers Macromolecules,Vol.34.No.16.2001 be expressed conveniently using a parameter 0s十1 四 gtothietirproYiCestcnkhehrke mode nd the e dyna of (mic antom ion of fluctu ons due to entan excitations uw=k,+网T k+1 (23 tonian of the CMM (15) the modes hoth he exact calculation of averages r the qu g we (24 summarize the r ults nd give gen ral expres In particular mations arhi of the po r net of the 42-7 25 nonvanishing mean excitation U with 4 (16) (2》=∑(u2》Sanm2+a2Rm2(26) hal-2闭+园un-ar+ sum 2 L因+kunm27 kgT 1 -1 v 1-12 (upe) (27 2 due to the e quen D)etc. io D).(v order 因=∑k,《um》-(uu》 =2-1)g0)+(1-1-g0-1% (28 Both distributions are Gaussian and their widths kT Gconstr=g(1) (29) um2》 。+团 (19) (20) of the confining potential.One plausible choice is 7A)=7a=1) (30 follow from the Hamiltonian and the condition that the state of preparation.In particula that this choice leads to a situation a3=eu3+u3-7 (21) a%e also the ion indepth a2elhtsoheenge1ateeagDetn ean excitations.on the o and varyhe o ly with relation for the deformation dependence of the totalThus, the introduction of the single node springs does not change the eigenvectors of the original Kirchhoff matrix. The derivation of eq 14, which is the Hamilto￾nian of the Constrained Mode Model (CMM),44 is a central result of this work. It provides the link between the considerations of Eichinger,11 Graessley,45 Mark,46 and others on the dynamics of (micro) phantom net￾works and the ideas of Edwards and Flory on the suppression of fluctuations due to entanglements. C. Solution and Disorder Averages: The Con￾strained Mode Model (CMM). Since the total Hamil￾tonian of the CMM is diagonal and quadratic in the modes, both the exact solution of the model for given bvp and the subsequent calculation of averages over the quenched Gaussian disorder in the bvp are extremely simple.44 In the follow￾ing we summarize the results and give general expres￾sions for quantities of physical interest such as shear moduli, stress-strain relations, and microscopic defor￾mations. Consider an arbitrary mode bup of the polymer net￾work. Under the influence of the constraining potential, each Cartesian component R will fluctuate around a nonvanishing mean excitation UBp with Using the notation δBup ≡ bup - UBp, the Hamiltonian for this mode reads Expectation values are calculated by averaging over both the thermal and the static fluctuations, which are due to the quenched topological disorder (in order to simplify the notation, we use l ≡ l(λ ) 1), 〈vpR 2〉 ≡ vpR 2(λ ) 1)〉 etc.) Both distributions are Gaussian and their widths follow from the Hamiltonian and the condition that the random introduction of topological constraints on the dynamics does not affect static expectation values in the state of preparation. In particular Eq 21 relates the strength l of the confining potential to the width of P(vpR). The result, 〈vpR 2〉 ) (1/γp)(kBT/ kp), 〈UpR 2〉 ) γp(kBT/kp), 〈δupR 2〉 ) (1 - γp)(kBT/kp) can be expressed conveniently using a parameter which measures the degree of confinement of the modes. As a result, one obtains for the mean square static excitations Quantities of physical interest are typically sums over the eigenmodes of the Kirchhoff matrix. For example, the tube diameter is defined as the average width of the thermal fluctuations of the nodes: In particular More generally, distances between any two monomers rnmR ) rnR - rmR in real space are given by For the discussion of the elastic properties of the different tube models it turns out to be useful to define the sum Using eq 27, the confinement contribution to the normal tension2,44 and the shear modulus can be written as D. Model A: Deformation independent strength of the Confining Potential. To completely define the model, one needs to specify the deformation dependence of the confining potential. One plausible choice is i.e., a confining potential whose strength is strain independent. The following discussion will make clear that this choice leads to a situation which mathemati￾cally resembles the phantom model without constraints. Using eq 30 the thermal fluctuations (and therefore also the tube diameter eq 25) are deformation indepen￾dent and remain isotropic in strained systems. The mean excitations, on the other hand, vary affinely with the macroscopic strain. This leads to the following relation for the deformation dependence of the total 0 e γp≡ l kp + l e 1 (22) 〈UpR 2 (λ)〉 ) λR 2 ( lR(λ) kp + lR(λ)) 2kp + l l kBT kp (23) dTR 2 (λ) ) 1 M ∑ p 〈δupR 2 (λ)〉 (24) dTR 2 ) kBT Ml ∑ p γp (25) 〈rnmR 2 (λ)〉 ) ∑ p 〈upR 2 (λ)〉Sp,nm 2 + λR 2 RnmR 2 (26) g(λ) ) kBT V 1 1 - λR 2 ∑ p ( 〈upR 2 〉(λ) 〈upR 2 〉 - 1 ) (27) σT(λ) ) 1 V ∑ p kp (〈up| 2 (λ)〉 - 〈up⊥ 2 (λ)〉) ) (λ2 - 1)g(λ) + (1 - λ-1 )g(λ-1/2) (28) Gconstr ) g(1) (29) 6l A(λ) ) 6l A(λ ) 1) (30) H ) Hph + Hconstr (15) UpR(λ) ) lR(λ) kp + lR(λ) vpR(λ) (16) HpR[vpR] ) kp 2 UpR 2 (λ) + lR(λ) 2 (UpR(λ) - vpR(λ))2 + lR(λ) + kp 2 δupR 2 (17) 〈Ap(λ)〉 ) ∫ dvp ∫ dδup Ap[vp,δup] P(vpR)P(δupR) (18) 〈δupR 2 (λ)〉 ) kBT kp + lR(λ) (19) 〈vpR 2 〉 ) kp + l kpl kBT (20) 〈upR 2 〉 ) 〈δupR 2 〉 + 〈UpR 2 〉 ≡ kBT kp (21) 5678 Mergell and Everaers Macromolecules, Vol. 34, No. 16, 2001