Improved model of nonaffine strain measure

Transcription

1 Improved model of nonaffine strain measure S. T. Milner a) ExxonMobil Research & Engineering, Route 22 East, Annandale, New Jersey (Received 27 December 2000; final revision received 3 April 2001) Synopsis A new calculation of the strain measure for entangled polymers is presented, in which the entanglement network is modeled as a set of entanglement points to which are attached four entanglement strands, randomly oriented in equilibrium. The network deforms nonaffinely to maintain a net zero force on each entanglement point, following a recent suggestion of Marrucci. The resulting strain measure in the case of uniaxial and biaxial extension as well as simple shear is well described by Q C /tr(c ) with 0.7, where C 1 is the Finger tensor. The resulting second normal stress ratio N 2 /N 1 is (1 )/ The original Doi Edwards strain measure is well described except for the second normal stress by this same function with The Society of Rheology. DOI: / Recently, Marrucci et al have proposed that the strain measure Q E of Doi and Edwards 1978a, 1978b, 1986 should be modified, to take into account the concept that the forces on each entanglement point must balance before and after a deformation. These authors argued that because entanglement points are massless, the putative four entanglement strands leading away from a given entanglement must carry forces that sum to zero. Since the magnitude of the force carried by an entanglement strand is equal to the thermal tension 3k B T/a absent chain stretching effects, the zero-force requirement at each entanglement point constrains the unit tangent vectors of the four participating entanglement strands to sum to zero. During a step deformation, all entanglement points would presumably move affinely initially. However, this will not guarantee that the new locations of the entanglement points satisfy the zero-force requirement. That is, the entanglement points must adjust their positions nonaffinely in some way after the initial step strain. Marrucci et al. proposed a rather idealized model of this nonaffine motion, a model in which the entanglement strands are constrained always to be parallel to one of the three principal axes of the extensional deformations they consider. Within this highly idealized model, they demonstrated that the strain measure Q E is not the celebrated Doi Edwards expression Q ij E E u i E u j / E u, 1 E u but becomes instead Q E C 1/2 tr C 1/2, 2 a Electronic mail: stmilne@erenj.com 2001 by The Society of Rheology, Inc. J. Rheol. 45 5, September/October /2001/45 5 /1023/6/$

2 1024 MILNER where C 1 E E T is the Finger tensor in terms of the deformation tensor E. The authors emphasize that their new proposed strain measure gives a much improved prediction of the normal stress ratio in shear, namely N 2 /N 1 1/4 in the limit of small strains as compared to the Doi Edwards value of 1/7. However, given the idealized nature of their model, and the sensitivity of such quantities as the normal stress ratio to details of the deformation process, a calculation more faithful to the physical proposal made by the authors seems warranted. This note presents the results of a more realistic calculation of the strain measure Q E resulting from nonaffine motion of entanglement points after a step strain. The present model is conceptually simple: 1 Repeatedly choose from an isotropic distribution initial unit tangent vectors u of the four strands emanating from an entanglement point. These will not in general sum to zero, of course. 2 While fixing the endpoints of the four vectors, adjust the location of the entanglement point e so that the four forces taken always to have constant magnitude sum to zero. The lengths of the four vectors are rescaled to unity after the adjustment; thus they are u u e u e. 3 The idea is that on the average, the endpoints of the four vectors, which are four other entanglement points, will be adjusting in some uncorrelated way to satisfy their own zero-force requirement. We are hence neglecting correlated motions of nearest-neighbor entanglements. 3 Now affinely deform the four vectors, with the origin taken at the entanglement point. Thus the four vectors become E u. In general, the corresponding forces always with constant magnitude equal to the thermal tension will no longer sum to zero. Again adjust the location of the entanglement point e, such that the sum of the forces vanishes. Now the displacement vectors with respect to the shifted entanglement point are u E u e. 4 4 Compute the corresponding stress tensor by averaging over the choice of the initial four tangent vectors, using the general expression T ij F i R j, 5 where is the number of strands per unit volume, F is the force on the th strand, and R is the end-to-end vector of the th strand. Assuming a constant tube diameter, the average volume per strand after deformation is R, hence 0 / R. The density of entanglement strands decreases after a deformation due to the retraction process. Hence the stress tensor is T ij 0 3k B T/a u i u / u j, 6 u where the averages are carried out over the initial isotropic set of values of u. The expression Eq. 6 is the same as for Doi and Edwards, but the deformation process, being stochastically nonaffine, is different from the usual affine u E u. It is then a simple matter to generate a set of entanglements and subject them to uniaxial, biaxial, and simple shear deformations for a range of strain amplitudes. The

3 NON-AFFINE STRAIN MEASURE 1025 FIG. 1. Normal stress T 11 T 22 vs uniaxial strain ; points are from stochastic average of nonaffine model, line is from Eq. 7 with 0.7. necessary averages over 1000 initial configurations were performed using MATHEMATICA in a few minutes of CPU time on a Macintosh G3. One important technical remark: when large shear deformations are applied, a few percent of the random initial configurations become degenerate, in that the entanglement point becomes coincident with one of the attachment points at some deformation. Rather than invent some continuation of the network and its deformation beyond this point, such conformations were eliminated entirely from the average which implies some small conformational bias in the initial random configurations. Surprisingly, the result of the calculation described above turns out to be simple: for volume-preserving uniaxial, biaxial, and shear strains, the strain measure within the present model turns out to be very well described by the form found in Marrucci et al. 2000, Q E C tr C, 7 but with an exponent of 0.7 rather than 1/2. The numerical results for the three different types of deformations are shown in Figs. 1 5, together with stresses predicted from Eq. 7 with 0.7. Unfortunately, Eq. 7 implies a normal stress ratio in the limit of low shear strains of N 2 /N 1 (1 )/2. For 0.7 this gives 0.15 rather than 1/4, and so spoils the perhaps fortuitous agreement between experiment and the results from the idealized model of Marrucci et al. FIG. 2. Normal stress T 11 T 33 vs biaxial strain ; points are from stochastic average of nonaffine model, line is from Eq. 7 with 0.7.

