Counterion Valence-Induced Tunnel Formation in a System of

In this work, the Brownian dynamics simulation with the open source software LAMMPS(46) is performed to study the system coupled with a Langevin therm...
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J. Phys. Chem. B 2009, 113, 11625–11631

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Counterion Valence-Induced Tunnel Formation in a System of Polyelectrolyte Brushes Grafted on Two Apposing Walls Juan Yang and Dapeng Cao* DiVision of Molecular and Materials Simulation, Key Lab for Nanomaterials, Ministry of Education, Beijing UniVersity of Chemical Technology, Beijing, 100029, P.R. China ReceiVed: May 11, 2009; ReVised Manuscript ReceiVed: July 10, 2009

A Brownian dynamics (BD) simulation is performed to investigate the effect of counterion valence on the properties of polyelectrolyte (PE) brushes grafted on two apposing walls. By increasing the counterion valence from monovalence to divalence and further to trivalence, the PE brushes on two separate walls begin to shrink to produce a tunnel in the center of the confinement, which provides a path for the nanoparticle to pass the PE brushes in this system. That is to say, the added counterions can act as a switch-controller of the opening-closing behavior of the PE chains grafted on the two walls. By exploring the dynamic properties of nanoparticles, it is found that the mobility of nanoparticles increases with the addition of the higher valent counterions; i.e., the nanoparticle can diffuse along the tunnel induced by high valent counterions. In addition, we also investigate dependences of the thickness and distribution of the PE brushes on the grafting density, concentration, and valence of counterions. It is expected that this work can provide a good insight into the design of formation of a tunnel for nanoparticle access in the PE brushes controlled by the counterion valences. 1. Introduction Polyelectrolyte brushes (PEBs) consist of charged polymers grafted to the surfaces of various geometries1,2 such as planar, cylindrical, or spherical walls, by chemical or physical means.2,3 In the last decades, PEBs have received a lot of attention because of their potential technological applications in colloidal stabilization,4 surface modification protection,5 and lubrication.6,7 Researchers from the fields of biology, materials science, and soft matter8,9 have made great efforts to study the PEB grafted on a single wall by means of theory,10-12 simulation,13-15 and experiments.16-18 A recent review19 explored most of the progress on the topic of a grafted polyelectrolyte (PE). The properties and structure of the PEBs depend on a wide range of parameters, such as the grafting density and charge of the PE chain, the counterion valence, the extra ion concentration, and solvent conditions. The PEBs in different surrounding parameters have been identified by different brush regimes. Given the length and grafting density of a PE chain, a transition from the nonlinear osmotic brush to the salted brush will appear with the increase of the salt concentration. In this osmotic brush regime, the counterions are mainly trapped inside the brush. Theoretical studies predicted that the brush thickness does not depend on the PE grafting density in this regime. In the salted brush regime, the brush thickness scales with the grafting density by a power of 1/3. However, Romet-Lemonne et al.20 and Seidel et al.21,22 reported that in the osmotic brush regime the brush thickness slightly increases with the grafting density. In addition, Jiang et al.23 predicted the same dependence of brush thickness on the grafting density in the osmotic regime by using the nonlocal density functional theory (DFT). The swelling behavior of the PEs in a salt solution has been investigated through a number of molecular simulations, theories, and experiments by exploring the influences of cation valence and concentration. For example, a recent experimental * Corresponding author. E-mail: [email protected] or cao_dp@ hotmail.com. Fax: 86-10-64427616.

