Polymer crystallization in the presence of “sticky” additivesAshok Kumar Dasmahapatra, Hemant Nanavati, and Guruswamy Kumaraswamy Citation: J. Chem. Phys. 131, 074905 (2009); doi: 10.1063/1.3174449 View online: http://dx.doi.org/10.1063/1.3174449 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v131/i7 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Polymer crystallization in the presence of “sticky” additivesAshok Kumar Dasmahapatra,1,2 Hemant Nanavati,1,a�,b� andGuruswamy Kumaraswamy2,a�,c�

1Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 400076, India2Polymer Science and Engineering Division, National Chemical Laboratory, Pune 411008, India

�Received 23 January 2009; accepted 18 June 2009; published online 21 August 2009�

The effect of “sticky” additives �viz., those that have attractive interactions with the polymer� onpolymer crystallization, has been investigated by dynamic Monte Carlo �DMC� simulations.Additive-polymer attractive interactions result in a slowing down of the polymer chain diffusivity inthe melt state. Our results show that with increasing additive stickiness, polymer crystallinitydecreases monotonically, and thinner crystallites form, viz., crystallization is inhibited by thepresence of sticky additives. Unusually, the observed “specific heat” peak at the phase transitionshows nonmonotonic behavior with additive stickiness, and exhibits a maximum for intermediatevalues of additive stickiness. While the origins of this unexpected behavior are not clear, we showthat it correlates with a large interchange between crystalline and amorphous states of themonomers, in the vicinity of the additives. At this intermediate additive stickiness, we also find thatcrystallization follows a qualitatively different route—crystallinity shows a non-Avrami-likeevolution, unlike the case at low or high additive stickiness. © 2009 American Institute of Physics.�DOI: 10.1063/1.3174449�

I. INTRODUCTION

Additives are a class of materials that are added to aparent polymer, typically during compounding, so as to im-part desired properties without modifying the molecular ar-chitecture of the polymer.1 All commodity polymers, withoutexception, are supplied in additivated form. Additives areused to impart a variety of functionalities. For example, an-tioxidants are used in almost all polymers, to inhibit theiroxidative degradation; lubricants are used to promote flow;antistatic compounds are used to prevent build-up of staticcharge on the surface of the polymeric part; nucleatingagents are used to provide surfaces for heterogeneous nucle-ation of semicrystalline polymers; plasticizers are used todepress the polymer glass transition temperature, Tg, etc. Ad-ditives that modify bulk properties of a polymer, such asantioxidants, nucleating agents, etc., need to be uniformlydispersed in the polymer to be effective, while additives suchas antistatic agents, that modify surface properties, need to“bloom” to the surface during processing.

Additives that are used to rigidify polymers are calledantiplasticizers. We classify antiplasticizers into two kinds,depending on the mode of rigidification. Antiplasticizers ofthe first kind are bulky molecules that disperse in a polymer,and decrease the available free volume. This reduction infree volume in the vicinity of antiplasticizers inhibits chainmobility2,3 and typically increases the polymer Tg, thuschanging the relaxation spectrum and mechanical propertiesof the additive-polymer “composite.” For example, addition

of a large fraction �14% by weight� of nonpolar mineral oilto 1,4-polybutadiene results in a shift of the �-relaxation tolower frequencies �viz., an increase in Tg�, but no change inthe �-relaxation, suggesting that local, noncooperative mo-tions are not influenced by the additive. Further, there is nochange in the shape of the �-relaxation spectrum and nochange in the fragility, suggesting that rigidification of thepolymer is effected by an increase in the local friction uni-formly along the chain.4

Antiplasticizers of the second kind are “sticky” mol-ecules, viz., there are attractive interactions between the ad-ditive and polymer. For example, Don et al.,5 reported thathydrogen bonding between the carbonyl group of an additive�a polycaprolactone-polycarbonate blend� and the hydroxylgroup of the epoxy matrix results in an increase in the modu-lus of the epoxy that correlates with an increase in the acti-vation energy for the � relaxation. Thus, in this system,rigidification of the epoxy is effected by hydrogen bondingof 2-hydroxypropyl ether groups, localized along the chain.Similarly, Perepechko and Yakovenko6 reported that when asmall amount of water is added to polycaproamide, the for-mation of “short cross-linking bridges” due to water-amideinteractions restricts polymer chain motions and results inantiplasticization.

There are only very few literature reports that examinethe influence of antiplasticizers. Further, mostexperiments,4–15 such as those described above, andsimulations16 investigate rigidification of amorphous poly-mers by antiplasticizers. To the best of our knowledge, thereare no reports that describe the effect of antiplasticizers onstructure formation in semicrystalline polymers.

Here, we describe lattice Monte Carlo �MC� simulationsthat probe how crystallinity develops when sticky additives�viz., wherein there is an attractive interaction between the

a�Authors to whom correspondence should be addressed.b�Electronic mail: hnanavati@che.iitb.ac.in. Telephone: 91-22-25767215.

Fax: 91-22-25726895.c�Electronic mail: g.kumaraswamy@ncl.res.in. Telephone: 91-20-25902182.

Fax: 91-20-25902618.

THE JOURNAL OF CHEMICAL PHYSICS 131, 074905 �2009�

0021-9606/2009/131�7�/074905/10/$25.00 © 2009 American Institute of Physics131, 074905-1

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additive unit and the polymer� are dispersed in the bulk of asemicrystalline polymer. Intuitively, one would imagine thatadditives that “dilute” a polymer melt and that decreasechain mobility in the melt would inhibit crystallization andsignificantly slow down crystallization kinetics. Accordingly,our results show a monotonic decrease in crystallinity andcrystallite dimensions, with an increase in additive-polymerattraction. However, surprisingly, the change in crystalliza-tion kinetics and the magnitude of the “specific heat” peak atcrystallization are nonmonotonic with additive-polymer in-teractions. Thus, our preliminary investigations on the influ-ence of sticky additives on polymer crystallization alreadyreveal interesting, nonintuitive behavior.

