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Langmuir 2009, 25, 3808-3813
Dissipative Particle Dynamics Simulations of Complexes Comprised of Cylindrical Polyelectrolyte Brushes and Oppositely Charged Linear Polyelectrolytes Li-Tang Yan*,†,‡ and Xinjun Zhang† Physikalische Chemie II, UniVersita¨t Bayreuth, D-95440 Bayreuth, Germany, and Department of Chemical Engineering, Tsinghua UniVersity, Beijing 100084, P. R. China ReceiVed NoVember 18, 2008. ReVised Manuscript ReceiVed January 22, 2009 Dissipative particle dynamics (DPD) approach is used to investigate the conformational behaviors and interactions of the complex between a cylindrical polyelectrolyte brush (CPB) and linear polyelectrolytes (LPs) with opposite charges. The effective complex between CPB and LPs and its dependence on the amount and length of LPs are examined. It is found that the CPB conformation presents collapse and reswelling with the increasing amount of LPs. The collapse is caused by the replacement of monovalent CPB counterions by LPs and the condensation of LPs on the CPB which reduce the osmotic pressure inside the brush. The swelling of the collapsed CPB is induced by the excluded volume effects of additionally absorbed LPs and LP counterions. The results show that the addition of LPs can not enhance the effective complex between the CPB and LPs when the total charge of LPs exceeds that of CPB. Our simulation also demonstrates that the increase of the LP length leads to a shrinking of the CPB which consequently exhibits rod-like or spherical conformations. The most effective complex between a CPB and LPs can be reached only when the contour length of LPs is not less than that of the CPB side chain.
1. Introduction Polyelectrolytes (PEs), i.e., polymers being equipped with ionizable groups, have received a great deal of attention in recent years.1-6 Anchoring PE chains by one end sufficiently dense to a surface they form PE brush. The PE brushes can be generated in different geometries, such as planar7-9 or spherical PE brushes,10-13 depending on the geometry of the surface. In the past 2 decades, the investigation on PE brushes has become one of the most active fields in polymer science due to the entirely new properties of PE brushes compared with brushes of uncharged polymers.14-16 The main feature of PE brushes is the strong confinement of the counterions within the brush layer, which causes the high osmotic pressures inside the brushes.17-19 The conformational dependence on ionic strength and valency of counterions was investigated in detail for planar or spherical * Corresponding author. † Physikalische Chemie II, Universita¨t Bayreuth. ‡ Department of Chemical Engineering, Tsinghua University.
(1) Osawa, F. Polyelectrolytes; Marcel Dekker Press: New York, 1971. (2) Kantor, Y.; Kardar, M. Phys. ReV. Lett. 1999, 83, 745. (3) Yin, D.; Yan, Q.; de Pablo, J. J. J. Chem. Phys. 2005, 123, 174909. (4) Netz, R. R. Phys. ReV. Lett. 2003, 90, 128104. (5) Jusufi, A.; Likos, C. N.; Lo¨wen, H. Phys. ReV. Lett. 2002, 88, 018301. (6) Zhulina, E. B.; Borisov, O. V.; Birshtein, T. M. Macromolecules 1999, 32, 8189. (7) Csajka, F. S.; Seidel, C. Macromolecules 2000, 33, 2728. (8) Ahrens, H.; Fo¨ster, S.; Helm, C. A. Macromolecules 1997, 30, 8447. (9) Biesalski, M.; Ru¨he, J. Macromolecules 1999, 32, 2309. (10) Guo, X.; Ballauff, M. Phys. ReV. E 2001, 64, 051406. (11) Mei, Y.; Lauterbach, K.; Hoffmann, M.; Borisov, O. V.; Ballauff, M.; Jusufi, A. Phys. ReV. Lett. 2006, 97, 158301. (12) Mei, Y.; Ballauff, M. Eur. Phys. J. E 2005, 16, 341. (13) Mei, Y.; Hoffmann, M.; Ballauff, M.; Jusufi, A. Phys. ReV.E 2008, 77, 031805. (14) Ru¨he, J.; Ballauff, M.; Biesalski, M.; Dziezok, P.; Grohn, F.; Johannsmann, D.; Houbenov, N.; Hugenberg, N.; Konradi, R.; Minko, S. AdV. Polym. Sci. 2004, 165, 79. (15) Zhou, F.; Huck, W. T. S. Phys. Chem. Chem. Phys. 2006, 8, 3815. (16) Jain, P.; Dai, J.; Grajales, S.; Saha, S.; Baker, G. L.; Bruening, M. L. Langmuir 2007, 23, 11360. (17) Pincus, P. Macromolecules 1991, 24, 2912. (18) Borisov, O. V.; Birshtein, T. M.; Zhulina, E. B. J. Phys. II 1991, 1, 521. (19) Groenewegen, W.; Egelhaaf, S. U.; Lapp, A.; van der Maarel, J. R. C. Macromolecules 2000, 33, 3283.