investigation24 reported that the replacement of Na+ with La3+ at the fixed ionic strength results in a collapse of the polystyrene sulfonic acid brush tethered on spherical particles. Similarly, the variation of cation valence can manipulate the conformation of a star PE,25 which has been demonstrated theoretically in our previous work.26 In a solution with divalent or trivalent cations, the PE size is drastically reduced, compared to that in a monovalent one. The conformation can be restored when the multivalent cations are removed. Jiang and Wu27 used the nonlocal DFT to find that the charge correlation in the presence of multivalent cations results in a collapse of a PEB at an intermediate PE grafting density. Zhulina etc.28 adopted a scaling theory to demonstrate that at low ionic strengths of the solution the PEB swells with the increase of the salt concentration. When multivalent cations are added, the scaling exponents describing the dependences of the brush thickness on salt concentration become a function of the cation valence. Moreover, the generic experiments29-32 and simulations33 have shown that the added cations with higher valence are more effective in compacting a long DNA molecule (or a polyanion in general), implying a dominant electrostatic mechanism for a PE collapse. In addition, the cation valence may also affect the distributions of PE segments and ions within the brush.34 By considering the effect of the salt concentration, Kumar and Seidel35 presented that in the nonlinear osmotic brush regime the brush thickness decreases with the salt concentration, and the exponent is -1/3, which is in agreement with the scaling prediction by Pincus.10 In our previous work,36 we explored a macroion-PE system and observed the collapse and re-expansion of a PE chain with the increase of the salt concentration in the PEs-macroions systems in trivalent solutions. Furthermore, the mobility of macroions and PEs increases monotonically with increasing salt concentration in monovalent and trivalent solutions, and the mobility in trivalent solutions is always larger than that in corresponding monovalent salt solutions. Most of the investigations on the PEBs were subjected to the PE grafted on a single wall. Unfortunately, a few surface

10.1021/jp904367b CCC: $40.75  2009 American Chemical Society Published on Web 08/05/2009

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Yang and Cao

Figure 1. Schematic illustration of a primitive model.

force measurements7,37,38 and only several simulations39-41 were for the case of two parallel walls. A surface force experiment can only measure the pressure of a brush on the surfaces but cannot determine the density profiles of the brush. Sjostrom and co-workers39 employed a Monte Carlo simulation to study the conformation of the PE-bound surface and found that the interaction between the PEs on two walls increases, compared to a usual double layer force. The increased repulsion is due to an expansion of the chains perpendicular to the surfaces, which in turn is driven by the counterion entropy. Hehmeyer and Stevens40 used molecular dynamics (MD) simulations to study a bead-spring model of the PEs attached to two apposing walls in a salt-free solution and explored the effects of the surface density of the grafted polymer, the chain length, and the gap width on the structure and pressure of the PEB system. They found that the density profiles exhibit a maximum, which is absent in the case for a single wall. Moreover, a maximum of density profiles appears as the separation distance decreases, due to the fact that the two apposed brushes try to avoid an overlap by shrinking. This result might contribute to the lubrication properties.7 Kumar and Seidel41 reported a MD simulation on completely charged PE brushes grafted on two parallel surfaces and discussed the different physical natures of the various regimes. They also observed that the brushes shrink as they approach each other to avoid interpenetration due to an electrostatic interaction. The excluded-volume effect becomes important at high compressions. Huang and co-workers42 used dissipative particle dynamics (DPD) to study the solvent flow through a slit channel grafted with stimuli-responsive polymer brushes and found that the coated channel could regulate solvent permeability when the grafted polymer brush underwent a conformational change. In this work, we use Brownian dynamics simulations to investigate the effects of counterion valence on the extension and collapse of the PE chains grafted on two separate walls. The emphasis is placed on the control of the counterion valence to the formation of the tunnel between two grafted PE layers, which might be used in drug delivery and other related applications. That is to say, the added counterions might act as a switch-controller of the opening-closing behavior of the PE chains grafted on the two walls. This work will also explore the effects of a wide range of surrounding parameters, such as the grafting density and the counterion concentrations, on the properties of PE chains and the nanoparticles in the system. 2. Models and Simulation Details We used a coarse-grained model (CGM) to investigate the diffusion of uncharged nanoparticles in a solution confined between two parallel surfaces grafted by charged PEs. The

