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J. Phys. Chem. B 2001, 105, 11468-11473
Molecular Dynamics Simulation of Collinear Polymer Nanoparticle Collisions: Reaction and Scattering B. C. Hathorn,*,† B. G. Sumpter,† M. D. Barnes,‡ and D. W. Noid† Computer Science and Mathematics DiVision, and Chemical and Analytical Sciences DiVision, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831 ReceiVed: July 11, 2001; In Final Form: September 21, 2001
Collinear collisions of polyethylene nanoparticles are studied, with a particular emphasis on the transition between sticking and reactive scattering trajectories. The former are characterized by collisions where the two polymer particles form a particle dimer, the latter by collisions resulting in the gross change of particle structure, such as particle fragmentation or transfer of polymer chains from one particle to another. It is observed that in the collinear case there are no scattering trajectories where the particles undergo nearly elastic collisions and rebound with small energy change. This property is attributed to the fact that during the collision process the particles undergo reorganization, and the energy involved is difficult to recover. The result is that at all but extremely high velocities, particles undergo rapid and irreversible energy transfer from the relative translational coordinate into other internal coordinates, resulting in dimerization of the polymer particles.
I. Introduction There has been much involvement in the experimental production1-9 and theoretical investigation10-22 of polymer nanoparticles and their properties. It is anticipated that these novel structures may open the door to materials with novel and specifically tailored electrical and optical properties.9 Recently, investigations of polymer micro- and nanoparticles have demonstrated that well-defined aggregate structures can be formed, including chains of polymer particles.23 These structures have long lifetimes and are being investigated for their possible applications. In the present work, we seek to investigate the process of formation of structures by modeling the collisional dynamics of the particles that result in the superparticle structures. One notable phenomenon is that the particle agglomerates produced in the experiments do not fully coalesce, the individual particles retain much of their “molecular character”. In the past, molecular dynamics has been employed by numerous authors in the study of polymer systems, to extract such important information as morphology, chain mobility, stress relaxation, phase separation and other macroscopic properties (see, for example refs 24-30). We believe the present investigations comprise the first molecular dynamics studies of collisions of particles composed of polymer chains, and the resulting outcomes. Particle sintering has been a well-studied phenomenon. Essentially, particle growth occurs via particle aggregation followed by sintering. A number of previously employed models to study particle agglomeration have focused on phenomenological models.31-34 More recently, molecular dynamics simulations have been employed to study atomistic nanoparticles;35-40 however, these particles consisted of small clusters of atoms * Corresponding author. † Computer Science and Mathematics Division. ‡ Chemical and Analytical Sciences Division.
that would have relatively free migration within the final composite structure and allow for (relatively) rapid coalescence, which minimizes the surface energy with respect to the volume. It is clear that in the present case, where the polymer nanoparticles retain much of their initial structure,23 the transport of individual units within the cluster must be limited. As such, in the present calculation, we expect that the collisions should represent the middle ground between scattering type collisions and particle agglomeration. Collisions between molecules are an inherently important phenomenon in chemistry. Ultimately, collisions control the chemical kinetics and dynamics observed in molecular interactions. Much of the focus in the past has been in the close interactions between small molecules, and a large amount on information has been obtained, including such information as the translation to vibration energy transfer, and the lifetimes of collision complexes formed.41 In an inelastic close encounter of two molecular species there are ultimately three possible outcomes: the molecules can “stick” and remain tightly bound after the collision if enough of the energy is removed from the translational coordinate so that it has insufficient energy to redissociate, the molecules can scatter, when the amount of energy transferred from the relative translational coordinate to the internal coordinates leaves enough energy in the relative translational coordinate for the molecule to redissociate, and the molecules can react or undergo a fundamental change in structure prompted by the near interaction and the surplus of energy obtained from the relative translational coordinate. Numerous experimental and theoretical examples of these phenomena in small molecules are available in the literature,41 and in the present paper we extend the discussion to the theoretical treatment of larger molecules consisting of polymer particles made up of thousands of atoms. In the case of collisions of small molecules, which have been studied in exhaustive detail, the likelihood of a sticking collision is small, since there are limited avenues for dissipation of the
10.1021/jp012646z CCC: $20.00 © 2001 American Chemical Society Published on Web 10/24/2001
