Molecular Dynamics Simulations of Polyelectrolyte Multilayering on a

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Langmuir 2005, 21, 1118-1125

Molecular Dynamics Simulations of Polyelectrolyte Multilayering on a Charged Particle Venkateswarlu Panchagnula,† Junhwan Jeon,‡ James F. Rusling,†,§ and Andrey V. Dobrynin*,‡,| Polymer Program, Institute of Materials Science, Departments of Chemistry and Physics, University of Connecticut, Storrs, Connecticut 06269, and Department of Pharmacology, University of Connecticut Health Center, Farmington, Connecticut 06032 Received September 9, 2004. In Final Form: November 2, 2004 Molecular dynamics simulations of polyelectrolyte multilayering on a charged spherical particle revealed that the sequential adsorption of oppositely charged flexible polyelectrolytes proceeds with surface charge reversal and highlighted electrostatic interactions as the major driving force of layer deposition. Far from being completely immobilized, multilayers feature a constant surge of chain intermixing during the deposition process, consistent with experimental observations of extensive interlayer mixing in these films. The formation of multilayers as well as the extent of layer intermixing depends on the degree of polymerization of the polyelectrolyte chains and the fraction of charge on its backbone. The presence of ionic pairs between oppositely charged macromolecules forming layers seems to play an important role in stabilizing the multilayer film.

1. Introduction Layer-by-layer assembly of charged molecules is a versatile route to structured and robust ultrathin films, which have evoked tremendous interest in a variety of potential applications ranging from biosensors to nanoengineering and drug delivery.1 The simplicity of the electrostatic assembly technique with practically no limitations on the shape or identity of the macromolecular charge-bearing species allows the fabrication of multilayer films to user design specifications from synthetic polyelectrolytes,2 DNA,3 proteins,4 nanoparticles,5 and so forth. First reported in 1966 and rediscovered in the early 1990s,6,7 this technique of film assembly is based on the long-range electrostatic attraction of oppositely charged molecules. The key to successful deposition of multilayer assemblies in a layer-by-layer fashion is the inversion and subsequent reconstruction of surface properties. A typical experimental procedure involves immersing a solid * Corresponding author. E-mail: [email protected]. † Department of Chemistry, University of Connecticut. ‡ Institute of Materials Science, University of Connecticut. § University of Connecticut Health Center. | Department of Physics, University of Connecticut. (1) (a) Lvov, Y. In Protein Architecture: Interfacing Molecular Assemblies and Immobilization Biotechnology; Lvov, Y., Mo¨hwald, H., Eds.; Marcel Dekker: New York, 2000; pp 125-167. (b) Lvov, Y. In Handbook Of Surfaces And Interfaces Of Materials, Vol. 3. Nanostructured Materials, Micelles and Colloids; Nalwa, R. W., Ed.; Academic Press: San Diego, CA, 2001; pp 170-189. (c) Rusling, J. F.; Zhang, Z. In Handbook Of Surfaces And Interfaces Of Materials, Vol. 5. Biomolecules, Biointerfaces, And Applications; Nalwa, R. W., Ed.; Academic Press: San Diego, CA, 2001; pp 33-71. (d) Rusling, J. F.; Zhang, Z. In Biomolecular Films; Rusling, J. F., Ed.; Marcel Dekker: New York, 2003; pp 1-64. (2) Ruths, J.; Essler, F.; Decher, G.; Riegler, H. Langmuir 2000, 16, 8871-8878. Arys, X.; Laschewsky, A.; Jonas, A. M. Macromolecules 2001, 34, 3318-3330. (3) Sukhorukov, G. B.; Mohwald, H.; Decher, G.; Lvov, Y. Thin Solid Films 1996, 285, 220-223. (4) Rusling, J. F. In Protein Architecture: Interfacing Molecular Assemblies and Immobilization Biotechnology; Lvov, Y., Mohwald, H., Eds.; Marcel Dekker: New York, 2000; p 337. (5) Lvov, Y.; Ariga, K.; Onda, M.; Ichinose, I.; Kunitake, T. Langmuir 1997, 13, 6195-6203. (6) Iler, R. K. J. Colloid Interface Sci. 1966, 21, 569-594. (7) Decher, G. Science 1997, 277, 1232-1237.

