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Melting of polymer nanocrystals: a comparison between experiments andsimulation†

Noureddine Metatla,a Samuel Palato,a Basile Commarieu,b Jerome P. Claverie*b and Armand Soldera*a

Received 28th July 2011, Accepted 19th September 2011

DOI: 10.1039/c1sm06446k

Polymer nanocrystals have attracted considerable attention because of their potential applications in

future technology and their fascinating properties which differ from those of corresponding bulk

materials. The essential influence of the nanointerface in nanocrystals is apparent in the linear

dependence of the melting temperature with the inverse sheet thickness, i.e. the Gibbs–Thomson

behaviour. Yet, few experimental and theoretical works have been attempted to highlight the influence

of nanointerfaces on the thermal properties of nanocrystals. In this work, simulations were used to

evaluate the melting temperature of crystalline polymer nanosheets. Ensuing results were compared

favourably to experimental melting temperatures stemming from alkane chains and functional

polyolefins, thus validating our simulation approach. Both experimental and simulated results followed

Gibbs–Thomson behaviour and a procedure was devised to extract the heat of melting as well as the

surface energy from these results. Thus, surface energy of various nanocrystals was found to be widely

different for various experimental systems, demonstrating the significance of the environment on

thermal properties of nanocrystals.

Introduction

The properties of nanomaterials often differ from those of the

corresponding bulk materials, and the extrapolation of consti-

tutive or phenomenological macroscopic laws to the nanoscale is

often not straightforward, and vice versa.1,2 One of the main

reasons for the poor applicability of macroscale laws to the

nanoscale domain is the predominance of interfacial phenomena

in the latter case. Fundamental properties of crystals at the

nanoscale are significantly different to those of bulk crystals, and

melting and crystallization temperatures rarely correspond to

those of the bulk.3 Properties of nanocrystals are also strongly

influenced by the shape and size.4,5 For example, the dissolution

rate and the pharmacokinetic profile of drug nanocrystals, which

find wide acceptance as a means to deliver poorly soluble drugs,

are in part controlled by the rich variety of allomorphic forms

which are accessible to these crystals.6,7An efficient way to clarify

experimental observations relies on the use of appropriate

computational tools to probe the very nature of interactions at

stake.8 In these regards, atomistic simulation is well-suited to

study nanomaterials, as it probes comparable dimensions.9,10

aQuebec Center for Functional Materials, Dept of Chemistry, Universit�e deSherbrooke, Sherbrooke, J1K 2R1, QC, Canada. E-mail: Armand.Soldera@USherbrooke.ca; Fax: +1-819-821-8017; Tel: +1-819-821-7650bQuebec Center for Functional Materials, NanoQAM, Dept of Chemistry,UQAM, Succ Centre Ville, CP8888, Montreal, H3C 3P8, QC, Canada.E-mail: Claverie.Jerome@uqam.ca

† Electronic supplementary information (ESI) available: DSC traces. SeeDOI: 10.1039/c1sm06446k

This journal is ª The Royal Society of Chemistry 2012

Comparing simulation results with experimental data thus

contributes to our understanding of the intricate structure–

property relationships of nanomaterials.

In this context, with the ever increasing importance of poly-

meric nanocrystals in a variety of domains, accurate prediction

of size-dependent melting temperature and enthalpy is of para-

mount importance.11,12 We and others have been interested in

accessing such thermal transitions by a variety of simulation

techniques.13–17 However, to our knowledge, the atomistic

prediction of the melting transition of a polymer crystal is still in

its infancy.18–23 Tsuchiya et al. have used a NPT ensemble, i.e.

number of atoms, pressure and temperature are kept constant, to

predict the melting point of n-alkanes in bulk. For a simple

alkane (C16H34), the calculated melting point is 31 K below the

experimental value.22 Recently, Romanos and Theodorou, still

using a NPT ensemble, have proposed an original procedure to

determine the melting temperature of isotactic polypropylene

using atomistic simulation.19 Although this approach gives

accurate predictions, it is long, and difficult to implement for

most polymers.24 Moreover, to our knowledge existing predictive

models are only valid for bulk systems, and do not take into

account compartmentalization at the nanoscale level.

