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Effect of interfacial interactions on the crystal growth rate in model PE/Si[O.sub.2] nanocomposites: comparing experiments with Lauritzen-Hoffman model and MD simulation.

INTRODUCTION

Polymer crystallization is kinetically controlled under conditions far from thermodynamic equilibrium involving molecular motions on various length and time scales. Many plastics processing technologies utilize this feature of plastics for in-situ formation of property-specific super-molecular structure. Nanocomposites with engineered surface chemistry represent novel means of both controlling the kinetics of morphogenesis in semicrystalline polymers and enhancing its long-term stability. However, the literature on crystallization kinetics of polymer nanocom-posites with varying interfacial adhesion often provides contradictory conclusions. This is due to the experimental difficulty in achieving sufficient control over the interfacial energetics, inability to eliminate primary nucleation by adhering nanoparticles and by utilizing vastly different experimental protocols. With some exceptions (1-5), many of the studies vary multiple structural variables (particle size, shape, surface treatment, chain molecular weight, chain rigidity and polarity, nucleating agents, etc.) and test parameters (nonisothermal, specimen size, sample preparation, characterization technique, etc.) at the same time. This prevents separation of the various contributions to the overall crystal growth rate, often exhibiting opposite dependence on the given variable.

Effect of spherical nanoparticles on the isothermal spherulite growth rate was investigated for iPP filled with nanometer-sized non-nucleating silica (1-3), high-density polyethylene (HDPE) filled with CaC[O.sub.3] (6) and silica nanoparticles (5) and low density polyethylene (LDPE) filled with CaC[O.sub.3] particles of varying particle size and size distribution (7). The reduction of the crystal growth rate with increasing filler content observed in isotactic polypropylene (iPP) nanocomposites with weakly attractive interface was ascribed to the geometrical constraints to the lamellae stacking by the rigid nanoparticles. In addition, strong nucleation activity of the nanoparticles was observed as well as formation of smaller spherolites and increased fraction of the [beta]-phase (8). On the other hand, acceleration of the overall crystallization kinetics was observed for maleic anhydride grafted iPP (iPP-MA) modified iPP/Si[O.sub.2] nanocomposites with increasing the iPP-MA content (9), (10). In HDPE/Si[O.sub.2] nanocomposites with weakly attractive interface, the substantial reduction of [G.sub.II] was ascribed to the retarded reptation of chains reeling to the crystal growth front (5). Enhancing the interfacial attraction in HDPE filled with nanoscale CaC[O.sub.3] caused enhanced primary nucleation activity and reduced both secondary surface nucleation and chain transport to the growth front (11). In contrary, investigating the effect of CaC[O.sub.3] with varying particle size and size distribution on crystallization of LDPE revealed relatively weak effect of CaC[O.sub.3] on its isothermal crystallization kinetics (7).

In the study of high surface to volume ratio platelet-shaped nanoparticles filled iPP, the authors speculated that the overall crystallization rate resulted from competiton between enhanced nucleation and reduced chain difusion rate. The enhanced nucleation rate was assumed due to the large nucleating iPP-filler interface area and the reduced difusion rate of chains was ascribed to their mobility retarded due to confinement between the exfoliated montmorillonite (MMT) platelets (11-13). In another study, the MMT/iPP nanocomposite, the crystal growth rate was a nonmonotonous function of MMT content, its degree of exfoliation (14) and nanoparticle-chain interaction strength (15-18). In contrary, reducion of the crystallization rate from quiescent melt to 1/2. of that for the neat iPP by adding up to 1 wt% of graphene nanosheets into iPP was attributed to their increased primary nucleation activity (19).

Adding fibrous multi-wall carbon nanotubes (MWCNTs) into poly-[epsilon]-caprolactone (PCL) resulted in accelerated isothermal crystallization compared with the neat PCL (20) similarly to the effect of octadecylamine modified single wall carbon nanotubes (SWCNT) in iPP (21). In another article, the observed decrease of the crystallization rate in functionalized MWCNT filled iPP compared to the priscine MWCNT filled iPP was attributed to the weakened nucleation activity of the functionalized carbon nanotubes (CNTs) (19). In a comprehensive study (4), the various CNTs used induced thicker lamellar crystals in in-situ polymerized PE compared with the neat polyethylene (PE) and CNTs exhibited strong primary nucleation activity. At low [v.sub.f], the crystallinity increased due to enhanced nucleation. At high [v.sub.f], dramatic reduction of the crystallinity was observed and ascribed to chain confinement by large specific interface area CNTs.

