Glass transition phenomena observed in stereoregular PMMAs using molecular modeling

Composites: Part A 36 (2005) 521–530 www.elsevier.com/locate/compositesa

Glass transition phenomena observed in stereoregular PMMAs using molecular modeling Armand Soldera*, Noureddine Metatla De´partement 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 viewpoint 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 * Corresponding author. Tel.: C1 819 8217650; fax: C1 819 8218017. E-mail address: [email protected] (A. Soldera). 1359-835X/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2004.10.019

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

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Fig. 1. PMMA repeat unit with the two vectors, u~ CH and u~ CO , which reveal the backbone and side-chain/backbone local dynamics, respectively.

that stem from the simulation in the bulk that can explain the PMMA interactions with the surface.

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 configuration [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). VZ

X

2. Simulation description

b

C 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 simulation. 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 intermolecular 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

KðbKb0 Þ2 C

X

HðqKq0 Þ2

q X ½V1 ½1KcosðfKf01 ÞCV2 ½1Kcosð2fKf02 Þ f

CV3 ½1Kcosð3fKf03 Þ X qi qj X  sij 12  sij 6  C C 43 K 3rij i!j ij rij rij iOj

ð1Þ

K, H, V1, V2, V3, b0, q0, f01 , f02 , 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. VZ

X ½K2 ðbKb0 Þ2 CK3 ðbKb0 Þ3 CK4 ðbKb0 Þ4  b

X C

½H2 ðqKq0 Þ2 CH3 ðqKq0 Þ3 CH4 ðqKq0 Þ4 

q

X C

½V1 ½1KcosðfKf01 ÞCV2 ½1Kcosð2fKf02 Þ

f

CV3 ½1Kcosð3fKf03 ÞC XX C b

C C C C

Fbb0 ðbKb0 Þðb0 Kb00 Þ Fqq0 ðqKq0 Þðq0 Kq00 ÞC

q0

XX b

f

b0

f

XX XX q

ðb0 Kb00 Þ½V100 cosfCV200 cos2fCV300 cos3f ðqKq0 Þ½V1000 cosfCV2000 cos2fCV3000 cos3f

f

q

q

3rij

Kfqq0 cosfðqKq0 Þðq0 Kq00 Þ

0

X qi qj iOj

Fbq ðbKb0 ÞðqKq0 Þ

q

ðbKb0 Þ½V10 cosfCV20 cos2fCV30 cos3f

C

C

XX b

XXX f

Kc c2

c

b

XX q

X

X C i!j

" 3ij

rij* 2 * rij

!9

rij* K3 rij

!6 # ð2Þ

A. Soldera, N. Metatla / Composites: Part A 36 (2005) 521–530 Table 1 Parameters for non-bonding energetic term; (a) for AMBER; (b) for pcff force fields ˚) si (A 3i (kcal molK1) Ref.   

 sij 12 qij 6 (a) AMBER 43 rij K rij with 3ij Z ð3i 3j Þ1=2 and sij Z ðsi sj Þ1=2 Atoms

CT C OS O HC

3.500 3.750 3.000 2.960 2.500 ˚) ri (A

Atoms

a

0.066 0.105 0.170 0.210 0.030

a a a a

3i (kcal molK1)

 9

r * 6  r* ðr * Þ3 ðr * Þ3 with 3ij Z 2ð3i 3j Þ1=2 ðri* Þ6 ðrj* Þ6 and (b) pcff: 3ij 2 rijij K 3 rijij i j  6 1=6 * Þ Cðr * Þ6 ðr rij* Z i 2 j CT C OS O HC

4.010 3.810 3.420 3.535 2.995

0.054 0.120 0.240 0.267 0.020

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

K2, K3, K4, H2, H3, H4, V1, V2, V3, V0 1 V0 2, V0 3, V00 1, V00 2, V00 3, V000 1,V000 2, V000 3, b0, q0, f01 ; f02 , f03 , 3ij, rij* , 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

523

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 configurations have been initially generated. Comparing the end-toend 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 minimization), 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

Table 2 Partial charges for AMBER, and in bracket for pcff force fields Atoms C1 H1C1 H2C1 C2 C3 a

q K0.120 (K0.106) 0.066 (0.053) 0.066 (0.053) 0.000 (0.000) 0.510 (0.702)

Atoms C4 H1C4 H2C4 H3C4 O1

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

q K0.180 (K0.159) 0.066 (0.053) 0.066 (0.053) 0.066 (0.053) K0.330 (K0.396)

Atoms O2 C5 H3C5 H1C5 H2C5

q

Ref.

K0.430 (K0.531) 0.160 (0.066) 0.030 (0.053) 0.030 (0.053) 0.030 (0.053)

a a a a a

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Table 3 Parameters for bonding energetic term; (a) for AMBER; (b) for pcff force fields ˚) b0 (A

Bonds

(a) AMBER: K (bKb0)2 CT–CT 1.526 CT–C 1.522 CT–HC 1.090 CT–OS 1.410 C–O 1.229 C–OS 1.323 Bonds

˚) b0 (A

k2 (kcal ˚ K2) molK1 A

k (kcal molK1 ˚ K2) A

Ref.

