Static scaling behavior of a crystallizable polymer melt model
We present results from molecular-dynamics simulations for semiflexible polymer melts of the coarse-grained Polyvinyl alcohol (CG-PVA) model that is commonly used for simulations of melt-crystallization. To establish the properties of its high temperature liquid state as a reference point, we characterize in detail the structural features of equilibrated polymer melts with chain lengths . Furthermore, we examine the validity of Flory’s ideality hypothesis for this model system. For sufficiently long polymers , the chain length dependence of the end-to-end distance and the gyration radius follow the scaling predictions of ideal chains and the probability distributions of the end-to-end distance, and form factors are in good agreement with those of ideal chains. Inspection of the intrachain correlations reveals evidences for incomplete screening of self-interactions in long chains. However, the observed deviations are small. Finally, we provide a detailed characterization of primitive paths of long equilibrated polymer melts in the entangled regime and we compare them to the original polymer conformations.
Polymer melts are dense liquids consisting merely of macromolecular chains. The main characteristic of polymer melts is their high-packing density which leads to overlapping of pervaded volume of their chains Rubinstein and Colby (2003). As a result, density fluctuations in a melt are small and similar to a simple fluid every monomer is isotropically surrounded by other monomers that can be part of the same chain or belong to other chains. Nevertheless, the crystallization of polymer melts remarkably differs from monomeric liquids due to chains connectivity constraints that must be compatible with lattice spacings of a crystalline structure. Upon cooling of a crystallizable polymer melt, a semicrystalline structure emerges that comprises of regularly packed, extended chain sections surrounded by amorphous strands.
Despite intensive research, the mechanism of polymer crystallization is still poorly understood as polymer crystallization is determined by an intricate interplay of kinetic effects and thermodynamic driving force Keller (1968); edited by J.-U. Sommer and Reiter (2003). From theoretical perspective, attempts have been made to understand thermodynamics of polymer crystallization. These efforts include lattice-based model Flory (1956), Landau-de Gennes types of approach Olmsted et al. (1998) and density functional theory Sushko et al. (2001); McCoy et al. (1991). Among these, the density-functional theory is a particularly a promising approach as it can predict equilibrium features of inhomogeneous semicrystalline structure from the homogeneous melt structure provided that a suitable free-energy functional is employed and accurate information about chain conformations and structure of polymeric liquids is supplied Oxtoby (2002). Such an input can be obtained from simulations of a crystallizable polymer melt that allow to compute the intrachain and interchain correlation functions.
A commonly used crystallizable polymer model to simulate polymer crystallization is the coarse-grained Polyvinyl alcohol (CG-PVA) model Meyer and Muller-Plathe (2001). This model is a semiflexible bead-spring model that is obtained by a systematic coarse-graining of the atomistic simulations of polyvinyl alcohol. The main distinctive feature of the model is its anharmonic intrachain bending rigidity that leads to crystallization from the melt upon cooling. Recently, CG-PVA model has been employed to study the dependence of polymer crystallization on the chain length Jabbari-Farouji et al. (2017); Triandafilidi et al. (2016). To understand the influence of chain length and entanglements on the melt crystallization Luo and Sommer (2013, 2016), the properties of its high-temperature liquid state need to be firmly established as a reference point.
On the account of above discussion, the main goal of the present paper is to provide a comprehensive characterization of static properties of the CG-PVA polymer melts. The prior studies of structural properties of this system have been limited to short chains Vettorel et al. (2007). Here, we present the results for CG-PVA polymer melts with that includes chain lengths in the entangled regime. Characterizing the conformational features of longer polymer melts, we accurately determine the persistence length, the Flory’s characteristic ratio and the entanglement length of CG-PVA polymers. We also carefully examine the consequences of anharmonic bending potential on polymers conformations. Our results provide a basis for rationalization of the chain-length dependent crystallization of CG-PVA polymers Jabbari-Farouji et al. (2017); Triandafilidi et al. (2016) that will be discussed in detail in a future work.
