Structural properties of crystallizable polymer melts : intrachain and interchain correlation functions

Structural properties of crystallizable polymer melts:

Intrachain and interchain correlation functions

Thomas Vettorel,*Hendrik Meyer,^†and Jörg Baschnagel

Institut Charles Sadron, CNRS-UPR22, 6 Rue Boussingault, 67083 Strasbourg, France Matthias Fuchs

Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany 共Received 5 October 2006; published 10 April 2007兲

We present results from constant pressure molecular-dynamics simulations for a bead-spring model of a crystallizable polymer melt. Our model has two main features, a chemically realistic intrachain rigidity and a purely repulsive interaction between nonbonded monomers. By means of intrachain and interchain structure factors we explore polymer conformation and melt structure above and below the temperatureT_crys^homof homo- geneous crystallization. Here, we do not only determine average spatial correlations, but also site-specific correlations which depend on the position of the monomers along the polymer backbone. In the liquid phase aboveT_crys^homwe find that this site dependence can be well-accounted for by known theoretical approximations, the Koyama distribution for the intrachain structure and the polymer reference interaction site model共PRISM兲 for the interchain structure. This is no longer true in the semicrystalline phase. BelowT_crys^homshort chains fully extend upon crystallization, whereas sufficiently long chains form chain-folded lamellae which coexist with amorphous regions. The structural features of these polymer crystals lead to violations of premises of the Koyama approximation or PRISM theory so that both theoretical approaches cannot be applied simultaneously.

Furthermore, we find a violation of the Hansen-Verlet freezing criterion; our polymer melt crystallizes more easily than a simple liquid. This hints at the importance of the coupling between conformation 共backbone rigidity兲and density共packing constraints兲for polymer crystallization.

DOI:10.1103/PhysRevE.75.041801 PACS number共s兲: 61.25.Hq, 61.20.Ja

I. INTRODUCTION

When supercooling a crystallizable polymer melt the melt undergoes a transition to a semicrystalline state 关1兴. This polymer crystal is characterized by a hierarchy of ordered structures. The basic units are chain-folded lamellae, consist- ing of regularly packed, extended chain sections. The lamel- lae interleave with amorphous regions to form layers, which in turn organize themselves in larger scale superstructures 共spherulites兲.

It is commonly believed that this hierarchical structure is the result of kinetic effects which predominate crystallization of the supercooled polymer melt关2,3兴. This point of view is supported by various observations. For instance, the lamellar thickness共⬇10 nm兲is found to be much smaller than values predicted by equilibrium considerations关1,4,5兴; the melting behavior of the crystal depends on thermal history关6兴; and true thermodynamic coexistence of the polymer liquid and crystalline phase does not seem to exist because the crystal- lization temperature is generally observed to be lower than the melting temperature关1兴.

It is thus not surprising that great effort has been invested in exploring the features of the metastable polymer melt共see, e.g., Refs.关2,7兴 for reviews兲. Important problems that have been addressed include the crystallization mechanism in the very early stage 共nucleation-and-growth 关4兴, spinodal-

assisted crystallization 关8兴兲, the criteria for spontaneous se- lection of the lamellar thickness关4,5兴, or the molecular pro- cesses responsible for the growth of the crystal front关3兴.

In addition to these studies of the metastable melt, at- tempts have also been made to tackle the problem of poly- mer crystallization “from the liquid”—that is, to understand the thermodynamic driving force leading to the liquid-solid transition on cooling. For instance, density-functional theo- ries 共DFTs兲 were developed 关9,10兴. Quite generally, DFT 关11兴is a method to predict equilibrium features of an inho- mogeneous system 共the crystal兲 from information about the structure of the homogeneous system共the liquid兲, provided an accurate free-energy functional is known for the consid- ered system and the required structural input is available with sufficient precision. DFT approaches to polymer crys- tallization关9,10兴utilized an approximate description of the conformation and structure of the polymer liquid. A critical test of the proposed free-energy functional could thus be achieved only partially. Here, we demonstrate that this ap- proximate description of the liquid structure is not 共or no longer兲necessary. We carried out molecular-dynamics simu- lations of a coarse-grained model for a crystallizable polymer melt and show that the pertinent intramolecular and intermo- lecular correlation functions can be determined over a large range of wave vectors with high statistics. Our work could thus inspire further developments of DFT, eventually leading to an accurate equilibrium free energy functional for polymer liquids. Such an accurate functional is a prerequisite for an extension of DFT to dynamical problems, such as the growth of new phases or other out-of-equibirium processes关11,12兴. The outline of this paper is as follows. First we discuss the simulation model and present some details about the

*Present address: Max-Planck Institut für Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany.

†Corresponding author. Email address: hmeyer@ics.u-strasbg.fr

1539-3755/2007/75共4兲/041801共14兲 041801-1 Konstanzer Online-Publikations-System (KOPS) ©2007 The American Physical Society URL: http://www.ub.uni-konstanz.de/kops/volltexte/2007/3214/

simulation technique 共Sec. II兲. This is followed by a brief introduction to the quantities used to explore the structure of our model above and below the temperature of homogeneous crystallization. Section IV then discusses the corresponding results obtained for—averaged and monomer-resolved—

intrachain and interchain correlation functions. Section V summarizes the main findings of our work.