We organize our paper as follows: we describe the modeland simulation technique in Sec. II. We discuss our results inSec. III followed by a summary in Sec. IV.

II. MODEL AND SIMULATION TECHNIQUE

In recent years, several MC based rapid chain-growthtechniques have been developed, such as pruned enrichedRosenbluth method �PERM�,17 CBMC,18 random end-switch-configuration biased Monte Carlo �RES-CBMC�,19

etc. These are suitable for single chain simulations and can-not be employed for dense systems representing melts. Otheralgorithms such as the wormhole moves algorithm20 are alsonot suitable for simulation of crystallization of a dense melt,where the same polymer chain could exist in amorphous andcrystalline regions. Lattice simulations of dense systems arepossible by the cooperative motion algorithm �CMA� byPakula and co-worker21,22 and the single site version of thebond fluctuation model23–26 by Hu.27,28 CMA has been em-ployed for lattice systems with 100% occupancy. However,the time evolution of the morphology cannot be unambigu-ously mapped to the simulated MC steps �MCSs, which aredescribed later in this section�.23 In contrast, Hu et al.29 usedMC simulations to examine copolymer crystallization andwere able to successfully reproduce experimentally ob-served, nontrivial trends such as the influence of comonomercontent on the development of crystallinity on cooling, etc.Therefore, we use a dynamic MC �DMC� methodology simi-lar to that of Hu et al.27–29 to address the crystallization ofhomopolymer melts containing additives. We now describeour simulation technique in detail.

Our simulations probe dense polymer melts with a latticeoccupation density �fraction of lattice sites occupied by ad-ditives and polymers� of 0.9375 on a cubic lattice�coordination number=26� of size 32�32�32. Monomersand additives occupy one lattice site each. Thus, the simula-tion box for our unadditivated polymer melt comprises 480chains, each having 64 monomer units. Additivated systemscomprise �480−n� chains and 64n additive units. Thus, amelt containing 1.67% additives is simulated by placing512�=64�8� additive units and 472�=480−8� chains on thelattice.

Initially, we place the additives and polymer chains in aregular fashion in the simulation box. We employ a microre-laxation algorithm, viz., a combination of single site bondfluctuation and slithering diffusion to move the chain units

along the lattice sites with periodic boundary conditions.27,30

Specifically, our algorithm randomly �random numbergenerator, MT19937 �Ref. 31�� selects a vacant site and at-tempts to move a monomer or additive from one of the near-est neighboring sites to this vacant site. Depending on theposition of the unit along the backbone chain, our algorithmselects the appropriate microrelaxation move. For example,if the selected unit is at a chain end, then it will attempt tomove either by slithering diffusion24 or by a single site bondfluctuation25 �cf. end bond rotation� with equal probability.On the other hand, if the selected unit is a nonterminal one,it will attempt to move by a single site bond fluctuation.25

The moves chosen for the polymer chain are physically re-alistic; evidence from literature indicates that such MC simu-lations are capable of faithfully describing polymer chaindynamics in a melt, from large length scales down to aneffective bond.23,32,33 For additives, only a single site move isperformed. The combination of these moves is known toproduce faster relaxation in such a highly dense system.26 Weenforce excluded volume interactions and disallow bondcrossings.32,33 Such a scheme satisfies detailed balance andprecludes “locked-in” states, which chain-growth self-avoiding walk algorithms may face.24,34

The success of attempted moves is governed by theMetropolis35 scheme with a probability p=exp�−�E /kT�,where �E is the change in energy for the transition from theold to the new conformation, k is the Boltzmann constant,and T is the absolute temperature. The change in energy, �E,for a move in our simulation is calculated as �E /kT=�NpUp+�NcUc+�NmaUma, where �Np, �Nc, and �Nma

are the net changes in the numbers of nonbonded parallelbonds, collinear bonds, and monomer-additive contacts, re-spectively; Up, Uc, and Uma are the driving energies �normal-ized by kT� for pairs of nonbonded parallel bonds, collinearbonds, and monomer-additive contacts, respectively. Uma canbe considered an additive-monomer exchange energyand can be expressed using the quasichemicalapproximation36 as Uma=�ma− ��mm+�aa� /2, where �mm, �aa,and �ma are the pairwise interactions between m-m, a-a, andm-a, respectively.

Thus, in our simulations, polymer crystallization isdriven by locally stiffening the chains �by making bonds col-linear, via the Uc term� and by arranging polymer chainsparallel to each other37 �via the Up term�. Thus, Uc is used tocoarse grain the bond torsion potential while Up is used tocoarse grain the enthalpic interactions that drive packing in acrystalline unit cell. Since the atomistic origins of both ofthese coarse grained interactions are the same, we assumehere that the energies for a pair of bonds to be collinear andto be parallel are equal, i.e., Uc /Up=1 �increasing Uc /Up

from 1 to 10 in simulations of unadditivated polymers didnot significantly influence either the crystallinity or thelamellar thickness of the crystals. Therefore, we have per-formed all our simulations at Uc /Up=1�. For simplicity, weset the monomer-additive attraction, Uma=�Up, where � isthe “stickiness” parameter, specifying the strength of themonomer-additive attraction relative to the parallel bond in-teraction energy. In our work, ��0 and higher � impliesstronger monomer-additive attraction.

