The Molecular Structure and Dynamics of Short Oligomers of PMDA

Yoichi Ogata , Kenji Fujita , Megumi Nakanishi , Motoki Okaniwa , Manabu Sekiguchi. Journal of Polymer Science Part B: Polymer Physics 2012 50 (18), 1...
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J. Phys. Chem. B 2002, 106, 4617-4631

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The Molecular Structure and Dynamics of Short Oligomers of PMDA-ODA and BCDA-ODA Polyimides in the Absence and Presence of Water S. Neyertz,* D. Brown, A. Douanne, C. Bas, and N. D. Albe´ rola Laboratoire Mate´ riaux Organiques a` Proprie´ te´ s Spe´ cifiques (LMOPS), UMR CNRS 5041, Baˆ timent IUT, UniVersite´ de SaVoie, Campus Scientifique, 73376 Le Bourget-du-Lac, France ReceiVed: August 15, 2001; In Final Form: January 29, 2002

Four constant-pressure molecular dynamics (MD) simulations were undertaken at room temperature to compare, at the molecular level, the structure and dynamics of short model oligomers of two polyimides based on the same diamine, 4,4'-diaminophenyl ether (ODA). The dianhydrides were, respectively, pyromellitic dianhydride (PMDA) and bicyclo(2.2.2)-oct-7-ene-2,3,5,6-tetracarboxylic dianhydride (BCDA). The systems under study were first analyzed in the pure state. In a second stage, 3.3% water, by weight, was inserted into the simulation boxes to mimic the highly hygroscopic nature of polyimides. The models are in the glassy state and reproduce many features found experimentally. The PMDA-ODA and BCDA-ODA oligomers have similar intramolecular flexibility, similar trends in the energy and density variations when water is added, as well as identical probeaccessible volumes and oligomer-water interaction sites. However, the choice of the dianhydride clearly affects the density, the cohesion, the intermolecular interactions, the morphology of the void space, and the degree of water clustering in these short-chain systems. The diffusion of water in the anomalous regime is characterized by two distinct behaviors.

1. Introduction Polyimides are high-performance polymers, which exhibit several technologically interesting characteristics, such as excellent mechanical properties, electric insulation, and good chemical, thermal, and long-term stabilities. Their applications range from electronics, separation membranes, and coatings to foams and fibers.1 Most efforts in the field of polyimides are currently directed toward their synthesis and characterization.2 However, the synthesis of a new polyimide and the analytical laboratory work will often be very time-consuming and expensive. Within this context, other approaches such as molecular dynamics (MD) simulations,3 which integrate Newton’s equations of motion for all the atoms in a system, can be used to complement experimental evidence. MD provides a structural and dynamical picture of the system under study and attempts to link its characteristics at the atomistic level to properties of interest. It can also be used as a predictive tool for designing new materials or for interpreting various phenomena. Simultaneously, experiment serves as a check on simulations, so that both approaches will strengthen each other.4 The number of parameters required for models of complex polymers such as polyimides is large and makes these systems very difficult to simulate. In addition, the huge variety in chemical motifs allows for several possible combinations of diamines and dianhydrides1 with a wide range of physical properties. Some of these aspects have already been studied by MD simulations, such as carbon dioxide, oxygen, nitrogen, and hydrogen gas diffusion in fluorinated 6FDA-based short- or long-chain polyimides,5-8 or oxygen and nitrogen gas diffusion in BTDA-DMDA.9 MD was also used to complement scanning tunneling microscopy images of polyimide Langmuir-Blodgett films10 and X-ray diffraction studies.11,12 Other work was * To whom correspondence should be addressed.

concerned with the alignment of liquid-crystal molecules on a polyimide oligomer monolayer13,14 or with the poling of piezoelectric APB-ODPA-based five-units chains.15 Zhang et al.16,17 and Kang et al.18 have reported structureproperty relationships of oligomers and polyimides based on the BTDA dianhydride and different diamines. They were able to obtain properties such as conformational statistics, characteristic ratios, persistence lengths, solubility parameters, elastic moduli, and yield strains. MD of single-chain oligomers of PMDA-PFMB and BPDA-PFMB in a vacuum were used to estimate persistence lengths for these polymers.19 Intra- and intermolecular nematic order in ten-units perylene-containing polyimides was also characterized using this technique.20 In the present work, our goal is to compare two model oligomers of polyimides based on the same diamine, 4,4'diaminophenyl ether (ODA), at the molecular level. The dianhydrides are, respectively, pyromellitic dianhydride (PMDA) and bicyclo(2.2.2)-oct-7-ene-2,3,5,6-tetracarboxylic dianhydride (BCDA). The study of relatively “short” oligomers (between 150 and 200 atoms per chain) can be justified by the fact that they are in the vitreous state, and, as such, are a convenient first approximation for their glassy long-chain counterparts. Although the available experimental data on the real oligomer systems only deal with their crystalline phases,21,22 the simulations will show that the amorphous oligomers are indeed in an arrested state. This observation can be related to the very high glass-transition temperatures found in [long chain] polyimides, that is, typically several hundred degrees above room temperature.23 Such stiff molecules are unlike many short-chain oligomers which would be liquids instead of amorphous solids at room temperature. Another specificity of these oligomers is that their individual monomers have significantly larger lengthto-breadth ratios than polyethylene, poly(vinyl chloride), or polypropylene, for example.

10.1021/jp0131525 CCC: $22.00 © 2002 American Chemical Society Published on Web 04/17/2002

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Neyertz et al.

Figure 1. The monomer structure of PMDA-ODA and BCDA-ODA polyimides. End groups are hydrogens.