II. MODEL AND SIMULATION

Our studies utilize a bead-spring model obtained by a controlled coarse-graining procedure 关13兴 from atomistic simulations of poly共vinyl alcohol兲 共PVA兲. In this section, we briefly describe this coarse-grained共CG兲 model, referred to as “CG-PVA model” in the following, and discuss its strengths and limitations. More details may be found in Refs.

关14–17兴.

A. The CG-PVA model

Figure 1 illustrates the link between the CG-PVA and PVA models. A monomer of the coarse-grained chain corre- sponds to a monomer of PVA. This identification forces con- secutive monomers along the CG-PVA chain to overlap. The average bond lengthb₀must thus be smaller than the mono- mer diameter of PVA. Fluctuations of the bond length b about b₀=␴/ 2 = 0.26 nm—␴= 0.52 nm roughly agrees with the monomer diameter of PVA—are restricted by a harmonic potential,

U_bond共b兲=1

2k_bond共b−b₀兲². 共1兲 The fluctuations of b are small because the force constant k_bondis large,k_bond= 2704k_BT/␴^2.

Monomers of distinct chains and of the same chain if they are three bonds or more apart interact by a 9-6 Lennard- Jones共LJ兲potential,

U_LJ共r兲=

冦

^{冋 冉}

^冊

^冉

^冊

^册

冣

Here,␴0= 0.89␴^{= 0.46 nm,}␧0= 1.511k_BT, andC= 4␧0/ 27 is chosen such that U_LJ共rcut兲= 0 for the cutoff distance r_cut

=共3 / 2兲^1/3␴0 共⯝1.02␴兲, r_cut being the minimum of the LJ potential. With the latter choice, nonbonded interactions are purely repulsive.

In addition to bond and excluded-volume interactions, a potentialU_ang共␪兲is also associated with the angle␪^between consecutive CG bonds.U_ang共␪兲 is determined directly from atomistic simulations by Boltzmann-inversion of the prob- ability distribution P_ang共␪兲, that is, by posing P_ang共␪兲

⬀sin␪^exp关−U_ang共␪兲/k_BT兴. Figure 1 depicts the resulting bond angle potential. U_ang共␪兲 clearly reveals the fingerprint of the underlying PVA chain. Since the bond angle of CG- PVA is determined by two successive torsions at the atomis- tic level共1兲,U_ang共␪兲 has three minima corresponding to the three energetically favorable states, trans–trans, trans–

gauche, and gauche–gauche. U_ang共␪兲 thus confers a semi- flexible character to the CG-PVA chains with a flexibility mechanism such as in the rotational-isomeric-state 共RIS兲 model关18,19兴. This feature becomes important for the struc- ture of the melt in the semicrystalline state.

B. Discussion of the model 1. Units

The coarse-graining procedure described in Ref.关13兴and further discussed in Ref.关17兴allows one to adjust the param- eters of the CG potentials so as to reproduce共some兲confor- mational and structural features of the underlying atomistic model. The optimization is carried out at a specific thermo- dynamic state point, characterized, e.g., by some temperature T and pressure p. For PVA, T= 550 K and p= 1 bar were chosen at the atomistic level. This corresponds toT= 1 and p= 8 for CG-PVA. Here, we employed reduced units, that is, energies are measured in units ofk_BT 共with the Boltzmann constantk_B= 1兲, lengths in units of␴共=1兲, and time in units of ␶⁼␴

冑

2. Strengths and limitations of the model

The CG-PVA model has advantages and drawbacks. A drawback certainly is that chemical specificity is lost. The identification “1 monomer of PVA⇔1 bead of CG-PVA”

does not allow one to describe features which are directly related to monomer properties of PVA. This implies, for in-

C C^H

C^OH

H H

C

H OH H

H

CHC

OH H H H H

C C C

OH OH H

H

Interaction Angular Excluded−Volume Interaction

Bonded Interaction

σ/2 _Torsions

(Atomistic level)

C C C C

C

0 5 10

60 90 120 150 180

Uang(θ)

θ

tt tg

gg (a)

(b)

FIG. 1. Schematic representation of the correspondence between the CG-PVA model and the underlying atomistic PVA chain. At the coarse-grained level, the interaction potentials for chain connectivity, excluded volume, and chain stiffness are indicated.

The angular potentialU_ang共␪兲 共in units ofk_BT兲is shown as a func- tion of the bond angle␪共in degrees兲. The minima ofU_ang共␪兲reflect specific states of two successive torsions at the atomistic level 共tt=trans-trans, tg=trans-gauche, and gg=gauche-gauche兲.

stance, that the crystal structure of PVA cannot be repro- duced. Stretched parts of the CG-PVA chains have cylindri- cal symmetry which leads to hexagonal ordering 共see Sec.