On an experimental point of view, we are interested in

preparing functional polyolefins bearing polar side chains

randomly distributed along the chain.25–30 We recently demon-

strated that such polymers exhibit unexpected melting transitions

when dispersed under the forms of nanoparticles.31 This

prompted us to investigate the influence of the size of the

nanoparticle on the melting point of the polymer. In this paper,

Soft Matter, 2012, 8, 347–352 | 347

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we thus consider a model system constituted of a crystal having

one nano dimension in the z axis, but infinite lateral dimensions.

In a later report, we will consider crystals having nanosizes in all

three dimensions. A polyethylene crystal having a nanosize in the

z dimension only (i.e. a sheet) can be generated by crystallizing

alkane chains or a polymer whereby the chain-packing is peri-

odically disrupted by defects, the distance between two defects

corresponding to the thickness of the sheet. Defects include CH3

groups (chain termination) or pendant groups which are suffi-

ciently bulky not to be incorporated into the crystal.

We thus report a straightforward and rapid approach to model

the melting point of nanocrystals of polyethylene, and we

compare the ensuing results with experimental data of both

alkane chains and polyolefins. Differences between results

obtained from alkane chains and functional polyolefins are

outlined. Experimental and simulated series of data are discussed

through the use of the Gibbs–Thomson equation.

Experimental section

All copolymers were obtained through published proce-

dures.26,30–32 All linear n-alkanes were purchased from Sigma-

Aldrich. Melting points were measured by modulated differential

scanning calorimetry (DSC) using a Mettler Toledo DSC823e

(TOPEM modulation) equipped with an FRS5 sample cell,

a sample robot, a Julabo FT400 intracooler and an HRS7 sensor.

Samples were heated from 20 �C to 140 �C at a rate of 1 �Cmin�1

and data were analyzed using the STAR software. The amplitude

of TOPEM modulation was 0.025 K, using switching times

comprised between 15 and 30 seconds. All reported values are for

samples which have been slowly cooled from the melt at a rate

of 1 �C min�1.

Simulation section

Simulations were carried out in the canonical statistical ensemble

(constant number of particles, volume and temperature (NVT)).

The Nose–Hoover algorithm was employed to keep constant

volume and temperature.33,34 The equation of motion was inte-

grated using the velocity-Verlet algorithm with an integration

time step of 1 fs.35 The cut-off radius for short-range intermo-

lecular interactions was fixed to 10 �A, while the Ewald summa-

tion method was used to take into account long range

electrostatic interactions.23 Molecular dynamics (MD) simula-

tions were performed using the open source LAMMPS package36

with the second generation force field, pcff.37

To build polyethylene nanocrystals, the Accelrysª software

package was first used to generate the orthorhombic cell of

crystalline polyethylene using lattice parameters a, b, and c,

respectively equal to 7.388, 4.929, and 2.539 �A.38 The interface

region was represented by an H atom excluded from the crystal.

Thus, each crystalline strand between two folds in a real crystal

was modeled by a linear alkane chain constituted of N carbon

atoms. For a given N, the z direction was elongated by N/2 times

c, the x and y directions were replicated by n (resp. m) times the

a (resp. b) cell parameter. Periodic boundary conditions were

then eliminated from this super cell, yielding a nanocrystal

constituted of 2 nm alkane chains ofN carbon atoms, the factor 2

coming from the number of chains per unit crystal cell. The

348 | Soft Matter, 2012, 8, 347–352

nanocrystals built through this procedure (an example is shown

in Fig. 1a), and referred below as nambN, are consigned in

Table 1. These nanocrystals were used as starting configuration

to perform molecular dynamics (MD) at temperatures ranging

from 200 K to 400 K with 10 K increments. The MD duration

was decided sufficient when the melting temperature Tm, as

determined below, reached a stationary value.