MD Studies of Polymer Chain Collisions relative translational energy into the small number of vibrational modes. For collisions that form bound complexes, it is typically assumed that there must be a third body present to remove the excess energy via a stabilizing collision in the LindemannHinshelwood sense.42,43 On the other hand, in small molecules, scattering and reactive collisions are well documented, the former frequently involving removal of an amount of relative translational energy corresponding to a vibrational excitation of one of the molecules, the latter involving chemical changes of the molecular species. Such reactions, particularly those of triatomic systems, have been studied in great detail, both theoretically and experimentally, for decades, with a number of systems such as the F + H2 system being elevated to the status of “model system”.41 In the present case, we depart from typical small molecule collisions and investigate collisions of nanoparticles made up of thousands of atoms. In contrast to the scattering events involving small molecules, one does not expect to observe fine detail of vibrational energy transfer corresponding to excitation of a single quantum state, but rather, a continuum of energy transfer, due to the huge density of vibrational states in the large molecules.15,16,18,21 At the same time, however, the amount of energy exchanged in a single collision may be substantially larger, when the lifetime of the collision is sufficiently long to allow for equilibration between the energy in the collision coordinate and the internal vibrational modes. The transfer of energy from the translational coordinate to internal vibrational modes provides an opportunity for the collision complex to persist, due to dissipation of the localized collision coordinate into the bath of other vibrational modes. As a result, one expects that the probability of a persistent collision complex forming should be greatly enhanced with the increase in the number of internal vibrational modes. In addition, one might also expect that, analogous to small molecule collisions, there might be regimes that demonstrate inelastic and reactive scattering. The case of inelastic scattering should be demonstrated by two colliding particles that rebound with a significant change in the relative translational energy. The concept of a reactive collision is somewhat different in the case of a nanoparticle and is considered in the context of a reactive multibody collision, where the particles have a dramatic change in their structure, typically evidenced by a loss of particle integrity or a transfer of polymer chains from one particle to the other. We note that the present study investigates collisions at substantially higher velocities than have been studied elsewhere, where the intent was to investigate the evolution of aggregate morphology and potentials of agglomerated particles under near thermal conditions.36,44 In the present case, where we are interested in the reactive interactions between the particles, substantially more energy must be introduced to surmount the large energetic barriers posed by chain transfer and particle fractionation. The results in the present paper represent an extrapolation beyond the lower velocity conditions of the current experimental regime to a range where particles must undergo substantial acceleration prior to collisions. We expect that in the next several years experiments will be designed to study collisions under these and similar conditions. In section II we review the basic principles and methods utilized in the molecular dynamics simulation of particle collisions. In section III we describe the results of the simulations performed, and in section IV we discuss the results in the context of experimental observations and theoretical expectations.
J. Phys. Chem. B, Vol. 105, No. 46, 2001 11469 TABLE 1: Potential Parameters for Polyethylene Particle Systems two-body bonded constantsa D ) 334.72 kJ/mol re ) 1.53 Å R ) 199 Å-1 three-body bonded constantsb γ ) 130.122 kJ/mol θe ) 113 degrees four-body bonded constantsa a ) - 18.4096 kJ/mol b ) 26.78 kJ/mol two-body nonbonded constantsa ) 0.4937 kJ/mol σ ) 4.335 Å a
References 52 and 53. b References 50 and 51.
II. Simulation Methodology For the purposes of the present study, we have examined motions and interaction forces between individual polymer particles both before and after interactions take place. The method of the calculations corresponds to a classical mechanical simulation, and the details of the computations using the geometric statement function approach are described more fully elsewhere,45 and are presented concisely here. For simplicity of the polyethylene model, we have collapsed the CH2 and CH3 units into a single monomer of mass 14.5 amu. By neglecting the internal structure of these groups, the number of coordinates is greatly reduced and thus the equations of motion for the system are simplified. The model has been shown to be useful to study the low-temperature behavior of the system where the effects of the hydrogens have little effect on the heat capacity and entropy of the system.49 Polymer particles have been treated with a molecular dynamics approach,46,47 integrating Hamilton’s equations of motion in time. In the present case, we have treated coordinates and momenta in the Cartesian frame, where the total kinetic energy is diagonal. The Hamiltonian for the system is specified as9,49
H)T+
∑V2b + ∑V3b + ∑V4b + ∑Vnb
(1)
where T is the kinetic energy component, expressed in terms of Cartesian coordinates, the terms V2b, V3b, and V4b represent the 2-, 3-, 4- body terms for monomer units in an individual polymer strand, and Vnb is the nonbonded interaction between individual monomer units separated by 4 or more monomer units along the chain and within a spherical cutoff of 10 Å. The functional forms of the potentials are given by49-53
V2b ) D{1 - exp[-R(rij - re)]}
(2)
1 V3b ) γ(cos θ - cos θe)2 2
(3)
V4b ) 8.77 + a cos τ + b cos3 τ
(4)
Vnb ) 4
[( ) ( ) ] σ rij
12
-
σ rij
6
,
(5)
with the values of the constant terms given in Table 1. The distances between the various monomer units, rij are given by the standard Cartesian relation distance. Although a number of other united atom potentials are available, typically they contain fixed bond lengths between adjacent carbon atoms (see, for example, refs 54-56). The technique we currently employ for
11470 J. Phys. Chem. B, Vol. 105, No. 46, 2001
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Figure 2. Classification of sticking and reactive trajectories as a function of relative velocity (in [IMAGE ]) and internal particle temperature.