IV C 3兲, instead of to a monoclinic crystalline phase charac- teristic of PVA关20兴. In some cases, a further drawback might be that attractive or electrostatic interactions, present at the atomistic level, are not explicitly incorporated in the model;

they only enter implicitly into the potential parameters through the mapping procedure. So, situations where these interactions are important cannot be explored 共for instance, attractive interactions must be included for the simulation of polymer-air interfaces兲.

On the other hand, we feel that, for the study of polymer crystallization, the CG-PVA model has two main strengths, one computational, the other related to the physics of poly- mer crystallization. Computationally, the model is conve- nient. The simple monomer structure and the short-range po- tentials allow for an efficient simulation of large systems.

The study of large systems is important to minimize finite- size effects on the structure formation upon crystallization 共for further discussion see Ref.关15兴兲. From a physical point of view, the model provides the interesting insight that we can suppress monomer-monomer attractions and still observe key features of polymer crystallization关14–16兴. The compe- tition of packing constraints and the RIS-like chain stiffness, both of which increase on cooling, is sufficient to reproduce the following, experimentally well-known facts 关4,7兴. 共i兲 Short chains fully extend when crystallizing, while long chains fold back on themselves, forming lamellarlike struc- tures共cf. Fig.2兲. Detailed analysis of the chain-folded lamel- lae reveals the fingerprint of the angular potential U_ang共␪兲 关15兴. The folds consist oftrans–gaucheandgauche–gauche states. Apparently, the secondary minima ofU_ang共␪兲 kineti- cally stabilize these states and thus assist chain folding.共ii兲 The thickness d of the lamellae is inversely proportional to the crystallization temperatureT_crys of isothermal relaxation experiments 共i.e., an amorphous melt is quenched to T_crys

below the melting point and then relaxed isothermally兲.

Lamellae of a given thickness melt at a specific temperature T_m共d兲⬎T_crys共d兲. T_m共d兲 is also inversely proportional to d.

共iii兲For continuous cooling and heating cycles, the homoge- neous crystallization and melting temperatures increase with chain lengthN. The qualitative features of this increase agree well with experimental results on short alkanes关16,17兴.

C. Simulation aspects

We analyze polymer melts ofnlinear chains consisting of N monomers each. More precisely, the studied values of N and n are N= 10 共n= 288兲, N= 50 共n= 72兲, and N= 100 共n

= 192兲. Our results are obtained from constant pressure simu- lations 共p= 8; Berendsen barostat 关21兴兲 at various tempera- turesT共Langevin thermostat关21兴兲.

We explore structural properties of the liquid phase for Tⱖ0.8 and of the crystalline phase at T= 0.4. The lower bound,T= 0.8, of the liquid phase is determined by the tem- perature T_crys^hom where homogeneous nucleation occurs. For N= 100, Refs. 关15,16兴 found T_crys^hom= 0.78. As we expect T_crys^homⱕ0.78 for N⬍100, the choice Tⱖ0.8 for the liquid phase should be appropriate for allN studied. We thus refer toT_crys^hom= 0.78 as “onset temperature of homogeneous crystal- lization” in the following.

In the liquid phase, we analyze time series of fully equili- brated configurations at each T. The configurations of the crystalline phase are obtained by continuous cooling from T= 1 to 0.4 with the rate 5⫻10⁻⁶, followed by isothermal relaxation at temperature T. A systematic variation of the cooling rate⌫, from 2⫻10⁻⁵to 5⫻10⁻⁷, was carried out for N= 10. As expected, with decreasing ⌫ the transition from the liquid to the crystal becomes sharper and the structure of the crystalline phase better ordered. For N= 100, however, the impact of the cooling rate appears to be weak共at least for the rates studied here兲. Certainly, a crystal of fully stretched chains, as forN= 10 共see Fig.2兲, is not easily accessible in that case, even for slow cooling, andv共T兲always exhibits a rather smooth transition to the共semi-兲crystalline phase.

III. ANALYZED QUANTITIES AND TECHNICAL DETAILS

A. Static structure factors

One way to characterize the structure of a melt is to ex- plore spatial correlations between two monomersaandbvia density-density correlation functions. In this approach, the basic correlation function is the monomer-monomer共or site- site兲structure factor S_ab共q兲 which is defined by

Sab共q兲= 1 n

兺

i,j=1 n

具exp关−iq·共ri

a−r_j^b兲兴典. 共3兲 Here,qis the wave vector in reciprocal space,r_i^adenotes the position of monomera共=1 , . . . ,N兲in chaini共=1 , . . . ,n兲, and the angular brackets represent the average over all configu- rations of the melt.

