Effect of the Bridging Conformation of Polyelectrolytes on the Static

Aug 9, 2008 - A Brownian dynamics (BD) simulation is performed to investigate the effect of the bridging conformation of a polyelectrolyte (PE) with t...
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Langmuir 2008, 24, 10138-10144

Effect of the Bridging Conformation of Polyelectrolytes on the Static and Dynamic Behavior of Macroions Yiyu Hu, Ran Ni, Dapeng Cao,* and Wenchuan Wang DiVision of Molecular and Materials Simulation, Key Laboratory for Nanomaterials, Ministry of Education, Beijing UniVersity of Chemical Technology, Beijing 100029, P.R. China ReceiVed June 21, 2008 A Brownian dynamics (BD) simulation is performed to investigate the effect of the bridging conformation of a polyelectrolyte (PE) with two charged heads (two-heads PE) on the radial distribution function (RDF) and diffusion behavior of macroions on the basis of the coarse grained model. For comparison, the system containing macroions and the PE with only one charged head (one-head PE) is also investigated. The simulation results indicate that, at low concentrations, the bridging effect of the two-heads PE chain leads to correlation of macroions. The reason is that at low concentration the gyration radius of the PE chain is less than the average distance between two macroions. When the two-heads PE chains are adsorbed on different macroions, the bridging effect of the PE chain dominates the RDF and diffusion behavior of the macroions. With the increase of the concentration of the system, when the gyration radius of the PE chain is greater than the average distance between two macroions, the bridging effect of the PE chain becomes trivial. By investigating the mechanism of the two-heads PE chain affecting the static and dynamic properties of the macroions, we can provide useful information for the synthesis of stabilizers and destabilizers of colloidal particles.

1. Introduction Macroion-polyelectrolyte (PE) complexes have recently motivated a great amount of computational, analytical, and experimental work. The reason for this is not only the interesting behavior seen in such systems but also the great importance for many industrial processes, such as water treatment as flocculant/ water insoluble mixtures, powder processing as dispersion agents, or food technology as rheology modifiers. Many biomacromolecules, for instance, DNA, are also PEs, which can form complexes with proteins or membranes. These complexes are expected to play key roles in biological regulation processes with applications in therapeutic delivery systems. For example, the behavior of a DNA-histone complex is thought to be one of the crucial factors in the packing of DNA in living cells. Polyelectrolytes are charged molecules that display a high solubility in water and strong adsorbing capacity on surfaces bearing an opposite charge. An interesting feature is that they can act as not only a stabilizing agent but also a destabilizing agent in particle suspensions. A central aspect in the analytic studies by Gurovitch and Sens,1 Park et al.,2 and Mateescu et al.3 was the possibility to overcharge a spherical macroion by oppositely charged PEs. Considering the attraction between the two oppositely charged species, Goeler and Muthukumar4 determined the conditions for the adsorption of PEs on spherical and cylindrical surfaces by means of variational calculation. Some research groups5-9 studied the effect of added salt and bare stiffness of the polymer on the shapes of the macroion-PE * To whom correspondence should be addressed. Email: caodp@ mail.buct.edu.cn; [email protected]. (1) Gurovitch, E.; Sens, P. Phys. ReV. Lett. 1999, 82, 339. (2) Park, S. Y.; Bruinsma, R. F.; Gelbart, W. M. Europhys. Lett. 1999, 46, 454. (3) Mateescu, E. M.; Jeppesen, C.; Pincus, P. Europhys. Lett. 1999, 46, 493. (4) Goeler, F. v.; Muthukumar, M. J. Chem. Phys. 1994, 100, 7796. (5) Nguyen, T. T.; Shklovskii, B. I. J. Chem. Phys. 2001, 115, 7298. (6) Netz, R. R.; Joanny, J. F. Macromolecules 1999, 32, 9026. (7) Schiessel, H.; Bruinsma, R. F.; Gelbart, W. M. J. Chem. Phys. 2001, 115, 7245. (8) Kunze, K. K.; Netz, R. R. Phys. ReV. E 2002, 66. (9) Nguyen, T. T.; Shklovskii, B. I. Physica A 2001, 293, 324.

