Conformational behaviors of a charged-neutral star micelle in salt-free solution

Transcription

1 PAPER Physical Chemistry Chemical Physics Conformational behaviors of a charged-neutral star micelle in salt-free solution Mingge Deng, ab Ying Jiang, ab Xuejin Li,* ab Lei Wang ab and Haojun Liang* ab Received 18th November 2009, Accepted 5th March 2010 First published as an Advance Article on the web 20th April 2010 DOI: /b924281c The conformational behaviors of charged brushes on a micelle self-assembled by charged-neutral diblock copolymers in salt-free solution are extensively analyzed using a coarse-grained dissipative particle dynamic (DPD) simulation. When only monovalent counterions exist, the brush conformation of the corona in the micelle is exactly consistent with the predictions from the blob-scaling theory based on the spherical polyelectrolyte brush model, which differentiates the system into three distinct regimes: (I) quasi-neutral regime, (II) Pincus regime, and (III) osmotic regime. For multivalent counterions such as divalence and trivalence, however, the strong electrostatic correlations lead the micelle structures to deviate obviously from those of scaling predictions. The collapse of the brush appears to be due to the drop in the osmotic pressure inside the corona region of the micelle. 1 Introduction The conformations and physical properties of polyelectrolyte brushes are of great interest because of their fundamental significance and potential applications in the field of bio- and nanotechnology. 1 3 In particular, spherical polyelectrolyte brushes (SPBs) with long polyelectrolyte chains grafted onto a solid core in the size of colloidal dimensions have been widely utilized as novel carrier particles for functional biomolecules. 4 6 Generally, photo-emulsion polymerization 7 and controlled radical polymerization 8 are two normal methods of manufacturing the SPB. Recently, an efficient and easy approach has been developed by simply dissolving chargedneutral diblock copolymer in appropriate solvents, 9 whereby micelles with neutral cores and charged hairs, so-called charged-neutral star micelles, were produced. The distinctively responsive properties of this type of polymer brush and its potential applications in industry have been extensively investigated previously in terms of transition mechanisms in a controlled environment. 10 It is very likely that the comprehensive investigations on the SPBs, in comparison with neutral polymeric brushes, may offer us an opportunity to understand deeply the manner of conformational transformation of a polymeric brush. In the past decades, there has been active interest in this field, and it has attracted the attention of many experimental and theoretical scientists However, owing to fact that the performances of the charged chains are governed mutually by multiple parameters such as ionic strength, electrostatic interaction, and valence of counterions, the system is expected to respond in a complicated manner a CAS Key Laboratory of Soft Matter Chemistry, Department of Polymer Science and Engineering, University of Science and Technology of China, Hefei, Anhui , People s Republic of China. xjli7@ustc.edu, hjliang@ustc.edu.cn b Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui , People s Republic of China during stimulations emanating from the circumstance. Up to this date, an understanding of this sort of system is far from complete, and many problems are still left to challenge us. To comprehend the conformational behaviors of this brush system, we studied the conformational transitions of charged-neutral star micelles built with charged-neutral diblock copolymers using a dissipative particle dynamics (DPD) simulation. 2 Model and method 2.1 Dissipative particle dynamics formulation We study the conformational transitions of charged-neutral star micelles in salt-free solution with the help of the DPD simulation technique. DPD is a simple but intrinsically promising simulation method that allows the study of the conformational behaviors of charged-neutral block copolymers. 21 In DPD simulation, a particle represents the center of mass of a cluster of atoms, and the position and momentum of the particle is updated in a continuous phase but spaced at discrete time steps. Particles i and j at positions r i and r j interact with each other via a pairwise additive force, consisting of three components: (i) a conservative force, F C ij ; (ii) a dissipative force, F D ij ; and (iii) a random force, F R ij. All forces are non-zero within a cut-off radius r c. Hence, the total force on particle i is given by F i ¼ X F C ij þ F D ij þ F R ij iaj where the sum acts over all particles within r c. Specifically, in our simulations F i ¼ X a ij oðr ij Þn ij go 2 ðr ij Þðn ij v ij Þn ij þ soðr ij Þz ij Dt 1=2 n ij iaj where a ij is a maximum repulsion between particles i and j, r ij is the distance between them, with the corresponding unit vector ð1þ ð2þ This journal is c the Owner Societies 2010 Phys.Chem.Chem.Phys., 2010, 12,