Figure 1. Initial configuration of polymer particles.
rapid integration of the equations of motion does not restrict us to a system where the bonded interactions have fixed distances. In the present case, initial conditions for the trajectories comprised the coordinates of individual polymer particles, which had previously been obtained by an annealing,14 with momenta chosen randomly in the radial coordinate so as not to excite any internal angular momentum. The randomly chosen momenta were rescaled so as to produce the appropriate temperature for the simulation,
1
∑i 2m
i
p2i
3
) NkBT, 2
A number of properties were tracked during the course of the simulation. Principal among them were the positions and velocities of the centers of mass of the particles, the center of mass separation of the two particles, and the moments of radial distribution of the particles. The first two give the time development of the positions and distances of the particles; the last gives a measure of the particle integrity (when polymer chains are exchanged between the particles or become detached from the parent particle, the average distance of the monomer units from the center of mass becomes large). We note that in the following discussions, the relative velocity is the velocity between the two particle centers of mass. III. Results
(6)
where T is the absolute temperature and kB is Boltzmann’s constant. The thermalized particles were offset in the z-coordinate and given an initial relative velocity in the z-coordinate so as to produce a collinear (with respect to the centers of mass) collision. The initial offset was chosen so as to give at least 1 ps of time for the particles to thermally equilibrate before coming within the maximum range of forces with the other particle (in the present case, 10 Å), Figure 1. Integration of the equations of motion was accomplished by use of novel symplectic integrators developed in our laboratory.48 Simulations were continued for a minimum of 100 ps, and longer in the case of trajectories leading to dimerization of the particles to ensure that the final outcome did not allow for additional changes in the unified particle morphology.
A number of trajectories were run for collisions of particles with 30 chains, each composed of 100 monomer units. Various initial conditions for the internal temperature and relative velocities of the particles were assumed, with internal temperatures being chosen to be near absolute zero, and above and below the melting temperature of the particles, up to a temperature of 1000 K. The classification scheme for the results of the trajectories is outlined below. A summary of the trajectory outcomes is represented in Figure 2. In the present paper we focus our attention principally to collisions at high relative velocities, so as to investigate the possibility of reactive collisions. Low velocity collisions do not show behavior in the context of reaction and scattering described in the present paper, and result in uniformly “sticking” type trajectories. During the course of the simulation, trajectories were deemed to result in particle sticking if the resultant separation in the center of mass of the two initial particles became bounded and
MD Studies of Polymer Chain Collisions
J. Phys. Chem. B, Vol. 105, No. 46, 2001 11471 (the cutoff of the Lennard-Jones nonbonded interactions), leading to no possibility of return. At the same time, however, the particles must maintain their integrity, so 〈r2〉 must remain bounded for both particles. In our simulations, no such trajectories were observed. IV. Discussion
Figure 3. Time dependence of 〈r2〉 for typical reactive and unreactive trajectories. Initial relative collision velocities are 10 and 20 Å/ps, respectively. The curve for the 20 Å/ps collision has been scaled by a factor of 0.1 so as to fit on the same axes. Only one particle is shown for each collision.
