Langmuir 1997, 13, 1173-1181
1173
Theoretical Studies of the Structure and Dynamics of Semicrystalline PPTA/Water Vapor Interfaces† D. A. Mooney and J. M. D. MacElroy* Department of Chemical Engineering, University College Dublin, Belfield, Dublin 4, Ireland Received March 11, 1996. In Final Form: June 10, 1996X Monte Carlo (MC) and molecular dynamics (MD) simulation techniques have been employed to study the equilibrium and dynamical properties of water vapor sorbed in two different grain boundaries (110/ 110, 200/200) of a semicrystalline aromatic polyamide, poly(p-phenyleneterephthalamide) (PPTA). Henry’s law constants (H) and isosteric heats of adsorption (qst) were obtained from Monte Carlo (NVT) simulations, and the influence of polymer mobility on the transport behavior of sorbed water particles was investigated using molecular dynamics (NVE, NVT) simulation techniques. It is shown that the diffusive motion of the water particles is related to a cooperative interaction between the water molecule and polymer conformational dynamics at the surface of the PPTA crystallites. In particular the reorientational motion of the phenyl rings plays a central role in the diffusive mechanism, and an approximate expression for the frequency of the water molecule diffusive jumps was derived in terms of an effective phenyl ring torsional energy barrier. Fair agreement was obtained between this model and exact diffusion results computed from the simulated trajectories. These results indicate that it may be possible to quantify the affect of polymeric group substitution on the transport properties of sorbed particles in similar polymers.
1. Introduction In the last twenty years theoretical advances in the physics and chemistry of polymer systems have been assisted to a significant extent by developments in computer simulation at the atomic level. The seminal work of Ryckaert, Ciccotti, and Berendsen1 on molecular simulation via constraint dynamics and more recent studies (summarized by Roe2 ) employing simulation of fully flexible large scale molecular structures have demonstrated the feasibility of conducting atomistic simulations of macromolecular systems on contemporary computers. These techniques and associated refinements of the procedures involved have provided the impetus for detailed studies in this area including the mechanical properties of bulk polymers,3 interfacial conformational states in thin polymer films,4 and diffusion of small penetrants in polymer structures.5-17 * To whom all correspondence should be addressed. E-mail:
[email protected]. † Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland/Slovakia, September 4-10, 1995. X Abstract published in Advance ACS Abstracts, September 15, 1996. (1) Ryckaert, J. P.; Ciccotti, G.; Berendsen, H. J. C. J. Comput. Phys. 1977, 23, 327. (2) Roe, R. J., Ed. Computer Simulation of Polymers; Prentice-Hall: Englewood Cliffs, NJ, 1991. (3) Theodorou, D. N.; Suter, U. W. Macromolecules 1985, 18, 1467. (4) Mansfield, K. F.; Theodorou, D. N. Macromolecules 1991, 24, 6283. (5) Rigby, D.; Roe, R. J. J. Chem. Phys. 1987, 87, 7285. (6) Trohalaki, S.; Rigby, D.; Kloczkowski, A.; Mark, J. E.; Roe, R. J. Polym. Prepr. 1989, 30, 23. (7) Sonnenburg, J.; Gao, J.; Weiner, J. H. Macromolecules 1990, 23, 4653. (8) Takeuchi, H.; Okazaki, K. J. Chem. Phys. 1990, 92, 5643. (9) Takeuchi, H. J. Chem. Phys. 1990, 93, 2062. (10) Takeuchi, H. J. Chem. Phys. 1990, 93, 4490. (11) Takeuchi, H.; Roe, R. J.; Mark, J. E. J. Chem. Phys. 1990, 93, 9042. (12) Trohalaki, S.; Kloczkowski, A.; Mark, J. E.; Rigby, D.; Roe, R. J. In Computer Simulation of Polymers; Roe, R. J. Ed.; Prentice-Hall: Englewood Cliffs, NJ, 1991; Chapter 17. (13) Muller-Plathe, F. J. Chem. Phys. 1992, 96, 3200. (14) Muller-Plathe, F.; Rogers, S. C.; van Gunsteren, W. F. Chem. Phys. Lett. 1992, 199, 237. (15) Gusev, A.; Suter, U. W. Polymer Preprints; ACS Division of Polymer Chemistry: Washington DC, 1992; Vol. 33, p 631. (16) Sok, R. M.; Berendsen, H. J. C.; van Gunsteren, W. F. J. Chem. Phys. 1992, 96, 4699.
