Glass transition phenomena observed in stereoregular PMMAs using molecular modeling

Glass transition phenomena observed in stereoregular

PMMAs using molecular modeling

Armand Soldera*, Noureddine Metatla

Departement de Chimie, Faculte des Sciences, Universite de Sherbrooke, Sherbrooke, Que., Canada JIK 2R1

Abstract

The way surface interactions could affect thermal behavior of a polymer is of relevant importance in the polymer-nanocomposite studies.

A complete description of such microscopic interactions is not yet reached. To give a better understanding of such a behavior poly(methyl

methacrylate), PMMA, offers a particular regard since according to the tacticity of its chain, different physical properties are exhibited.

Among the properties of interest is the glass transition temperature, Tg whose difference between the two configurations is more than 60

degrees. This difference tends to cancel out as the thickness of the PMMA film on a specific substrate decreases. Accordingly the Tg of the

isotactic chain, i-PMMA increases while that of the syndiotactic chain, s-PMMA, decreases. Different attempts have been carried out to

explain this different behavior. In this article, results stemming from molecular simulations in the bulk are reviewed, and used to give

possible explanation of both behaviors.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Stereoregular PMMA

1. Introduction

Poly(methyl methacrylate), PMMA, whose repeat unit is

displayed in Fig. 1, is a widely used polymer principally due

to its high transparency property in visible light. Moreover,

PMMA is also a matrix of importance in nanocomposites [1].

Actually, the commercially used PMMA is a statistical

copolymer composed of meso and racemic dimer. In fact,

according to the tacticity of its chain, different behaviors are

observed: different solubilities, crystal structures [2], .Accordingly, these variations in the properties can be

interpreted in changes in molecular characteristics only. As

a matter of fact, if molecular modeling can deal with these

differences a better understanding of the reason that gives rise

to these changes can be undertaken. Finally, a better

description from a phenomenological or theoretical view-

point can be addressed to the property of interest. Among

these properties the glass transition phenomenon offers a real

center of study since it is still a source of debate. From

experimental measurements, the isotactic chain, i-PMMA,

exhibits a glass transition temperature, Tg, near 60 8C, while

1359-835X/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compositesa.2004.10.019

* Corresponding author. Tel.: C1 819 8217650; fax: C1 819 8218017.

E-mail address: [email protected] (A. Soldera).

the syndiotactic chain, s-PMMA, shows a Tg, near 130 8C [3].

Moreover, this difference in Tgs tends to cancel out in

confined films, or at the surface, when the film thickness

decreases [4]. Different approaches exist to deal with these

glass transition behaviors. In this article molecular modeling

is used and offers a new regard to these tricky problems.

A review of simulations of stereoregular PMMAs in

regard to the glass transition phenomenon is presented. The

purpose of this review is to try to address the problem of

glass transition behavior in confined films of PMMAs

through results stemming from a simulation in the bulk. In

fact several questions have to be tackled before any

simulations of the glass transition of stereoregular

PMMAs at the surface, or in confined films, can be carried

out. Does the simulation in the bulk give the accurate

difference in Tgs? Can specific parameters be identified that

clearly exhibit a different behavior at the Tg and according

to the PMMA configuration? Can these parameters be used

to observe the Tg in confined films? Moreover, can the

results stemming from simulation in the bulk be used to

explain the behavior of PMMA films? To answer these

questions, the article is divided into three parts. The first part

deals with the force field used to carry out the simulations.

The second part presents the properties obtained from the

simulation in the bulk. The final part reports the conclusions

Composites: Part A 36 (2005) 521–530

www.elsevier.com/locate/compositesa

Fig. 1. PMMA repeat unit with the two vectors, ~uCH and ~uCO, which reveal

the backbone and side-chain/backbone local dynamics, respectively.

A. Soldera, N. Metatla / Composites: Part A 36 (2005) 521–530522

that stem from the simulation in the bulk that can explain the

PMMA interactions with the surface.

2. Simulation description

2.1. Force fields

Due to the great number of atoms involved in the

simulation, empirical methods, molecular mechanics and

dynamics, are currently used to deal with polymer simu-

lation. Consequently, the quality of the force field used for

the simulation has a major impact on the final results. It

actually expresses the average electronic interaction between

atoms [5]. For that purpose, it includes mathematical terms

whose parameters are fitted for each kind of atoms to

accurately express the different interactions. These terms

tend to mimic the different forces that exist between the

atoms. They are usually divided into two major groups: the

intramolecular terms that take into account the connectivity

and the flexibility of the polymer chains, and the inter-

molecular terms, or non-bond terms, that are constituted by a

repulsive, a dispersive and an electrostatic terms. Numerous

force fields exist. They differ from their mathematical

expression, their fitting parameters, the number of potential

associated with an atom, the coordinates, and the purpose of

the force field (biological interest, polymers .) [6]. They are

generally classified into two categories according to the form

of their mathematical expression. More precisely the second

category force fields posses off-diagonal intramolecular

terms, or cross-terms. They actually express possible

interactions between different internal parameters. The use

of these additional terms offers to the force field a greater

transferability between compounds, and allows a better

representation of some properties such as the normal modes.

However, the presence of these terms increases Central

Process Unit (CPU) time; and their physical significance is

still source of debate [7]. The use of a first category force field

that does not possess cross-terms, shows several advantages:

less CPU time is required, each mathematical term can be

attributed to a specific physical observation. Consequently a

greater portion of the phase space can be explored with the

same amount of CPU used for a second category force field.

In this paper the two force fields are compared regarding to

the difference in results stemming from molecular simulation

of the glass transition.

