Crystallization in polymer melts: extensive DPD study

Crystallization in polymer melts: extensive DPD study

Crystallization in polymer melts: extensive DPD study P.I. Kos,1, a) A.A. Gavrilov,1 V.A. Ivanov,1 and A.A. Chertovich1 Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, Russia

arXiv:1708.09499v1 [physics.chem-ph] 30 Aug 2017

(Dated: 1 September 2017)

In the present work, crystallization in polymer melts with different poor solvent concentration was observed by dissipative particle dynamics simulation technique. We simulate systems close to equilibrium states. Full analysis consists of three main parts: calculating systems with different polymer concentrations using dissipative particle dynamics technique, defining crystalline fraction using our own rules and splitting it into separate clusters using simple cluster analysis. Crystalline fraction weakly depends on concentration and differs due to dynamical effects. Average cluster size has two high values at 10% and 90%. Average crystalline segment length is small because of flexibility organic molecules. PACS numbers: 61.25.Hq Keywords: Polymers, Dissipative particle dynamics, Crystallization, Cluster analysis

I.

INTRODUCTION

The crystallization of polymer materials play an important role in many applications, including its mechanical and optical properties. Understanding of polymer crystallization on the molecular level is one of the most challenging problem in modern polymer physics. Nowadays the detailed knowledge of polymer crystallization mechanism stay unclear: in many cases it is even unknown what is the general process scenario and how external conditions and microscopic chain characteristics affect the crystallization behavior. From the physical point of view the main difference of polymer crystallization from standard low-molecular inorganic crystallization is only the fact that monomers are constrained by connecting into the chains and there is an introduction of polymer conformation concept. As a conclusion, the network of entanglements is formed in the melt of long non-phantom chains. This dramatically slow down the system dynamics and give polymer materials many important mechanical features. In addition this network of entanglements leads to the fact that the full crystallization and accomplishing process is almost impossible in system with long polymer chains, we can talk only about partly crystallized polymer systems. The polymer crystallization was discovered almost 60 years ago1–3 , but still there is no generally accepted analytical description of polymer crystallization based on statistical physics approach. There is a classical theory of Lauritzen and Hoffmann4–7 , but this theory based on nucleation theory in low-molecular crystallization and do not consider polymer conformations at all. As it was shown in8 , there is great overestimation (several orders of magnitude) of lamellar thickness in Lauritzen and Hoffmann theory. The other problem with this theory is the independence of lamellar thickness on crystallization temperature, which contradicts with the experimental results9 . Actually almost all other modern approaches to describe polymer crystallization does not correspond to some microscopic view of polymer crystallization at the level of chain conformation and its statistical behavior. Some recent exception is the work of Stepanow10 who proposed a simple kinetic model of polymer crystalline lamella formation, based on competition between coil and rod-like conformations in overcooled polymer melt. This theory predicts correct

a) Electronic

mail: [email protected]


2

lamellar thickness versus temperature scaling, similar to more phenomenological approach by Strobl11 . Nowadays both analytical and numerical studies8,12–14 agreed that the equilibrium conformation of stiff polymer chain in crystalline material is somehow crumpled fragmented conformation, not a rod-like. In12,13 , similar to classical theory of Lauritzen and Hoffmann, the main driving force of crystallization is the intermonomer attraction. But several recent papers15–19 suppose that intermonomer interaction will play major role only at late stages, during crystallites contraction and system glass transition. During initial stages of crystallization the most important role should play chain segments orientation, concentration and mobility. Computer simulations probably could push forward the construction of microscopic model of polymer crystallization. The polymer crystallization was studied by simulations both in Molecular Dynamics (MD)8,16–21 and Monte Carlo scheme14,22,23 . Recognized leaders in this field are groups of Yamamoto18,23 , Rutledge17,24,25 , Sommer15,19,26,27 and Schilling16,28,29 . All these groups use standard MD approach and coarse-grained models of poly(ethylene) or poly(vinilalcohol). At this research we firstly propose to use Dissipative Particle Dynamics (DPD)30 for the simulation of polymer crystallization. We believe, that DPD allow us to undertake broader research with larger time and space scales. That is because DPD has a special features like soft potentials (which provide to enhance integration step) and increased numerical density. This makes DPD considerably faster than standard MD scheme during studies of condensed polymer materials at large time scales. Because of soft nature of used potentials there are no strong restrictions on beads excluded volume and thus makes chain partly phantom. This is of course not suitable to mimic crystallization, but on studying equilibrium for thermodynamic properties like block-copolymer microphase separation. Recently shown31 that it is possible to choose simulation parameters keeping both chains non-phantom and comfortably large integration step. So thereby we provide computer simulation to understand basic principles of crystallization in polymer melts at a coarse-grained level. Main purpose of this research is to understand fundamental principles of many neighboring crystallites growth in polymer melt and the influence of polymer concentration on the system behavior. All studied features of our crystallized polymer system caused by few very general properties: connectivity of monomer beads into chains, presence of intrachain stiffness (memory along the chain about orientation bonds between beads) and topologically restrictions (entanglements of non-phantom chains). Although details of some specific polymer chemical structure may change individual features of the crystallization process we believe that general trends and several common regularities of melt crystallization process should stay alive. II.