S0743-7463(96)00233-8 CCC: $14.00
Although the simulations reported in each of the studies cited above have been primarily restricted to amorphous polymer models, it is generally recognized that polymers most frequently exhibit a wide degree of crystallinity or order. This is demonstrated by the large number of structural models which have been proposed as descriptions of bulk polymers which vary from the fringed micelle model for polymers of low crystallinity18 to the paracrystalline model for polymers of high crystallinity.19 Recent experimental and theoretical work on extended chain polymers20,21 has also led to the development of a grain boundary theory for highly crystalline polymers in which the amorphous component is associated with intercrystallite domains generated during the packing of crystallites. The influence of the interfacial heterogeneities present in the grain boundaries of semicrystalline polymers on the properties of sorbed materials is, however, poorly understood at this time, and a microscopic theory is required for this purpose. In this paper, molecular modeling techniques are employed to investigate the transport behavior and nature of water sorbed at low-to-moderate loadings within the interfacial domains generated by the lateral chain invariant (LCI) grain boundaries20 between crystallites of a stiff chain polyamide, poly(p-phenyleneterephthalamide) (PPTA). This polymer is chosen for investigation in this work in view of both the relatively simple structure of the unit cell of the PPTA crystallites and current evidence for the predominance of the LCI grain boundary subclass in the bulk polymer. The latter conclusion is supported by the lengths of the PPTA polymer chains (∼200 nm)22 and the observation via spectroscopic methods that the crystallites generated are approximately hexagonal in shape with a diameter of ∼3.5 nm.23 Self-diffusion coefficients and configurational properties for the water molecules are computed via molecular dynamics and Monte Carlo (17) Gusev, A. A.; Arizzi, S.; Suter, U. W.; Moll, D. J. J. Chem. Phys. 1993, 99, 2221. (18) Hermann, K.; Gerngross, O.; Abitz, W. Z. Phys. Chem. B. 1930, 10, 371. (19) Hosemann, R. Z. Phys. 1950, 128, 1. (20) Martin, D. C.; Thomas, E. L. Philos. Mag. A 1991, 64, 903. (21) Martin, D. C.; Thomas, E. L. Macromolecules 1991, 24, 2450. (22) Morgan, R. J.; Pruneda, C. O.; Steele, W. J. J. Polym. Sci., Polym. Phys. Ed. 1983, 21, 1757. (23) Jackson, C. L.; Schadt, R. J.; Gardner, K. H.; Chase, D. B.; Allen, S. R.; Gabara, V.; English, A. D. Polymer 1994, 35, 1123.
© 1997 American Chemical Society
1174 Langmuir, Vol. 13, No. 5, 1997
Mooney and MacElroy Table 1. Dreiding Inter- and Intramolecular Potential Functions Employed for PPTA
∑
EVdW )
pairs (i,j)
[( ) ] Bij
Aij
-
r12 ij
r6ij
(i)
vdW
where
Aij ) 4ijσ12 ij Bij ) 4ijσ6ij
∑
Eel )
pairs (i,j)
simulation, and the results obtained provide a unique insight into the role played by the configurational and dynamical structure of the heterogeneous grain boundary region both on the equilibrium conformational states of the water/PPTA system and on the permeation rates of the sorbate within the semicrystalline medium.
Eφ )
qiqj
4π0rij
(ii)
Coul
[( ) ( ) ]
(iii)
1 Eθ ) kθ(θ - θ0)2 2
(iv)
Eψ ) K(1 - cos ψ)
(v)
Ehb ) Dhb 5
Figure 1. (a) Segment of the rodlike PPTA macromolecule. (b) Unit cell of crystalline PPTA.
[( ) ]
Rhb rDA
∑
12
-
Rhb rDA
10
(cos θDHA)4
kφ[1 + cos(nφ - φ0)]
(vi)
dihedrals
2. Molecular Model and Simulation Technique 2.1. Structural Model and Potential Energy Functions for PPTA. PPTA consists of alternating para-linked and amide linkages. In its commercial form as a fiber, the rodlike macromolecules are orientated in the axial direction and are essentially fully extended (see Figure 1a). This is due to the energy barrier (≈20 kcal/mol) associated with rotation of the N(H)-C(O) bonds which have a partial double-bond characteristic due to delocalization of charge. Rotation about the aryl N(H) and aryl C(O) bonds is less restricted (≈2 kcal/mol) although there is still a preference for the lower energy, more stable form coinciding with the linear conformation. In the fiber, molecular chains form crystallites through hydrogen bonding and the arrangement of the crystallites is such that the hydrogen-bonded planes are in the radial direction with respect to the fiber axis. The crystal structure of PPTA is well established, existing in either one of two crystal modifications (I, Northolt;24 II, Haraguchi et al.25 ). Crystal modification I is predominately associated with commercial fibers with the unit cell being monoclinic (pseudo-orthorhombic), as shown in Figure 1b. In this work molecular mechanics (MM) simulation studies were initially conducted to verify the minimum energy conformation of the unit cell reported by Yang et al.26 In their simulations Yang et al. employed the DREIDING potential energy model,27 in which the total potential energy of the polymer is expressed as
Etotal ) EVdW + Eel + Ehb + Eθ + Eψ + Eφ
(1)
where EVdW, Eel, and Ehb are London-van der Waals, electrostatic, and hydrogen-bonding terms, respectively, and Eθ, Eψ, and Eφ are intramolecular contributions arising from bending and torsion. We have also employed the DREIDING model and the (24) Northolt, M. G. Eur. Polym. J. 1974, 10, 799. (25) Haraguchi, K.; Kajiyama, T.; Takayanagi, M. J. Appl. Polym. Sci. 1979, 23, 915. (26) Yang, X.; Hsu, S. L. Macromolecules 1991, 24, 6680. (27) Mayo, S. L.; Olafson, B. D.; Goddard, W. A., III. J. Chem. Phys. 1990, 94, 8897.