The main shortcoming of the Avrami model (22-24), used to analyze experimental crystallization kinetics data in most of the nonisothermal investigations, consists of the fact that it averages the spherulite growth rate over the entire specimen rather than analyzing the growth rate of an individual spherulite. Hence, the Avrami model cannot capture the local nanoscale peculiarities of the crystallization process as affected by the presence of rigid heterogeneities with their length scale, D, similar to that for the chain radius of gyration, [R.sub.g].

For isothermally crystallized linear polyethylene, phenomenological description of the temperature dependence of the crystal growth rate, G, for the kinetically dominant {110} growth front is described satisfactorily by the Lauritzen-Hofmann (LH) model (5), (25):

G = [G.sub.0] exp[- [G*.sub.D.sup.0]/R(T - [T.sub.v])] exp [- K/T[DELTA]T]. (1)

The [G.sub.0] is assumed temperature independent pre-exponential factor strongly dependent on the molecular characteristics, [DELTA]T = ([T.sub.m.sup.0] - T) is the under-cooling, [T.sub.m.sup.0] = 408 K is the PE equilibrium melting temperature, T is the crystallization temperature, [Q*.sub.D.sup.0] is the activation energy for the chain self-diffusion in the melt, [T.sub.v] is the Vogel temperature equal to ~103 K for the common PE (25).

During the crystal growth, the internal bond rotation is virtually the only way for a segment to move, hence, the chain motion is a consequence of the successive internal bond rotations of all the segments resulting in the chain reptation primarily in the lengthwise direction. Hoffman and Miller (25) rearranged Eq. 1 for the growth rate in neat PE, [G.sub.II.sup.0], considering chain reptation as the mechanism for chain transport to the crystal growth front (25):

[G.sub.II.sup.0] [approximately equal to] [kappa][C.sub.0.sup.1/2]/N {exp[- [G*.sub.D.sup.0]/R(T - [T.sub.v])] exp [- K/T[DELTA]T]}. (2)

The [C.sub.0] is the configurational path degeneracy expressing the variation in the length of the attached stems and ranging from 5 to 25 for the PE and the growth rate reduction factor. [kappa], was expressed as (25):

[kappa] [approximately equal to] 1/[xi]0 exp ([Q*.sub.D.sup.0]/R[T.sub.0]). (3)

The ethylene monomer friction coefficient, [xi]0, equals to 2.2 x [10.sup.-12] kg/s at the reference temperature [T.sub.0] = 449 K (25). For the entangled neat PE melt, the activation energy for chain reptation, [G*.sub.0.sup.0] varies from 25 kJ/mol to 29.3 kJ/mol (5). Combining Eqs. 2 and 3 results in the [G.sub.II.sup.0] being inversely proportional to the chain friction coefficient, [[xi].sub.c].

The term dynamic fragility is frequently used to express the strength of the temperature dependence of dynamical properties of glass forming liquids (the breath of the glass transition region) (26), (27). Investigating crystallization kinetics of supercooled polymer melts, Sanz et al. (28) suggested that less fragile polymers, such as PE, exhibit far less strongly intermolecularly cooperative segmental dynamics compared to highly fragile stiff backbone polymers. Adding large specific surface area nano-particles can substantially alter short distance coupling of motions of different parts of the polymer chain and its surrounding chains. Hence, segmental motions in the nanocomposite melt become increasingly decoupled both along the chain and in the neighboring chains, thus, making formation of aggregates of chain segments (stems) exhibiting higher local orientation order in the melt less probable. Thus, one can predict that nanoparticles, greatly modifying both dynamics of individual chains and colleclive segmental motion in polymer melts, will slow down the crystal growth rate significantly under the condition of their negligible nucleation effect.