PE brushes (SPBs).6,10–13,20 The theoretical analysis based on the mean-field approaches predicted a collapse of planar PE brushes depending on the valency and the concentration of the added salt.6 A collapse of SPB also occurs upon addition of multivalent counterions because the replacement of monovalent counterions by multivalent ones reduce the number of counterions inside the brushes.12,13 Recent advances in synthetic polymer chemistry have made it possible to produce cylindrical polyelectrolyte brushes (CPBs), i.e., polymer chains, densely grafted with multiple PE side chains, which have potential applications in nanotechnology and biomedicine, e.g., as drug carriers.21-26 In contrast to the uncharged cylindrical polymer brushes, CPB is more complicated due to the additional length scales set by the long-range Coulomb interaction. Furthermore, compared to the planar and spherical PE brushes, more conformational degrees of freedom from the main chain (backbone) render the study for CPB more difficult. In addition to experiments, computational simulation is really an effective method to investigate the propertied of CPB.27,28 Understanding the complexes between large macromolecules is also a subject of great interest in physics, chemistry, and biology because such reactions are ubiquitous in nature. Examples include the macroion-PE complex, protein-DNA interaction, and the DNA-dendrimer complex.29-32 The electrostatic interaction of (20) Santangelo, C. D.; Lau, A. W. C. Eur. Phys. J. E 2004, 13, 335. (21) Xu, Y.; Bolisetty, S.; Drechsler, M.; Fang, B.; Yuan, J.; Ballauff, M.; Mu¨ller, A. H. E. Polymer 2008, 49, 3957. (22) Xu, Y.; Drechsler, M.; Bolisetty, S.; Fang, B.; Yuan, J.; Harnau, L.; Ballauff, M.; Mu¨ller, A. H. E. Soft Matter 2009, 5, 379. (23) Li, C.; Gunari, N.; Fischer, K.; Janshoff, A.; Schimidt, M. Angew. Chem. int. Ed. 2004, 43, 1101. (24) Stephan, T.; Muth, S.; Schimdt, M. Macromolecules 2002, 35, 9857. (25) Kroeger, A.; Belack, J.; Larsen, A.; Fytas, G.; Wegner, G. Macromolecules 2006, 39, 7098–7106. (26) Lienkamp, K.; Noe, L.; Breniaux, M.-H.; Lieberwirth, I.; Gro¨hn, F.; Wegner, G. Macromolecules 2007, 40, 2486–3502. (27) Polotsky, A.; Charlaganov, M.; Xu, Y.; Leermakers, F. A. M.; Daoud, M.; Mu¨ller Axel, H. E.; Dotera, T.; Borisov, O. Macromolecules 2008, 41, 4020. (28) Yan, L.-T.; Xu, Y.; Ballauff, M.; Mu¨ller A. H. E.; Bo¨ker, A. J. Chem. Phys. B, 2009, in press. (29) Maiti, P. K.; Bagchi, B. Nono Lett. 2006, 6, 2478.