individual contributions to the total potential energy functions are summarized in Table 1. Nonbonded interactions are described by Lennard-Jones (12-6) repulsion-dispersion functions (eq i); Coulombic interactions are determined by the partial charges, qi and qj, on each of the atoms, which are estimated using the method proposed by Rappe´28 (eq ii); and by hydrogenbonding (12-10) terms (eq iii). In the DREIDING force field the latter potential involves three atoms: (1) the donor (D), a highly electronegative atom, which is bonded to (2) a hydrogen (H) atom which forms a hydrogen bond with (3) another electronegative acceptor atom (A). The intramolecular bonded interactions include (i) bond angle bending (eq iv); out-of-plane bending (or inversion) (eq v), and dihedral angle torsion (eq vi). The outof-plane bending potential is sometimes deemed necessary to maintain planarity and arises, for example, in the case of sp2hybridized four-atom structures e.g. when an atom I is bonded to three other atoms J, K, and L. Assuming ψ denotes the angle between the IL bond and the JIK plane, then the expression provided in Table 1 for Eψ is obtained. Using the potential functions provided in Table 1 and the potential parameters and bond lengths cited in Tables 2 and 3, we have confirmed that the proposed unit cell (see Figure 1b) conforms well to the experimentally determined dimensions of crystal modification I (theoretically (a, b, c) ) (0.834, 0.5, 0.1298 nm); experimentally (a, b, c) ) (0.787, 0.518, 0.1296 nm)). Moreover, the symmetry center of the polymer chain at the center of the unit cell was also found to be located at (0.417, 0.25, 0.6 nm), in good agreement with results reported by Rutledge and Suter,29 who cite (0.415, 0.25, 0.59 nm). 2.2. Simulation of LCI Grain Boundaries in PPTA. Recent theoretical and experimental work with rigid rod polymers20,21 has led to the classification of four types of grain boundaries (in order of increasing energy): (i) lateral chain invariant (LCI), (ii) lateral chain rotation (LCR), (iii) axial chain invariant (ACI), and (iv) axial chain rotation (ACR). Independent experimental work by Jackson et al.23 provides support for the assumption that the LCI grain boundary is predominant in bulk (28) Rappe´, A. K.; Goddard, W. A., III. J. Phys. Chem. 1991, 95, 3358. (29) Rutledge, G. C.; Suter, U. W. Macromolecules 1991, 24, 1921.
Semicrystalline PPTA/Water Vapor Interfaces Table 2. Potential Parameters for the Dreiding Interand Intramolecular Potential Functions Lennard-Jones 12-6 Parameters atoma
σ (Å)
/k (K)
H H HB C N O
2.8464 21 2.8464 21 3.472 99 3.262 56 3.033 154
7.654 0.051 47.888 38.975 48.190
Hydrogen Bonding Parameters Rhb (Å) ) 2.75 Dhb (kcal/mol) ) 7.0 Angle Bending kθ (kcal/(mol rad2)) ) 100 Inversion K (kcal/mol) ) 40 Torsion (I-J-K-L) (a) A dihedral (partial) double bond involving two sp2 atoms in the J, K position: kφ (kcal/mol) ) 45 (b) A dihedral resonance bond involving two resonant atoms in the J, K position: kφ (kcal/mol) ) 25 (c) A dihedral single bond involving two sp2 or resonant atoms in the J,K position: kφ (kcal/mol) ) 5 a H ) phenyl hydrogen. H HB ) hydrogen atoms capable of hydrogen bonding.
Table 3. Polymer Bond Lengths for Simulations atom 1
atom 2
bond length (Å)a
atom 1
atom 2
bond length (Å)a
1 1 1 2 2 3 3 4 4 5 5 6 7 7 8 8 10 10 11 11 12 12 13 13 14 14 15 16 16 17
2 6 45 3 19 4 20 5 7 6 21 22 8 9 10 23 11 15 12 24 13 25 14 16 15 26 27 17 28 18
1.4394 1.4134 1.4043 1.4049 1.0022 1.4293 1.0465 1.3979 1.4127 1.4170 1.0342 1.0289 1.3280 1.1903 1.3779 0.8563 1.4210 1.4028 1.4156 1.0629 1.3885 1.0053 1.4186 1.3309 1.3706 0.9929 1.0369 1.2785 0.9131 1.2294
17 29 29 30 30 31 31 32 32 33 33 34 35 35 36 36 38 38 39 39 40 40 41 41 42 42 43 44 44 45
29 30 34 31 47 32 48 33 35 34 49 50 36 37 38 51 39 43 40 52 41 53 42 44 43 54 55 45 56 46
1.4108 1.4314 1.4221 1.3925 1.0255 1.4406 1.0272 1.4697 1.4145 1.4306 1.0019 0.9999 1.2892 1.2382 1.3351 0.9200 1.4321 1.4134 1.4248 1.0229 1.4260 0.9916 1.4500 1.3496 1.3967 1.0304 1.0094 1.2654 0.8970 1.2241
a The atoms on the dimer of PPTA shown on the right are numbered with reference to the bond lengths given in the table.