The two force fields used to simulate the glass transition

phenomenon are AMBER [8], a first category force field,

and pcff [9], a second category force field especially built up

to work with a great variety of polymers. Other first class

force fields exist, such as DREIDING that was used to

simulate mechanical properties of the isotactic configur-

ation [10]. The AMBER force field has been selected since

it was used to study the difference in solubility between

the two PMMA configurations [2]. The mathematical

expressions of the two classes of force fields are presented

in the following equations: mathematical expression of the

AMBER force field in Eq. (1), and that of pcff in Eq. (2).

V ZX

b

KðbKb0Þ2CX

q

HðqKq0Þ2

CX

f

½V1½1KcosðfKf01Þ�CV2½1Kcosð2fKf0

2Þ�

CV3½1Kcosð3fKf03Þ��

CXiOj

qiqj

3rij

CXi!j

43ij

sij

rij

� �12

Ksij

rij

� �6� �ð1Þ

K, H, V1, V2, V3, b0, q0, f01, f0

2, f03, 3ij, sij, qi, qj, are potential

parameters included into the force field. b, q, f, rij, are bond

length, valence angle, dihedral angle, and non-bonding

distance between two atoms i and j, respectively. These

parameters are got during simulation, and give the actual

energy value.

V ZX

b

½K2ðbKb0Þ2CK3ðbKb0Þ

3CK4ðbKb0Þ4�

CX

q

½H2ðqKq0Þ2CH3ðqKq0Þ

3CH4ðqKq0Þ4�

CX

f

½V1½1KcosðfKf01Þ�CV2½1Kcosð2fKf0

2Þ�

CV3½1Kcosð3fKf03Þ��C

Xc

Kcc2

CX

b

Xb

Fbb0 ðbKb0Þðb0Kb0

0Þ

CX

q

Xq0

Fqq0 ðqKq0Þðq0Kq00ÞC

Xb

Xq

FbqðbKb0ÞðqKq0Þ

CX

b

Xf

ðbKb0Þ½V01cosfCV 0

2cos2fCV 03cos3f�

CX

b0

Xf

ðb0Kb00Þ½V

001 cosfCV 00

2 cos2fCV 003 cos3f�

CX

q

Xf

ðqKq0Þ½V0001 cosfCV 000

2 cos2fCV 0003 cos3f�

CX

f

Xq

Xq0

Kfqq0cosfðqKq0Þðq0Kq00Þ

CXiOj

qiqj

3rij

CXi!j

3ij 2r*

ij

r*ij

!9

K3r*

ij

rij

!6" #ð2Þ

Table 1

Parameters for non-bonding energetic term; (a) for AMBER; (b) for pcff

force fields

Atoms si (A) 3i (kcal molK1) Ref.

(a) AMBER 43sij

rij

�12

Kqij

rij

�6� �

with 3ij Z ð3i3jÞ1=2 and sij Z ðsisjÞ

1=2

CT 3.500 0.066 a

C 3.750 0.105 a

OS 3.000 0.170 a

O 2.960 0.210 a

HC 2.500 0.030 a

Atoms r�i (A) 3i (kcal molK1)

(b) pcff: 3ij 2r*

ij

rij

�9

K3r*

ij

rij

�6� �

with 3ij Z2ð3i3jÞ1=2 ðr*

iÞ3ðr*

jÞ3

ðr*iÞ6ðr*

jÞ6

and

r*ij Z

ðr*iÞ6Cðr*

jÞ6

2

� �1=6

CT 4.010 0.054

C 3.810 0.120

OS 3.420 0.240

O 3.535 0.267

HC 2.995 0.020

For clarity, the mathematical expression is presented.a W.L. Jorgensen et al., J. Am. Chem. Soc., 118 (1996) 11225.

A. Soldera, N. Metatla / Composites: Part A 36 (2005) 521–530 523

K2, K3, K4, H2, H3, H4, V1, V2, V3, V01 V0

2, V03, V00

1, V002, V00

3,

V0001,V000

2, V0003, b0, q0, f0

1; f02, f0

3, 3ij, r*ij, qi, qj, are potential

parameters included into the force field. b, q, f, rij, are bond

length, valence angle, dihedral angle, and non-bonding

distance between two atoms i and j, respectively. These

parameters are got during simulation.

The different fitting parameters, except for cross-terms,

used for the simulations are presented in Table 1 for the

non-bonding term, Table 2 for the partial charges of the

electrostatic term, Table 3 for the bonding term, Table 4 for

the valence angle term, and Table 5 for the torsion term. The

AMBER type convention for atoms is employed: CTZsp3

carbon, CZcarbonyl carbon, HCZhydrogen attached to

carbon, OZcarbonyl oxygen, OSZether and ester oxygen.

The non-bonding parameters used with AMBER force

field actually come from the OPLS force field [11]. These

parameters have been determined to fit experimental density

and vaporization enthalpies. They actually show a great

success to model liquids. Comparatively the non-bonding

parameters (the partial charges) used in the AMBER force

field are computed from the crystal structures. It is the reason

why no accurate description of the glass transition have been

reached. It has to be mentioned that the intramolecular

Table 2

Partial charges for AMBER, and in bracket for pcff force fields

Atoms q Atoms q

C1 K0.120 (K0.106) C4 K0.180 (K0

H1C1 0.066 (0.053) H1C4 0.066 (0.0

H2C1 0.066 (0.053) H2C4 0.066 (0.0

C2 0.000 (0.000) H3C4 0.066 (0.0

C3 0.510 (0.702) O1 K0.330 (K0

a W. L. Jorgensen et al., J. Am. Chem. Soc., 118 (1996) 11225.

parameters used with the non-bonding parameters of OPLS

are used with the AMBER or CHARMM force fields [11].

2.2. Simulation

All the simulations have been carried out using the

periodic boundary conditions [12]. One polymer chain with

one hundred repeat units (RU), propagate into the periodic

box according to the self-avoiding walk [13] technique with

the long-range non-bonded interactions described by

Theodorou and Suter [14]. The procedure is implemented

in the Accelrys Amorphous-Cellq software. The initial

density was 1.115 g cmK3, as experimentally found [15].