SIMULATION METHODOLOGY

DPD is a method of coarse-grained molecular dynamics adapted to polymers and mapped onto the classical lattice FloryHuggins theory32–35 . In short, macromolecules are represented in terms of the bead-and-spring model, with particles interacting by conservative force (repulsion), dissipative force (friction), and random force (heat generator). A soft repulsive potential enhances stability of the numerical scheme for integrating the equations of motion, makes it possible to increase a time step and thus access large times scales when complex polymeric structures are studying. It was shown that such model describes well the dynamics of polymer melts at different time scales36,37 . Consider an ensemble of particles (beads) obeying Newtons equations of motion: dri = vi , dt mi

dvi = fi , dt

(1)

(2)


fi =

X

3

(Fijb + Fijc + Fijd + Fijr ) .

(3)

i6=j

where ri is coordinate, mi is mass and vi is velocity of an i-th bead, respectively, fi is the force acting on it. The summation is performed over all other beads within the cut-off radius, which we choose to be the length unity, rc = 1. Below we assume that all quantities entering Eq. 1-3 are dimensionless and we set mi for any i to unity. First two terms in the sum are conservative forces. Macromolecules are represented in terms of the bead-and-spring model. Fijb is a spring force describing chain connectivity of beads: Fijb = −K(rij − l0 )

rij , rij

(4)

where K is a bond stiffness, l0 is the equilibrium bond length. If beads i and j are not connected Fijb = 0. Fijc is a soft core repulsion between i- and j-th beads:

Fijc =

  aij (1 − rij )rij /rij , rij ≤ 1 

,

(5)

0, rij > 1

where aij is a maximum repulsion between beads i and j attained at ri = rj . Since Fijc has no singularity at zero distance, a much larger time step than in the standard molecular dynamics could be used. Other constituents of fi are random force Fijr and dissipative force Fijd acting as a heat source and surrounding media friction, respectively, they are taken as dictated by the Groot-Warren thermostat35 . More detailed description of our simulation methodology could be found somewhere else38 . It was shown that the DPD method is consistent with both the scaling theory of polymers (e.g., it gives correct relationships between the average radius of gyration of a coil and the number of units in the coil) and the Rouse dynamics39,40 . Since we study polymer systems with different concentration of polymer: C = 10%, 20%, 50%, 70%, 90%, 95% and 100%, and the DPD scheme suppose to use explicit solvent particles, the remaining from the polymer volume was filled by solvent beads. The repulsion parameter between polymer particle and solvent particle was chosen to be ∆a = 10, unless otherwise specified in the text. Such condition corresponds to a poor solvent case, the Flory-Huggins parameter of polymer-solvent interaction could be calculated using common expression χij = 0.306∆aij from the work35 and occurs to be χP S ≈ 3. The use of soft volume and bond potentials leads to the fact that the bonds are formally phantom capable of self-intersecting in three dimensions. The phantom nature of chains does not affect the equilibrium properties (for example, the chain gyration radius or the phase behavior of the system); moreover, it greatly speeds up the equilibration of the system. However, if studying crystallization behavior, that require explicit consideration of the presence of entanglements between chains, it is necessary to introduce some additional forces that forbid the self-intersection of the bonds. These forces are usually quite cumbersome and considerably slow the computation. Nikunen et al.31 described a method for turning chains nonphantom in DPD without any additional forces. It is based on geometrical considerations: if any two units in the system cannot be located closer than rmin /2, every unit in the system effectively has an excluded radius of rmin /2. If it assumed that each √ bond has a maximum length lmax , the condition of self-avoiding chains is 2rmin > lmax . Although particles in DPD are formally point-like, they have an excluded volume due to the presence of the repulsive potential governed by the aij value. Similarly, the existence of a bond potential causes the bond to have a maximum possible length. In our study, we chose aii = 150, l0 = 0.2, k = 150, and with this set of parameters we can be sure that none chain crossover events could be happen during our simulations. In addition the value l0 = 0.2 force beads, which are the neighbors along the chain, located considerably closer