Another aim that we pursue is to examine the credibility of the Flory’s ideality hypothesis for this semiflexible polymer model. Flory’s hypothesis states that polymer conformations in a melt behave statistically as ideal random-walks on length scales much larger than the monomer’s diameter Flory (1969); Doi and Edwards (1986). This ideality hypothesis is a mean-field result that relies on the negligibility of density fluctuations in polymer melts. Therefore, it validity is not taken for granted. Indeed, the computational studies of fully flexible long polymers for both lattice (bond fluctuation) and continuum (bead-spring) models have revealed noticeable deviations from the ideal chain behavior Wittmer et al. (2004, 2007a, 2007b); Beckrich et al. (2007); Hsu (2014). The theoretical calculations show that these deviations result from the interplay between the chain connectivity and the melt incompressibility which foster an incomplete screening of excluded volume interactions Wittmer et al. (2007a); Beckrich et al. (2007); Semenov (2010). However, a recent study of conformational properties of long semiflexible polymer melts demonstrates that the deviations diminish as the chains bending stiffness increases and the conformations of sufficiently stiff semiflexible chains are well described by the theoretical predictions for ideal chains Hsu and Kremer (2016).
Investigation of the static scaling behavior of semiflexible CG-PVA polymer melts confirms that CG-PVA polymers display globally random-walk like conformations. Notably, for chains with , the results of the end-to-end distance, gyration radius, their the probability distribution functions and the chain structure factor are in good agreement with the theoretical predictions for ideal chains. However, examining the intrachain correlations, we find some evidences for deviations from the ideality. The mean-square internal distance of long chains are slightly swollen compared to ideal chains due to incomplete screening of excluded volume interactions. Additionally, the second Legendre polynomial of angle between bond vectors exhibits a power law decay for curvilinear distances larger than the persistence length providing another testimony for self-interaction of chains. However, we note that these deviations although visible are small.
The remainder of the paper is organized as follows. In Sec. II, we briefly review the CG-PVA model and provide the simulations details. We present a detailed analysis of conformational and structural features of polymer melts in Sec. III and we compare simulation results to the theoretical predictions for ideal chains. We investigate conformational properties of the primitive paths of long chains in section IV where we determine the entanglement length of fully equilibrated CG-PVA chains. Finally, we summarize our main findings and discuss our future directions in section V.
Model and Simulations details
We equilibrate polymer melt configurations of the coarse-grained Polyvinyl alcohol (CG-PVA) model using molecular dynamics simulations. In the following, we first briefly review CG-PVA model and then provide the details of simulations.
Recap of the CG-PVA model
The CG-PVA model is obtained by a systematic coarse-graining of atomistic simulations of Polyvinyl alcohol (PVA) Meyer and Muller-Plathe (2001). It is a bead-spring model in which each bead of the coarse-grained chain with diameter nm corresponds to a monomer of the PVA polymer. The fluctuations of the bond length about its average value are restricted by a harmonic potential
The bond stiffness constant is large and leads to bond length fluctuations with a size comparable to monomer diameter. Monomers of distinct chains and the same chain that are three bonds or farther apart interact by a soft 6-9 Lennard-Jones potential,
in which and . Here, K is the reference temperature of the PVA melt Meyer and Muller-Plathe (2001). We truncate and shift the Lennard-Jones potential at in our simulations. Note that our choice of is different from initial studies where the non-bonded interactions were truncated at the minimum of LJ potential and thus were purely repulsive. However, as we will see, the structural properties are CG-PVA polymer melts remain unaffected.
The distinguishing characteristic of the CG-PVA model is its anharmonic angle-bending potential Meyer and Muller-Plathe (2001) as presented in Fig. 1. This bond angle potential is determined directly from atomistic simulations by Boltzmann inversion of the probability distribution of the bond angle . The minima of reflect the specific states of two successive torsion angles at the atomistic level and they correspond to three energetically favorable states, trans-trans, trans-gauche and gauche-gauche. Therefore, the bending potential retains semiflexibility of chains originated from the torsional states of the atomistic backbone Meyer and Muller-Plathe (2001).
Units and simulation aspects
We carry out molecular dynamics simulations of CG-PVA polymers with chains lengths using LAMMPS Plimpton (1995). We report the distances in length unit nm. Meyer and Muller-Plathe (2001). The time unit from the conversion relation of units is with the monomer mass . The starting melt configurations are prepared by generating an ensemble of (number of chains) self-avoiding random walks composed of monomers with an initial density of . Following a fast push off to remove the interchain monomer overlaps, we equilibrate disordered melt structures in the NPT ensemble using a Berendsen barostat and Nose-Hoover thermostat. The temperatures and pressures are reported in reduced units and , equivalent to K and bar in atomistic simulations. The time step used through all the simulations is .
The polymer configurations for were equilibrated until the average monomers mean-square displacement is equal or larger than their mean square end-to-end distance . The time for which is a measure the relaxation time of chains end-to-end vector and it is comparable to the Rouse time for the short chains and the disengagement time for the entangled chains. Polymer melts with , were equilibrated until is comparable to their mean-square gyration radius . Table 1 provides a summary of configurations of polymers and the simulation time in units of before characterization of their static properties begins.