The use of oligomers also prevents the well-known problems related to the generation of initial structures for long-chain polymers prior to MD simulations.24,25 Although it is now possible to generate dense samples of stiff long-chain polymers,6-9,26 the method used is most often based on the growth into an ever decreasing volume and at low density with subsequent densification.24 The density of the final systems can be as good as within 2% of the experimental value,7 but the equilibration procedure can be rather intricate,6,7 and it has been shown that growing at lower densities could introduce a bias in the chain statistics.27,28 Our approach is to use another technique which allows samples to be grown directly at a density close to the desired value and with conformations expected from the equilibrium melt at the required temperature.29 It has been thoroughly tested for polyethylene, polyoxyethylene, and poly(vinyl chloride) on chain lengths for which full relaxation can be achieved in a reasonable MD simulation time.25,30-33 We have also very recently successfuly tested it for polycyclic polymers,34 and we intend to apply it to long-chain polyimides in a subsequent piece of work. In the case of our oligomers though, they can be decorrelated at high temperatures within the nanosecond time scale routinely accessible to classical MD simulations and are thus largely free of any potential artifacts due to the method of preparation. We hereby attempt to get a first insight into the way the chemical structure of the dianhydride will affect the microscopic and macroscopic properties of these two molecules. Schematic chemical structures of PMDA-ODA and BCDA-ODA are shown in Figure 1, while the planar PMDA and kinked BCDA motifs are displayed in perspective in Figure 2. As one of the main drawbacks of polyimides is their hygroscopic nature, which results in the absorption of up to a few percent water under ambiant environmental conditions and which degrades their dielectric properties,35 both models were studied in the absence and presence of water. The inserted water was fixed at 3.3% in weight, since the solubility of water vapor in films of PMDA-ODA at 100% relative humidity room temperature has been found by means of a quartz crystal microbalance to correspond at saturation to a water uptake of 3.3 ( 0.3% in weight.36 This technique has the advantage of being very convenient with a vacuum system containing controlled amounts of water vapor.36 We did not attempt to calculate the equilibrium uptake of water of the two model systems in contact with air at a relative humidity of 100%. Our goal in this work is to characterize the oligomer/water interactions given an experimental estimate of the water content. We do note though that it is now also possible to calculate the equilibrium proportion of water in dense polymer systems submerged in liquid water using a combination of the thermodynamic integration approach and Widom’s test particle insertion method.37-39

Figure 2. Schematic representation in three dimensions of the PMDA and BCDA motives linked to phenyl carbons (stripes ) carbons, gray ) nitrogens, plain ) oxygens, and hydrogens are omitted for clarity).

In Section 2 we give the computational details while the results are given and discussed in Section 3. 2. Computational Details Four different simulations were carried out to compare the PMDA-ODA and BCDA-ODA model oligomers both with and without water. They will be referred to hereafter as the PMDAODA, the PMDA-ODA + water, the BCDA-ODA, and the BCDA-ODA + water systems. The MD simulations were performed using the gmq program40 either in its scalar form on COMPAQ alpha servers or in its parallel form, ddgmq,41 on the multiprocessor Cray T3E of the IDRIS supercomputing center (Orsay, France) and on the IBM SP2 and the SGi ORIGIN 2000 of the CINES supercomputing center (Montpellier, France). 2.1. Potential. The potential, or force field, describes the potential energy of a system as a superposition of simple analytical functions which are presented in this section. High-frequency vibrational bond-stretching modes were removed according to the SHAKE routine,42 with a relative tolerance of 10-6, to allow for the use of a reasonable time step. All bond lengths were rigidly constrained to standard values.43 The functional form of the potential used in this study is

Upot )

∑θ Ubend(θ) + ∑τ Utors(τ) + ∑ Uoop(i) + ∑

i-sp2

(i,j)nb

ULJ(|rij|) +

∑ Ucoul(|rij|)

(1)

(i,j)nb

where (A) the first term describes the polymer angle-bending deformations by a harmonic function in the cosine of the bond angles, θ:

Ubend(θ) )

kθ (cos θ - cosθ0)2 2

(2)

(B) the second term represents the torsional motions around

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the dihedral angles τ by a sixth-order polynomial in cos τ: 6

Utors(τ) )

an cosnτ ∑ n)0

(3)

(C) the third term keeps sp2 structures planar by using a harmonic function in the distance d from the i-sp2 atom to the plane defined by its three attached atoms:

Uoop(i-sp2) )

koop 2 d 2

(4)

(D) the fourth term describes excluded-volume interactions between atoms belonging to the same molecule (but separated by more than two bonds) as well as atoms belonging to different molecules through the Lennard-Jones form of the van der Waals potential:

(( ) ( ) )

ULJ(|rij|) ) 4ij

σij |rij|

12

-

σij |rij|

6

(5)

(E) the fifth term accounts for Coulombic interactions in the same conditions as (D) with qi and qj being charges on atoms i and j, and 0 being the vacuum permittivity:

Ucoul(|rij|) )

qiqj 4πo|rij|

(6)

The long-range effect of the electrostatic potential was taken into account by the Ewald summation method.44,45 The use of Ewald sums is a time-consuming technique, but, to our knowledge, it remains the only formally correct solution to the problem of performing the infinite sum of partial charges distributed in a periodic system. Other techniques such as the fast multipole methods (FMM) or particle-mesh methods contain inherent approximations and the consensus is that they do not become faster than Ewald until system sizes become very much larger than those used here.46,47 A well-optimized scalar Ewald routine is generally accepted to spend equal amounts of CPU time in real and reciprocal space48 so in fact the overhead is only a factor of 2; the van der Waals interactions are calculated in real space in any case. In addition, we will show that there is a very significant contribution of the intermolecular Coulombic energy, especially when we add polar water molecules to our oligomer systems. As such, we prefer to be cautious and to treat the Coulombic interactions in the most accurate way. Parameters for eqs 2-5 were directly derived from the TRIPOS 5.2 force field,43 since this force field has been tested on a large number of cyclic compounds. The charges used in eq 6 were obtained by performing ab initio calculations on four representative fragments of the PMDA-ODA and BCDA-ODA structures (see Figure 3) with Gaussian9449 at the B3LYP/631G** level. Partial charges, qi/e, were then extracted by an electrostatic-potential (ESP)-fitting procedure.50 Only those charges in the central moieties of the model fragments were used in the subsequent MD simulations. All partial charges are given in Figure 3. Water potential parameters, including charges, were taken from the simple point-charge model, SPC/E,51 which is known to provide a reasonable representation of real bulk water.52 The ij and σij cross-terms for the van der Waals parameters (eq 5) were obtained from the geometric mean of ii and jj and from the arithmetic mean of σii and σjj. 2.2. Packing Models for the Pure Oligomers. PMDA-ODA and BCDA-ODA all-atom model oligomers with n ) 4 (see