10.1021/la803825x CCC: $40.75 2009 American Chemical Society Published on Web 02/18/2009
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PE with oppositely charged macroions is important in several technological applications such as targeted drug delivery, paper making and producing pigment coating.29–31 However, previous investigations on the properties of CPB were mainly focused on that of CPB in simple salt (i.e., monovalent and multivalent salt) solutions.21–28 There is no study regarding the structural behavior of CPB in the presence of linear PEs (LPs) with opposite charges, which, consequently, is the issue that we will address in the present paper by the simulation method. Atomistic simulations, e.g., molecular dynamics (MD), may be selected to deal with this system. However, because of their accurate nature, atomistic simulation models are computationally intensive, meaning that the time and length scales feasible for atomistic approaches are very limited. This is a major problem especially in charged systems where equilibration and residence time scales of various ions can be exceedingly large. In the present simulations, we use the dissipative particle dynamics (DPD) technique which is a coarsegrained molecular dynamics (MD) approach and can capture the hydrodynamics of complex fluids.33,34 For the complicated problem considered here DPD offers an approach that can be used for modeling physical phenomena occurring at larger time and spatial scales than typical MD as it utilizes a momentumconserving thermostat and soft repulsive interactions between the beads representing clusters of molecules. Moreover, the method can be extended to the investigation of PEs by incorporating electrostatic interactions.35,36 In this work, the conformational behaviors and interactions of the complex between a CPB and LPs with opposite charges are investigated by employing a DPD method. We find and analyze the collapse and reswelling of the CPB with increasing amount of oppositely charged LPs. The effective complex between CPB and LPs and its dependence on the amount and length of LPs are also examined.
2. Model and Simulation Details In DPD, the pairwise interactive force acting on a bead i by the bead j contains three parts: the conservative force (FCij ), the dissipative force (FDij ), and the random force (FRij). The conservative forceFCij , which is derived from a soft interaction potential within a certain radius a, is given by
FijC ) RijωC(rij)eij
(1)
where rij) ri- rj, rij)| rij| and eij ) rij/rij. Rij is the maximum repulsion between bead i and bead j, which can relate to the Flory-Huggins χ-parameter.33 In the present simulations, the interaction between like species Rii is chosen to be 78, corresponding to the size of 3 water molecules per bead.34 The hydrophilic nature of the side chains is captured by setting the repulsion parameterRij between water and the monomer ions of the side chains to be 72 (corresponding to a negative value of χ), while a repulsion parameter between water and the monomers of the backbone with 83 (corresponding to a positive value of χ) is selected to specify the hydrophobic nature of the backbone. The counterions and other ions are set to have the interaction parameters of water. In order to ensure the conservative force soft and repulsive, the weight function ωC(rij) is chosen as (30) Lyulin, S. V.; Vattulainen, I.; Gurtovenko, A. A. Macromolecules 2008, 41, 4961. (31) Ni, R.; Cao, D.; Wang, W.; Jusufi, A. Macromolecules 2008, 41, 5477. (32) Kelly, C. V.; Leroueil, P. R.; Nett, E. K.; Wereszczynski, J. M.; Baker, J. R.; Orr, B. G.; Holl, M. M. B.; Andricioaei, I. J. Phys. Chem. B 2008, 112, 9337. (33) Groot, R. D.; Warren, P. B. J. Chem. Phys. 1997, 107, 4423. (34) Groot, R. D.; Rabone, K. L. Biophys, J. 2001, 81, 725. (35) Groot, R. D. J. Chem. Phys. 2003, 118, 11265. (36) Sirchabesan, M.; Giasson, S. Langmuir 2007, 23, 9713.