PPTA, and furthermore, since hydrogen bonding exists in certain crystallographic directions and van der Waals bonding exists in others, one would expect different grain boundary energetics within this LCI subset. On the basis of the hexagonal crystallite model proposed by Jackson et al.23 (see Figure 2), the two LCI grain boundaries composed of the high-energy interfacial combinations 110-110 and 200-200, with the individual faces in
Langmuir, Vol. 13, No. 5, 1997 1175 perfect misregistry prior to thermal relaxation, are investigated in this work. The simulation sequence proceeds as follows: (i) Two semiinfinite crystals are generated and are initially relaxed via a combination of molecular mechanics (MM) and molecular dynamics (MD)30 (in all of the MD simulations conducted in this work we have employed the RATTLE constraint dynamics algorithm31 with a time step of 1.25 fs). The 110 or 200 face at the outer edge of the crystallite is exposed to vacuum with the polymer interfacial region modeled as shown on the left hand side of Figure 2. The outer two layers of polymer chains in each individual crystal face are fully mobile with the exception of bond length constraints, and a third (inner) layer is fully constrained though atomistically modeled. The deep interior of the crystallite is treated as a continuum, and the potential functions provided in Table 4 are employed to characterize the interactions between this region and the atoms of the mobile surface chains of PPTA.32 Furthermore, for the purpose of simulating a semi-infinite crystal, periodic imaging30 is employed in the x and y directions shown in Figure 2 with the periodic cell dimensions Lx and Ly equal to 2.0 and 2.596 nm for the 200 grain boundary system and 1.944 and 2.596 nm for the 110 grain boundary system, respectively. The nonbonded London-van der Waals and hydrogen-bonding interaction potentials between the atomistically modeled polymer chains were truncated beyond a spherical cutoff equal to 0.85 nm, and the electrostatic interactions between all of the partial charges within the fundamental cell depicted in Figure 2 were computed in the nearest image approximation.30 During the relaxation process the local atom density, bond angle distribution, and torsion angle distribution within the first two polymer layers next to the vacuum are monitored. (An initial 0.5 ns equilibration period was employed for each crystallite face with subsequent 0.25 ns production runs to accumulate conformational and dynamical data for the equilibrium state in each case.) (ii) Pairs of the relaxed interfaces are next selected and brought into close contact, at which point further relaxation is undertaken subject to the same x-y periodic imaging and atom/atom interaction potential truncation described above. All grain boundary simulations reported here correspond to a density of 0.75 g/cm3 (h ) 1.729 nm) with the volume of the grain boundary region (hLxLy) defined in the manner shown on the right hand side of Figure 2. These simulations demonstrate the influence of vicinal crystallite surfaces on the local chain motions (in particular the reorientational motion of the phenyl rings on the polymer chains) and hence provide information on the prevalent conformational states present in the amorphous phase of PPTA. Among the properties evaluated to monitor these effects were the torsional autocorrelation functions (TAFs) for the five primary torsion angles in the polymer chains depicted in Figure 3. The TAF is defined for torsion angle φ by the expression9
Rφ(t) )
(〈cos φ(0) cos φ(t)〉 - 〈cos φ(0)〉2) (〈cos φ(0)2〉 - 〈cos φ(0)〉2)
(2)
Torsion angle averages and their standard deviations were also computed and the frequency of torsional flips was determined during the simulations. 2.3. Simulation of Sorbed Water within the Grain Boundaries in PPTA. The potential energy function for water, defined here as Φ(r,φ′,θ′,ψ′,θ), where r is its center-of-mass position and φ′, θ′, and ψ′ are the Euler angles describing its orientation, was determined using the nonbonded interaction potentials given by eqs i-iii in Table 1 as well as an intramolecular bond-bending potential defined by the angle θ (eq v). Both O-H bond lengths were constrained, and the relevant potential parameters employed in this case are provided in Table 2. The Henry’s law adsorption equilibrium constant H and the isosteric heat of adsorption for water were determined for the free PPTA 110 and 200 crystal faces as well as for the 110/110 and 200/200 grain boundaries using
H)
1 LxLyh
∫〈exp(-
)〉
Φ(r,φ′,θ′,ψ′,θ) kBT
φ′θ′ψ′θ
dr
(3)
1176 Langmuir, Vol. 13, No. 5, 1997
Mooney and MacElroy
Figure 2. Model crystallite and grain boundary structure. The fundamental cells for the free crystallite surface simulations are those indicated by the “eye” symbol on the left hand side, and the fundamental cells for the grain boundaries are within the boxed regions shown on the right: (a) 200/200 and (b) 110/110 face and grain boundary. the influence of energetic heterogeneity on this kinetic process. The properties computed during the evolution of the water molecule trajectory included the water center-of-mass velocity autocorrelation function and the mean-square displacement which are required in the evaluation of the diffusion coefficient33,34
and
lim qst
NH2Of0
(
) RT 1 -
[
)
d ln (H - 1) d ln T
∫〈Φ*(r,φ′,θ′,ψ′,θ) exp(-Φ*(r,φ′,θ′,ψ′,θ))〉 ) RT 1 ∫[〈exp(-Φ*(r,φ′,θ′,ψ′,θ))〉 - 1] dr
φ′,θ′,ψ′,θ
φ′,θ′,ψ′,θ
]
DR )
dr
(4)
where Φ*(r,φ′,θ′,ψ′) ) Φ(r,φ′,θ′,ψ′)/kBT. The terms in the angular brackets in both of these equations were estimated by Monte Carlo30 insertion of water molecules, placed at different centerof-mass positions and orientations during the course of a molecular dynamics trajectory of the “dry” polymer. The sampling frequency corresponded to 5 × 104 insertions every 103 time steps. In addition to H and qst the dynamical properties of water molecules confined within the LCI grain boundaries of PPTA were determined via the RATTLE constraint dynamics algorithm.31 The MD computations were conducted by randomly placing water molecules, at a specified occupancy (4, 7, or 10 molecules), within the grain boundary domains described in section 2.2. The length of a production run for a given loading was typically 0.5 ns or longer. In these simulations the principal objective was to determine those particular sites or functional groups within the grain boundaries which give rise to ratelimiting transition-state barriers during diffusion and to assess (30) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987. (31) Anderson, H. C. J. Comput. Phys. 1983, 52, 24. (32) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974.