Figure 3. Ab initio calculated partial charges for model fragments of PMDA-ODA and BCDA-ODA. Part (a) gives the charges on the dianhydride moiety of PMDA-ODA, (b) the charges for the diamine moiety of PMDA-ODA, (c) the charges for the dianhydride moiety of BCDA-ODA, and (d) the charges for the diamine moiety of BCDAODA. Charges are assigned to their nearest atom and are symmetrized throughout the plane perpendicular to the indicated structures. Arrows point to charges carried by hydrogen atoms.

Figure 1) were created using equilibrium bond lengths and bend and dihedral angles.43 Twenty-seven molecules for each system under study were placed randomly within a given simulation box of approximate size 40 × 40 × 40 Å3. Each PMDA-ODA oligomer contained 158 atoms, thus giving a total number of 4266 atoms in its simulation box. Each BCDA-ODA oligomer contained 190 atoms, thus giving a total number of 5130 atoms in its simulation box. Although three isomeric forms are possible for the BCDA moiety (boat-boat, boat-chair, chair-chair), these are noninterconvertible once the monomer is formed. To our knowledge, no experimental evidence exists concerning the proportions of the different isomeric forms. Molecular mechanics calculations on the BCDA moiety alone using TRIPOS and a variety of other commonly used force fields all gave the boatboat conformer as the lowest energy form. The simulations presented here thus used the boat-boat form. The MD simulations were generally carried out under NPT constant-pressure conditions where the on-diagonal and offdiagonal components of the pressure tensor are maintained at ∼1 bar and ∼0 bar, respectively, by loose-coupling with a coupling constant of 5 ps.53 The simulation boxes are thus allowed to relax toward their equilibrium density and box shape. The leapfrog algorithm was used to integrate equations of motion with a time step of 1 fs and the temperature was maintained at ∼300 K by loose-coupling to a heat bath54 with a coupling constant of 0.1 ps. Both pure systems were first energy-minimized to remove high-energy overlaps and then were run at ∼1000 K at constant volume to decorrelate the chains. The extent of configurational decorrelation was monitored by following the normalized auto-

4620 J. Phys. Chem. B, Vol. 106, No. 18, 2002 correlation function for the square end-to-end distances of the chains.32 These correlation functions decayed to zero after about 400 ps for both oligomers under study at 1000 K. Although both PMDA-ODA and BCDA-ODA oligomers are relatively stiff molecules, they exhibit conformational degrees of freedom around the dianhydride-diamine and diamine-diamine linkages. At such a high temperature, the probability density for the square end-to-end distance (not shown) has a broad classical r2-weighted Gaussian form and does not show peaks for specific conformers. Nevertheless, checks were made on individual oligomers to confirm that decorrelation at 1000 K was indeed due to significant changes in the end-to-end distances of the chains. Each simulation box was subsequently cooled toward its target temperature of 300 K at a rate of -1 K/ps under NPT conditions. The simulations were then run at ∼300 K until the density settled around a constant value, which happened after about 800 ps. We used the criterion for equilibration of Suter and co-workers in their work on hydrated polyamides38 and bisphenol A-polycarbonate and poly(vinyl alcohol),39 that is, the absence of any drifts in density larger than 3 × 10-4 g cm-3 ps-1 during the last 100 ps of the equilibration procedure. This allows for equilibration of the systems in their local minima in configuration space.38 2.3. Insertion of the Water Molecules. Both oligomer + water simulation boxes were built directly from the pure oligomer boxes, following their production run. As explained in the Introduction, the value of 3.3% in weight 36 was used as the standard water percentage for our hydrated systems. This specific value was experimentally measured on films of longchain PMDA-ODA at 100% relative humidity.36 To our knowledge, there are no experimental data available on the solubility of water in BCDA-ODA. Thus, 78 molecules of water for the PMDA-ODA oligomers and 84 molecules of water for the BCDA-ODA oligomers were inserted into the corresponding simulation boxes. The PMDAODA + water simulation box contained 4500 atoms in total, while the BCDA-ODA + water box contained 5382 atoms in total. Although water can be inserted randomly into the polymer microstructures,37,39 we used a slightly more complicated procedure. A box of water of the same size as the pure oligomer samples was first prepared at a temperature of 300 K and at a density of ∼1 g cm-3. The two boxes were then superimposed and the relevant number of water molecules were added by selecting those which overlapped least with the atoms in the oligomers. This procedure clearly avoids high water-water overlap energies which could result from random insertions and reduces as much as possible the initial water-chain repulsions and hence the disruption to the carefully prepared matrix. However, it is unclear whether this method is really preferable to a completely random insertion. Given the level of mobility of the water (see later), we suspect that both methods will give similar results. The hydrated systems were then run at ∼300 K until the density settled around a constant value, applying the same criterion for equilibration as above. This occurred within 500 ps for both PMDA-ODA + water and BCDA-ODA + water systems. 2.4. Other Simulation Details. A given simulation box was decomposed by the parallel code into 2 × 2 × 2 ) 8 domains each of ∼20 Å side length and each containing two link cells per dimension with a length greater than the cutoff plus the shell width (1.5 Å) used in the creation of the neighbor table.