074905-2 Dasmahapatra, Nanavati, and Kumaraswamy J. Chem. Phys. 131, 074905 �2009�

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We begin by equilibrating the system at Up=0. As Up,Uc, and Uma are normalized by kT, setting Up=0 represents ahigh temperature state, viz., an athermal melt. At Up=0, wecalculate squares of the instantaneous end to end distances ofsegments of length i, Ri

2 �i=N, N /2, N /4, and N /8�, as func-tions of MCSs �one MCS is defined as 0.9375�323 MCmoves, viz., on the average one attempted MC move for eachunit, monomer, as well as additive present in the simulationbox�. We did not observe any appreciable change in the valueof Ri

2 beyond 5000 MCS. To verify that the athermal melt isat equilibrium after 5000 MCS, we calculate the scaled seg-mental mean square end to end distance, �Rj

2� / j, for segmentlengths j �j=1 to N−1� along the chain, and observe that thisreaches a plateau with increase in segment length.38 We havechecked the scaling relation of �RN

2 � �N=16, 32, 64, and 128,averaged over 5000 MCS� with chain length, N, and havefound that the exponent is �0.5, characteristic of a polymermelt.

After equilibration at Up=0, we progressively increasethe value of Up in steps of 0.05. An increase in Up can beconsidered as a decrease in temperature—in our work, wepresent data as a function of Up. As at Up=0, we allow thesample morphology to develop over 5000 MCSs andcalculate properties as an average over the subsequent5000 MCSs. In our simulations, we calculate the fractionalcrystallinity, Xc, specific heat, Cv �described later�, additive-monomer pair distribution function, and mean square dis-placement of center of mass as described below.

Increasing Up in uniform steps of 0.05 corresponds to anonlinear cooling profile �since Up�1 /T�. Nonisothermalcrystallization is cooling rate dependent. On changing the Up

step size from 0.05 to 0.02 �shorter steps correspond toslower cooling�, there is a shift in the crystallization pointtoward lower Up �viz., higher temperature�—this behavior isconsistent with experimental observations. To further exam-ine the dependence of our results on cooling rate, we haverun simulations for homopolymer crystallization, where Up

is increased in steps of 0.02, as well as at a constant coolingrate �viz., increasing Up in nonuniform steps at constant��1 /Up�=0.2�. We observe that there are no differences inthe qualitative trends for crystallization �viz., increase incrystallinity on cooling or for the appearance of a specificheat peak� in the two cooling profiles. Data from simulationswhere the temperature �1 /Up� is decreased linearly with astep size of 0.2 closely match the data from simulationswhere Up is increased with a step size of 0.02. Thus, thequalitative trends in our results are generally valid and thecooling profile employed in our simulations does not affectthe generality of our conclusions.

To monitor the phase transition from a disordered moltenstate to an ordered crystalline state, we calculate Xc as afunction of Up. We arbitrarily define a bond as crystalline ifit is surrounded by more than five nearest nonbonded parallelbonds in a manner similar to that of Hu et al.29 Crystallinityis defined as the ratio of the number of crystalline bonds tothe total number of bonds in the system. A crystallite is acontiguous aggregate of crystalline bonds with the same ori-entation. We characterize crystallites by their size �S� andtheir thickness �L�. The size of a crystallite is the number of

crystalline bonds that comprise it. We calculate the distribu-tion of the crystallite sizes and the average crystallite size asa function of Up. We also calculate the thickness of a crys-tallite as the average fold length in the chain direction. Theaverage thickness is calculated over all crystallites present inthe system and is expressed as the average number of mono-mer units in the crystal thickness direction.

We calculate the specific heat like property, Cv, in amanner similar to that for the equilibrium specific heat, viz.,from the total energy fluctuations �normalized by the totalnumber of monomer units and additives in the simulationbox�. Our simulations represent a system that is out ofequilibrium—therefore, the energy fluctuation term that wecalculate is not the equilibrium specific heat. However, sincewe calculate properties over a much longer number of MCSthan is required for structural relaxation, we continue to callthe calculated energy fluctuation term the specific heat anddenote it as Cv for convenience. At the phase transition, Cv

shows a peak as fluctuations in energy39 become large. TheUp value at which we observe a peak in Cv on cooling fromthe melt state is termed the crystallization Up, in tune withexperimental �calorimetry� convention, even though crystal-lization takes place over a range of Up.

The spatial distribution of additives in the system ischaracterized by calculating the additive-additive �a-a� andadditive-monomer �a-m� pair distribution functions, g�r�.These are the fractions of additives and monomers, at a dis-tance r from an additive �averaged over all additives�, re-spectively. Similarly, to characterize the local environment�crystalline/amorphous� in the vicinity of an additive, we cal-culate additive-crystalline �a-c� monomer and additive-noncrystalline �a-nc� monomer pair distribution functions atr=1.

We have simulated crystallization of polymers contain-ing x=0.4%, 0.83%, 1.67%, 3.125%, and 6.25% of stickyadditives. At each additive fraction, we have investigated arange of stickiness parameters, �=0, 5, 10, 20, 30, and 40. Inthis parameter range, we encounter attempted moves with a�E of up to 45kT. However, the majority of the moves are ofless than 10kT �at least �60% of possible moves and �99%of accepted moves in all simulations�.

