Theory of Crystals and Interfaces in Polyethylene and Isotactic

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Theory of Crystals and Interfaces in Polyethylene and Isotactic Polypropylene Jianguang Feng, Hongfu Zhou, Xiangdong Wang, and Jianguo Mi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b00376 • Publication Date (Web): 04 Apr 2016 Downloaded from http://pubs.acs.org on April 9, 2016

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The Journal of Physical Chemistry

Theory of Crystals and Interfaces in Polyethylene and Isotactic Polypropylene Jianguang Fenga,b, Hongfu zhoub, Xiangdong Wangb*, and Jianguo Mi a*

a

State Key Laboratory of Organic-Inorganic Composites, Beijing University of Chemical Technology, Beijing, China b

School of Materials and Mechanical Engineering, Beijing Technology and Business University, Beijing, China

We present a three-dimensional density functional method to investigate the crystallization thermodynamics of polyethylene and isotactic polypropylene. Within the theoretical framework, the effects of excluded volume, dispersive attraction, chain connectivity, and conformation are integrated into the nonlocal free-energy functional. Under the condition of crystal–melt phase equilibrium, the crystal lattice parameters as a function of temperature are firstly determined through the restricted and full free-energy minimizations of crystal cell. The interfacial density profiles and interfacial tensions are then calculated via the restricted and full free-energy minimizations of interfaces. Accordingly, the free-energy barriers and critical sizes during the formation of crystal nuclei are predicted to evaluate the structures and properties of such crystallites. Some results are in good agreement with the available experimental data, indicating that the present theoretical approach could be a good candidate for understanding the molecular mechanism of polymer crystallization.

*

Corresponding author: [email protected], [email protected]

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1. Introduction

It is known that the structures of crystal and interface provide an important key factor in regulating the properties and functionalities of polymeric materials.1-3 The microstructure of polymer crystal can be effectively modulated by controlling the process of nucleation and growth, which correlate to the interfacial thermodynamic and kinetic properties.4-6 Thus, higher levels of control over polymer crystallization cannot be achieved without understanding the fundamentals of nucleation. Up to now, it is still an open issue to understand microscopically the polymer crystal nucleation with the intermonomer interactions and the thermodynamic boundary conditions as the inputs. The essential difficulty of studying nucleation arises from the fact that the critical nucleus size typically fall in the range of dozen to a few dozen nanometers, which is hardly accessible to most of the current experimental methods.7 Moreover, it is difficult to study the relations of the pure crystalline and amorphous regions due to the close proximity and interpenetration of different phase. In this regard, a microscopic theory of freezing is undauntedly necessary. Since the 1950s one of the main goals of polymer crystallization theory has been to understand the mechanism of crystal–melt phase transition and nucleation process. The well-known methodology of classical researches is the lattice-based theory.8 The theory considers polymer chains with hard-sphere sites and represents the polymer on a regular lattice in order to describe the phase behavior of the lattice model polymer by a simple mean-field theory. The theory suggests that polymer inflexibility and entropic packing forces are main factors to influence the amorphous-to-crystalline phase transition. With the accumulation of experimental data and the advent of new experimental techniques, the weakness of the standard model is coming to be exposed. Since the fine architecture of polymer chains has been overlooked, it is therefore difficult to accurately predict a density jump at the phase transition when the fraction of lattice sites occupied by the polymer chains is set equal to unity.


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Phase field crystal (PFC) model has been extended to polymer crystallization.9-12 The model was originally proposed for small molecule systems, where an asymmetric local free-energy density has been adopted to represent the metastable melt and the stable crystal. To capture various metastable polymer crystals, the order parameter in PFC at the solidification potential has been treated to be supercooling dependent such that it can assume an intermediate value, reflecting imperfect polycrystalline nature of polymer crystals. However, the stability solution of the single phase field equation yields a square or a circular interface of the crystal. Classical Density functional theory (DFT) is another candidate to study the structure and thermodynamic properties of inhomogeneous systems. Starting from the interparticle potential and the fluid correlations, DFT is able to predict the crystallization transition and the full crystal structure. Polymer is non-rigid structure in which bond constraints need approximately fix distances and angles between neighboring monomers. The developments of the classical DFT are conducive to describe the structure and properties for polymer crystallization. Recently, PFC combined with DFT has been developed and applied to deal with the thermodynamics properties of soft condensed matter in a high efficient way. The connection between DFT and PFC has been examined and indicated that the combined approach is an ideal tool to study nucleation and growth process.13-15 In above DFT and PFC approaches, the inhomogeneous free-energy functional is unexceptionally derived from the direct correlation function with a second-order expansion form. It is well known that such free-energy functional is unable to correctly describe high density systems, especially for highly ordered polymer crystal. The deviation is mainly attributed to the inaccurate direct correlation function for hard-sphere repulsion. To solve this problem, the fundamental measure theory (FMT) was specially proposed for hard-sphere mixture,16 and there have been a few modified versions of FMT since it was proposed.17-20 The FMT not only succeeds in describing the properties of hard-sphere fluid and solid, and ACS Paragon Plus Environment


