A Monte Carlo Study of Spherical Electrical Double Layer of

Nov 3, 2006 - density and the chain length of the polyelectrolytes are two key factors that affect ... The understanding of the properties of the elec...
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J. Phys. Chem. B 2006, 110, 26232-26239

A Monte Carlo Study of Spherical Electrical Double Layer of Macroions-Polyelectrolytes Systems in Salt Free Solutions Ran Ni, Dapeng Cao,* and Wenchuan Wang Lab of Molecular and Materials Simulation, Department of Chemical Engineering, Beijing UniVersity of Chemical Technology, Beijing 100029, P. R. China ReceiVed: September 27, 2006; In Final Form: NoVember 3, 2006

A canonical Monte Carlo simulation is performed to investigate the microstructure and the electrical double layer (EDL) of polyelectrolytes around macroions in the bulk systems based on the primitive model. We explore the influences of particles size, chain length, and charge density of polyelectrolytes on the microscopic behavior of the macroions-polyelectrolytes systems. The simulation results show that the surface charge density and the chain length of the polyelectrolytes are two key factors that affect the microstructure of polyelectrolytes around the macroions and potential of mean force between the macroions as well as the ζ potential of the spherical EDL constructed by polyelectrolytes. The high surface charge density of a polyelectrolyte leads to the polyelectrolyte acting as a bridge for the aggregation of macroions, causing the presence of the attraction between macroions. The polyelectrolytes with a long chain length present a cooperativity effect for the adsorption of the polyelectrolytes on the surface of the macroions. Furthermore, the two key factors both induce the overcharge of the macroions. The longer the chain length and the higher surface charge density of the polyelectrolytes, the stronger is the overcharge.

1. Introduction The understanding of the properties of the electrical double layer (EDL), which is the structure formed by electrolyte ions and a charged surface, is important for colloidal and biological sciences because of a lot of related applications, such as colloidal stability,1 electrokinetics, viscosity of the suspension,2 and the complexations of biological systems.3,4 An EDL may be of different geometries such as planar,5-8 cylindrical,9-14 spherical,15-18 or ellipsoidal19,20 depending on the geometries of the charged surfaces. In the past two decades, a series of experimental and theoretical investigations have been performed on the properties of the EDLs of different geometries. The properties of EDLs of simple electrolytes on the charged surfaces of different geometries have been extensively investigated using computer simulations,5-8,11 hypernetted chain/mean spherical approximation (HNC/MSA),9,12,21,22 and densityfunctional theories (DFTs).14,17,23,24 However, there were relatively few studies on the EDLs of polyelectrolytes on the charged surfaces,25-27 especially for the spherical EDLs of polyelectrolytes around the oppositely charged particles. On the basis of the restricted primitive model, the spherical EDLs of simple electrolytes around the oppositely charged spherical surface were investigated by using HNC/MSA,16,18 the Monte Carlo simulations,15 and the DFTs,17 and the charged inversion phenomena for the colloidal particles were observed. By considering the ζ potential of the spherical EDLs, which is directly related to the measured mobility in electrophoresis experiments,18 they found that the ζ potential is a non-monotonic function of the surface charge density of the colloidal spherical surface. Moreover, Yu et al.17 predicted a bilayer structure of counterions around the surface of the highly charged spherical particle. For the EDLs of polyelectrolytes on the charged surfaces, researchers only investigated the planar EDLs by using * Corresponding author. Phone: 86-10-64443254. Fax: 86-10-64427616. E-mail: [email protected] and [email protected].

experiments,27 Monte Carlo simulations,25 self-consistent field theory,28 and DFTs26. Åkesson et al.25 found that the chain connectivity of polyelectrolyte chains leads to significant differences, compared with the system of unconnected monomers. However, to our knowledge, few investigations are found on the properties of the spherical EDLs of polyelectrolytes around oppositely charged spherical particles. In the solution containing polyelectrolytes and oppositely charged macroions, the polyelectrolytes would wrap on the surface of macroions to form complexations due to the electrostatic attraction between the two oppositely charged species. Considering the interactions between the oppositely charged particles, Goeler and Muthukumar29 determined the conditions for the adsorption of polyelectrolytes on spherical and cylindrical surfaces by means of variational calculations. Some research groups30-34 studied the effects of added salt and bare stiffness of the polymer on the shapes of the macroionpolyelectrolytes complexations, even emphasized on the experimentally relevant DNA-histone systems.33,35 When the polyelectrolytes adsorbed on an oppositely charged macroion, the polyelectrolytes overcharged the macroion, leading to the charge inversion. Within the framework of self-consistent field theory, Gurovitch and Sens36 found that chain connectivity between the beads of polyelectrolytes enhances the intensity of the charge inversion. When the charged DNA chains are added to oppositely charged colloidal water solutions, the phase separation occurs between neutral DNA-macroion complexations and non-neutral complexations.30 Nguyen and Shklovskii30 used the phenomenological theory to obtain the phase diagram. Recently, many researchers used DFT37,38 to investigate the solutions containing polyelectrolytes and oppositely charged macroions.17,39,40 Considering molecular simulation to be a powerful tool investigating the microstructures and thermodynamics properties of the complexations formed by polyelectrolytes and macroions, many researchers used molecular simulation techniques to