Natural obtaining of the bond direction

Calculation of the direction for each bead i-1

4

i

beads to bonds → 𝑢𝑢𝑖𝑖

i+1

𝒖𝒖𝒊𝒊 : (𝑥𝑥𝑖𝑖+1 − 𝑥𝑥𝑖𝑖−1 ; 𝑦𝑦𝑖𝑖+1 − 𝑦𝑦𝑖𝑖−1 ; 𝑧𝑧𝑖𝑖+1 − 𝑧𝑧𝑖𝑖−1 )

i-1

i

→ 𝑣𝑣𝑖𝑖

i+1

𝒗𝒗𝒊𝒊 : (𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑖𝑖−1; 𝑦𝑦𝑖𝑖 − 𝑦𝑦𝑖𝑖−1; 𝑧𝑧𝑖𝑖 − 𝑧𝑧𝑖𝑖−1 )

FIG. 1. Direction of the polymer subchains is much more important than its coordinates. So during determination of crystallization in our coarse-grained simulation we switch the description from beads coordinated to bonds orientation. u~i - direction of i−th bead, v~i - direction of i−th bond.

to each other than non-linked beads. Repulsive volume interactions between i and i ± 2, 3... beads provide effective chain stiffness, similar to the so-called tangent hard spheres model41 . Thus we kill two birds with one stone because we have a melt of stiff non-phantom chains, which is naturally has a tendency to undergo crystallization transition only by simple excluded volume interactions. The obtained average distance between linked monomer units is < l >= 0.48, the chain persistent length b is about 10 − 15 monomer units, this could be confirmed from Figure 5a bellow in the text. The other parameters are: DPD number density ρ = 3, noise parameter σ = 3 and integration time step ∆t = 0.02. We use rather long chains N = 103 monomer units each. The initial system conformation was a set of random walk (Gaussian) chains with equilibrium < l >= 0.48 bond length. As soon as we reach the desired polymer concentration the remaining volume is randomly filled with the solvent particles up to final density. Thus we study model crystallization from the overcooled state caused by the rapid change in both chain stiffness and the system temperature. During this selection of simulation parameters and start conditions the experimental system in our mind was the PAN fibre formation in coagulation bath. The simulation box was set to be cubic with 50 × 50 × 50 DPD units in each axis, with periodic boundary conditions in all directions. There are 375 000 DPD beads in simulation box and it is up to 375 polymer chains in case of 100% concentration. The average size 3 of Gaussian √ chain with N = 10 and < l >= 0.48 is equal to square root from contour length l N ≈ 15, for stiff chain we have to multiply contour length to the persistent length √ N lb ≈ 76. Thus on average each polymer chain located only in single simulation box and does not interact with itself via periodic boundaries. Because all monitored parameters were smooth and gradual with time and concentration, we perform simulation of only one system for each concentration value. But all the simulated systems more or less self-averaged. For calculations we use original domain-decomposition parallelized DPD code and performed simulations at MSU supercomputer facilities42 . The maximum time we reach is 108 DPD steps for each polymer concentration. Before cluster analysis we have to separate crystalline and amorphous fractions. There are two obvious ways to analyze simulated systems: study set of beads or study set of bonds. In case of bonds we naturally obtain the direction, whereas in case of beads we have to calculate it as a vector from (i − 1)-th bead to (i + 1)th. The case of bonds is most straightforward and precise, so we choose to analyze a set of bonds. Coordinates of bonds calculated as average mean of appropriate connected beads: i − 1 and i. To recognize whether the particular bond undergo crystalline state we use the following rules (detailed description of the choice of parameters is given in Supplementary): 1. Calculate neighbors for each bond inside the sphere with radius 1.5 dpd units. 2. Define how many neighboring bonds are collinear with the selected one, collinearity threshold criteria is: ± 17 degrees.


