Diblock Polyampholytes Grafted onto Spherical Particles: Monte Carlo

Diblock Polyampholytes Grafted onto Spherical Particles: Monte Carlo Simulation and Lattice Mean-Field Theory. Anna Akinchina*, Nadezhda P. Shusharina...
8 downloads 0 Views 397KB Size
Download PDF
Langmuir 2004, 20, 10351-10360

10351

Diblock Polyampholytes Grafted onto Spherical Particles: Monte Carlo Simulation and Lattice Mean-Field Theory Anna Akinchina,*,† Nadezhda P. Shusharina,‡ and Per Linse† Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, Box 124, SE-221 00 Lund, Sweden, and Chemistry Department, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3290 Received April 16, 2004. In Final Form: August 4, 2004 Spherical brushes composed of diblock polyampholytes (diblock copolymers with oppositely charged blocks) grafted onto solid spherical particles in aqueous solution are investigated by using the primitive model solved with Monte Carlo simulations and by lattice mean-field theory. Polyampholyte chains of two compositions are considered: a copolymer with a long and a short block, A100B10, and a copolymer with two blocks of equal length, A50B50. The B block is end-grafted onto the surface, and its charge is varied, whereas the charge of the A block is fixed. Single-chain properties, radial and lateral spatial distributions of different types, and structure factors are analyzed. The brush structure strongly depends on the charge of the B block. In the limit of an uncharged B block, the chains are stretched and form an extended polyelectrolyte brush. In the other limit with the charges of the blocks compensating each other, the chains are collapsed and form a polyelectrolyte complex surrounding the particles. At intermediate charge conditions, a polyelectrolyte brush and a polyelectrolyte complex coexist and constitute two substructures of the spherical brush. The differences of the brush structures formed by the A100B10 and A50B50 polyampholytes are also analyzed. Finally, a comparison of the predictions of the two theoretical approaches is made.

1. Introduction Polyelectrolyte systems have been an important subject in polymer science for many years. Because of their unique properties, polyelectrolytes find a variety of applications in industry and medicine. Many polymers of biological origin, including DNA, are polyelectrolytes. The presence of long-range electrostatic interactions makes them attractive for experimental and theoretical studies. Solutions containing mixtures of oppositely charged polyelectrolytes are of particular interest, demonstrating rich aggregation behavior.1-4 Despite a large number of experimental works, the complete theory of these systems is not yet developed. Polymer/solvent phase separation in a symmetric polyelectrolyte mixture, that is, when the total negative charge on polymers is equal to the total positive charge, has been considered.5,6 The formation of soluble complexes in dilute solutions of an asymmetric mixture of positively and negatively charged polyions, that is, when there is a nonzero uncompensated charge on the polymers, has been investigated by Monte Carlo (MC)7-9 and molecular dynamics simulations.10 Another rapid and expanding area that involves mixtures of polyelectroytes * To whom correspondence should be addressed. E-mail: [email protected]. † Lund University. ‡ University of North Carolina at Chapel Hill. (1) Kabanov, A. V.; Bronich, T. K.; Kabanov, V. A.; Yu, K.; Eisenberg, A. Macromolecules 1996, 29, 6797. (2) Philipp, B.; Dautzenberg, H.; Linow, K. J.; Ko¨tz, J.; Dawydoff, W. Prog. Polym. Sci. 1989, 14, 91. (3) Pogodina, N. V.; Tsvetkov, N. V. Macromolecules 1997, 30, 4897. (4) Kabanov, V. A. Fundamentals of polyelectrolyte complexes in solution and in the bulk. In Multilayer thin films; sequential assembly of nanocomposite materials; Decher, G., Schlenoff, J. B., Eds.; Wiley: New York, 2003. (5) Borue, V. Y.; Erukhimovich, I. Y. Macromolecules 1990, 23, 3625. (6) Castelnovo, M.; Joanny, J. F. Eur. Phys. J. E 2001, 6, 377. (7) Srivastava, D.; Muthukumar, M. Macromolecules 1994, 27, 1461. (8) Hayashi, Y.; Ullner, M.; Linse, P. J. Chem. Phys. 2002, 116, 6836. (9) Hayashi, Y.; Ullner, M.; Linse, P. J. Phys. Chem. B 2003, 107, 8198.