In order to analyze the static properties of CG-PVA polymers, we extract from the polymers configurations the normalized probability distribution functions of the internal distances, the gyration radius, the bond length and angle of polymers as well as the bond length and angle of their primitive paths. We acquire the numerical probability distribution of a desired observable by accumulating a histogram of a fixed width . Then, we obtain the normalized probability distribution function as .
Structural features of CG-PVA polymer melts
We first present our results on the chain-length dependence of the mean square end-to-end distance and the gyration radius for chain sizes in the range . The mean square end-to-end distance is defined as
and the mean square gyration radius is given by
where is the position of th monomer of chain number and is the center of mass position of the th polymer chain in a sample. Here, includes an averaging over equilibrated configurations that are apart for the shorter chains and apart for the longer chains with . Fig. 2 shows and as a function of chain length . Here, denotes the mean-square bond length that is independent of the chain length. The longer chains with follow the relation valid for ideal chains de Gennes (1979). For shorter chains the ratio is in the range . Additionally, chains with follow the scaling behavior of ideal chains . The extracted scaling exponent from fitting versus with a power law gives that is identical to the value for the ideal chains. The observed scaling behavior for the mean square end-to-end distance and the gyration radius of CG-PVA polymers with suggests that long semiflexible polymers behave like ideal chains. In the following, we investigate in more detail the conformational statistics of individual chains, and compare them to the theoretical predictions for ideal chains de Gennes (1979); Rubinstein and Colby (2003) .
We begin by characterizing the intrachain correlations for both monomer positions and bond orientations. To quantify the positional intrachain correlations, we calculate the mean-square internal distances (MSID) for various chain lengths defined as
where is the curvilinear (chemical) distance between the th monomer and the th monomer along the same chain. MSID is a measure of internal chain conformation that can be used to evaluate the equilibration degree of long polymers.
In Fig. 3a, we present the rescaled mean square internal distance, obtained by averaging over polymer melt configurations that are ( ) apart for short (long) chains. Up to , we find a good collapse of all the MSIDs. However, for larger curvilinear distances, the MSIDS of longer chains are slightly larger but all follow the same master curves suggesting that longer chains are a bit swollen due to chains self-interactions. Note that the deviations of of chains for curvilinear distances from other long chains are due to their poor equilibration. These results show that the chains are equilibrated at shorter length scales but their large-scale conformation still needs much longer equilibration time where is the entanglement length as will be discussed in the section IV.
From the asymptotic behavior of mean square end-to-end distances of long CG-PVA chains, we can extract their characteristic ratio and Kuhn length . The characteristic ratio is defined by the relation where is the average bond length. From MSID of longer polymers, and , we obtain . The Kuhn length gives us the effective bond length of an equivalent freely jointed chain which has the same mean square end-to-end distance and the same maximum end-to-end to distance Rubinstein and Colby (2003). For a freely jointed chain with Kuhn segments with bond length , we have and . For CG-PVA polymers, we find and . Equating and of the CG-PVA chains with those of the equivalent freely jointed chain, we obtain .
Next, we compare the mean square internal distance of CG-PVA polymers with of the generalized freely rotating chain (FRC) model Flory (1969); Honnell et al. (1990a) for idea chains. If the excluded volume interactions between different parts of a certain polymer are screened, one expects the FRC model to provide a good description of CG-PVA polymer melts. of FRC model only depends on the value of where is the angle between any two successive bonds in a chain. It is given by
The value of for the CG-PVA model can be obtained from as
where is presented in Fig. 1. Doing the integration in Eq. (6) at numerically, we obtain . We can also directly infer from the MD simulations results as where is the th unit bond vector of the th chain and the averaging is carried out over all the chains and several polymer melt configurations. From MD simulations, we deduce the universal value of independent of the chain length. This value agrees well with the Boltzmann-averaged mean value.
In Fig. 3a, we have also included the MSID of an equivalent freely rotating chain with . We find that MSIDs of short chains fully agree with that of the freely rotating chain model whereas the MSIDs of longer chains present noticeable deviations from the FRC theory for . Hence, FRC model slightly underestimates the MSID of longer chains. Likewise, the characteristic ratio of the FRC model, given by is slightly lower than the estimated from the simulation results. These small deviations are most-likely due to the correlation hole effect that stems from incomplete screening of interchain excluded volume interactions and leads to long-range intrachain correlations de Gennes (1979).