For spatially homogeneous and isotropic systems, such as a polymer melt in the liquid phase forT⬎T_crys^hom,Sab共q兲only

0.4 0.6 0.8 1

0.85 0.9 0.95 1

v/v(T=1)

0.4 0.6 T 0.8 1

0.85 0.9 0.95 1

v/v(T=1)

T N=10 N=100

0.48 0.5

0 0.1

v(T=1)

1 / N 0.48

0.5

0 0.1

v(T=1)

1 / N

FIG. 2. Main figure: Phase diagram of CG-PVA, i.e., specific volumevversus temperatureT, for two chain lengths,N= 10共full line兲and 100共dotted line兲.共The volume is normalized by its value at T= 1.兲 Crystallization manifests itself by a drop of v共T兲. The phase transition is sharper forN= 10 than for N= 100. For short chainsN= 10, crystallization leads to the formation of a crystal of fully stretched chains共cf. lower snapshot兲. ForN= 100, on the other hand, a lamellarlike structure is obtained 共cf. upper snapshot—

crystalline domains are emphasized by the darker lines兲. Inset: De- pendence ofv共T兲on chain length forT= 1共fluid state兲. We find that the volume varies linearly with 1 /N; this chain length dependence may be associated with the chain ends共see, e.g.,关45兴兲.

depends on the modulus q of the wave vector. In order to compare liquid and crystalline phases we will also, for T⬍T_crys^hom, only determine structure factors that are spherically averaged over all wave vectors with the same modulusq共as in powder diffraction experiments兲.

S_ab共q兲can be split into intrachain and interchain contribu- tions

Sab共q兲=wab共q兲+␳chhab共q兲, 共4兲 where

␳ch= n

V 共5兲

is the chain density关V⫽volume of the共periodic兲simulation box兴. The intrachain contribution is given by

w_ab共q兲=1 n

兺

i=1 n

具exp关−iq·共r_i^a−r_i^b兲兴典, 共6兲 and the interchain contribution by

␳chhab共q兲=1 n

兺

⫽j n

具exp关−iq·共ri

a−r_j^b兲兴典. 共7兲 The static structure factor measured in scattering experi- ments is recovered by averaging over all monomer pairs 共a,b兲

S共q兲= 1 N

兺

a,b=1 N

Sab共q兲=w共q兲+␳mh共q兲, 共8兲 where␳m=N␳chis the monomer density and

w共q兲= 1 N

兺

a,b=1 N

wab共q兲, h共q兲= 1 N²

兺

a,b=1 N

hab共q兲. 共9兲 Here, w共q兲 is the intrachain structure factor 共form factor 关22兴兲 of a chain and h共q兲 is the Fourier transform of the site-averaged intermolecular pair-correlation function关23兴.

Commonly only the average structure factors S共q兲 and w共q兲 are considered. In the following, we also discuss site- resolved quantities, such asS_ab共q兲 and w_ab共q兲. One aim of our study is to understand to what extent specific monomer correlations deviate from the average behavior.

B. Technical details

The computation of structure factors from simulation re- quires a large number of共independent兲configurations in or- der to minimize statistical errors. This is particularly impor- tant for site-resolved quantities 关24兴 where averages along the chain cannot be carried out due to the explicit depen- dence on the monomer indices共a,b兲. ForT⬎T_crys^homwe aver- aged the structure factors over 360 configurations for N= 10 共the system consists of 288 chains兲, 400 configura- tions for N= 50共72 chains兲, and 1000 configurations forN

= 100 共192 chains兲. For T⬍T_crys^hom about 300 configurations were used for statistical averaging.

The structure factors were calculated as functions of modulusq of the wave vectorq. Only wave vectors compat-

ible with the periodic boundary conditions of the cubic simu- lation box were used in this calculation. For boxes of linear dimensionL this implies the following lower bounds on the q value: q_min= 0.56 共L= 11.31 for N= 10兲, q_min= 0.53 共L= 11.98 forN= 50兲, andq_min= 0.30共L= 20.89 forN= 100兲.

IV. STATIC PROPERTIES OF THE LIQUID AND CRYSTALLINE STATES

A. Average structure factors at high temperature:

An overview

Figure3compares the structure factorS共q兲with the form factor w共q兲 for three chain lengths, N= 10, 50, and 100, at T= 1. At this temperature,S共q兲displays features characteris- tic of the liquid state. For allNthe structure factor is small at low q, reflecting the weak compressibility ␬T of the melts.

With increasing q, S共q兲 increases toward the “amorphous halo,” the first maximum occurring at q^*, and then exhibits oscillations whose amplitude decreases and eventually con- verges to 1 asq→⬁. Figure3shows that this large-qbehav- ior is entirely intramolecular. Forqⲏ20,S共q兲agrees with the form factorw共q兲and so the intermolecular correlation func- tionh共q兲vanishes共see Fig.4兲. On the other hand, intermo- lecular correlations strongly contribute for smallerq, in par- ticular forqⱗq^*. The different importance of intrachain and interchain contributions above and belowq^*is a property of the CG-PVA polymer liquid, which bead-spring models with- outU_ang共␪兲do not necessarily exhibit共for q⬎q^*both intra- chain and interchain contributions are important关25兴兲. CG- PVA shares this property with atomistic models of polymer melts共see, e.g.,关26,27兴兲.

Further insight into the interchain structure of the CG- PVA model can be obtained by comparingh共q兲 with an ap- proximation motivated by the theory of simple liquids. Since the particles of a simple liquid have no internal structure, we havew共q兲= 1 and so␳mh共q兲=S共q兲− 1关23兴.