While during the simulation using the pcff force field, three

configurations have been generated, in the case of

simulation using the AMBER force field, a different

procedure have been adopted in order to give a better

description of the configurational space. Thirty configur-

ations have been initially generated. Comparing the end-to-

end distance of each configuration and the predicted one,

using the RIS model, makes a first selection. The second

selection was made through an energetic viewpoint: after a

relaxation procedure (molecular dynamics and minimiz-

ation), the configurations with the highest energy are

rejected. Finally ten configurations for each PMMA

configurations have been selected for the simulation of the

glass transition.

To get the Tg, the simulated dilatometry technique was

employed [16]. As in the experimental dilatometry

technique, the specific volume, i.e. the inverse of the

density, is reported with respect to the temperature

(Fig. 2). The intercept of the lines joining the points of

the two phases, the vitreous and rubbery, yields the value

of the Tg. In order to acquire the density at a desired

temperature, molecular dynamics, MD, simulations are

performed in the NPT statistical ensemble, i.e. constant,

number of particles, pressure and temperature. The

simulated density directly stems from the knowledge of

the periodic box volume.

While for pcff force field, codes from Accelrys have been

employed, in the case of the AMBER force field, the

DL_POLY code was used [17]. The integration of the

Newtonian equations of motion to carry out the MD

simulation using the two force fields have been done

through the same pattern: the leap-frog variant of the

velocity Verlet algorithm with an integration step of

Atoms q Ref.

.159) O2 K0.430 (K0.531) a

53) C5 0.160 (0.066) a

53) H3C5 0.030 (0.053) a

53) H1C5 0.030 (0.053) a

.396) H2C5 0.030 (0.053) a

Table 4

Parameters for valence energetic term; (a) for AMBER; (b) for pqff force

fields

Angles q0 (deg) h kcal

molK1 8K2)

Ref.

(a) AMBER: H(q–q0)2

CT–CT–CT 109.50 37.0 a

CT–CT–C 111.10 63.0 b

HC–CT–HC 109.50 28.5 a

CT–CT–HC 109.50 37.0 a

CT–C–O 120.40 80.0 b

CT–C–OS 115.00 80.0 c

O–C–OS 125.00 80.0 c

C–OS–CT 117.00 60.0 c

OS–CT–HC 35.00 109.5 b

Angles q0 (deg) k2 (kcal

molKl

degK2)

k3 (kcal

molK1

degK3)

k4 (kcal

molK1

degK4)

(b) pcff:H2ðqKq0Þ2 CH3ðqKq0Þ

3 CH4ðqKq0Þ4

CT–CT–CT 112.67 39.516 K7.443 K9.5583

CT–CT–C 108.53 51.9747 K9.4851 K10.9985

HC–CT–HC 107.66 39.641 K12.921 K2.4318

CT–CT–HC 110.77 41.453 K10.604 5.129

CT–C–O 123.145 55.5431 K17.2123 0.1348

CT–C–OS 100.318 38.8631 K3.8323 K7.9802

O–C–OS 120.797 95.3446 K32.2869 6.3778

C–OS–CT 113.288 61.2868 K28.9786 7.9929

OS–CT–HC 107.688 65.4801 K10.3498 5.8866

a J. Wang, et al., J Comput. Chem., 22 (2000) 1219.b S.J. Weiner et al., J. Comput. Chem., 7 (1986) 230.c J. Wang, et al., J Comput. Chem., 21 (2000) 1049.

Table 3

Parameters for bonding energetic term; (a) for AMBER; (b) for pcff force

fields

Bonds b0 (A) k (kcal molK1

AK2)

Ref.

(a) AMBER: K (bKb0)2

CT–CT 1.526 310.0 a

CT–C 1.522 317.0 a

CT–HC 1.090 340.0 b

CT–OS 1.410 320.0 a

C–O 1.229 570.0 a

C–OS 1.323 450.0 c

Bonds b0 (A) k2 (kcal

molK1 AK2)

k3 (kcal

molK1 AK3)

k4 (kcal

molK1 AK4)

(b) pcff: K2ðbKb0Þ2 CK3ðbKb0Þ

3 CK4ðbKb0Þ4

CT–CT 1.53 299.67 K501.77 679.81

CT–C 1.5202 253.7067 K423.037 396.9

CT–HC 1.1010 345.0 K691.89 844.6

CT–OS 1.43 326.7273 K608.5306 689.0333

C–O 1.202 851.1403 K1918.4882 2160.7659

C–OS 1.3683 367.1481 K794.7908 1055.2319

For clarity, the mathematical expression is presented.a S.J Weiner et al., J. Comput. Chem., 7 (1986) 230.b W.D. Cornell et al., J. Am. Chem. Soc., 117 (1995) 5179.c J. Wang, et al., J. Comput. Chem., 21 (2000) 1049.

A. Soldera, N. Metatla / Composites: Part A 36 (2005) 521–530524

0.001 ps, and a duration of 100 ps. During MD simulations

with the pcff force field, the pressure was controlled

according to the Parrinello–Rahman algorithm [18], while

the temperature was imposed, in the primary step by the

velocity-scaling algorithm, and then through the Andersen

algorithm [19]. In the case of AMBER, the Berendsen

thermostat and barostat were used to keep the system at

prescribed temperatures and pressures [20].

Configurations at 307, 267, 247, 227, 167, 127, 67 and

27 8C, were used with a prolonged MD simulation time of

1 ns at the corresponding temperature. Configurations are

saved every 0.5 ps. This procedure was actually carried out

only with simulations using the pcff force field [21].