remained so for the whole of the simulation time scale. At the same time, the particles maintained their integrity, with no parts of the particles becoming dissociated from the parent, as would characterize a “reactive trajectory”, described below. A large number of trajectories were observed to be “reactive” in that one or both of the particles lost its structure, and fragments of the particle became dissociated from the parent during the course of the trajectory. A straightforward measure of the reaction of the particles is evidenced by the average value of the square of the radial distribution,
〈r2〉 )
1
N
(xi - xCM)2 + (yi - yCM)2 + (zi - zCM)2 ∑ N i)1
(7)
where (xi, yi, zi) are the positions of the monomer units making up the particle, and (xCM, yCM, zCM) are the positions of the centers of mass. Reaction of a particle leads to a value of 〈r2〉, which is unbounded as time progresses. A comparison of the time dependence of 〈r2〉 is illustrated in Figure 3 for typical reactive and unreactive trajectories. Trajectories were deemed to be reactive using this measure if 〈r2〉 exceeds 1000 Å2, and were explicitly visualized to determine the nature of the deformation. In these cases, it was observed that the particles fragmented into smaller bodies (typical fragmentations resulted in two smaller particles being formed, due to the large surface energy required to make small particles). A determination was made that a trajectory could be characterized as a scattering trajectory if at the conclusion of the simulation, center of mass separation of the particles was increasing and the distance of separation was more than 10 Å
The results obtained in the present trajectory studies for collinear collisions are striking. It is apparent that there are essentially no elastic or near elastic collinear collisions, where a small amount of energy is transferred from the relative translational energy into internal degrees of freedom for the particles. The transfer of energy into internal modes is apparently quite efficient, and given the extremely high density of states for internal motions, it is quite unlikely to be localized into a center of mass separation mode, which is required for the subsequent redissociation of the particles. As a result, at low collision velocities, the particles invariably stick, with the majority of the relative motion converted into internal motion. At higher collisional velocities, there is a sufficient amount of energy to concentrate it in one of the modes corresponding to a redissociation; however, the energy is randomized sufficiently rapidly as to show little preference for the dissociation of any individual part of the molecule. As a result, the integrity of the particles is lost and small pieces, corresponding to individual chains of 100 monomers, break off. Ultimately, in our simulations there is little sensitivity of the transition from sticking to reactive trajectories on the internal energy of the polymer particles. Effectively, all relative translational energy and thermal energy of the particles is accessible for the reactive fragmentation; however, relative translational energy is more efficient in producing reactive collisions. In the present simulations, the particles were treated as having thermal temperatures in the range T e 1000 K, the total internal energy per monomer unit of the particles being small compared to the amount of relative kinetic energy required for a reactive trajectory (relative velocities on the order of 15-20 Å/ps). The relative translational energy imparted to the system is much more effective in producing reactions than is the initial thermal energy. We observe that at low temperatures 9500 kcal/mol of relative translational energy is sufficient to produce a reactive scattering event, while this amount of internal vibrational energy in the particles requires the addition of approximately 8000 kcal/mol of additional relative translational energy to produce a reactive trajectory. The presence of the reactive collision at a velocity of 17 Å/ps and 300 K in Figures 2 and 4 demonstrates that the partition dividing reactive and unreactive collisions is not, in fact, a sharp boundary, but rather an approximate one. Closer inspection of this trajectory reveals that in this collision, only one chain is lost from one of the two particles, the other particle remaining intact. The increase in total surface area, and hence the surface energy cost, is not so large as to break a particle into larger pieces. We expect, however, that since a single trajectory was run at each of the initial conditions a small modification of initial conditions, for instance a particle rotation, would likely produce a sticking trajectory. In all of the other reactive trajectories, both particles lose their integrity, indicating that enough energy is available to overcome the substantial surface energy that is converted to kinetic energy during the collision process. One might expect that the particles at very low temperatures would be sufficiently cold as to be closer to rigid spheres, and thus more apt to undergo elastic collisions, but this is not found to be the case. One finds that the majority of the interparticle
11472 J. Phys. Chem. B, Vol. 105, No. 46, 2001
Figure 4. Classification of sticking and reactive trajectories in terms of the partition of thermal and relative translational energy.