10.1021/jp0663412 CCC: $33.50 © 2006 American Chemical Society Published on Web 12/21/2006

Macroions-Polyelectrolytes Systems investigate the system. Stoll and co-workers41,42 investigated the effects of the stiffness of polyelectrolyte chains, charge density of macroions, and solution pH on the adsorption and desorption behavior of polyelectrolytes on the surface of macroions. They found that, for the fixed chain length, the decreases of chain stiffness and salt concentration would enhance the adsorption of polyelectrolytes on the macroions.41 As the polyelectrolytes can be adsorbed on the surfaces of different macroions, the polyelectrolytes would act as the bridge between macroions. This was considered as the bridging effect of polyelectrolytes. By exploring the forces between two charged particles mediated by polyelectrolytes, Jonsson and co-workers43,44 found that there existed two types of interactions between macroions: the entropic bridging and energetic bridging interactions. Therefore, the polyelectrolytes-mediated interaction potential was strongly not monotonic and showed pronounced variation with respect to the chain length of the polyelectrolytes. Granfeldt et al.45 investigated the interactions between two charged colloidal particles grafted by oppositely charged polyelectrolytes and found that, when the distance between the two particles is beyond twice the diameters of the particle, no apparent interaction appears between two particles; that is, the interaction approaches zero. This observation was also presented in the investigation of Podgornik and Jonsson.46 In addition, many researchers used the cell model to explore the effects of the flexibility,47,48 linear charge density,49-51 and concentration52-54 of the polyelectrolyte on the microstructure of the macroionspolyelectrolytes complexations. On the basis of the primitive model (PM), in this work, we intend to use Monte Carlo simulations to investigate the spherical electrical double layer of polyelectrolytes around oppositely charged colloidal particles in salt-free solutions. Considering that, in the macroions-polyelectrolytes solutions, the macroions may be overcharged by the oppositely charged polyelectrolytes and the overcharged complex would adsorb other macroions,55 meaning that the polyelectrolytes act as a bridge for the aggregation of the macroions, we plan to investigate the potential of mean force between macroions. In addition, we also explore the effects of particle size, chain length, and charge density of polyelectrolytes on the macroionspolyelectrolytes conformations and electrical properties of the EDLs formed by polyelectrolytes around the macroions. The remainder of this article is organized as follows. We first depict the computational method of Monte Carlo simulation, including molecular models and simulation details. Then, we explore the effects of segmental size, chain length, and charge density of the polyelectrolytes on the microstructures and the electrical properties of the EDLs formed by polyelectrolytes around the spherical macroions. Finally, some discussion is addressed. 2. Models and Simulation Methods 2.1. Model. We use the PM to investigate the solutions containing polyelectrolytes and oppositely charged macroions. Within the PM framework, the macroion is modeled as a charged hard sphere, and the polyelectrolyte is represented by the tangentially connected charged sphere beads, while the solvent (water) enters the model only through its relative dielectric permittivity. Therefore, the system studied only contains two components of the charged macroions and the oppositely charged polyelectrolytes. In the calculations, we use Mb to represent the number of spherical beads of the chain and σb and Zb for the diameter and charge of each bead, respectively. Obviously, the total charge of one polyelectrolyte is Zp ) ZbMb

J. Phys. Chem. B, Vol. 110, No. 51, 2006 26233 in units of elementary charge. In addition, we employ σM and ZM to denote the diameter and charge of the macroion, respectively. The total potential energy (U) of the system can be expressed as

U ) UUnbond + UBond + UAngle

(1)

As the polyelectrolyte is regarded as a freely flexible chain in our model, the total energy is free of the angle energy, that is

UAngle ) 0

(2)

In eq 1, the nonbonding energy is a summation of the hard sphere repulsion and Coulomb interaction, given by

UUnbond ) UH-S + UCoul )