schematic illustration of the model is shown in Figure 1. Following our previous work26,43 and the one from Kremer and co-workers,40,44,45 the PEB is modeled as Nbrush bead-spring chains of length of Mlength, in which one end of these chains is anchored to one of the two uncharged planar surfaces and the center of the end segment is located at z ) -14.5σ and z ) 14.5σ, respectively. Throughout our calculations, the length of the PE chains is set as Mlength ) 28. The charge of each segment in the PEB is qbrush ) -1. All the PEBs were randomly generated on the two slabs, which were modeled by a periodic sphere packed lattice of beads with the diameter σ located at Z ) (15.5σ. Thus, the separation of the confinement is D ) 30σ. This selection for the separation distance plus the chain length of Mlength ) 28 is to guarantee that the PE chains on two planes do not interpenetrate into each other.41 Moreover, the PEBs can well display the collapse and reswelling behavior at this distance. Within the simulation box of L · L · LZ, the grafting density of the PE on each planar surface is given by Fg )(0.5Nbrush)/(L · L). The charges of the added counterions (i.e., cation) qCation are set as mono-, di-, and trivalent according to different conditions. The number of the added counterions NCation of different valences is governed by the charge ratio β ) |MlengthNbrushqbrush/NCationqCation|, which is related to the total charge of the PEB and that of the added counterions. Extra ions always ensure the charge neutrality in the system. All the ions in the solution are described as charged Lennard-Jones (LJ) spheres. In the calculations, we employ NCation and NE to denote the numbers of counterions and extra ions, while qCation and qE denote the charges of counterions and extra ions, respectively. We put Nnano nanoparticles into the solution and study the effect of the added counterion on the diffusion of nanoparticles. In our model, the diameters of all the PE segment, planar surface, added couterions, and extra ions are σ at the LJ length scale, except that the diameter of the nanoparticle is 2σ. In this work, the Brownian dynamics simulation with the open source software LAMMPS46 is performed to study the system coupled with a Langevin thermostat. In the simulations, the reduced temperature is T* ) kBT/ε ) 1.2 and the Bjerrum length is lB ) 3σ ) 0.714 nm, corresponding to the value of water at room temperature. All the particles were enclosed in a cubic box with the size of LBox ) 30σ, and the simulation box is periodic only in x and y directions, while in the direction perpendicular to the grafting surface, the system is restricted between two slabs. During the simulation, a canonical ensemble (i.e., NVT ensemble) was adopted. The solvent (i.e., water) effects are taken implicitly into account through its dielectric permittivity and Brownian motion. The Langevin equation for the motion of particle i holds

System of PE Brushes Grafted on Two Apposing Walls

mi

d2ri 2

dt

) -∇iU - miγ

dri + Wi(t) dt

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(1)

where mi is the mass of particle i, and it is proportional to its volume, which means the mass of nanoparticle Mnano is 8 times in units of LJ mass scale, m, i.e., the mass of other particles in the system. U denotes the total potential energy of the system. γ and Wi(t) are the friction coefficient and stochastic force of the solvent, respectively, which are linked through the dissipation-fluctuation theorem 〈Wi(t) · Wi(t′)〉 ) 6mγkBTδijδ(t - t′), with kBT being the thermal energy. Within the model, the Coulomb potential for the pair ij is treated by the particle-particle particle-mesh (PPPM) method,47 where i and j denote either a PE segment, a counterion, or an extra ion, and is given by

U(rij) ) kBTlB

qiqj rij

(2)

where qi is the charge of particle i; rij is the center-to-center distance between charged species; lB ) e2/4πε0εrkBT denotes the Bjerrum length, where ε0 is the permittivity of vacuum and εr is the relative permittivity of the solvent; and e is the elementary charge. In addition, it is assumed that the chains are in a good solvent modeled by a purely repulsive short-range interaction that is described by a shifted LJ potential, ULJ

ULJ(r) )

{

[( r -σ ∆ ) - ( r -σ ∆ )

4εLJ 0

12

6

+

1 4

]

for r < 21/6σ + ∆ for r g 21/6σ + ∆ (3)

where ∆ ) 0 is for the particles of the same size, i.e., PE-PE, PE-counterion, and PE-extra ion interactions, while ∆ ) 0.5σ is for the particles with different diameters, that is, the nanoparticle-PE and nanoparticle-ion interactions. The connectivity of beads in a PE chain is maintained by the finite extensible nonlinear elastic (FENE) potential developed by Kremer and co-workers.40,44,45