3. Results and discussion

3.1. Simulated dilatosmetric results

The dilatometric graphs for the two PMMA configur-

ations and for the two force fields are reported in Fig. 2. It

has to be pointed out, that with a view to clarity, standard

deviations are not displayed (in order of 0.005 g cmK3

below Tg and 0.01 g cmK3 above Tg). The difference in Tgs

between the two PMMA configurations is clearly exhibited

whatever the force field is. It is found to be 54 8C for pcff,

and 64 8C for AMBER. These values have to be compared

with the 69 8C experimentally obtained. It has to be pointed

out that the experimental difference takes into account

the mass of the polymer through the use of the Fox–Flory

equation [22]. A possible explanation of the better

representation of this difference by the AMBER force

field could stem from the better depiction of the phase space.

However, the absolute values of the Tgs using the AMBER

force field are clearly found superior than the experimental

ones. Such a difference could be explained from two

viewpoints. Firstly, the quenching rate is in order of 109

times more rapid than an experimental quenching rate.

According to the time–temperature superposition principle,

the Tg has to be higher than the experimental one. Secondly,

as mentioned by Boyd et al. the value of the dihedral

potential has a great impact on the value of the Tg: higher is

the dihedral potential higher is the Tg [23]. Such a behavior

is in agreement with the value of the dihedral potentials of

the two force fields, presented in Table 5. For instance,

the AMBER force field exhibits a cross-barrier energy for a

torsion along the backbone in order of 0.50 kcal molK1

higher than for the same torsion in the pcff force field.

Moreover, volumetric thermal coefficients, a, directly

obtained from dilatometric curves and experimental ones

are compared in Table 6 [22]. In the vitreous state, a

coefficient is slightly underestimated in the case of

simulation using the AMBER force field. In the rubbery

state, it is also slightly under the experimental value, but in

the case of the pcff force field, the coefficient is two times the

experimental value. Finally, the difference in specific

Table 5

Dihedral angle energetic term for AMBER, and in bracket for pcff force fields

Dihedral angles V1 (kcal molK1) f1 (deg) V2 (kcal molK1) (f2 (deg) V3 (kcal molK1) f3 (deg) Ref.

CT–CT–CT–CT 0.22 (0.00) 180.0 (0.0) 0.34 (0.0514) 180.0 (0.0) 0.195 (K0.1430) 0.0 (0.0) a

CT–CT–CT–C 0.00 (0.0972) 0.0 (0.0) 0.00 (0.0722) 0.0 (0.0) 0.1555 (K0.2581) 0.0 (0.0) b

HC–CT–CT–CT 0.00 (0.0000) 0.0 (0.0) 0.00 (0.0316) 0.0 (0.0) 0.155 (K0.1681) 0.0 (0.0) a

HC–CT–CT–HC 0.00 (K0.1432) 0.0 (0.0) 0.00 (0.0617) 0.0 (0.0) 0.130 (K0.1083) 0.0 (0.0) a

CT–CT–C–O 0.00 (0.0442) 0.0 (0.0) 0.00 (0.0292) 0.0 (0.0) 0.067 (0.0562) 180.0 (0.0) c

CT–CT–C–OS 0.00 (1.8341) 0.0 (0.0) 0.00 (2.0603) 0.0 (0.0) 0.000 (K0.0195) 0.0 (0.0) c

CT–OS–C–CT 0.00 (2.5594) 0.0 (0.0) 2.70 (2.2013) 0.0 (0.0) 0.000 (0.0325) 0.0 (0.0) d

O–C–OS–CT 1.40 (0.0000) 180.0 (0.0) 2.70 (2.2089) 180.0 (0.0) 0.000 (0.0000) 0.0 (0.0) a

C–OS–CT–HC 0.00 (0.0000) 0.0 (0.0) 0.00 (0.0000) 0.0 (0.0) 0.3833 (K0.1932) 0.0 (0.0) c

V1½1KcosðfKf01Þ�CV2½1Kcosð2fKf0

2Þ�CV3½1Kcosð3fKf03Þ�:

a J. Wang, et al., J. Comput. Chem., 22 (2000) 1219.b W.D. Cornell et al., J. Am. Chem. Soc., 117 (1995) 5179.c S.J. Weiner et al., J. Comput. Chem., 7 (1986) 230.d J. Wang, et al., J. Comput. Chem., 21 (2000) 1049.

Fig. 2. Simulated dilatometric curves for the two PMMA configurations

according two force fields: i-PMMA (;), and s-PMMA (6) through the pcff

force field simulation, and i-PMMA (+), and s-PMMA (*) through the

AMBER force field simulation.

A. Soldera, N. Metatla / Composites: Part A 36 (2005) 521–530 525

volume between the two PMMA configurations obtained

using the AMBER force field, 0.02 cm3 gK1 is in the same

order than the experimental one, 0.018 cm3 gK1 [24].

In conclusion, molecular modeling can deal with the

difference in the glass transition temperatures between

the two PMMA configurations. Specific volume is actually

the property that shows the difference in the Tgs. However,

this parameter could not be used in the simulation of

polymers in confined films. Specific properties have to be

regarded. Nevertheless, it has been shown that a first

category force field gives accurate results. To confirm this

finding, comparisons in the properties issue from the

simulations using the two force fields have to be performed.

3.2. Properties

Since the difference in Tgs has been clearly observed

between the two PMMA configurations using the pcff force

field, further investigations were carried out to explain such

variation. Two kinds of properties have been envisioned:

energetical and dynamical properties. Using the AMBER

force field, only energetical properties are presented. A

complete description of the local dynamics pictured by this

force field will be the subject of a subsequent article.

3.2.1. Energetical properties

The different energetic terms except the cross terms have

been regarded through MID simulations and compared

between the two force fields, pcff and AMBER.