310.0 317.0 340.0 320.0 570.0 450.0

a a b a a c

k3 (kcal ˚ K3) molK1 A

k4 (kcal ˚ K4) molK1 A

(b) pcff: K2 ðbK b0 Þ2 C K3 ðbK b0 Þ3 C K4 ðbK b0 Þ4 CT–CT 1.53 299.67 K501.77 CT–C 1.5202 253.7067 K423.037 CT–HC 1.1010 345.0 K691.89 CT–OS 1.43 326.7273 K608.5306 C–O 1.202 851.1403 K1918.4882 C–OS 1.3683 367.1481 K794.7908

679.81 396.9 844.6 689.0333 2160.7659 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.

Table 4 Parameters for valence energetic term; (a) for AMBER; (b) for pqff force fields Angles

q0 (deg)

(a) AMBER: H(q–q0)2 CT–CT–CT 109.50 CT–CT–C 111.10 HC–CT–HC 109.50 CT–CT–HC 109.50 CT–C–O 120.40 CT–C–OS 115.00 O–C–OS 125.00 C–OS–CT 117.00 OS–CT–HC 35.00 Angles

q0 (deg)

k2 (kcal molKl degK2)

h kcal molK1 8K2)

Ref.

37.0 63.0 28.5 37.0 80.0 80.0 80.0 60.0 109.5

a b a a b c c c b

k3 (kcal molK1 degK3)

(b) pcff:H2 ðqK q0 Þ2 C H3 ðqK q0 Þ3 C H4 ðqK q0 Þ4 CT–CT–CT 112.67 39.516 K7.443 CT–CT–C 108.53 51.9747 K9.4851 HC–CT–HC 107.66 39.641 K12.921 CT–CT–HC 110.77 41.453 K10.604 CT–C–O 123.145 55.5431 K17.2123 CT–C–OS 100.318 38.8631 K3.8323 O–C–OS 120.797 95.3446 K32.2869 C–OS–CT 113.288 61.2868 K28.9786 OS–CT–HC 107.688 65.4801 K10.3498 a b c

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

k4 (kcal molK1 degK4)

K9.5583 K10.9985 K2.4318 5.129 0.1348 K7.9802 6.3778 7.9929 5.8866

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 configurations 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

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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 CT–CT–CT–C HC–CT–CT–CT HC–CT–CT–HC CT–CT–C–O CT–CT–C–OS CT–OS–C–CT O–C–OS–CT C–OS–CT–HC

0.22 (0.00) 0.00 (0.0972) 0.00 (0.0000) 0.00 (K0.1432) 0.00 (0.0442) 0.00 (1.8341) 0.00 (2.5594) 1.40 (0.0000) 0.00 (0.0000)

180.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 180.0 (0.0) 0.0 (0.0)

0.34 (0.0514) 0.00 (0.0722) 0.00 (0.0316) 0.00 (0.0617) 0.00 (0.0292) 0.00 (2.0603) 2.70 (2.2013) 2.70 (2.2089) 0.00 (0.0000)

180.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 180.0 (0.0) 0.0 (0.0)

0.195 (K0.1430) 0.1555 (K0.2581) 0.155 (K0.1681) 0.130 (K0.1083) 0.067 (0.0562) 0.000 (K0.0195) 0.000 (0.0325) 0.000 (0.0000) 0.3833 (K0.1932)

0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 180.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0)

a b a a c c d a c

V1 ½1K cosðfK f01 ÞC V2 ½1K cosð2fK f02 ÞC V3 ½1K cosð3fK f03 Þ: 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.

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.

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

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

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.

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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

Vitreous

i-PMMA At-PMMA s-PMMA i-PMMA at-PMMA s-PMMA

Rubbery

Experimental

2.4!10K4 KK1 11.0!10K4 KK1

2.0!10K4 KK1 4.2!10K4 KK1

10.0!10K4 KK1

5.2!10K4 KK1

5.7!10K4

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: 3hðuðtÞ,uð0ÞÞ2 i K 1 2

AMBER 2.0!10K4 KK1

2.7!10K4 KK1

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 equilibrium 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.

P2 ðtÞ Z

pcff 3.0!10K4 KK1

(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:   t 1 tc Z G (4) bKWW bKWW 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 parameters, tc, and bKWW, are determined from one MD trajectory, they have to be computed at different temperatures. 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]:   B tc ðTÞ Z A exp (5) T K T0 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].

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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, u~ CH (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.

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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.

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).

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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.

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, u~ CO (Fig. 1), since three kinds of motion are considered through the autocorrelation function analysis: # the librational motion; # the motion due to the side-chain; # 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 interpretation 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

Fig. 7. Neperian logarithm of the transition frequency of the side-chain of iPMMA (;), and s-PMMA (6), with respect to the Tg/T ratio.

temperatures. Such a behavior has to be compared to the backbone mobility in this temperature range that is comparable for both configurations. Accordingly, at high temperatures, 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 sidechain and the backbone therefore exists. Comparatively, looking at the logarithm of the correlation 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. 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.

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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).

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]. Accordingly 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 correlation 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 configurations exhibit a variation of the Tg when casted on a surface [4]. From ellipsometry measurements, the Tg of

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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 equivalence 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. Accordingly, 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 syndiotactic configurations, respectively. The conformational energy of s-PMMA is clearly superior to that of i-PMMA. The same tendency in experimental and simulated data

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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 confinement 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 cooperativity 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 intradiade 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 Re´seau Que´be´cois de Calcul Haute Performance, the Canada Foundation for Innovation, and the Institut Supe´rieur des Mate´riaux et Me´caniques Avance´s 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.

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