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thereby acting as a suitable reference for systems with additional soft repulsive or attractive interactions, but also gives insight into problems of statistical physics of excluded volume from a more fundamental point of view.21,22 Investigations in recent years in soft condensed matter have given unambiguous evidence that DFT integrated with FMT is a versatile and powerful tool to study the microstructure and properties of inhomogeneous systems.23-27 Very recently, we have used the DFT to study the crystal−liquid phase transitions and interfacial properties of Lennard−Jones (LJ) fluid,28 carbon dioxide,29 and water.30 Good agreements of theoretical results with corresponding experimental or computational data are quite remarkable, indicating that the DFT is a powerful methodology for studying crystallization mechanism on a microscopic level. Greatly desire is the invention of a new standard model for polymer crystallization by taking more realistic molecular pathways into account. In this work, we present a density functional method to study the crystal–melt phase transitions, the structure and properties of crystallites and interfaces, and the nucleation attributes of polyethylene (PE) and isotactic polypropylene (iPP). PE is important because it is the simplest polymer that forms a crystalline structure, and it serves as a model system for testing our understanding of polymer crystallization in general. iPP has also been frequently chosen as model system since its wide applications as conventional plastics. In the theoretical model, a modified FMT is employed to describe the free-energy functional of hard-sphere repulsion, a weighted free-energy functional is integrated to represent dispersive attraction, a direct correlation function derived from the polymer reference interaction site model (PRISM) integral equation is applied to construct the free-energy functional arising from polymer configuration, and the contribution of chain connectivity is concerned in the limit of infinitely strong association. With the universal force field parameters, the crystal densities at crystal−melt equilibria, the density profiles in crystal cell and at interfaces, nucleation free-energy barriers, and critical sizes of nuclei are quantitatively predicted by minimalizing the free-energy of crystals and interfaces. These thermodynamic properties are expected to ACS Paragon Plus Environment

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provide a solid basis to understand the complex structures of crystal and interface during the nucleation process.

2. Theory and Equations We consider a polymer molecule in consist of M monomers bonded together to form chains, where each monomer contains N interaction sites. In general, the grand potential can be expressed as the following form Ω[{ρ m , n (r )}] = F [{ ρ m , n (r )}] − ∑ ∑ ∫ [ µ m , n ρ m , n (r )]dr m

∀m = 1, M ; n = 1, N

(1)

n

where ρ m,n (r ) is the density of the nth site in the mth monomer. F [{ ρ m ,n ( r )}] is the Helmholtz free-energy, µ m , n is the chemical potential for the nth site in the mth monomer in the bulk state. Generally, without an external potential, the density distribution of a homogeneous melt is regarded as a constant. The crystal, however, should be viewed as a self-sustained inhomogeneous system, even in absence of any external potential. The initial density distribution in a crystal cell can be constructed through a sum Gaussians at each lattice site 3

2 −ν R −r  ν 2 ρm,n (r ) =   ∑ e m ,n  π  {Rm,n }

(2)

where ν is a measure of the width of Gaussian peaks, the vectors of lattice sites are R m ,n with

R m ,n = h1a1 + h2a 2 + h3a3 for integer h1 , h2 , h3 , and a1 , a 2 , a3 are primitive translation vectors of the real-space crystal lattice. The value of ν and the lattice parameters involved in a1 , a 2 , and a3 are determined via a restricted free-energy minimization. The free-energy in eq. 1 includes the ideal and excess contributions. In presenting the different contributions of chain chemistry and geometry, we have

F [{ρ m,n (r )}] = ∑∑ k BT ∫ drρ m,n (r )[ln( ρ m ,n (r )) − 1] + F ex [{ρ m,n (r )}] m

n

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(3)


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F ex [{ρ m ,n (r )}] = F hs [{ρ m ,n (r )}] + F att [{ρ m ,n (r )}] + F chain [{ρ m ,n (r )}] + F stiff [{ρ m ,n (r )}]