3.2.1.1. Non-bond energy. The intermolecular energy of the

syndiotactic chain is found higher than in the case of the

isotactic chain for both force fields: the difference is

15 kcal molK1 in the case of the AMBER force field, and

35 kcal molK1 in the case of the pcff force field [21]. This

variation between the two force fields explains the

difference in the density between the two configurations

(Fig. 2). Such a behavior agrees with the free volume theory.

Higher is the non-bond energy, lower is the ability of

the chain to crossover the energetic barrier, and thus to

move more easily; and thus higher is the Tg. However, this

difference could not explain such a great difference between

the two Tgs.

3.2.1.2. Intramolecular energy. As in the case of the pcff

results, the dominant difference in intramolecular energy

terms obtained using the AMBER force field comes from

the valence angle energy [21]. The two other non-cross

terms, the bonding and dihedral energy functions, do not

exhibit significant differences. The variation in the valence

energy is different for both force fields: 0.75 and

0.35 kcal molK1 RUK1 in the case of pcff and AMBER

force fields, respectively. However, the difference in energy

between the two configurations stems from the same reason:

a greater aperture of the intra-diade angle q 0 (Fig. 1) to

lessen the interactions between the side-chains [25]. Using

the AMBER force field, the difference in the intradiade

Table 6

Experimental volumetric thermal expansion coefficients, a, and by simulations using the AMBER and pcff force fields, for both PMMA configurations, in the

vitreous and rubbery phases

State Configuration Experimental pcff AMBER

Vitreous i-PMMA 3.0!10K4 KK1 2.0!10K4 KK1

At-PMMA 2.7!10K4 KK1

s-PMMA 2.4!10K4 KK1 2.0!10K4 KK1

Rubbery i-PMMA 11.0!10K4 KK1 4.2!10K4 KK1

at-PMMA 5.7!10K4

s-PMMA 10.0!10K4 KK1 5.2!10K4 KK1

A. Soldera, N. Metatla / Composites: Part A 36 (2005) 521–530526

angle is in order of 0.68 while it is of 1.18 with pcff. This

difference has to be compared to the difference in the

valence energy, reported above. From Table 4, it is seen that

for a CT–CT–CT (CT corresponds to an sp3 atom in the

AMBER notation of atomic potentials) angle, the equili-

brium value is different: 112.678 in the case of pcff, and

109.50 for AMBER. Consequently, to compensate the same

amount of energy, the intradiade angle aperture is less

pronounced using the AMBER force field.

Dilatometric and energetic results clearly show that the

elimination of cross terms in the force fields does not greatly

affect the interpretation of the difference in Tgs between the

two PMMA configurations. One advantage of using a first

category force field is that energetic representation is

simplified. Accordingly, less parameters are needed for

further simulations of PMMA on a surface interaction.

Moreover, a better representation of the phase space is done

with AMBER; and thus the entropic contribution is better

represented and the quasi-ergodic-hypothesis, although not

satisfied, is better addressed.

3.2.2. Local dynamics

Primary results concerning the local dynamics have been

obtained using the AMBER force fields. For duration of

100 ps the same tendency in the results as in the case of the

pcff force field results is observed [26]. However, 100 ps is

not sufficient to correctly represent an autocorrelation

function, due to the lack of points at higher times. MD

simulations of 1 ns long have been carried out only with

the pcff force field, and therefore results with this force field

are presented in this paper.

3.2.2.1. Procedure. To specifically study the local dynamics

of a polymer chain, the correlation times have to be

computed. They are accessed through a molecular dynamics

trajectory by extracting the autocorrelation function,

hu(t)$u(0)i. This function corresponds to the memory

function of a molecular segment, represented by the vector

u, at a time t with respect to its initial orientation at tZ0.

The second term of the Legendre polynomial, P2(t), is then

directly obtained using the following equation:

P2ðtÞ Z3hðuðtÞ,uð0ÞÞ2iK1

2(3)

The integration of P2(t) over the entire time domain gives

numerically the correlation time tc; this procedure could not

be easily done most of the time. The use of a function that

fits the P2(t) curve bypasses this problem. Usually, a

stretching exponential function, a Kohlrausch–Williams–

Watts (KNW) function is employed [27]: exp½Kðt=tÞbKWW�,

where t is the averaged relaxation time, and bKWW

represents the departure from a simple exponential decay

(0!bKWW!1). The purpose of the use of such function is

that the correlation time is directly determined from the two

fitting parameters t and bKWW:

tc Zt

bKWW

G1

bKWW

� �(4)

The physical interpretation of the correlation time tc is

that it refers to the time spent for a molecular segment to

loose its ‘memory’. The fitting parameter, the KWW

exponent bKWW, is also a relevant parameter to study the

detailed mechanisms of the local dynamics. In fact bKWW

can be directly correlated to the cooperativity [28], or the

fragility of the compound according to the Coupling Model,

CM: lower is bKWW, higher is the fragility, and higher is

thus the cooperativity [29]. This KWW exponent captures

the cooperativity motion of the different geometrical

parameters it refers to. However, since these two par-

ameters, tc, and bKWW, are determined from one MD

trajectory, they have to be computed at different tempera-

tures. Accordingly, an accurate picture of the local

dynamics associated with a specific molecular segment is

exhibited. The graph of the correlation times with respect to

the temperature can then be expressed through the use of

the Vogel–Fulcher Tammann, VFT equation, or the

Williams–Landel–Ferry, WLF equation. Both equations

are mathematically equivalent. Since the VFT equation

exhibits an explicit term, the apparent activation energy B, it

is used in the following of the text [26]:

tcðTÞ Z A expB

T KT0

� �(5)

T0 is the temperature at which the configurational

entropy vanishes. B is referred as the apparent activation

energy. Actually, the B parameter is not an activation energy

since the equation clearly shows a non-Arrhenian behavior

at temperatures above Tg [30].