is the formation of alternating layers of cations and anionic polyelectrolytes at solid surfaces, originally developed by Decher and co-workers.11 Also of great interest are the properties of polymers grafted onto solid surfaces. If the repulsive interaction among the polymers is sufficiently strong, the chains become stretched and the structure obtained is referred to as a polymer brush. When the grafted chains are charged, the electrostatic interaction between the chains can be substantial already at a low grafting density, making it easy to access the brush regime. Such polyelectrolyte brushes have been subjected to extensive theoretical investigations involving both (semi-)analytical,12-14 numerical,15 and simulation16 methods. Also, more complex brushes have been considered.17-19 From what is being discussed, it is anticipated that oppositely charged polyelectrolytes attached to a surface will demonstrate a very interesting behavior. For example, a mixed polyelectrolyte brush made of weak polyacid [carboxyterminated poly(tert-butyl acrylate)] and weak polybase [poly(2-vinylpyridine); P2VP] grafted onto a planar surface has been recently studied experimentally.20 The authors analyzed pH-responsive behavior of the brush and observed a nonmonotonic pH dependence of the brush thickness with the minimum at the isoelectric point: they (10) Winkler, R.; Steinhauser, M. O.; Reineker, P. Phys. Rev. E 2002, 66, 021802. (11) Multilayer thin films; sequential assembly of nanocomposite materials; Decher, G., Schlenoff, J. B., Eds.; Wiley: New York, 2003. (12) Pincus, P. Macromolecules 1991, 24, 2912. (13) Borisov, O. V. J. Phys. II 1996, 6, 1. (14) Zhulina, E. B.; Wolterink, J. K.; Borisov, O. V. Macromolecules 2000, 33, 4945. (15) Wolterink, J. K.; van Male, J.; Stuart, M. A. C.; Koopal, L. K.; Zhulina, E. B.; Borisov, O. V. Macromolecules 2002, 35, 9176. (16) Csajka, F. S.; Seidel, C. Macromolecules 2000, 33, 2728. (17) Zhulina, E. B.; Borisov, O. V. Macromolecules 1998, 31, 7413. (18) Shusharina, N. P.; Linse, P. Eur. Phys. J. E 2001, 4, 399. (19) Shusharina, N. P.; Linse, P. Eur. Phys. J. E 2001, 6, 147. (20) Houbenov, N.; Minko, S.; Stamm, M. Macromolecules 2003, 36, 5897.

10.1021/la0490386 CCC: $27.50 © 2004 American Chemical Society Published on Web 10/12/2004

10352

Langmuir, Vol. 20, No. 23, 2004

have observed stretched positively charged P2VP brush at low pH (pH < 3.2), stretched negatively charged poly(acrylic acid) brush at high pH (pH > 6.7), and neutral brush at pH ) 4.9 with two to three times reduced brush thickness. Polyelectrolytes carrying charges of both signs on a chain are referred to as polyampholytes. The positive and negative charges can be distributed randomly along the chain, or charges of one sign can be arranged in long sequences (blocks). The behavior of diblock polyampholytes is expected to be similar to the behavior of the mixtures of oppositely charged polyelectrolytes in a sense that they can precipitate or form soluble aggregates under certain conditions. They are more amenable to theoretical consideration, because the concentrations of oppositely charged blocks are not independent. Theoretical studies21 and computer simulations22,23 of diblock polyampholytes in solution have been performed. Uncompensated charges were found to be a stabilization factor for micelle formation.21 The goal of the present study is to describe the behavior of diblock polyampholytes grafted onto solid spherical particles at different charge conditions. The diameter of the particle is small as compared to the polyampholyte length, so we are dealing with spherical polymer brushes. Planar brushes of the same type have been studied by lattice mean-field theory.18,19 In the present study, we examine the structure of the brushes by means of the primitive model solved by MC simulation and a lattice self-consistent mean-field theory. The mean-field theory is based on a random mixing approximation where fluctuations are neglected. Our focus is on the structural changes of the spherical brushes as the charge of the grafted block ranges from zero to full charge compensation of the other block of the diblock polyampholyte. These changes are expected to be similar to those for a mixed polyelectrolyte brush.20 In particular, in the zero-charge limit of the grafted block we recover the polyelectrolyte brush, whereas for a compensating charge a polyelectrolyte complex surrounding the particles is found. At intermediate conditions, a coexistence of both types of structures appears, and here the spherical brush is composed of two substructures. There is a close similarity of the predictions of the two approaches; hence, the less computer intensive lattice meanfield theory is useful for extensive parameter variations, whereas the primitive model solved by MC simulations provides a more detailed description. 2. Model and Methods We consider a system comprising diblock polyampholytes grafted onto a spherical particle in a dilute aqueous solution containing a small amount of simple salt. Two closely related models are invoked: (i) the primitive model solved by MC simulations and (ii) a lattice mean-field model solved numerically. Common aspects of the two models are provided in section 2.1, while specific details are given in sections 2.2 and 2.3, respectively. 2.1. General. The main aspects of the system are schematically illustrated in Figure 1. A spherical particle with radius Rsph is placed in a spherical cell with radius Rcell containing an aqueous solution with a small amount of simple salt. Flexible diblock polyampholytes are end(21) Castelnovo, M.; Joanny, J. F. Macromolecules 2002, 35, 4531. (22) Imbert, J. B.; Victor, J. M.; Tsunekawa, N.; Hiwatari, Y. Phys. Lett. A 1999, 258, 92. (23) Baumketner, A.; Shimizu, H.; Isobe, M.; Hiwatari, Y. 2001, 13, 10279.

Akinchina et al.

