ВЫСОКОМОЛЕКУЛЯРНЫЕ СОЕДИНЕНИЯ, Серия C, 2013, том 55, № 7, с. 1021-1028

УДК 541.64:539.2


© 2013 г. Athanasios K. Morozinis"' b, Christos Tzoumanekas"' b, Stefanos D. Anogiannakis", and Doros N. Theodorou"' b

a School of Chemical Engineering, Department of Materials Science & Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, Zografou Campus, 15780Athens, Greece b Dutch Polymer Institute, P.O. Box 902, 5600AX Eindhoven, The Netherlands e-mail: tzoum@central.ntua.gr (Christos Tzoumanekas)

Abstract—A molecular-level understanding of cavitation in polymer networks upon imposition of mechanical stress is still lacking. Molecular Dynamics simulations of crosslinked amorphous Polyethylene (PE) were conducted in order to study cavitation as a function of the prevailing stress. We first show that the characteristic relaxation times related to tube confinement and chain connectivity can be obtained by examining the mean square displacement of middle chain monomers. Then, we present a methodology for predicting the cavitation strength and understanding its dependence on cohesive interactions and entropic elasticity. Our simulations show that experimental observations and predictions of continuum mechanics analysis, which relate the critical stress for cavitation to the Young's modulus of the rubber, are in agreement with the observed tensile triaxial stress below which a pre-existing cavity cannot survive in a cavitated sample.

DOI: 10.7868/S0507547513050103


Cavitation is a ubiquitous phenomenon with tremendous technological significance. It signifies the development of cavities (voids) within a material under external load. It is relevant to the mechanical performance of elastomers [1] and constitutes the first step in the process of debonding of surfaces kept together by pressure-sensitive (soft) adhesives [2]. Hence, a molecular-level understanding of cavitation in amorphous polymers that would enable predicting the cavitation strength, and therefore enhancing it by appropriate modification of the chemical constitution and composition of the material, would be highly desirable.

Cavitation can be described as a phase separation phenomenon in terms of nucleation and growth processes, wherein a cavity develops within a condensed phase that is being subjected to an isotropic tensile (hydrostatic) stress field (negative pressure) [3—5]. Below a certain negative pressure the liquid phase becomes unstable and cavitation is spontaneous (the free energy barrier is reduced to zero). From thermody-namic considerations, the stress at the limit of stability of an amorphous polymer above its glass temperature Tg, calculated from an Equation of State (EoS) [6], is one or two orders of magnitude higher than cavitation strengths observed experimentally. Clearly, the initial emergence of cavities within a stretched polymer is not fully understood.

On the other hand, a continuum mechanics analysis [7] of heterogeneous nucleation (or cavity inflation) predicts the growth of a preexisting ^m-sized cavity when the far field pressure is more tensile than Pc = —5E/6, where E is the Young modulus, in good agreement [1, 7—9] with experimental observations. Recent work [9—13] describes in more detail current approaches to cavitation and fracture in unfilled rubbers and gels. Our aim here is to provide a simulation protocol and a molecular level understanding of cavitation initiation in amorphous polymers, above Tg, in the light of existing continuum approaches and experimental findings.


We analyzed a perfect network system, composed of Polyethylene (PE) subchains linked with tetrafunc-tional crosslinks. The system consists of 64 crosslinks and 128 chains, of 201 united atoms each, not counting the crosslinks at chain ends. United atoms correspond to CH2 methylene units, except for tetrafunc-tional crosslinks which correspond to carbon atoms. The crosslinks are initially situated at the sites of an underlying diamond lattice and all subchains are attached to crosslinks at both their ends. Thus, the networks are free of all kinds of defects (pendant chains, chain loops, etc.).

Molecular dynamics (MD) simulations were performed in the NPT ensemble, at temperature 450 K and pressure 1 Atm, by employing the united atom TraPPE force field [14]. The cutoff in the range of non-bonded pairwise interactions was set to 5.5a, where a is the hardcore bead diameter. All MD simulations reported in the paper have been conducted using the Large-scale Atomistic/Molecular Massively Parallel Simulator (LAMMPS) software [15] using three dimensional periodic boundary conditions.