(4)

where k B is the Boltzmann constant, T is the absolute temperature, F hs [{ ρ m ,n ( r )}] , F att [{ρ m ,n (r )}] , F chain [{ρ m , n ( r )}] , and F stiff [{ ρ m , n ( r )}] correlate to the effects of hard-sphere repulsion, dispersive

attraction, chain connectivity, and chain conformation, respectively. For the contribution of hard-sphere repulsion, the fundamental measure theory16 is postulated to have the form

β F hs [{ρ m ,n (r )}] = ∫ Φ hs [nτ ( r )]dr

(5)

with β = 1 / k BT . To adequately describe the crystal, the tensor version of White Bear II18 for hard-spheres system is applied to modify the free-energy density 3 ( −n2nV 2 ⋅ nV 2 + nV 2 ⋅ nT ⋅ nV 2 − tr(nT ) + n2 tr(nT ) ) n1n2 − nV 1 ⋅ nV 2 (6) ϕ1 ( n3 ) + ϕ2 ( n3 ) 2 1 − n3 16π (1 − n3 ) 3

Φ hs  nτ ( r )  = −n0 ln (1 − n3 ) +

2

where the definitions of n0 , n1 , n2 , n3 , ϕ 1 ( n 3 ) , ϕ 2 ( n3 ) , nV 1 , n V 2 , and n T can be seen elsewhere.18,31 The free-energy density for the dispersive attraction can be expressed with the weighted density approximation form32

β F att [{ρ m ,n ( r )}] = ∑ ∑ ∫ ρ m ,n (r ) {a1 {ρ m ,n ( r )} + a 2 {ρ m ,n ( r )}}d r m

(7)

n

where a1 {ρm,n (r)} and a2 {ρm,n (r)} are given by the first-order mean spherical approximation33   1 + z1, R   1 + z2 R   z R z2 R a1 {ρ m,n (r )} = −2πρ m,n (r ) β ∑∑∑∑ xm,n xm ',n 'ε  k1  G0, ( z1 )e 1, −  − k2  G0 ( z2 )e −  2 z1  z22   m m' n n'   

(8)

+8πρ m,n (r ) β ∑∑∑∑ xm,n xm ',n 'ε R I ∞ − 8πρ m,n (r ) β ∑∑∑∑ xm,n xm ',n 'ε g 0 ( R) R I1 3

m

m'

n'

n

3

m

m'

n

n'

a2 {ρ m ,n (r )} = −πρ m ,n (r ) β ∑∑∑∑ xm ,n xm ',n 'ε  k1 ( G1, ( z1 ) e z1R ) − k 2 ( G2 ( z2 )e z2 R )  m m' n n' −4πρ m ,n (r ) β ∑∑∑∑ xm ,n xm ',n 'ε g1 ( R ) R 3 I1 m

m'

n

(9)

n'

where xm ,n is the mole fraction, ε , R , k1 , k 2 , I1 , I ∞ , G0 , and G0 are the interaction parameters between the two sites ( m, n ) and ( m ', n ') , and their detailed definitions can been seen in Ref. 33. ρ m,n (r ) is the weighted density, which is defined as ACS Paragon Plus Environment

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ρ m ,n (r ) = ∑ ∑ ∫ ρ m′,n ' (r')ω(attm ,n ),( m′,n ') ( r − r' ) dr' m'

(10)

n'

with ω(attm ,n ),( m′,n ') ( r ) = u(attm ,n ),( m′,n ') ( r ) / ∫ u(attm ,n ),( m′,n ') ( r )dr , and u(attm ,n ),( m ′,n ') ( r ) is the attractive part of LJ potential between site n of monomer m and site n ' of monomer m ' with a distance of r( m, n ), ( m′,n ') . The polymer chain can be treated as a sequence of tangentially bonded sites enforced by giving each site a label and allowing the sites to exclusively bond to their specific matching sites. As a result, the free-energy functional induced from chain connectivity is given by21,34

β F chain [{ρm,n (r)}] =

   δ ( r′ − r′′ − σ (m,n),(m′,n ') )  1 ′ ′ ′′ ′′ 1 ln ({ }) d r r − d r y r ρ ξ ρ ( ) ( )     (11) ∑∑ ∑ ′ m , n ( m , n ),( m , n ') χ m , n ∫ 2∫ 4πσ 2(m,n),(m′,n ') m n A∈Γ( m,n )     

where Γ( m,n ) means the set of all the associating points on the site n in monomer m . ( m ', n ') is the neighboring site which is connected to site ( m, n ) . y(m,n), (m′,n ') ({ξχ }) is the cavity correlation function, and