The effect of particle-chain interfacial interactions is accounted for by enhanced both chain friction coefficient, [[xi].sub.c], and the increased activation energy of chain self-diffusion, [Q*.sub.D], compared with that in the neat polymer, [Q*.sub.D.sup.0]. The [[xi].sub.c] for the neat PE chain with N monomer units, each exhibiting monomer friction coefficient [[xi].sub.0], equals to N[[xi].sub.0]. By adding particles exhibiting attractive interfacial interaction with chain segments to the polymer melt, the [[xi].sub.c] is enhanced substantially due to formation of trains on the particle surface. Considering [N.sub.a] the average number of monomer units in the train, the [[xi].sub.c] can be expressed as (29), (30):

[[xi].sub.c] = ((N - [N.sub.a])[[xi].sub.0] + [N.sub.a][[xi].sub.a]). (4)

The [[xi].sub.a] is the friction coefficient of segments in the adsorbed trains (30). The only [[xi].sub.a] data available in literature so far were measured for polystyrene (PS) containing weakly attractively interacting nanoparticles and 5.6 x [10.sup.3] [[xi].sub.0] (30). One can assume that for the PE, the difference between [[xi].sub.0] and [[xi].sub.a] may be of the same order (31-33). For weak and strong attraction, respectively, [N.sub.a] = [N.sup.1/2] and [N.sub.a] = 1/2N. respectively (29), and, [N.sub.a] = 0 for noninteracting interface. For chains with N < [10.sup.6], the [N.sup.1/2]/[[xi].sub.0] can be considered negligible compared with [[xi].sub.a], thus, the [[xi].sub.c] [approximately equal to] [N.sup.1/2][[xi].sub.a] and [[xi].sub.c] [approximately equal to] 1/2 N [[xi].sub.a] for the weak and strong attraction, respectively. For the noninter-acting interface, [[xi].sub.c] = N[[xi].sub.0] similarly to the neat PE. Substituting for [kappa] (Eq. 3) in Eq. 2, the [G.sub.II.sup.2] for the neat PE can be expressed as:

[G.sub.II] [approximately equal to] [([C.sub.0]/N)].sup.1/2 1/[N.sup.1/2] [[xi].sub.0] exp ([Q*.sub.D]/R[T.sub.0]) x

{exp [[Q*.sub.D]/R(T - [T.sub.v])] exp [- K/T[DELTA]T]}. (8)

Similarly, the [G.sub.II] for the nanocoinposite with strong inter-afacial attraction can be expressed as:

[G.sub.II] [approximately equal to] [([C.sub.0]/N)].sup.1/2 1/[N.sup.1/2] [[xi].sub.a] exp ([Q*.sub.D]/R[T.sub.0]) x

{exp [[Q*.sub.D]/R(T - [T.sub.v])] exp [- K/T[DELTA]T]}. (6)

for the weak interfacial attraction:

[G.sub.II] [approximately equal to] [([C.sub.0]/N)].sup.1/2 1/[[xi].sub.a] exp ([Q*.sub.D]/R[T.sub.0]) x

{exp [[Q*.sub.D]/R(T - [T.sub.v])] exp [- K/T[DELTA]T]}. (7)

and for the noninleracting interface:

[G.sub.II] [approximately equal to] [([C.sub.0]/N)].sup.1/2 1/[N.sup.1/2] [[xi].sub.0] exp ([Q*.sub.D]/R[T.sub.0]) x {exp [[Q*.sub.D]/R(T - [T.sub.v])] exp [- K/T[DELTA]T]}. (8)

Equations 6 and 7 suggest direct proportionality between [G.sub.II] and [[xi].sub.a.sup.l] in the nanocomposite melts and weakening of the [M.sub.n] dependence of [G.sub.II] in nanocomposite which is in agreement with experimental observations of [M.sub.n] scaling of the [G.sub.II] for monodisperse PE tilled with fumed Si[O.sub.2] (5). For the former prediction, reliable experimental data do not exist in the literature so far. Equation 8 suggest the only effect of nanoparticles in noninteracting system, is formation of additional topological constraints to the chain transport expressed in the enhanced [Q.sub.D]*.