The density of the equilibrated system p = 0.78 g/cm3 and the characteristic ratio = 8.5 are in good agreement with corresponding experimental estimates and previous simulations of PE melts [16]. The characteristic ratio was estimated from the plateau of

R \n)) / nl as a function of n (not shown), where

{r 2(n)^ is the mean square end-to-end distance of a sub-

chain strand consisting ofn bonds, while l = 1.54 A is the average bond length along network subchains. For an

unperturbed strand of n bonds CrXl = (r2(n)j /nl2 is n -independent in the limit of large n, a behavior which is observed here as n increases towards the subchain length of 200 bonds.

The model system was created and equilibrated according to the following steps. At the nodes of a diamond lattice we first placed tetrafunctional carbon atoms. The lattice constant was then adjusted so that the carbon atoms were bonded with PE chains at full extension (all-trans state). This was an unen-tangled state. Then, the network was let to relax at 1 Atm. Due to entropic elasticity and cohesive interactions it contracted to the melt monomer density. At this stage the system was not equilibrated confor-mationally. To impose equilibration, a phantom chain simulation at the melt density (NVT ensemble) of duration 8 ns was performed. Following this phantom chain simulation, bead overlaps were eliminated by the gradual push-off method [17]. Interchain interactions were gradually restored in the course of an NVT MD run of duration 1 ns [17]. Then, the local density was relaxed by an NPT MD run for 1 ns, so that the system ultimately became an entangled PE network.


Topological Analysis and Entanglement Density

The generated MD trajectory was subjected to topological analysis, frame by frame, by using the CReTA (Contour Reduction Topological Analysis) algorithm [18]. CReTA reduces a dense system of polymer chains to the corresponding system of Primitive Paths (PPs) [19, 20], constructed as the shortest paths under the same 'Topological Constraints' (TCs) as the origi-

nal chains. By fixing chain ends in space and by prohibiting chain crossing, the contour lengths of all chains are simultaneously minimized (shrunk), until they become piecewise linear objects coming together at the nodal points of a network. The PPs are ultimately reduced to very thin objects consisting of fused spherical beads of diameter 0.4 A, which can be mapped to the initial chain monomers. Explanatory videos and other details can be found elsewhere [18].

The entanglement molar mass Me, can be estimated from the Kuhn segment of the PP, i.e., by mapping the PP conformations to equivalent Random Walks (or freely jointed chains) [18]. Thus, the Kuhn segment of the PP, measured in number of united-atom carbon beads, is given by

Ne = N(R2)/(Lpp)2, (1)

where {ris the mean squared end-to-end distance, (Lpp) is the average PP contour length, and N = 201 is the subchain length in CH2 methylene units. We find that Ne = 45.44, which corresponds to Me = 636 g/mol, leading to an entanglement density which is approximately 1.65 times that of a PE melt, where Ne = 75 [18].

This finding is compatible with a comparatively high shear modulus (~5.9 MPa) measured via MD for the PE network (see next sections), in relation to the plateau modulus of a long chain PE melt, which is approximately 2 MPa. Since our model network is well equilibrated and its density and conformational stiffness, which control [21] entanglement density in flexible polymer melts, are similar to those of a PE melt, the higher degree of entanglement is most probably due to the short subchains and the artificial preparation conditions of the network. In contrast to reality, entanglements here were introduced after the chains were linked to form a network, by letting them interpenetrate during a fixed density phantom chain simulation stage. This issue will be discussed in more detail elsewhere [22]. A different method for the generation of defect-free networks, involving the use of many interpenetrating diamond networks, has been discussed by Everaers [23].

Mean Square Displacement and Characteristic Relaxation Times

In this section we focus on the distances spanned dynamically by the atomistic chains of our system. By construction of our system, large scale diffusion is pro-

hibited and trapped entanglements are expected to apply an effective tube constraint [20, 23—26] on real chain conformations.

In Fig. 1a we present the monomer mean square displacement (msd), defined as

Ne N

gi(t)=N~NËE(mo - MO)]2)>

' ch »


i=l n=l

where r;-„ (t) is the position vector of the «-th unitedatom bead of chain i, at time t. To reduce chain end effects the msd was averaged over the central 10% beads of each chain. Moreover, to improve the statistics, multiple time origin averaging was performed. We observe that at long times gl(t) reaches a plateau, indicating that reptation cannot take place, as expected for a crosslinked system.

A log-log plot of the reduced msd g1(t)/t °'5 as a function of time is presented in Fig. 1b. We can observe four different scaling regimes. At very short times there is a regime, usually called 'ballistic' (though here the

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