Figure 1. Schematic illustration of a system comprising diblock polyampholytes grafted onto a spherical particle (large sphere) centered in a spherical cell. The outer A blocks are composed of negatively charged segments (small light spheres), and the inner B blocks are composed of positively charged segments (small dark spheres). Simple ions included in the system are omitted for simplicity. All radial distances are measured from the surface of the particle.

grafted onto the surface of the particle with the grafting density σ ) Nchain/Asph, where Nchain is the number of chains and Asph ) 4πRsph2 is the surface area of the particle. Throughout, Rsph ) 18 Å and Nchain ) 10 have been used. The diblock polyampholyte consists of NA negatively and NB positively charged segments, each carrying the fractional charge -|e|τA and +|e|τB, respectively. The composition of the diblock polyampholyte is ANABNB with the B end grafted onto the surface. Two different block compositions are considered: (i) long negatively charged A block (NA ) 100) and short positively charged B block (NB ) 10) denoted by A100B10 and referred to as the asymmetric length composition and (ii) blocks of equal length (NA ) NB ) 50) denoted by A50B50 and referred to as the symmetric length composition. An important variable in this study is the charge compensation parameter

γ≡

NBτB NAτA

(1)

quantifying the compensation of the negative charge of the A block by the positive charge of the B block. At γ ) 0, the charge of the B block is zero, so the diblock consists of a charged A block and an uncharged B block. At γ ) 1, the absolute values of the block charges are equal. The fractional charge per A segment is kept constant (τA ) 0.2), and, hence, the variation of γ is achieved by changing the fractional charge per B segment (τB) or, alternatively viewed, the linear charge density of the B block. The simple ions are monovalently charged (Z( ) (1) and the concentration of added salt (Fsalt ) 4.7 × 10-8 Å-3) is comparable to the counterion concentration. For simplicity, the counterions of the brush and the cations of the simple salt are identical in nature. We assume that the permittivity (r) is constant throughout the whole system, that is, the permittivity of the spherical particle is equal to that of the solution, and the permittivity of the solution is not affected by the presence of polymers or counterions. Throughout, a temperature T ) 298 K and r ) 78.5 have been used. Furthermore, athermal solvent conditions for the polymer segments and the simple ions are assumed. The surface of the particle is treated as a hard wall with no other energetic parameters. Values of important parameters used are compiled in Table 1. 2.2. Primitive Model. Within this model, the particle, chain segments, and simple ions are represented as hard

Polyampholytes Grafted onto Spherical Particles

Langmuir, Vol. 20, No. 23, 2004 10353

Table 1. Specification of the Systems Investigateda General Rcell ) 600 Å (PM); 1800 Å (theory) d ) 6 Å (theory) T ) 298 K r ) 78.5

cell radius layer thickness temperature relative permittivity

Spherical Particle radius

Rsph ) 18 Å

Grafted Diblock Polyampholytes number of polyampholyes Nchains ) 10 NA ) 100; NB ) 10 NA ) NB ) 50

fractional charge per segment asymmetric length composition symmetric length composition

τA ) 0.2; τB ) 0, 0.4, 0.8, 1.2, 1.6, 2.0 τA ) 0.2; τB ) 0, 0.04, 0.08, 0.12, 0.16, 0.2

block charge ratio segment radius interaction parameters

γ ) 0, 0.2, 0.4, 0.6, 0.8, 1 RA ) RB ) 2.0 Å (PM) χA,solvent ) χB,solvent ) 0.0 (theory)

a

Ui )

Simple Ions Fsalt ) 4.7 × 10-8 Å-3 R( ) 3.7 Å (PM) Z( ) (1

PM: primitive model. Theory: lattice mean-field theory.

spheres, of which the chain segments and the simple ions are charged. The chain segments have the radius RA ) RB ) 2.0 Å and the simple ions R( ) 3.7 Å. The chain segments are connected by harmonic bonds, the same bonds for both types of segments. The grafted B segments are in hardsphere contact with the particle, but otherwise they possess unrestricted mobility on the surface of the particle. The solvent enters the model only by its relative permittivity. In more detail, the total potential energy of the system U is

U ) Uhc + Uel + Ubond + Ucell

(2)

where the hard-core repulsion Uhc and the Coulomb interaction Uel are given by

Uhc + Uel )

{

Uij ∑ i Rcell

(7)

Ucell )

degree of polymerization asymmetric length composition symmetric length composition

salt number density ion radii ion charges

Taking into account all interactions in the system, the root-mean-square (rms) segment-segment separation becomes 〈R2bb〉1/2 ≈ 6.0 Å. The chains are assumed to be completely flexible. Simulation of a single uncharged chain gave a bare persistence length of lP ) 7.3 Å.24 The hardcore repulsion between chain segments makes lP slightly larger than 〈R2bb〉1/2. Finally, the confinement potential energy Ucell in eq 2 is given by

(5)

where Nbond is the number of bonds in the system [Nbond ) Nchain(NA + NB - 1)], rm,bond is the bond length of bond m, r0 is the unperturbed equilibrium separation (r0 ) 5.4 Å), and kbond is the force constant (kbond ) 0.415 N/m).