The observed swelling of long chains suggests that an evidence of remnant long-range bond-bond correlations should be detectable. Thus, we examine the intrachain orientational bond-bond correlations as a function of the internal distance . Fig. 3b represents the orientational bond-bond correlations for different chain lengths. We find that the bond-bond correlation functions of all the chain lengths decay exponentially up to and can be well described by . We extract the so-called persistence length from the bond-bond orientational correlation functions. is defined by their decay length, more precisely:
Using and , we obtain . Alternatively, we can estimate the persistence length from as which leads to comparable to the estimated value from fitting the bond-bond orientational correlation functions with an exponential decay. Notably, the relation valid for worm-like chains roughly holds for semiflexible CG-PVA polymers.
For larger values, similar to the MSIDs, bond-bond correlations deviate from exponential decay but we do not observe any sign of long-range power law decay reported for 3D melts of the fully flexible chains Wittmer et al. (2004) and the semiflexible polymers with a harmonic bending potential Hsu (2014). A careful examination of the bond-bond correlations of longer chains for which we have sufficient statistics (inset of Fig. 3b), shows that becomes slightly negative for larger values with a peak around before it decays to zero. A negative value of implies that bonds that are farther apart are anti-correlated due to local backfolding of the chains, similar to the case of 2D polymer melts Meyer et al. (2010). This anti-correlation is a signature of non-monotonic angle bending potential presented in Fig. 1 and it is further amplified in semicrystalline polymers obtained from cooling of the melt Meyer and Muller-Plathe (2002); Jabbari-Farouji et al. (2015).
To further inspect the origin of these small but systematic deviations from FRC model, we investigate the non-Gaussian parameter of CG-PVA polymers and compare it to that of the FRC model prediction. The non-Gaussian parameter is defined in terms of second and fourth moments of internal distances and it vanishes if the internal distances are Gaussian distributed. The fourth moment of internal distances of FRC model depends only on and Honnell et al. (1990b) where is the second Legendre polynomial. Fig. 4a presents the non-Gaussian parameter of CG-PVA polymers of different chain lengths that is compared to that of the FRC model evaluated with and extracted from simulation results. Overall, we find a good agreement between the of CG-PVA polymers and that of the FRC model. However, we notice small deviations from FRC model for long chains and .
In Fig. 4b, we have plotted for different chain lengths. According to the FRC theory, should also decay exponentially as Porod (1953). We find that the initial decay of up to is well described by the FRC model predictions. However, for larger we observe important deviations from the FRC model and exhibits a clear power law behavior over more than 1 decade of similar to the 2D melts of fully flexible polymers Meyer et al. (2010). The observed power law behavior is a manifestation of long-range bond-bond correlations along the chain backbone that result from incomplete screening of excluded volume interactions. As noted by Meyer et. al. in the case of 2D polymer melts, it is related to the return probability after bonds and the local nematic ordering of nearby bonds Meyer et al. (2010).
To understand better the origin of this power law decay, we investigate the intrachain nematic ordering by calculating the for all bond pairs that belong to the same chain and their midpoints are a distance apart. is almost independent of chain length for . In the inset of Fig. 5, we have presented the as a function of distance for . As can be seen, the orientational correlations oscillate and decay rapidly with . This behavior shows that for a fixed curvilinear distance only bonds which are spatially close to each other with separations contribute to . Therefore, one expects that will be directly proportional to the return probability of monomer after bonds Meyer et al. (2010) that we denote by . More precisely
where is the probability distribution function of the end-to-end vector of all the chain segments (subchains) with bonds.
For an ideal self-similar chain for any subchain of size follows a Gaussian distribution of the form
where and Rubinstein and Colby (2003); Doi and Edwards (1986). Hence, the return probability scales as . For semiflexible CG-PVA polymers with local correlations, we expect that follows Eq. 10 for . The scaling exponent that we obtain from fitting of with power law is , that is not so far from the prediction for the Gaussian chains. This small discrepancy is most likely due to insufficient statistics for large values. To test the validity of Gaussian distribution for the internal distances we examine the behavior of intrachain distributions in the subsequent subsection.