∫ 〈v (0) v (t)〉 dt ∞
0
R
R
(R ) x, y)
(5)
or
DR )
1 d 〈|R(t) - R(0)|2〉tf∞ 2 dt
(R ) x, y)
(6)
Furthermore, to correlate the motion of the water molecule with the dynamics of the polymer chains, we have locally mapped the torsion angles undergoing significant change during large scale displacements of the water molecules. It is anticipated that by tracking the local water/polymer dynamics in this manner it will be possible to determine those chain segments which are primarily responsible for sorbate diffusive jumps within the grain boundary.
3. Results and Discussion 3.1. Free Surface of Semi-infinite PPTA Crystallites. Results for the atomic number density profiles for thermally relaxed 200 and 110 crystallite faces are shown in Figure 4 (the z coordinate is normal to the crystallite surface, as shown in Figure 2). An important point to (33) Green, M. S. J. Chem. Phys. 1952, 20, 1281; 1954, 22, 398. (34) Kubo, R.; Toda, M.; Hashitsume, N. Statistical Physics II. Nonequilibrium Statistical Mechanics; Springer-Verlag: Berlin, 1985. (35) Fukuda, M.; Ochi, M.; Miyagawa, M.; Kawai, H. Text. Res. J. 1991, 61, 668. (36) Barrer, R. M. In The Solid-Gas Interface; Flood, E. A., Ed.; Marcel Dekker, Inc.: New York, 1967; Vol. 2.
Semicrystalline PPTA/Water Vapor Interfaces
Langmuir, Vol. 13, No. 5, 1997 1177
Table 4. Long Range Interactions for Semi-infinite Crystals of PPTA
[ ()
EVdW,LR ) 4πijnjσij2 0.2
a
σij rij
10
- 0.5
()
4
σij rij
σij -
]
6∆(rij + 0.61∆)3
(i)
b
Eel,LR )
2π LxLy
NMP
∑q ∑
exp(-gzi)
i
g(1 - exp(-g∆)) xig1 yig2 f(g1,g2) cos 2π + Lx Ly
{
i)1
g1,g2*0
({
×
}) ({
h(g1,g2) sin 2π
where
({
NB
c
f(g1,g2) )
j
j)1
({
NB
h(g1,g2) )
∑q exp(gz ) sin 2π j
j
j)1
g ) 2π
+
Lx
Ly
(ii)
})
yjg2 +
Lx
Ly
})
yig2
xig1 + Lx
})}
yig2
xig1
xjg1
∑q exp(gz ) cos 2π j
+
Ly
x( ) ( ) g1 Lx
2
+
g2 Ly
2
a ∆ ) crystal plane spacing normal to the surface of interest. n j ) atomic surface density for crystal atoms of type j. b NMP ) the c number of mobile atoms in the system. NB ) the number of background atoms.
Figure 4. Atomic density (s), Fa, and Henry’s law profiles (b) for the isolated crystallite faces: (a) 200 face; (b) 110 face.
lim H ) 1 + hf∞
)1+
Figure 3. Five primary torsion angles in the PPTA amide linkages: angle 1, OdCamsNsH; angle 2, CarsCarsCamdO; angle 3, CarsCarsNsH; angle 4, CamsNsCarsCar; angle 5, CarsCarsCamsN.
observe is the broadening and flattening of the density profiles for the atoms of the polymer chains at the crystallite surfaces due to the relatively free mobility of the chains on the crystallite periphery (the interior of the crystallite corresponds to z coordinates less than 0.0). Also shown in Figure 4 are the normalized density profiles for adsorption of water vapor under Henry’s law conditions on the individual crystallite faces. Note the degree of water penetration (to within 0.2 nm of the center of the surface layer of polymer chains), in agreement with the inference from experimental observations of deuterated PPTA that water has relatively free access to most of the atoms of the surface groups but not to the crystallite interior.23 Another point to note from the results shown in Figure 4 is the distinct difference between the Henry’s law density profile for water adsorption on the 200 crystallite face (Figure 4a) and the profile for adsorption on the 110 crystallite face (Figure 4b). The areas under these two profiles are simply related to the two-dimensional Henry’s law constants, B, for the individual faces, which are defined by
1 LxLyh B h
∫(〈exp(- Φ(r,φ′,θ′,ψ′,θ) )〉 k T B
φ′,θ′,ψ′,θ
)
- 1 dr
(7)
and it is observed that the value of B for the 110 face is over two orders of magnitude larger than the coefficient for the 200 face (due primarily to H-bonding between the water molecule and the 110 face). This demonstrates that the primary sites for water adsorption are on the 110 crystallite faces (and not the 200 faces) within PPTA, and this conclusion may also be shown to be in excellent agreement with the deuterium exchange results of Jackson et al.23 The isosteric heats of adsorption reported in the legends of Figure 4a and b also illustrate the much greater affinity of adsorbed water for the 110 crystallite face with the low value of qst for the 200 face, actually implying that this surface is hydrophobic. The TAFs for torsion angles 1-5 (see Figure 3) for the 110 surface PPTA chains of the semi-infinite crystallite are plotted in Figure 5 for the time period 0.0 ps < t < 0.5 ps. The most important points to note from these results are the following: (i) angle 1 decorrelates very rapidly due to the freedom of movement of the hydrogen and oxygen atoms at the ends of the 4-atom group; (ii) angles 2 and 5 both decorrelate at essentially the same rate due, primarily, to the oscillatory motion of the nearby terephthalamide ring; (iii) angles 3 and 4 also decorrelate at a similar rate, in this case due to the oscillatory motion of the nearby p-phenyldiamine ring; (iv) the decay of the TAFs for angles 2 and 5 is accompanied by large amplitude oscillations which slowly die away in the pico to nanosecond time range. In contrast, the amplitude of the oscillations for the TAFs for angles 3 and 4 is approximately a factor of two smaller, which suggests that the torsional motion of the p-phenyldiamine ring decorrelates more rapidly than the corresponding motion of
1178 Langmuir, Vol. 13, No. 5, 1997
Mooney and MacElroy
Figure 5. TAFs for the 110 isolated face. The torsion angles are as defined in Figure 3: (a) (b) 1, (O) 2, (4) 5; (b) (b) 3, (O) 4.