complexation, and even emphasized on the experimentally relevant DNA-histone systems.8,10 Using appropriate parameters for DNA-histone complexes, Kunze and Netz10 found complete wrapping for intermediate salt concentrations only, in agreement with experiments. On the basis of a cell model, Linse and coworkers used a molecular simulation to explore the effects of the flexibility,11,12 linear charge density,13,14 surfactant tail length,15 and concentration16-18 of the PEs on the microstructure to the macroion-PE complexation. Sokolowski et al.19-23 used the density functional theory (DFT) to describe the adsorption and capillary condensation of nonuniform ionic fluids on fluid walls and slitlike pores, in which the ionic fluids were composed of chain molecules with selected segment charges. They found that there are three factors, for example, packing effects, electrostatic forces, and chain connectivity, influencing the structure of a fluid near the wall, and the role of electrostatics forces becomes less significant at higher temperatures. As we know, in the bulk systems, PEs may be adsorbed on the surfaces of different macroions, and the PEs would act as a bridge between macroions. This was considered as the bridging effect of a PE chain. By exploring the forces between two charged particles mediated by PEs, Jonsson and co-workers24,25 found that there existed two types of interactions between two charged macroions: the entopic bridging and energetic bridging interac(10) Kunze, K. K.; Netz, R. R. Phys. ReV. Lett. 2000, 85, 4389. (11) Wallin, T.; Linse, P. Langmuir 1996, 12, 305. (12) Jonsson, M.; Linse, P. J. Chem. Phys. 2001, 115, 10975. (13) Wallin, T.; Linse, P. J. Phys. Chem. 1996, 100, 17873. (14) Akinchina, A.; Linse, P. Macromolecules 2002, 35, 5183. (15) Wallin, T.; Linse, P. J. Phys. Chem. B 1997, 101, 5506. (16) Wallin, T.; Linse, P. J. Chem. Phys. 1998, 109, 5089. (17) Jonsson, M.; Linse, P. J. Chem. Phys. 2001, 115, 3406. (18) Skepo, M.; Linse, P. Macromolecules 2003, 36, 508. (19) Bryk, P.; Pizio, O.; Sokolowski, S. J. Chem. Phys. 2005, 122, 174906. (20) Pizio, O.; Bucior, K.; Patrykiejew, A.; Sokolowski, S. J. Chem. Phys. 2005, 123, 214902. (21) Bryk, P.; Sokolowski, S.; Pizio, O. J. Chem. Phys. 2006, 125, 024909. (22) Tscheliessnig, R.; Billes, W.; Fischer, J.; Sokolowski, S.; Pizio, O. J. Chem. Phys. 2006, 124, 164703. (23) Bryk, P.; Pizio, O.; Sokolowski, S. J. Chem. Phys. 2005, 122, 194904. (24) Podgornik, R.; Akesson, T.; Jonsson, B. J. Chem. Phys. 1995, 102, 9423. (25) Sjostrom, L.; Akesson, T.; Jonsson, B. Ber. Bunsen-Ges. 1996, 100, 889.

10.1021/la801957j CCC: $40.75  2008 American Chemical Society Published on Web 08/09/2008