To check that our algorithm is robust against aphysicalfrustration of morphology, we have run test simulationswith an artificial Hamiltonian that resets �E�10kT to�E=10kT. Thus, in these test simulations, attempted moveswith energies greater than 10kT are accepted with an artifi-cially increased probability, corresponding to 10kT. We notethat this procedure affects only 1% of the accepted movesand observe that evolution of semicrystalline morphology inthese test simulations is similar to that in our MetropolisDMC simulations—thus, indicating that we do not freeze inaphysical structures in our simulations, that real systems canfree themselves from. Further, we always have a healthyfraction of accepted moves in our simulations. In the ather-mal state �Up=0�, moves are accepted with a probability ofaround 18%. As Up increases, the acceptance rate decreasesbut is always above 1%, which is reasonable for DMC oflarge topologically connected dense systems such as polymermelts.32

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Polymer crystallization is a nonequilibrium phenomenonand results in a coexistence of amorphous and crystallinephases over a range of temperatures. The work of Hu et al.29

demonstrated that several experimental details of copolymercrystallization can be accurately reproduced by DMC simu-lations with a similar methodology. Specifically, previouswork demonstrated that DMC simulations faithfully capturethe dynamical evolution of semicrystalline microstructureduring chain folding,40 sectorization of solution crystals,41

and melting of lamellar polymer crystals.27 In our work, we“cool” the polymer by reducing Up in steps of 0.05. Speci-fying the step size is equivalent to specifying a cooling ratein real experiments. Thus, a step size of 0.01, for example,represents experiments at a slower cooling rate, relative toour work. Simulations of the crystallization of unadditivatedhomopolymers at different cooling rates �Up step sizes of0.05, 0.02, and 0.01� show that the crystallinity �defined ear-lier in this section� increases abruptly and then saturates asthe polymer is cooled. Further, the crystallinity at saturationis a function of cooling rate—higher cooling rates result inlower crystallinity. Thus, the trends from “control experi-ments” using our simulation system match reported experi-mental results.42

III. RESULTS AND DISCUSSION

We begin by describing in detail the crystallization of asystem containing 1.67% additives �viz., 512 additive par-ticles and 472 chains of N=64; additive units per chain�O�1�� and contrasting it to the behavior of unadditivatedpolymer. At first, we describe nonisothermal crystallizationon cooling from an equilibrated “high temperature” state�Up=0� to Up=0.5 with a step size �viz., “cooling rate”� of0.05.

A. Development of crystallinity on cooling

As the polymer is “cooled” from an athermal melt state�viz., Up is increased from 0�, chain segments start “stiffen-ing” �viz., becoming collinear� and begin aligning parallel toeach other. This is a physically realistic representation of

polymer crystallization36 since it is believed that polymerchains first “stiffen” locally into all-trans or helical structuresthat then pack into crystalline unit cells, with chain axesparallel. The additives are well dispersed in the polymer,both in the melt state �Fig. 1, Up=0� and on cooling into thesemicrystalline state �compare Fig. 1 with Fig. 6�. For theunadditivated polymer �H� as well as for the polymer con-taining 1.67% additives, there is an abrupt increase in crys-tallinity at Up=0.3 for all �=0, 5, 10, 20, 30, and 40 �Fig. 2�indicating that within the resolution of our simulations, de-spite the presence of additives, we find no change in the Up

value for the onset of crystallinity; i.e., the additives do notchange the “temperature” of crystallization.

For unadditivated polymer and for additivated polymerswith low values of � �� up to 10�, the rapid increase incrystallinity reaches a plateau �by Up=0.4�. For higher val-ues of � ��20�, the initial increase in crystallinity at theonset Up is less abrupt and the crystallinity continues to in-crease gradually as the sample is cooled to higher Up �up toUp=0.5�. The saturation crystallinity, Xc

sat, decreases mono-tonically as the additives become stickier from �0.77 for theunadditivated polymer and for �=0 to 0.765, 0.74, 0.63,0.46, and 0.43 for �=5, 10, 20, 30, and 40, respectively.Thus, as the additives become stickier, they inhibit crystalli-zation: Crystallinity decreases and the onset of crystallizationis less sharp relative to unadditivated polymer. Also, reminis-cent of copolymers, polymers containing sticky additives���20� continue to increase their crystallinity on cooling tohigher Up. However, unlike in copolymers, we find nochange in the Up for onset of crystallization with increasing� � within the resolution of our simulations�. Reheating fromUp=0.5 to Up=0 with a heating step size of 0.05 shows ahysteresis on melting �Fig. 3 for H�, a characteristic signatureof a nonequilibrium, first order transition in bulk polymers.

Polymer crystals are out of equilibrium and the crystal-linity that develops in a polymer depends largely on the con-ditions of crystallization. To understand how crystallizationconditions determine sample crystallinity in the presence ofadditives, we compare samples where crystallinity developsisothermally after quenching, with the samples discussedpreviously, viz., where crystallinity evolves on “slow” cool-ing �in steps of �Up=0.05�.

FIG. 1. Snapshot of the simulation box at Up=0 showing the dispersion ofthe additive in polymer matrix. The filled circles represent the additives�concentration=1.67%�.

FIG. 2. Change in crystallinity with Up for unadditivated �H� and additi-vated polymers ��=0, 5, 10, 20, 30, and 40 ; additive concentration=1.67%�. The lines joining the points are meant only as a guide to the eye.

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�1� In one simulation, after equilibration at Up=0, wequench a sample containing 1.67% additives with�=0 to Up=0.5. After annealing for 2.5�105 MCSs, acrystallinity of 0.77 is reached, and there is little furtherevolution of the crystallinity. At Up=0.5, we abruptlychange the value of � from 0 to 40 for this isothermallycrystallized sample and anneal for a further 2.5�105

MCSs. On annealing, we observe “melting” and thecrystallinity decreases from 0.77 only down to 0.71�compared to �i� Xc

sat=0.77 for a sample with �=0,cooled slowly to Up=0.5, and �ii� Xc

sat=0.43 for asample with �=40, cooled slowly to Up=0.5�.