σ(m,n),(m′,n') = (σ(m,n) + σ(m',n') ) / 2 is the interaction diameter. The cavity correlation function is expressed as a set of weighed nonlocal densities ξ χ .34 A flexible chain is an ideal freely jointed chain with random bond angles and identical bond lengths, whereas a semiflexible chain is regarded as a continuous space curve with a local energy density associated with bending. For a semiflexible polymer chain, due to the effect of bending potential, it has a larger size than the flexible one with the same degree of polymerization. As the chain becomes stiffer, it tends to pack more order, which results in a larger loss of configurational entropy. Therefore, the free-energy functional related to the polymer configurational entropy is positive, and can be written as35

β F stiff [{ρ m ,n (r )}] = −

1 ∑ ∑ ∑ ∑ dr ′ drρ m,n (r ) ρ m ',n ' (r ′)c(stiffm,n ),( m′,n ') ( r′-r ) 2 m m' n n' ∫ ∫

(12)

semiflexible flexible with the approximation c(stiff . Here c(semiflexible and c(flexible m , n ),( m ′ , n ') ( r ) = c ( m , n ),( m ′ , n ') ( r ) − c ( m , n ),( m ′ , n ') ( r ) m , n ),( m ′ , n ') ( r ) m , n ),( m ′ , n ') ( r )

stand for the direct correlation functions of semiflexible and flexible polymer chains, respectively. According to the configurational entropy analysis,

F stiff [{ρ m ,n (r )}] should be positive. c(semiflexible and m , n ),( m ′ , n ') ( r )

are obtained through the PRISM integral equation c(flexible m , n ),( m ′ , n ') ( r )

36



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r r r r r r h(r) = ∫ dr ' ∫ dr '' ω(| r − r ' |)c(| r '− r '' |)[ω(r '') + ρh(r '')]

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(13)

where h (r ) , ω( r ) , and c ( r ) are M × N matrices, and the elements are h( m,n ), ( m′,n ') (r ) , ω( m,n),( m′,n ') (r ) , and c( m,n ),( m′,n ') ( r ) , respectively. h( m,n ),( m′, n ') (r ) is the total correlation function, ω( m,n ),( m′,n ') (r ) is the intramolecular correlation function, which can be represented by Koyama model for flexible or semiflexible chains.37,38 The Kovalenko–Hirata approximation39 is adopted to solve the equation. An essential task of DFT is to determine ρ m,n (r ) in a crystal cell through the full minimization of the grand potential, or equivalently the intrinsic Helmholtz free-energy density 

ρ m ,n (r ) = exp  βµm ,n − β 

δ F ex [{ρ m ,n (r )}]   δρ m ,n (r ) 

(14)

The process of free-energy minimization for polymer crystal can be performed as follows: (1) Guess the crystal cell parameters, determine the value of ν in eq. 2 through a restricted minimization of the free-energy functional described in eq. 3. (2) Use ν to determine the new crystal lattice parameters via another restricted minimization of the free-energy functional. (3) Repeat the iteration procedure until the average differences between the old and new cell parameters are less than 10−4 . (4) The finally crystal profile ρ m ,n (r ) is then determined via a full free-energy minimization with eq.14. During the full minimization iteration, we use δ Fmex,n (r ) / δρ m ,n (r ) instead of ρ m,n (r ) to ensure the convergence for solving the equation. The iteration is repeated until the average fractional difference over any three-dimensional grid point between the old and new δ Fmex,n (r ) / δρ m ,n (r ) is less than 10-3. In the numerical calculation, the system is discretized into 256 × 256 × 256 grids with a cuboid box of lengths, Lx , L y , and Lz in all three directions. The widths Lx , L y , and Lz are chosen to contain one unit crystal cell. Fast Fourier transform is applied to compute the convolution of the free-energy functional, which enables a high-efficient algorithm in three-dimensional space. After the determination of ρ m,n (r ) , the crystal−melt phase equilibrium can be calculated by the grand canonical method40 ACS Paragon Plus Environment

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 F ( ρ + t ) + Fi ( ρ − t )  Fi +1 ( ρ ) = min  i  0< t < t max 2  