{

with Rcell ) 600 Å. Formally, the confinement potential acts on all species but in practice only on the simple ions because (i) the particle onto which the chains are grafted is fixed in the center of the cell and (ii) the largest chain rms end-to-end distance, appearing for A100B10 at γ ) 0, is 〈R2ee〉1/2 ) 300 Å, corresponding to 〈R2ee〉1/2/Rcell ) 0.5. Simulations with a larger cell radius or without a simple salt have also been performed with only marginally different results. For example, for A100B10 at γ ) 0 the rms end-to-end distance 〈R2ee〉1/2 increases by 3% as the cell radius is increased to Rcell ) 1000 Å. The primitive model was solved by performing MC simulations in the canonical ensemble (constant number of particles, volume, and temperature) according to the standard Metropolis algorithm.25 The potential energies were evaluated using eqs 2-7, and all interactions in the spherical cell were included; hence, no potential cutoff was applied. Initial configurations were generated by first placing the grafted segments in contact with the particle and then successively and randomly placing the remaining segments of the chains at a separation r0 from the previous one. The simple ions were placed randomly. Throughout, checks for hard-sphere overlap were made. Two types of trial displacements were applied: (i) translational move of a single chain segment or a simple ion and (ii) pivot rotation of a section of a chain. In the last case, the rotation of the nonattached part was performed on the basis of a randomly selected pivot segment. Special care was taken for the trial displacement of grafted segments to retain their hard-sphere contact with the particle. The frequency of the pivot rotations was 1/100 of a single segment trial displacement. The chain segment translational displacement parameter was ∆segment ) 3 Å, the simple-ion translational displacement parameter ∆ion ) 10 Å, and the maximal rotational pivot angle ∆pivot ) 10-180°. The equilibration runs involved 5 × 105 MC passes (trial moves per particle), and the production runs involved 106 MC passes. All the simulations were performed using the integrated MC/molecular dynamics/Brownian dynamics simulation package MOLSIM.26 As will be discussed in detail in section 3, the diblock polyampholytes attain two classes of conformations, stretched and collapsed. At intermediate γ both conformations are present. It is conjectured that transitions between collapsed and stretched conformations constitute the bottleneck of the present simulations. Therefore, such transitions for individual chains have been monitored. (24) Akinchina, A.; Linse, P. Macromolecules 2002, 35, 5183. (25) Allen, M. P.; Tildelsley, D. J. Computer simulation of liquids; Oxford: New York, 1987. (26) Linse, P. MOLSIM, 3.4 ed.; Lund University: Lund, Sweden, 2003.

10354

Langmuir, Vol. 20, No. 23, 2004

Akinchina et al.

For the more demanding case with the A100B10 polyampholyte, at γ ) 0.2, 0.4, and 0.6 the distributions of, for example, the end-to-end distance are bimodal for 5, 8, and all 10 chains, respectively, showing the appearance of both collapsed and stretched conformations for single chains. The remaining chains were only stretched. These and similar findings illustrate that the transition frequency between collapsed and stretched conformations is satisfactory. 2.3. Lattice Mean-Field Theory. The lattice meanfield theory used was initially developed by Scheutjens and Fleer27,28 and later generalized to polyelectrolytes by Bo¨hmer et al.29 and by Israels et al.30,31 Central to the theory is the division of the space adjacent to a spherical surface into concentric shells and the division of each shell into lattice cells of equal size. The Bragg-Williams approximation of random mixing is applied within each layer separately, and, hence, all the lattice sites in a shell are equivalent. One lattice cell contains either one polymer segment, solvent, or a simple ion. The model contains five different species: positively and negatively charged polymer segments, positively and negatively charged ions, and solvent molecules. Because a mean-field approximation is applied within each layer, we can write the volume fraction profiles of the species as functions of layer number i only. The possibility of obtaining radial concentration profiles is coupled to the existence of radial-dependent potentials. The species potential uSi is determined as the derivative of the free energy with respect to the species concentration. The potential is species-dependent and is defined to be zero in a homogeneous bulk solution far away from the surface. It can be expressed as a sum of three terms according to el uSi ) ui′ + uint Si + uSi

(8)

where ui′ is a “hard-core” contribution, uint Si is a contribution from short-range interactions, and uel Si is a contribution from long-range Coulomb interactions. The species-independent term ui′ ensures that the space is completely filled in layer i according to ∑SφSi ) 1, i ) 1, 2, ..., M, where φSi is the volume fraction of species S in layer i. The value of ui′ is related to the lateral pressure in a continuous model, and in the bulk ui′ becomes zero. The short-range contribution uint Si is expressed by

uint Si

χSS′(〈φS′i〉 ∑ S′

) kBT

b φS′ )

(9)

where χSS′ is the Flory-Huggins interaction parameter for the S-S′ pair, kB is the Boltzmann constant, and T is the temperature. In eq 9, the angular brackets denote an b is the free averaging over three adjacent layers, and φS′ (bulk) volume fraction of species S′. In bulk, uint Si ) 0. Finally, the long-range Coulomb interaction energy is given by

uel Si ) qSψi

(10)

where qS is the charge of species S and ψi is a mean (27) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619. (28) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980, 84, 178. (29) Bo¨hmer, M. R.; Evers, O. A.; Scheutjens, J. M. H. M. Macromolecules 1990, 23, 2288. (30) Israels, R.; Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1993, 26, 5405.