[ ( )]

r 1 UFENE(r) ) - KFENER02 ln 1 2 R0

2

(4)

where the spring constant KFENE is equal to 30kBT/σ2; kB is the Boltzmann constant; T is the absolute temperature; and the maximum bond length R0 between the beads is set to 2σ. The simulations started from randomly generated initial configurations. Each simulation contains 3 × 106 steps for equilibration of the system with the time step of ∆t ) 0.0005τ and the following 1 × 106 steps for production. 3. Results and Discussion 3.1. Effect of Grafting Density on the Brush Thickness. First of all, we study the dependence of the brush thickness on the grafted density of PE chains on the parallel walls. Monovalent extra ions are added for electrical neutrality of the system. In this section, the number of PEs (Nbrush) on each wall varies from 30 to 120, which means that the corresponding grafting

Figure 2. Thickness of the PE brush as a function of the grafting density Fg.

density Fgσ2 increases from 0.017 to 0.067 approximately. The brush thickness (H) of the PEB calculated from the density profile of the PE chain is given by

H)

2

∫0box/2 zFb(z)dz ∫0box/2 Fb(z)dz

(5)

where H is the height of the brush; z is the perpendicular distance of a PE segment from the surface; and Fb(z) is the density profile of the PE chains. Figure 2 shows the dependence of the brush thickness on the grafting density. The brush thickness exhibits a steady upward trend when the grafting density increases, which is due to the excluded-volume effect of the PE segments and the shortrange repulsion between the neighboring segments. In addition, it is also found that the brush thickness (H/σ) increases from 14.27 at Fg ) 0.017 to 15.66 at Fg ) 0.067. The varied range is only about 1.39 (the extent of extension is about 9.7%), which indicates that the PEB swells slightly with increasing grafting density; i.e., the thickness of the PEB depends weakly on the grafted density. This observation agrees basically with the previous theoretical and experimental work.48,49 However, there are some deviations from the nonlinear osmotic brush behavior at Fg < 0.04. In the work of Csajka and Seidel,17 they demonstrated that the small lateral system size may influence the large scale properties such as brush thickness due to possible self-interaction of the chains. When the grafting density Fg is less than 0.04, the finite-size effect might take effect and ultimately lead to the deviation of our results from the nonlinear osmotic brush behavior at small Fg. 3.2. Effects of Concentration and Valence of Counterion on the Brush Thickness. Adding the counterions into a bulk solution will lead to the redistribution of PE chain segments and consequently causes the change of PEB extension behavior. In this section, the emphasis is placed on the influences of concentration and valence of the counterions on the brush thickness by changing the charge ratio β between the counterions and the PEB from 0.3 to 2.0, where the number of grafted PE chains is Nbrush ) 36 on each plane. Figure 3 shows the relationship of PEB thickness and β. Clearly, at β e 1, the thickness of the PEB in monovalent counterion solution remains constant because the variation of β does not change the composition of the system. When more monovalent counterions are added (here it reaches β ) 1.5 and β ) 2.0), additional

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Figure 3. Thickness of PE brushes at different counterion valences as a function of β.

anions must be added to guarantee the electric neutrality of the system. The addition of more counterions makes some extra ions into the bulk solution except for those confined in the brush layers and subsequently reduces the osmotic pressure between the brush layer and the bulk solution, which thereby diminishes the driving forces to stretch the PE chains. Consequently, as shown in Figure 3, the increase of the monovalent counterions leads to the shrinking of the PEB. However, for divalent and trivalent counterions, more multivalent counterions were added into the solution, and the brush thickness decreases and reaches a minimum at the stoichiometric point β ) 1 and then increases beyond the stoichiometric point. Furthermore, the brush thickness in the divalent counterion solution is always larger in the trivalent one, which suggests that the trivalent solution is more efficient for the shrinking of a PEB. Meanwhile, the brush thickness of the PEB in the divalent counterion solution is always smaller than that in the monovalent one. This observation is caused by the following two main factors. One is the positively charged counterion-mediated electrostatic interaction among the PE segments, and the other is the excluded-volume effect of all particles in the solution. For a given β, the number of particles in the system becomes smaller with the increase of the counterion valence from +1 to +2 and further to +3. That is to say, the excluded-volume effect can partly explain the decrease of the brush thickness. Another factor to explain the shrinking of the PEB is the counterion-mediated electrostatic interaction. When the multivalent counterions are added, the electrostatic attraction between the multivalent counterion and the PE segment became stronger than that with the monovalent counterion,36 which consequently leads to the shrinking of PEBs. To better clarify the mechanism, we will further discuss it in Figure 4. As mentioned above, the PEB thickness in the divalent and trivalent solutions decreases at β < 1, which may be attributed to the competition of the excluded-volume effect and the electrostatic interaction. At β < 1, when more divalent (or trivalent) counterions are added, the divalent (or trivalent) ions will replace the monovalent counterions within the brush layer for its stronger attraction to the PE chains, leading to the shrinking of a PEB. However, at β g 1, the PE in a monovalent solution collapses due to the competition between the excludedvolume effect and the electrostatic interaction between the added counterions and the anions for charge compensation. In this regime, the electrostatic interaction between a part of counterions and the anions dominates the competition. This case is different from these in divalent and trivalent solutions. At β g 1, with more multivalent counterions added, the chain will reswell because the multivalent counterions are attracted tightly on the PE chains, which would cause the charge inversion.36 To