potential is not due to interactions between roughly spherical polymer particles but, rather, is due to deformations of the particles. The comparison between the interaction potential for deformed (after 160 ps of interaction) and undeformed particles, against that which is actually observed during the course of a slow collision event is represented in Figure 5. As a result, when two cold particles approach each other, the lowest energy state results in substantial deformation of the particles, as in Figures 6 and 7, and the particles relax to that configuration. The characteristic feature of the minimum energy configuration is to maximize the enclosed volume with respect to the surface area. The energy obtained from this relaxation is invariably difficult to recover, and the result is a collision where the two particles coalesce into a large agglomerated particle, irreversibly sticking together. In addition, the minimization of the potential energy is accompanied by an increase of the kinetic energy (e.g., thermal heating) of the particles in the constant energy system. Elsewhere, the conversion of this potential energy into kinetic energy has been implicated in the rapid reorganization of the particles.35 In the present case, however, the presence of polymer chains inhibits the free transport of matter and retards the coalescence rate. The deformation of the particles also plays a significant role in the ultimate fate of reactive collisions. Typically, the deformation leads to evolution of polymer strings extending from the main body of the polymer particle in the direction of its collision partner, as in Figure 6. As previously noted, the deformations produce a structure in which the initial particles are deformed from a nearly spherical morphology, to one where the morphology of the unified particle approaches a spherical structure. In a collision with sufficiently high energy, it is these segments of the initial particles that undergo the most severe
Hathorn et al.
Figure 5. Interaction potential between polymer particles. The upper curve represents the interaction between two rigid polymer particles that are not allowed to relax to a more energetically favorable configuration during the collision process. The lower curve represents the interaction energy during the course of a simulated collision, where the particles are allowed to relax. The trajectory shown is for a comparatively low velocity collision, with initial relative velocities of 1 Å/ps.
Figure 6. Deformation of particles. The trajectory is for particles with initial internal temperature of 5 K and relative velocity of 12 Å/ps. For clarity, a rotation and displacement has been introduced to separate the particles.
deformation that continue in the direction of their initial momentum and ultimately become dissociated from the parent particle. Last we note the lower sensitivity of the outcome of the particle collisions to the initial thermal energy of the particles than to the initial relative translational energy. The energy required to fracture or otherwise change the composition of the
MD Studies of Polymer Chain Collisions
Figure 7. Same as Figure 6, but without the artificial separation between the particles.
particles is quite large, because entire chains must be removed from the parent structure. This differs from atomistic calculations of particles with atomic or small molecule substructures, where the transfer of a single subunit from one particle to another does not experience a high energetic barrier. As a result, the relative collision velocities required to induce reactive trajectories are orders of magnitude higher than normal thermal velocities. The most efficient manner to remove a chain from the particles is by a directional momentum of all of the subunits of a chain. This mechanism is achieved by introduction of additional coordinated relative velocity, whereas introduction of thermal energy into the chains produces randomly oriented velocities, which lack the concerted character required for particle fragmentation. Such a picture accounts for the inefficiency of thermal energy to produce reactive collisions, in comparison to translational energy. Acknowledgment. We thank the anonymous reviewer for helpful comments. This work was sponsored by the Division of Computer Science and Mathematics and the Division of Materials Sciences, Office of Basic Energy Sciences, U.S. Department of Energy under Contract DE-AC05-00OR22725 with UT-Battelle at Oak Ridge National Laboratory (ORNL), using resources of the Center for Computational Sciences at Oak Ridge National Laboratory. One of us (B.C.H.) has been supported by the Postdoctoral Research Associates Program administered jointly by ORNL and the Oak Ridge Institute for Science and Education. References and Notes (1) Kung, C.-Y.; Barnes, M. D.; Sumpter, B. G.; Noid, D. W.; Otaigbe, J. Polym. Preprint 1998, 39, 610. (2) Barnes, M. D.; Kung, C.-Y.; Sumpter, B. G.; Noid, D. W.; Otaigbe, J. Optics Lett. 1999, 24, 121. (3) Barnes, M. D.; Ng, K. C.; Fukui, K.; Sumpter, B. G.; Noid, D. W. Mater. Today 1999, 2, 25. (4) Barnes, M. D.; Ng, K. C.; Fukui, K.; Sumpter, B. G.; Noid, D. W. Macromolecules 1999, 32, 7183. (5) Ford, J. V.; Sumpter, B. G.; Noid, D. W.; Barnes, M. D. Chem. Phys. Lett. 2000, 316, 181. (6) Ford, J. V.; Sumpter, B. G.; Noid, D. W.; Barnes, M. D. J. Phys. Chem. B 2000, 104, 495. (7) Ford, J. V.; Sumpter, B. G.; Noid, D. W.; Barnes, M. D. Polymer 2000, 41, 8075. (8) Ford, J. V.; Sumpter, B. G.; Noid, D. W.; Barnes, M. D.; Otaigbe, J. U. Appl. Phys. Lett. 2000, 77, 2515. (9) Otaigbe, J.; Barnes, M. D.; Fukui, K.; Sumpter, B. G.; Noid, D. W. AdV. Polym. Science 2001, 154, 1.
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