-1 0 1 2 3

0 10 20 30 40 50

S(q),w(q)

q N=10

N=50 N=100 q^∗ 1/R_g

-1 0 1 2 3

0 10 20 30 40 50

S(q),w(q)

q N=10

N=50 N=100 q^∗ 1/R_g

-1 0 1 2 3

0 10 20 30 40 50

S(q),w(q)

q N=10

N=50 N=100 q^∗ 1/R_g

-1 0 1 2 3

0 10 20 30 40 50

S(q),w(q)

q N=10

N=50 N=100 q^∗ 1/R_g

-1 0 1 2 3

0 10 20 30 40 50

S(q),w(q)

q N=10

N=50 N=100 q^∗ 1/R_g

-1 0 1 2 3

0 10 20 30 40 50

S(q),w(q)

q N=10

N=50 N=100 q^∗ 1/R_g

S(q) w(q)

FIG. 3. Structure factor S共q兲 共lines兲 and form factor w共q兲 共circles兲 versus q at T= 1 for N= 10, 50, and 100. The data for N= 10 and 100 are shifted for clarity共by −1 forN= 10, by +1 for N= 100兲. The vertical arrow indicates the value ofqassociated with the amorphous halo q^* and the vertical dashed lines the radii of gyration,R_g⯝1.08 forN= 10, R_g⯝3.11 forN= 50, andR_g⯝4.46 forN= 100. The position and the amplitude of the amorphous halo slightly depend onN:q^*⯝6.65,S共q^*兲⯝1.61 forN= 10;q^*⯝6.55, S共q^*兲⯝1.71 forN= 100.

Figure4compares␳mh共q兲toS共q兲− 1 to stress similarities and differences of our polymer melt compared to a simple liquid. We find that␳mh共q兲 and S共q兲− 1 closely agree with one another around the amorphous halo, i.e., for q=q^*

⬇6.6. This finding is not unreasonable. The first peak ofS共q兲 characterizes the packing of monomers in the melt. It is dominated by interchain contribution and should thus be similar to that of a simple liquid of monomers. Interestingly, the agreement between ␳mh共q兲 andS共q兲− 1 appears to im- prove as the chain length increases.

For qq^*, however, S共q兲 is fully determined by intra- chain effects, as pointed out before in the discussion of Fig.

3. Thus the simple-liquid approach, ␳mh共q兲=S共q兲− 1, must deviate from the simulation data asq increases beyond q^*. The approach must also break down in the opposite limit of small q due to a polymer-specific effect, the “correlation hole”关28兴. The term “correlation hole” means that the prob- ability of finding a monomer of another chain in the typical volume occupied by a particular chain is decreased. Thus the correlation between monomers of different chains drops with increasing distance. This drop largely compensates the in- crease ofw共q兲for smallqso that the sum of both intra- and interchain contributions yields a small finite value forS共q兲as

q→0, corresponding to weak compressibility of the melt.

Indeed, Fig.4reveals that␳mh共q兲⬇−w共q兲.

In the next sections we will address the intrachain and interchain correlation functions in more detail, focusing in particular on their site-dependence in the liquid and crystal- line phases of the melt. Such detailed information about the structure might prove important for the further development of density-functional approaches to polymer crystallization 关9,10兴.

B. Intrachain structure factors

Following Sec. III A site-resolved structure factors are the basic quantities from which averages, such asw共q兲, may be obtained. So we begin our discussion of the intrachain struc- ture by an analysis of the site-dependent form factor,w_ab共q兲. Figure 5 depicts wab共q兲 at T= 1 for N= 100 and various combinations of the monomer pair共a,b兲. We find thatwab共q兲 does not depend explicitly on the site index, but only on the number of bonds, 兩a−b兩, between the sites a and b. This feature is found for all chain lengths studied共N= 10, 50, and 100兲 and also for other bead-spring models 关24,29兴. It im- plies that the position of the monomer pair with respect to the chain end has a negligible influence on the spatial corre- lation between the sitesaandb.

For the further interpretation ofwab共q兲two limits are in- teresting, the limit of small and large contour length兩a−b兩. Spatial correlations of adjacent monomers along the chain 共兩a−b兩= 1兲are determined by the bond potential of the CG- PVA model 关Eq. 共1兲兴. Since this potential is rather stiff, it forces the bond length to remain close to the mean valueb₀. So we expectw_ab=a+1共q兲 to be given by

w_aa+1共q兲=

冓

冔

共Here and in the remainder of this section on intrachain prop- erties we suppress the chain index i on the monomer posi-

-2 -1.5 -1 -0.5 0 0.5 1

0 5 10 15 20

ρmh(q)

q ρmh(q)(N=10)

S(q)-1(N=10)

ρmh(q)(N=100)

S(q)-1(N=100)

-50 -40 -30 -20 -10 0

0 0.5 1 1.5 2

ρmh(q)

q 1/R_g(N=10) 1/R_g

(N=100)

N=10 N=50 N=100 -w(q) (a)

(b)

FIG. 4. Interchain structure factorh共q兲 versus q forT= 1 and variousN. The upper panel compares␳mh共q兲 共symbols兲forN= 10 and 100 to the “simple-liquid approximation”S共q兲− 1共lines兲. The lower panel magnifies the behavior of␳mh共q兲 共symbols兲for smallq.