Effect of the Bridging Conformation of PEs

tions. Theoretical work provided a clear mesoscopic picture of the bridging interaction between macroscopic surfaces and elucidated the effects of salt and nonelectrostatic excluded volume on the strength and range of this interaction.26-30 In a salt-free system, where PEs act as counterions to two infinite charged planar walls, Akesson27 found that the repulsive double layer interaction seen in monovalent electrolyte solution disappears completely in presence of PEs, and the PEs instead give rise to a strong short-ranged attraction. The attraction is due to the bridging conformation of the PE chains from one charged wall to the other. This means that the attraction stems from a “coiling back” of the chain. Cao and Wu31,32 also observed this phenomenon in the investigations of the telechelic polymer brushes and the colloidal stabilization. Besides, Podgornik and co-workers24,26,28,33-35 also found the attraction between charged macromolecules. Granfeldt et al.36 investigated the force between two charged colloid particles grafted by oppositely charged PEs and found that, when the distance between the two particles is beyond twice the diameter of the particles, no apparent interaction appears between two particles. This observation was also found by Podgornik and Jonsson28 and Ni et al.37 In addition, Podgornik and Saslow38 investigated the PE-mediated bridging interactions between the fixed cylindrical macroions in a two-dimensional hexagonal crystalline array. Ilekti et al.39 experimentally found that, in the presence of polyions, the micelles attract each other. There are two possible mechanisms for this attraction.27 The first one is that the PEs bind to the micelles, compensate their charge, and bridge them together.40 This effect is similar to the bridging flocculation of colloidal particles.41,42 The other is that the PEs bind to the micelles, compensate their charge, and cause an electrostatic attraction between them. This attractive force may be the same as the attraction caused by ionic correlations between charged surfaces. The two mechanisms are difficult to distinguish. In most of the studies, they are not distinguished, but both are called the “bridging effect”. A simulation study by Granfeldt et al.36 confirmed the existence of an attractive interaction at low ionic strengths, which is compensated by a repulsive osmotic term as the salt concentration is increased. At low salt concentration, the attraction is insensitive to the salt content. However, for high salt concentration, the attraction decreases with increasing salt concentration.43 By simulations and a meanfield theory, Woodward et al.40 also found this behavior, but they believed that the origin of this behavior is not due to changes in the bridging component of the force which stems from the stretching of chains spanning the gap between the surfaces. Instead, the decrease of attraction is because of a repulsive osmotic (26) Podgornik, R. J. Phys. Chem. 1992, 96, 884. (27) Akesson, T.; Woodward, C.; Jonsson, B. J. Chem. Phys. 1989, 91, 2461. (28) Podgornik, R.; Jonsson, B. Europhys. Lett. 1993, 24, 501. (29) Borukhov, I.; Andelman, D.; Orland, H. Europhys. Lett. 1995, 32, 499. (30) Chatellier, X.; Joanny, J. F. J. Phys. II 1996, 6, 1669. (31) Cao, D.; Wu, J. Langmuir 2006, 22, 2712. (32) Cao, D.; Wu, J. Langmuir 2005, 21, 9786. (33) Podgornik, R. J. Chem. Phys. 1993, 99, 7221. (34) Podgornik, R.; Jonsson, B. Biophys. J. 1994, 66, A292. (35) Podgornik, R. J. Chem. Phys. 2003, 118, 11286. (36) Granfeldt, M. K.; Joensson, B.; Woodward, C. E. J. Phys. Chem. B 1991, 95, 4819. (37) Ni, R.; Cao, D.; Wang, W. J. Phys. Chem. B 2006, 110, 26232. (38) Podgornik, R.; Saslow, W. M. J. Chem. Phys. 2005, 122. (39) Ilekti, P.; Martin, T.; Cabane, B.; Piculell, L. J. Phys. Chem. B 1999, 103, 9831. (40) Woodward, C. E.; Akesson, T.; Jonsson, B. J. Chem. Phys. 1994, 101, 2569. (41) Cabane, B.; Wong, K.; Wang, T. K.; Lafuma, F.; Duplessix, R. Colloid Polym. Sci. 1988, 266, 101. (42) Lafuma, F.; Wong, K.; Cabane, B. J. Colloid Interface Sci. 1991, 143, 9. (43) Huang, H. H.; Ruckenstein, E. J. Colloid Interface Sci. 2004, 275, 548.

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Figure 1. (a) Schematic diagrams of the two-heads and one-head PEs. (b) Snapshot of the macroion-PE system, where the PE chain has two charged heads, denoted by red spheres. The large white spheres stand for the macroions, and the yellow spheres are the neutral monomer of the PE chain.

pressure contribution due to the small ions. They found that the bridging component of the force is insensitive to added salt. Motivated by the above investigations, we focus our attention on the effect of the bridging conformation of the PEs with only two charged heads, corresponding to a telechelic polymer with two charged heads in experiments, on the static and dynamic behavior of the macroions. Because the two heads of the PEs hold the opposite charge to the macroion, the two heads of the PE may be adsorbed on different macroions, forming a bridge between two macroions. Therefore, the formation of the bridge will affect the static and dynamics properties of the PE-macroion system significantly. In this work, the bridge effects of the PE on the behavior of the macroions are explored systematically by using Brownian dynamics simulations. The remainder of this paper is organized as follows. We first depict the computational method of Brownian dynamics simulations, including molecular models and simulation details. We then explore the effects of the concentration of the system and the length of the PE chain on the static and dynamic behavior of the macroions. Finally, some discussion is addressed.