the terephthalamide moiety. The latter is believed to be due, in part, to differences in the amide hydrogen and oxygen interactions with their respective environments. The principal conclusion implied by these results is that for short periods of time (∼1.0 ps) both the terephthalamide and p-phenyldiamine rings undergo significant rocking motions due to the relatively small energy barriers to torsional motion in both cases (see Table 2), and this should have a direct bearing on the dynamics of adsorbed water at short times as it diffuses along the 110 face of the PPTA crystallite. Similar conclusions are reached from an analysis of the TAFs for the relaxed 200 crystallite face. An interesting perspective of the temporal dependence of the sorption sites of the free surface of the PPTA crystallite is demonstrated in Figure 6, where the minimum potential energy surface maps for water are reported for the 110 crystal face. Figure 6a is the unrelaxed 110 face, displaying large potential wells and barriers to diffusion. Thermal relaxation of this virgin crystal face provides the surface maps shown in Figure 6b and 6c, and an important point to note is the manner in which this surface varies with time. Figure 6c is a snapshot of the potential energy surface for the same crystal face as Figure 6b but 0.25 ns later. The crests and troughs of the water molecule potential energy have shifted, and one can observe a “dynamic heterogeneity” in the surface. Surface maps of the type shown in Figure 6 are reminiscent of scanning tunneling micrographs, and it would be of interest in future work to make a comparison between these simulation results and STM experimental observations. 3.2. Properties of Water Sorbed at Low Loadings within the LCI Grain Boundaries of PPTA. The Henry’s law properties for water in both the 200/200 and 110/110 grain boundaries at a polymer density of 0.75 g/cm3 (as defined by the polymer mass within the volume hLxLy) are reported in Figure 7. The relative magnitude of the peaks in the water density profiles for both grain
Figure 6. Surface map for the 110 PPTA crystallite face generated by energy minimization of the potential for a single water molecule placed on the individual sites of a 40 × 40 grid: (a) unrelaxed face; (b) thermally relaxed face at 300 K at a reference time of 0.0 ns; (c) same face as in (b) 0.25 ns later.
boundaries follows trends similar to those observed in Figure 4 for the unbounded crystallite surfaces. The corresponding Henry’s law constants and the zero loading isosteric heats of adsorption, shown in the legends, verify conclusions reached earlier in the discussion of the data for the free crystallite surfaces that water vapor sorption occurs primarily within the 110/110 grain boundary. Recent simulation results37 for both parameters, at densities approaching that found in commercial PPTA (grain boundary density ∼ 1.32 g/cm3), yielded values of H110 ) 99 800 and qst ) 74.16 kJ/mol, which are in good agreement with experimental data provided by Fukuda et al.35 (H110 ∼ 99 000 and qst ∼ 70 kJ/mol). Results for the 110/110 grain boundary TAFs for the five primary torsion angles illustrated in Figure 3 are presented in Figures 8a and b. Both parts of this figure clearly suggest that the proximity of adjacent crystallite faces within the grain boundary region has only a minor affect on the torsional autocorrelation functions. This is to be expected if we assume that the short-time TAFs are primarily determined by the small-angle, high-frequency rocking motions of both the terephthalamide and pphenyldiamine rings. As we will see below, however, the (37) Mooney, D. A.; MacElroy, J. M. D. Work to be presented at the National Annual AIChE Meeting, Chicago, Nov. 11-15th, 1996.
Semicrystalline PPTA/Water Vapor Interfaces
Figure 7. Atomic density (s), Fa, and Henry’s law profiles (b) for the PPTA grain boundaries: (a) 200/200 grain boundary; (b) 110/110 grain boundary.
Figure 8. TAFs for the 110/110 grain boundary (0.75 g/cm3). The torsion angles are as defined in Figure 3: (b) 1, (O) 2, (0) 3, (9) 4, (4) 5.
confinement of the polymer chains due to crystallite/ crystallite interactions has a substantial effect on the lowfrequency phenyl ring reorientational dynamics, which
Langmuir, Vol. 13, No. 5, 1997 1179
Figure 9. Diffusion coefficients in the x direction (O), and y direction (b) for the fully mobile PPTA grain boundaries. Corresponding diffusivities for the rigid polymeric system are shown for the (0) x direction and (9) y direction: (a) 200/200 grain boundary; (b) 110/110 grain boundary.
