5.1. Complex Modulus
We show the log-log plot of the Elastic Storage Modulus
and Viscous Loss Modulus
for all the infinite and open weave polymeric materials as the shear response frequency is varied in Figure 5
. The frequency of oscillation is normalized by
is a time-scale on the order of the diffusion time
(see Table 4
for the infinite weaves we see that, in the range of frequencies studied, we have
for all the simple weaves. The crossover of
is absent for those systems within this range of frequencies, which indicates no behavioral transition in the samples which exhibit solid properties. When
is larger than
the elastic response is dominant indicating there is relatively few polymer rearrangements (reptation) within the network structure. This indicates that energy is mainly stored elastically in the stretching and bending of bridging polymeric chains. This can be verified by our Writhe
quantity for the chains as it reaches a minimum at the extrema of the oscillatory strain period within this regime (see Section 5.2
). The systems with large
behave like stiff materials having strong entanglements similar to imperfect networks having transient covalent crosslinking [51
]. This indicates that polymer solutions of long linear semiflexible chains can behave like crosslinked networks, even in the absence of explicit crosslinks. A similar observation was reported in [58
are independent of the frequency of oscillation and we see a crossover at frequency
that corresponds to period
. At frequencies higher than
(period times shorter than
) there is a significant dependence of moduli on the frequency which increases with increasing topological complexity. This is in agreement with predictions for polymeric networks [49
]. The line segments shown in the figure indicate a scaling between ∼
, respectively, to be compared with that of Rouse chains.
The alternate interlacing weave, wIII, is the only infinite weave for which
intersect and for which
both seem to scale as
in the intermediate frequencies. Moreover, for wIII,
for low frequencies. We find that the original configuration of wIII is not favorable to the stiffness of the chains and the chains need to stretch resulting in a larger
. This causes extra collisions with other chains which results in larger values of
as well. At high frequencies, we notice a shift from filament bending to stretching which results in higher values of
. Such transitions have also been observed in networks of actin filaments [59
for the open systems we find that both
are initially constant up to
and then increase and intersect at
. We have
. This suggests two critical times in the polymer chain dynamics. The first is
and the second is
at which we find have a trend of slightly increasing with decreasing density of the systems as predicted in [61
]. We find that with increasing frequency the response tends to become dissipative.
At low frequencies
we find the trends follow the Rouse model. For larger frequencies we find that
tend to a plateau value. Similar scalings were reported in [62
] and in [47
] for polymer solutions of linear FENE chains of similar molecular weight, which further shows that the use of harmonic bonds does not significantly influence our findings. We find a decreasing slope of
for increasing entanglement which suggests a slower relaxation mechanism.
We show the loss tangent as a function of the frequency of oscillation in Figure 6
. We remark that
can be interpreted as reflecting the strength of what is sometimes called “colloidal forces”. In other words, if
then the particles are highly associated and sedimentation could occur. If
, the particles are highly unassociated. The loss tangent is almost constant, close to 0, for all the simple infinite weaves (w0,wI,wII). The values of the open weaves are greater than one and then decrease to the values of the corresponding infinite weaves. The asymptotic ordering of the phase lag of the systems is
We find that, our data at larger frequencies that all the materials behave like elastic solids, as is often seen in large frequency responses. The inset graph shows the corresponding log-log plot only for the open systems. It reveals a cross-over at approximately
, which corresponds to times on the order
. This time-scale could be related to the entanglement time as in [49
]. This characteristic time-scale, seems to decrease with the topological complexity of the weave. The large frequency tail of
decreases more slowly with increasing topological complexity and density indicating a substantial dissipation effect related to entanglement.
5.2. Conformational Analysis
We show configurations of the polymer weaves at different times during deformation in Figure 7
. For the small oscillation frequencies, the infinite chains follow the deformation of the defining box, attaining an s-shape conformation. The open chains, significantly rearrange in time and tend to avoid the boundary by aligning with the orientation of the deformation. This process happens more slowly for wI and even more slowly for wII and wIII systems due to topological obstacles. We note that the chains tend to form bundles of chains, giving an inhomogeneous material, suggesting that the inhomogeneity decreases with increasing density and entanglement complexity. Similar phase separation of polymer solutions in oscillatory shear has been observed experimentally in [64
We find the transition from bundle-dominated structures to entanglement dominated structures is related to the entanglement of the chains as has been also reported in [65
]. A possibility for the bundle formation is finite-size effects. To examine that, we performed similar simulations in equilibrium in the NVE and in the NPT ensemble for the w0 infinite system. In both cases, bundle formation occured rapidly in the simulation. This indicates that the bundle formation is not a finite size effect.
We propose that the chains bundle together in order to decrease their deformation which occurs due to thermal fluctuations and due to the deformation of the cell. As the chains bundle, they form tubes of chains. A larger diameter tube resists the deformation stronger than the individual chains. This larger tube structure is apparent in the w0 infinite system. The same happens in two and three directions respectively for wI, wII and wIII. This also happens transiently at initial times for the open systems. At first the open chains form these bundles then they keep rearranging and entangling further until finally forming globules.
Our topological methods can be used to more precisely characterize the conformational features of the chains. We do this by measuring the Writhe
and the Periodic Linking Number
topological quantities that we introduced in Section 2
. We remark that the chains in our system are loosely entangled relative to highly knotted systems yielding, as a consequence, Periodic Linking Numbers and Writhe that are less than one. While the quantities appear small relative to those of highly knotted systems, they still provide a useful characterization of the collective configurations of the polymeric chains of the material and their rearrangements. As our results indicate these are useful in understanding the connection between topology and mechanical responses even in systems of low molecular weight (below that of the entanglement length).
