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ical studies in this field encounter large problems when constructing the models of hybrid materials because it is difficult for ... ear expansion of the material.

ISSN 19950780, Nanotechnologies in Russia, 2010, Vol. 5, Nos. 5–6, pp. 333–339. © Pleiades Publishing, Ltd., 2010. Original Russian Text © P.V. Komarov, Y.T. Chiu, S.M. Chen, L.V. Zherenkova, Yu.N. Kovalenko, 2010, published in Rossiiskie nanotekhnologii, 2010, Vol. 5, Nos. 5–6.

ARTICLES

Effect of Inorganic Filler on the Thermal Properties of Polymer Nanocomposite: Atomistic Computer Simulation P. V. Komarova, b, Y. T. Chiuc, S. M. Chenc, L. V. Zherenkovab, and Yu. N. Kovalenkob a

Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28, Moscow, 119991 Russia b Tver State University, ul. Zhelyabova 33, Tver, 170100 Russia c Industrial Technology Research Institute, 31040 Hsinchu, Chung Hsing rd., 195, R.O.C email: [email protected] Received October 27, 2009; in final form February 10, 2010

Abstract—The effect of inorganic filler on the thermal properties of a polyimide–SiO2 nanocomposite was studied using atomistic molecular dynamics simulation. The increase in the weight content of the filler above a particular threshold (about 20%) was shown to cause a sharp drop in the temperature coefficient of the lin ear expansion of the material. DOI: 10.1134/S1995078010050083

INTRODUCTION Interest in the study and development of new mate rials based on polymer matrices and different nano sized inorganic fillers results from the wide variety of technical uses for these systems [1]. The main contri bution to the strength and thermal stability of nano composites comes from the scaffold of nanoparticles, while the polymer matrix is responsible for the conser vation of the shape of the material and imparts plastic ity to it [2, 3]. As a matter of fact, such materials are a kind of disperse systems. The effect of nanoparticles on the macroscopic properties of polymer nanocomposites is one of most poorly studied issues in nanosystem physics. Theoret ical studies in this field encounter large problems when constructing the models of hybrid materials because it is difficult for nanoparticles subsystem to simulta neously take into account such important factors as their shape, surface properties, and volume distribu tion. Numerous recent studies allow one to consider the research of thin polymer films in the context of research on nanocomposite materials where the poly mer matrix fills the space between nanoparticles. The polymer matrix in nanocomposites and thin polymer films does not form a continual phase O

because of the dimensional restrictions for the domain of matrix localization in these materials, which are comparable in order of magnitude with the size of macromolecules. Experimental studies show that sur face tension, stability, and other properties of poly mers forming thin films on solid surfaces differ consid erably from those of the continual phase [4, 5]. Dimensional effects have the most influence on the change in glass transition temperature (Tg), depending on the polymer layer thickness [6–8]. It is interesting that, according to the work by Forrest et al. [7], there was very little change in the glass transition tempera ture for a film applied to a surface and for a film con fined between two substrates. Keddie et al. [9] per formed the first studies on the influence of two differ ent substrates (gold and silicon oxide) on the glass transition temperature of thin films of poly(methyl methacrylate). It was found that Tg for poly(methyl methacrylate) decreases on a gold substrate and increases on a silicon oxide substrate. The authors explained this phenomenon by the change in the mobility of polymer molecules on account of surface forces. In [10] a detailed study on the effect that the energy of interphase interaction and polymerfilm thickness have on the thermal properties of polymer

O

N

O

N O

O n

O

Fig. 1. The chemical structure of polyimide prepared from BPDA and 134APB monomers.