ωC(rij))1-rij for rij < 1 and ωC(rij))0 for rij g 1.33 The dissipative and the random forces are
FijD ) -γωD(rij)(Vij · eij)eij
(2)
FijR ) σωR(rij)ξij∆t-1 ⁄ 2eij
(3)
where Vij ) Vi - Vj. γ is the dissipative strength which controls the heat dissipated in a time step, and σ is the noise strength with σ ) 3.67. ξij is a random number which has zero mean and unit variance. The weight functions ωD(rij) and ωR(rij), which couple together to form a thermostat, have the relations as33
ωD(rij) ) [ωR(rij)]2
(4)
σ2 ) 2γkBT
(5) 33
D
A simple function form for ω (rij) is chosen as
{
ωD(rij) ) [ωR(rij)]2 )
(1 - rij)2 (r < 1) 0 (r g 1)
(6)
When modeling polymers, the chains are constructed by connecting the adjacent particles via an extra harmonic spring33
FijS )
∑ Crij
(7)
j
where the constant C is -4.0. The electrostatic force FEij between charges beads i and j is analyzed as reported in Groot’s work.35 According to this study, the electrostatic field is solved by smearing the charges over lattice grid whose size is determined by a balance between the fast implementation and the correct representation of the electrostatic field. In the present simulations the grid size is set to a. For each charged bead, a charge proportional to fn(ri) ) 1 - rin/Re is assigned to every grid node within a radius Re ) 1.6a in such a way that the sum of all charged nodes is equal to the charge of the bead, where rin denotes the distance between the grid node n and the ion i.35 The electric field is solved according to the following equation:
∇
(
)
ε ∇ φn ) -Fje,nΓ εrε0
(8)
where ε, εr and ε0 are the values of the dielectric permittivity of the medium, the vacuum, and the water, respectively.35 Fje,n is the averaged charge density, and ∇φn is the electric gradient at grid node n. Γ ) 13.87 is the coupling constant which corresponds to the Bjerrum length of water at room temperature. The same smearing distribution, fn(ri), which is used to solve the electrostatic field, is consistently used to evaluate electrostatic force on an ion. Then, the electrostatic force on ion i can be calculated from
FijE ) -qi
∑ fn(ri) ∇ φn
(9)
n
where qi is the charge of the ion i.35 Again the sum over n is limited to grid nodes within the distance Re from ri. Similar to MD simulations, DPD capture the time evolution of many-body system through the numerical integration of Newton’s equation of motion. Here we use a modified velocity-Verlet algorithm due to Groot and Warren33 to solve the motion equation. In the simulations, the radius of interaction, the bead mass, and the temperature are set as the unit, i.e., a ) m ) kBT ) 1. A characteristic time scale is then defined as τ ) [ma2/kBT]. The time step ∆t ) 0.06τ and a total bead number density of 3/a3 are selected. The physical length and time scales in the present simulations are a ≈ 0.646 nm and τ ≈ 88 ps.34
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Figure 2. Radius of gyration of the CPB, Rg, as a function of β with L ) 20. The inset illustrates the plot of the root-mean-square end-to-end distance of side chains, λ, against β.
CPB will swell again with further addition of LPs at β > 1 (Figure 1d). In order to quantify the size of the CPB, the radius of gyration Rg is measured, which is defined as
R2g ) Figure 1. Conformations of CPB with L ) 20 and different β: (a) β ) 0, (b) β ) 0.5, (c) β ) 1, and (d) β ) 2.5. The red, yellow and blue spheres indicate the beads of the backbones, side chains, and LPs.
In this study, the CPB used in the simulations consists of the backbone with 80 beads (contour length Mc ) 80a) of which every second carry a side chain with 9 beads (contour length Nc ) 9a). Although the model-CPB is much smaller than the real one the essential physical effects are captured.12,13,28 Each bead in the side chain carries a charge of +1, forming the monomer ions. The LP consists of L beads and each bead carries a charge of -1. Various L are selected to consider the effects of LP length on the complex between CPB and LPs. The number of LP, Nl, is obtained with β ) |2LNl/MN|, where β is the ratio between the total charge of CPB and that of LP. The other ions, i.e., the monovalent counterions from CPB and the monovalent positive ions from LP, always ensure the neutrality in the system. The volume of the simulation box is set as (32a)3 such that the CPB can be considered to be independent and the finite size effects can be avoided. For comparison, the box with volume (15a)3 is used for a PE chain with 50 beads in other simulations.35 The system is equilibrated for 2 × 105 time steps; the desired quantities are produced during a further 2 × 105 time steps.
