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Calculation of Silberberg’s Polymer Segmental Adsorption Energy by a Free Space Molecular Modeling Technique B. J. Clifton, T. Cosgrove, and M. R. Warne* School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, U.K. Received February 26, 1999. In Final Form: June 17, 1999 Free space molecular modeling has been employed to calculate the segmental heat of adsorption (χs) for poly(ethylene) and poly(vinyl acetate) adsorbing from a carbon tetrachloride solvent onto a graphite surface. The technique manipulates contact energies and coordination numbers which are calculated using the commercial software packages Cerius2 and Polygraf. The estimation of the segmental heat of adsorption is extended to a range of poly(ethylene-vinyl acetate) (P(E-VAc)) copolymers by use of a two-state model. Both the ethylene and vinyl acetate segments are predicted to adsorb onto graphite from carbon tetrachloride; however, the value of χs for ethylene was found to be significantly higher than that of vinyl acetate. Investigation of the energetic terms revealed that the van der Waals contribution was significantly greater than the Coulombic contribution in the adsorption process of either of the polymer segment types. The effective χs for P(E-VAc) random copolymers of varying composition allowed adsorbed amounts to be calculated by use of the Scheutjens-Fleer lattice model.
Introduction
χs ) χsdo + ln φcr + χsc -
Adsorbed polymers may be considered as tools by which one may manipulate the properties of an interface. Consequently, the strength of polymer-interface interactions is relevant when considering processes such as flocculation and stabilization of colloidal dispersions. A key parameter in polymer-interface interactions is the polymer segmental adsorption energy (χs), divided by kT, which was first defined by Silberberg.1 Dimensionless, this is written as
χs ) -(µpol - µsol)/kT
(1)
where µpol and µsol are interaction free energies of the solid surface with polymer and solvent molecules, respectively, with respect to a suitable reference (e.g., bare solid + polymer melt/bulk solvent) and k is the Boltzmann constant. Polymer adsorption occurs for positive values of χs, while negative values imply that the polymer does not adsorb. Previous work2-5 regarding the calculation of the strength of adhesion of polymers at interfaces has determined an experimental value for χs by assuming a lattice model. Using low molecular weight displacer molecules, a critical composition (φcr) of the displacer is determined and corresponds to the full displacement of the polymer from the surface. The resulting critical desorption displacer concentrations are then used to evaluate χs (assuming strongly anchored segments) from2f,g (1) Silberberg, A. J. J. Chem. Phys. 1968, 48, 2835. (2) (a) van der Beek, G. Langmuir 1989, 5, 1180. (b) van der Beek, G. Ph.D. Thesis, University of Wageningen, The Netherlands, 1991. (c) van der Beek, G. J. Phys. (Paris) 1988, 49, 1449. (d) van der Beek, G. Langmuir 1991, 7, 327. (e) Van der Beek, G. P.; Stuart, M. A. C.; Fleer, G. J. Macromolecules 1991, 24 (12), 3553. (f) Stuart, M. A. C.; Fleer, G. J.; Scheutjens, J. M. H. M. J. Colloid Interface. Sci. 1984, 97, 526. (g) Stuart, M. A. C.; Fleer, G. J.; Scheutjens, J. M. H. M. J. Colloid Interface Sci. 1984, 97, 515. (h) Dijt, J. C.; Stuart, M. A. C.; Fleer, G. J. Macromolecules 1994, 27 (12), 3229. (3) Gritsenko, O. T.; Nesterov, A. E. Eur. Polym. J. 1991, 27 (4-5), 455. (4) Wohlfart, C. Acta Polym. 1991, 42 (10), 503. (5) Chen, C. H.; Riffle, J. S.; et al. Macromolecules 1994, 27 (22), 6376.
{λ1χpd - (1 - φcr)(1 - λ1)∆χdop} (2) where po, pd, and do superscripts correspond to polymersolvent, polymer-displacer, and displacer-solvent contacts, respectively.