(15)

where Fi ( ρ ) denotes the free-energy density after ith transforms,

ρ

is the average density of melt or

crystal, tmax corresponds to min( ρ max − ρ , ρ ) , and ρ max is the maximum of density allowed by the given free-energy functional. The chemical potential is related to the free-energy through µ = ∂F ( ρ ) / ∂ρ . In the grand canonical method, the process is designed following the volume combination step by step. The combination Fi +1 ( ρ ) is developed from Fi ( ρ ) by taking into account density variation. The chosen criteria for Fi +1 ( ρ ) is to find the global minimum of Fi ( ρ ) . This transform needs to be performed successively until volume size is infinite, at which Fi +1 ( ρ ) is stationary. After ith transforms, the phase equilibrium can be achieved at ∆µ = 0 , here ∆µ is the chemical potential difference between two phases. At equilibrium state, the initial density distribution ρ m ,n (r ) in a crystal−melt interface is assumed as the following form   

ρ m ,n (r ) = ρ m ,n +  (ν / π )

3/2

∑e

{R m ,n }

−ν R m , n − r

2

 − ρ m ,n  Θ ( z )  

(16)

where ρ m ,n is the melt density at equilibrium state, Θ ( z ) represents the interface varying from crystal to melt, and can be written as  1, z < z0  z − z0  1  ) , z0 ≤ z ≤ zR Θ R ( z ) =  1 + cos(π ∆zR  2   0, z > zR 

(17)

∆zR = zR − z0 = ( R1 / R )λ ∆z , 0 ≤ λ ≤ 1, R1 ≤ R

(18)

with

where R1 is the magnitude of the smallest nonzero lattice vectors, ∆ z is the width of the interface, and the parameter λ controls the rate of broadening of the solid density peaks. ∆ z and λ are determined via the restricted free-energy minimization method. The free-energy functional is full-minimized with eq. 14 to



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determine the final interfacial density distribution.41 In the numerical calculation, the system is discretized into 256 × 256 × 512 grids to include the integrate interface. After minimization of the grand potential, the interfacial tension can be calculated by

γ = ( Ω − Ωb ) / A = ( F [{ρ m ,n ( r )}] − ∑ ∑ ∫ µ m ,n ρ m ,n (r )dr + pV ) / A m

where Ωb = − pV , Ωb is the grand potential of melt, region, and

(19)

n

p is the pressure, V is the volume of the interface

A is the interface area.

For polymers, homogeneous nucleation can be as a cylinder against two surfaces including side surface and end surface.42 For this reason, a cylinder model is assumed to describe the shape of crystallites in polymer systems. According to the classic nucleation theory, the free-energy difference for nucleation between homogenous melt and cluster can be written as

∆Ω = 2π R 2γ e + 2π RH γ s − π R 2 H ( f cell − f liq )

(20)

where γ s and γ e are the interfacial tensions under the condition of solute supersaturation for the side and end surfaces of a cylindrical nucleus of radius R and length

H , f cell is the free-energy density of bulk

crystal cell， f liq is the free-energy density under the condition of solute supersaturation. During the nucleation, the free-energy barrier can be turned into an unconstrained saddle-point problem. The lowest maximum of free-energy as a function of

R and H must be crossed to get to larger R and H . Such

saddle point corresponds to critical radius Rc = 2γ s ( f cell − f liq )

(21)

and critical height H c = 4γ e ( f cell − f liq )

(22)

The free energy at the saddle point which corresponds to the nucleation barrier is thus

∆Ωc = 8πγ s2γ e / ( fcell − f liq ) 2

(23)


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In the theoretical calculations, the force field parameters for PE and iPP, including bond length, bond angle, and nonbonded LJ parameters, are taken directly from the universal TraPPE-UA model.43, 44 In order to consider the overlap of different sites in real polymer, the van der Waals volume is calculated to represent the sphere packing, where the packing fraction is equal to the effective packing fraction of actual polymer chain. In this case, the integrity of the lattice structure in crystal phase can be protected. In this study, there are 100 monomers in each chain in PE or iPP. The molecular weights of the PE and iPP are about 3000g/mol and 4000g/mol, respectively. Application of DFT demands a selection of crystal lattice type as the prerequisite to describe the structures and free-energies of crystal and interface. The PE crystal stabilizes in a simple base-centered orthorhombic crystal structure as observed by X-ray and neutron scattering experiments,45 whereas iPP has several structural polymorphism, namely, α (monoclinic), β (hexagonal) and

γ (triclinic) forms.