Fig. 4. KWW exponent, bKWW, of the i-PMMA (;), and s-PMMA (6), of

the C–H bond vector orientation, with respect to the Tg/T ratio.

A. Soldera, N. Metatla / Composites: Part A 36 (2005) 521–530 527

The VFT equation accurately expresses the link between

the motions responsible for the local dynamics with

molecular processes accountable for the glass transition.

Two other investigations are particularly useful to address

the local dynamics, and more precisely the cooperativity.

The first representation is to report the KWW exponent with

respect to the temperature. This exponent actually expresses

the departure from an Arrhenian behavior. If bKWW equals 1,

the motion of the molecular segment is independent of the

others; it corresponds to the characteristic temperature, T*.

This temperature can be associated to the temperature of the

a–b bifurcation [31]. If bKWW equals 0, the T0 temperature

is reached; it is usually referred to the isentropic Kauzmann

temperature, TK. Consequently, more bKWW is distant from

1, greater is the cooperativity between the molecular

segments it deals with. The second way of investigation of

the cooperativity is to report the inverse of the KWW

exponent with respect to the logarithm of the correlation

time. 1/bKWW is actually related to the cooperativity length

inside a Cooperatively Rearranging Region, CRR. This

length is different from the cooperative length computed

according to the Donth’s equation [32]. It has to be pointed

out that in such representation the temperature does not

intervene directly.

3.2.3. Backbone chain investigation

The motion of the backbone is captured through the

analysis of the C–H bond vector, ~uCH (Fig. 1), that is directly

linked to the polymer backbone. Actually information about

its motion gives insight to the backbone dynamics [26]. The

correlation times associated with this vector for the two

configurations, with respect to the inverse of temperature are

displayed in Fig. 3. It has to be pointed out that during all the

text both configuration behaviors are reported relatively to

their Tg. From Fig. 3, it is clearly seen that the two backbone

exhibit the same mobility at a particular temperature above

Fig. 3. Correlation times, tc, of the i-PMMA (;), and s-PMMA (6), of the

C–H bond vector orientation, with respect to the Tg/T ratio.

the Tg. This behavior is also confirmed by the approximately

same apparent activation energy of both configurations:

11.8 kcal molK1 for s-PMMA, and 12.5 kcal molK1 for

i-PMMA, and by the same cooperativity behavior (Fig. 4).

However, since the Tgs are different, mobility at an absolute

temperature has not to be the same for the two configurations.

The two linear relationships of bKWW with respect to the

inverse of the temperature observed in Fig. 4, and the ln(tc)

versus 1/bKWW represented in Fig. 5 are expected as

mentioned by Rault [31].

The three figures, Figs. 4–6, display the same behavior

for the two backbones of different PMMA configuration [26].

The same cooperativity along the backbone chain is observed

at a relative temperature above Tg, and the same CRR length

dependence on the correlation time is seen. There is thus no

backbone mobility difference. However, a clear difference in

Fig. 5. Neperian logarithm of the correlation times, tc, of the C–H bond

vector orientation with respect to the inverse of the KWW exponent,

1/bKWW, for i-PMMA (;), and s-PMMA (6).

Fig. 6. Correlation times,tc, of the i-PMMA (;), and s-PMMA (6), of the

CZO bond vector orientation, with respect to the Tg/T ratio.

Fig. 7. Neperian logarithm of the transition frequency of the side-chain of i-

PMMA (;), and s-PMMA (6), with respect to the Tg/T ratio.

A. Soldera, N. Metatla / Composites: Part A 36 (2005) 521–530528

Tgs definitely exists. Based on experimental observations

that showed that the side-chain motion influences the

backbone mobility [33], the side-chain mobility is

investigated.

3.2.4. Side-chain investigation

The motion of the side-chain could not be directly

captured through the analysis of the CZO bond vector, ~uCO

(Fig. 1), since three kinds of motion are considered through

the autocorrelation function analysis:

#

#

#

Fig. 8. KWW exponent, bKWW, of the i-PMMA (;), and s-PMMA (6), of

the CZO bond vector orientation, with respect to the Tg/T ratio.

the motion due to the backbone.

The librational motion that occurs at very low times is

actually not considered through the fit of the P2(t) graph by the

stretching function. Since the two other motions are difficult to

separate, the analysis of the CZO bond motion is analyzed

through the cooperativity between the side-chain and the

backbone. However, attention has to be paid to the interpret-

ation of such a motion because at small times only side-chain,

motion is envisioned: the b relaxation (side chain motion)

occurs at a lower temperature than the a relaxation (backbone

motion). Nevertheless, the stretching exponential considers the

entire time domain. In fact, it will be argued that the analysis

could only be carried out at high temperatures [34].

Considering the VFT equation applied to the correlation

times behavior of the two PMMA configurations, the apparent

activation energies are found different: 11 and 5 kJ molK1 for

s-PMMA and i-PMMA, respectively [26]. The different

behavior of the correlation time for the two configurations is

reported versus the inverse of temperature in Fig. 6. The

activation energy only associated with the side-chain motion is

directly obtained from the slope of the frequency of transitions

between the ‘up’ and ‘down’ states [35] versus the inverse

temperature (Fig. 7). From Fig. 7, one can see that the number

of transitions is greater for i-PMMA than for s-PMMA.

However, this difference tends to cancel out at high

temperatures. Such a behavior has to be compared to the

backbone mobility in this temperature range that is compar-

able for both configurations. Accordingly, at high tempera-

tures, backbone mobility of both PMMA configurations is

equal, and the rotation frequency of the side-chains is also

comparable. Therefore, nothing can be deduced from these

two behaviors when they are looked at separately. However,

the KWW exponent behavior versus the temperature is

different in this temperature range (Fig. 8). The inverse of

this exponent can actually be considered as a number of

individual units participating to the a motion: it thus reveals a

cooperativity [31]. Considering Fig. 8, at high temperatures,

an undoubtedly difference in bWWW is observed between the

two PMMA configurations. A real coupling between the side-

chain and the backbone therefore exists.