2 n ij, v ij is the difference between the two velocities, z ij is a random number with zero mean and unit variance, and g and s are parameters coupled by s 2 =2gk B T. The weight function o(r ij ) is given by oðr ij Þ¼ 1 r ij=r c r ij or c ð3þ 0 r ij r c The standard values s = 3.0 and g = 4.5 are used in our study. By joining consecutive particles with a spring force, we can construct coarse-grained models of polymers. 22,23 The harmonic spring force with a spring constant k s = 10.0 and an equilibrium bond length a 0 = 0.86 in our simulations has the form, F S ij = k s (1 r ij /a 0 )n ij. (4) The total force can also have an electrostatic contribution, which is derived from the electrostatic field solved locally on a grid. For the dimensionless Poissons equation which is scaled with DPD length and energy, r(p(r)rc(r)) = l B r(r), (5) where l B = e 2 /(k B Te) is the Bjerrum length that measures the distance at which two charged particles interact with each other with thermal energy k B T, and p(r) is the dielectric permittivity related to the value in pure water. With an iterating algorithm, 24,25 we can solve the field equation successfully, then the electrostatic force on charged particle i is given by ( " #) X F E 1 jr j r i j=r e i ¼ q i rcðr j Þ P j 0 ð1 jr j 0 r ð6þ ij=r e Þ j where R e is the smearing radius and r j is the grid position for smearing out this point charge. Details on the grid method that we used are available elsewhere. 25 The simulations are performed using a modified version of the DPD code named MYDPD. 26,27 Time integration of motion equations is calculated by a modified velocity Verlet algorithm 22 with l = 0.65 and time step Dt = Mesoscopic model for charged-neutral block copolymers Within the DPD approach, some molecules of the system are coarse-grained by a set of particles. In our simulations, the polyelectrolyte is modeled as a block copolymer with section of hydrophilic and hydrophobic blocks. Specifically, we considered a subsystem containing m diblock copolymer chains in a salt-free solution, each with a hydrophilic block A built with N A charged monomers and a hydrophobic block B with N B neutral monomers. Each of the charged monomers carries one unit of positive charge q = +1, and the total charges carried by block copolymers are Q = mn A q. The valence n of counterions is constrained to n = Q/N C (where N C is the number of counterions) with the electro-neutrality requirement in this presumed salt-free system. In our specific case, polyelectrolyte chains each having twelve charged monomers and four neutral monomers are encapsulated into a cuboid cell (subsystem) of length d = 40. These diblock polyelectrolytes self-aggregate into a spherical micelle with a Fig. 1 Schematic representation of a charged-neutral star micelle. d is the length of the simulation box, R A is the width of the corona formed by the charged blocks, and R B is the radius of the core formed by neutral blocks. neutral core radius of R B and a charged corona thickness of R A, as shown in Fig. 1. We used the simple model to characterize the dilute micelle solution with volume fraction f ¼ 4 3 pðr A þ R B Þ 3 =d 3. The scaling approaches based on a simple SPB model 5,11 are briefly elucidated in the following. Following Shusharuna et al., 5 the elastic force per chain related to the conformational entropy losses in the stretched chain is, F elast k B T R3=2 A N 3=2 A a5=2 0 In our simple SPB model, the neutral monomers (hydrophobic particles) are collapsed into the core of the charged-neutral micelle, the elastic force of this part can be balanced by the hydrophobic interactions between the hydrophobic particles and the hydrophilic particles/solvent particle. Thus, in our opinion, it can be ignored in the simple SPB model. In our simulations, the charged monomer A and neutral monomer B are set to have the same bond length. When most of the counterions are outside the micelle, the elastic force F elast is balanced mainly by the unscreened electrostatic force F elec k B T l BQ 2 R 2 A m This is referred to as the Pincus regime identified by Shusharina. 5 Then, the equilibrium thickness of the corona is ð7þ ð8þ R A B N A 3/7 au 2/7 m 2/7 Q 4/7, (9) where u = l B /a is the relative electrostatic interaction strength. As we have stated earlier, the total charges carried by the block copolymers are Q = mn A q, thus, the equilibrium thickness of the corona can be rewritten as: R A B N A au 2/7 m 2/7. (10) In the osmotic regime, 28 most of the counterions are inside the corona, and the elastic force F elast is balanced mostly by the osmotic pressure of the counterions inside the corona F osm k B T N A nr A ð11þ 6136 Phys. Chem. Chem. Phys., 2010, 12, This journal is c the Owner Societies 2010