{

Uij ∑ i,j

σi + σj 2 Uij ) Z Z e2 σi + σj i j r g 4π0rrij ij 2 ∞

(3)

rij
1, the total charge densities of polyelectrolytes around the macroion within the distance σM + σb/2 overcharge the macroion. Figure 3 shows the adsorption amount as a function of the reduced size of macroions to the segments of polyelectrolytes. Apparently, with the increase of the reduced size (i.e., σM/σb), the adsorption amount increases then gradually approaches the horizon. Because the adsorption amount is larger than 1, we believe that the overcharge induced by polyelectrolytes occurs. As a result, the overcharged macroion would attract other macroions, which is distinct evidence for the bridging effect of the polyelectrolytes for macroions.

W(r) ) -ln[gM-M(r)] kT

(9)

where k is the Boltzmann constant, T is the system temperature, and r is the distance between the mass centers of macroions. Figure 4 shows the potential of mean force between macroions as a function of distance between the centers of macroions. As shown in Figure 4, when the reduced size is relatively small (σM/σb ) 2.0), the interaction between macroions is dominated by repulsion. With the increase of the reduced size of σM/σb, the attraction between macroions becomes more significant and even dominates the interaction between macroions at the reduced sizes of σM/σb ) 8 and 10. This attraction between two macroions is induced by the bridging effect of the polyelectrolytes. Furthermore, we found that, when the distance between two macroions is beyond 2-fold diameters (2σM ) 8 nm) of the macroion, no noticeable interaction between macroions can be observed for all systems studied. This observation agrees well with the results of Granfeldt et al.45 about the interactions between two charged micelles grafted by oppositely charged polyelectrolytes. Additionally, we also calculate the ζ potential, ζ, of the spherical electrical double layer constructed by polyelectrolytes, which is defined as the mean electrostatic potential at the closest separation between the macroion and the segments of polyelectrolytes. In terms of the ionic density distribution, the ζ potential can be calculated by

ζ)

e 0r

∫σ∞ +σ M

2

b

[CM gM-M(r) ZM +

[

CSeg gM-b(r) Zb] r -

]

2r2 dr (10) σM + σb

We present in Figure 5 the ζ potential as a function of the relative surface charge density, Q*, where Q* is the surface charge density ratio between the macroion and the segments of the polyelectrolyte, that is, Q* ) QM/Qb. It can be seen from Figure 5 that the ζ potential does not monotonically increase with the increase of the relative surface charge density Q* but has a maximum value between Q* ) 1.5 and 2.5. The behavior of the ζ potential versus the relative surface charge density

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Figure 5. ζ potentials as a function of the relative surface charge density of the macroions to the segment.

Figure 6. Radial distribution functions between the macroion and the segments of polyelectrolytes as a function of the distance between the centers of the two ionic species. The detailed parameters are Zb ) -1.0, σM ) 4.0 nm, σM/σb ) 4.0.

qualitatively agrees with that of the spherical double layer of simple electrolytes around an isolated macroion.17 3.2. Influence of the Length of Polyelectrolytes. To investigate the influence of cooperativity between the connected segments in the polyelectrolyte on the macroions-polyelectrolytes systems, we perform a series of MC simulations for the different lengths of polyelectrolytes. In these simulations, we only vary the chain lengths of the polyelectrolytes from Mb ) 10 to 80, and other parameters remain fixed. For example, the reduced size of macroions to the segments of polyelectrolytes is set as σM/σb ) 4.0, and the charge of the segment of polyelectrolytes is fixed at Zb ) -1.0. Figure 6 shows the RDFs of the macroion-polyelectrolyte system. When the chain length of the polyelectrolyte is relatively small (Mb ) 10), there exists a significant second layer structure of polyelectrolytes around the macroion. However, as the chain length of the polyelectrolytes increases from 10 to 20, the second layer becomes gradually blurred. Correspondingly, the contact value gains a great jump. It may be due to the cooperativity effect between the connected segments of the polyelectrolyte, which weakens the electrostatic repulsion between the connected segments of the polyelectrolyte and makes the EDL formed by long chains tighter than that by short chains. When the chain length of the polyelectrolytes is larger than 20, the cooperativity

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Figure 7. Adsorption amount as a function of the chain length of the polyelectrolyte.