Comparatively, looking at the logarithm of the corre-

lation time versus the cooperativity length, i.e.1/bWWW

(Fig. 9), a clear difference is observed in the low time range

and cooperativity length. A difference of 0.6 in the

cooperativity length is actually significant since according

to Fig. 5, and for higher values, both configurations exhibit

Fig. 9. Neperian logarithm of the correlation times, tc, of the CZO bond

vector orientation, with respect to the inverse of the KWW exponent,

1/bKWW, of i-PMMA (;), and s-PMMA (6).

A. Soldera, N. Metatla / Composites: Part A 36 (2005) 521–530 529

the same behavior, i.e. a linear relationship. Considering a

particular correlation time in this region, the cooperativity

length of i-PMMA is found higher than that of s-PMMA; in

agreement with Donth’s cooperativity length [32]. Accord-

ingly the influence of the side-chain motion to the mobility

of the backbone is found more extensive in i-PMMA chains.

Looking only at the motion of the backbone for both

configurations the cooperativity length versus the corre-

lation time is identical since the mobility is comparable.

However, the way to make such mobility is different. The

influence of the side-chain motion of the i-PMMA chain on

the mobility of the backbone chain is greater.

The analysis to demonstrate the correlation between the

backbone and the side-chain is based on the important

mobility encountered at high temperatures. However, it

could not be pursued as the temperature approaches the Tg:

the number of side-chain rotations reduces more for

s-PMMA than for i-PMMA, while for the KWW

exponents, they tend to be equal for the two stereoisomers.

Therefore, as the temperature is decreasing this exponent

does not reveal the cooperativity between the backbone and

the side-chain any more since the backbone becomes more

and more rigid. Consequently, the influence of each kind of

relaxation cannot be well documented unless the different

motions can be undoubtedly separated. Further studies are

presently carried out in this way using the AMBER force

field.

3.3. Impact on the surface interaction

To correlate results obtained in the bulk to confined

geometries, a close link with experimental data has to be

undertaken. From experimental data, both PMMA configur-

ations exhibit a variation of the Tg when casted on a

surface [4]. From ellipsometry measurements, the Tg of

s-PMMA is found to decrease while Tg of i-PMMA

increases. It has to be pointed out that a decrease of the

i-PMMA Tg is observed when the a relaxation is reported

versus the film thickness [36].

Different approaches have been envisioned to explain the

variation of the Tg with the film thickness: Tg is different

according to the film depth (the multilayer approach) [37];

different sliding motions could explain the evolution of Tg

(reptation model) [38]; existence of percolation [39]. In this

article, we pursue a molecular approach by considering bulk

conclusions stemming from MD simulation run in the bulk,

and experimental observations at the interface.

From experimental data, it has been shown that the p

fraction of CZO bonded to the surface is equivalent for

both PMMA configurations. The increase of p with the

temperature has been reported at different film thicknesses,

h [4]. The increase of the slope, dp/dT, with h reveals that

mobility is not restricted on the surface. At a certain

temperature, T*, this slope decreases. This temperature has

been shown to be equivalent to the Tg. Thus at T!T*,

motions are principally due to the b motion, i.e. motion of

the side-chain. At TOT*, translational motions begin. From

simulation, it has been observed that the side-chain of

i-PMMA influences more the backbone mobility than in the

case of s-PMMA. Data stemming from simulation reveal

that this side-chain influence the backbone mobility in a

longer length for i-PMMA than in the case of s-PMMA.

Comparing with experimental data that reveal the equival-

ence of the fraction of bonded carbonyl on the surface for

both configurations, a freezing of bonded carbonyls for the

isotactic configuration will have a greater impact on the

backbone mobility than for the syndiotactic one. Accord-

ingly, the increase in Tg with the decrease in the film

thickness of the isotactic configuration can be understood

through a freezing of the side-chain that dampens the

backbone mobility. However, two experimental results have

to be clarified. Firstly, with regard to our conclusions the Tg

of s-PMMA has also to increase. However, this increase is

observed when interactions with the surface are strong.

Weak interactions have greater impact on i-PMMA than on

s-PMMA chains. The decrease of Tg of s-PMMA is due to

conformational rearrangement that seems to be more

relevant for this configuration; in fact s-PMMA is more

sensitive to entropic effect, i.e. confinement effects [4]. Such

results cannot be envisioned through a molecular simulation

in the bulk. However, the conformational energy has been

computed through the knowledge of g and t population. The

difference between the conformational energies of both

configurations is in agreement with experimental data: a

value of 471 cal molK1 [40] or 680 cal molK1 [15] is found

for i-PMMMA, and 879 cal molK1 [40] or 1227 cal molK1

[15], while from molecular modeling values 735 and

1035 cal molK1 are obtained for the isotactic and syndio-

tactic configurations, respectively. The conformational

energy of s-PMMA is clearly superior to that of i-PMMA.

The same tendency in experimental and simulated data

A. Soldera, N. Metatla / Composites: Part A 36 (2005) 521–530530

is preserved; this energy will thus be looked at in confined

films.

Secondly, experimentally the relaxation a decreases with

the thickness of the film [39]. From time–temperature

superposition principle, this relaxation a is equivalent to the

glass transition temperature, Tg. However, due to this

difference in the behavior of both properties, the confine-

ment induces a difference in the way a property has to be

regarded. More precisely, Tg has been identified to T*

determined from ellipsometry. Consequently, the properties

have not been observed in the same length scale. Molecular

modeling gives insight into the local dynamics, therefore

through molecular viewpoint of the Tg. Further analyses

have to be pursued in this way.