Here, three chain lengths are shown,N= 10, 50, and 100. For allN the dashed lines represent the negative of the corresponding intra- chain structure factor, −w共q兲 共cf. Fig.3兲. The vertical dotted lines indicate theq-values corresponding to 1 /R_g共R_g⯝1.08 for N= 10 andR_g⯝4.46 forN= 100兲.

-0.2 0 0.2 0.4 0.6 0.8 1

0 2 4 6 8 10

wab(q)

q sin(qb₀)/(qb₀)

1/R_g

w₁₂(q) w_{25 26}(q) w_{50 51}(q) w_{1 11}(q) w_{25 35}(q) w_{50 60}(q) w^G_{1 11}(q)

FIG. 5. Site-resolved intrachain structure factorw_ab共q兲versusq forN= 100 andT= 1. The figure illustrates the behavior ofw_ab共q兲 by examples for the monomer pair共a,b=a+n兲 withn= 1 and 10.

For eachnthree pairs are shown, corresponding to the end mono- mer 共a= 1兲, the middle monomer 共a= 50兲, and a monomer in- between共a= 25兲. The dotted lines represent the Gaussian approxi- mation, Eq.共11兲. The nearest-neighbor approximation, Eq.共10兲, is also indicated; it is indistinguishable from the simulation data for w₁₂共q兲. The dashed vertical line marks theq-value associated with the radius of gyration共R_g⯝4.46兲.

tionsr_i^a, see Sec. III A.兲Indeed, Fig.5 reveals that Eq.共10兲 is in very good agreement with the simulation data. On the other hand, for large兩a−b兩microstructure effects, reflecting some stiffness of the chains on local scales, should no longer be dominant. We then expect that the result for ideal Gauss- ian chains关22兴,

wab^G共q兲= exp共−q²兩a−b兩ᐉ²/6兲, ᐉ²=6R_g²

N , 共11兲 represents a viable approximation for wab共q兲, provided length scales much larger than 1 /q^*are considered.

Figure 5 compares w_ab^G共q兲 to the simulatedw_ab共q兲. As a representative example for internal correlations with 兩a−b兩⬎1, the figure shows data for兩a−b兩= 10. We find that Eq.共11兲is in reasonable agreement with the simulation only forqⱗ1. For largerq, deviations occur. Whilew_ab^G共q兲decays monotonically, the simulation results display oscillations, similar to the nearest-neighbor correlations w_aa+1共q兲, but strongly damped in the present case. Since the oscillations of w_aa+1共q兲result from the rodlike behavior of a bond, i.e., from the “rigidity” of the chain on this length scale, we may speculate that the behavior of wab共q兲 for large contour length, which is not captured by the Gaussian model, is re- lated to the intrinsic stiffness of the CG-PVA chains. We pursue this idea in the following section.

1. Koyama distribution and w_ab„q…

The bond angle potential共Fig. 1兲 confers a semiflexible character to the CG-PVA model. Inspired by work on similar models 关30,31兴 we thus attempt to describe w_ab共q兲 by the Koyama distribution关27,32,33兴

wab^K共q兲=sinBabq

Babq exp共−A_ab² q²兲, 共12兲 where共rab=兩r^a−r^b兩兲

A_ab² =1 −C_ab

6 具r_ab²典, B_ab² =C_ab具r_ab²典,

C_ab² =1

2

冋

册

Originally, the Koyama distribution was proposed as an ap- proximation to the distribution of internal distancesrab for the wormlike chain model 关32,33兴. As noted by Mansfield 关33兴, the Koyama distribution is not limited to wormlike chains, but can be applied to any semiflexible chain model for which the second and fourth moments ofrab are known 共providedC_ab² ⱖ0兲.

We determined these moments and compared Eq.共12兲to our simulation data. Figure 6 illustrates the results of this comparison forN= 100 andT= 1. Clearly, the Koyama dis- tribution represents a substantial improvement over Eq.共11兲;

it provides a very good description ofw_ab共q兲for all兩a−b兩in the studiedqrange. Small deviations were only observed for N= 10 in the case of the most remote monomers, i.e., for 兩a−b兩→N− 1. This suggests that the intrachain structure of

CG-PVA in the melt can be accurately predicted by using only the first two even moments,具r_ab² 典and具r_ab⁴ 典, of the dis- tance distribution between monomers.

Since具rab

4 典is always larger than or equal to具rab

2典², the key parameterC_abof the Koyama distribution obeys the inequal- ity C_ab² ⱕ1. The equality holds if 具rab

4典=具rab

2典². This is the case for rodlike behavior whererab=兩a−b兩b₀. Then, we re- cover Eq.共10兲from Eqs.共12兲 and共13兲for b=a+ 1. On the other hand, Cab vanishes if the internal distances rab are Gaussian distributed. Therefore Cab measures deviations from Gaussian behavior, and we may refer to it as a non- Gaussian parameter 关23兴. If Cab vanishes, Eq. 共12兲 gives back Eq. 共11兲 provided we write 具rab2 典=兩a−b兩ᐉ². The Koyama distribution thus interpolates between the expected small-scale关Eq.共10兲兴and large-scale behaviors关Eq.共11兲兴of our model.