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Si

O

Fig. 2. Unit cell of βcrystobalite.

was reported. Two different polymers were used in the study for the formation of thin films on the surface of a layer of octadecyltrichlorosilane, which, in turn, was exposed to different doses of X rays with the aim of changing the properties of its surface. The results unambiguously indicate the correlation between the physical properties of polymers forming thin films and both the film thickness and the properties of the sub strate surface. Direct evidence of the similarity in the physical properties was obtained recently for ultrathin films on solid surfaces and polymer matrices with nanoparticle filler. First, using computer simulation, Starr et al. [11] predicted that a system composed of a polymer matrix filled with regularly distributed nanoparticles should have the same physical properties as a thin polymer film on the surface of a solid substrate. It is implied that the substrate and the nanoparticles are made of the same material, while the polymer film thickness equals the average distance between nanoparticles. The conclusions were later confirmed experimentally in the work by Bansal et al. [12] by a detailed compar ison of the properties of a polystyrene–SiO2 nano composite and a polystyrene thin film applied on a flat SiOx substrate. The analogy between the two types of materials is explained by the fact that their physical properties are determined considerably by the proper ties of the polymer–inorganic subsystem interface (a boundary between two phases) [3] because a consider able portion of the polymer matrix is located within the area of surface forces of substrate (thin films) or nanoparticles (hybrid materials). We conclude from this fact that the main parameters that manage changes in the internal structure of hybrid materials are (1) the weight fraction of inorganic filler (wt) and (2) the energy of the interphase interaction. The first parameter is related, depending on system, to either the polymer matrix thickness or the average distance

between nanoparticles. The second parameter charac terizes the properties of the solid surface. The presence of two main parameters determining the behavior of hybrid systems is an attractive feature because it substantially simplifies the task of develop ing theoretical models of composite materials. Other factors, such as the shape of nanoparticles, the non uniformity of the surface, and the volume distribution features can be introduced in the model either as small perturbations or as limiting cases. Available theoretical approaches used for studying the properties of nanocomposites can be convention ally divided into three groups: (i) the molecular level of consideration based on the use of atomistic molecular dynamics; (ii) continual approaches, including micro mechanics and a finite element analysis; and (iii) hybrid schemes that combine different methods for considering materials on small and large temporal and spatial scales [13–15]. The advantage of the molecularlevel consider ation is that it is based on the chemical structure of the initial components. Its disadvantage is the impossibil ity to consider the large volumes of substance over suf ficiently long time intervals. In its turn, the continual models allow operations with macroscopic volumes of substance, but they make no allowance for the speci ficity of interatomic interactions. The hybrid methods make it possible to overcome the disadvantages of the described approaches; however, there is no recognized calculation scheme on their basis to date. Among available works, we can note those by Odegard et al. on the calculation of mechanical properties of nanocom posites [16]. In this paper we have studied the properties of a model of an organic–inorganic nanocomposite com posed of a layer of polymer matrix and a silica layer. Such a system, on the one hand, is equivalent to the thin polymer film on a solid substrate; on the other hand (due to the above analogy [11, 12]), it should correspond to a filled polymer matrix. The con structed model has been used to study the effect of the weight fraction of inorganic filler on the linear expan sion coefficient (LEC) of hybrid material. All calcula tions were performed by atomistic molecular dynam ics using DL POLY 2 [17] on the SKIF–MSU super computer complex [18]. MODEL The atomistic model of a polyimide–SiO2 nano composite was constructed as a nanolaminate (a lay ered nanomaterial composed of interlacing layers of polymer (matrix) and an inorganic component (a solid substrate or filler)). To develop the model, we used materials whose physical properties are well studied. To construct a polymer matrix, we employed polyim ide (PI) obtainable from two monomers: 3,3',4,4' biphenyltetracarboxylic dianhydride (BPDA) and 1,3bis(4aminophenoxy)benzene (134APB) [19].