Neyertz et al. As we used parallel processors, the Ewald parameters have to take into account the virtual decomposition of the system into domains as required by our general purpose parallel MD code. For this reason, the optimization does not correspond to the scalar case as we have to deliberately reduce the real space truncation, that is, increase R, at the expense of a larger overhead in reciprocal space. This allows us, however, to spread the calculation across eight processors rather than just using one. The reciprocal space part of the Ewald sum in fact parallelizes very simply in a domain decomposition framework so this does not incur any extra penalty over and above that of the scalar code. Within the restriction on the cutoff and acceptable CPU costs,55 the optimal convergence of the Ewald sum was hereby obtained for the PMDA-ODA systems using R ) 0.32 Å-1 and Kmax ) 11, while the real-space potential was truncated at 7.5 Å. Corresponding parameters for the BCDA-ODA systems were R ) 0.28 Å-1 and Kmax ) 10, while the real-space potential was truncated at 8 Å. The van der Waals potentials were also truncated at 8 Å and long-range corrections to the energy and the pressure were calculated on the basis of the radial distribution functions being equal to unity beyond the cutoff. The production time was 1600 ps for the pure oligomer systems and 2200 ps for the oligomer + water systems, during which configurations were stored at 5-ps intervals, and thermodynamic and conformational data were stored every 0.5 ps for post-analysis. The PMDA-ODA + water oligomer system is shown in Figure 4. Periodic boundary conditions were applied in all three dimensions. For comparison, a pure SPC/E51 water simulation was undertaken for a system of 256 water molecules at 300 K. This pure water simulation run was carried out for 150 ps under atmospheric pressure using Ewald parameters of R ) 0.3 Å-1 and Kmax ) 6, while the real-space and van der Waals potential were truncated at 9 Å. All other details were identical to those given above. 3. Results and Discussion 3.1. The Glassy State. When cooled to 300 K, the oligomer systems effectively settle into an arrested state showing little mobility afterward. The upper bound for the self-diffusion coefficient of both PMDA-ODA and BCDA-ODA oligomers at 300 K was estimated at 10-8 cm2 s-1. This order of magnitude is low and confirms that our oligomers are indeed in the glassy state.56 This does not prevent some degree of motion and relaxation in the simulations but, as has been found in other simulations of glassy polymers,37-39 the state for both systems is clearly a nonequilibrium one. The same phenomenon actually affects real laboratory measurements where physical “aging” of glassy polymers is well-known.57 Although the use of short-chain oligomers will affect global properties such as density and radii of gyration, their amorphous structure and low degree of mobility suggest that local order may closely resemble that in their long-chain counterparts. To avoid explicit end effects, the first dianhydride and the last diamine in a given chain were omitted from the local order analyses. In the case of those properties which are clearly affected by the chain length, we will examine the trend as a function of the chemical structure and compare this to the trend found in available experimental data. As was pointed out by Knopp and Suter,38 the ability of the water to totally sample its preferred sites in the polymer matrice is also limited in the glassy systems. However, as will be shown later, water molecules can show significant mobility in the

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Figure 4. A schematic representation of the PMDA-ODA + water simulation box. The oligomer molecules are displayed by their wire-frames and water molecules are shown in a space-filling representation. The box is fully periodic in its three dimensions so atoms outside the box will have images in the central cell.

TABLE 1: Results of the Simulations Undertaken at 300 K for the Four Systems under Study Plus Those for the Pure SPC/E Water Simulationa properties Vexpans diamine-diamine. They reflect once again the strong tendency of these systems to pack through parallel stacking of the dianhydride unit either with itself or with a diamine ring. The order of the interactions in the BCDA-ODAbased structures are consistent with the strong steric repulsion brought about by the dianhydride motif: diamine-diamine > dianhydride-diamine > dianhydride-dianhydride interactions. In all cases, water seems to have a very limited effect on chain-chain interactions. Both types of oligomers are obviously too stiff to undergo major rearrangements of their local structures on the nanosecond time scale. This confirms the picture brought about by the angle and radii of gyration analyses and suggests once again that water will not disrupt the chain structure but rather will occupy sites that are already available in the pure phase. 3.4. Morphology of the Void Space. As both oligomers have different microstructural characteristics, it is interesting to study the amount of “void space” in a chain matrix and the manner in which it is distributed. These quantities are not uniquely

Neyertz et al. definable but estimates can be obtained from empirical calculations,61,72 positron annihilation experiments,73,74 labeling and probes,75 as well as from atomistic simulations.76-78 Although values obtained from different methods applied to the same polymer are unlikely to be the same, they can in principle complement each other when applied to the same selection of polymers from a particular family. In this section, we use this approach to try and correlate the effect of the different dianhydride units. The fractional free volume (FFV) available in the dry oligomers can be empirically estimated according to the various group contributions by the Bondi method. 72 From the simulated values, we obtain an FFV of 0.126 for the PMDA-ODA oligomer and of 0.190 for the BCDA-ODA oligomer. These values fall within the experimentally reported values for polyimides which are usually in the range 0.1 to 0.223,79 and suggest that there is more free volume available in the bulky BCDA-ODA oligomer than in the planar PMDA-ODA oligomer. However, it remains to be seen whether this extra free volume in the BCDA-ODA oligomer can be accessed by molecules of the size of water. We thus implemented a Monte Carlo estimation method to further characterize the void space in both oligomer systems with or without water. The general principle is similar to the “phantom sphere approach” described in refs 76 and 78. The space accessible to water molecules is approximated by a procedure in which probes are randomly inserted into MD configurations previously stored. These probe insertions are done independently and as such do not “see” the other probes. There are thus no assumptions made about the shape of whatever cavities might be present. The radius of these spherical probes was first fixed at 2.45 Å, as this corresponds to the closest water O‚‚‚O distance found in the simulations and is thus a good estimate of the minimum interaction distance between two water molecules. Those probes which are further than 2.45 Å from all the atoms in the configuration are counted and their positions are stored. A total of 200 000 trial insertions per configuration was found to give converged results for the probability of accepting an insertion. The amount of probe-accessible volume in the simulation box is then simply defined as the product of this probability of accepting an insertion with the volume of the simulation box. To characterize the free volume distribution, we used the stored positions of all accepted probes and then defined an interaction limit Rp which links those probes having been inserted into the same “hole” for a given configuration. The subsequent “clusters” of linked probes that are formed are then used to define the volumes of the individual holes from the proportion of the total number of accepted probes in the holes and the total probeaccessible volume of the configuration. As ever, this kind of method of hole-size characterization is somewhat subjective as the distribution depends very strongly on the choice of Rp. For large Rp, we obtain “percolation” as all probes belong to one large cluster; for small Rp, each probe forms its own separate hole. Either of these two extremes leads to rather uninteresting distributions. We found by trial and error that a value of Rp ) 0.7 Å leads to distributions with a range of hole sizes that can be used to compare the different systems in a relative way. All results were averaged over all the stored configurations for a given system. The average probe-accessible volumes (PAVs) for the 2.45 Å probe in the pure oligomers were 933 ( 3 Å3 for the PMDAODA and 1115 ( 3 Å3 for the BCDA-ODA systems. In fact, this corresponds in both cases to a percentage of accepted probe