3 And the equilibrium thickness of the corona is R A B n 2/5 N A a. (12) Together with the scaling approaches, we carried out DPD simulations to study the conformational behaviors of a charged-neutral star micelle in salt-free solution. 3 Results and discussion For simplicity, the charged-neutral micelle in monovalentcounterion solution is firstly chosen as our model. Three typical regions corresponding to the conformations of brush block as a function of relative electrostatic interactions u are displayed in Fig. 2 and Fig. 3. Within the region of the small value of u, most of the counterions distribute into the solution without entering into the interior of the corona region because of the extremely weaker attractions of charged segments on counterions, which are incapable of overcoming the large entropy aroused by fluctuation of the ions (see Fig. 2a). The conformation of the charged blocks on the micelles is essentially dominated by the elastic force similar to those in the neutral micelles made of the amphiphilic diblock copolymers. The swelling of the PE corona of the charged-neutral micelle is not remarkable in comparison with the neutral corona. The region is referred to as the quasi-neutral regime. The remarkable character of this quasi-neutral regime is that the thickness of corona R A is independent of u. Our simulation results soundly demonstrated this point, as indicated in u o 0.01 in Fig. 3. Fig. 2 Density profile of polyions (black line), counterions (red line) and net charges (blue line) as a function of the radius from the center of the micelle R with different relative electrostatic interaction (a) u = 0.001,(b) u = 0.1 and (c) u = 2.5, respectively. The morphologies on the right (solvent are omitted for clarity; Block A, green; Block B, blue; Counterions, red) correspond to each density profile. Fig. 3 Micelle corona thickness R A as a function of relative electrostatic interaction u. In the intermediate regimes, i.e., the Pincus regime, with the enhancement of attractive force on the counterions by charged segments, a part of the ions are adsorbed into the micelles, while a large number of ions still remain in solution due to the entropy effect. Owing to the lack of sufficient quantities of counterions within the corona region of the micelles, only a small part of the charges on the brush block is screened, and the majority of charges remain unscreened. The conformation of the block in this circumstance is balanced by two factors: repulsion among charges on the block and recoiling force with the requirement of maximum conformational entropy of chain. The scaling theory indicates the exponent relation of R A B u 0.28 (eqn (10)) for the growth of corona thickness with u value. Our present calculation results are consistent with the scaling theoretical prediction (Fig. 3). As for the system having larger values of u, more ions are absorbed into the corona of the micelle, which results in the rise of osmotic pressure and screening of the charges on the polyelectrolyte. The two effects are mutually responsible for the conformation of charged corona blocks. As indicated in Fig. 3, the growth of the thickness of the corona starts to deviate from the Pincus regime elucidated by eqn (10). When the fluctuation entropy of counterions are effectively hurdled, the ions involved in the micelle inside of the micelle reach saturation, and most parts of the charges on the polyion (polyelectrolyte) are screened by counterions. The charged blocks gain a maximum extension in length and then remain constant (Fig. 3), coinciding with the scaling prediction in eqn (12) This region is designated in terms of osmotic regime (Fig. 2c). Despite agreement of scaling theory with our calculations in the case of the monovalent counterion, the behavior of R A for divalent and trivalent counterion is perceived to deviate from the scaling predication in both Pincus and osmotic regimes. For instance, the corona thickness of the charged blocks in a di- or trivalent counterion circumstance exhibits a tiny collapse in osmotic regime for very large u value, unlike in the monovalent counterion where the thickness remains steady. A detailed explanation of this phenomenon will be given later on. Next, the details on the conformation and behavior of the micellar corona are elucidated in Fig. 4 via a calculation of the This journal is c the Owner Societies 2010 Phys.Chem.Chem.Phys., 2010, 12,