substrate into dilute solutions of anionic or cationic polyelectrolytes for a period of time optimized for adsorption followed by a rinsing step to remove any unadsorbed material. Further film growth is achieved by alternating the deposition of polyanions and polycations from their aqueous solutions. After a few dipping cycles, experiments show a linear increase of multilayer thickness, indicating that the system reaches a stationary regime. Several important features have been established for the layer-by-layer film formation. The thickness of each adsorbed layer shows almost a linear dependence on the salt concentration.8,9 Flexible polyelectrolytes in twocomponent multilayers are known to intermix over several adjacent layers. This layer intermixing can be suppressed by using more rigid blocks for the assembly.7,10 Intrinsic charge compensation by polyions accompanied by overcharging and the kinetically irreversible nature of deposition has also been reported.9 It was also shown that the layer thickness and molecular organization of adsorbed polymers could be tuned by adjusting the pH of the dipping solutions.11 In this case, the solution pH controls not only the adsorbing polyelectrolyte but also the previously adsorbed layers. There is an ionization threshold below which there is no multilayer formation.12 The measurements of the Young’s modulus of polyelectrolyte multilayers yielded an estimate that is very close to that expected for ionomers.13 It could therefore be possible that the structure of multilayer films is close to that of ionomers, (8) Losche, M.; Schmitt, J.; Decher, G.; Bouwman, W. G.; Kjaer, K.; Macromolecules 1998, 31, 8893-8906. (9) Schlenoff, J. B.; Ly, H.; Li, M. J. Am. Chem. Soc. 1998, 120, 76267634. Dubas, S. T.; Schlenoff, J. B. Macromolecules 1999, 32, 81538160. (10) Kleinfeld, E. R.; Ferguson, G. S. Science 1994, 265, 370-373. Glinel, K.; Laschewsky, A.; Jonas, A. M. Macromolecules 2001, 34, 52675274. (11) Shiratori, S. S.; Rubner, M. F. Macromolecules 2000, 33, 42134219. (12) Steitz, R.; Jaeger, W.; Klitzing, R. v. Langmuir 2001, 17, 44714474. Voigt, U.; Jaeger, W.; Findenegg, G. H.; Klitzing, R. v. J. Phys. Chem. B 2003, 107, 5273-5280. (13) Vinogradova, O. I.; Andrienko D.; Lulevich, V. V.; Nordschild, S.; Sukhorukov, G. B. Macromolecules 2004, 37, 1113-1117.

10.1021/la047741o CCC: $30.25 © 2005 American Chemical Society Published on Web 01/07/2005

Polyelectrolyte Multilayering on a Charged Particle

reflecting strong ionic interactions between the polycations and polyanions within multilayers. Although layer-by-layer assembly has been widely used on charged planar surfaces, there has been increasing interest in their spherical counterparts. Multilayer assembly of polyelectrolytes on spherical particles has been successfully performed to obtain hollow nano- and microsized structures.14-16 This process alters the physiochemical properties of the spherical substrates, which have many potential applications in drug delivery, catalysis, composites, surface coatings, and so forth. Despite extensive experimental studies, the theoretical models of electrostatic self-assembly are still in their infancy. Netz and Joanny17 considered the formation of multilayers in a system of semiflexible polymers, assuming that the deposited layer structure was fixed, providing a solid charged substrate for the next layer. Mayes et al.18 applied a similar idea of a solid substrate to flexible polyelectrolytes. Flexible polyelectrolytes were assumed to form a brushlike layer at the solid substrate of previously adsorbed chains in which the loop size and layer thickness are controlled by the strength of the ionic pair interactions formed between oppositely charged chains in neighboring layers and by electrostatic repulsion between polyelectrolytes within the same layer. Strong charge overcompensation occurs in the case of highly ionized flexible polymer chains adsorbed in a brushlike manner, leading to the formation of multilayers. This model predicts a square root dependence of polymer surface coverage on salt concentration. Both these models neglect interpenetration and chain complexation between the layers. An opposite limit of strong intermixing of polyelectrolytes between neighboring layers was considered by Castlenovo and Joanny,19 by incorporating the complex formation between oppositely charged polyelectrolytes into selfconsistent field equations. Analysis of these equations was limited to solutions of high ionic strength, where the electrostatic interactions are exponentially screened and can effectively be treated as short-range interactions. Another route for multilayer formation was shown by Solis and de la Cruz.20 They found that stratification of polyelectrolytes near charged surfaces could also be achieved by increasing the incompatibility between oppositely charged polymers. It is difficult to directly test all assumptions of theoretical models in experiments. However, computer simulations can help to elucidate factors involved in the multilayer assembly process and verify theoretical assumptions made in models of electrostatic film assembly. This, for example, can be done by changing the interaction parameters, by changing the length of the simulation runs, and by analyzing the layer structure, which is easily accessible through the polymer density profiles, counterion distributions, and so forth. (14) (a) Caruso, F.; Carsuo, R. A.; Mohlwald, H. Science 1998, 282, 1111-1114. (b) Gittins, D. I.; Caruso, F. J. Phys. Chem. B 2001, 105, 6846-6852. (15) Caruso, F.; Fiedler, H.; Haage, K. Colloids Surf., A 2000, 169, 287-293. (16) Usha, A. S.; Caruso, F.; Rogach, A. L.; Sukhorukov, G. B.; Kornowski, A.; Mohwald, H.; Giersig, M.; Eychmuller, A.; Weller, H. Colloids Surf., A 2000, 163, 39-44. (17) Netz, R. R.; Joanny, J.-F. Macromolecules 1999, 32, 9013-9025. (18) Park, S. Y.; Rubner, M. F.; Mayes, A. M. Langmuir 2002, 18, 9600-9604. (19) Castlenovo, M.; Joanny, J. F. Langmuir 2000, 16, 7524-7532. (20) Solis, F. J.; de la Cruz, M. O. J. Chem. Phys. 1999, 110, 1151711522.