Short Oligomers of PMDA-ODA and BCDA-ODA Polyimides TABLE 2: The Average Percentages of Probe-Accessible Volume (%PAV) in the Pure Oligomer Systems as a Function of the Probe Radius. Errors in the %PAV Are (1 in the Last Figure Quoted probe radius/Å

PMDA-ODA % PAV

BCDA-ODA % PAV

1.70 1.85 2.00 2.15 2.30 2.45

20.44 13.38 8.46 5.21 3.15 1.88

21.17 14.22 9.11 5.60 3.31 1.88

insertions of 1.88 ( 0.01 and thus, in a relative sense, there is no difference in the PAVs. To see whether this was always the case, we repeated the same procedure with probe radii ranging from 1.70 Å (i.e., roughly the van der Waals radius of a carbon) to 2.45 Å. Results are given in Table 2 and show that the average probe-accessible volumes, expressed as a percentage of the total volume, are marginally larger in the BCDA-ODA system for probe sizes less than 2.45 Å. We do not find the same level of differences as the FFVs predicted by the Bondi group method (see above). However, both characterizations do show the same trend. The underlying distributions of the amount of probe-accessible volume as a function of hole size are shown in Figure 7 for both types of oligomers without (Figure 7a and 7b) and with (Figure 7c and 7d) water present. Figure 7e and 7f gives the distributions for the oligomer + water systems with the actual water molecules being ignored when the probes are inserted. This allows us to see how the hole-size distribution is affected by the inclusion of the water. Although all analyses presented were carried out with the 2.45 Å probe, a check with smaller radii probes gave similar trends. The percentage of accepted probe insertions was 0.87 ( 0.01% for PMDA-ODA + water and 1.38 ( 0.01% for BCDA-ODA + water simulations. If the water is omitted for the purpose of the virtual probe insertions, the corresponding values are 2.11 ( 0.02% and 2.53 ( 0.01% for PMDA-ODA + water and BCDA-ODA + water, respectively. In general, these distributions show the same monotonically decreasing form found by Nagel et al.78 using their “sphere hole model”. The exception is the pure BCDA-ODA oligomer which displays a bimodal form having a dearth of holes in the 70 Å3 range and a second maximum near 120 Å3. However, this tendency disappears when the water is added (see Figure 7f). We do find in all cases that BCDA-ODA-based systems (where the BCDA moiety leads to poor packing in its vicinity) tend to favor smaller holes with respect to their PMDA-ODA counterparts; the logarithmic scale of Figure 7 does not facilitate the visualization of these differences in the distributions but they are very consistent. Following the inclusion of water, the mid-size and large holes disappear in the PMDA-ODA + water system (Figure 7c), as they are very easy to access directly. The trend is much less pronounced in the BCDA-ODA + water system (Figure 7d) where the holes are smaller and their accessibility to the water molecules are more uniform. In both water-excluded distributions (Figure 7e and 7f), an increase in the accessible volume is seen for all hole sizes. The “swelling” brought about by the inclusion of water appears thus quite uniform and seems independent of the chemical structure of the dianhydride. Both systems have clearly different distributions for the accessible volume to our probe. In a later section, we discuss a cluster analysis on the actual water molecules inserted, and we will examine the correlation between both approaches.