4. Conclusion

This article shows that the difference in the Tgs between

the two PMMA configurations can be revealed by a first or

second category force field. From an energetical analysis,

simulation using these two force fields reveal comparable

behavior: a higher non-bond energy for the syndiotactic

configuration, in agreement with the free volume theory,

and a higher valence energy that is due to a greater aperture

of the intra-diade angle for i-PMMA to lessen the

interactions between the side-chains. A local dynamics

investigation has then shown that both configurations

exhibit the same backbone behavior. The analysis of the

side-chain actually revealed that it is the greater coopera-

tivity between the side-chain and the backbone that is at the

origin of the difference in the Tgs. This result could explain

only partially the behavior of both stereoregular PMMAs

films on a surface. However, the analysis on the bulk has

displayed factors that are important in the molecular

simulation of PMMA on a surface: aperture of the intra-

diade angle, side-chain mobility, and conformational

energy. Such knowledge would give insight to specific

interactions of PMMA in nanocomposites.

Acknowledgements

The work has been possible through the financial support

of the Natural Sciences and Engineering Research Council

of Canada and Universite de Sherbrooke, the computer

facilities of the Reseau Quebecois de Calcul Haute

Performance, the Canada Foundation for Innovation, and

the Institut Superieur des Materiaux et Mecaniques Avances

du Mans. The authors also wish to thank Pr. Y. Grohens for

fruitful scientific discussion, and Pr. A.-M. Tremblay for the

use of his computer cluster, Elix2.

References

[1] Ash BJ, Siegel RW, Schadler LS. Macromolecules 2004;37:1358.

[2] Apel UM, Hentschke R, Helfrich J. Macromolecules 1995;28(6):

1778.

[3] Karasz FE, MacKnight WJ. Macromolecules 1968;1:537.

[4] Grohens Y, Hamon L, Reiter G, Soldera A, Holl A. Eur Phys JE 2002;

8:217.

[5] Jensen F. Introduction to computational chemistry. Chichester: Wiley;

1999.

[6] Osawa E, Lipkowitz KB. In: Boyd DB, editor. Reviews in

computational chemistry, vol. 6. New York: VCH Publishers; 1995.

p. 6.

[7] Gelin BR. Molecular modeling of polymer structures and properties.

Munich: Hanser Publishers; 1994.

[8] Pearlman DA, Case DA, Caldwell JW, Ross WS, Cheatham III TE,

DeBolt S, Ferguson D, Seibel G, Kollman P. Comp Phys Commun

1995;91:1.

[9] Maple JR. In: Schleyer PvR, editor. Encyclopedia of computational

chemistry, vol. 2. Chichester, UK: Wiley; 1998. p. 1015.

[10] Sane SB, Cagin T, Knauss WG, Goddard WA. J Comput Aided Mat

Des 2002;8:87.

[11] Jorgensen WL, Maxwell DS, Tirado-Rives J. J Am Chem Soc 1996;

118:11225.

[12] Allen MP, Tildesley DJ. Computer simulation of liquids. Oxford:

Clarendon Press; 1987.

[13] Meirovitch H. J Chem Phys 1983;79(1):502.

[14] Theodorou DN, Suter UW. Macromolecules 1985;18:1467.

[15] O’Reilly JM, Mosher RA. Macromolecules 1981;14:602.

[16] Rigby D, Roe RJ. J Chem Phys 1987;87(12):7285.

[17] Smith W, Forrester TR. J Mol Graphics 1996;14:136.

[18] Parrinello M, Rahman A. J Appl Phys 1981;52:7182.

[19] Andersen HC. J Chem Phys 1980;72:2384.

[20] Berendsen HJC, Postma JPM, Van Gunsteren WF, DiNola A,

Haak JR. J Chem Phys 1984;81:3684.

[21] Soldera A. Polymer 2002;43:4269.

[22] Soldera A. Macromol Symp 1998;133:21.

[23] Gee RH, Boyd RH. Comput Theor Polym Sci 1998;8:91.

[24] Lub J, Van Vroonhoven FCBM, Van Leyen D, Benninghoven A.

J Polym Sci, Part B: Polym Phys 1989;27:2071.

[25] Vacatello M, Flory PJ. Macromolecules 1986;19:405.

[26] Soldera A, Grohens Y. Macromolecules 2002;35:722.

[27] Antoniadis SJ, Samara CT, Theodorou DN. Macromolecules 1998;31:

7944.

[28] Ngai KL. J Chem Phys 1998;109:6982.

[29] Bbhmer R, Ngai KL, Angell CA, Plazek DJ. J Chem Phys 1993;99:

4201.

[30] Ferry JD. Viscoelastic properties of polymers. New York: Wiley;

1970.

[31] Rault J. J Non-Crystal Sol 2000;271:177.

[32] Donth E. J Polym Sci: Part B: Polym Phys 1996;34:2881.

[33] Kuebler SC, Schaefer DJ, Boeffel C, Pawalzik U, Spiess HW.

Macromolecules 1997;30:6597.

[34] Soldera A, Grohens Y. Polymer 2004;45:1307.

[35] Sundararajan PR. Macromolecules 1986;19:415.

[36] Hartmann J, Gorbatschow W, Hauwede J, Kremer F. Eur Phys JE

2002;8:145.

[37] Forrest JA, Dalnoki-Veress K. Adv Coll Interf Syst 2001;94:167.

[38] de Gennes PG. Eur Phys JE 2000;2:201.

[39] Arndt M, Stannarius R, Groothues H, Hempel E, Kremer F. Phys Rev

Lett 1997;79:2077.

[40] Grohens Y, Hamon L, Spevacek J, Holl Y. Macromol Symp 2003;

203:155.