4 1026 MILNER FIG. 3. Shear stress vs shear strain ; points are from stochastic average of nonaffine model, line is from Eq. 7 with 0.7. One may relax the one key assumption of the present model, that the entanglement points to which the given entanglement point is attached themselves move strictly affinely, as follows: simply extend the model by adding one more more generations to the fourfold tree of entanglement strands. In other words, replace the four affinely moving points by four entanglement points that are themselves connected by entanglement strands to three other points that are assumed to move affinely. Having done this, one can carry out a similar stochastic averaging procedure to that described above, wherein not only the central entanglement point but also the four entanglement points to which it is connected move nonaffinely to satisfy the zero-force condition. Finally, the stress is evaluated only from contributions at the central entanglement point. In this way, the entanglement point can be progressively removed from any direct connection to affinely moving points. Although the numerical procedure is now an overnight job, the results are the same: Eq. 7 with 0.7 describes the results to within the noise. At this point, the main unphysical feature of the model so generalized is that the entanglement network, represented as a regular fourfold tree structure, contains no loops. However, the inclusion of loops in the structure poses conceptual problems within the model. If the topological structure were regular, then the starting state would be, e.g., a diamond lattice. Applying the zero-force condition at equilibrium would require the lattice to be regular, leading to a nonstochastic model much like that of Marrucci et al. Or, one may suppose the starting state to be fourfold connected but of irregular topology. It is hard to imagine how to construct such a state, especially without introducing trapped FIG. 4. First normal stress T 11 T 22 vs shear strain ; points are from stochastic average of nonaffine model, line is from Eq. 7 with 0.7.

5 NON-AFFINE STRAIN MEASURE 1027 FIG. 5. Second normal stress T 22 T 33 vs shear strain ; points are from stochastic average of nonaffine model, line is from Eq. 7 with 0.7. entanglements, which may become stressed when the system is deformed. Confronting these difficulties is left to future work. Of course, one can always take the point of view that the idealized model has suggested a simple family of strain measures, which may be used phenomenologically without reference to the idealized model. In fact, the Doi Edwards strain measure itself can be reasonably approximated by Eq. 7 with 0.8, except that the magnitude of the second normal stress is underpredicted for 0.8 we have 0.1 rather than the Doi Edwards 1/7. However, if one is inclined to take the physical model of nonaffine deformation more seriously, the present results show at least that the strain measure is sensitive to the details of the nonaffine deformation, and that the favorable results for of Marrucci et al. may be fortuitous. As a final remark, it is worth remembering that the most serious shortcoming of the Doi Edwards theory in the nonlinear regime is catastrophic shear thinning, leading to the prediction that uniform simple shear flows are unstable to shear-banding which is not observed. It can be shown that the strain measure Eq. 7 for either 1/2 or 0.7, in an independent alignment approximation IAA calculation of the nonlinear steady shear stress T 0 3kT/a 0 d Q E 8 still shear thins with an asymptotic exponent of 3/2. This result depends only on two properties of Q E): 1 for small shear strains gamma, Q xy (E) is linear in the strain ; and 2 for large shear strains, Q xy (E) falls off as 1/. Then, the large-shear behavior of T xy, is determined by the short-time behavior of the stress relaxation function (t). For the Doi Edwards stress relaxation function neglecting contourlength fluctuations, (t) goes as t 1/2 for short times. Experimentally, the behavior is more like t 3/4 corresponding to a power-law decay of G ( ) above the terminal peak with an exponent of 1/4. But for (t) going as t for any between zero and unity corresponding to a power law in G ( ) above the terminal time of 1, IAA leads to a steady state shear stress going as 1 for large shear rates, and hence catastropic shear thinning for any reasonable power law in (t), regardless of the details of Q E). The resolution of this conundrum lies elsewhere Marrucci 1996, Likhtman et al

6 1028 MILNER References Doi, M. and S. F. Edwards, Dynamics of concentrated polymer systems. Part 2: Molecular motion under flow, J. Chem. Soc., Faraday Trans. 2 74, a. Doi, M. and S. F. Edwards, Dynamics of concentrated polymer systems. Part 3: The constitutive equation, J. Chem. Soc., Faraday Trans. 2 74, b. Doi, M. and S. F. Edwards, The Theory of Polymer Dynamics Clarendon, Oxford, 1986, Chap. 7. Likhtman, A., S. T. Milner, and T. C. B. McLeish, Microscopic theory for the fast flow of polymer melts, Phys. Rev. Lett. 85, Marrucci, G., Dynamics of entanglements: A nonlinear model consistent with the Cox-Merz rule, J. Non- Newtonian Fluid Mech. 62, Marrucci, G., F. Greco, and G. Ianniruberto, A simple strain measure for entangled polymers, J. Rheol. 44,