Since it is rather difficult to control the strength of inter-factial interactions and preserving the negligible primary nucleation activity of the nanoparticles experimentally, use of computer simulation can provide useful insight into the effect of these variables on crystallization kinetics. In this paper, simple molecular dynamics (MD) simulation of the crystal growth rate, [G.sub.II], was performed for a system of 100 chains each containing 100 C[H.sub.2]-C[H.sub.2] segments and nano-particle volume fraction, [v.sub.f], equal to 0.02 and 0.04. Monte Carlo (MC) method was used to generate the initial equilibrium chain configuration in the model system. The MD simulation variables included chain-particle interfacial interaction strength, [v.sub.f] and degree of topological constraint to chain folding/stem deposition at the crystal growth face. The crystallization kinetics was simulated over the time period of 100 ns. The isothermal [G.sub.II] measured for low [M.sub.n] monodisperse PE filled with fumed Si[O.sub.2], at 401 K was used for comparison with results of MD simulations.

Experimental and Simulation Details

Materials and Methods. Nearly monodisperse model polyethylene was used as the matrix. Number average molecular weight ([M.sub.n]) and polydispersity index ([M.sub.w]/[M.sub.n]) were determined employing high temperature gel-permeation chromatograph (Waters, USA) at 140[degrees]C in decalin using polystyrene standards. The [M.sub.n] was 3 X [10.sup.4] g [mol.sup.-1] and the polydispersity index. [M.sub.w]/[M.sub.n], was 1.01. As received fumed silica (Sigma Aldrich, USA) with specific surface area of 390 [m.sup.2] [g.sup.-1] and mean primary particle diameter D = 8 nm and average size of the string-like clusters of primary particles of 200 nm was used as the nano-filler. The Si[O.sub.2] volume fraction, [v.sub.f], investigated was 0, 0.02, and 0.04. Method of Dobreva showed no measurable nucleation activity of the filler used (5).

Nanocomposites were prepared by adding silica and Irganox 1076 stabilizer (Ciba, Switzerland) into xylene solution of the polyethylene at 130[degrees]C under ultrasonic vibrations followed by vigorous stirring for 1 h and drying at 70[degrees]C for 10 h in the vacuum oven. Dried posite powder was compression molded at 170[degrees]C using a two hot plate TP 400 hydraulic press (Fontijne, NL) into 1 mm thick sheets. For the reference, the neat polyethylene underwent the same processing procedure as the nanocomposites. Approximately 10 [micro]m thick slices cut out from the pressed sheets were placed between two glass slides. Before each measurement, the specimen was melted at 170[degrees]C in the LTS 350 hot stage (Linkam, UK). After 5 min at 170[degrees]C. the specimen was quickly cooled down to the crystallization temperature.

Simulation Details. United atom force field has been employed (34) for polyethylene melts, in which polymer chains interact via bonding and nonbonding interactions. The total interaction energy, [E.sub.TOT], can be expressed as a sum of all the contributions in the form:

[E.sub.TOT] = [E.sub.BOND] + [E.sub.ANGLE] + [E.sub.TORSION + [E.sub.NONBOND] (9)

The harmonic bond length potential is expressed as (34):

[E.sub.BOND] = [k.sub.b][(b - [b.sub.O]).sup.2], (10)

where [k.sub.b] = 1470 kJ [mol.sup.1] and [b.sub.0] = 1.53 [angstrom]. The three body bond angle bending potential is given by (34):

[E.sub.ANGLE] = [k.sub.0][([theta] - [[theta].sub.0]).sup.2]), (11)

where [k.sub.0] = 252 kJ [mol.sup.1] [rad.sup.2] and [[theta].sub.0] = 1.91 rad (109.5[degrees]). The torsional potential is (34):

[E.sub.TORSION] = 1/2[K.sub.1] (1 - cos([empty set])) + 1/2[k.sub.2](1 - cos(2[empty set])) + 1/2[k.sub.3](1 - cos(3[empty set])), (12)

where [k.sub.1] = 6.804 kJ [mol.sup.1], [k.sub.2] = 3.6414 kJ [mol.sup.1], and [k.sub.3] = 13.608 kJ [mol.sup.1]. These parameters yield a gauche energy of 2.1 kJ [mol.sup.1]. a gauche-trans barrier of 12.6 kJ [mol.sup.1] and gauche-gauche barrier of 21 kJ [mol.sup.1]. The non-bonded potential for atoms separated by more than three bonds follows the Lennard-Jones (LJ) relation (34):

[E.sub.NONBOND] = 4[epsilon][[([sigma]/r).sup.12] - [([sigma]/r).sup.6]], (13)

where [epsilon] = 0.0504 kJ [mol.sup.1] and equilibrium reference distance a = 4.01 [angstrom] with a given cut off distance of 12 [angstrom]. By variation in the LJ particle-chain interactions, the strength of interactions has been set up (Table 1).