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The substrate model was created on the basis of silicon dioxide as βcrystobalite with a hydroxylated surface (hereinafter called SiO2) [20]. Figure 1 shows the chemical structure of the initial PI monomer unit, while Fig. 2 displays the unit cell of βcrystobalite. All forces in the system were calculated using the Amber 96 valence force field [21] extended due to the use of additional force constants to determine the interactions involving silicon atoms [22]. The total potential energy of the PI–SiO2 system is a sum of contributions of the following summands: (i) defor mation potentials of covalent bonds, (ii) deformation potentials of valence angles, (iii) deformation poten tials of torsion angles, (iv) van der Waals interactions, and (v) electrostatic interaction. Oligomeric chains of polyimide molecule were constructed from 12 base monomers (Fig. 1). The number of molecules of polymer matrix N was the main calculation parameter, which specifies the weight fraction wt of silicon oxide in the system. A pure polymer matrix and substrate containing no polymer were considered as separate cases to verify the ade quacy of the valence force field. The optimal parame ters of structure and the partial atom charges of base PI monomer were calculated by AM1 and PM3 semiem pirical methods of molecular orbitals [23]. Both meth ods give similar values of calculated characteristics. A model of inorganic substrate was constructed as an infinite layer on the basis of a unit cell of βcrysto balite (UC) propagated along three noncoplanar crys tallographic directions. The total number of UC that form the substrate is n × m × k. Boundary atoms with free valence bonds along two directions were con nected through periodic boundary conditions. The remaining valence bonds on the “surface” of the sub strate were filled with hydroxyl groups. To calculate the partial charges on Si (O atoms and –OH groups in substrate), we used the same methods as were used for the assessment of atom charges in polyimide molecule, i.e., AM1 and PM3. Since these methods work well only for a small number of atoms, the calculations were performed for the model of silica as a spherical nanoparticle (17 Å in diameter) with a hydroxylated surface. The total number of atoms in the model is 183. The diameter of nanoparticle is close to the thickness of the substrate used upon the genera tion of nanocomposite models. For further applica tion, we used charges calculated by the AM1 method, because they agree well with the results of Hartree– Fock calculations in the 621G* basis for pure silicon oxide [24]. The models of PI–SiO2 materials were constructed in a parallelepiped cell with periodic boundary condi tions. Cell dimensions were defined by three vectors Ai = (Ai · ax(i), Ai · ay(i), Ai · az(i)), i = 1, 2, 3, where Ai are integers that define the cell size along three main directions a(1), a(2), and a(3). The substrate was con structed from 6 × 6 × 2 unit cells of βcrystobalite. To NANOTECHNOLOGIES IN RUSSIA

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produce a system with the prescribed number of chains N, a simulation cell (SC) with a parallelepiped shape with a square base equal to the size of the sub strate (i.e., A1 = A2) was created. The height of SC was adjusted in such a manner that it provides a sufficient space to dispose the prescribed number of polymer chains with an initial density of 0.5 g/cm3. A low initial density was used to facilitate the construction of the polymer chains with gauss conformation within the cell volume. The layer of polymer matrix was con structed with activated periodic boundary conditions along the substrate surface (the directions defined by the main vectors a(1) and a(2)). When a pure polymer matrix was constructed, the periodic boundary condi tions were activated along all directions of the simula tion cell. During simulation, cell geometry can vary dynam ically, because its parameters are associated with the components of isotropic strain tensor σαβ and pre scribed average pressure P (NσT ensemble). Tempera ture and pressure were controlled with a Nose– Hoover thermostat and barostat [25]. Electrostatic interactions were calculated using the Ewald’s method [26]. The dielectric permittivity of the ε medium was set to be 1, because all partial charges on atoms were taken into account in an explicit form. The Verlet scheme was used for the integration of the equation of motion. The values of the main parameters of molec ular dynamics were taken from [27], where a similar scheme was used to prepare an atomistic system. The initial procedure for preparing a material model includes several stages. Initially, each model is equilibrated under conditions of NVT ensemble at temperature T = 600 K during a short time interval of 100 ps to remove the internal strain in the system. Then, a compression procedure is performed under conditions of NσT ensemble at P = 1000 atm and T = 600 K. When the system density adopts an equilibrium value, the pressure in the system gradually decreases to normal (P = 1 atm) over 300 ps and system annealing is accomplished. At this stage, a stepwise temperature switch occurs within the 300 to 600 K range with a step of 50 K. The annealing procedure is repeated three times with an overall duration of 1.5 ns. At the final stage of preparation of each system, a final relaxation under normal conditions (P = 1 atm and T = 300 K) was performed until the system reached equilibrium density ρ. This was controlled by the fulfillment of 2 2 condition ( 〈 ρ 〉 – 〈 ρ〉 )1/2 < 0.02 g/cm3 over a time interval of 100 ps. The total duration of the last stage of relaxation depends on the size of the system, which is determined by the number of atoms comprising the system; on average, it is ~2 ns. For each value of parameter N, three independent models of material were generated to avoid the influence of initial condi tions on the final state of the system. Thus, the results below were obtained on the basis of averaging the cal culated characteristics for each parameter in the set. 2010