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Polyelectrolyte multilayering on a spherical particle was studied using the Monte Carlo simulations21-23 based on the assumption that multilayering occurs whether one proceeds in a stepwise fashion, as envisaged in a real experiment, or when the oppositely charged polyelectrolytes are added together. In such a situation, it was noted that the polyelectrolyte multilayering on a spherical surface requires a sufficiently strong extra short-ranged attractive interaction between the macroion and the polyelectrolyte. Such Monte Carlo simulations assume that the multilayering is in an equilibrium state. Thus, its preparation is independent of the route chosen to reach equilibrium. Although surface charge reversal is essential in the assembled layer growth, several fundamental questions bearing importance for understanding these systems remain unanswered. Recently, we showed that electrostatic interactions are indeed the leading driving force for the multilayer assembly.24 In this paper, we use molecular dynamics simulations to address the following: (a) the effect of the chain degree polymerization and charge fraction on surface charge reversal, leading to polyelectrolyte multilayering around a charged spherical particle, and (b) the intermixing between the layers, which has been widely observed in experiments.1a,b The sequential deposition process implemented in our simulations resembles a real experimental situation unlike the method adopted earlier.23 Our simulation results emphasize the importance of electrostatic interactions and short-range attractive interactions as the driving force in multilayer formation with surface overcharging accompanying layer growth at every deposition step. The simulation model and method are described in section 2. In section 3, we present the simulation results along with a detailed analysis of the layer structure and dynamics. Finally, in section 4, we summarize our findings. 2. Model and Simulation Methodology The molecular dynamics simulations of multilayer assembly were performed from solutions of polyelectrolyte chains with degrees of polymerization N ) 32, 16, and 8 monomers. For each chain length, the fractions of charged monomers on a chain are equal to f ) 1, 0.5, and 0.25 corresponding to every one, second, and fourth bead in the chain carrying a charge. Multilayers were formed around a charged spherical particle that has Nsphere ) 80 negatively charged Lennard-Jones beads with the diameter 1σ (see Figure 1a). The central particle with 80 overlapping beads on its surface has a 3-D structure similar to a bucky ball of C80. The radius of this particle, measured as the distance between its center and centers of mass of the forming particle beads, is equal to 1.5σ. Such a structure is compact enough to restrict the adsorption onto the outer surface of the particle. This will enable multilayer growth only on the outside of the core of the central charged particle. When the radius of this central particle is increased beyond 1.5σ, the beads on the surface no longer overlap and leave spaces in between. These spaces are wide enough to permit chain penetration to adsorb onto its interior. The connectivity of beads in (21) Messina, R.; Holm, C.; Kremer, K. Langmuir 2003, 19, 44734482. (22) Messina R. Macromolecules 2004, 37, 621-629. (23) Messina, R.; Holm, C.; Kremer, K. J. Polym. Sci., Part B: Polym. Phys. 2004, 42, 3557-3570. (24) Panchagnula, V.; Jeon, J.; Dobrynin, A. V. Phys. Rev. Lett. 2004, 93, 037801-1.

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Figure 1. Evolution of the multilayer structure around the spherical macroion during the adsorption of fully charged (f ) 1) polyelectrolytes with N ) 32. The snapshots were taken after the completion of deposition steps from 1 through 5 with a time duration of 2.0 × 105 MD steps per each deposition step. The macroion is shown in black in part a, followed by the buildup of layers 1-5 sequentially color-coded as red, blue, magenta, cyan, and orange beaded chains (b-f).

the chains is maintained by the finite extension nonlinear elastic (FENE) potential:25

UFENE(r) ) -

(

1 r2 kspringR2max ln 1 - 2 2 R

max

)

(1)

lBqiqj rij

ULJ(rij) )

{

(2)

where the Bjerrum length, lB ) e2/kBT, defined as the length scale at which the Coulomb interaction between two elementary charges e in a dielectric medium with a dielectric constant  is equal to the thermal energy kBT, describes the strength of the electrostatic interactions. In our simulations, the value of the Bjerrum length lB is equal to 1.0σ. All charged particles in our simulations are monovalent ions with a valency qi ) (1. The particleparticle particle-mesh (PPPM) method implemented in LAMMPS26 has been used to calculate the electrostatic interactions with all periodic images of the system. Also, (25) Stevens, M. J.; Kremer, K. J. Chem. Phys. 1995, 103, 16691690.