Figure 8. Potential of mean force between the macroions as a function of the distance between the macroions.

effect becomes inconspicuous. To better understand the cooperativity, we also calculate the microstructure of the monomer around the macroion, and the RDF of the monomer-macroion system is also added into Figure 6 for comparison. Compared to the microstructure of polyelectrolytes of Mb ) 10, the RDF of the monomer ion around the macroion presents a smaller contact value. Furthermore, the bilayer structure formed by the segments of polyelectrolytes disappears; that is, the cooperativity of chain connectivity is prohibited for monomer ions. To further explore the overcharge of the macroion due to the cooperativity of the polyelectrolytes with long chain lengths, we present in Figure 7 the adsorption amount changing with the chain length of the polyelectrolytes. Similar to the case of adsorption amount versus the reduced surface charge density, the adsorption amount of polyelectrolytes on the surface of the macroion increases with the chain length of the polyelectrolytes. The adsorption amount is larger than 1, meaning the presence of the overcharge is induced by polyelectrolytes, which is consistent with the behavior induced by the bridging effect. In addition, the influence of the chain length of the polyelectrolytes on the interaction between macroions is also investigated. We present in Figure 8 the potential of mean force between macroions versus the distance between macroions. Clearly, the interaction between macroions is dominated by repulsion when the polyelectrolytes are relatively short (Mb )

Macroions-Polyelectrolytes Systems

Figure 9. ζ potential of the spherical electrical double layer constructed by polyelectrolytes changing with the chain length of the polyelectrolyte.

10). With the increase of the length of polyelectrolytes, the bridging effect caused by polyelectrolytes becomes more significant, and at the range of 5∼7 nm, the attraction induced by the cooperativity effect of long chains appears. Interestingly, because of the prohibition of chain connectivity for monomer ions, the interaction between macroions mediated by monomer ions is globally repulsive. However, compared to that induced by the high surface charge density, the attraction is weak and cannot cause the aggregation of the macroions. Similarly, we are also interested in the influence of the chain length on the ζ potential of the EDL constructed by polyelectrolytes. We show in Figure 9 the ζ potential of the spherical EDL changing with the chain length of polyelectrolytes. From the expression of the ζ potential in eq 10, we can see that, with the increase in chain length (i.e., the adsorption amount of the polyelectrolytes on the macroion), the value of the ζ potential decreases. However, with the increase in the bridging effect of polyelectrolytes for macroions, the value of the ζ potential is elevated. As a result, the ζ potential is closely dependent on two factors: adsorption amount and bridging effect. When the chain length of polyelectrolytes increases from 10 to 20, the ζ potential exhibits a sharp decrease. This is because the adsorption amount of polyelectrolytes exhibits a significant increase but the bridging effect of polyelectrolytes does not increase significantly. The global tendency of the ζ potential decreases with the increase in the chain length of polyelectrolytes, which means that the increase in the length of the polyelectrolytes enhances the adsorption of polyelectrolytes on macroions more significantly than the bridging effects of polyelectrolytes for macroions. 3.3. Influence of the Charge Density of Polyelectrolytes. In this section, we investigate the influence of the charge density of polyelectrolytes on the macroions-polyelectrolytes systems. Here, we only vary the charge density (Zb) of the segments of polyelectrolytes and keep the chain length and the reduced size of macroions to polyelectrolytes fixed. So, in the simulations, the concentration of the polyelectrolyte and the number of the polyelectrolyte change with Zb. The detailed parameters are σM ) 4.0 nm, Mb ) 20, σM/σb ) 4.0, CSeg ) |(ZMCM)/Zb|, and Np ) (Cseg‚LBox3)/(Mb), and the charge densities of the segments of polyelectrolytes are changed from -0.5 to -4.0. Figure 10 shows the RDF of the macroion-polyelectrolyte system. Under the condition where the segmental diameter of the polyelectrolytes is fixed, the change of the charge density in the segments of the polyelectrolyte is equivalent to that of

J. Phys. Chem. B, Vol. 110, No. 51, 2006 26237

Figure 10. Radial distribution functions between the macroion and the segment of polyelectrolytes. The detailed parameters are σM ) 4.0 nm, Mb ) 20, σM/σb ) 4.0.