Compared with two other structures, the α form is by far the most common, being found in normal crystallization from solution or melt. In consideration of the evaluation of small free-energy differences among the three types, we focus on the monoclinic crystal structure. For a selected unit cell, it is necessary to determine the identity period along the axes and interaxial angles. Due to the bonding constraint, the lattice parameter c along the chain back bone is fixed based on the TraPPE-UA force field, whereas the lattice parameters a and b is independent of the force field. Meanwhile, the interaxial angles are determined according to the defined crystal type. As such, the geometry of PE crystal cell is c = 2.58 Å , θ x = θ y = θ z = 90 ° , and the angle of the projection of the chain backbone in the a - b plane with the b axis is 45°; whereas the configuration of iPP cell is c = 6.50 Å ，θ x = θ z = 90° and θ y = 90 °20′ . In a unit cell, the number of particles should be satisfied with Ncell = ρcrystalVcell , where the average density is

ρ crystal = ∫

Vcell

0

ρ (r )dr / Vcell , and Vcell is calculated from the lattice lengths and angles. For PE crystal, the ACS Paragon Plus Environment


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volume is Vcell = a ⋅ b ⋅ c . While for iPP crystal, the volume is Vcell = a ⋅ b ⋅ c ⋅ sinθ y . In a perfect unit cell, the number of particles is a constant. Thus, for a given average density, the volume is fixed. Therefore, only a and b need to be determined with respect to the restricted and full free-energy minimization. Figure 1 presents the temperature-dependent crystal lattice parameters for PE and iPP based on the experimental densities. As the temperature increases, the calculated lattice parameters for PE (Figure 1a) and iPP (Figure 1b) increase and differ from the experiment values46, 47 by no more than about 5%. In order to maintain the integrity of the lattice structure, the positions of individual sites are slightly changed, which corresponds to the variation of the lattice parameters. The interaction energy between neighboring chains mainly determines the packing mode. Thus, the variation can be attributed to the interactions of polymer chains. It is now generally accepted that the variation must in some way depend on the interchain distances, which would affect the degree of interactions. In addition, the nonlinear nature of the expansion of both a and b dimensions are easily apparent, and therefore it is possible to obtain the values of the thermal expansion coefficients of a and b axes. We find that that the optimized ratio of b: a is almost unaltered as the temperature changes. As a consequence, the optimal ratios for PE and iPP are 0.63 and 3.32, respectively. Such optimal values are set as constants for the two polymer crystals in the subsequent calculations.

3. Results and Discussion We first evaluate the capability of the current theoretical model to predict the crystal–melt phase transitions. Figure 2 shows the averaged amorphous and crystalline densities of PE (Figure 2a) and iPP (Figure 2b) under different equilibrium states and compares the predicted curves with the corresponding experimental data.48, 49 As evident from the theoretical results, the agreements of the phase-coexistences with previous experiments are very good with errors about 5%. The comparison indicates that the current ACS Paragon Plus Environment

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theoretical model is possible to describe the specific intermolecular interactions and microscopic structure, and thereby to predict the crystal–melt phase transitions of PE and iPP with reasonable accuracy. The accurate prediction of phase transitions provides great convenience for the structure analysis of crystal and interface. Nevertheless, we note that the calculated crystalline density errors are consistently higher than amorphous, which suggests that there may be some systematic error from extending the TraPPE-UA intermolecular attractions (optimized for liquids) to a solid. One sees that, at same pressure, iPP has a higher phase transition temperature than PE. In general, the more flexible compact chains have the lower melting temperatures. The primary difference between the molecular architectures of iPP and PE is the methyl side group on every second site of the polypropylene backbone. Thus, it is indicated that iPP has stronger backbone stiffness than PE and the local architecture can affect their phase behaviors. With the crystal–melt coexistence data, we proceed to analyze the spatial density distributions and interfacial tensions of the two polymer crystals. Given the lack of spatial and temporal resolution of current experimental techniques, theory offers a valuable alternative to improve our understanding of the molecular mechanism of crystallization. One of the main advantages of the current model is the ability to provide the spatial structure. As shown in Figure 3, the crystal structure of PE is a simple base-centered orthorhombic crystal structure. A complete crystal is composed of a few of repeated unit cells along the crystal axes. Here we present two unit cells, which contain one unit cell on the x axis, one unit cell on the repeats on the

y axis, and six

z axis. It is shown in Figure 3a that the crystal cell is composed of stacked layers, each layer

being assembled of chain molecules with identical helical structure. A polymer chain is connected by intramolecular covalent force on the

z axis. However, there is only intermolecular van der Waals force

between each polymer chain on the other axes. In our calculations, the effect of chain length on crystal structure has been considered, and no discernible difference can be observed by making the chains longer due to the periodicity. Figures 3b depicts the optimal spatial density distribution of CH2 sites in the (1×1×2) ACS Paragon Plus Environment