[( ) ( ) ( ) ( ) ]

4LJ

0

The spring constant kspring is equal to 15kBT/σ2, where kB is the Boltzmann constant, T is the absolute temperature, and the maximum bond length Rmax between the beads is set to 2σ. Counterions were explicitly included in our simulations. Interaction between any two charged particles, bearing charge valences qi and qj and separated by a distance of rij, is described by the Coulomb potential:

UCoul(rij) ) kBT

all the charged and uncharged particles interact through the truncated/shifted Lennard-Jones (LJ) potential:

σ rij

12

-

σ rij

6

-

σ rc

12

+

σ rc

6

r e rc r > rc (3)

where rij is the distance between any two interacting beads i and j and σ is the bead diameter chosen to be the same regardless of the type of beads. A cutoff distance rc ) 2.5σ was chosen for the macroion/polymer-polymer interactions and rc ) 21/6σ for the polymer-counterion as well as counterion-counterion interactions. The interaction parameter LJ is set to 0.75kBT. As the depth of the energy well, given by LJ, cannot be directly compared with experimental values, we investigated a range of values for LJ from good to poor solvent conditions that are known to be realistic for soft matter systems.27 The effective interaction for θ-solvent condition is given by LJ ) (0.34 ( 0.02)kBT for the uncharged system. The value LJ ) 0.75kBT, thus, is that of a poor solvent condition and is sufficiently strong for the assembly of multilayers. We also found that the multilayers did not form under good solvent conditions. The combination of FENE and LJ potentials prevents the chains from crossing each other during the simulation run. We can map our system onto an experimental system, such as sodium poly(styrene sulfonate) (NaPSS), in water. (26) Plimpton, S. J. Comput. Phys. 1995, 117, 1-19. LAMMPS website: http://www.cs.sandia.gov/∼sjplimp/lammps.html. (27) Micka, U.; Holm, C.; Kremer, K. Langmuir 1999, 15, 40334044.

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Table 1. Parameters Used in the Simulations

rinsing step. To separate adsorbed chains from the rest of the polyelectrolytes we used a cluster algorithm with the cutoff radius equal to 1.2σ. A chain is considered to belong to a cluster if it has at least one monomer within a distance of 1.2σ from any monomer belonging to a chain forming the cluster. The cluster analysis was performed by analyzing the matrix of distances between all monomers in the system. After each simulation run, only the counterions needed for the compensation of excess charge of the particle-adsorbed chains in the aggregate were kept in the simulation box to maintain system electroneutrality. At the beginning of the second layer deposition step, the simulation box is refilled with M2 ) M1 oppositely charged polyelectrolytes together with their counterions. This is followed by the simulation run (dipping cycle). We repeated the dipping and rinsing cycles to simulate the buildup of nine layers and performed two different simulations with durations of 2.0 × 105 and 2.0 × 106 integration steps for every deposition cycle. This translates to about 2.0 Rouse relaxation times for short simulation runs and 20 relaxation times for the longer runs. Before collecting the data, the system was allowed to equilibrate for a half time of the total simulation run during each dipping cycle.

parameter

value

N Nsphere f qi σ Rmax ) 2σ rc ) 2.5σ kspring ) 15kBT/σ2 T  Lbox ) 26σ lB ) 1.0σ

32, 16, and 8 80 1, 0.5, and 0.25 (1 7.14 Å 14.28 Å 17.85 Å 0.729 kJ/mol/Å2

degree of polymerization of the chain no. of beads on the macroion fraction of charged monomers charge valence on a monomer/bead diameter of each bead maximum bond length cutoff distance spring constant

298 K 78.5 184 Å 7.14 Å

temperature dielectric constant of water length of the cubical simulation box Bjerrum length

If we assume that our Bjerrum length, lB ) 1.0σ, is equal to the Bjerrum length in aqueous solutions at room temperature (T ) 298 K), lB ) 7.14 Å, the monomer size is equal to σ ) 7.14 Å. This corresponds to approximately 2.9 monomers of NaPSS with a monomer size 2.5 Å. Compared with a fully charged chain (f ) 1), this corresponds to a fraction of charged monomers f ) 0.35. As we are interested in the general class of polyelectrolyte multilayer adsorption, each bead in the coarse-grained bead spring model employed in our simulations represents several chemical units. This method of mapping has been routinely used in the literature earlier.28 The values of parameters and their relation to the corresponding real units used in the simulation are summarized in Table 1. For simplicity, reduced units will be used for discussion in the rest of the manuscript. Simulations were carried out in a constant number of particles, volume, and temperature ensemble (NVT) with periodic boundary conditions. The constant temperature is achieved by coupling the system to a Langevin thermostat. In this case, the equation of motion of the ith particle is

m

dv bi (t) ) F Bi(t) - ξv bi(t) + F BiR(t) dt

(4)