5

FIG. 2. Typical snapshots of several systems for analysis at t = 108 (in DPD steps). Only polymer beads are shown, solvent beads are removed. All chains are colored by red or blue color accordingly to which kind of phase each bead belongs to. (a) - 70%, (b) - 90%, (c) - 100%

3. Bond crystallization criteria is the ratio of collinear neighbors to total number of neighbors. Bond is considered as crystalline, if this ratio is greater than 0.4, otherwise - amorphous. To characterize the crystallization state of the system we perform standard clustering procedure43 to collect neighboring in space crystalline bonds into clusters. We use the same cutoff radius 1.5 dpd units for this cluster analysis procedure. To characterize the chain conformations we calculate dependence of squared spatial distance between two units on the distance along the chain R2 (n) separately for crystalline and amorphous segments (see Figure 5). Such dependence proved to be very useful to test polymer conformational state at different length scales44,45 . Also we study these segments length distribution at different timescales. III.

RESULTS AND DISCUSSION

Several snapshots of the simulation box with three polymer concentrations at final time are shown in Figure 2, solvent particles are hidden, red and blue colors show different bead types: crystalline and amorphous accordingly. It is easy to recognize a set of crystallites intercalated by solvent (empty regions) or amorphous phases. The crystallite size is much smaller than the simulation box size and consists of many different chains. This conclusion is rather obvious and Fig. 2 mainly give readers a bird’s-eye view on the system under study and could be a starting point for further data mining. At first glance it seems that, the crystallite size in more concentrated systems is much smaller than in more sparse systems. This seems true even for close concentrations 70% and 90%. But below we will show that this conclusion works only up to 90% polymer fraction: the average crystallite size in Fig. 2b (90%) is even smaller than in Fig. 2c (100%), but increased fraction of amorphous beads shadows that vision. Figure 3 represents the crystalline fraction for different times and concentrations. We may see a typical dependence of crystalline fraction on the total polymer fraction in the system. In Fig. 3 the dynamics of crystalline fraction depending on time is not the same for each concentration and as polymer fraction is higher so the dynamics is slower because of entanglements. But finally we see, that dependence is not so strong and crystalline fraction values are in the interval (0.2; 0.45). Also there is surprisingly flat dependence of final crystallization level on the polymer concentration (Fig. 3a). Note, that this crystalline fraction is among all chains, excluding solvent. Figure 4a depicts the overview of the typical system (50% polymer fraction) at the final time of 108 DPD steps. We color by red all beads which belong to crystalline phase (see section 2. Methodology) and by blue color all non-crystalline (i.e. amorphous) beads. One


6

FIG. 3. (a) Average cluster size on polymer concentration for all observed systems. (b) Dependence of crystalline fraction on polymer concentration for all observed systems.