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unit cells. Five PE chains are located about their lattice positions on the cells and the chains are parallel with each other in a zigzag arrangement. By a comprehensive observation, the density profile of CH2 sites correlates to the steric configuration of the PE chains in Figure 3a. Figure 4 describes the crystal structure of iPP with α type, which consists of one unit cell on the x axis, one unit cell on the

y axis, and two repeats on the z axis for a total of two unit cells and six polymer

chains. In Figure 4a, the iPP chain molecule has isoenergetic right-hand and left-hand helices due to the presence of the asymmetrically substituted methyl sites. On the other hand, each chain contains three-folded helices with alternating CH2 and CH sites, and a pendent CH3 site is added to each CH residue in the isotactic fashion. The specific property of the polymer chain is the binding forces, with covalence forces in the chain backbone direction and van der Waals forces in the two other directions. Figure 4b shows the optimized density profile of the three sites in the (1×1×2) unit cells. The value peak of sites on the right handed helical chain is equal to the value peak depicted sites on the left handed helical chain since the sites are overlapped, which corresponds to Figure 4a. The structure of polymer crystal–melt interface has long been a subject of great interest and intensive debate. It is believed that the structures and properties of side and end interfaces are important issues to deal with the process of crystal nucleation and growth. Figure 5 displays the density profiles of CH2 sites in PE between the restraints of the (100)–melt (Figure 5a) and (010)–melt (Figure 5b) interfaces. The two side interfaces are similarly represented with appropriate changes in the region and directions of the periodicity. There are three and four molecular layers in the interfacial regions of (100)–melt and (010)–melt, respectively. The distributions show a continuous and distinct decline from the bulk crystal to the melt, with an intervening transition zone or interphase. The density profiles of iPP at different crystal–melt interfaces are shown in Figure 6, where the minimized density field at the (100)–melt and (010)–melt interfaces can be seen clearly. There are four and ACS Paragon Plus Environment

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three structural repeating units at the two interfaces. The relative widths and positions of the density peaks in the interlayer spacing are different. Near the two bulk crystal surfaces, the monomer densities exhibit ordered fluctuation. Difference in the peak height within the two interfacial regions reflects the tighter packing within crystal layers. Once the density profiles on the interfaces have been determined, the corresponding interfacial tensions can be easily calculated to evaluate the stable or metastable states during the crystallization process. Such interfacial tensions are the major hindering effect or conversely driving force for polymer crystal growth. The temperature-dependent side and end interfacial tensions are show in Figure 7. As can be seen in Figure 7a, the interfacial tensions of PE crystal increase with increasing temperature, which are different from most liquids. It is believed that the interchain spacing (bulk crystal volume) and configuration are the essential factors to affect interfacial tension. When interface is formed, chains in the interface lose considerable intermolecular energy with increasing temperature, but need energy to maintain the stability of chain configuration. It seems that the configuration plays a more important role than interchain spacing. At 400K, the values at the (100), (010), and (001) interfaces are 13.04, 13.65, 23.06 erg/cm2, which are close to the reported data.50-52 While for iPP, the interfacial tensions decrease with increasing temperature, as can be seen from Figure 7b. The methyl sites in the surface layer gain considerable freedom, and there can be larger interchain spacing propagating inward from the interface with increasing temperature. In this case, the main contribution arises from the interchain spacing, since crystalline iPP has a tighter packing. The interfacial tension in the (010) direction is slightly lower than it in the (100) direction, which could be induced by the tighter chain packing at the (010) interface. iPP is forced to pack in tight helices because of its methyl side sites. In contrast, PE packs somewhat more loosely as linear chain in zigzag configuration. It is noteworthy that the (001) interface has significantly higher free-energy, which could be attributed to the chain folding.



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According to the density profiles in the crystal and crystal–melt interfaces, the corresponding nucleation free-energy barriers and critical sizes are calculated to evaluate the difficulties in homogenous crystallizations of PE and iPP. It is known that the crystallization behavior of polymer chains is very complicated. Unlike small molecules, polymers can form crystals only under the essential that the cooling process from the molten state occurs slowly enough to enable the necessary rearrangements of the chains. In particular, homogenous nucleation is quite difficult to observe in experiment for bulk system. The morphology and properties during crystallization depend on the detailed microstructure of the polymer crystal interfaces. A crystal nucleus is thermodynamically metastable if it is surrounded by a local supersaturated state, which is the driving force for the nucleus to grow. The supersaturation ratio is defined as S = p p 0 , where p is the pressure of the supersaturated liquid and p 0 is the saturation liquid pressure. Due to trivial differences between the (100) and (010) interfaces, we consider to use