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ρ, g/cm3

1.30

1.25 Tg 1.20 300

350

400

450

500

550

600 T, K a(3)

Fig. 3. Temperature dependence of density ρ of BPDA– 134APB polyimide. The intersection of solid lines obtained by the linear approximation of ρ(T) profile on two characteristic sections corresponds to the glass transi tion temperature of the polymer.

a(1)

RESULTS AND DISCUSSION At the preliminary stage of computations, the ther mal characteristics of polymer matrix and substrate were reproduced to verify the adequacy of the used valence force field. Glass transition temperature Tg and linear expansion coefficient (LEC) were calcu lated for polyimide, while only LEC was calculated for silicon oxide. To perform test computations, a polymermatrix was constructed from nine polyimide molecules and prepared according to the scheme described in the previous section. Thermal characteristics were calcu lated based on the analysis of the temperature depen dence of system density ρ within the 250–600 K range. The position of the first break on the ρ(T) dependence shown in Fig. 3 is identified with the glass transition temperature of the polymer [27]. The assessment of glass transition temperature gives 495 ± 5 K, which agrees well with the experimental value of 493 reported in [19]. Let us note that all points of Fig. 3 were obtained by the averaging of the results of three inde pendent initial configurations of the system. The cal culation of the linear expansion coefficient was accomplished by the formula –1

β = ( 3ρ 0 ) ( Δρ/ΔT ) P = const

nz

(1)

for T = 300 K, where ρ0 is an average value of system density for the given temperature. The magnitude of linear expansion coefficient β is 74.4 ± 9.1 × 10–6/K, which agrees well with the range for LEC of 40–60 × 10–6/K typical for polyimides with allowance made for the calculation error [28].

Fig. 4. A snapshot of the simulation cell manifolded along the main axes a(1) and a(3). The cell contains an example of a nanocomposite model constructed from 9 polyimide chains composed of 12 base monomers (Fig. 1) and substrate involving 6 × 6 × 2 unit cells of silicon oxide as βcrystobalite. The system was obtained at T = 300 K and P = 1 atm, the weight fraction of inorganic component was wt = 0.38.

The linear expansion coefficient for silicon oxide was calculated for the system composed of 5 × 5 × 5 unit cells of βcrystobalite. Upon constructing this system, all atoms with free valences were connected through periodic boundary conditions along three main crystallographic directions (the continual phase model). To study the temperature response of the sys tem, the model of substrate was relaxed at two temper atures, 300 and 350 K, under conditions of the NσT ensemble (P = 1 atm) for 500 ps. The established val ues of system density were used for to calculate LEC by formula (1). The obtained result of 0.8 ± 0.5 × 10–6/K is close to the experimental value for βcrystobalite [29]. Thus, the results of an assessment of the glass tran sition temperature and linear expansion coefficient allow us to suppose that the variant of the valence force field [21, 22] gives the opportunity to predict with acceptable accuracy the thermal properties of materi als simulated in this work. To study the effect of inorganic filler on the linear expansion coefficient of PI–SiO2 nanocomposite, we constructed a number of model samples of the mate rial at T = 300 and 350 K (P = 1 atm). The number of polymer chains N varied within the range of 3 to 25, whereas the number of unit cells of βcrystobalite (n ×

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ρ, g/cm3 (a)

80 4

I

60

II

3 40 2 20 III

1

0 0

0.2

0.4

0.6

0.8

1.0 wt

Fig. 5. Dependence of the linear expansion coefficient of the PI–SiO2 system on the weight fraction of silicon oxide (P = 1 atm).

