Macromolecules.Vol.34.No.16.2001 Tube Models for Rubber-Elastic Systems 568 sent,entia Sq)=0dx0dr× har in the elow.the o四aas for ert in On the ot In the present case,eg 26 reduces to te e is oft the osionarhd tions characterized a2a》=表dp(um2Mepr-pr(54 h tu 8hnedvaecatio2.8 In the undeformed state (20=1》=2R21x-1 (55) s under dis 把hntre or欧ven by the Debe ions 0.As a consequence.t "intrinsic"pha modeled as s A.The Warner-Edwards Model.Warner and statistics of long pa ths order to recov ver the hantom network shear modulus as to cros the Rouse model matr takes the ratio simple tridiagonal form ined tube-like by sh hat出 -210 0 ad and M he lines alculation.where we evaluate model A for linear K- 1-2 transformation matrix T (57刀 s=p=六广plr月 (50) d2-2 (ra2》 2R2 =21x-1+ k。=4ksin偎 51) 1-3 1-eR-0)(58 R If we consider a path with given radius of) te that the ltterqnbe the tube diameter btans pprox a2-10Kra2(》 4a2 d2=六人dp+ 0W=1+ey-1 (59) 、 T 4k+)2√匠 (52) sure which could be further simplified.since in this limit the co g to y d and tube,ie,k多 y1,where fy)tends to one. to the ambiguous use of the term “tube” in the literature (including the present paper). A real tube is a hollow, cylindrical object, suggesting that in the present context the term should be reserved for the confining potential described by quantities such as Bêi,bvp, l. It is in this sense that we speak of an “affinely deforming tube”. However, a harmonic confining tube potential is a theoretical construction which is difficult to visualize. For example, in the continuum chain limit used below, the forces exerted “per monomer” become infinitely small corre￾sponding to Bêi f ∞, l f 0. On the other hand, the term tube is often associated with the tube “contents”, i.e., the superposition of the accessible polymer configura￾tions characterized via a locally smooth tube axis (the equilibrium positions UBp) and a tube diameter dT (defined via the fluctuations Bδup). This second definition refers to measurable quantities.49 Which kind of tube we are referring to, will hopefully always be clear from the context and the mathematical definition of the objects under discussion. In the case of linear polymers, the phantom model reduces to the Rouse model with vanishing equilibrium positions RBi ≡ 0. As a consequence, there are no strain effects other than those caused by the confinement of thermal fluctuations. In particular, the “intrinsic” phan￾tom modulus vanishes (see eq 5). Since the networks are modeled as superpositions of independent linear paths, we have to introduce confinement of type A in order to recover the phantom network shear modulus Gph in the absence of entanglements. In the Rouse model, the Kirchhoff matrix takes the simple tridiagonal form and, depending on the boundary conditions, is diago￾nalized by transforming to sin or cos modes using the transformation matrix The eigenvalues of the diagonalized Kirchhoff matrix (K˜ )pp ) (S-1 K S)pp ) kp are given by If we consider a path with given radius of gyration Rg 2, the basic spring constant is given by k ) (N kBT/2Rg 2). In the continuous chain limit (N f ∞), sums over eigenmodes can be approximated by integrals. For example, one obtains from eq 25 an expression for the tube diameter which could be further simplified, since in this limit the springs representing a chain segment between two nodes are much stronger than the springs realizing the tube, i.e., k . l. For normally distributed internal distances brxx′ be￾tween points x ) n/N, x′ ) m/N on the chain contour the structure factor is given by In the present case, eq 26 reduces to In the undeformed state so that the structure factor is given by the Debye function: A. The Warner-Edwards Model. Warner and Edwards15 used the replica method to calculate the conformational statistics of long paths through ran￾domly cross-linked phantom networks. The basic idea was to represent the localization of the paths in space due to their integration into a network by a coarse￾grained tube-like potential. Recently, it was shown by Read and McLeish34,35 that the same result could be obtained along the lines of the following, much simpler calculation, where we evaluate model A for linear polymers. Evaluation of the integrals in eqs 25 and 26 yields for the deformation independent tube diameter and the internal distances We note that the latter equation can be rewritten in the form with a universal scaling function fA(y) which does not depend explicitly on the deformation. Equation 59 measures the degree of affineness of deformations on different length scales. Locally, i.e., for distances inside the tube with Rg 2| x - x′| , dT2 corresponding to y , 1, the polymer remains undeformed. Thus, limyf0 f(y) ) 0. Deformations become affine for Rg 2| x - x′| . dT2 and y . 1, where f(y) tends to one. K ) k ( -2 1 0 ... 0 1 -2 1 0 ... ··· 0 ... 1 -2 ) (49) S ) (S)jp ) 1 xN exp(iπ jp N) (50) kp ) 4k sin2 (pπ 2N) (51) dTR 2 ) 1 N ∫0 N dp 1 kp + l ) kBT xl(4k + l) ≈ kBT 2xlk (52) S(bq, λ) ) ∫0 1 dx ∫0 1 dx′ × exp (- 1 2 ∑R ) 1 3 qR 2 〈rxx′R 2 (λ)〉 ) (53) 〈rxx′R 2 (λ)〉 ) 1 N ∫-∞ ∞ dp 〈upR 2 (λ)〉|e iπpx - e iπpx′ | 2 (54) 〈rxx′R 2 (λ ) 1)〉 ) 2Rg 2 | x - x′| (55) S(bq, λ ) 1) ) 2N q4 Rg 4 (exp(-q2 Rg 2 ) - 1 + q2 Rg 2 ) (56) dTAR 2 ) kBT 2 xklA (57) 〈rxx′R 2 (λ)〉 2Rg 2 ) λR 2 |x - x′| + (1 - λR 2 ) dTAR 2 Rg 2 (1 - e-(Rg 2|x-x′|)/(dTAR 2) ) (58) 〈rxx′R 2 (λ)〉 - 〈rxx′R 2 (1)〉 (λR 2 - 1)〈rxx′R 2 (1)〉 ) fA( Rg 2 |x - x′| dTA 2 ) fA(y) ) 1 + e-y - 1 y (59) Macromolecules, Vol. 34, No. 16, 2001 Tube Models for Rubber-Elastic Systems 5681