TABLE 1. Lennard-Jones potential parameters used.

Parameter  Weak         Strong       No           Repulsive
           attractive   attractive   interaction  interaction
           interaction  interaction

[sigma]    4.01         4.01         4.01         4.01
           [Angstrom]   [Angstrom]   [Angstrom]   [Angstrom]

[epsilon]  0.05O4       0.504        0 kJ/mol     0.0504
           kJ/mol       kJ/mol                    kJ/mol
                                                  From 0.2
                                                  [Angstrom]
                                                  : 0 kJ/mnl


The system of 100 chains, each one containing 100--C[H.sub.2]--C[H.sub.2]--segments, was generated by the Monte Carlo method using the potential field described above. The bond lengths and bond angles were kept constant and the torsional angles were distributed according to the Boltzmann distribution at the given temperature. The Nose-Hoover temperature coupling was utilized with time constant, t = 0.1 ps. Velocities at 500 K were generated by the Maxwell distribution and the first 5 ns of the simulation was assumed to equilibrate the system. The size of the starting simulation box was 2.7 nm X 8.0 nm X 7.0 urn (V = 151 n[m.sup.3]).

The pre-existing crystal surface was defined as aligned chains in the xy plane with the interlayer spacing of 3.78 nm which corresponds to the crystal facet {100} (34) and only growth from this pre-existing surface was simulated. The global bond orientation parameter, S. at time t, was used as a standard measure of order. To calculate the spatial distribution of the order within the simulation box, a local orientation order was defined. The method implemented by Waheed (35) based on the convolutions of the local chain orientation vectors with spatial box functions was utilized. The growth rates were calculated from the order parameter extrapolation to the completion of the crystalline layer. Only formation of the first and the second layer from the preexisting crystal surface was considered. The pre-existing crystal surface served as a reference for the order parameter value and no other primary nucleation was considered.

Each nanoparticle consisted of 46 Lennard-Jones particles bound to form icosahedral shape (Fig. 1). The diameter of the particle was 2 nm. The particle was inserted into the simulation box prior to generating the polymer melt. Consequently, the sample was allowed to equilibrate at 500 K for 1 ns. A single nanoparticle occupied 1.81 vol% and two particles 3.62 vol% of the simulation box which was in the range of silica volume fraction used in the experiments. The particle distance, C, from the preexisting crystal surface and the interparticle distance, [D.sub.p], were varied (Fig. 2).

The simulation was conducted in the NVT ensemble [constant number of particles (N), volume (V), and temperature (T)] at 400 K (127[degrees]C) to fall into the temperature Regime II of PE crystallization. The molecular simulation package Gromacs 4.0.5 was used and Visual Molecular Dynamics was utilized for visualization of the results. The computational cost was 25.797 Mnbf [s.sup.-1], simulations were conducted for 100 ns.

RESULTS AND DISCUSSION

Many simplifying assumptions such as short chains, short times, etc. were considered in the MD simulation. Hence, the results are primarily used to analyze the trends rather than to calculate the actual growth rates. At first, isothermal crystallization of the ensemble of 100 chains was performed and compared with available experimental data for PE to validate the simulation algorithm used. The calculated characteristic ratio, [C.sub.N], was equal to 5.7 which is in a fair agreement with the [C.sub.N] = 6.3 published for the chain length of 100 segments in literature (36), (37). The radius of gyration, [R.sub.g], was calculated equal to 3.74 nm in a reasonable agreement with literature data (37). The ratio of the chain radius of gyration, [R.sub.g], and the end-to-end distance, [check]<[r.sup.2]>, was greater than 6, thus, the short chains used in our simulation should not be treated as Gaussian coils.

The crystal growth rate, [G.sub.II], was determined in two consecutive layers from the pre-existing crystal surface, introduced prior to the start of the simulation, utilizing the order parameter development (35) related to density fluctuation (Fig. 3). The first peak on the left side of Fig. 3 represents the pre-existing crystal surface. The second and further peaks are signifying the newly developed crystalline order. The symmetry of the graph in Fig. 3 is due to the periodic boundary condition used. The time dependence of the obtained order parameters was similar to that published by Waheed et al. (35) utilizing the Steel type potential field for the pre-existing crystal surface.