2. Models and Simulation Methods 2.1. Model. We use a coarse grained model (CGM) to investigate the solutions containing PEs and oppositely charged macroions. Within the framework of the CGM, the macroion is represented by a charged Lennard-Jones (LJ) particle; the PE is modeled as a flexible bead-spring chain consisting of LJ particles represented by the model of Kremer and co-workers,44-46 in which only one or two heads are charged (one-head, two-heads). The one-head and two-heads PEs are shown in Figure 1a. The solvent (water) enters the model only through its relative dielectric (44) Grest, G. S.; Kremer, K.; Witten, T. A. Macromolecules 1987, 20, 1376. (45) Grest, G. S. Macromolecules 1994, 27, 3493. (46) Stevens, M. J.; Kremer, K. J. Chem. Phys. 1995, 103, 1669.

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permittivity. Therefore, the systems studied only contain two components: the charged macroions and the oppositely charged PEs. All the systems are under electroneutrality conditions. In the calculations, we use Nb, σb, and Zb to represent the number of segments of a chain, the diameter of the segment, and the charge of each charged head segment, respectively. In addition, we employ σM and ZM to denote the diameter and charge of the macroion, respectively. Also, the numbers of PE chains and macroions are denoted as Np andNM, respectively. Excluded volume interactions are introduced via a pure shortranged repulsive LJ potential, given by

ULJ(r) )

{

[( r -σ ∆ ) - ( r -σ ∆ ) + 41 ]

4εLJ

12

6

r < 21 ⁄ 6σ + ∆ rg2 σ+∆ (1) 1⁄6

0

where ∆ ) 0 is for the macroion-macroion and monomermonomer interaction, ∆ ) 2.5σ for the macroion-monomer interaction, r is the separation distance, εLJ is the strength of the LJ interaction, and σ is the LJ collision diameter. The interaction parameter εLJ is set to 1 kBT. It is assumed that all the particles have an identical interaction (εLJ). The PE chain connectivity is modeled by using a standard finitely extensible nonlinear elastic (FENE) potential:46

[ ( )]

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

2

(2)

The spring constant KFENE is equal to 30kBT/σ2, where kB is the Boltzmann constant, T is the absolute temperature, and the maximum bond length R0 between the beads is set to 2σ. Interaction between any two charged particles i and j, where i and j denote either a macroion or a charged monomer, is described as

Ucoul(r) ) kBT

lBZiZj rij

(3)

where rij is the center-to-center distance between charged species. The Bjerrum length is defined as lB ) e2/4πε0εrkBT, where ε0 and εr are the permittivity of a vacuum and the relative permittivity of the solvent, respectively. We are interested in the behavior of macroions in aqueous solutions at room temperature, that is, 298 K. In our simulations, the reduced temperature of the system is T* ) kBT/εLJ ) 1.0, and the value of the Bjerrum length lB, which is equal to 2.0σ, is 0.71 nm, corresponding to the value of water at room temperature. In this work, the macroion charge is fixed at ZM ) 20, while the charge of each charged head is fixed at Zb ) -5. The coulomb interaction is treated by the particle-particle particle-mesh (PPPM) method.47 2.2. Simulation Details. We perform Brownian dynamics simulations (BD) by using the open source software-LAMMPS package.48 The system is studied in a canonical ensemble (NVT) where the temperature is controlled using a Langevin thermostat, and three dimension periodical boundary conditions are adopted in all the directions. The length of the simulation box is denoted as lbox. The equation of motion of any mobile particle gives

mi

dV bi Vi(t) + b FiR(t) (t) ) b Fi(t) - ξ b dt

(4)

where b Vi is the ith particle velocity and b Fi is the net deterministic force acting on the ith particle of mass mi. b FRi is the stochastic (47) Frenkel, D.; Smit, B. Understanding Molecular Simulation-From Algorithm to Applications; Academics: New York, 1996. (48) Plimpton, S. J. Comput. Phys. 1995, 117, 1.

Table 1. Parameters Used in the Simulations parameter σ T* ) kBT/εLJ ) 1.0 εLJ ) kBT lbox σM ) 5σ σb ) σ ZM ) 20 Zb ) -5 Rmax ) 2σ lB ) 2.0σ KFEEN ) 30kBT/σ2