Figure 11. Potential of mean force between the macroions versus the distance between the macroions.

the surface charge density of the segments. In earlier discussion, we found that the effect of the segmental size of the polyelectrolyte on gM-M(r) and gM-b(r) was closely related to the surface charge density of the segments. As a result, the charge density in the segments of the polyelectrolyte would have the same effect on gM-M(r) and gM-b(r) as on the segmental size of the polyelectrolyte, because their key factors are both the surface charge density of the segments of the polyelectrolyte. With the increase of the charge density of the segments, the contact value of gM-b(r) becomes significantly large, which shows the same behavior with the increase of σM/σb in Figure 1. The difference is that the changing intensity of gM-b(r) is larger than that shown in Figure 1. Figures 11 and 12 show the potential of mean force between macroions and the adsorption amount of polyelectrolytes on the macroion, respectively. Analogically, the potential of mean force between the macroions and the adsorption amount of polyelectrolytes on the macroion also hold behavior similar to that in Figures 3 and 4. Figure 13 shows the ζ potential of the EDL of polyelectrolytes around the macroion as a function of the absolute value of the reduced charge density of polyelectrolytes to macroions. Clearly, the ζ potential monotonically decreases with the increase of the charge density of polyelectrolytes. This means that the influence of the charge density of the polyelectrolytes on the adsorption of polyelectrolytes on the surface of the macroion is more significant than that on the bridging effects of poly-

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Figure 12. Adsorption amount changing with the reduced charge density of the polyelectrolyte to the macroions.

Figure 13. ζ potential changing with the reduced charge density of the polyelectrolyte to the macroions.

electrolytes. As an illustration, here we also calculate the mean electrostatic potential, Ψ(r), by the expression

Ψ(r) )

e 0r

∫r∞ [CM gM-M(r′) ZM +

[

CSeg gM-b(r′) Zb] r′ -

]

r′2 dr′ (11) r

According to eq 11, we know that ζ ) Ψ[(σM + σb)/2]. Figure 14 shows the mean electrostatic potential changing with the distance from the center of the macroions. When the charge density of the polyelectrolytes is relatively small (Zb ) -0.5), the mean electrostatic potential monotonically decreases to zero with the increase of the distance. With the increase of the charge density of the polyelectrolytes, the mean electrostatic potential presents a negative minimum value and a positive maximum value. The negative minimum value is due to the charge inversion, and the positive maximum is due to the macroions bridged by polyelectrolytes. Moreover, the negative minimum decreases with the increase of the charge density of the polyelectrolytes, meaning that there exists a more significant charge inversion with the increase of the charge density of the polyelectrolytes. 4. Conclusions On the basis of the primitive model, we have investigated the spherical EDL of polyelectrolytes around oppositely charged macroions and the bridging effects of polyelectrolytes for

Figure 14. Mean electrostatic potential versus the distance from the center of the macroion.

macroions in salt-free solutions using Monte Carlo simulations. Mainly, we investigate the effects of the reduced size of the macroion to the polyelectrolytes, the chain length, and charge density of the polyelectrolytes on the microstructures and EDLs of the macroions-polyelectrolytes systems. The simulation results indicate that the effects of the reduced size of the macroions to the polyelectrolytes and the charge density of the polyelectrolytes on the microstructures and EDLs of the macroions-polyelectrolytes systems can reduce to the effect of the surface charge density of the polyelectrolytes, because their determined factor is the surface charge density of the polyelectrolytes. As a result, the surface charge density and the chain length of the polyelectrolytes are two key factors that affect the microstructures and EDLs of the macroionspolyelectrolytes systems. The high surface charge density of the polyelectrolytes leads to the polyelectrolytes acting as a bridge for the aggregation of the macroions, causing the presence of the attraction in the potential of mean force between two macroions. This behavior is the same as that observed by Prausnitz and coauthors for the monomer-macroion systems.58 The polyelectrolytes with long chain lengths present a cooperativity effect for the adsorption of the polyelectrolytes on the surface of the macroions. Furthermore, the two key factors both induce the overcharge of the macroions. The longer the chain length and the higher charge density of the polyelectrolytes, the stronger the overcharge is. In addition, we also found that, when the distance between two macroions is beyond 2-fold diameters (2σM ) 8 nm) of the macroion, no noticeable interaction between macroions can be observed for all systems studied. This observation agrees well with the results of Granfeldt et al.45 about the interactions between two charged micelles grafted with oppositely charged polyelectrolytes. Acknowledgment. This work is supported by the National Natural Science Foundation of China (20646001), Beijing Novel Program (2006B17), and Excellent Talent Fund of the Beijing University of Chemical Technology. References and Notes (1) Cao, D. P.; Wu, J. Z. Langmuir 2005, 21, 9786. (2) Kimura, H.; Niimi, H.; Tsuchida, A.; Okubo, T. Colloid Polym. Sci. 2005, 283, 1079. (3) Evans, D. F.; Wennerstrom, H. The Colloidal Domain: Where Physics, Chemistry, Biology and Technology Meeting; Wiley-VCH: New York, 1994.

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