Results and discussion

The value of the melting temperature (Tm) was established as the

midway temperature between the temperatures separating the

jump in the potential energy, as shown in Fig. 2.19 Once Tm was

locatedwithin a 5Kerror, amore precise determination ofTmwas

achieved by reducing the temperature increment to 5 K, giving

a 2.5 K error. The value of Tm was also determined as the value

between the temperature at which the heat capacity peaked and

the subsequent data (Fig. 2b). The heat capacity was computed

from energy fluctuations.15 It was alternatively obtained as the

value between the temperatures atwhich the trans rotameric state,

t, reached an almost constant value and its preceding data

(Fig. 2c). For the amorphous phase, the plateau value is around

60%.39 For all systems, the three determinations yield identical

values (within 2.5 K, which corresponds to the mesh size of the

temperature sweep in MD). It has to be mentioned that the same

crystalline structure is taken for alkane chains and polyethylene.40

In order to neglect the effects of the lateral dimension of the

crystal, both the x and y dimensions must be expanded to infinite

length. The melting point of the crystal is captured in a variant of

the Gibbs–Thomson equation (eqn (1)):41

Tm ¼ Tom

�1� 2se

Dhm

1

l� 2s

Dhm

1

x

�(1)

where Tom and Dhm are respectively the melting temperature and

the melting enthalpy per unit volume of bulk polyethylene; seand s are the interfacial tensions of the crystal, in the plane

normal to z and in the planes normal to x and y, respectively; �x is

the harmonic mean of the edge distances x and y; l is the crystal

thickness. By fixing l and letting x and y to vary, eqn (1) can be

rewritten under the following form:

Tm ¼ a1

xþ bðlÞ (2)

where

a ¼ �Tom

2s

Dhm(3)

and

bðlÞ ¼ Tom

�1� 2se

Dhm

1

l

�(4)

b(l) corresponds to the melting temperature of sheets with finite

thickness, l. It is actually equivalent to the well-known Gibbs–

Thomson equation (eqn (5)):

Tm ¼ Tom

�1� 2se

Dhm

1

l

�(5)

Fig. 3 reveals a linear relationshipbetweenTmand1/�x forN¼ 40

as suggested by eqn (1) or (2). Other chain lengths, containing 16,

This journal is ª The Royal Society of Chemistry 2012

Fig. 1 (a) Nanocrystal designated by 4a4b24 in a perspective view (see text for more details). The unit cell is also displayed (in red). (b) The same system

at 320 K, just after melting.

Table 1 Nanocrystals with N ¼ 16, 24, 32 and 40. Volume is annotatedonly for systems which have been studied

ModelNumber ofchains

Crystal volume/nm3

16 24 32 40

3a4bN 24 8.88 13.31 17.75 22.194a4bN 32 — 17.75 23.67 29.595a6bN 60 22.19 33.29 44.48 55.486a5bN 60 — 33.29 — —7a8bN 112 41.42 62.13 82.84 103.558a8bN 128 — — — 118.359a10bN 180 66.57 99.86 133.14 —11a12bN 264 97.64 — — —

Fig. 2 Determination of the melting temperature for the nanocrystal

3a4b32 by MD (duration 80 ns). (a) Potential ( ) energies per alkane

chain, (b) heat capacity at constant volume ( ), and (c) percentage of the

trans rotameric state, t, (-) with respect to temperature.

Fig. 3 Simulated Tm from eqn (2) for l ¼ 4.95 nm (N ¼ 40) with respect

to 1/�x. See the text for more details.

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24 and 32 carbon atoms, show similar behaviour (simulated curves

are displayed in the ESI†). The values of a and b(l) stemming from

fitting (eqn (2)) are reported in Table 2. b(l) corresponds to Tm for

This journal is ª The Royal Society of Chemistry 2012

a polyethylene sheet of thickness l. It is expected to vary linearly

with 1/l according to eqn (4). Fig. 4 illustrates such simulated and

experimental variations of Tm for systems containing alkane

chains of different lengths (Fig. 4 and Table 2).