Bi is the net deterministic where b vi is the bead velocity and F force acting on the ith bead of mass m. F BiR is the stochastic force with zero average value 〈F BiR(t)〉 ) 0 and δ-functional BiR(t′)〉 ) 6ξkBTδ(t - t′). The friction correlations 〈F BiR(t) F coefficient ξ was set to ξ ) m/τLJ, where τLJ is the standard LJ time τLJ ) σ(m/LJ)1/2. The velocity-Verlet algorithm with a time step of ∆t ) 0.01τLJ was used for integration of the equations of motion (eq 4). The solvent effect is modeled by using a dielectric medium with an effective dielectric constant. The dielectric continuum captures the solvent effects on the strength of electrostatic interactions. By coupling the system to the Langevin thermostat, the stochastic effects of the solvent are taken into account. Simulations were performed by the following procedure. The spherical particle remains fixed at the center of a cubic simulation box with a box size Lbox ) 26σ during the whole simulation run (Figure 1). Counterions from the charged particle were uniformly distributed over the box volume. Then, M1 positively charged polyelectrolytes together with their counterions were added into the simulation box. For our longest chains with N ) 32, there were 16 chains in the simulation box. During each deposition step for chains of different lengths, we have maintained the same polymer concentration c of newly added polyelectrolyte chains at c ) 0.03σ-3. After completion of the first simulation run (dipping cycle), unadsorbed polyelectrolyte chains were removed, corresponding to the (28) Bright, N. J.; Stevens, M. J.; Hoh, J.; Woolf, T. B. J. Chem. Phys. 2001, 115, 4909-4918.

3. Results and Discussion 3.1. Adsorption of Polyelectrolytes on the Spherical Particle. We now describe the sequential deposition process of fully charged polyelectrolytes with a degree of polymerization N ) 32. Simulations corresponding to nine deposition steps have been performed, and the snapshots in Figure 1 represent the first five deposition steps. Each layer of oppositely charged polyelectrolytes has been displayed in a different color for clarity. In the first snapshot (Figure 1a), the negatively charged spherical macroion is seen placed in the center of a cubical cell containing positively charged polyelectrolytes before the start of the simulation run. Counterions are not shown in the snapshots for clarity. Subsequent snapshots (Figure 1b-f) represent adsorption events as seen at the end of each simulation run consisting of 2.0 × 105 MD steps each. The simulation of the first layer results in an adsorption of four polyelectrolyte chains (128 monomers) onto the surface of the spherical particle made up of 80 negative beads. Thus, in addition to charge neutralization, an excess of 48 charges is accumulated around the particle that allows further adsorption of oppositely charged polyelectrolyte chains. The four chains wrap around the charged particle, covering it completely. Simulation of the next layer deposition, as described above in the simulation methodology, led to the second layer reversing the surface charge. Further dipping cycles are repeated to assemble layers of polyelectrolytes onto one another onto the spherical core of adsorbed chains. Unlike the first layer, subsequent layers tend to form what can be considered as “patches” or “islands” of layers. This difference arises as the size of the adsorbed core increases while the charge density is less than that of the spherical particle alone. As the deposition steps are repeated, the incoming chains have access to interior chains which they can displace or adsorb onto, depending on their respective charges, causing desorption or intermixing to occur. The snapshots in Figure 1e and f show intermixing among various types of the adsorbed chains. Thus, at the outset, the multilayer structure seems to be much more complicated than the simplistic, “stratified”, and frozen picture that one is tempted to imagine. Simulations of much longer

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Figure 2. Dependence of the number of adsorbed monomers Nads on the number of deposition steps N for a time duration of 2.0 × 105 MD steps per each deposition cycle.

durations (2.0 ×106 MD steps each) resulted in a similar overcharging and layer buildup. The evolution of the layer structure can be monitored by plotting the number of adsorbed monomers Nads during the assembly as a function of the number of deposition steps. This parameter shows strong dependence on the chain degree of polymerization N and fraction of charged monomers f.24 Figure 2 represents data for the charged chains with f ) 1 and 0.5. The adsorbed amount is consistently high for chains with N ) 32 (f ) 1 and 0.5) followed by N ) 16 (f ) 1) over others. For chains with degrees of polymerization N ) 8, the number of adsorbed chains is significantly less and seems to approach a steady state as the number of deposition steps increases. An initial repetitive pattern of adsorption and desorption with no gain in the number of adsorbed chains is found for these chains. For short chains with N ) 8 (f ) 0.5) and weakly charged chains with f ) 0.25 for all the chain lengths (N ) 32, 16, and 8), although the first layer is formed, it is unable to achieve the charge reversal essential for growing additional layers. This trend can be understood by considering the concept of charge overcompensation due to charge fractionalization.29 When a polyelectrolyte chain adsorbs electrostatically on a charged surface, the stoichiomentry is greater than 1:1, more often than not, with the formation of loops and tails. The charge on these loops and tails builds up the excess charge essential for the growth of the multilayers. This effect is more pronounced in longer chains, with higher degree of ionization than in the ones that are either short or partially charged. In the case of shorter chains, the loose ends that form tails or loops are less probable, and in chains that are not fully charged, the residual charge is less even if loops and tails are formed. It is worthwhile to discuss the effect of first layer interactions with the macroion on the layer adsorption. When purely repulsive nonbonded (Lennard-Jones) interactions between the central macroion and the polyelectrolyte chains were used in the simulations, multilayering was not observed, although the first layer forms with charge reversal due to attractive electrostatic interactions with the macroion. The first layer is subsequently peeled off during the second deposition cycle, as the oppositely charged chains prefer complexation with (29) Grosberg, A. Yu.; Nguyen, T. T.; Shklovskii, B. I. Rev. Mod. Phys. 2002, 74, 329-345.