The simulated crystal growth rate, [G.sub.II.sup.0], for the neat model polymer was 4 X [10.sup.-4] nm [ns.sup.-1]. This value agrees with the theoretical predictions of Strobl (37) and is in agreement with the simulations data of Waheed (35). The maximum experimentally obtained [G.sub.II.sup.0] was of the order of [10.sup.-5] nm [ns.sup.-1]. The slower growth rate obtained experimentally compared to the values obtained by MD simulations was ascribed to the higher molecular weight of the chains in the experiments and their slower chain dynamics. Also, simulation techniques operate on the length scale of individual molecules while the experiments provide data on the microscale at the best, thus, effectively averaging the nanoscale events.

The nanocomposite growth rate, [G.sub.II], was normalized to the growth rate in the neat polymer, [G.sub.II.sup.0], to obtain the relative growth rate, [G.sub.II.sup.rel]. In Fig. 4, the simulated relative crystal growth rate, [G.sub.II.sup.rel] = [G.sub.II]/[G.sub.II.sup.0], is plotted as a function of the particle distance from the original crystal surface, C. for a constant [v.sub.f] [approximately equal to] 0.02. Horizontal lines represent the experimental [G.sub.II.sup.rel] values obtained for the monodisperse PE with Mr, = 3 x [10.sup.4] containing 2 vol% of fumed Si02 crystallized isothermally at 401 K. Two distinct regions can be identified in the plot. For C 1.6 nm, attractive interfacial interaction reduced the [G.sub.II] proportionally to the interaction strength. No significant perturbation of the [G.sub.II.sup.rel] by the nanoparticle was observed for noninteracting interface and slightly greater [G.sub.II.sup.rel] was obtained for the repulsive interface. This seems in agreement with the fragility-dependent decoupling of the crystal growth rate from viscosity observed for a range of supercooled liquids (38). This decoupling is a result of the heterogeneity of segmental dynamics in supercooled liquids which is reduced by added nanoparticles substantially (39). The trends obtained for the repulsive interfacial interactions are in agreement with the accelerated chain dynamics predicted for repulsive interfactial interactions (39-41). However, no experimental data on isothermal crystallization of repulsive interface nanocomposite has been published so far.

For C < 1.6 nm, the topological constraints to chain mobility became dominant causing additional reduction of the Gr for all the interactions simulated. This seems to be in agreement with experimental results on retarded crystallization kinetics in chains highly constrained by exfoliated clay sheets (11-13) or large specific interface area CNTs (4). In addition, the threshold distance of 1.6 nm equals to approximately four times the stem thickness for PE (25) and, thus, agrees reasonably well with simple theoretical predictions of the distance from a crystal surface at which chain conformation becomes unaffected by surface immobilization (43). Hence, one may envision that the nanoparticles localized closer than about 1.6 nm to the growth face hinder chain folding and/or stem attachment to the crystal growth front. This is also in agreement with the coarse grain and Monte Carlo simulations of PE nanocomposites (44) suggesting significant effect of both interfacial attraction and interparticle distance on the substantial slowing of the chain dynamics compared to the neat PE (45).

To account for the effects of chain confinement and volume fraction, two particle simulations were performed ([v.sub.r] [approximately equal to] 0.04). For interfacial attraction, the effect of increased [v.sub.f] on the extent of [G.sub.II.sup.rel] reduction was much stronger than the effect of topological constraints due to particle distance from the growing crystal surface (Figs. 5 and 6). When moving particles closer to each other while keeping their distance from the crystal surface constant, the chain confinement was enhanced (46) Simulation results showed (Fig. 7) that varying the inter-particle distance had significant effect on the crystal growth rate only in the case of attractive interactions and largest value of C. This result can be interpreted as either multiple particle bridging retarding further chain dynamics or formation of additional topological constraint to hinder chain folding and/or slow down stem attachment. In the regime II, the crystal growth rate is dependent upon both the initial and subsequent stem placement paths, thus, two different expressions for the LH transport term are required for the first versus all the subsequent stems (45), (47).