electrostatic potential in layer i. In line with the randommixing approximation of the short-range interaction, it is reasonable to let the electrostatic potential depend only on the radial distance. Moreover, we assume (arbitrarily) ψb ) 0 in the bulk. We relate the electrostatic potential to the charge density through the Poisson equation

0r∇2ψi ) -Fi

(11)

where 0r is the permittivity of the medium, ∇2 is the Laplacian, and Fi ) ∑SqSφSi is the charge density in layer i. The charges of the species are located at spherical surfaces in the middle of each lattice shell, and the space between the charged surfaces is free of charge. To calculate the volume fraction of species in layer i, it is convenient to introduce weighting factors, which are Boltzmann weights of the species potentials according to

GSi ) exp(-uSi/kBT)

(12)

If uSi and, thus, GSi are known, the relative weights of all the possible conformations can be calculated and, hence, also the volume fraction profiles can be evaluated. The species volume fraction φSi is simply related to nxsi, the number of sites in layer i occupied by segments of rank s (the sth segment in a chain) belonging to component x according to

φSi )

1 Li

Nx

∑x s)1 ∑δS,t(x,s)nxsi

(13)

where Li is the number of lattice sites in the layer i and Nx is the number of segments of component x (Nx ) 1 for the simple ions and solvent and Nx > 1 for the polymer). The Kronecker δ selects only segments of rank s of component x if they are of type S. The expression for the segment distribution is more complex. The correct weights of all conformations, as well as the connectivity of the chains, have to be taken into account. With the partition function as the origin, nxsi is obtained by the use of a matrix method and is then given by32 s+1

∏ s′)N

nxsi ) Cx{∆Ti ‚[

x

s

(Wt(x,s′))T]‚ ∏ s′)2

(Wt(x,s′))T]‚s}{∆Ti ‚[

p(x, 1)} (14) where Cx is a normalization factor related to the bulk volume fraction of component x, Wt(x,s′) a tridiagonal matrix comprising elements which contain factors describing the lattice topology as well as weighting factors for each segment of rank s belonging to component x, p(x, 1) a vector describing the distribution of the first segment of component x among the layers, and ∆ and s are elementary column vectors. Because uSi is needed to obtain φSi (eqs 12-14) and because uSi depends in turn on φSi (eqs 8 and 9), eqs 8, 9, and 12-14 need to be solved self-consistently. In addition, the electrostatic potential (entering in eq 8 via eq 10) has to fulfill Poisson’s equation (eq 11). All lattice mean-field calculations were performed with a lattice spacing d ) 6 Å. The choice of lattice spacing was directed to have the same bond lengths in the two theoretical approaches. The spherical particle was, hence, represented by a hard spherical surface with a radius of (31) Israels, R.; Leermakers, F. A. M.; Fleer, G. J. Macromolecules 1994, 27, 3087. (32) Linse, P.; Bjo¨rling, M. Macromolecules 1991, 24, 6700.

Polyampholytes Grafted onto Spherical Particles

Langmuir, Vol. 20, No. 23, 2004 10355

three lattice layers, and a reflecting boundary at i ) M with M ) 300 has been used. All interaction parameters were exactly 0 (χSS′ ) 0), and the salt concentration corresponded to the salt volume fraction φsalt ) 2 × 10-5 (φ- ) 10-5). The results of the lattice mean-field calculations have been converted back to angstroms using d ) 6 Å. 3. Results and Discussion 3.1. Overview. We start by providing an overview of some of the brush structures obtained. Figure 2 displays final configurations obtained from the MC simulations of the polymer structures formed by the A100B10 diblock polyampholyte at γ ) 0, 0.4, and 1. At γ ) 0, when the outer A blocks are charged and the inner B blocks are uncharged, the chains are stretched and form an extended and dilute polyelectrolyte brush around the particle (Figure 2a). At γ ) 1, when the charges of the A blocks and B blocks exactly compensate each other, the polyampholytes are collapsed and a dense polyelectrolyte complex is formed around the particle (Figure 2c). Finally, at intermediate γ, when the charge of the A blocks is only partially compensated by the charge on the B blocks, a coexistence of both stretched and collapsed chains occurs (Figure 2b). The polyelectrolyte complex appears to involve all B segments and the A segments of four chains, and, hence, the complex has a net charge near zero. The corresponding snapshots for the A50B50 diblock polyampholyte are shown in Figure 3. The general features are similar; however, it appears that at γ ) 0 the chains are less stretched and that the polyelectrolyte complex formed at γ ) 1 is less dense. Hence, the snapshots suggest that (i) the conformations (states) of the chains depend on the charge of the inner B block and (ii) at an intermediate charge compensation two different types of states are present. In the following, we will quantify properties of single chains and of spherical polymer brushes, compare brush properties formed by the two polyampholytes, and compare the predictions of the two theoretical approaches. 3.2. Single-Chain Properties. Probability distributions of the (i) end-to-end distance Ree, (ii) radius of gyration RG, (iii) shape ratio R2ee/R2G, and (iv) asphericity33 of individual chains at different charge compensations have been determined from the MC simulations of the primitive model. Below, we will focus on the end-to-end distance and the shape ratio, two properties providing information on size and shape, respectively, of the chains. Regarding the shape ratio, it becomes 12 for a rigid rod, 6 for a Gaussian chain, and 2 for a homogeneous sphere with two randomly inserted points. Figure 4 displays the normalized probability distribution of the end-to-end distance P(Ree) and the shape ratio P(R2ee/R2G) for the A100B10 diblock polyampholyte at different γ (solid curves). At γ ) 0, P(Ree) shows that the end-to-end distance of A100B10 varies between 200 and 300 Å, whereas at γ ) 1 the distribution peaks at 50 Å (Figure 4a). At all intermediate values of γ except γ ) 0.8, P(Ree) displays a bimodal behavior. The general features of P(R2ee/R2G) are similar (Figure 4b). At γ ) 0, a distinct maximum appears at R2ee/R2G ) 10, whereas at γ ) 1 the maximum is shifted to R2ee/R2G ≈ 2.5. At intermediate γ, again a bimodal behavior appears with the exception for γ ) 0.8. Both types of distributions given in Figure 4 confirm the existence of two states of the chains. At γ ) 0, the (33) Zifferer, G.; Olaj, O. F. J. Chem. Phys. 1994, 100, 636.