3. Results and Discussions 3.1. Influences of Amount of Linear Polyelectrolytes. We first consider the influences of LP amount on the conformational behaviors of the complex between a CPB and LPs. For the purpose, the length of LP is fixed at L ) 20 and various amounts of LPs are selected which is indicated by β, i.e., the ratio between the total charge of the CPB and that of LPs. A. Conformational BehaViors of CPB. Figure 1 illustrates the conformations of the CPB with various amounts of LPs indicated by differentβ. For the case in the absence of LP, i.e., β ) 0, the CPB exhibits a horseshoe-like conformation and the side chains extend sufficiently because the monomer ions will repel each other due to their charges (Figure 1a). With increasing β, CPB presents more compact states. A spherical conformation even occurs when the total charge of LPs is equal to that of CPB, i.e., β ) 1 (Figure 1c). Interestingly, one can find that the collapsed
〈
N
∑
1 (b r -b r CM)2 N i)1 i
〉
(10)
where the summation runs over the N positions b ri of every bead of the CPB and b rCMis the center of mass of the CPB. The rootmean-square end-to-end distance of the side chains, λ, is also calculated to characterize the size change of the side chains.37 Figure 2 and its inset illustrate respectively Rg and λ as functions of β. Clearly, both the size of the CPB and that of side chains decrease with the increasing β when β e 1. This is the conformational collapse that has previously been observed for PEs38 and SPB11,13 with multivalent counterions under salt-free conditions. The smallest values of Rg and λ occurs forβ ) 1 at which the total charge of LPs neutralizes that of the CPB. For the cases of β > 1, these sizes, however, get larger for a higher β, demonstrating that the CPB starts to swell again, in close analogy with PE chains39,40 and SPB41 affected by the exceeding salt or oppositely charged LPs. To understand the collapse of the CPB in the presence of LPs at β e 1, we calculate the densities of different ions, F(r), with respect to the nearest bead that belongs to backbone, r. Figure 3 is an example showing these profiles with β ) 1 and L ) 20. From Figure 3, it can be found that the peak of LP ions is much higher than that of CPB counterions, revealing that, near the backbone of the brush, the density of LP ions is higher than that of monovalent CPB counterions although every bead of them (LP ions and CPB counterions) carries the same charge, i.e., -1. In fact, a LP can be seen as a multivalent ion consisting of L charged beads. In terms of the simulations and experiments for CPB22,28 or SPB11,13 in the presence of multivalent counterions, a replacement of monovalent counterions by the multivalent ones will take place within the brushes. The replacement can induce the deduction of the total number of counterions inside the brushes, leading to the collapse of the CPB. The replacement of monovalent counterions by the multivalent ones can be examined further through the correlation effect between different ions.11,13 To consider this effect, we calculate (37) Saariaho, M.; Subbotin, A.; Szleifer, I.; Ikkala, O.; ten Brinke, G. Macromolecules 1999, 32, 4439. (38) Winkler, R. G.; Gold, M.; Reineker, P. Phys. ReV. Lett. 1998, 80, 3731. (39) Hsiao, P.-Y.; Luijten, E. Phys. ReV. Lett. 2006, 97, 148301. (40) Hsiao, P.-Y. J. Phys. Chem. B 2008, 112, 7347. (41) Jusufi, A. J. Chem. Phys. 2006, 124, 044908.
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Figure 3. Density profiles of different ions, i.e., monomer ions (black line), LP ions (red line), and CPB counterions (green line), with respect to the backbone of the CPB at β ) 1 and L ) 20.
Figure 5. Plots of the relative net charge, V (solid circle), and the charge ratio between absorbed LPs and CPB, η (open circle), against β with L ) 20.
Figure 4. Pair correlation function g(r) between monomer ions and LPs (red line), CPB counterions (green line) with β ) 1 and L ) 20.
Figure 6. Density profiles of LP counterions with respect to the backbones of the CPBs at β > 1 and L ) 20.