J. Phys. Chem. B, Vol. 106, No. 18, 2002 4625 3.5. Structure and Mobility of the Water. We did not study the sorption process. This has been done for dense polymer simulation boxes by Suter and co-workers,37-39 who showed that a combination of the thermodynamic integration approach and Widom’s particle insertion method can give a very good estimate of the excess chemical potential of water in such systems. In this paper, we do not try to calculate the expected uptake of water by the different oligomers but rather try to characterize the oligomer/water interactions given an experimental estimate of the water content inside a given oligomer simulation box. 3.5.1. Radial Distribution Functions. Radial distribution functions for oligomer/water and water/water interactions were obtained from the PMDA-ODA + water and BCDA-ODA + water simulations. Two sites on the oligomer chains are markedly favored for coordination with water molecules, namely, ketone oxygens on the dianhydride moieties and ether oxygens on the diamine moieties. Well-characterized peaks were found at ∼2.50 Å for (ketone O...water O) and ∼2.65 Å for (ether O...water O), while the closest peaks were at ∼1.55 Å for (ketone O...water H) and ∼1.65 Å for (ether O...water H). This suggests that water forms hydrogen bonds with the oxygens on the oligomer chains. The average numbers of water oxygens (i.e., of water molecules) around a ketone or an ether oxygen are displayed in Figure 8. The preferred sites for water coordination are the dianhydride ketone oxygens for both oligomers. As shown in Figure 2, the ketone oxygens are directed outward and are thus less affected by steric effects. They also carry a relatively large partial negative charge (see Figure 3). Water coordinates less with diamine ethers in both PMDA-ODA and BCDA-ODA oligomers. The ratio between the n(r) for ketone O...water O and the n(r) for ether O...water O interactions were calculated at a distance of 3.5 Å, which corresponds to the first trough in all gO‚‚‚O(r). They are respectively 4.4 for the PMDA-ODA oligomer and 2.6 for the BCDA-ODA oligomer. Although the vicinity of the ketone O is always the preferred site for coordination, a larger total available space in the BCDA-ODA oligomer and smaller occupancy sites tend to decrease this selectivity and make the ether oxygens more accessible to the water molecules. This suggests that steric effects are indeed a key factor in the occupancy sites of the water molecules. Water-water radial distribution functions also show strong peaks at ∼2.75 Å for O...O and ∼1.75 Å for O...H interactions. They correspond to the formation of water clusters within the systems and are markedly favored in the PMDAODA oligomer with respect to the BCDA-ODA oligomer (see Figure 8). The ratios between the n(r) for water O...water O and water O...ketone O at 3.5 Å are 4.2 for the PMDA-ODA oligomer and 3.6 for the BCDA-ODA oligomer. The corresponding ratios between the n(r) for water O...water O and water O...ether O at 3.5 Å are 18.5 for the PMDA-ODA oligomer and 9.2 for the BCDA-ODA oligomer. The water thus globally appears to favor clusters with respect to chain coordination. The existence of two basic types of water molecular sites in polyimides is in full agreement with experimental characterizations. Van Alsten et al. analyzed infrared spectra for a variety of polyimides at saturation and concluded that water exists both in an isolated form, that is, hydrogen-bonded to the chain backbone, and in clusters consisting of small numbers of water molecules.60 This two-sites picture is confirmed by the dielectric relaxation and deuteron NMR studies of Xu et al.80 Li et al. studied the influence of water in long-chain PMDA-ODA by

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Figure 7. Distributions of the amount of volume accessible to trial insertions of a 2.45 Å radius probe as a function of the hole size. (a) and (b) are for the PMDA-ODA and BCDA-ODA dry oligomers. (c) and (d) are for the PMDA-ODA + water and BCDA-ODA + water systems. (e) and (f) are the corresponding distributions for the oligomer + water systems with the actual water molecules being ignored when the probes are inserted.

proton, deuteron, and oxygen-17 NMR and showed that the first water site could be assigned to single molecules dispersed within the polymer matrix and probably hydrogen-bonded to oxygens along the polymer chains, while the second water site corresponds to small water clusters or chains of water molecules.81 Another carbon-13 NMR study identified the carbonyl group of the imide ring as a preferential site for the aggregation of water molecules.82 3.5.2. Water Clusters Analysis. The probability density of clusters can be obtained by identifying all water oxygens within 3.5 Å of another water oxygen (i.e., the first trough in the gO‚ ‚‚O(r)) and by labeling them as being “connected” to that oxygen. The water oxygens forming a connected group are then defined as being members of the same cluster. The water cluster

probability densities over the production runs are shown in Figure 9. As the water molecules were inserted into their respective oligomer matrices, the same analyses were performed on the initial oligomer + water configurations. Applying the same criterion as above, it was found that the initial insertions had resulted in 75% (PMDA-ODA) and 78% (BCDA-ODA) water molecules being single, 8% (PMDA-ODA) and 15% (BCDAODA) forming dimers, and the rest contributing to larger clusters. The water molecules underwent a redistribution in the early stages of both simulations before achieving a steady state. As such, the influence of the initial insertion on the average water cluster probability densities is rather limited.

Short Oligomers of PMDA-ODA and BCDA-ODA Polyimides

Figure 8. Average numbers of water oxygens as a function of distance around a chain ketone oxygen, a chain ether oxygen, or another water oxygen in the PMDA-ODA + water and BCDA-ODA + water simulations.

Figure 9 shows that there is a larger degree of water aggregation in the PMDA-ODA oligomer. The free-energy difference between isolated water molecules and water molecules in clusters has been evaluated from the vibrational spectrum of PMDA-ODA as 5.56 kJ mol-1 for clusters of 2, 7.95 kJ mol-1 for clusters of 3, 10.38 kJ mol-1 for clusters of 4, and 12.76 kJ mol-1 for clusters of 5.60 The cluster distribution is directly correlated to the morphology of the void space, with a larger number of small holes in the BCDA-ODA oligomer and consequently a larger probability for single water molecules to occupy these gaps. Indeed, this is the case for 42% of the water molecules in the PMDA-ODA + water simulation, whereas on average 54% of water molecules are single in the BCDA-ODA + water simulation. The bimodal form of the voidspace distribution and the dearth of holes in the BCDA-ODA oligomer 70 Å3 range (Figure 7b) explains the existence of more dimers in the PMDA-ODA + water system (29% of the water molecules vs 23%). The second maximum in the bimodal distribution would correspond to trimers and tetramers which appear very similar in both systems. Larger clusters are slightly more numerous in the PMDA-ODA system, which reflects the existence of larger holes in the latter (see Figure 7a). It is thus obvious from the clusters analyses that the degree and nature of water aggregation will be totally dependent on the features of the void space in the different oligomers. 3.5.3. Water Mobility. The most common way to evaluate diffusion coefficients of penetrant molecules in MD simulations is through Einstein’s equation:

1 tf∞ 6t

D ) lim

(10)

where the angle brackets imply an average over all penetrant molecules and all possible time origins, to. However, in atomistic simulations of necessarily limited duration, it is essential for a converged estimate of D that the diffusion has reached the Einstein limit, that is, is linearly proportional to t and the motion of the penetrant molecule is a random walk. It is now well established that anomalous, that is, non-Fickian, diffusion of small molecules in polymers occurs at short time scales before the crossover to Einstein behavior, that is, is proportional to tn with (n < 1).83 One will typically need very long trajectories to establish unambiguously Einstein’s behavior.84 Although the mean-square displacement of water oxygens indicate diffusion up to several