�2� If, instead, we start with a sample containing 1.67% of�=40 additives, and quench from the high temperaturestate to Up=0.5, annealing for 2.5�105 MCSs yields acrystallinity of 0.412 �as compared with 0.43 obtainedfor slow cooling�. Now, making the additives lesssticky, viz., abruptly changing the value of � from 40 to0, results in a dramatic increase in crystallinity to 0.767after annealing for a further 2.5�105 MCSs.

Thus, the formation of semicrystalline microstructure isa strong function of the crystallization conditions. Increasingthe particle stickiness after crystallinity has developed,has only a limited influence on polymer melting for adeeply quenched polymer �Up=0.5�, but decreasing con-straints by decreasing stickiness results in significant furthercrystallization.

B. Crystallite structure

Polymers crystallize into two-dimensional objects �viz.,lamellae� in our simulations, viz., the crystallite thickness�average fold length in the chain direction� is smaller thanthe lateral dimensions in the plane of the crystallite. Figure 4shows a part of a large crystallite at Up=0.5 from a simula-tion of unadditivated polymer, crystallized by cooling fromUp=0 to Up=0.5 in steps of 0.05. The geometry observed forthe crystallite here is also typical of structures obtained insimulations of additivated systems. We can clearly identifychain folded sections, including a few crystalline stems thatshow adjacent re-entry into the crystallite.

We now examine the influence of additive stickiness onthe dimensions of the crystallites formed. We observe that

the decrease in crystallinity for increasing values of � �Fig.2� is associated with a decrease in the average size and thick-ness of the crystallites, as might be intuitive. In all our simu-lations, we observe a very wide distribution of crystallitesizes—therefore, we note that a comparison of the values of�S� provides only a limited perspective of structure. Thecrystallite thickness, however, shows a relatively narrow dis-tribution, and the average thickness, �L�, is close to the peakvalue of the distribution. �L� grows abruptly at the onset ofcrystallization and then rapidly approaches a �-dependentplateau value for higher Up �Fig. 5�, similar to the trendobserved for crystallinity �see Fig. 2�. As the additives be-come stickier �viz., as � increases�, the average thickness ofthe crystallites decreases monotonically.

For unadditivated polymer, we have observed that forisothermal crystallization when quenched to various values,Up

q, from the high temperature Up=0 state, the average crys-tallite thickness, �L�, decreases with increased Up

q �data notpresented� as expected. Higher undercoolings result in less-stable, thinner crystallites. Additivation with sticky additivesalso results in the formation of thinner crystallites in ournonisothermal simulations.

FIG. 3. Change in crystallinity with Up for crystallization and melting of anunadditivated polymer, showing hysteresis. The lines joining the points aremeant only as a guide to the eye.

FIG. 4. A snapshot of a typical crystallite from a simulation of an unaddi-tivated polymer at Up=0.5. This crystallite contains around 650 crystallinemonomers.

FIG. 5. Change in average crystallite thickness with Up for unadditivated�H� and additivated ��=0, 5, 10, 20, 30, and 40; additive concentration=1.67%� polymers. The lines joining the points are meant only as a guide tothe eye.

074905-5 Polymer crystallization with sticky additives J. Chem. Phys. 131, 074905 �2009�

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The decrease in crystallinity and crystallite size with in-crease in � can be directly visualized in snapshots of oursimulation box at Up=0.5 for the unadditivated polymer andfor �=5, 20, and 40 �Fig. 6�. Near the edges of the box, it isalso possible to observe parallel stacks of chains formingcrystallites similar to those for the unadditivated polymershown in Fig. 4. As can be observed, the additives remainwell dispersed in the simulation box even after polymer crys-tallization for all �. This is quantitatively confirmed by cal-culating the value of the additive-additive �a-a� pair distri-bution function at the nearest neighbor position, gaa �data notpresented�.

To estimate the local polymer crystallinity in the vicinityof the additives, we calculate g�1�ac, the pair distributionfunction for crystalline monomers in contact with additivesat Up=0.5 and present these data, averaged over all additivesand over 5000 MCS �Fig. 7�. We observe that for �=0, voidsare expelled preferentially to the vicinity of the additive asthe polymer crystallizes leading to a low value of g�1�ac.For ��0, the additives attract monomers, and therefore,g�1�ac+g�1�anc�1 for Up�0 �g�1�anc represents the fractionof nearest neighbor sites to the additive that are occupied bynoncrystalline monomers�, viz., all the nearest neighboringsites to the additives are occupied by monomers to increaseadditive-monomer contacts. For low ��=5, 10�, we observethat polymer crystallinity in the vicinity of the additives isvery high at Up=0.5 �g�1�ac�0.9, Fig. 7, compared toXc

sat=0.77 and 0.75 for �=5 and 10, respectively, Fig. 2�.Further increase in � results in a decrease in g�1�ac. Interest-ingly, the ratio of g�1�ac to the bulk crystallinity is higherthan 1 for �=5, 10, and 20 and lower than 1 for higher ��Fig. 7, inset�. Thus, the vicinity of the additives is morecrystalline than the bulk for �20 suggesting that localiza-tion of the monomer by the additive stickiness enhancescrystallinity in the vicinity of the additive for low to inter-

mediate stickiness. For higher stickiness ���20�, the localcrystallinity in the vicinity of the additive, viz., g�1�ac, islower than the bulk crystallinity suggesting that in this re-gime, crystallization in the vicinity of the additive is inhib-ited. At all Up, g�1�ac /Xc shows a maximum at �=10. Wewill discuss the significance of this in Sec. III D.