4 Fig. 4 Pair correlation function g(r) between charged segments and charged segments in osmotic regime at u = 2.5. Inset: MSD (hdr 2 i)of counterions as a function of reduced time at u = 2.5. The self-diffusion coefficient D 0 (n =+1)=0.83,D 0 (n =+2)=0.46andD 0 (n =+3)= (The black, red and blue lines represent the valence of the counterions as +1, +2 and +3, respectively.) pair correlation functions, g(r), between charged segments and the counterions of mono-, di-, and tri-valence, respectively. In a simulation, it is straightforward to measure g(r), which is the ratio between the average particle number density r(r) at a distance r from a reference particle and the density at a distance r from a particle in an ideal gas at the same overall density. It has been previously proposed that the following equation can be used to calculate the g(r), 29 gðrþ ¼ rðrþv N ¼ nðrþv 4pr 2 DrN ð13þ where n(r) is the number of particles that are a distance between r and r + Dr away from the reference particle, and N and V are the total number of particles in the system and the volume of the system, respectively. The relative higher peak of g(r) for trivalent counterions reveals the existence of a relatively strong correlation between charged segments and the ions with multiple valences. Furthermore, an obvious appearance of the second peak in the cases of di- and trivalence indicates the establishment of somewhat ordered structures of the ions around the charged segments, implying a strong condensation of multivalent counterions yielded by the strong electrostatic correlations. The drop of osmotic pressure in the system can be attributed primarily to the condensation of the counterions Moreover, the reduction in the number of multivalent ions with the charged-neutral requirement in the system is another subordinate reason for the drop of osmotic pressure. This clarifies why the brushes formed by these charged blocks collapse in multivalent ion solution for the larger value of u (Fig. 3). It is reasonable to conceive that the strong correlation between the charged segments and the counterions can account for not only the localization of ions in the space around the charged segments but the suppression of the mobility of these ions in the solution as well. The self-diffusion coefficient D 0 of the ions is calculated based on the function of the time dependence of average mean-square displacement (MSD) hdr 2 i =6D 0 t (inset of Fig. 4). Compared with the mono- and di-valence cases, the mobility of trivalence is significantly hindered, implying the freeze effect of ions inside the corona of the charged-neutral micelles. Based on the above analysis, conformations of the charged blocks, stretched out or collapsed on the spherical core built in hydrophobic chains, is intimately correlative to the strength of electrostatic interaction of two charged segments and that of ions and segments. Upon the increase in the relative electrostatic interaction u, the thickness of brush R A increases in an S style as the system goes from the quasi-neutral to the Pincus regime and then to osmotic regime (Fig. 3). To provide a deeper insight into the process, the dependence of electrostatic potential energy E on the relative electrostatic interaction u is presented. Z E ¼ rðrþcðrþ pðrþ jrcðrþj 2 dr: 8pl B ð14þ To calculate the electrostatic potential energy, we follow the iterating algorithm of Beckers et al. 24 and the grid method of Groot, 25 where the electrostatic field is solved on a grid. This method has been used to evaluate the electrostatic force F E in our simulations, it is consistently used to evaluate the electrostatic potential energy E. Calculation strategies adopted here permit the consideration of the inhomogeneity of the electrostatic permittivity in the system. In the quasi-neutral regime, as the majority of the counterions are distributed in the solution without being contained inside of the micellar corona, positive and negative charges do not screen each other at all, and the electrostatic potential energy is expected as E p u, as shown in Fig. 5. In the Pincus regime, although the electrostatic potential energy still increases with the relative electrostatic interactions u, it deviates from the linear relationship of E p u, due to the screening effect produced by the counterions adsorbed inside of micelles. In the osmotic regime, more counterions intrude into the coronal part of the micelle due to strong relative electrostatic interactions. The screening effects are also strengthened heavily, especially on the multivalent counterions. 30 Thus, we can observe an obvious decline of Fig. 5 Electrostatic potential energy E as a function of relative electrostatic interaction u Phys. Chem. Chem. Phys., 2010, 12, This journal is c the Owner Societies 2010