in turn has a direct bearing on the diffusional properties of water within the grain boundaries of PPTA. The individual diffusion coefficients for water in the x-y directions within the model PPTA grain boundaries are plotted in Figure 9 as functions of water loading. The results obtained for the 200/200 boundary (Figure 9a) are typically an order of magnitude higher than those for the corresponding conditions within the 110/110 grain boundary (Figure 9b) due, primarily, to the much weaker interactions between the water molecules and the 200 crystallite face. Diffusion in each grain boundary is also observed to be anisotropic. The trend exhibited by the diffusivity Dx,200 (x direction, 200/200 grain boundary) is typical of unhindered diffusion in one-dimensional channels while the results for Dx,110 are characteristic of severely hindered diffusion caused by near-normal alignment of alternating phenyl rings in this face. Both Dy,200 and Dy,110 display a maximum in the vicinity of NW ∼ 8, which is believed to be attributable to the saturation of high-energy surface sites in both cases. Simulation data not reported in this paper confirm that the high-energy sites for preferential sorption of water are located between pairs of amide groups, and, with reference to Figure 2, it is observed that, due to steric effects, the maximum number of accessible sites is one per PPTA monomer repeat unit. This result is also in good qualitative agreement with sorption data reported by Fukuda et al.35 Their results suggest that, in the noncrystalline region of PPTA fibers, the number of water molecules adsorbed per amide site in a monolayer is approximately 0.4 mol/mol. Below NW ) 8, the water molecules are subject to the energetically heterogeneous structure of the relaxed crystallite surface, and for NW > 8, sorption is primarily in the form of molecular clusters around the primary sites. Also shown in Figure 9b are Dx and Dy for diffusion within a rigid 110/110 grain boundary at a loading of 10
1180 Langmuir, Vol. 13, No. 5, 1997
Mooney and MacElroy
Figure 11. Locally mapped torsion angle for the conditions reported in Figure 10.
Figure 10. (a) Characteristic water particle trajectory for the fully mobile PPTA 110/110 grain boundary. (b) Water particle energetics and differential square displacement.
water molecules, and it is observed that these values are approximately an order of magnitude smaller than the results obtained for the fully mobile system. The underlying reasons for the different behavior in both cases may be best demonstrated with reference to Figure 10a, where we provide a trace of the complete trajectory of one of the water molecules confined within the mobile 110/110 grain boundary, and to Figure 10b, where the energetic characteristics and differential square displacements of this molecule (in which ∆t ) 0.6 ps) during the simulation run are reported. Up to a period of t ∼ 60 ps the particle oscillates in a low energy site on the surface, and at t ∼ 60 ps it leaves the surface and executes a jump across the grain boundary. As may be seen from Figure 10b, this correlates with a large increase in the total energy of the water molecule which, in turn, is due to a large transfer of thermal energy from the polymer to the diffusing water particle over a very short period of time. We have carefully scanned the polymer dynamical data (particularly the dynamics of all torsion angles in the vicinity of this event) to obtain a deeper insight into the underlying mechanism for such jumps, and the only event detected in the surface polymer chains which can be directly linked with the dynamics of the water molecule is depicted in Figure 11. This event corresponds to a medium-to-large scale torsional rotation of an aromatic ring (in this case a diamine ring). An extensive study of all of the water molecule trajectories in the mobile PPTA grain boundaries revealed that at least 85% of the diffusive jumps executed by the particles were directly correlated with phenyl ring rotations corresponding to an average sweep angle from the equilibrium position of 26.7°. All such jumps within the 110/110 grain boundary also coincided with an average minimum thermal energy for the water molecule of 10.1 kJ/mol in contrast to the lower energy of 6.7 kJ/mol for the 200/200 boundary, and each of these observations may be quantified by the following simplified model.
Figure 12. Partial flip frequency as a function of sweep angle: (0) average flip frequency for torsion angles 2-5 as defined in Figure 3; (b) flip frequency from eq 11; (s) flip frequency obtained from eq 12.
Two-dimensional random walk theory provides the following expression for the diffusion coefficient
1 Dxy ) νλ2 4
(8)
where λ2 represents the mean-square jump length for a diffusive step and ν corresponds to the frequency with which the molecule jumps. If we assume this frequency is simply the product of the phenyl ring partial flip frequency required for a critical sweep angle of θc ) 26.7° and the Boltzmann probability that the molecule has sufficient energy ∆E to complete the jump, then eq 8 may be written as
1 ∆E 2 Dxy ) ν(θc) exp λ 4 RT
(
)
(9)
In Figure 12 the exact (molecular dynamics) simulation results for the average of the partial flip frequencies of the four phenyl ring torsion angles depicted in Figure 3 for the 110/110 grain boundary are plotted as a function of sweep angle θ (we note here that, statistically, the results for the individual torsion angles were indistinguishable). For θc ) 26.7°, ν(θc) ) 0.53 ps-1 and using ∆E ) 10.1 kJ/mol and the value λ2 ) 0.343 nm2 obtained directly from the simulation results for a water loading of 10 molecules, we find that eq 9 predicts Dxy ) 0.8 × 10-5 cm2/s. This estimate is in good agreement with the value 1.2 × 10-5 cm2/s computed via MD simulation using eq 5 (or 6) and suggests that eq 9 could form the basis for an analysis of the influence of the structural parameters of the polymer on the magnitude of the diffusivity. It should be noted that the interpretation of the frequency term in
Semicrystalline PPTA/Water Vapor Interfaces