Yang and Cao

Figure 4. RDFs between brush segments and the added cations for different counterion valences as a function of the distance at β ) 1.0.

Figure 5. Mean-square displacement (MSD) of counterions of different valences.

maintain the charge neutrality in a solution, the extra added ions carrying negative charge would be absorbed onto the chains. Therefore, the brush reswells under the effect of the counterionmediated electrostatic interaction. This observation agrees well with the results of experiments.24 From the variation of the brush thickness and the results from the previous work,10,11 it is found that the brushes underwent a transition from an osmotic brush to a salted brush with increasing valence. To better illustrate the role of electrostatic interaction in the acting mechanism, we studied a simple case of β ) 1 with Nbrush ) 36 on each wall and Mlength ) 28, in which the counterions are able to fully compensate the charges of the PEB and no extra anions are required. By calculating the radial distribution functions (RDFs) between counterions and PE segments, we can explore the effect of counterion valence on the attraction between the counterions and PE segments. Figure 4 shows the RDFs between PEB segments and counterions. It can be found that the main peaks of all three curves are located at r ) 1.05σ. However, the height of the peak falls from 18.8 in the trivalent solution to 11.9 in the divalent solution and further to 5.3 in the monovalent solution. This means that the binding of PEB to oppositely charged cations depends strongly on the valence of the added counterions. With increasing counterion valence, the electrostatic attraction between the cations and the PE chain segments becomes stronger. That is to say, the binding between trivalent counterions and PE chain segments is more intense than that between monovalent counterions and chain segments. The decreasing trend of the peaks at small distances with the increase of counterion valences also verifies the fact that the multivalent counterions are absorbed and condensed on the PE chains. Thus, we believe that besides the excluded-volume effect, the electrostatic interaction and the consequent couterion

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Figure 6. Snapshots of simulated systems (β ) 1), where blue spheres denote the anchored walls; green spheres denote PE brushes; red spheres denote the added counterions; and yellow spheres denote the nanoparticles.

condensation also play an important role in the decrease of the PEB thickness when the counterion valence increases. To better clarify this phenomena, we also plot the mean-square displacement (MSD) of monovalent, divalent, and trivalent counterions in Figure 5. It is found that the slope of MSD vs time sharply decreases with the increase of counterion valence. This is due to the stronger electrostatic interaction of multivalent counterions and PE segments, compared with that of monovalent counterions. The strong multivalent counterion-PEB correlation leads to the condensation of multivalent cations on the PEBs, which, in turn, causes a reduced mobility of added cations, as expressed by their MSD (see Figure 5). The phenomena of counterion condensation caused by the strong electrostatic interaction lead to the decrease of brush thickness. Furthermore, besides the factor of the smaller number of multivalent counterions, the condensation of multivalent counterions might also lead to the fact that the osmotic pressure of multivalent counterions within the brush is smaller than that of monovalent counterions, which might be another cause of the shrinking of the PEBs. 3.3. Effect of Counterion Valence on the Mobility of Nanoparticles. To give a visual representation of the structural transition of the PEB, we presented in Figure 6 the typical configurations of these systems under different counterion valences and β ) 1. It is clear that with the increase of counterion valence from mono- to di- and further to trivalence the PEBs on two apposed walls start to collapse onto their anchored walls. It is also found that the PEB in Figure 6(c) is more compacted than that in Figure 6(b). Furthermore, nanoparticles (yellow balls) are emerged in the PEBs in the monovalent solution while they are squeezed into the tunnel formed between the two brush layers in the di- and trivalent solutions, which will affect the diffusion of nanoparticles significantly. Therefore, we also investigate the dynamical behavior of the nanoparticles in these systems. The diffusion coefficient, DS, is given by