5 electrostatic potential energy E, as shown in the inset of Fig. 5, and the transition from Pincus regime to osmotic regime appears. 4 Conclusion In this paper, we have made a systematical analysis of the conformation transition of spherical polyelectrolyte brushes in salt-free solution of charged-neutral micelles under different valent counterions using the dissipative particle dynamic simulation. Our calculation results indicate that the scaling predictions can well match our simulation results for the case of monovalent counterions in the systems but deviate from those for multiple valence counterions. The deviation implies that the scaling analysis fails for the treatment of complex circumstances such as those existing in the strong correlation between the charged segments and counterions in the multiple valence counterions system. In our simulation, we found that the trivalent counterions can condense to the charged segments when electrostatic interactions are extremely strong. This condensation may suppress the osmotic activity of the trivalent counterions inside the micelle corona and lead to the collapse of the corona. Moreover, the transitions from quasi-neutral to Pincus regime and from Pincus to osmotic regime are clearly understood in terms of electrostatic potential energy. Acknowledgements We would like to thank the two anonymous referees whose critical comments helped us in improving the quality of our manuscript. We are grateful for the financial support provided by the Program of the National Natural Science Foundation of China (Nos , and ), NBRPC (Nos. 2005CB and 2010CB934500). X. Li would like to acknowledge the financial support provided by China Postdoctoral Science Foundation (No ) the Fundamental Research Funds for the Central Universities, the K. C. Wong Education Foundation, Hong Kong. Parts of the simulations were carried out at the Shanghai Supercomputer Center. References 1J. Ru he, M. Ballauff, M. Biesalski, P. Dziezok, F. Grohn, D. Johannsmann, N. Houbenov, N. Hugenberg, R. Konradi, S. Minko, M. Motornov, R. R. Netz, M. Schmidt, C. Seidel, M. Stamm, T. Stephan, D. Usov and H. N. Zhang, Adv. Polym. Sci., 2004, 165, F. Zhou and W. T. S. Huck, Phys. Chem. Chem. Phys., 2006, 8, M. Ballauff and O. Borisov, Curr. Opin. Colloid Interface Sci., 2006, 11, L. Zhang and A. Eisenberg, Polym. Adv. Technol., 1998, 9, N. P. Shusharina, P. Linse and A. R. Khokhlov, Macromolecules, 2000, 33, K. Kegler, M. Salomo and F. Kremer, Phys. Rev. Lett., 2007, 98, X. Guo, A. Weiss and M. Ballauff, Macromolecules, 1999, 32, J. Pyun, T. Kowalewski and K. Matyjaszewski, Macromol. Rapid Commun., 2003, 24, L. Zhang, K. Yu and A. Eisenberg, Science, 1996, 272, S. Minko, Polym. Rev., 2006, 46, O. Borisov and E. Zhulina, J. Phys. II, 1997, 7, J. K. Wolterink, F. A. M. Leermakers, G. J. Fleer, L. K. Koopal, E. B. Zhulina and O. V. Borisov, Macromolecules, 1999, 32, M. Roger, P. Guenoun, F. Muller, L. Belloni and M. Delsanti, Eur. Phys. J. E, 2002, 9, F. Muller, P. Guenoun, M. Delsanti, B. Dem, L. Auvray, J. Yang and J. W. Mays, Eur. Phys. J. E, 2004, 15, Y. Mei, K. Lauterbach, M. Hoffmann, O. V. Borisov, M. Ballauff and A. Jusufi, Phys. Rev. Lett., 2006, 97, Y. Mei, M. Hoffmann, M. Ballauff and A. Jusufi, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 77, R. Ni, D. P. Cao, W. C. Wang and A. Jusufi, Macromolecules, 2008, 41, L.-T. Yan and X. J. Zhang, Langmuir, 2009, 25, L.-T. Yan, Y. Y. Xu, M. Ballauff, A. H. E. Mu ller and A. Bo ker, J. Phys. Chem. B, 2009, 113, W. D. Tian and Y. Q. Ma, Macromolecules, 2010, 43, P. J. Hoogerbrugge and J. M. V. A. Koelman, Europhys. Lett., 1992, 19, R. D. Groot and P. B. Warren, J. Chem. Phys., 1997, 107, R. D. Groot and T. J. Madden, J. Chem. Phys., 1998, 108, J. V. L. Beckers, C. P. Lowe and S. W. de Leeuw, Mol. Simul., 1998, 20, R. D. Groot, J. Chem. Phys., 2003, 118, G. De Fabritiis, M. Serrano, P. Espanol and P. V. Coveney, Phys. A, 2006, 361, M. Serrano, G. De Fabritiis, P. Espanol and P. V. Coveney, Math. Comput. Simul., 2006, 72, N. P. Shusharina, I. A. Nyrkova and A. R. Khokhlov, Macromolecules, 1996, 29, D. Frenkel and B. Smit, Understanding Molecular Simulation-From Algorithms to Applications, Academic Press, Orlando, A. G. Moreira and R. R. Netz, Eur. Phys. J. E, 2002, 8, 33. This journal is c the Owner Societies 2010 Phys.Chem.Chem.Phys., 2010, 12,