Intrachain distribution functions
Having examined the intrachain correlations, we investigate conformational behavior of chains by extracting the probability distribution of the internal distances, i.e from the polymer configurations. As before, denotes the distance between any pair of monomers and that are bonds apart. Let us first consider the probability distribution function of bond length corresponding to . In Fig. 5a, we have shown the distribution of bond length that is independent of the chain length . Interestingly, exhibits two peaks. The normalized distributions of bond length can be well described by a sum of two Gaussian distributions:
in which represent the standard deviations for each Gaussian distribution. The main peak corresponds the average bond length . However, there exists a very small but visible fraction of bonds that are stretched with their peak centered at . This tiny peak is a manifestation of incomplete screening of excluded volume interactions. We find that the stretched bonds can lie at any position along the chain backbone. As the Lennard-Jones interactions between two adjacent monomers are switched off, the stretching of a bond can arise due to return of a monomer after steps and this effect is linked to the long-range bond correlations discussed in the former subsection.
Next, we examine the probability distributions of bond angles and compare them with the form expected from the Boltzmann distribution where is a normalization constant such that . Fig. 5b presents the obtained from accumulating the histograms of bond-angles as well the Boltzmann distribution prediction. Overall, we find a good agreement between the two probability distribution functions for all the chain lengths. Although, the number of stretched bonds in simulations is slightly higher than the theoretical prediction. Consistent with the existence of a secondary peak in the bond length distribution, a slightly larger fraction of stretched bonds is linked to the incomplete screening of excluded volume interactions.
Next, we focus on the normalized probability distribution of internal distances for and compare them to the theoretical distribution functions. The exact segmental size distribution functions of semiflexible polymers for for an arbitrary are not known. However, Koyama has proposed approximate expressions the probability distribution functions of of wormlike chain models Schmidt and Stockmayer (1984) that are applicable to any semiflexible polymer model for curvilinear distances larger than the persistence segment Mansfield (1986). The Koyama distribution is constructed in such a way that it reproduces the correct second and forth moments of internal distances, i.e. and and it interpolates between the rigid-rod and the Gaussian coil limits Mansfield (1986). It is found to account rather well the site-dependence of the intrachain structure of short CG-PVA polymers Vettorel et al. (2007).
The Koyama distribution can be expressed in terms of scaled internal distances as
in which and . As one would expect, at sufficiently large for which the non-Gaussian parameter vanishes, the Koyama distribution becomes identical to the Gaussian distribution valid for fully flexible ideal chains. For ideal chains, the probability distribution function of the is given by Eq. 10. As a result, the corresponding probability distribution function for the reduced internal distances, , follows from
where . Particularly, one expects that the distribution functions of the end-to-end distance of sufficiently long chains should follow this distribution.
Let’s first look at the distribution of internal distances for short subchains (segments). We verified the distribution of subchains does not depend on the chain length. Therefore, we focus on subchains of polymers with monomers. Fig. 6a presents the for subchains comprising of and 5 bonds. As can be seen, for such short segments the features of angular potential are dominant and the Koyama distribution can not provide an accurate description of segmental size distribution although it agrees well with in the central region of the distribution and it captures accurately the hight of the peaks.
Fig. 6b shows for subchains with and 29 bonds that are larger than the persistence segment size. We note that the signature of bending potential is still visible for . Thus, the Koyama distribution does not provide a good description. For , the Koyama distribution agrees quite well with the extracted from simulations whereas Gaussian distribution exhibits a poorer agreement. Additionally, we have included the distribution of the scaled end-to-end distance of short chains with and monomers in Fig. 6b. They coincide with the distributions of subchains with the same length. These results show that the chain-end effects, if any, are negligible and only depends on . For longer subchains and 100, depicted in Fig. 6c displays a perfect agreement with the Koyama distribution. For such long subchains, the distribution functions approach to that of Gaussian given by Eq. (13), as one would expect.
Finally, we present the normalized probability distributions of the scaled end-to-end distance for in Fig. 7a. We find that all the data for collapse on a single master curve. We note that data present a larger scatter from the master curves in the central regions of distribution functions. These deviations are most likely due to a poor equilibration of chains. We have also plotted the corresponding theoretical predictions for the -independent normalized distribution function given by Eq. (13) in Fig. 7a. We find a very good agreement between the master curves and the theoretical predictions for the ideal chains.