the pair correlation function between monomer ions and counterions, g(r). Figure 4 illustrates the profiles of the quantity for the LPs and CPB counterions. It can be found that the first correlation peak of LPs is higher than that of CPB counterions, revealing that LPs have a stronger condensation on the monomer ions. Also, small bumps, i.e., the harmonic peaks, occur on a large scale of the profile of LPs. We thus conclude that a stronger correlation between the monomer ions and charged beads, corresponding to strong condensation, takes place for LPs, which leads to a decrease of the osmotic pressure inside the brush.11,13,22,28 In order to gain insight into the reswelling behavior of the CPB shown in Figures 1 and 2 at β > 1, the relative net charge, V, and the local charge ratio between absorbed LPs and CPB, η, are calculated. For the calculation of V, we account for the summation of all charges which are located inside the rootmean-square end-to-end distance of side chains, qnet, and V is defined as V ) qnet/qm, where qm is the total charge of the monomer ions. The local charge ratio η ) NsL/qm is set to consider the amount of absorbed LPs where Ns is the number of absorbed LPs. The LP is considered as being absorbed only when the distance between the mass centers of LP and CPB is less than λ. Figure 5 illustrates these both parameters as functions of β in the same scale. As shown by the plot of V, a small net charge is observed over the whole range of β. However, it is evident that a clear charge inversion occurs with a crossover at β ) 1. The plot of ηreveals that almost all LPs are absorbed onto CPB at β e 1. However, a crossover takes place on further addition of LPs (β > 1). One can find that, at β > 1, the total charge of absorbed LPs is more than that of monomer ions, i.e., η > 1.
Although the CPB is completely neutralized by absorbed LPs at β > 1, local correlation effects will attract more LPs. For neutralization purposes, LP counterions are absorbed as well, which can be discerned from Figure 6. The figure illustrates the density profiles of LP counterions with respect to the backbone at β > 1. Obviously, the peaks in the profiles increases with increasing β, indicating that the LP counterions will also be absorbed on addition LPs. The exclude volume effects of absorbed LPs and LP counterions, in turn, increase the CPB thickness, causing a reswelling of the brush. B. The EffectiVe Complex Between the CPB and LPs. To clarify the role of the LP amount on the effective complex between CPB and LPs in more detail, we calculate the fraction of LP beads condensed onto the CPB, fc. For condensation we use the following simple criterion: a LP bead is considered to be condensed on the CPB if it has a CPB bead in its first coordination shell. Similar to previous studies,42 to evaluate the size of the shell, we first calculate the correlation functions for beads of monomer ions and LPs. The radius of the first coordination shell is then extracted from the position of the first minima in the correlation functions. In Figure 7, we present fractions of LP beads that have condensed onto a CPB with variousβ. One can see that fc increases with the increasingβ when β e 1, demonstrating that a higher β corresponds to a more effective complex between the CPB and LPs at β e 1. However, a crossover occurs at β ) 1 and, above β > 1, the fraction of LP beads condensed onto the CPB is almost maintained due to an approximate saturation of absorbed LPs. It reveals that the addition of LPs can not enhance the effective complex between the CPB and LPs at β > 1. (42) Bo¨ckmann, R A.; Hac, A.; Heimburg, T.; Grubmu¨ller, H. Biophys. J. 2003, 85, 1647.
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Figure 7. Average fraction of LP beads condensed onto the CPB in complex, fc, as a function of β with L ) 20.
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Figure 9. Density profiles of monomer ions with respect to the backbones of the CPBs at different L and β ) 1.
Figure 10. The gyration radius of the CPB, Rg, as a function of L at β ) 1. The inset illustrates the plot of the end-to-end distance of side chains, λ, against L.
Figure 8. Conformations of the CPB withβ ) 1 and different L: (a) L ) 2, (b) L ) 4, (c) L ) 9, and (d) L ) 40. The red, yellow and blue spheres indicate the beads of the backbones, side chains and linear PEs.