We find a very different behavior of the Writhe for the weave of type w0 between the open and infinite systems (the situation is similar for wI and wII). In the case of the infinite periodic systems, the mean absolute Writhe of the chains shows a sinusoidal behavior. This is seen most clearly when the frequency of oscillation is small. During the shearing cycle of the unit cell in our simulations, the Writhe is maximum when the shear deformation is the least and the Writhe reaches a minimum when the shear deformation is at its greatest. This behavior is indicative of the chains stretching at the maximum deformation and relaxing to a more entangled state when the shear deformation is relaxed.
In the case of open systems, the mean absolute Writhe of the chains also follows a sinusoidal behavior, but it changes significantly in time. This happens because the chains are free to attain any possible configuration and tend to disentangle and relax to configurations similar to those of random coils. Indeed, the final values are similar to those of a semiflexible random coil of comperable length as reported in [36
]. This behavior becomes less pronounced as we increase the density and complexity of the weave because the disentanglement time increases and the chains do not have sufficient time to rearrange.
5.3. Topology and Mechanical Responses
We investigate both closed and open chain systems. The closed chain systems were found to be less interesting since the values of the Writhe and the Periodic Linking Number did not change significantly as a function of the frequency of oscillation. This is as expected, since without breakage of bonds the topology must remain close to that of the original configuration. Similarly, the loss tangent of the infinite systems does not change significantly throughout the experiment.
Here, we focus on the open chains systems where there is the potential for significant rearrangements of polymer chain configurations and topology. We show the mean absolute Writhe of open chains as a function of the loss tangent for small frequencies of oscillation in Figure 8
We find a decay of the mean absolute Writhe with the loss tangent and a clustering of the data for each system. It is notable that the proposed Writhe measure groups together the systems of similar material response. The clustering observed for these systems indicates that the global entanglement of the chains imposed by the original conformation affects the response of the material significantly. The responses at small frequencies are clustered and ordered with their Writhe decreasing as .
We remark that as the frequency increases we found the Writhe of the open systems tended to meet the values of their corresponding closed (infinite) chain versions. This occurs since the chains cannot escape their original configuration as readily at large oscillation frequencies. For the case, approximately the same values for open and closed chains were found throughout.
Previous studies have shown that there is an almost linear relationship between the number of kinks per chain and the mean absolute Writhe of a chain in the case of a melt of linear FENE chains [36
]. The relationship emerges as a function of the molecular weight in equilibrium conditions.
In non-equilibrium conditions, the relation becomes more complex [37
]. The viscosity can be obtained as the limit
. Because the frequencies studied here are relatively large, however, we can examine how the ratio
depends on topology as a reference. Our results for the smallest frequencies indicate that the Writhe of the open chains decreases with
, suggesting a decrease of the Writhe with viscosity, see Figure 9
. The number of kinks has been shown to increase with viscosity [9
]. A similar inverse behavior between the Writhe of the chains and the number of kinks was observed in the initial time of an elongation of the chains [37
We show the mean absolute Periodic Linking Number of the open systems at small frequencies as a function of the loss tangent in Figure 10
. We find that the mean absolute Periodic Linking Number decreases with the loss tangent for
. We find significant clustering of the data corresponding to the different systems. The clustering of the responses indicates that the imposed global topology of the initial configuration significantly affects the response of the material. We remark that with increasing frequency we found the open systems tended to the values of the corresponding infinite systems. This occurs because, at large frequencies, the open chains cannot escape their original configurations.
The increase of the linking number implies the presence of persistent entanglements. Persistent entanglements are tight contacts between chains that significantly restrict their motion [66
]. Such contacts are likely to cause significant bond stretching under deformation that is followed by a decrease of the Loss Tangent. These results indicate that interactions underlying mechanical responses can be effectively captured by the Periodic Linking Number.
From our results we obtain information about the equilibrium modulus. The equilibrium modulus is defined as the limit
. We show the storage modulus for the smallest frequencies in Figure 9
. Our results suggest that the equilibrium modulus increases with the linking of the chains. Interestingly, a similar linear relation between the entanglement density as obtained from the Gauss linking integral and the shear modulus for ring polymers was reported in [32
Our results on the Writhe and Periodic Linking Number show a competing relation between these two with respect to the Loss Tangent. We provide a brief explanation for this effect. If the Writhe is large and the Periodic Linking Number is small, this can be interpreted as meaning the chains attain random conformations with no significant topological constraints. This would result in a behavior that is primarily dissipative. This suggests that inter-chain contributions to stress dominate through collisions of molecules induced by the Brownian motion.
In contrast when the Writhe is small and the Periodic Linking Number is large, we expect that the chains get stretched by the presence of persistent entanglements. In this case, intra-chain contributions to stress would dominate. As the ratio
increases, the persistence of entanglements increases. This implies that the bond stretching increases, which decreases the Loss Tangent. Therefore, we expect the loss tangent to increase with the decreasing ratio of Periodic Linking Number versus Writhe. In fact, this is confirmed in our results as seen in Figure 11
Our results show, for the open systems at small frequencies, such a trend of the ratio of the Periodic Linking Number over the Writhe as a function of the Loss Tangent. This indicates that one can control the viscoelastic properties of a material by controlling the ratio of the Writhe and Periodic Linking Numbers of the constituent chains. This also suggests that
is a measure of the inter-chain contribution versus the intra-chain contribution to the stress. This finding may contribute to our understanding of the interplay between these two contributions [22