In contrary, for the repulsive and noninteracting interface, the [G.sub.II.sup.rel] reduction was dominated by topological constraints and the effect of [v.sub.f] was far less significant (Fig. 5). One can conclude that the effect of interfacial attraction dominates the [G.sub.II.sup.rel] reduction over the topological constrains for interfacial attraction while for noninteracting and repulsive interface, topological constraints dominate the reduction of [G.sub.II.sup.rel] (Fig. 6). Interestingly, the simple MD simulation used showed slight [G.sub.II.sup.rel] increase for the repulsive interfacial interaction, compared with noninteracting system. Since the is no nucleation activity of the nanoparticles, this can be attributed to enhanced mobility of the chains reeling from the melt to the crystal growth front which is in agreement with the accelerated dynamics of chains in the vicinity of repulsive surface predicted by Starr and Douglas (40).

The activation energy of retarded chain reptation in the vicinity of the nanoparticle surface, [Q.sub.D.sup.i*], was estimated as [Q.sub.D.sup.i*] = 10 X [Q.sub.D.sup.*0] (54), where [Q.sub.D.sup.*0] was the value for the unperturbed bulk reptation. The distance to which the particle surface affects given chain directly around a single particle was assumed to have the thickness of the order of 1 reptation tube diameter (~2 NM for the PE used in our experiments). The average activation energy of the chain reptation from the melt to the growing crystal front, c"I'Qi'), was calculated using the simple bi-modal distribution of chain dynamics expressed in Eq. 5:

[[cale.sup.Q]*.sub.D] = (1 - [v.sub.f] - [v.sub.i])[Q.sub.D.sup.*0] + vi[Q.sub.D.sup.i*] (14)

where the [[cale.sup.Q]*.sub.D] is the resulting average activation energy, [Q.sub.D.sup.*0] is the average activation energy of reptation in unperturbed bulk (24 kJ/mop, [Q.sub.D.i*] (240 kJ [mol.sup.l]) is the average activation energy of the retarded reptation of chains in the layer 1 reptation tube diameter thick on the surface of the Si[O.sub.2] particle (5), [v.sub.f] is the filler volume fraction and vi is the volume fraction of the chains exhibiting retarded repta-tion calculated assuming uniform layer thickness on every particle and neglecting particle clustering (5).

The relative crystal growth rate for attractively interacting interface can be predicted dividing Eqs. 6-8 by Eq. 5. This yields a rough estimate of the [G.sub.II.sup.rel] reduction with varying the interfacial interactions in the form of the following expression for strong attraction:

[G.sub.II.sup.rel] [approximately equal to] 2[[xi].sub.0]/[[xi].sub.a] exp ([Q.sub.D*] - [Q.sub.D.sup.*0]/R[T.sub.0]) (15)

for weak attraction

[G.sub.II.sup.rel] [approximately equal to] [N.sup.1/2][[xi].sub.0]/[[xi].sub.a] exp ([Q.sub.D*] - [Q.sub.D.sup.*U]/R[T.sub.0]) (16)

and for noninteracting interface:

[G.sub.II.sup.rel] [approximately equal to] exp ([Q.sub.D*] - [Q.sub.D.sup.*0]/R[T.sub.0]) (17)

By substituting for R = 8.3 x [10.sup.3] J[K.sup.1] [mol.sup.l], [T.sub.0] = 408 K, [[xi].sub.0] = 2.2 x [10.sup.12] kg [s.sup.l], [[xi].sub.a] = [10.sup.9] kg [s.sup.l], [Q.sub.D.sup.*0] = 2 x [10.sup.4] J [mol.sup.l] and [Q*.sub.D] = 6 x [10.sup.4] J [mol.sup.l] for [v.sub.r] = 0.04 (5), (25), the [G.sub.II.sup.rel] was estimated using Eqs. 15-17 equal to 5 x [10.sup.3], [10.sup.2], and 0.96 for strong and weak attraction and noninteraction interface, respectively. The simple MD simulation yielded [G.sub.II.sup.rel] for [v.sub.f] = 0.04 equal to 3 x [10.sup.3], 7 x [10.sup.2] and 0.62 for the strong and weak attraction and noninteracting interface, respectively. Value of [G.sub.II.sup.rel] = 8 x [10.sup.2] was obtained from experimentally measured growth rates for the monodisperse PE with [M.sub.n] = 3 x [10.sup.4] g [mol.sup.l] filled with 4 vol% of weakly attractively interacting fumed [Si0.sub.2] which is in the same order with the simple model (Eq. 16) predicted [G.sub.II.sup.rel] = [10.sup.2] and the simple MD simulated [G.sub.II.sup.rel] = 7 x [10.sup.2] values (Fig. 8).