0

−30

20

10

0

10

20

30 z, A

ρ, g/cm3 (b) 4

m × k) in the substrate was fixed to be 6 × 6 × 2. The total number of atoms in the system varied from 4326 to 23882. Figure 4 exemplifies the visualization of one of the prepared nanocomposite models with N = 9. For clarity, the figure shows a manifolded snapshot of a simulation cell in the projection on vectors a(1) and a(3). It is obvious that our model has anisotropic ther mal properties; therefore, its LEC was calculated only for one specified direction, which was defined by per pendicular nz to the substrate surface (see Fig. 4). To calculate LEC, all atom coordinates of each sys tem were stored every 10 ps in the course of a produc tive run for 0.5 ns. The average thickness L of the model sample was calculated on the basis of the trajec tory for two temperatures (T = 300 and 350 K) along the normal nz. The linear expansion coefficient in this case was calculated by the formula –1

(2) β = ( L 0 ) ( ΔL/ΔT ) P = const , where L0 corresponds to T = 300 K. The calculated dependence of LEC on the weight fraction of inorganic filler wt is shown in Fig. 5. The value of wt was calculated as the ratio of the substrate weight to the total weight of the system for the specific magnitude of N. The figure shows that the dependence of the linear expansion coefficient for the simulated system has a very nonlinear character. We can mark in the plot three typical sections denoted by Roman numerals. On the first section, when wt Ⰶ 0.2, LEC is virtually constant. Further, when wt exceeds a thresh old value of about 0.2, a sharp drop in LEC is observed, which gradually goes to saturation in the third section. The predicted behavior for the LEC of PI–SiO2 nanocomposite (Fig. 5) agrees qualitatively with the observed profile in [30], which presents measurements of the thermal properties of thin polymer films based on polyimide with a different weight fraction of silica NANOTECHNOLOGIES IN RUSSIA

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30

−20

−10

0

10

20

30 z, A

Fig. 6. The profile of local density of the PI–SiO2 system in the direction perpendicular to substrate surface (com posed of 6 × 6 × 2 unit cells of βcrystobalite): (a) N = 3 and (b) N = 9 (P = 1 atm, T = 300 K). Dotted lines correspond to the density of free polymer matrix (1.28 g/cm3) and sil icon oxide (2.3 g/cm3).

nanoparticles. The abovementioned experimental study also noted a region with sharp drop in the LEC of the material, after which the variation in β(wt) is stabilized. Let us note that the nanocomposite described in [30] is not completely equivalent to the system studied in the present work, because another polyimide was used for its preparation. Moreover, the computer model under study had obvious dimensional restrictions both in the length of oligomer chain and in the total simulation size cell. The qualitative agree ment of our results indicates the existence of general mechanisms for the formation of the physical proper ties of filled nanocomposites and thin polymer films. To explain the dependence of LEC on wt, let us analyze the profiles of local density ρ(z) (see Fig. 6) for N = 3 and 9, which correspond to a filler weight frac tion of 0.65 and 0.38. Each profile is drawn along the 2010