∆χdop ) χpd + χdo - χpo φcr is the critical displacer volume fraction and λ1 is the fraction of contacts that a lattice site has with sites in one of the adjacent layers, e.g., for a hexagonal lattice, λ1 ) 3/12 ) 0.25. However, this is not a trivial task. To calculate a value for χs, it can be seen from eq 2, that χsdo, χsc, φcr, and all the Flory-Huggings χ parameters must first be determined. Solvent strength data as derived by Snyder6 and displacer adsorption isotherms are used to calculate χsdo. χsc can be estimated from the loss of configurational entropy (assuming rotational entropy is small). Desorption isotherms yield φcr, and literature data on polymer-solvent interactions can be used to deduce χpd. By combining results for pairs of displacers from two different solvents, four simultaneous equations are solved, which provide a value for ∆χdop. Inserting these results into eq 2 results in a value of χs. One result is obtained for each pair of displacers. Hence at least two displacers and two solvents are required for each segment type. Because of the large amount of chromatographic data available in the literature, research in this area has mainly centered on silica and alumina surfaces. Many interesting quantities, for example, the adsorbed amount, the fraction of bound segments, and the segment density profile, are a function of χs. Therefore the ability to obtain accurate χs values is a prerequisite for further theories. Clearly a simple method which does not involve a lattice structure and can be applied to any solvent/ substrate combination has enormous advantages. This study examines if a more “in depth” model is required and (6) Synder, L. R. Principals of Adsorption Chromatography; Marcel Dekker: New York, 1968.
10.1021/la9902284 CCC: $18.00 © 1999 American Chemical Society Published on Web 09/25/1999
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Superscript p denotes that the central molecule for the coordination number calculation is a polymer segment (s for solvent). Hence, calculating coordination numbers provides a way of determining χsc ignoring any contribution from specific solvents. The Weighted Average for Random Copolymers. If two segment types of a copolymer are considered and the polymer in question is statistically random, then individual χs terms may be combined to give an averaged “effective” χsAB. This two-state model gives the following equation7
exp(χsAB) ) νA exp(χsA) + νB exp(χsB)
Figure 1. Contact energy description for χs.
whether such a model provides an increase in accuracy and insight. The adhesion strength for a particular polymer segment type on a graphite surface is computed by a free space molecular modeling as a function of chain length. Coordination numbers are explicitly determined, with segment/ solvent volumes taken into consideration. Homopolymers of poly(ethylene) and poly(vinyl acetate) adsorbing onto graphite from carbon tetrachloride are considered. The simulation yields a χs parameter in a matter of hours. As comparable experimental values are difficult to obtain, only a few studies in the literature by a small number of research groups2-5 are available for comparison. Graphite was used in this work because of its well-known and simple structure. Computer Simulations The Contact Energy Model. We can interpret χs in terms of contact energies, and this is shown in Figure 1. This leads to the expression (per mol of contacts)
χs ) -[Ap-s - o-s + 1/2(o-o - p-p)]/RT
(3)
where p ) polymer segment, o ) solvent molecule, and s ) substrate, p-s, o-s, o-o, p-p ) contact energy (kJ mol-1 contact-1), R ) gas constant, and a ) scaling area ) area of solvent molecule/area of adsorbed polymer segment. In this case, the constant A is used to standardize the polymer-surface interaction for the case of comparing different polymer segment types, for example, copolymers. This is the distinct advantage of free space molecular modeling compared to a lattice model. A lattice model considers solvent molecules and polymer segments to have equal volume and, hence, equal coordination numbers. From a molecular modeling point of view, p-s and o-s can be obtained directly from molecular energy calculations, while o-o and p-p may be found in the literature as enthalpies of vaporization, ∆Hvap. The Critical Segmental Adsorption Energy. A critical value of χs exists (χsc) which must be exceeded for adsorption to occur. This can be approximated by the configurational entropy change on adsorption for a single isolated chain1 assuming the rotational entropy loss is small. Clearly the configurational entropy loss per monomer incurred on adsorption will be dependent on differences in the coordination numbers (Z) between a chain in two dimensions (adsorbed) and that of a chain in three dimensions (free):
χsc ≈ -ln(Z2Dp/Z3Dp)
(4)
(5)