Figure 2. Snapshots of spherical particles with end-graphed A100B10 diblock polyampholytes at charge compensation parameters (a) γ ) 0, (b) γ ) 0.4, and (c) γ ) 1.0 from MC simulations of the primitive model. The radius of the visible spherical regions is 315 Å. Color code: particle (green), A segments (blue), B segments (red), positively charged ions (white), and negatively charged ions (yellow).

extension of the chain is 〈R2ee〉1/2 ) 239 Å, that is, 36% of its contour length L ) Nchain〈R2bb〉1/2 ) 660 Å, and the average shape ratio is near 10; hence, at this condition the chains are strongly stretched and this state will be referred to as the stretched state. Nevertheless, the A blocks are not completely stretched (see Figure 2a), consistent with a weak (≈0.05kBT) Coulomb interaction between

10356

Langmuir, Vol. 20, No. 23, 2004

Akinchina et al.

Figure 4. (a) Normalized probability distribution of the endto-end distance P(Ree) and (b) normalized probability distribution of the shape factor P(R2ee/R2G) for the A100B10 diblock polyampholyte at the charge compensation parameters γ ) 0, 0.2, 0.4, 0.6, 0.8, and 1 obtained from MC simulations of the primitive model (solid curves). Corresponding probability distributions obtained by linear superposition according Pγ ) (1 - γ)Pγ)0 + γPγ)1 for γ ) 0.2, 0.4, 0.6, and 0.8 are also shown (dotted curves). The distributions change regularly between γ ) 0 and γ ) 1.

Figure 3. Same as Figure 2 but for the A50B50 diblock polyampholyte.

bonded segments. In contrast, at γ ) 1 we have 〈R2ee〉1/2 ) 2 〉1/2 ) [〈R2bb〉Nchain]1/2 ) 63 56 Å as compared to 〈Ree,Gaussian Å and the shape ratio peaks at 2, showing that at this condition the chains have a smaller extension than a Gaussian coil and are nearly spherical [although P(R2ee/R2G) shows considerable shape fluctuations]. In the following, this state will be referred to as the collapsed state. The clear bimodal behavior of the distribution functions at intermediate γ shows that at these conditions the stretched and collapsed states are coexisting. The nature

of the coexistence can furthermore be addressed by comparing the probability distributions at intermediate γ with relevant linear superpositions of the probability distributions at γ ) 0 and 1 (dotted curves in Figure 4). Besides confirming the coexistence of states, such a comparison points out some second-order effects. First, the positions of the peaks at γ ) 0.2, 0.4, and 0.6 for the stretched state are shifted to shorter Ree and R2ee/R2G, respectively, and the peaks at γ ) 0.8 are absent, as compared to the linear superpositions. That implies that the stretched chains are less extended as they become fewer. Second, the amplitudes of the peaks for the collapsed state are smaller, showing that the collapsed chains become more extended and less spherical-like as there are fewer of them. The corresponding normalized probability distributions for the A50B50 diblock polyampholyte are displayed in Figure 5. Instead of making a full analysis of these distributions, we focus on the differences between the two types of polyampholytes. In general, single-chain properties of A50B50 are similar to those of A100B10 (cf. Figures 4 and 5). The main differences appearing are that (i) the position of the peak for the stretched state in P(Ree) is shifted to shorter Ree, 180 Å as compared to 250 Å (the difference is still significant after taking into account the slightly different contour lengths ≈600 and ≈660 Å, respectively), (ii) the end-to-end distance distributions are broader with a bimodal behavior only at γ ) 0.4, (iii) the shape-ratio distributions are slightly broader with a bimodal behavior only at γ ) 0.6, and (iv) the results at intermediate γ are closer to the distributions obtained by linear superposition. Hence, although qualitatively the same picture emerges for the A50B50 diblock polyampholyte, issues ii and iii imply that the distinction between the two states becomes less accentuated as compared to the intrinsic fluctuations of the two states, and issue iv implies that the investigated properties of the stretched