CONCLUSIONS

Our simple MD simulation confirmed that, at constant [v.sub.r], the crystal growth rate, [G.sub.II], in the linear flexible polymer filled with non-nucleating nanoparticles (NPs) depends strongly on the particle-matrix interfacial interactions and NP-induced topological constraints to chain attachment to the crystal growth front. Two distinct regions were identified in respect to the NP distance from the crystal growth face, C. For C [great than or equal to] 1.6 nm, attractive interfacial interaction reduced the [G.sub.II] proportionally to the interaction strength in qualitative agreement with the reduction of crystal growth rate observed experimentally for non-nucleating NP-filled PE. No significant reduction of the [G.sub.II.sup.rel] by the NP was observed for noninteracting interface and slightly greater [G.sub.II.sup.rel] was obtained for the repulsive interface. The trends obtained for the repulsive interfacial interactions are in agreement with the accelerated chain dynamics predicted in literature. Below 1.6 nm, topological constraints to stem attachment became the dominant mechanism controlling the crystal growth rate regardless of the strength of the interfacial interactions. The threshold [C.sub.erit], = 1.6 nm was about four times the stem thickness of -0.4 nm published in literature for PE.

For interfacial attraction, the effect of increased [v.sub.f] on the [G.sub.II.sup.rel] reducion was much stronger than the effect of topological constraints due to NP distance from the growing crystal surface. In contrary, for the repulsive and noninteracting interface, the [G.sub.II.sup.rel] reduction was dominated by topological constraints and the effect of [v.sub.f] was far less significant. Fair agreement was found between experimen-tal results for monodisperse PE filled with nanosized Si02 and those obtained using the simple modified LH model and/or MD simulation described in this work. Hence, one may conclude that the simple LH model and MD simulation can qualitatively correctly describe fundamental processes affecting the isothermal crystallization kinetics in the presence of NPs with their size of the order of the radius of gyration of the chains.

The importance of the findings summarized above is several fold. First, simple theoretical model was validated relating the NP content, dispersion and NP-chain interfacial adhesion to the isothermal crystal growth rate in linear flexible chain polymers. Second, theoretical leads were provided for the design and use of function specific NPs in processing technologies based on in-situ generating of controlled polymer morphology in advanced polymer packaging production technologies. For example, adding small amount of spherical NPs with repulsive interfacial interaction should accelerate isothermal crystallization of linear flexible polymers while retaining negligible NP nucleation activity. By adding very small amount of NPs ([v.sub.f] < 0.02) with interfacial attraction, crystallization rate can be reduced by 2 orders of magnitude due to interface retarded chain diffusion. To reduce the crystal growth rate to the same extent, much larger [v.sub.f] of repulsive or noninteracting NPs must be added, since the mechanism is only topological constraints to chain mobility by neighboring NPs. Third, our findings may partly explain the nonmonotonous dependence of the isothermal crystal growth rate on the interfacial adhesion reported for polymer nanocomposites in literature. Most probably, the observed initial increase of the crystal growth rate with addition of adhesive agent is due to enhanced primary nucleation by rising the NPs nucleation activity due to reduced interfacial energy barrier while the secondary nucleation and growth remain unaffected. After activating all the available NPs, all the interface retarded chain transport and reduced secondary nucleation due to increasingly decoupled segmental motions, both along the chain and in the neighboring chains, become dominant reducing the overall crystal growth rate significantly.

Correspondence to: J. Jancar; e-mail: jancar@fch.vutbr.cz and jancar@ceitec.vutbr.cz

Contract grant sponsor: Czech Grant Agency; contract grant number: P205/10/2259.

DOI 10.1002/pen.23524

Published online in Wiley Online Library (wileyonlinelibrary.com).

[c] 2013 Society of Plastics Engineers

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J. Jancar, (1), (2) R. Balkoval (1)

(1) Institute of Materials Chemistry, School of Chemistry, Brno University of Technology, Brno, Czech Republic

(2) Central European Institute of Technology (CEITEC), Brno University of Technology, Brno, Czech Republic
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