Lennard-Jones length units reduced temperature Lennard-Jones energy units length of the cubical simulation box diameter of each macroion diameter of each bead charge valence on a macroion charge valence on a charged head maximum bond length Bjerrum length spring constant

bRi (t) force with zero average value and δ-functional correlations 〈F R b Fi (t′)〉 ) 6ξkBTδ(t - t′). The friction coefficient was set to ξ ) m/τLJ, where τLJ is the standard LJ time. δ(t - t′) is the time step. Here, choosing δ(t - t′) ) 0.05 and the time is measured in units of the LJ time τLJ ) σ(m/εLJ)1/2, where m is the particle mass. In this paper, the mass of monomer (mb) is equal to 1, and the mass of the macroion (mM) is related to the monomer mass mb through the ratio of their volumes as mM/mb ) (σM/σb)3 ) 125. Typical simulation parameters are summarized in Table 1. In all the simulations, we set σM ) 5σ, σb ) σ. All the simulations started from a randomly generated initial configuration. Each simulation contains 2 × 106 steps for equilibration of the system, and a following 1 × 106 steps for the ensemble average.

3. Results and Discussion 3.1. Effect of Concentration. Although our goal is to explore the bridging effect of the PE with two charged heads (twoheads) on the behavior of the macroions, for comparison we also consider the macroion-PE system in which the PEs with only one charged head (one-head). The one-head PE model is very similar to the surfactant-like polymer in experiments. Correspondingly, the length of the one-head PE is the half of that of the two-heads PE, just like the cutoff in the middle of the PE of the two-heads. Obviously, for the system containing the PEs of one head and the macroions, no bridging conformation is expected between macroions. To have an intuition in understanding, we show a snapshot of our simulation results in Figure 1b, where we can observe the bridge confirmation clearly. To consider the influence of concentration on the static and dynamic properties of macroion-PE systems, we performed a series of BD simulations. In these systems studied, the length of the box is fixed at lbox ) 40σ, meanwhile the length of the PEs (two-heads (one-head)) is fixed at Nb ) 20 (10). Here, the concentration is denoted by the volume fraction: φ ) π(NMσM3 + NbNpσb3)/(6lbox3). We change the volume fraction φ ) 0.04, 0.054, 0.081, 0.0121, and 0.162, corresponding to NM ) 30, 40, 60, 90, and 120, and the number of chains (two-heads (onehead)) Np ) 60 (120), 80 (160), 120 (240), 180 (360), and 240 (480), respectively. 3.1.1. Static Properties. Figure 2 shows the radial distribution function (RDFs) of macroions at different concentrations, denoted by gM-M(r). In each panel from (a) to (e), the concentration of the system containing the one-head PE is the same as that of the system containing the two-heads PE. At low concentrations, say, φ1 ) 0.040 and φ2 ) 0.054, we found that the RDFs of the macroions show a prominent difference for the one-head PE and two-heads PE. The curves corresponding to the two-heads PE show a higher peak value, whereas the curves corresponding to the one-head PE have no marked peak, as shown in Figure 2a and b. With the increase of the concentration, the difference in the RDF of the macroions becomes gradually blurred (see Figure 2c), and finally the RDFs of the macroions are overlap evenly

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Figure 2. Radial distribution functions of macroions at different volume fractions: (a) φ1 ) 0.040; (b) φ2 ) 0.054; (c) φ3 ) 0.081; (d) φ4 ) 0.121; and (e) φ5 ) 0.162.

(see Figure 2d and e). In the systems containing the two-heads PE and macroions, the PE can be adsorbed on two different macroions and cause the correlation attraction between two macroions. This correlation attraction was referred to as the bridging interaction by Podgornik.49 The correlation leads to the formation of the PE-mediated bridge, as was seen in the snapshot in Figure 1, and the attraction persists as long as the chains gain configurational entropy by “coiling back”. The situation is similar to that in a salt-free system, where PEs act as counterions to two infinite charged planar walls.27 However, in the systems containing the one-head PEs and macroions, there are no bridging conformations between macroions. Correspondingly, no marked peak appears in this case, as mentioned earlier. Actually, besides the bridging effect in the systems studied, the packing effect always exists and the degree of the packing effect is only determined by concentration. When the concentration is small (φ1,φ2), the average distance between the macroions is comparably large. Accordingly, the two-head PE mediated bridge causes the correlation of macroions and leads to a strong bridging effect on the macroions. As a result, a high peak value appears in the curve corresponding to the two-heads PE, but there is no evidence of a peak in the one corresponding to the one-head PE. With the increase of the concentration, the packing effect of macroions becomes dominative gradually. The increase of the packing effect causes that the bridging effect can be ignored when φ4 ) 0.121, φ5 ) 0.162. As shown in Figure 2d and e, the corresponding two curves overlap with each other. Additionally, from Figure 2, we can find that the peak value of the RDF of macroions in the one-head PE solution increases monotonically with φ, whereas that in the two-heads PE solution shows nonmonotonic behavior with φ. With the increase of φ, the peak of the RDF of macroions in the two-heads PE solution decreases during φ1 ∼ φ3 and then increases from the transition point, that is, φ3 ) 0.081. Obviously, at low concentrations, the bridging effect of the two-heads PE dominates the distribution of macroions, leading to the high peak of gM-M(r) at φ1 ) 0.04. On the contrary, in the system containing the one-head PEs and macroions, no marked peak appears at the low concentration, because no bridge conformation exists in the system. In order to investigate the microstructure of PEs in the complex systems, we also calculate the root-mean-square end-to-end distance (Reed) of PEs. In the system containing the one-head PEs and macroions, the Reed value decreases monotonically with (49) Podgornik, R.; Licer, M. Curr. Opin. Colloid Interface Sci. 2006, 11, 273.