Polyampholytes Grafted onto Spherical Particles

Figure 5. Same as Figure 4 but for the A50B50 diblock polyampholyte.

and collapsed states depends less on γ. The reason is most likely the reduced A-block charge (NAτA) and the distribution of the B-block charge (NBτB) over a larger number of segments (larger NB and smaller τB). At this stage, we cannot distinguish the relative importance of these two contributions. Thus, the presented single-chain data support the notion provided by the snapshots that the chains appear in two distinctly different states, one stretched and one collapsed. At the uncompensated charge condition only stretched chains are present, at the compensated charge condition only collapsed chains appear, whereas at intermediate conditions both types of states coexist. The distinction between the two types of states is stronger for the A100B10 diblock polyampholyte. The distribution functions of the radius of gyration and the asphericity (data not shown) fully support this picture. 3.3. Spatial Probability Distributions. The brush structure will now be considered by examining the distribution of segments near the surface of the particle. We will use the probability of finding (i) an A segment

Langmuir, Vol. 20, No. 23, 2004 10357

PA-seg(r), (ii) a B-segment PB-seg(r), (iii) a free end segment of the A block PA-end(r), and (iv) a junction between the A and B blocks PAB-junc(r) at the distance r from the surface. The radial integrals of these probability distributions are unity. In more detail, the probability distribution Px(r), x ) A-seg, B-seg, A-end, or AB-junc, was obtained from (i) the MC simulations by discrete sampling of the frequency at which x appeared at distance r using a bin width d ) 6 Å and (ii) the lattice mean-field theory calculations by transformation of the volume fraction φxi to Pxi according to Pxi ) nxi/∑inxi, where nxi ) Liφxi is the number of x in layer i with Li ) (4π/3)[(i + R)3 - (i - 1 + R)3] being the number of lattice cells in layer i with R ≡ Rsph/d ) 3, and thereafter conversion of length units through Px(r) ) (1/d)Pxi with r ) (i - 0.5)d. The general features of the spatial probability distribution functions will first be considered, and thereafter the differences of the results obtained from the primitive model and the lattice mean-field theory will be discussed. In most cases, the differences are small. Properties of the Spherical Brushes. The four probability distributions for the A100B10 diblock polyampholyte at γ ) 0, 0.4, and 1 are shown in Figure 6. When the short inner B block is uncharged (γ ) 0), the chains are negatively charged and the probability distribution of the A segments is almost uniform over a broad distance range (Figure 6, panels a and b), which is similar to the segment distribution in a polyelectrolyte star.13 At compensating charge (γ ) 1), the A segments form a dense layer extending ≈60 Å from the surface. At nonzero uncompensated charge (γ ) 0.4), the shape of the A segment profiles suggests the existence of two regions of different composition: one dilute and extending far from the surface and one dense remaining near the surface. The distributions of the B segments imply that there is a surface-induced depletion layer (Figure 6, panels c and d). The extension of the depletion layer grows when the B blocks become more charged. The distributions of the A ends (Figure 6, panels e and f) are very similar to those of the end-to-end distance (Figure 4a). Because the A block is much longer than the B block, these similarities imply that the stretched chains are predominantly oriented perpendicular to the surface, as also was illustrated in Figure 2. The distribution of the AB junctions displays only a moderate dependence on γ (Figure 6, panels g and h). The

Figure 6. (a and b) Probability distributions of A segments PA-seg(r), (c and d) probability distributions of B segments PB-seg(r), (e and f) probability distributions of free A ends PA-end(r), and (g and h) probability distributions of AB junctions PAB-junc(r) as a function of the radial distance r from the particle surface for the A100B10 diblock polyampholyte at the indicated charge compensation parameters. Panels a, c, e, and g show data obtained from MC simulations of the primitive model, and panels b, d, f, and h provide results obtained from the lattice mean-field theory; see text for further details. The lower values of the ordinates are 0, and the upper ones are (a and b) 0.03, (c and d) 0.04, (e and f) 0.03, and (g and h) 0.06.

10358

Langmuir, Vol. 20, No. 23, 2004

Akinchina et al.

Figure 7. Same as Figure 6 but for the A50B50 diblock polyampholyte. The upper values of all ordinates are 0.02.