can easily see the separation of crystalline phase into different clusters - independent crystallites, which could be easily segregated via standard clustering procedure (see Figure 4b). As we see clusters have complicated surface but almost the same size. Involved in crystallites subchains have hexagonal architecture and it corresponds to nematic phase of liquid crystal. Because even in first coordination sphere there is no any crystal lattice. Effective tube diameter is bigger in crystallite than in melt. It happens because single chain has small mobility perpendicularly to the backbone. But anyway we see in Figure 4c perfect hexagonally packed rods so we see typical structure factor dependence in Figure 4d. In order to better understand the structure of crystalline clusters we study the behavior of average cluster size (see Figures 3a). We can conclude that there is again one general scenario of crystallites growth for all polymer concentrations. Random crystallites in starting conformation break down because they are non-equilibrium. Then two processes happen in our system. First, subchains, which aim to join crystallite, get penalty in surface energy to win in volume energy. The displacement of subchains is mostly parallel in crystallite. Second, each subchain may move it’s own tube of excluded volume to form crystallite with other subchains to decrease volume interaction energy. First process is much easy for dynamics but requires enough solvent to be notable, the second process is dominant at high concentrations of polymer and becomes the only one in melt. Actually this is surprisingly interesting - dependence on polymer concentration the average crystallite size has two extremes: at the low concentration and at 90%. We think that it may happen because at 10% there are so little polymer, as it is profitably to create the only one crystallite. Number of crystallites grows with the increasing of concentration and


7

FIG. 4. (a) - system after separating all beads in two groups: red beads are crystalline and blue beads are amorphous. (b) - the same system after cluster analysis, different colors correspond to separate clusters. Both are 50% polymer. Solvent particles are removed from both, amorphous – only from (b). (c) - single crystallite from (b). (d) - structure factor for the

they start to partly combine each other. By the way at high concentrations number of solvent particle become less and the dynamic of chains becomes much more slower to form big cluster. To study the structure of chain segments inside and outside crystallites we plot the squared spatial distance R2 versus distance along the chain n separately for crystalline and amorphous segments in the middle and at the end of simulation time for intermediate concentration 0.7, see Fig. 5a. The weight distributions of these segments is presented in Figure5b. Similar to Fig. 5, these curves looks very alike for all studied concentrations. It is clear that the crystalline segments have a rod-like conformation, with power-low dependence R2 (n) = n2 . Contrary to that amorphous segments have a stiff conformation at the average length, up to persistent length b = 12 and then turn to much more crumpled conformations and have a scaling R2 (n) ≈ n2/3 unless it reaches Gaussian conformation line n. We remind the reader here that the initial starting conformations were Gaussian and curve lies exactly along the R2 (n) = n line. So at the large scales (n > 500) the chain conformation did not change at all and x2/3 scaling means only transient effective crumpling45 . Looking to the weight distribution of crystalline and amorphous segments again exhibit (Fig. 5) sharp difference between them. While amorphous segments have a Flory-like distribution with some density increasing in short subchains, crystalline segments have much more narrow distribution with well-defined maximum rod length about 30 monomer units. It looks like that this value corresponds to the maximum thickness of the growing crystalline lamella and further crystallite growth and agglomeration will continue mostly in perpendicular direction. In addition we vary chain stiffness and solvent quality for the case 20% polymer fraction (the system with fast relaxation dynamics) and look at the average cluster size, number of clusters and crystalline fraction. To choose inconsistency between chain and solvent we have checked several various values of Flory-Huggins parameter χ. It was surprisingly interesting, that average cluster size, number of clusters and crystalline fraction weakly


8

FIG. 5. (a) All curves are plotted for 70% polymer fraction. Separately crystalline and amorphous parts is plotted for different times. Additionally there is start conformation distribution to show random walk behavior. (b) Average length of crystalline block is 3 beads. This happens because of strict conditions at analyzing.

depend on quality of solvent, but thought we use in our analysis middle value χ = 3. We also test internal parameter equilibrium bond length (l0 ), here we mean that elastic force equal zero at this length. As soon as it is key contribution into stiffness mechanism the influence on distribution is huge. In Fig. 6 case l0 = 0.4 dpd units can not be plot on average cluster size or number of clusters because the crystalline fraction decreases with the simulation time. In two other cases l0 = 0.2 and l0 = 0.1 crystalline fraction increase with the time, but in case with lowest l0 system tend to freeze, due to the Kuhn segment significantly increase. From the other side we see that average cluster size is almost the same and therefore there is a critical value for l0 from 0.2 to 0.4. If value less than critical we may observe crystallization of subchains in the system, otherwise we get amorphous melt or dispersion. It is clear that we do not reach exact equilibrium state during our simulations, because of strict computational limitations. Most probably during further simulations the overall crystallization level will grow little but the main trends will be the same and clusters rearrangement is possible in such systems. We define several main stages from the beginning of simulation. In case of 90% polymer fraction and less the influence of solvent is significant in crystallization process. Because the entropic gain is much bigger that temporary loss in surface energy and chain quite easy rearrange to form crystallites. In melts (polymer fraction 95% and more) solvent contribution is not significant and only