Figure 3. Root-mean-square end-to-end distance (Reed) of PEs at different volume fractions: (a) two-heads PE and (b) one-head PE.

the increase of φ (also see Figure 3b). Apparently, this is caused by the packing effect of the high concentration. However, in the system containing the two-heads PEs and macroions, Reed shows nonmonotonic behavior with the increase of φ, as shown in Figure 3a. Reed increases with φ and reaches a maximum at φ ) 0.081. Further increasing φ leads to a decrease of Reed. The changing trend of Reed with φ is analogous to the change of the peak of gM-M(r). As discussed above, in the range of 0.040 e φ e 0.081, the average distance between macroions may be greater than the gyration radius of the two-heads PE chain. With the increase of φ, the two-heads PEs can therefore stretch well, because many bridging conformations are formed. As a result, in the range of 0.040 e φ e 0.081, Reed increases with φ. At φ > 0.081, the average distance between two macroions may be smaller than the gyration radius of the PE chain. In this case, the bridging effect of the PE chain becomes trivial. In addition, due to the packing effect, at comparably high concentrations (0.081 e φ e 0.162), Reed decreases with the increase of φ. 3.1.2. Dynamic Properties. In this section, we study the diffusion behavior of the macroions. The general idea is to extract the corresponding diffusion coefficient from the mean square displacement (MSD) of the macroions. This is also suitable for Brownian dynamic behavior of colloidal macroions. The MSD is a measure of the average distance that a molecule travels. From the slope of the MSD curve of the center of mass of the particles versus time, the diffusion coefficient D can be evaluated using the Einstein equation:

D ) lim tf∞

1 1 msd(t) ) lim 〈|r(t) - r(0)2| 〉 tf∞ 6t 6t

(5)

where r(t) stands for the position vector of a particle at time t and the bracket denotes an ensemble average. Figure 4 shows five examples of the MSD (0.040 e φ e 0.162) of the macroions as a function of time. Similarly, in each panel from (a) to (e), the concentration of the system containing the one-head PE is the same as that of the system containing the two-heads PE. The corresponding diffusion coefficient D is presented in Figure 5. In order to explore the influence extent

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Figure 4. Mean square displacement (MSD) of macroions at different volume fractions: (a) φ1 ) 0.040; (b) φ2 ) 0.054; (c) φ3 ) 0.081; (d) φ4 ) 0.121; and (e) φ5 ) 0.162.

Figure 5. Diffusion coefficients of macroions at different volume fractions. D1 and D2 denote the diffusion coefficients of macroions in the one-head and two-heads PE solutions, respectively. Inset: D1/D2 as a function of the volume fraction.

of bridging conformations on the diffusion of macroions, we also plot D1/D2 versus volume fraction φ in the inset of Figure 5, where D1 is the diffusion coefficient of macroions in the onehead PE solution and D2 is that in the two-heads PE solution. For the one-head PE solution, no bridging conformation exists, and it is only the packing effect that affects the diffusion of macroions. Therefore, D1 decreases monotonically with the increase of φ. However, for the two-heads PE solution, there are two factors (bridging effect and packing effect) affecting the diffusion of macroions. As a result, D2 is insensitive to φ and shows a slight decrease globally. Actually, the difference between D1 and D2 is induced entirely by the bridge conformation of the two-heads PE. The bridge effect will lead to the slowing of macroion motion. Comparing D1 with D2, it can be found that, at the same φ value, D1 is always greater than D2 (see Figure 5), and the deviation reduces with the increase of φ. To observe the difference clearly, D1/D2 is shown in the inset of Figure 5. At φ > 0.121, D1/D2 approaches 1. That is to say, at φ > 0.121, the effect of the bridge conformation of the PE on the diffusion of macroions vanishes. Correspondingly, in Figure 4d (φ4 ) 0.121) and e (φ5 ) 0.162), the two MSD curves also overlap.