where νA and νB are the fractions of monomer segments A and B, respectively, and χsA and χsB are the surface affinities for segments A and B. It then follows that the random copolymer is effectively a homopolymer with a χs that is a weighted average of the χs for its individual components. Simulation Technique. Individual polymer segments are studied by computer simulation. The total potential energy consisting of bonding, angle, angular torsion, inversion, van der Waals, hydrogen bonding and Coulombic terms is calculated from parameters included in the Dreiding II8 semiempirical force field. We use the Dreiding II force field because of its usefulness in predicting structures for a wide range of molecules containing carbon, including organic polymers networked surface structures. Nonbonded van der Waals interactions are modeled using the Lennard-Jones 12-6 potential. From the calculated intermolecular and intramolecular forces the optimum geometry and dynamic behavior of the systems are evaluated. To calculate the optimum structure the conjugate gradient9 minimization method is used. During an energy minimization, the forces acting on atoms are calculated and the atom coordinates are adjusted in order to reduce the molecular energy. The minimization is terminated when the root-mean-square (RMS) energy per atom is 0.004 kJ mol-1. Several molecular dynamics ensembles may be considered useful for studying the adsorption of polymers at interfaces. We consider isothermal molecular dynamics as a useful way to sample conformational space. Energy can be taken from a heat bath to overcome polymeric rotational barriers. This is achieved using the simple nonHamiltonian approach of Berendsen et al.10 Also, for alkane chains, the most favored conformation is known to be linear from experimental results. Therefore, quenched dynamics is used for the adsorption simulation onto graphite. Quenched dynamics operates with periods of dynamics followed by structural minimization enabling subtle differences in adsorbed conformational energies to be explored. However, rotational barriers are apparent for vinyl acetate oligomers due to the acetate groups protruding from the alkane backbone. In this case all vinyl acetate oligomers are simulated by annealed molecular dynamics followed by quenched dynamics. This prevents an intermediate structure from being caught in a local energy minimum. (7) (a) Fleer, G. J.; Cohen Stuart, M. A.; Scheutjens, J. M. H. M.; Cosgrove, T.; Vincent, B. Polymers at Interfaces; Chapman and Hall: London, 1993. (b) Brorling, M.; Linse, P.; Karlstrom, G. J. Phys. Chem. 1991, 95 (17), 6706. (8) Mayo, S. L.; Olafson, B. D.; Goddard, W. A. J. Phys. Chem. 1990, 94, 8897-8909. (9) Fletcher, R.; Reeves, C. M. Comput. J. 1964, 7, 149. (10) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; Di Nola, A.; Hack, J. R. J. Chem. Phys. 1984, 81, 3684.
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Figure 2. Annealed dynamics run for 50 VAc segments on graphite. Three cycles of 300-500 K dynamics are shown.
Calculation of Contact Energies: Polymer-Surface and Solvent-Surface. For polymer-surface interactions, the p-s energy term is determined by first generating a minimized random conformation of n polymer segments (1 < n e 50). Initial modeling experiments consisted of a large previously minimized planar graphite surface (154 Å × 114 Å), with five randomly oriented and minimized segments placed approximately 10 Å away. Periodic boundary conditions were not implemented. The graphite atoms are held rigid, as defined by the atomic potentials. This is achieved by a combination of energy minimization and molecular dynamics. The total potential energy is determined and then recalculated without the surface present. The difference yields p-s. It has been well established that alkane chains adsorb in a linear conformation lying flat against the graphite surface. As such following minimization until convergence, quenched dynamics (NVT, minimization at t ) 0.1 ps intervals) was run on each linear alkane chain oligomer. The time step, ∆t, remained at 0.001 ps, while the length of each dynamic run, t, depended on the molecule being studied. For an alkane with n Et segments, n < 50, t ) 10 ps, n g 50, t ) 30 ps. For VAc segments, as well as the quenched method, we employed annealed dynamics: initial temperature ) 300 K, intermediate temperature ) 500 K, with 100 steps of NVT dynamics per 10 K increment. This annealing cycle was repeated three times for each oligomer of VAc, enabling rotational barriers to be overcome. After annealing, a further period of NVT dynamics was run, followed by minimization until convergence. The length of NVT dynamics was set at 20 ps. Figure 2 shows the annealing cycles for 50 segments of VAc, while Figure 3 indicates that 20 ps of NVT dynamics is sufficient to reach equilibrium, i.e., the average potential energy remains constant. It is often argued that significantly longer time scales are required for the minimization of polymer species. However, in this work the time scales appear sufficient within the constraints of molecular dynamics, where time scales are of the order of picoseconds, as opposed to micoseconds. Figure 3 also shows that the initial dynamics scaling technique of assigning atom velocities at double the simulation temperature has equilibrated within 2 ps. The simulation was repeated after the first final conformation was reached. This second set of energies
Figure 3. NVT dynamics of 50 VAc segments on graphite after initial annealed run.