104

102

Χ=9 Χ=3 Χ=1

9

100

Χ=9 Χ=3 Χ=1

102

Crystallyzation fraction

Number of clusters

Average cluster size

103

101

101

100 100

1000

10000

100000

100

1x106

100

Number of simulation steps

1000

10000

100000

Χ=9 Χ=3 Χ=1

10-1

10-2

10-3

10-4

1x106

100


1000

10000

100000

1x106


FIG. 6. All the data performed for the 20% polymer fraction. From left to right: average cluster size in number of bonds, number of clusters and crystalline fraction versus time in DPD steps.

102

L0=0.1 dpd units L0=0.2 dpd units

100

L0=0.1 dpd units L0=0.2 dpd units

Proportion of crystallyzation phase

104

Number of clusters

Average cluster size

103

102

101

101

100 100

1000

10000

100000


1x106

100 100

1000

10000

100000


1x106

L0=0.1 dpd units L0=0.2 dpd units L0=0.4 dpd units

10-1

10-2

10-3

10-4 100

1000

10000

100000

1x106


FIG. 7. All the data performed for the 20% polymer fraction. From left to right: average cluster size, number of clusters and crystalline fraction versus time in DPD steps. On the first and second graph there is no case l0 = 0.4 because we do not analyse when cluster consists of less than 5 beads and there is not even one, satisfied this condition.

reptations and diffusion play role in crystallization process. But both cases have the same behavior: system fall down into the nearest minimum instantly after the start simulation and then find states with less free energy and if they kinetically available, system goes into this new state.

IV.

CONCLUSIONS

DPD probably is the most suitable method to study general aspects of crystallization in polymers. Here we perform record simulations of crystallizing semi-flexible polymer chains 1000 monomer units long with various concentrations and in poor solvent. All systems under study express similar behavior of melt crystallization, but with nontrivial concentration dependence. Observed non-equilibrium crystallization from the overcooled state has the following general properties: 1. There is one general scenario of crystallization for this model independently of the chain concentration: after some relaxation delay, which depends on the initial polymer concentration, the crystallization happens very sharply (in log time scale) and then saturates having a plateau. The average cluster size depends strongly on the chain stiffness - the main crystallization driving force - and almost independent on polymer-solvent interaction. During its growth the single crystallite cannot pass the surrounding entanglements by neighboring chains and other crystallites and its


10

natural size is limited probably close to the Ne value. To overcome this limitation the crystallite surface became crumpled and this could be the driving force for fractal-like spherulite formation during further crystal growth. 2. Final average crystallinity level is only weakly depends on the concentration: about the half of polymer material (40-50%) undergo crystalline phase, while the rest stay amorphous and located between crystallites. That is: for a selected inter-chain potential there is always a preferable (in average) crystalline lamella height, consisting of many rod-like chain segments with the rod size about 20-30 monomer units, that is at least twice larger that the equilibrium persistent length b and come close to the melt entanglement length Ne . These rod-like segments have hexagonal-like packing inside single crystallite. Each crystallite consists of different chains and each chain participate in several crystallites. 3. In contrast to average crystallinity level there is well pronounced non monotonous mean crystallite size dependence on polymer concentration. That is: the average cluster size at 90% is almost 3 times larger than at 50% and 100%. We believe that this fact originated from the competition of two opposite processes that limit the crystallite size: at small concentrations there is lack of surrounding polymer material and no large clusters could be formed (i), while at high concentration the number of entanglements from surrounding chains is so high that it refuse the segments to migrate to large distances and consolidate small crystallites into larger one. 1 P.

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