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the first layer and ultimate desorption. This happens as the chains from the second deposition cycle encounter repulsive interactions with the macroion along with the attractive electrostatic interactions with the first layer. Therefore, the presence of short-range attractive interactions between the macroion and the polyelectrolytes is essential for the first layer to be intact and the multilayer assembly to progress. This aspect has also been observed in previously reported simulations on charged spherical21 and planar surfaces,22 which underline the need of sufficiently strong extra short-range interactions for the multilayering. The occurrence of chain exchange can be observed by monitoring the evolution of the total number of adsorbed monomers in the aggregate. For example, in the case of the deposition of fully charged (f ) 1) polyelectrolyte chains with a degree of polymerization N ) 16, after completion of six deposition steps, there are 496 monomers in the polymeric film covering the charged particle. After completion of the seventh deposition step, the total number of monomers grows up to 608, out of which only 464 monomers were adsorbed during the first six deposition steps. This difference of 32 monomers accounts for the desorption of two chains. This indicates an exchange that took place between the incoming polyelectrolytes and the ones of similar charge previously present in the aggregate. Nevertheless, the total number of adsorbed chains increased substantially, indicating a steady layer growth. This dynamic exchange that occurred during the assembly confirms the accessibility of interior layers to the newly coming chains. It is worthwhile to note that Schlenoff and co-workers reported slow displacement or exchange of adsorbed chains along with the kinetically irreversible nature of the deposited layers.9 Thus, desorption or chain exchange evidenced in this simulation attempts to capture what might transpire in an actual experiment and indicates the possibly dynamic nature of the multilayer assembly process. The faster than linear increase in the total number of adsorbed monomers on the charged particle observed in Figure 2 can be attributed to the increase in the available accessible area for the incoming chains as the mass of the adsorbed layer increases. Assuming spherical layer structure around the macroion, the radius of gyration of the adsorbed layers (Rg; Rg2 is defined as the mean square distance between the monomers in a given conformation and the center of mass of the aggregate) of polyelectrolyte chains increases with the total number of adsorbed monomers Nads. Thus, by plotting Nads/Rg2 as a function of the number of deposition steps (Figure 3), the polymer surface coverage shows linear dependence. For each system, the steady state regime is reached after the deposition of the first few layers. Simulations of much longer durations (2.0 × 106 MD steps each) result in overcharging and layer buildup, as seen in shorter simulation runs. In these simulations, the chain exchange occurs more frequently. However, variations in the chain degree of polymerization reveal different regimes of the adsorption process at different time scales (Figure 3). For the longest chains, N ) 32 with f ) 1, the total amount of adsorbed monomers after nine deposition steps differs only by two chains (64 monomers) for the simulation durations of 2.0 × 105 and 2.0 × 106 MD steps. In the case of N ) 16 (f ) 1), this difference in adsorbed monomers for the two different simulation durations increases to 13 chains containing 208 monomers. When a longer chain (N ) 32) adsorbs, there are more available adsorption sites than in the case of shorter chains. In this case, desorption of chains is less likely even in the longer

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Figure 4. Dependence of the overcharging fraction |∆Q|/fNads(() on the deposition step for different fractions of charged monomers, f ) 1 (filled symbols) and f ) 0.5 (open symbols), with degrees of polymerization N ) 32 (diamonds), 16 (squares), and 8 (triangles).

Figure 3. Dependence of the polymer surface coverage Nads/ Rg2 on chain degrees of polymerization N ) 32 (diamonds), 16 (squares), and 8 (triangles), for the fully charged chains (f ) 1), as the number of deposition steps (Nl) increased for a time duration of (a) 2.0 × 105 MD steps and (b) 2.0 × 106 MD steps per deposition step.