Remarkably, both simulated and experimental values scale

linearly with 1/l in perfect agreement with Gibbs–Thomson

equation (eqn (5)) but the simulated melting temperatures are

always higher than the experimental ones by a relatively constant

offset of 30 K (Fig. 4). From Fig. 4, we can thus first conclude

that even a single layer of n-alkanes reproduces well the melting

behaviour of real alkane. Such phenomenon was already noted

by H€ohne.42 Second, the difference between simulated and

experimental values is principally due to the choice of force field

and to kinetic effects.43 It will probably be possible to reduce it by

choosing an appropriate force field, such as COMPASS,44 but

such work would be beyond the scope of this paper. However,

since the offset between simulated and experimental Tm is rela-

tively constant, the results were plotted again using Tm/Tom as

Soft Matter, 2012, 8, 347–352 | 349

Table 2 Slope (a) and the ordinate at the origin (b(l)/K) of eqn (2)stemming from different l

N l/nm a b(l)/K

16 1.91 �43.9 31324 2.92 �61.4 36232 3.94 �60.5 38940 4.95 �75.9 401

Fig. 4 Gibbs–Thomson equation representation reporting experimental

( ), and simulated ( ) Tm versus 1/l, for alkane chains. Linear fits are also

displayed.

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ordinate, respectively using the value of Tom found for each data

set (Fig. 4). The overlap between experimental and simulated

data (Fig. 5) is noteworthy. This of course indicates that the

value of the slope of the straight line, i.e. se/Dhom eqn (5), is

correctly predicted by our atomistic simulations. The value of

Dhom is a volumic value for a polymer crystal of infinite length. In

a nanocrystal, the value of the melting enthalpy is affected by the

considerable number of interfacial chains in comparison to bulk

chains. Nevertheless, determining the value of the interfacial

tension of the crystal with the melt, se, can be addressed.

The actual energy that is extracted from simulations carried

out in the canonical ensemble corresponds to the internal energy,

u. The melting energy Dum for each nanocrystal was then

Fig. 5 Modified Gibbs–Thomson equation representation reporting

experimental ( ) and simulated ( ) Tm/Tom versus 1/l, for alkane chains.

Linear fits are also displayed.

350 | Soft Matter, 2012, 8, 347–352

obtained as the jump in potential energy at the melting temper-

ature (Fig. 1a).‡ The bulk value of the melting energy, Duom, was

then extrapolated following a procedure analogue to what was

previously described for the computation of the melting

temperature. The link between Dum and Dhm is not straightfor-

ward at the microscopic scale since the volume change and the

Laplace pressure are not easy to grasp, as can be seen in Fig. 1b

where the volume of the droplet is ill-defined. However, at

infinite edge distances x and y, and infinite crystal thickness, l, i.e.