The dependence of C_ab² and 具rab

2 典 on the contour length 兩a−b兩 is presented in Fig.7 for N= 100 and T= 1. With in- creasing 兩a−b兩, C_ab² continuously decreases and the mean- square internal distance tends towards 具r_ab² 典⬀兩a−b兩b₀². De- viations from Gaussian behavior thus diminish as larger contour lengths are considered. However, for N= 100, they do not fully vanish, even if兩a−b兩=N− 1.

In the present case, these deviations can be interpreted as a microstructure effect. By invoking the Flory ideality hy- pothesis关18,28兴we can assume that intrachain excluded vol- ume forces are screened in the CG-PVA melt. It should thus be possible to calculate the intrachain structure from an

“ideal” single chain model 关18兴. Here, a generalized freely rotating chain 共GFRC兲model appears to be appropriate be- cause it resembles our CG-PVA model in the following re- spect 关34兴: its conformational properties are determined by the bond angle␪ which fluctuates according to some distri- bution P共␪兲. For the GFRC model, the second and fourth moments depend only on具cos␪典and具cos²␪典 共see Appendix B兲. We determined these averages from the simulation and calculatedC_ab² and 具r_ab² 典 according to Eq. 共13兲 and the for- mulas of Appendix B. The results of this calculation are in- cluded in Fig. 7 共solid lines兲. In fact, good agreement be-

-0.2 0 0.2 0.4 0.6 0.8 1

0 5 10 15 20

wab(q)

q 1/R_g

w₁₂(q) w₁₃(q) w₁₄(q) w₁₅(q) w^K_ab(q)

FIG. 6. Comparison of site-resolved structure factors w_ab共q兲 共lines兲with the Koyama approximation, Eqs.共12兲and共13兲 共circles兲. The examples shown are representative for all monomer pairs, al- though they always involve the end monomer共a= 1兲. The simula- tion data refer to N= 100 and T= 1. The vertical dashed line indicates the q-value associated with the radius of gyration 共R_g⯝4.46兲.

tween the GFRC predictions and the simulation data is found 关35兴.

2. Intrachain properties at lower temperature

The preceding discussion was concerned with tempera- tures much larger than the onset temperature of homoge- neous crystallizationT_crys^hom共=0.78兲. In this section, we focus on temperatures closer to and belowT_crys^hom.

共a兲 Non-Gaussian parameter Cab. Figure 8 depicts the non-Gaussian parameterC_ab² for variousNand illustrates its temperature dependence. As expected, deviations from Gaus- sianity are stronger for shorter chains—compare the results for N= 50 and N= 100 at T= 1—and become more pro- nounced on cooling toward T_crys^hom. However, the features of C_ab² discussed in the previous section, in particular the good agreement between the GFRC prediction and the simulation data, remain the same so long asT⬎T_crys^hom.

This state of affairs changes when the melt crystallizes.

Previous work关14–16兴revealed that CG-PVA chains shorter thanN⬇50 fully extend on crystallization, whereas longer chains fold back on themselves, forming lamellarlike crystal- line structures that coexist with amorphous regions共cf. Fig.

2兲. For N= 10 we thus expect and find—see inset of Fig.

8—that C_ab² exhibits rodlike behavior, that is,C_ab² = 1 for all 兩a−b兩. ForN= 100, on the other hand, the situation is differ- ent. C_ab² is negative for 0.38ⱗ兩a−b兩/Nⱗ0.88. A negative value of C_ab² implies that internal distances larger than ex- pected from ideal chains occur so that 具rab

4 典 exceeds the Gaussian result 5具rab

2 典²/ 3. This finding is not unreasonable.

Large internal distances can result from stretched chain sec- tions which participate in the crystalline lamellae关36兴.

共b兲Form factors.From Fig. 8 we may conclude that the form factorw共q兲forN= 100 in the crystalline state cannot be described by the Koyama approximation because C_ab² is negative. The approximation could work, however, for

N= 50 at T⬎T_crys^hom where C_ab² ⬎0. This is indeed the case.

Figure9compares simulation results forw共q兲in theTinter- val 0.8ⱕTⱕ1 with

w^K共q兲= 1 N

兺

a,b=1 N

wab^K共q兲, 共14兲 where we used Eq.共9兲andwab^K共q兲is given by Eqs.共12兲and 共13兲. We find thatw^K共q兲provides a good description ofw共q兲 forT⬎T_crys^hom.