3.2. Influences of LP Length. Next we consider the influences of LP length on the conformational behaviors of the complex between a CPB and LPs. For the purpose, various LP lengths L are considered whereas the ratio between the total charge of the CPB and that of LPs, β ) 1, is kept unchanged. A. Conformational BehaViors of CPB. Figure 8 illustrates the conformations of a CPB in the presence of LPs with various lengths at β ) 1. It is observed that the CPB exhibits a rod like conformation at L ) 2. With the increasing L, the CPB collapses and more LPs are absorbed. When L ) 9, it turns to a spherical conformation and almost all LPs are absorbed by the CPB. The collapse of side chains can be examined by the density profile of monomer ions. Figure 9 shows the profile from the CPB backbone with L from 1 to 30. With the increasing length of LP, the peak of F(r) grows. Simultaneously, the tail of the profile becomes shorter; i.e., the increasing of the LP length leads to a shrinking of the brush. We calculate the gyration radius of the CPB, Rg, and the rootmean-square end-to-end distance of side chains, λ, to quantify the collapse of the CPB with the increasing LP length. Figure 10 and its inset show Rg and λ as functions of L. In accordance with the above observation of the density profiles, the sizes of
the CPB and its side chains decrease with increased LP length. Up to L < 4, Rg shrinks dramatically from 7.9a to 5.4a. This is in line with the CPB in the presence of multivalent counterions.22,28 Furthermore, the chainlike characteristic of LPs with more charges cause a further shrinking. A similar change can also be seen from the plot of λ in the inset. However, above L > 9, one can note that both Rg and λ are almost maintained, revealing that the brush reaches an approximate saturation at about L ) 9. Figure 11 illustrates the pair correlation function between monomer ions and LPs, g(r), and the MSD of LPs with various L. Clearly, the height of the first peak in g(r) is higher for a longer LPs (Figure 11a). Moreover, the small bumps, i.e., the harmonic peaks, also increase significantly with increased LP length. This reveals that the correlation between the CPB and LPs increases with the LP length. The strong CPB-LP correlation, in turn, affects the mobility of the LPs as well, which is expressed by their MSD in Figure 11b.The slope of MSD versus time sharply decreases with the increasing L, resulting in a reduced mobility and a stronger condensation. Consequently, the stronger condensation of longer LPs will decrease the osmotic pressure inside the brush, leading to the collapse of the brush with the increasing L.11,13,22,28 B. The EffectiVe Complex between the CPB and LPs. The fraction of LPs condensed onto the CPB, fc, is also used here to consider the influence of the LP length on the effective complex between the CPB and LPs. In Figure 12, we plot fc as a function of LP length at β ) 1. It is observed that fc grows gradually with the increasing LP length when L e 9. However, a crossover occurs in the plot at L ) 9 above which the effective CPB-LP complex almost keeps unchanged. Note that the bead number,
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Figure 12. Average fraction of LP beads condensed onto the CPB in complex, fc, as a function of LP length L at β ) 1.
Figure 11. (a) Pair correlation function g(r) between monomer ions and LPs with different L at β ) 1. (b) MSD 〈∆r2〉 of LPs with different L at β ) 1.
i.e., the contour length, of each side chain of the CPB is set to Nc ) 9. Thus, we conclude that the most effective complex between a CPB and LPs can be reached only when the contour length of LPs is not less than that of the CPB side chains.
are also examined. It is found that the CPB conformation presents collapse and reswelling with the increasing amount of LPs. The collapse is caused by the replacement of monovalent CPB counterions by LPs and the condensation of LPs on the CPB which reduce the osmotic pressure inside the brush. The swelling of the collapsed CPB is induced by the exclude volume effects of additionally absorbed LPs and LP counterions. The results show that the addition of LPs can not enhance the effective complex between the CPB and LPs when the total charge of LPs exceeds that of CPB. Our simulation also demonstrates that the increase of the LP length can lead to a shrinking of the CPB which consequently exhibits rod-like or spherical conformations. It is concluded based on the simulation result that the most effective complex between a CPB and LPs can be reached only when the contour length of LPs is not less than that of the CPB side chain. Upon the fact that the LP can be seen as the simple model of DNA or protein, these results should promote the understanding and application of CPBs in nanotechnology and biomedicine, etc.
4. Conclusions Using a DPD approach, we investigate the conformational behaviors and interactions of the complex between a cylindrical polyelectrolyte brush (CPB) and linear polyelectrolytes (LPs) with opposite charges. The effective complex between the CPB and LPs and its dependence on the amount and length of LPs
Acknowledgment. L.-T.Y acknowledges the hospitality of Professor Alexander Bo¨ker at the University of Bayreuth and the valuable discussion with Dr. Youyong Xu. The financial support from Alexander von Humboldt Foundation is highly appreciated. LA803825X