DS )

〈∆r(t)2〉 1 lim 4 tf∞ t

(6)

where 〈∆r(t)2〉 is the mean-square displacement (MSD) of nanoparticles. Figure 7 shows the diffusion coefficients of nanoparticles as a function of β in mono-, di-, and trivalent counterion solutions, respectively. Obviously, the diffusion coefficient of nanoparticles in the trivalent counterion solution is always greater than that in the divalent solution, and the diffusion coefficient in the divalent solution is always greater than that in the monovalent

Figure 7. Diffusion coefficients of nanoparticles as a function of the charge ratio β.

solution at the same β, which means that the mobility of the nanoparticles is enhanced by the increase of the counterion valence. Compared to monovalent counterions, multivalent counterions have the stronger electrostatic attraction with PE chains, which causes the collapse of PEBs and consequently leads to the larger tunnel for the faster diffusion of nanoparticles. In addition, we also observe that at β < 1 the diffusion coefficients of nanoparticles in the di- and trivalent solutions increase, and they reach the maximum at β ) 1 and finally decrease at β > 1. The diffusion behavior of nanoparticles is just the reverse of the PEB thickness because the larger the PEB thickness, the smaller the tunnel. 3.4. Effects of Valence and Concentration of Counterions on the Distributions of PEB and Counterions. In this section, we explore the microstructure of the PEB by the distributions of PE segments and added counterions in mono-, di-, and trivalent solutions. The density profile is always symmetric about the midplane of the confinement, so we only consider the region |z/D| e 1/2 to simplify the discussion, where z denotes the coordinates of the PEB segments and D is the separation of the two surfaces. Moreover, these results can be compared to single wall studies. Figures 8(a) and (b) present the density profiles of PE chain segments and counterions at different β in the monovalent counterion solution. It can be found that the first main peak near the surface (at z ) 14.5σ) arises for the PE segments directly connected to the surface. The other peaks correspond to the densities of consecutive segments. Similar high peaks can be identified in all other tethered chains and have been confirmed by recent molecular simulations35 and DFT studies.23 Here the counterion concentration is relatively low, and thus there exists a long “tail” in the density profiles of PE segments, which is quite different from that for neutral brushes11,50 and the case for PEB in high salt concentration

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Figure 8. Density profiles of (a) PE segments and (b) monovalent counterions as a function of the distance from the midplane between the apposed walls.

systems23 where the density distribution resembles a parabolic profile. For different β, the density profiles of PE segments exhibit a similar trend. The difference is that at β ) 1.5 the density profile of PE segments shows a slightly high third peak, compared to the case of β e 1. This is due to the extra addition of anions at β ) 1.5. With the increase of β, we did not observe an obvious brush collapse as shown in Figure 8(a). Apparently, the density of counterions grows with increasing β. In addition, the magnitude of the maxima in the density profiles of counterions exhibits an upward trend from 0.03 to 0.16 with β increasing from 0.3 to 1.5, which indicates that there are more counterions located within the brush with increasing counterion concentration. Moreover, it is found from Figure 8(a) that there exists a minimum in PE segment density around z ) 14.1σ, which just locates the maximum position of counterion density in Figure 8(b). Qualitatively, the structure of the PEB is significantly changed by increasing the counterion valence. Figures 9(a) and (b) show the density profiles of PE segments and counterions at different β in the divalent counterion solution. Obviously, with the increase of β from 0.3 to 1.0, the distribution range of PE segments decreases from 0-14.5 to 4-14.5; i.e., the PEB thickness decreases significantly, which leads to the formation of a tunnel between two brushes. Correspondingly, the density profiles of counterions exhibit a similar behavior with these of the PE segment; i.e., at β ) 1 all the divalent counterions distribute inside the PEB. Due to the excluded-volume effect, the main peak of the divalent counterions does not appear in the contact position with the surface. At β ) 1.5, the distribution ranges of PE segments and divalent counterions become more extensive than those at β ) 1. That is to say, the PEB exhibits the reswelling behavior at β g 1, which is consistent with the results shown in Figure 3.