J. Phys. Chem. B, Vol. 106, No. 18, 2002 4627

Figure 9. Histograms showing the probability densities of water O...water O clusters as a function of their size in the PMDA-ODA + water and the BCDA-ODA + water simulations.

nanometers for both PMDA-ODA + water and BCDA-ODA + water production runs, double logarithmic plots (not shown) of versus t give slopes of ∼0.6 for both simulations under study. The crossover to Einstein diffusive behavior has thus not yet occurred in our simulations and we cannot give converged values for the diffusion coefficients of water in these systems using this approach. We nevertheless attempted to characterize diffusion in the anomalous regime by calculating the distributions of the water oxygen coordinate displacements over a given time interval averaged over all time origins, F(R(t + to) - R(to)) with R ) x, y, and z. To further increase the statistics, only the absolute values of the displacements were used and displacements along the x, y, and z axes were averaged together for all water oxygens. These coordinate displacement distributions for a 100 ps time interval are given in Figure 10a for the PMDA-ODA + water system and in Figure 10b for the BCDA-ODA + water system. Clearly, these distributions cannot be fitted by a single Gaussian curve, as would have been expected from Fickian diffusion:

F(R(t + to) - R(to)) )

(

x

3 3 exp R2 2 2 2π 2

)

(11)

with ) + + over the whole distribution. However, the distributions do fit very well to a weighted (ω) combination of two Gaussian curves,

F(R(t + to) - R(to)) ) ω

(x

(

))

3 3 exp R2 2 2π 2

(1 - ω)

(x

(

+

3 3 exp R2 2 2 2π 2

))

(12)

thus suggesting that there are in fact two distinct types of behavior as far as the short time mobility is concerned. The first one, referred to in Figure 10 as type “1”, is evidently due to molecules which remain in (or return to) the very near vicinity of their position at to. The second one, referred to in Figure 10 as type “2”, is due to those molecules which manage to escape from their initial environments. It is ultimately this second type of behavior which leads to the limiting long-time diffusion coefficient. Indeed, ω decreases from 0.62 at t ) 10 ps to 0.35 at t ) 1 ns for the PMDA-ODA oligomers; the corresponding values are 0.63 to 0.39 for the BCDA-ODA oligomers. The difference between both systems is given by Figure 11, where an estimate of the diffusion coefficient for the type “2”

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Figure 11. The time evolution of the water oxygen diffusion coefficient for type “2” behavior, D(t), as defined in Equation 13. Circles are for the PMDA-ODA + water system and triangles for the BCDA-ODA + water system. The plain and dashed lines are fits to the algebraic form D(t)) λt-β (for the PMDA-ODA oligomer, λ ) 219.72 and β ) 0.59; for the BCDA-ODA oligomer, λ ) 213.58 and β ) 0.63).

Figure 10. Distributions F(R(t + to) - R(to)) with R ) x, y, z for the water oxygen coordinate displacements over the t ) 100 ps time interval averaged over all time origins, to, in the production runs. The actual curves are given in (a) for PMDA-ODA (circles) and in (b) for BCDAODA (triangles). The plain and dashed lines represent the two weighted Gaussians fitted to the actual data (cf Equation 12), i.e, behaviors of type “1” and type “2”. For PMDA-ODA, ω ) 0.56, ) 0.73 Å2, and ) 9.20 Å2. For BCDA-ODA, ω ) 0.57, ) 0.64 Å2, and ) 6.94 Å2.

behavior, D(t), is best obtained from the mean-square displacement of the fitted Gaussian curves at time interval t:

D(t) )

6t

(13)

Results spanning up to 1 ns (averaged over all time origins of the production runs) show an initial sharp decrease followed by a longer slower decrease which can be fitted very well, in both cases, to the algebraic form D(t) ) λt-β, with λ and β positive constants. It is obvious that this type of fit is not valid in the Einstein regime as it does not lead to a nonzero asymptote at long times. However, it characterizes the anomalous diffusion of the water oxygens in our models. The value at 1000 ps can be used as an upper bound of the true water diffusion coefficient in our oligomers, that is, Dwater e 3 × 10-7 cm2 s-1. The consistently higher D(t) for the PMDA-ODA system is possibly related to the more uniform distribution of the void space which could enhance diffusion. Single water molecules are more likely to get trapped in small holes in the BCDA-ODA structure. To get a picture of the local dynamics, the trajectories of individual water molecules were examined to characterize the mechanisms underlying water mobility in the oligomers. Some typical paths are shown in Figure 12. The two main types of behavior suggested above are found to occur. There are molecules remaining in a given site and fluctuating around their

Figure 12. Typical displacements of water molecules in the oligomer systems as a function of the simulation time.

equilibrium positions. The rest are molecules moving more or less smoothly away from their initial positions and even undergoing jumps between different sites, thus leading to overall displacements of up to 15-16 Å. The hopping time scale between the different sites is approximately 10-20 ps and the distance of a jump is typically 5-10 Å. To detail the evolution of a water molecule undergoing significant motion, all ketone, ether, and water oxygens that came within a radius of 3.5 Å of such a water oxygen were identified, and their positions with respect to this oxygen were monitored with time. It was soon clear that the local environment varies significantly with time. The coordination number within the 3.5 Å sphere fluctuated around an average of three oxygens, with a very rare occurrence of amine ether oxygens. The water molecules that undergo significant motion are thus apparently mostly linked to dianhydride ketone oxygens or other water oxygens. Behavior patterns include all four types of replacements in the coordination sphere of a given water oxygen, that is, ketone to water, ketone to ketone, water to ketone, and water to water oxygens. The progressive motion of a water molecule in the simulation boxes is best seen in Figure 13, which follows the evolution of a water coordination sphere over 2200 ps. The water molecule under study, taken from the PMDA-ODA + water run, is initially coordinated to two other water molecules and to one ketone oxygen. After about 300 ps, the water molecule dissociates both from its site on the oligomer chain and from its water cluster to move along dianhydride oxygens under a