were all within 5% of the first simulation. By recalculating the total energy with the graphite surface excluded from the energy expression, the difference is equal to the oligomer-surface contact energy. In this case, because of the effects of end groups which would not be present for a polymer, the energy calculations are repeated as a function of molecular weight. Linear regression yields the incremental contact energy per segment. Note, the intercept will not necessarily pass through the origin by this method due to the end group effects. For the solvent-surface contact energy, a minimized solvent molecule is placed in a random conformation 10 Å above the graphite surface. This is minimized until convergence and the energy noted. The total energy without the surface present is calculated and is subtracted from the average of 10 different initial solvent conformations. Technically, using a single solvent molecule is a limitation of the approach used here. However, when compared with data relating to heats of immersion, the rapid single molecule technique shows good agreement. Calculation of Contact Energies: Solvent-Solvent and Polymer-Polymer. For the case of the energy terms o-o and p-p, the experimental ∆Hvap/Z3D is used, rather than explicitly calculating the energy by molecular modeling. Where no experimental value for ∆Hvap exists, the corresponding Hildebrand/Hansen parameter yields ∆Hvap. This is described in the Appendix. From crystallographic data, a coordination number of 4 is used for the p-p term of Et and VAc. This assumes the polymer solution value is the same as for the melt and would appear reasonable. Coordination numbers necessary for the o-o term and for determining χsc are discussed below.
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Figure 5. Chain backbone orientations.
Figure 4. Example of carbon tetrachloride shell generated by the hand made cluster method.
Calculation of Coordination Numbers. Coordination numbers are calculated in two ways: either by packing a central solvent molecule manually or by use of the algorithms of Blanco.11 For the “Manual” method, an initial cluster is created by placing solvent molecules in a shell around a central solvent molecule. The outer shell is then minimized and repacked, while keeping the central molecule rigid. The limiting value is therefore a uniform compacted shell around a central solvent molecule, as determined by minimization. The total coordination number is judged by eye, i.e., when no more holes are apparent. For example, see Figure 4. For the Blanco set of algorithms,11 several steps are employed. The coordinates of a central solvent molecule remain unchanged while a particular orientation of the packing molecule is determined by randomly choosing three Euler angles. This approach potentially leads to an underestimation of the true Z3D: however its considerable advantage over other techniques is the speed. A random vector that points from the center of mass of the central molecule to the surface of a unit sphere is chosen. The packing molecule is then translated along the vector until the van der Waals surfaces of each molecule just touch each other. The process is then repeated. Each nearest neighbor added must just touch the central molecule while avoiding overlapping the other nearest neighbors. A specified number of packing trials are allowed before determining that no more will fit and the coordination number stored. In this case the number of packing trials was set to its maximum of 1000 attempts. The average value of 10 000 clusters yields the coordination number Z3D. Adsorbed Areas. The adsorbed areas per segment as viewed perpendicular to the surface were calculated by obtaining a printout of the overhead projection (perpendicular to the surface) of the final minimized adsorbed segments. A cut out of the adsorbed footprint was then weighed and calibrated against graphite hexagons obtained from the same plane parallel to the interface. This “hands on” approach proved to be significantly quicker than computer imaging with commercial software. The fact that n-alkane chains lie flat on the graphite surface and adopt a linear conformation as the most energetically favored position enables their adsorbed area (11) Blanco, M. J. Comput. Chem. 1991, 12 (2), 237.
Figure 6. Adsorbed area as a function of chain length, as viewed perpendicular from the surface.
to be easily determined and compared with experimental values. Two initial alkane chain conformations of segments were minimized near the interface. These corresponded to the chain backbone lying horizontal and perpendicular to the surface as shown in Figure 5. However, no corresponding area calculations for P(VAc) adsorbed onto graphite are available in the literature. As chains of vinyl acetate cannot adsorb completely flat, it is possible to overestimate the adsorbed area by including “overhangs” of atoms which may appear as part of the adsorbed footprint when viewed perpendicular from the surface. However, the final minimized projection was manipulated enabling overhangs to be identified and eliminated. The estimated vinyl acetate adsorbed area is therefore reasonable, and we estimate it to have an accuracy of 10%. Results and Discussion The Adsorbed Areas. Figure 6 gives the adsorbed area per segment as viewed perpendicular to the surface. The value of the scaling constant A was determined from the ratio of the two different segment areas compared to that of the solvent (CCl4 ) 0.294 ( 0.010 nm2). By this method AVAc ) 1.18 ( 0.010 and AEt ) 2.58 ( 0.17. The ratio of the two adsorbed areas (VAc area/mean Et area) gives a value of 2.19. The adsorbed area for parallel ethylene segments agrees well with that obtained by Clint12 for the adsorption of n-alkane vapors on graphite. Clint’s value (12) Clint, J. H. J. Chem. Soc. Faraday. Trans 1, 1970, 68, 2239-46.