In the crystalline state we can apply the Koyama approxi- mation only to those cases for whichC_ab² ⬎0. This condition is satisfied for short chains共N= 10兲 which form a crystal of rigid rods upon cooling. For rods, the Koyama approxima- tion becomes exact, and the form factor reads

0 0.2 0.4 0.6 0.8 1

0 20 40 60 80 100

0 1 2 3 4 5 6

C2 ab 〈r2 ab〉/(|a-b|b02)

|a-b|

〈r²ab〉/(|a-b| b0 2)

C²_ab 0

0.2 0.4 0.6 0.8 1

0 20 40 60 80 100

0 1 2 3 4 5 6

C2 ab 〈r2 ab〉/(|a-b|b02)

|a-b|

〈r²ab〉/(|a-b| b0 2)

C²_ab 0

0.2 0.4 0.6 0.8 1

0 20 40 60 80 100

0 1 2 3 4 5 6

C2 ab 〈r2 ab〉/(|a-b|b02)

|a-b|

〈r²ab〉/(|a-b| b0 2)

C²_ab 0

0.2 0.4 0.6 0.8 1

0 20 40 60 80 100

0 1 2 3 4 5 6

C2 ab 〈r2 ab〉/(|a-b|b02)

|a-b|

〈r²ab〉/(|a-b| b0 2)

C²_ab

Simulation Theory

FIG. 7. Non-Gaussian parameter C_ab² 关Eq. 共13兲兴 and mean- square internal distance 具r_ab²典 versus contour length 兩a−b兩. The simulation data forN= 100 and T= 1 are shown by circles. Since C_ab² and具r_ab²典depend only on兩a−b兩, the figure shows results from a gliding average over all monomers a separated by 兩a−b兩 from monomerb共i.e.,b=a+兩a−b兩兲.具r_ab²典 is divided by兩a−b兩b₀², where b₀= 1 / 2 is the average bond length, because Gaussian behavior, i.e., 具r_ab²典=C_⬁兩a−b兩b₀²共C_⬁ is the characteristic ratio关18兴兲, is expected for 兩a−b兩→N in the large-N limit. The solid lines indicate the theoretical results for C_ab² and 具r_ab²典 from the generalized freely rotating chain model共cf. Appendix B兲.

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

C2 ab

|a-b|/N T=1.0 T=0.8

T=1.0 0

0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

C2 ab

|a-b|/N T=1.0 T=0.8

T=1.0 0

0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

C2 ab

|a-b|/N T=1.0 T=0.8

T=1.0 0

0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

C2 ab

|a-b|/N T=1.0 T=0.8

T=1.0 0

0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

C2 ab

|a-b|/N T=1.0 T=0.8

T=1.0 0

0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

C2 ab

|a-b|/N T=1.0 T=0.8

T=1.0 0

0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

C2 ab

|a-b|/N T=1.0 T=0.8

T=1.0 0

0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

C2 ab

|a-b|/N T=1.0 T=0.8

T=1.0 N=100

N=50

0 0.5 1

0 0.5 1

T=0.4

0 0.5 1

0 0.5 1

T=0.4

0 0.5 1

0 0.5 1

T=0.4

0 0.5 1

0 0.5 1

T=0.4 Rod N=10 N=100

FIG. 8. Main figure: Non-Gaussian parameter C_ab² versus 兩a−b兩/NforN= 100 atT= 1 and forN= 50 atT= 0.8, 0.9, and 1. All temperatures are above the onset temperature of homogeneous crys- tallizationT_crys^hom= 0.78. As in Fig.7, the data shown represent gliding averages along the chain over all monomersawhich are separated by a contour length兩a−b兩from monomerb共i.e.,b=a+兩a−b兩兲. In- set:C_ab² versus兩a−b兩/NforN= 10 and 100 in the semicrystalline phase atT= 0.4. Perfect rodlike behavior,C_ab² = 1, is indicated by a horizontal solid line.

0.1 1 10 100

0 5 10 15 20 25

w(q)

q T=1.0

T=0.8 0.1

1 10 100

0 5 10 15 20 25

w(q)

q T=1.0

T=0.8 0.1

1 10 100

0 5 10 15 20 25

w(q)

q T=1.0

T=0.8 0.1

1 10 100

0 5 10 15 20 25

w(q)

q T=1.0

T=0.8 0.1

1 10 100

0 5 10 15 20 25

w(q)

q T=1.0

T=0.8 0.1

1 10 100

0 5 10 15 20 25

w(q)

q T=1.0

T=0.8 w(q) w^K(q)

0.6 0.8 1 1.2 1.4 1.6

5 10 15 20

0.6 0.8 1 1.2 1.4 1.6

5 10 15 20

T=1.0 T=0.8

FIG. 9. Form factorw共q兲 共lines兲for N= 50 chains at different temperatures, T= 0.8, 0.9, and 1, above the onset temperature of homogeneous nucleation, T_crys^hom= 0.78. Data for T⬍1 have been shifted downward by some factor for clarity关w共q兲/ 2 for T= 0.9, w共q兲/ 4 forT= 0.8兴. The circles represent the Koyama approxima- tion, Eq.共14兲. Inset: Comparison of the simulation results forw共q兲 atT= 1.0 and 0.8. The slight shift of the peak atq⬇15 toward lower q-values reflects the increase of chain stiffness upon cooling.