C. Influence of sticky additives on chain mobility

Crystallization is governed by both the driving force forphase change and polymer chain mobility. Having examinedthe structure of the crystallizing polymer/additive system, wenow investigate the influence of the sticky additives on poly-mer chain mobility. We measure the mean square displace-ment of the center of mass �dcm

2 , a measure of chain diffusiv-ity�, averaged over all chains as a function of Up �Fig. 8�without additive �H� and with additive ��=0, 5, 10, 20, 30,and 40�. As we cool the system �viz., as Up increases above0�, chain diffusivity decreases. In the melt state �Up0.3�,this decrease in chain diffusivity is greater for increasingvalues of � �Fig. 8�. Thus, increased additive stickiness re-sults in a decrease in polymer chain diffusivity in the meltstate. At Up=0.3, the polymer crystallizes �see Fig. 2� andthere is an abrupt decrease in chain diffusivity for all sys-tems. At Up�0.3, polymer chains in both additivated andunadditivated systems have a low value of diffusivity. For

FIG. 7. Change in average number of additive-crystalline monomer pairdistribution function at r=1, g�1�ac as a function of � at Up=0.5 �additiveconcentration=1.67%�. Error bars for g�1�ac are comparable to the size ofthe symbol. The ratio of g�1�ac to Xc is shown in the inset. The lines joiningthe points are meant only as a guide to the eye.

FIG. 8. Change in mean square displacement of center of mass averagedover all polymer chains as a function of Up for unadditivated �H� and addi-tivated systems ��=0, 5, 10, 20, 30, and 40; additive concentration=1.67%�. The lines joining the points are meant only as a guide to the eye.

(a)

(c)

(b)

(d)

FIG. 6. Snapshots at Up=0.5 from simulations for �a� unadditivated andadditivated systems �b, c, and d represent simulations at �=5, 20, and 40,respectively�. The yellow lines represent crystalline bonds, black lines rep-resent noncrystalline bonds, and empty circles represent additives.

074905-6 Dasmahapatra, Nanavati, and Kumaraswamy J. Chem. Phys. 131, 074905 �2009�

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Up�0.3, polymer chain diffusivity increases with an in-crease in �, probably due to the increase in amorphous con-tent with an increase in �. We have also calculated the mo-bility �viz., the mean square displacement� of monomers inthe neighborhood of the additives and observe that they showthe same trends with Up and �, as the chain diffusivity �datanot presented�.

The decrease in chain mobility with increasing � in themelt state �Up0.3� correlates with the observed decrease incrystallinity and in crystallite dimensions, as might be antici-pated. It is worth noting here that in these simulations, wehave 512 additive units and 472 chains, viz., around oneadditive unit per polymer chain. Therefore, the influence ofadditive particles on crystallization is readily apparent. Thesetrends are, however, similar even at lower additive loadingsand we will present data on systems with different additivefractions later in this paper.

D. Unexpected behavior at intermediate �

We measure the specific heat equivalent, Cv, from themean square fluctuations in total energy as a function of Up

for unadditivated and additivated systems as they crystallize�Fig. 9�. The melt-crystal phase transition is indicated by apeak in Cv. We observe a peak in Cv at the same Up�=0.3�for all additivated and unadditivated systems �Fig. 9� corre-sponding to the Up, at which Xc increases abruptly �Fig. 2�.Unexpectedly, the peak value of specific heat, Cv

�, exhibits anonmonotonic behavior with �: Cv

� increases from �=0 to amaximum at �=10, and then decreases monotonically forhigher � �Fig. 9, inset�. The data presented in Fig. 9 areaveraged over five simulations and the error bars in the mag-nitude of Cv

� represent the variability over these independentsimulations �Fig. 9, inset�.

It is clear that the maximum observed for Cv� at an inter-

mediate value of � is not within the error of the simulations.The limited Up resolution of our data makes it difficult tounambiguously comment on the area of the Cv peak—however, it is reasonable to state that to a first approxima-tion, the peak area follows the same trend as for the peakvalue, Cv

�. Thus, one might expect that Cv� would have the

same �monotonically decreasing� �-dependence as Xcsat.

Therefore, the nonmonotonicity observed for Cv� is indeed

surprising �see the inset of Fig. 9 and compare with Fig. 2�.While the crystallinity decreases monotonically with increas-ing additive stickiness, as is intuitive, the magnitude of en-ergy fluctuations at Up=0.3 is enhanced at intermediate val-ues of �.

In our simulations, the change in energy of the systemcomes from changes in the number of parallel and collinearbonds and from changes in additive-monomer contacts. Wehave calculated the magnitude of energy fluctuations basedonly on the changes in number of parallel and collinearbonds �viz., by ignoring energy fluctuations due to change inadditive-monomer contacts�, and have found that this domi-nates Cv. Thus, it is the fluctuations in the number of paralleland collinear bonds �rather than fluctuations in monomer-additive contacts� that determine Cv

� and that are, therefore,responsible for the maximum in Cv

� at intermediate �. Ascrystallinity arises from the stacking of bonds parallel toeach other, fluctuations in the number of parallel bonds sug-gest fluctuations of monomers between crystal and amor-phous states. Thus, the observed maximum in Cv

� at interme-diate ��=10� appears to result from an enhancement influctuation of monomers between crystal and amorphousphases, at that �.