Short Oligomers of PMDA-ODA and BCDA-ODA Polyimides

Figure 13. Time evolution of the local environment around a particular water oxygen undergoing motion between different sites. Diamonds and crosses indicate, respectively, dianhydride oxygens and other water oxygens that are found within a radius of 3.5 Å of the subject water oxygen over the 2200 ps simulation run. Different lines indicate different ligands. This water molecule belongs to the PMDA-ODA + water simulation box.

doublet form. It then reaches a stable dianhydride site at ∼400 ps. In the process, a new water molecule comes in its vicinity to reform the triplet attached to the oligomer chain. This geometry is stable for about 500 ps, after which time the water molecule under study appears to dissociate from its current ligands. Although it reaches another dianhydride site, it returns also to the former one, thus oscillating in a void between different chain oxygen sites. Water clusters easily associate and dissociate, as not less than 10 other water molecules enter its coordination sphere over the analysis time. However, only doublets or triplets have enough stability to last more than a few ps, that is, typically from about 300 to 500 ps. Successive dianhydride ligand sites can either be found on the same or on different oligomer chains. Because of the very low mobility of the oligomers, the water will coordinate to the closest available site on a chain, regardless of whether it is the same chain or not. Similar patterns can be seen in the BCDA-ODA + water run. For example, one of its water molecules is initially coordinated to two dianhydride ketone oxygens on two different chains and moves after 300 ps to another dianhydride site in the oligomer matrix, while forming at the same time a water doublet. This water also briefly coordinates with an ether oxygen but does not remain in its vicinity. The doublet dissociates after 500 ps, and the water molecule keeps moving from dianhydride site to dianhydride site on its own for about 1000 ps. At that time, two new water molecules come into its vicinity, thus forming a triplet which remains stable on the time scale of the simulation. Unlike chains in the rubber phase which participate actively to the transport phenomena,85 small molecules in glassy-state matrices will mainly undergo jumps in the free volume initially available. The possibility of a jump appears to be related to the presence of ketone oxygens, and to a much lesser extent to that of ether oxygens, in the near vicinity of the water molecules. 4. Conclusions Four MD simulations were undertaken to illustrate the structure and dynamics at the molecular level of PMDA-ODA and BCDA-ODA oligomers in the absence and presence of 3.3% water. At 300 K, each dynamic model was found to relax toward an arrested state thus showing that they are below their glasstransition temperatures. In agreement with the available experimental data, the average densities were quite a bit lower in the BCDA-ODA-based

J. Phys. Chem. B, Vol. 106, No. 18, 2002 4629 systems. Indeed, the intermolecular energies show a larger cohesion in the PMDA-ODA-based systems. The introduction of water leads in both cases to a slight increase in density which does not follow the linear combination rule for the volumes; the molecular volume of water is found to be significantly less in the chain + water systems than in the pure state at room temperature. The changes in intermolecular van der Waals energies are not explained by a simple additive model, and the simulations predict an enthalpically favorable hydration of both types of oligomers by water. BCDA-ODA oligomers are more coiled than their PMDAODA counterparts, which can be correlated directly to the threedimensional structure of the dianhydride unit and the existence of a kink in the BCDA moiety. Flexibilities and conformational transitions around nitrogen-phenyl and ether linkages are similar. Chain-chain radial distribution functions confirm that the average interchain spacing is smaller for PMDA-ODA than for BCDA-ODA oligomers. Dianhydride-diamine interactions, which can be linked to the existence of charge-transfer complexes, are markedly preferred in the PMDA-ODA-based systems. The same conclusion is reached for dianhydridedianhydride interactions, which are energetically favorable when the dianhydride structure is planar, while diamine-diamine interactions are very similar. Water has a very limited effect on chain-chain interactions, thus showing that the inserted molecules do not distort the stiff-chain structures. The analysis of the void space reveals significant differences between the distributions of “holes” in both structures under study. Indeed, BCDA-ODA oligomers will favor small holes whereas the hole-size distributions are much more uniform in the PMDA-ODA simulations. Despite a larger fractional free volume predicted empirically for the more bulky BCDA-ODA oligomers, a lot of this extra volume does not seem accessible to molecules of the size of water. The Monte Carlo estimation method designed to characterize the void space leads to a similar percentage of accepted insertions, that is, of relative probeaccessible volume, for both types of oligomers. In agreement with experimental observations, water molecules are able to coordinate to oxygens on the oligomer chains and to form clusters through hydrogen-bonding. While the oligomer-water coordination is relatively similar, the PMDA-ODA oligomer exhibits a higher degree of water aggregation, in agreement with the larger hole sizes in its void space. Two types of dynamic behavior are found, with most water molecules remaining and oscillating in a given site, while some of them undergo jumps between dianhydride sites and water clusters. Although the necessarily limited time scale of these simulations does not allow us to reach Einstein’s regime for the diffusion, we show that the diffusion of water in the anomalous regime can be characterized by an algebraic decrease with time. Despite the different geometries of the dianhydrides, the PMDA-ODA and BCDA-ODA oligomers have obviously quite a few features in common. Indeed, they exhibit similar intramolecular flexibility, similar trends in the energy and density variations, and identical probe-accessible volumes and oligomer-water interaction sites. However, they are clearly different as far as density, cohesion, intermolecular interactions, morphology of the void space, and the degree of water clustering (given a 3.3% insertion of water) are concerned. Finally, the same approach is currently under investigation for longer-chain polyimides and results will be compared to those found for the corresponding oligomers. However, our opinion is that, as the oligomers are clearly in the glassy state, their long-chain counterparts should exhibit a similar behavior.

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