Fig. 7b shows that the normalized probability distributions of the scaled gyration radius for different chain lengths. Similar to , the of chains with , collapse on a single master curve. The exact expression for the probability distribution of gyration radius is more complicated and does not have a compact form Fujita and Norisuye (1970). However, it is found that the formula suggested by Lhuillier Lhuillier (1988) for polymer chains under good solvent conditions provides a good approximation for ideal chains Vettorel et al. (2010); Hsu (2014); Hsu and Kremer (2016) too. The Lhuillier formula for the scaled gyration radius in -dimensions reduces to
in which the exponents and are related to the space dimension and the Flory exponent by and . and are system-dependent non-universal constants and is a normalization constant such that . We find that the data of of CG-PVA polymers can be well fitted by the -independent normalized distribution function given by Eq. (14) as plotted in Fig. 7b. Having investigated the conformational properties of CG-PVA polymers, we focus on their structural properties in the Fourier space in the next subsection Meyer et al. (2010).
Form factor and structure factor
A common way to characterize the structural properties of polymer melts is to explore their structure factor that can be measured directly in the scattering experiments. The structure factor encompasses the information about spatial correlations between the monomers via Fourier transform of density-density correlation functions. For spatially homogeneous and isotropic systems such as polymer melts at equilibrium, the static structure factor only depends on the modulus of the wave vector. The static structure factor measured in scattering experiments of amorphous melts is often spherically averaged over all the wave vectors with same modulus . This quantity can be computed as
where the angular brackets represent averaging over all the wave vectors with the same modulus and all the melt configurations.
given in Eq. (15) encompasses scattering from all the monomer pairs. It can be split into intrachain and interchain contributions
where (V volume of the simulation box) is the monomer density and
includes the contributions from intrachain pair correlations and it is called intrachain or single chain structure factor. Equivalently, known as the form factor Rubinstein and Colby (2003) is used to quantity the intrachain correlations in the Fourier space. The interchain contribution is given by that is defined as the Fourier transform of intermolecular pair correlation function Hansen and McDonald (1986) as
We present the behavior of the form and structure factors for different chain lengths. We first focus on the form factor as depicted in Fig. 8. The form factor of Gaussian chains, known as Debye function, is described by Rubinstein and Colby (2003)
In order to compare the behavior of CG-PVA polymers in the melt state with that of ideal Gaussian chains, in Fig. 9a, we have plotted the form factor of CG-PVA chains and Debye function versus . For all the chain lengths, we observe deviations from the ideal polymer behavior at high values. The onset of deviations shifts progressively to larger wave vectors for longer chains. For the longest chains , we have also presented the form factors in a Kratky-plot in Fig. 8b. This plot confirms the existence of a Kratky plateau in the scale free regime that extends up to for . The deviations at larger values reflect the underlying form of the bond angle potential that dominates the behavior of the form factor at length scales smaller or comparable to the Kuhn length.
Next, we present the structure factor of different chain lengths in Fig. 9. As we notice, the of all the polymer melts displays the characteristic features of the liquid-state. We find a very weak dependence on the chain length; for , the of various chain lengths are identical. We notice several important features in the structure factors. First, the structure factor at low is very small. By virtue of compressibility equation that relates the isothermal compressibility to the structure of the liquid, i.e. , we conclude that the polymer melts are almost incompressible. Second, the first peak of at characterizes the packing of monomers in the first nearest neighbor shell. The value of nearly agrees with reflecting that the first peak of is dominated by interchain contributions.
To gain more insight into the interchain correlations of CG-PVA polymers, we compare with that of simple liquids with no internal structure. For such a simple liquid, we have , hence Hansen and McDonald (1986). Fig. 10a shows both and for two chain lengths and . We find than in the region near the first peak and coincide confirming that the peak at is totally determined by the interchain correlations. In Fig. 10a, we have also included . We note that for very low wave vectors beyond the peak region, closely agrees with . This behavior shows that the correlation between monomers of different chains decreases with increasing distance. This decrease is concomitant by the increase of at low values such that the sum of both intrachain and interchain contributions yields a small finite value for as .
In the other extreme of , deviates from as the large behavior of structure factor is fully determined by the intrachain correlations due to correlation hole effect de Gennes (1979); Vettorel et al. (2007). The correlation hole effect leads to a decreased probability of finding a monomer of another chain in the pervaded volume of a particular chain. To illustrate this point, in Fig. 10b, we have shown and for and in the same plot. We see that the large- behavior is entirely dominated by intrachain contributions. These observations are in agreement with the prior investigations for short chain lengths Vettorel et al. (2007).
Primitive path analysis and entanglement statistics
Having investigated conformational and structural features of CG-PVA polymer melts, we focus on their topological characteristics, i.e. interchain entanglements. Entanglements stem from topological constraints due to the chains connectivity and uncrossability that restrict the movements of chains at the intermediate time and length scales. As first noted by Edwards Edwards (1967), the presence of neighboring strands in a dense polymer melt effectively confines a single polymer strand to a tube-like region. The centerline of such a tube is known as the primitive path (PP). A practical and powerful method for characterizing the entanglements is primitive path analysis (PPA). Such an analysis provides us with an operational definition of primitive path and allows to investigate statistics of chains entanglements.