We do not have an unambiguous explanation for the un-usual behavior of Cv

�. To gain insight into the behavior of Cv�,

we now examine structural attributes of the crystallizing bulkpolymer at the phase transition. For sticky additives, viz.,when ��0, the occupation density of monomers in the near-est neighbor positions to the additives is nearly 1 forUp�0. This is intuitive—when the additives are sticky,monomers occupy every lattice site in the vicinity of theadditives and exclude voids. Further, the mean squarefluctuations in the occupation density around the additive,�Gt= �g�1�2�− �g�1��2�, are small and decrease with increas-ing � �Fig. 10�. However, monomers in the vicinity of theadditives remain mobile �data not presented� and monomermobility increases with � �since the crystallinity decreaseswith increased ��.

We now examine the fluctuations in the number of crys-talline monomers in the vicinity of the additives. We calcu-late Gac= �g�1�ac

2 �− �g�1�ac�2 as a function of Up �Fig. 11�.Gac is significantly larger than Gt �compare Fig. 11 to Fig.

FIG. 9. Change in specific heat, Cv with Up for unadditivated �H� andadditivated ��=0, 5, 10, 20, 30, and 40; additive concentration=1.67%�systems. The inset shows the change in the peak value of Cv with �. Thelines joining the points are meant only as a guide to the eye.

FIG. 10. Mean square fluctuations in additive-monomer pair distributionfunction at r=1, as a function of Up for additivated ��=0, 5, 10, 20, 30, 40,and 50, with additive loading of 1.67%� system. The lines joining the pointsare meant only as a guide to the eye.

074905-7 Polymer crystallization with sticky additives J. Chem. Phys. 131, 074905 �2009�

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10� and exhibits a peak �having a magnitude of Gac� � at the

phase transition, for all values of � �Fig. 11, inset�. Similar toCv, Gac is small at all Up, other than at the transition point�Up=0.3� where it shows a maximum. Interestingly, atUp=0.3, the peak value is significant only at intermediate� �=10 and 20� and is small for all other �.

To summarize, additive stickiness results in the follow-ing: �a� the vicinity �nearest neighbor position� of the addi-tive has a monomer occupation density close to unity; �b� themonomers in the vicinity of the additive retain mobility atthe phase transition, Up=0.3, and monomer mobility islargely dictated by the crystallinity; �c� there is a large inter-change between amorphous and crystalline monomers in thevicinity of the additive at the phase transition for intermedi-ate values of �, correlating with the maximum in Cv

�; �d�finally, we note that the maximum in Cv

� at �=10 also corre-lates with the maximum in g�1�ac /Xc both at Up=0.5 �Fig. 7�and at the phase transition �Up=0.3, Fig. 12�.

Thus, while the maximum in Cv� correlates with several

structural attributes at the phase transition, these do not un-ambiguously suggest the physical origin for its unusualtrend. What then is the reason for the observed maximum inCv

� at intermediate �? In a physical experiment, the tempera-ture of a cooling polymer melt decreases continuously. We

model this in our experiments by increasing Up in discretesteps. The step size in Up determines the cooling rate with alarger step size corresponding to more rapid cooling. Further,the resolution to which we can determine the precise transi-tion point is also determined by the step size. This raises thequestion whether the value and location of the observedmaximum in Cv

� are artifacts of our simulation protocol, suchas the specific step size Up. To resolve this, we examine theisothermal crystallization of the additivated polymers, as afunction of MCSs on quenching from Up=0 to Up=0.5�close to the phase transition temperature, shallow quench�.If the route to crystallization is indeed modified at interme-diate values of �, then we would expect to see qualitativelydifferent crystallization kinetics, viz., development of crys-tallinity at intermediate ��=10,20� as compared with low orhigh �.

E. Isothermal crystallization

On quenching the polymer from the athermal state�Up=0� to Up�0.3, its crystallinity builds up as a functionof MCS. We observe the development of crystallinity for2.5�105 MCSs for all values of � and for H on quenchingto Up=0.5 �Figs. 13�a� and 13�b��. We examine the crystal-linity, Xc �Fig. 13�a�� and the scaled crystallinity,�Xc−Xc

0� / �Xc�−Xc

0�, that ranges from 0 to 1 �Fig. 13�b�� tocontrast the evolution of crystallinity at various �. Xc

0 repre-sents the crystallinity at the beginning of the annealing runwhile Xc

� is the crystallinity at the end of isothermal anneal-ing �i.e., Xc at the end of 2.5�105 MCSs�.

We observe that the scaled crystallinity evolves in thesame manner for unadditivated polymer as well as for addi-tivated polymer, with low ��5� and high ���30� values of�. For each of these systems, the evolution of the scaledcrystallinity superposes and follows the classical S-shaped,Avrami-like increase with MCS. However, at intermediate��=10,20�, wherein nonisothermal experiments showedanomalously large energy fluctuations, we observe an un-usual two-stage increase in crystallinity.

Thus, the evolution of crystallinity in an isothermal ex-periment is dramatically different for intermediate values of�. This unusual behavior correlates with the anomalous in-crease in fluctuations at Up=0.3 for intermediate �. Thus, theaccord in results from two independent experiments, viz., theincrease in fluctuations at intermediate � in nonisothermalcooling and the qualitatively different evolution of crystallin-

FIG. 11. Fluctuations in additive-crystalline monomer pair distribution func-tion at r=1, as a function of Up for additivated system ��=0, 5, 10, 20, 30,and 40 with additive concentration=1.67%�. The inset shows the change inthe peak value of the fluctuations with �. The lines joining the points aremeant only as a guide to the eye.

FIG. 12. Change in average number of additive-crystalline monomer pairdistribution function at r=1, g�1�ac as a function of � �with additive loadingof 1.67%�at Up=0.3. Error bars for g�1�ac are comparable to the size of thesymbol. The ratio of g�1�ac to Xc is shown in the inset. The lines joining thepoints are meant only as a guide to the eye.