Langmuir, Vol. 13, No. 5, 1997 1181
eq 9 differs from the more usual transition state expression for the diffusivity, which is generally cited as (see for example Barrer36)
1 ∆E 2 Dxy ) ν0 exp λ 4 RT
(
)
(10)
where ν0 is the vibrational frequency of the molecule on the adsorption site and ∆E is the activation energy for the diffusive jump. Typically ν0 ∼ 1.0 ps-1,36 and the activation energy is equivalent to the minimum thermal energy required for the jump to occur. It is observed that if the values cited above for ∆E and λ2 are employed in eq 10, then the estimated diffusivity is essentially the same as the value provided by eq 9 even though the underlying physical meaning of the frequency term is different. This is fortuitous however, and the merit of eq 9 lies in the possibility of expressing ν(θc) in terms of the interfacial properties of the polymer. An approximate expression may be derived for the partial flip frequency corresponding to an arbitrary sweep angle θ if it is assumed that the phenyl rings behave as rigid molecular groups undergoing oscillatory motion about the para axis. Furthermore, assuming that the restoring force for this motion is determined by the potential energy function φ ) A{1 - cos(2θ)}, in which A is treated as a fitting parameter, then it may be shown that ν(θ) is given by
ν(θ) )
{ ( (
x
)
A sin2 θ 2RT 1 exp πIR θ RT R exp(R)E1 R +
)
)}
A sin2 θ RT
(θ e π/2) (11a)
{ (
x
2RT 1 A exp πIR θ RT
) A R exp(R)E (R + RT)] 1
(θ > π/2) (11b)
where IR is the moment of inertia of the phenyl ring about the para axis, E1(x) is the exponential integral, and
A
R)
∞
∑ RTθn)1
11
n
(2k - 1)∫0 sin2n z dz ∏ nn! k)1 θ
2
For sweep angles less than π/4 eq 11a simplifies to
ν(θ) )
x
(
)
Aθ2 2RT 1 exp πIR θ RT
(12)
Equations 11 and 12 are compared in Figure 12 with the exact simulation results for the rotational dynamics of the phenyl rings, and a reasonable fit is obtained with A ) 19.95 kJ/mol (the differences at low θ are believed to be due to the flexibility of the phenyl rings in the MD simulations, and the lower values of ν at higher sweep angles are attributed to anharmonic repulsion interactions with neighboring PPTA chains). These expressions suggest that the diffusivity may be controlled by manipulating the effective torsion potential parameter, A, of the phenyl rings, e.g. by ring “stiffening” via functional group substitution to increase A and hence lower Dxy. This conclusion of course depends on two principal facts: (i) that the frequencies of the rate-controlling polymer vibrations (i.e. ring oscillations in the present case) are of the same order of magnitude or lower than the vibrational frequencies of the molecules on the adsorption sites and (ii) that the energy barriers associated with the diffusion pathways created by local polymer chain motion
are significantly lower than barriers existing in the quasistatic medium. The latter point is demonstrated by the results shown in Figure 9 for diffusion in the rigid 110/ 110 grain boundary. In this case it is expected that the admolecule vibrational frequency will determine ν, i.e. eq 10 should apply, and in support of this we note that the magnitude of ∆E obtained directly from an analysis of the water molecule trajectories for the static PPTA grain boundary was found to be approximately 15 kJ/mol. Using this value for ∆E in eq 10 gives Dxy ∼ 0.2 × 10-5 cm2/s, in fair agreement with the average of the values for Dx and Dy reported in Figure 9. However, with normal thermal vibrations and rotations in the realistically modeled polymer surface, the energy barriers are considerably lower and Dxy, as predicted by eq 9, is correspondingly larger. 4. Summary and Conclusions In this paper the properties of water vapor in two distinct grain boundaries (200/200, 110/110) of semicrystalline PPTA have been investigated via Monte Carlo and molecular dynamics techniques. Monte Carlo studies have demonstrated that the Henry’s law constants and isosteric heats for the 110/110 grain boundary were significantly greater than those for the 200/200 grain boundary, primarily as a result of the propensity for water to form strong hydrogen bonds with the 110 face. Molecular mechanics studies coupled with the trajectory information from molecular dynamics simulation of an isolated 110 face also illustrate the dynamic heterogeneity of the surface with significant changes in the energetic topography at different times. Tracer diffusion coefficients at various water loadings in each of the 110/110 and 200/200 grain boundaries were obtained using molecular dynamics techniques. Diffusion in the 110/110 grain boundary was significantly lower than that in the 200/200 grain boundary, primarily as a result of the higher barrier to diffusion on this surface. It is believed therefore that the diffusion-limited region is within the 110/110 grain boundary. Similarly, tracer diffusion results were obtained for water diffusing within a rigid 110/110 grain boundary region, and it was found that the magnitude of the diffusion coefficient was 5-10 times lower than that in the corresponding mobile polymer case. Extensive analysis of water particle trajectories and local polymer motion indicated a correlation between lowfrequency phenyl ring rotations within the grain boundary and water particle hops or jumps. On the basis of these observations a model has been proposed in which the rate a particle jumps is related to the product of a critical frequency for partial ring flips and a Boltzmann probability that the molecule has sufficient energy to complete the jump. An approximate expression for the partial flip frequency as a function of sweep angle was derived in terms of an effective phenyl ring torsional barrier, and fair agreement was obtained with diffusion results from the simulations. Future work will seek to assess the influence of temperature and grain boundary density on diffusion and the ability of the proposed model to predict this behavior. Acknowledgment. The Irish Science and Technology Agency, Forbairt, and Du Pont (U.K. Ltd.) are gratefully acknowledged for their partial financial support of this project. LA960233X