Figure 6. Radial distribution functions of macroions at different chain lengths: (a) Nb ) 8 (4); (b) Nb ) 10 (5); (c) Nb ) 20 (10); (d) Nb ) 30 (15); and (e) Nb ) 40 (20), where the first number is the length of the two-heads PE and the second number (in parentheses) is the length of the one-head PE.

This PE-mediated attraction between macroions was found by Podgornik et al.24 and also was found by Ni et al.37 via calculating the potential of mean force between the macroions. 3.2. Influence of Chain Length. In this section, we intend to investigate the effect of the length of PE chains on the static and dynamic properties of PE-macroion systems. We fixed the volume fraction at φ ) 0.054, the number of macroions at NM ) 40, and the number of chains (two-heads (one-head)) at Np ) 80 (160). The lengths of the PEs (two-heads (one-head)) are Nb ) 8 (4), 10 (5), 20 (10), 30 (15), and 40 (20). Correspondingly, the length of the box is changed with Nb so that the volume fraction is fixed. 3.2.1. Static Properties. Figure 6 shows the macroionmacroion RDFs at different PE lengths. In each panel from (a) to (e), although the length of the two-heads PE is twice that of the one-head PE, the number of the two-heads PE is half of that

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Figure 7. Mean square root gyration radius of the PE chain as a function of chain length.

of the one-head PE. Accordingly, in each panel, two systems contain the same number of PE beads. It can be observed from Figure 6 that, in panels (a)-(c), the prominent deviation of gM-M(r) appears for the two-heads PE and one-head PE, while in panels (d) and (e) the difference of gM-M(r) becomes trivial. In panels (a)-(c), the peak values of gM-M(r) in the two-heads PE solution are apparently larger than that in the one-head PE solution. The deviation of gM-M(r) in two systems reduces with the increase of the length of the PE chain. Obviously, it is due to the bridging effect mainly. In addition, it is also related to the packing fraction. At a fixed packing fraction, the average distance between two macroions remains unchanged basically. With the increase of the length of the PE chain, the gyration radius of the PE chain will exceed the average distance. As discussed earlier, once the gyration radius of the PE chain exceeds the average distance between two macroions, the bridge effect becomes trivial. Instead, the packing effect dominates the RDF of macroions. To clearly illuminate the above observation, we also calculated the mean square root gyration radius (Rg) of the two-heads PE chains, and the results are shown in Figure 7. With the increase of chain length, the mean square root gyration radius increases monotonically. The mean square root gyration radius Rg given by

Figure 8. Mean square displacements (MSD) of macroions at different chain lengths: (a) Nb ) 8 (4); (b) Nb ) 10 (5); (c) Nb ) 20 (10); (d) Nb ) 30 (15); and (e) Nb ) 40 (20), where the first number is the length of the two-heads PE and the second number (in parentheses) is the length of the one-head PE.

(6)

Figure 9. Diffusion coefficients of macroions at different chain lengths. D1 and D2 denote the diffusion coefficients of macroions in the one-head and two-heads PE solutions, respectively. Inset: D1/D2 as a function of the chain length (Nb).

where rcm is the center of mass position of the PE and ri is the position of the ith monomer of PE. 3.2.2. Dynamic Properties. The length of PE chain affects not only the RDF of macroions but also the diffusion behavior of the macroions. Figure 8 shows mean square displacement (MSD) of macroions at different chain lengths. The corresponding diffusion coefficient D is presented in Figure 9. In order to examine the influence extent of the length of the PE chain on the diffusion of macroions, we also plot D1/D2 versus chain length Nb in the inset of Figure 9. It can be observed from Figure 9 that the diffusion coefficients of macroions in the one-head PE solution are always greater than that in two-heads PE solution. This qualitatively agrees well with the behavior in Figures 4 and 5. Actually, the key factor affecting the static and dynamic behavior of macroions is the relationship between the gyration radius of the two-heads PE chain and the average distance of the macroions. The change of the PE chain length is related to the gyration radius, while the change of concentration is closely related to the average distance of two macroions. Accordingly, the behavior observed from Figures 8 and 9 is the same as that in Figures 4 and 5. When the one-head PEs adsorb on the macroions, it means that the macroions tow a long tail. This leads to the slowing of