There exists a couple of variants of PPA in the literature Kroger (2005); R. Hoy (2009); Everaers et al. (2004) that are all similar in spirit. Here, we implement the PPA method proposed by Everaers et al. Everaers et al. (2004) that identifies the primitive path of each polymer chain in a melt based on the concept of Edwards tube model Edwards (1967). The primitive path is defined as the shortest path between the chains ends that can be reached from the initial conformations of polymers without crossing other chains. In this analysis, topologies of chains are conserved, and chains are assumed to follow random walks along their primitive paths. Therefore, the primitive path is a random walk with the same mean square end-to-end distance but shorter bond length and contour length .
In practice, by extracting the average bond length of the primitive paths , we can determine all the other desired quantities. In particular, the Kuhn length of primitive path is obtained as
The so-called entanglement length , defined as the average number of monomers in the Kuhn segment of the primitive path, follows from
Operationally, we attain the primitive paths of polymers in a melt by slowly cooling the system toward while the two chain ends are kept fixed. During this procedure, the intrachain excluded volume interactions and bond angle potential are switched off. The system is then equilibrated using a conjugate gradient algorithm in order to minimize its potential energy and reach a local minimum. We perform primitive path analysis for the two longest chain lengths that are fully equilibrated, i.e. and 500 as it is known than poor equilibration affects the entanglement length R. Hoy (2009).
We first examine the probability distributions of the bond lengths of the primitive paths, i.e. as presented in Fig. 11. The distributions of bond lengths of the primitive paths are chain length dependent but both are centered at . Furthermore, the primitive path bond length fluctuations are considerably larger than those of their original paths. The normalized distributions of primitive path bond length can be well described by a Gaussian distribution of the form
where presents the -dependent standard deviation of .
Next, we investigate the statistical features of bond angles of the primitive paths. Fig. 12a presents the bond-bond orientational correlation function as a function of internal distance for the primitive paths of chain lengths and 500. For comparison, we have also shown the of the original polymer conformations. Similar to the original polymer conformations, the initial decay of for can be well described by an exponential decay. However, at short scales , bonds are slightly stretched out because of the constraints of fixed chain ends during minimization of primitive path length. Assuming an exponential function of the form , we can extract the persistence length of the primitive path . From the fit values, we find that is considerably larger than persistence length of original conformations .
We also examine the normalized probability distributions of bond angles of the primitive paths as displayed in Fig. 12b. Unlike the bond angle distributions of the original chain conformations, the bond angles of the primitive paths is unimodal with its peak centered around . Furthermore, the range of angles shrinks from for the original paths to for the primitive paths reflecting that the primitive paths are mainly in stretched conformations.
To explore the intrachain correlations of the primitive paths, we have plotted the mean square internal distances of the original and primitive paths in Fig. 13. As expected the values of for both paths approach the same value with increasing , since the chains endpoints during the primitive path analysis are held fixed. We find that results of for the primitive path can still be relatively well described by the generalized FRC model provided that we use extracted from bond-bond orientational correlations. Having confirmed that the mean end-to-end distance of the primitive paths remain identical to those of the original chains, we obtain for the Kuhn length of the primitive path. We note that the is larger than the Kuhn length of polymers .
Subsequently, we acquire the distribution of entanglement length as presented in Fig. 14a. We notice that has a narrow distribution and presents a weak dependence on possibly resulting from the finite size of the chains. Our estimated value of the average entanglement length is for and for . These results suggest that we are rather close to the asymptotic value of entanglement length . We have also plotted in Fig. 14b and we find that the position of the peak of coincides with our estimated value of . This observation is in agreement with the PPA analysis results for the Kremer-Grest (FENE) model Hsu and Kremer (2016).
We have investigated the static properties of the polymer melts of a semiflexible bead-spring model known as the CG-PVA model Meyer and Muller-Plathe (2001, 2002). The main distinctive feature of this model system is anharmonic bending potential that leads to polymer crystallization upon cooling the melt. Therefore, characterization of the structural properties in the liquid state provide a reference point for the crystalline states. We have equilibrated polymer melts with chain lengths . The results for the long chains allow us to determine the Kuhn length , the persistence length and the entanglement length of CG-PVA polymers accurately as summarized in Table II. We note that the relation holds for semiflexible CG-PVA polymers.