in the bulk, we can expect that the effect of the Laplace pressure

is negligible and change of volume is relatively modest, allowing

us to equate Dum and Dhm. A value of 258 � 26 J g�1 for Duom is

thus extrapolated which correlates well with the generally

accepted an experimental value of 280 J g�1 for Dhom.45 Using this

value, it is then possible to determine the interfacial tension from

the Gibbs–Thomson equation (eqn (5)), yielding 87 � 14 dyn

cm�1 for se. This value compares favourably with the value of

110 dyn cm�1 determined by Milner.46

In order to compare experimental and simulated values, two

kinds of experiments were actually considered (DSC thermo-

grams are displayed in the ESI†). The first series consisted of

melting points of linear alkanes. Such a system was used for

comparing simulation and experimental data, in order to report

any discrepancies in the Gibbs–Thomson behaviour. Constancy

in the difference between melting temperatures and comparable

slope, i.e. se/Dhom, has been revealed, making atomistic simula-

tion suitable to study more complex systems. For the second

series, we thus used the values of melting temperatures of linear

copolymers of ethylene with various polar monomers (methyl

acrylate, tert-butyl acrylate and N-vinyl pyrrolidinone). The l

value was calculated from the average number of CH2 units

between two polar groups (obtained from the molar composition

of the copolymer). Melting temperatures of functional olefins

and alkane chains are reported with respect to the inverse

thickness, l, in Fig. 6. The ordinate at origin for both kinds of

experimental data is equivalent: 399 � 2 K. Unexpectedly,

although the slope for simulated values matches the slope

obtained experimentally with linear alkanes, it is clearly different

from the one obtained with functional polyethylenes. Since Dhomis for a polymer with an infinite crystal length (i.e. for a pure

polyethylene devoid of polar groups), the difference in slope

reveals an interfacial tension of 61 dyn cm�1 for the copolymers,

which is only half the value found for alkanes. This lower

interfacial tension originates from the vastly different amor-

phous phases: CH3 groups for the alkanes versus a polyolefin rich

in which polar groups are concentrated for the copolymer.

Therefore, although both experimental systems follow a Gibbs–

Thomson behaviour, and although both systems lead to poly-

ethylene nanocrystals, their melting points are different, which

outlines the dominant influence of the environment of a nano-

crystal on its melting point, in stark contrast with bulk systems

‡ In order to compute the value of the melting energy in a systematicmanner, data above and below the melting temperature were fitted bytwo straight-lines. In order to eliminate distortion occurring attemperatures below the melting temperature, due to chain mobility andkinetic effects, only data for which more than 95% of the bonds werein dihedral trans conformation were selected. The melting energy wasthen calculated as the difference in energy between these twostraight-lines at the melting temperature.

This journal is ª The Royal Society of Chemistry 2012

Fig. 6 Gibbs–Thomson equation representation reporting experimental

data of alkane chains ( ) and functional olefins (-), and simulated ( )

Tm versus 1/l. Linear fits are also displayed.

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(the melting point of the polyethylene bulk crystal is not changed

by the environment).47–51

Conclusions

In this work we have demonstrated that by combining simulation

and experiment, molecular characteristics of nanocrystals were

evaluated. Simulations have been carried out in the canonical

ensemble to compute the melting temperature of sheets of

nanocrystals of alkane chains with different lengths. For all the

studied thicknesses, the simulated melting temperatures revealed

constancy in their difference with experimental data. Accord-

ingly, reporting the ratio of the melting temperature by the

melting temperature for an infinite chain length, i.e. polyethylene

crystal, yielded an accurate match between simulated and

experimental data. This similarity allowed the computation of

the heat of melting per CH2 in the polyethylene crystal in perfect

agreement with the experimental value. Interfacial tension was

thus computed and compared well with experimental data.

Melting transition phenomena have thus been captured in

atomistic simulation. Interestingly, we have demonstrated that

the melting point of a nanosheet is affected by its environment

(via the strong influence of the surface tension contribution).

Thus, unlike bulk materials for which melting points are

a canonical property (i.e. independent of environment), the

melting point of nanomaterials depends not only on the material

considered, on the size of the crystal, but also on its environment.

Thanks to the very good agreement between experimental and

simulated data for crystals of alkane chains, we believe that

atomistic simulations developed here offer an attractive and

convenient method to model nanocrystals and their

environment.

Acknowledgements

The present work was supported by the Centre Qu�eb�ecois des

Mat�eriaux Fonctionnels (CQMF), the Fonds Qu�eb�ecois de la

Recherche sur la Nature et les Technologies (FQRNT).

Computations have been made available thanks to the Canadian

Fund the Innovation (CFI), Calcul Qu�ebec (CQ), and Compute

This journal is ª The Royal Society of Chemistry 2012

Canada. SP thanks the National Science and Engineering

Research Council (NSERC) for a fellowship.

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