the motion of the macroions, and the longer the tail, the slower is the motion of the macroions. Accordingly, D1 decreases monotonically with the increase of Nb, while D2 shows nonmonotonic behavior with Nb, as shown in Figure 9. In this case of a short chain length of Nb ) 8, the two-heads PE chain may be adsorbed on different macroions, which causes the correlation between macroions, slowing the motion of the macroions. In this case, the diffusion coefficient is significantly predominated by the bridging effect. With the increase of Nb, the bridging effect becomes weak. When Nb is equal to 40, the influence of chain length on the diffusion of the macroions becomes trivial. We can also find from the inset of Figure 9 that D1/D2 decreases with the increase of Nb. That is to say the bridging effect of the twoheads PE chain on the motion of macroions becomes weaker with the increase of the length of the PE chain. 3.3. Effect of Temperature. Here, we also investigate the effect of temperature on the microstructure of macroions. Figure 10 presents the RDF of macroions at different reduced temperatures. It should be noted that we are interested in the behavior of macroions in aqueous solutions at room temperature, that is, T ) 298 K. In our simulations, it is T* ) 1.0. If we change the reduced temperature to T* ) 0.9, then the temperature of the system would be T ≈ 270 K, which is lower than the melting

Nb

Rg2 )

∑

1 2 〈 (ri - rcm) 〉 Nb i)1

10144 Langmuir, Vol. 24, No. 18, 2008

Figure 10. Radial distribution functions of macroions at different reduced temperatures: (a) two-heads PE and (b) one-head PE. All of these correspond to the same volume fraction φ2 ) 0.054.

point of ice, and the model would not be applicable. Therefore, we can only change a small range for the temperature, that is 0.9 < T* e 1.2. Clearly, we can observe from Figure 10 that in the narrow varying range of temperature the RDF of the macroions remains unchanged basically for both the two-heads and onehead PEs at the same volume fraction of 0.054. That is to say temperature does not show a significant effect on the RDF of the macroions in PE aqueous solutions.

4. Conclusions On the basis of the coarse grained model, we investigated the bridging effect of PE chains on the static and dynamic behavior of macroions in aqueous solutions by using Brownian dynamics (BD) simulations. Emphasis was placed on the effects of the concentration of the system and the length of the PE chain on the RDF and diffusion behavior of macroions. The simulation results indicated that, at low concentrations, the bridging effect of the two-heads PE chain led to aggregation of macroions. The reason can be explained as follows. At low concentrations, the

Hu et al.

gyration radius of the PE chain is less than the average distance between two macroions. Accordingly, when the two-heads PE chains were adsorbed on different macroions, the bridging effect of the PE chain dominated the RDF and diffusion behavior of the macroions. With the increase of the concentration of the system, when the gyration radius of the PE chain is greater than the average distance between two macroions, the bridging effect of the PE chain becomes trivial. Namely, at high concentrations, say, φ ) 0.162, the RDF and diffusion coefficients of the macroions in the two-heads PE solution are the same as those in the one-head PE solution, although the two-heads PE solution can form the bridging conformation between two macroions. By exploring the effect of the length of the PE chain on the diffusion of macroions, it was found that the diffusion coefficient of the macroions in the one-head PE solution decreases monotonically with the increase of the PE chain length, while that in the two-heads PE solution presents nonmonotonic behavior. The finding is mainly due to the bridging effect of the PE chain on the behavior of the macroions in the case of short chain length. When the length of the PE chain is very short, the gyration radius of the PE chain is always less than the average distance between two macroions. Therefore, the bridging effect of the two-heads PE chain dominates the static and dynamic behavior of macroions. By investigating the mechanism of the two-heads PE chain affecting the static and dynamic behavior of the macroions, we can provide useful information for the synthesis of stabilizers and destabilizers of colloidal particles. Acknowledgment. This work is supported by the National Natural Science Foundation of China (No. 20776005, 20736002), Beijing Novel Program (2006B17), NCET from the Ministry of Education (NCET-06-0095) and “Chemical Grid Program” and Excellent Talent Foundation from BUCT. LA801957J