We have also examined the validity of Flory’s ideality hypothesis for this model system. Our detailed examination shows that the conformations of CG-PVA polymers agree with many of the theoretical predictions for ideal chains. Notably, long polymer melts with follow the scaling relations and valid for ideal chains. The probability distribution functions of the reduced end-to-end distance and the reduced gyration radius for chain lengths also collapse on universal master curves that are well described by the theoretical distributions for the ideal chains.
Investigating the intrachain correlations, we find evidences for deviations from ideality. However, these non-Gaussian corrections are rather small and do not affect most of the large-scale conformational features. The mean square internal distances of short polymers up to show an excellent agreement with the predictions of the generalized freely rotating chain model Flory (1969). In contrast to short chains, the mean square internal distances of longer chains are slightly higher than those of the equivalent freely rotating chain model. The observed swelling of longer chains reflects an incomplete screening of excluded volume interactions and it most likely related to the correlation hole effect de Gennes (1979). We also compare the non-Gaussian parameter of CG-PVA polymers with that of the freely rotating chain model. The agreement is rather well and we only observe some small deviations for long chains.
A careful inspection of bond-bond correlations of long chains reveals that the non-monotonic angle-bending potential leads to a weak anti-correlations for curvilinear distances of about . The observed anti-correlation is a signature of chain backfolding in the melt. This feature is further enhanced in semicrystalline state and it is responsible for formation of chain-folded structures Meyer and Muller-Plathe (2002); Jabbari-Farouji et al. (2015). As a result of anti-correlations, we do not observe any long-range bond-bond correlations in CG-PVA polymers. However, the long-range intrachain correlations are visible in the second Legendre polynomial of the cosine of angles between the bonds, , that exhibit a power law decay similar to the reports for 2D polymer melts Meyer et al. (2008).
Moreover, we have investigated in detail the intrachain and interchain structure factors of different chain lengths. The interchain structure factor is almost independent of the chain length whereas the intrachain structure factor depends on as expected. We find that of sufficiently long semiflexible CG-PVA polymer are well-described by the Debye function for lengthscales larger than the Kuhn length. The agreement with the Debye function improves upon increase of . Notably, we observe a plateau in the Kratky plot for the range . Our results are in contrast with the findings for fully flexible chains that exhibit significant deviations from Debye function at intermediate wave-vectors Wittmer et al. (2007a); Beckrich et al. (2007); Hsu (2014). But, they agree with the recent findings that increasing the bending stiffness of the chains in a melt, irrespective of details, improves the agreement with the ideal-chain limit Hsu and Kremer (2016).
Using the primitive path analysis, we have determined the average entanglement length of long and equilibrated chains and we have compared the original polymer paths with their primitive paths.
Probing the bond-bond orientational correlation function and the mean square internal distance of primitive paths, we confirm the assumption that polymers behaves nearly as Gaussian chains along their primitive paths.
Notably, the Kuhn length of the primitive path is more than twice the of the original path.
The average bond length of primitive paths follows a Gaussian distribution and the peak of the first moment of the entanglement length probability distribution
agrees with the average entanglement length. Entanglements are known to strongly affect the dynamics and rheological properties of polymer melts Doi and Edwards (1986). To quantify the influence of entanglements on viscoelastic
properties of CG-PVA polymer melts, the dynamics of fully equilibrated chains is under investigation and it will be presented in a future work.
S. J.-F. is grateful to Jean-Louis Barrat for his support and insightful discussions. She also thanks Kurt Kremer and Hsiao-Ping Hsu for inspiring discussions. She also acknowledges financial support from the German Research Foundation (http://www.dfg.de) within SFB TRR 146 (http://trr146.de). The main part of the computations were performed using the Froggy platform of the CIMENT infrastructure supported by the Rhone-Alpes region (Grant No. CPER07-13 CIRA) and the Equip@Meso Project (Reference 337 No. ANR-10-EQPX-29-01). Additionally, the computing time granted on the supercomputer Mogon at Johannes Gutenberg University Mainz (hpc.uni-mainz.de) is gratefully acknowledged.
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We study the structural and topological features of crystallizable polymer melts of the coarse-grained Polyvinyl alcohol model. The main distinctive feature of this bead-spring model is its anharmonic bending potential that leads to a non-monotonic distribution of bond angles, , and chain-folding in supercooled melts. Investigation of the normalized bond angle distributions of primitive paths of entangled polymer melts shows that they are mainly in stretched conformations.