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normal toward the substrate surface nz with 0.5 Å spac ing. For clarity, each diagram shows average densities for the free polymer matrix (1.28 g/cm3) and substrate (2.3 g/cm3) as dotted lines. A sharp growth in density is observed for polyimide in the transition region, whose thickness is about 7.5 Å, on account of the strong interaction of two subsystems (polymer–sub strate). This agrees well with the results of other works related to the study of phase boundary formation [31, 32]. Since surface forces are characterized by fast attenuation (their amplitude is proportional to r–4 [33]), the perturbation of polymer density distribution caused by substrate quickly attenuates. Therefore, the average polyimide density at long distances from the surface equals the density of the free polymer matrix (see Fig. 6b). A comparison of two ρ(z) profiles in Figs. 6a and 6b demonstrates the increasing role of interphase interac tions in the distribution formation of polymer local density when the polymer matrix becomes thinner. It is obvious that polyimide molecules form domains with high degrees of ordering under the action of sur face forces. This is accompanied by the appearance of secondary peaks in ρ(z), which become comparable with the height of peaks in the interfacial zone. It is obvious that semicrystalline domains with high pack ing degrees, where the mobility of polymer molecules should decrease sharply, appear in the system [34]. Thus, when a small share of polymer molecules is within the range of action of surface forces, a polyim ide matrix located far from the substrate surface makes the largest contribution to the LEC formation of the material. Because for the most part it retains its former properties, the LEC of material remains close to the LEC of free polymer (Fig. 5, section 1). The volume fraction of polymer, which is affected by the surface forces of the substrate, increases with the weight frac tion of filler in the system. Since the properties of the polymer matrix in the interfacial zone differ from the properties of free polymer, LEC forms thanks to three contributions when the fraction of SiO2 in the system increases: the undisturbed polymer matrix (with high LEC), the phase boundary, and the substrate (whose LEC is low). Thus, the second and third contributions quickly become predominant when wt rises; this results in a substantial change in the physical proper ties of the nanocomposite (Fig. 5, section II). And, finally, when the whole polymer matrix is affected by surface forces, the properties of the hybrid system show a trend toward saturation (see Fig. 5, section III). CONCLUSIONS In this work, we developed an atomistic model of organic–inorganic nanocomposite based on polyim ide (polymer matrix) and silicon oxide in the form of βcrystobalite (solid substrate). We proceeded from the assumption that two kinds of nanomaterials are

equivalent—filled polymer matrices and ultrafine polymer films on solid surfaces—which was estab lished in [1, 2]. The model samples were used for studying the formation of the physical properties of hybrid material using the example of a molecular dynamics calculation of the linear expansion coeffi cient. Our calculations show that the dependence of the thermal properties of PI–SiO2 system has a strongly nonlinear character. The increase of the weight con tent of inorganic filler above a threshold value of ~0.2 leads to a sharp change in the LEC of the material. The results of this work confirm the conclusion that the polymer matrix located in the zone where the surface forces of an inorganic component act plays a leading role in the formation of the physical properties of polymer nanocomposites. Since the volume fraction of interface is dependent on both interphase interac tion energy and the area of the phase boundary, the properties of a nanocomposite can be readily con trolled by adjusting the properties of filler and its weight fraction in the system. It is important to determine the main parameters affecting the behavior of hybrid systems, because this makes it possible to substantially simplify the develop ment of theoretical models of nanocomposites and offers a methodological basis for applied studies on the development of structural materials of a new genera tion. ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research (project no. 09–03–00671a). We thank the Research Computer Center at Moscow State University for providing computational resources of the SKIF cluster to perform timecon suming computations. REFERENCES 1. N. G. Rambidi and A. V. Berezkin, Physical and Chem ical Foundations of Nanotechnologies (Fizmatlit, Mos cow, 2008) [in Russian]. 2. H. Fischer, “Polymer Nanocomposites: From Funda mental Research to Specific Applications,” Mater. Sci. Eng. 23, 763–772 (2003). 3. L. S. Schadler, L. C. Brinson, and W. G. Sawyer, “Poly mer Nanocomposites: A Small Part of the Story,” JOM 59 (3), 53–60 (2007). 4. J. A. Forrest, C. Svanberg, K. Revesz, M. Rodahl, L. M. Torell, and B. Kasemo, “Relaxation Dynamics in Ultrathin Polymer Films,” Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 58 (2), R1226–R1229 (1998). 5. B. Frank, A. P. Gast, T. P. Russell, H. R. Brown, and C. Hawker, “Polymer Mobility in Thin Films,” Macro molecules 29 (20), 6531–6534 (1996). 6. J. L. Keddie, R. A. L. Jones, and R. A. Cory, “Size Dependent Depression of the Glass Transition Tem

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