simulation run. Thus, the ease of desorption increases as the degree of polymerization decreases. As expected, when N ) 8, desorption is facilitated even at shorter simulation runs, as the adsorption sites per chain are far less than the longer chain counterpart. Hence, the adsorption scenario does not change much when the simulation proceeds to 2.0 × 106 MD steps. Interestingly, N ) 16 presents the intermediate case of a trapped state at 2.0 × 105 MD steps and a well equilibrated state when simulated for 2.0 × 106 MD steps, which accounts for the chain desorption. Although the number of adsorbed monomers varies with the fraction of charged monomers on the polymer backbone, the overcharging process is universal during the steady state growth. This can be confirmed by plotting the overcharging fraction |∆Q|/fNads(() as a function of the deposition steps (Figure 4). The overcharging fraction gives the ratio of the absolute value of layer overcharging |∆Q| to the net charge carried by the adsorbed chains fNads((). Nads(() represents the number of monomers adsorbed during a deposition step Nl and takes the corresponding desorbed chains into consideration. As the system reaches a steady state, the overcharging fraction has a value close to 0.5 and is independent of the chain’s degree of polymerization and fraction of charged monomers on the polymer backbone. However, the overcharging fraction has more fluctuations in the case of shorter chains with N ) 8 and f ) 1, which can be attributed to the fluctuations

in the number of adsorbed chains that are retained without being desorbed in the subsequent deposition steps. 3.2. Monomer Density Distribution. The local structure of the polyelectrolytes adsorbed onto the spherical particle can be analyzed using the radial monomer density distribution function F(r). These density distribution functions are shown in Figures 5-7, for different degrees of polymerization and duration of simulation runs. These distribution functions were averaged during deposition of the last layer independently for each set of chains adsorbed during different deposition steps. This separate data collection allowed us to analyze the evolution of the multilayer structure and the interpenetration of the layers during the deposition process. The plots clearly indicate a layered distribution of the polyelectrolyte chains around the spherical particle of radius 1.5σ. The first layer has more monomers present in it, and hence the sharp peak in the density profile. A decrease in polymer density of the subsequent layers is associated with the increase of the effective radius of the adsorbing aggregate. Figure 5a for the adsorption of chains with a degree of polymerization N ) 32 shows a multilayer arrangement with the peak for the seventh layer occupying a third region in the monomer density profile for the simulation run of 2.0 × 105 MD steps. Multilayering can also be seen in the case where the degree of polymerization is N ) 32 for a simulation run of 2.0 × 106 MD steps in Figure 5b and N ) 16 (Figure 6). The insets in these figures show the difference in radial density distribution of positively and negatively charged chains, ∆F(r) ) F+(r) - F-(r), representing the radial distribution of net charge. The multilayer nature of the structure can be clearly seen in the insets for N ) 32 and 16 in Figures 5 and 6. The asymmetric layer growth with an overall increase in the surface areas, discussed in the previous section, gives rise to a high probability of mistakes, and hence, the multilayering is limited to three layers only. Also, the appearance of chains from the last deposition steps (for example, the fifth to ninth layers) around the central particle can be attributed to this asymmetric layer growth and the “mistakes” propagating as the deposition proceeds. The layering is better stratified on charged planar surfaces and extends up to several layers.30 The adsorption on a spherical surface (as compared to a planar surface) is less favorable due to (30) Patel, P. A.; Jeon J.; Mather, P. T.; Dobrynin, A. V. Phys. Rev. E, submitted for publication.

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Figure 5. Density profiles of the fully charged (f ) 1) polyelectrolyte chains with N ) 32 after the completion of nine deposition steps with a duration of (a) 2.0 × 105 MD steps and (b) 2.0 × 106 MD steps each. The insets show the difference between the corresponding radial monomer densities of positively and negatively charged chains, ∆F(r) ) F+(r) - F-(r).

the significant chain conformational changes needed in the polymer to adsorb, resulting in a loss of entropy. Chains with like charges from different deposition steps exchange places, although this exchange is limited to immediate neighbors in the former case, while the penetration is almost complete and far reaching in longer simulation runs. Chains with N ) 16 show similar trends of intermixing. In the case of the shorter chains with N ) 8 (Figure 7), regardless of the duration of simulation runs, the intermixing is complete and the multilayer structure is not reached. Shorter chains have shorter relaxation times. Thus, even the shortest simulation run is sufficient to allow equilibration of the chains in a layer. Our simulation procedure corresponds to a Rouse dynamics of polymers for which the chain relaxation time increases with the chain degree of polymerization as N2. The chain relaxation time is 4 and 16 times shorter for chains with N ) 16 and 8, respectively, in comparison with that for chains having N ) 32 repeat units. Therefore, it is easier for shorter chains to penetrate into the interior layers even during shorter simulation runs. On the other hand, longer chains have a greater tendency to form multiple loops and tails that inhibit access to inner layers, at least temporarily. Therefore, multilayers are formed when the chains are sufficiently long and when the system is far from equilibrium. Thus, an intermediate partially trapped state is reached. Interestingly, this aspect of the simulation also gives us some insight about experimental time scales during the dipping cycles in an experimental situation. Ideally,

Panchagnula et al.

Figure 6. Same as Figure 5 for chains with a degree of polymerization N ) 16.