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Figure 7. Polymer-graphite (pol-surf) interaction energies as a function of molecular weight.
of 0.0547 ( 0.0025 nm2 per CH2 was calculated from BET data up to C12H26. An analogous regression with this work, only using data points up to C20H42 gives 0.0619 ( 0.007 nm2 per CH2 group. Contact Energy Results, Polymer-Surface. Polymer-surface contact energy results as a function of molecular weight are shown in Figure 7. The value of the individual slopes give p-s, per segment for Et and VAc segments as indicated. After Molecular Mechanics/Molecular Dynamics, both initial alkane chain geometries remained either perpendicular or horizontal to the surface and retained their linear structure. The segment-surface separation distance, as indicated above, also remained constant with increasing chain length. Only for C100H202 (Et50) lying perpendicular to the interface were distortions noticeable. The minimized structure showed twists along the carbon backbone, keeping some parts of the chain perpendicular while others were horizontal. The approximate ratio of perpendicular to horizontal configuration was 2:5, respectively. An explanation of this is the greater flexibility with reduced end effects for the longer chain, enabling rotational barriers to be overcome. As one aim of the study was to predict the energetic differences between horizontal and parallel carbon backbone adsorption, annealed dynamics was not considered necessary for alkane chains adsorbing onto graphite. A lower convergence limit or the use of annealed dynamics would have probably made all the segments horizontal for both types of initial conformations. The n-alkane p-s values of Figure 7 are in excellent agreement with other molecular dynamic simulations of physisorbed alkane chains. From Figure 7, p-s for perpendicular alkanes is -15.53 kJ mol-1 and horizontal alkanes is -16.63 kJ mol-1. Hentschke et al.,13 who undertook MD simulations using the Amber force field, reported a segmental value of -15.40 kJ mol-1 for the perpendicular conformation and -16.07 kJ mol-1 for the horizontal conformation. The modeled energies can also be directly compared to enthalpy of adsorption data for n-alkane vapors onto graphite. This is shown in Figure 8. The experimental study by Kiselev14 and others is in good agreement with the modeling technique. The data suggest that Kiselev observed a mixture of both parallel and perpendicular arrangements. (13) Hentschke, R.; Scho¨rmann, B. L.; Rabe, J. P. J. Chem. Phys. 1992, 96 (8), 6213. (14) Kiselev, A. V. Russ. J. Phys. Chem. 1961, 35 (2), 111.
Figure 8. Comparison of molecular modeling ethylenegraphite energies, with values from the literature.
From Figure 7 the horizontal alkane conformation is clearly favored with its lower interaction energy. This was more pronounced for the longer chain lengths. The energy per segment for the horizontal geometry was -1.26 kJ mol-1 (8%) more negative than for the corresponding perpendicular arrangement, and this difference is considered real. Many authors12-18 have observed that alkane chains adsorb onto graphite with their carbon backbone horizontal to the interface. The first was Kiselev15 in 1957. In fact, the alkane-graphite interaction is so favorable that the linear zigzag conformation of n-alkanes is thought to be compressed slightly to allow a fit between the hydrogen atoms attached to one side of the zigzag and the centers of the hexagons formed by carbon atoms in the basal plane of graphite. Confirmation of this by our molecular modeling work is shown in Figure 9. The fit gets progressively worse for longer chain lengths, and this may explain the discrepancies in adsorbed areas compared to Clint,12 who used relatively short chains. Other authors have also found large compressions on graphite. Livingston19 found an area per molecule for n-heptane 20% smaller on graphite than on other surfaces (TiO2, SiO2, BaSO4). Kiselev et al.20 calculated a 25% greater interaction for n-alkanes adsorbed with CH2 and CH3 groups located in the centers of hexagons than for adsorption at other sites. All the alkane chains considered in this study showed CH2 groups adsorbed on or near the center of the graphite hexagons. However, because of steric constraints, the adsorption of a random copolymer containing ethylene segments will have a mixture of both parallel and perpendicular adsorbed Et conformations. Therefore the average energy for the two orientations was used for further calculations. Comparing the annealed versus quenched dynamics for VAc oligomers in Figure 7 reveals that the quenched (15) Kiselev, A. V. Proc. Congr. Surf. Activity, 2nd 1957, 2, 168. (16) Hentschke, R.; Winkler, R. G. J. Chem. Phys. 1993, 99 (7), 5528. (17) Groszek, A. Proc. R. Soc. A 1970, 314, 473. (18) Ash, S. G.; Findenegg, G. H. Spec. Discuss. Faraday Soc. 1970, September. (19) Livingston, H. K. J. Am. Chem. Soc. 1944, 66, 569. (20) Kiselev, A. V.; et al. Izv. Akad. Nauk SSSR, Otd. Khim. Nauk 1957, 1314.
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Figure 9. Overhead view of final (minimized) adsorbed conformation of 10 Et segments. This conformation was used to calculate the polymer-surface contact energy.
structures have a slightly larger negative (