γ s = γ 100γ 010 to approximately substitute the side interfacial tension. Figure 8 shows a plot of predicted free-energy of homogenous crystal nucleation for PE at 400 K (Figure 8a) and iPP at 450 K (Figure 8b) in a supersaturating state S = 1.2 . As shown in figure 8, the free-energy difference as a function of radius and height is the sum of the bulk and surface terms, with a lowest maximum, corresponding to the ‘‘critical nucleus’’, beyond which further increases in sized actually decrease the total free-energy. The crystallites with the lowest nucleation barrier nucleates may grow, or convert to the most stable phase. The free-energy at the saddle point is the nucleation barrier. We get ∆Ω c = 23.62 kBT , Rc = 9.49 Å and H c = 32.82 Å for PE, and ∆Ω c = 42.27 k BT , Rc = 11.24 Å , and H c = 55.74Å for iPP. Figure 9 plots the predicted free-energy barrier (Figure 9a), radius (Figure 9b), and height (Figure 9c) as a function of temperature for the critical nucleation of PE and iPP crystals. In Figure 9a, one sees that the free-energy barrier needed for nucleation goes up linearly with temperature increases. For the two polymers, ACS Paragon Plus Environment

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their crystallizations become more difficult at higher temperature. If the temperature is higher than 250 K, it seems that iPP needs more energy to form crystallites. As can be seen in Figure 9b and Figure 9c, as the temperature increases, the critical radius and height of iPP nucleus increases quickly, whereas the sizes of PE nucleus is relatively insensitive to temperature. In this regard, the influence of temperature on iPP crystallization is more obvious.

4. Conclusion We have presented a DFT to investigate the spatial structure and thermodynamic properties of PE and iPP crystallites. Validated by available experimental data, the present theoretical approach is able to quantitatively characterize the crystal–melt phase transitions and crystal nucleation. In particular, we have presented the density profiles of PE and iPP chains in crystal cell and at crystal–melt interfaces. Detailed conformations of the chains traversing the crystalline and amorphous phases are expected to faithfully reflect the pathways of crystallizing chains and to give us some valuable information about the molecular mechanism of crystallization. Under the given supersaturation conditions, the nucleation free-energy barriers are predicted to evaluate the difficulties in formation of PE and iPP crystals. It is anticipated that this study can provide valuable perception for understanding of polymer crystallization mechanism and can be generalized to more complex polymer.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 21276010 and 21476007), and by Chemcloudcomputing of Beijing University of Chemical Technology.



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Figure Captions Figure 1 Determination of lattice parameters a and b as a function of temperature for the crystal cells of PE (a) and iPP (b). The lines are theoretical predictions, squares are the experimental values for PE,46 and triangles are the experimental ones for iPP.47

Figure 2 Crystal–Melt phase coexistent curves for PE (a) and iPP (b), respectively. The lines are theoretical predictions, squares are the experimental data for PE,48 and triangles are the experimental ones for iPP.49

Figure 3 (a) Schematics of PE crystal structure on the side view. The gray beads represent the CH2 sites. (b) Optimal geometry configuration of PE crystal at 400 K. The blue, yellow, and red bounded contour levels are 1.2 g/cm3, 1.0 g/cm3, and 0.8 g/cm3, respectively

Figure 4 (a) Schematics of iPP crystal structure on the side view. The cyan, gray and steelblue beads represent the CH, CH2 and CH3 sites. (b) Optimal geometry configuration of iPP crystal at 450K. The blue, yellow, and red bounded contour levels are 1.6g/cm3, 1.4 g/cm3, and 1.2 g/cm3, respectively.

Figure 5 Two-dimensional cuts of density profiles for PE at (100)–melt (a) and (010)–melt interfaces at 400K.

Figure 6 Two-dimensional cuts of density profiles for iPP with α form at (100)–melt (a) and (010)–melt interfaces at 450K.

Figure 7 Interfacial tension as a function of temperature on the (100), (010), and (001) directions for PE (a) and iPP (b).

Figure 8 Free-energy variation during the homogenous nucleation of PE (a) and iPP (b) crystals, respectively.

Figure 9 Free-energy barrier (a), radius (b), and height (c) as a function of temperature for the critical nucleation of PE and iPP crystals.


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Table of Contents Entry

Theory of Crystals and Interfaces in Polyethylene and Isotactic Polypropylene Jianguang Feng, Hongfu zhou, Xiangdong Wang, and Jianguo Mi


Theory of Crystals and Interfaces in Polyethylene and Isotactic

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