Figure 7. Same as Figure 5 for chains with a degree of polymerization N ) 8.

we should dip a solid substrate in a polyelectrolyte solution long enough so that the adsorption process reaches a

Polyelectrolyte Multilayering on a Charged Particle

Figure 8. Charge-charge correlation functions g((r) for the fully charged (f ) 1) polyelectrolyte chains with N ) 32, 16, and 8 after completion of nine deposition steps of duration of 2.0 × 105 steps per each deposition cycle.

steady state. In reality, some loosely adsorbed chains from a deposition step are propagated onto the subsequent layers even after the rinsing steps. Nevertheless, these mistakes carry on and govern the multilayer assembly process with an occasional loss of loose chains along its course. Thus, desorption or the process of chain exchange evidenced in this simulation closely resembles what happens in an actual experiment1a,b and indicates the dynamic nature of the assembly process. On the basis of the above results, multilayer structure can be understood as a partially trapped nonequilibrium state that given sufficient time will undergo complete intermixing. The idea of a frozen layer structure contradicts the results of our simulations. In our simulations, we found a strong intermixing of the layers even in the case of short deposition steps. The intermixing between polyelectrolyte chains deposited during different deposition steps continues during the whole simulation run. Thus, the assumption of the frozen layer structure17,18 should be reconsidered to obtain a more realistic theoretical model of electrostatic assembly. 3.3. Interchain Charge-Charge Correlation Functions. The reason for slow intermixing inside the adsorbed layer for long chains with a degree of polymerization N ) 32 and 16 is the formation of ionic pairs between oppositely charged monomers compensating each other. The local organization of such charged pairs in the adsorbed structure can be described by the charge-pair correlation function g((r)31 between positively and negatively charged monomers. Figure 8 shows the correlation functions g((r) between oppositely charged monomers for the duration of each deposition cycle equal to 2.0 × 105 MD steps. The correlation function g((r) is proportional to the probability of finding a pair of oppositely charged monomers at a distance of r. The correlation function sharply decreases at about 4σ in the case of N ) 8, indicating a very thin layer structure as compared to the slowly decaying g((r) for N ) 32 and 16 where the chargedpair correlations extend much further, indicating a more (31) Rapaport, D. C. The Art of Molecular Dynamics Simulation; Cambridge University Press: New York, 1995; Chapter 4.

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dense layering. g((r) has peaks around σ and 2σ, indicating the distance between the beads with predominant ionic correlations. The closest possible approach distance between the beads in any pair is σ. Such ionic pairs act as effective friction centers slowing the motion of the polymer chains. The polymer dynamics in this case is controlled by the association and dissociation of ionic pairs and can be described in the framework of the sticky Rouse model for short chains and by the sticky reptation model for chains above the entanglement threshold.32 Our simulations show that the formation of ionic pairs could lead to a 10 times increase of the chain relaxation time in the adsorbed layer in comparison with that of a chain in a solution. In the case of shorter chains, and also for chains with a smaller charge fraction of charged monomers, the number of ion pairs would be considerably low, allowing faster chain displacement than for the case of the fully charged chains with a higher degree of polymerization. 4. Conclusions In conclusion, multilayer film formation was found to be electrostatically driven with sequential charge reversal, giving rise to an assembly that is in a partially trapped state. Only strongly charged chains with degrees of polymerization N ) 32 and 16 having charge fractions of f ) 1 and 0.5 were able to regenerate the surface properties and to produce a steady buildup of multilayers. Chains that did not have a critical threshold of charge, namely, with degrees of polymerization N ) 32, 16, and 8 (f ) 0.25) and N ) 8 (f ) 0.5), failed to adsorb beyond the first layer. Among the chains that adsorb, a universal overcharging behavior was observed regardless of charge fraction and chain length. Our simulation results also emphasize the importance of the first layer in multilayer formation with a fine interplay of both electrostatic and nonbonded interactions. Further, the degree of intermixing arising from the exchange of chains during the course of the deposition cycle also varies with the chain length as well as the chain relaxation dynamics. Polyelectrolytes with a higher degree of polymerization have greater intermixing at longer simulation runs than in shorter runs. When the degree of polymerization is sufficiently low, the chains overcome the trapping barriers and equilibrate even during short simulation runs. Oppositely charged monomers form ion pairs that slow the chain motion. Thus, a successful theory of multilayer assembly should explicitly take into account the dynamic nature of chain adsorption and formation of ionic pairs. Acknowledgment. Funding for this research from the Petroleum Research Fund under grant PRF-39637AC7 (A.V.D.), from the National Science Foundation (DMR-0305203, A.V.D., and CTS-0335345, J.F.R.), and from NIEHS of the National Institutes of Health (PHS Grant No. ES03154, J.F.R.) is gratefully acknowledged. Supporting Information Available: A movie of the polyelectrolyte multilayer assembly containing snapshots of the simulation for five deposition steps. This material is available free of charge via the Internet at http://pubs.acs.org. LA047741O (32) Rubinstein, M.; Dobrynin, A. V. Curr. Opin. Colloid Interface Sci. 1999, 4, 83-87.