FIG. 13. Change in the �a� crystallinity and �b� normalized crystallinity withnumber of MCS for unadditivated �H� and additivated polymer �with addi-tive concentration=1.67%� for a quench depth, Up

q =0.50. Limited numberof data points shown by symbols in the trajectory, are meant as a guide tothe eye.

074905-8 Dasmahapatra, Nanavati, and Kumaraswamy J. Chem. Phys. 131, 074905 �2009�

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ity at those same intermediate � during isothermal coolingstrongly support our contention that unusual behavior is ob-served at intermediate �.

F. Influence of additive concentration

Finally, we describe the effect of additive concentrationon crystallization. Results for simulations at different addi-tive concentrations �viz., x=0.42%, 0.83%, 3.125%, and6.25%� are presented in Fig. 14. Here, the ratio of additiveunits to polymer chains varies from 0.27 �x=0.42%� by overan order of magnitude to 4.27 �x=6.25%�. However, the evo-lution in crystallinity shows qualitatively similar trends forall values of x. At each x, there is a decrease in Xc

sat with anincrease in �. This decrease is more pronounced at higheradditive loadings �compare Figs. 14�a�–14�d� at a fixed valueof ��. At x=0.42% and 0.83% �Figs. 14�a� and 14�b�, respec-tively�, appreciable change in crystallinity is observed onlyfor ��30 and even at �=40, Xc

sat decreases only to around0.6 �compare to 0.77 for an unadditivated system�. For in-creased additive fraction, x=3.125% �Fig. 14�c��, we observea decrease in crystallinity even at �=10, and at x=6.25%�Fig. 14�d��, we observe that Xc

sat decreases even for �=5.The Xc

sat at �=40 decreases to around 0.23 for x=3.125% andto around 0.1 for x=6.25%.

Thus, as might be anticipated, crystallization in the pres-ence of sticky additives is governed both by the loading �x�as well as the magnitude of stickiness, �. This is clearlyobserved in a plot of Xc

sat as a function of additive loading, xfor all � �Fig. 15�. At low value of �, Xc

sat is only weaklydependent on x �compare �=0, 5; Fig. 15�.

Interestingly, for high ��=30,40�, Xcsat depends only on x

and is almost independent of �. Thus, the effect of the addi-tive on crystallization saturates beyond a threshold value ofstickiness and Xc becomes dependent only on x. Since themonomer-additive interaction is a short-range contact inter-action, it is reasonable to expect that its effect saturates

beyond a threshold value of � and any further increase in thevalue of � produces no further change in the crystallization.

IV. SUMMARY

We describe the influence of sticky additives on the crys-tallization of a polymer melt. In our simulations, the attrac-tive monomer-additive interaction results in dispersion of theadditive in the polymer melt. An increase in additive loading,or in the stickiness at a given loading, results in a decrease inthe crystallinity and in crystallite dimensions. We believethat the effect of the additives arises from a decrease in poly-mer melt diffusivity due to attractive monomer-additive in-teractions. At low additive loading �approximately one addi-tive per four polymer chains� and at low additive stickiness,there is very little influence of the additive on crystallization.When the additives are very sticky, crystallization of mono-mers in the vicinity of the additives is highly inhibited. How-ever, the effects of the additive saturate above a thresholdstickiness and are then a function only of the loading.

The most interesting effect that we find is an anomalousincrease in energy fluctuations at the observed phase transi-tion for intermediate values of additive stickiness. Thesefluctuations correlate with monomers in the vicinity of theadditive undergoing a large interchange between crystallineand amorphous states. This change in the route to crystalli-zation dramatically influences the kinetics of crystal growthin isothermal crystallization experiments. At intermediate �,the polymer no longer exhibits the classical Avrami-likeS-shaped growth in crystallinity—rather, its approach to Xc

�

is slow and appears to happen in two stages.Our understanding of how sticky additives influence

polymer crystallization might be used to design new com-posite materials with tailored crystallization kinetics to opti-mize structure development during processing. For example,it might be interesting to systematically change the hydrogenbonding ability of additives �say, by comparing the influenceof primary, secondary, and tertiary amine surfactants�, and toinvestigate the influence of these additives on the crystalli-zation of polar polymers such as polyamides or polyesters.There is little experimental literature on such systematic in-vestigations and our results suggest a potentially useful av-enue for exploration. We believe that such studies might alsobe relevant in the case of more complex systems such as

FIG. 14. Change in crystallinity with Up for unadditivated �H� and additi-vated ��=0, 5, 10, 20, 30, and 40� systems for varying additive loading: �a�0.42, �b� 0.83, �c� 3.125, and �d� 6.25% respectively.

FIG. 15. Change in saturated crystallinity, Xcsat with additive loading, x for

�=0, 5, 10, 20, 30, and 40. The lines joining the points are meant only as aguide to the eye.

074905-9 Polymer crystallization with sticky additives J. Chem. Phys. 131, 074905 �2009�

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polymer nanocomposites, wherein attractive interactions areengineered between nanoparticulate additives and the matrixpolymer.

ACKNOWLEDGMENTS

A.K.D. acknowledges the financial support from theCSIR, India, SRF grant. We gratefully acknowledgeC-MMACS for providing computing resources. We grate-fully acknowledge Professor Sanat Kumar �Columbia�, Pro-fessor Sachin Shanbhag �Florida�, and Professor ChandanDasgupta �IISc, Bangalore� for useful discussions.

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074905-10 Dasmahapatra, Nanavati, and Kumaraswamy J. Chem. Phys. 131, 074905 �2009�

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