Yang and Cao

Figure 9. Density profiles of (a) PE segments and (b) divalent counterions as a function of the distance from the midplane between the apposed walls.

Figure 10. Density profiles of (a) PE segments and (b) trivalent counterions as a function of the distance from the midplane between the apposed walls.

Figures 10(a) and (b) show the density profiles of PE chain segments and counterions at different β in the trivalent counterion solution. Actually, the change trend in density profiles is the same as that in the divalent counterion solution. The difference is that the shrinking of the PEB in the trivalent

System of PE Brushes Grafted on Two Apposing Walls counterion solution is stronger, leading to the formation of a larger tunnel between two brushes. This observation is in agreement with the snapshot in Figure 6. 4. Conclusions On the basis of a coarse-grained model, we investigated the formation of a counterion valence-induced tunnel in the system of PEBs grafted on two parallel walls. The emphasis was placed on the effect of counterion valence on the microstructures and physical properties of PEBs. Our results show that the PE brushes are more extensive in monovalent solutions. With the increase of the counterion valence, the PE brushes shrink due to the stronger electrostatic interactions, which makes the PE brushes more compacted. The enhancement of the electrostatic attraction in the multivalent solution was manifested by the RDFs of PE-counterions, where the RDF curves display that the binding between the PE segment and the counterion is strengthened by raising the counterion valence. Meanwhile, the variation of the PEB thickness indicates that the PE brush underwent a transition from an osmotic brush to a salted brush. Furthermore, we also calculated the diffusion of nanoparticles confined between two PEBs, which is related to the effect of counterion valence on the shrinking of the PEB. The results indicate that when the counterion valence increases from monoto trivalent a large tunnel is formed in the center of the apposed walls, which provides an efficient path for the fast diffusion of nanoparticles. As a result, the diffusion coefficient of the naoparticles in the multivalent solution got a significant increase. In addition, we also explored the distributions of PE segments and the added counterions. It is found that the multivalent counterions make the PEB shrink significantly at β ) 1. Moreover, almost all the multivalent counterions distribute inside the PEBs at β ) 1. At β > 1, the PEBs reswell, which leads to the decrease of the formed tunnel. In short, to design a tunnel for nanoparticle access in the PEB system, it is a good choice to control the counterion valence and β. It is expected that this work would provide a good insight into the design of a pass tunnel for nanoparticles. Acknowledgment. This work is supported by the National Natural Science Foundation of China (No. 20776005, 20736002, 20874005), Beijing Novel Program (2006B17), NCET Program (NCET-06-0095), ROCS Foundation (LX2007-02), and Novel Team (IRT0807) from MOE of China, Chemical Grid Program and Excellent Talent of BUCT. J.Y. is thankful to Yiyu Hu for his helpful discussion. References and Notes (1) Halperin, A.; Tirrell, M.; Lodge, T. P. AdV. Polym. Sci. 1991, 100, 31. (2) Guenoun, P.; Argillier, J. F.; Tirrell, M. C. R. Acad. Sci. Ser. IV: Phys. Astrophys. 2000, 1, 1163. (3) Ruhe, J.; Ballauff, M.; Biesalski, M.; Dziezok, P.; Grohn, F.; Jonannsmann, D.; Houbenov, N.; Hugenberg, N.; Konradi, R.; Minko, S.; Motornov, M.; Netz, R. R.; Schmidt, M.; Seidel, C.; Stamm, M.; Stephan, T.; Usov, D.; Zhang, H. N. AdV. Polym. Sci. 2004, 165, 79. (4) Napper, D. H. Polymeric Stabilization of Colloidal Dispersions; Academic Press: New York, 1983.

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