largest probability appears at r ≈ 30-45 Å, as compared to contour length of the B block, ≈60 Å, showing that the B blocks are stretched at all conditions. Interestingly, the B blocks become more stretched as they become more charged; the A blocks displayed the opposite trend. In brief, from the analysis of the single-chain properties, we find that the diblock polyampholytes exist in stretched and collapsed states. The fraction of chains in the states depends on the charge compensation. From the spatial probability distributions, we furthermore conclude that (i) at the uncompensated charge condition the stretched chains form an extended polyelectrolyte brush, where the repulsion among the A segments stretches the A blocks as well as pulls out the B blocks also making them fairly stretched, and (ii) at the compensated charge condition the A blocks together with the B blocks form a dense polyelectrolyte complex, where the electrostatic attraction between A and B segments in conjunction with the large length asymmetry forces the B blocks to become even more stretched as compared to that at γ ) 0. At intermediate charge conditions, the polyelectrolyte brush and the polyelectrolyte complex coexist. The corresponding spatial segment distributions for the A50B50 diblock polyampholyte are shown in Figure 7. As now expected, the response of the brush structure on the variation of the charge compensation is qualitatively the same as for the A100B10 diblock polyampholyte, and again we will focus on the differences between the two types of polyampholytes. At γ ) 0, the probability distribution for the A segments is no longer uniform, because of the relatively long B block acting as an uncharged spacer between the A block and the surface (Figure 7, panels a and b). At γ ) 1, a dense layer again appears, but the thickness of the polyelectrolyte complex layer is almost twofold increased as compared to the A100B10 diblock polyampholyte (cf. Figures 2c and 3c). The A-end distributions (Figure 7, panels e and f) display much broader peaks at γ ) 0 and 1 and a less bimodal character at γ ) 0.4 as compared to those of the A100B10 chain. These results are consistent with the observations that the distinction between the extended and the collapsed states is smaller for the A50B50 diblock polyampholyte. Another marked difference between the two polyampholytes is how the radial extension of the region with B segments depends on γ. Whereas the B blocks in the A100B10 chains became more stretched as they were charged, the opposite appears for the B blocks in the A50B50 chains as documented by the B-segment distributions (Figure 7, panels c and d) and made even clearer by the AB-junction

distributions (Figure 7, panels g and h). The reason for this compaction is that the optimal thickness of the polyelectrolyte complex (in which the B blocks are engaged) is smaller than the extension of the B blocks when they are pulled by A blocks in the polyelectrolyte brush regime (see Figure 3). For the shorter B blocks in the A100B10 chains, the stretching in the polyelectrolyte complex is stronger than that in the polyelectrolyte brush. We now return to the difference in the thickness of the polyelectrolyte complexes at full charge compensation (γ ) 1). For both chain types, the electrostatic interaction between the two types of blocks is so strong that any extensive charge separation is enegetically unfavorable. For asymmetric chains with long A blocks and short B blocks with the higher linear charge density, the length of the very stretched B blocks controls the thickness of the complex. Figure 6g shows that the B blocks are stretched, and a comparison between panel a and panel c of Figure 6 shows that the A and B segment distributions were very similar. Regarding the A50B50 chains with equal block lengths, similar comparison between panel a and panel c of Figure 7 shows that also here the radial extensions of the two types of segments are very similar. However, Figure 7h shows that these similar radial Aand B-segment distributions appear without considerable stretching of the B blocks. Comparison between the Two Theoretical Approaches. In our analysis, we conjecture that the most important diversities between the two theoretical approaches, including assigned parameter values, are that in the lattice mean-field approach (i) the segment-excluded volume is larger (d ) 6 Å as compared 2RA ) 2RB ) 4 Å), (ii) the lateral spatial correlations are neglected, (iii) the chains possess full flexibility (a bare persistence length of lp ) 6.0 Å as compared to lp ) 7.3 Å), and (iv) direct backfolding of the chains is allowed. We anticipate that issues i and ii lead to a more extended and issues iii and iv to a less extended segment distribution. To further discriminate among the significances of these aspects, we will first consider a spherical brush of uncharged homopolymers with 50 segments without added salt at otherwise identical conditions as for the polyampholytes. Figure 8 displays the probability distributions of the segments of the spherical uncharged brush (dashed curves). It is noticed that the lattice mean-field theory predicts a slightly more extended brush as compared to the primitive model (peak position at r ≈ 40 Å as compared to ≈30 Å). Hence, we conclude that for the uncharged brush issues i and ii collectively slightly dominate over iii

Polyampholytes Grafted onto Spherical Particles

Langmuir, Vol. 20, No. 23, 2004 10359

Figure 8. Probability distribution of B segments for the A50B50 diblock polyampholyte at γ ) 1 (solid curves) and of segments of a grafted uncharged homopolymer with 50 segments obtained from MC simulations of the (now uncharged) primitive model and from the lattice mean-field theory. The homopolymer system contains no simple ions; otherwise the conditions are identical to those for the polyampholyte system when applicable.

and iv, and we believe that the importance of the excludedvolume difference exceeds that of the spatial correlations. Figure 8 also shows the probability distributions of the B segments for the A50B50 chains at γ ) 1 (solid curves) using the same data as in Figure 7, panels c and d. Here, the corresponding difference of PB-seg(r) is considerably larger with still the lattice mean-field theory predicting the more extended B-segment distribution. We propose that the increased discrepancy between the two theoretical approaches as going from the uncharged to a charged brush containing equal amount of positive and negative charges arises from the neglect of the lateral correlations in the lattice mean-field theory. In other contexts, it is known that the Poisson-Boltzmann equation predicts a smaller accumulation of simple ions near an oppositely charged surface as compared to MC simulations.34 Now, we return to Figures 6 and 7. A comparison of the results of the primitive model and the lattice mean-field theory given in Figures 6 and 7 shows an overall good qualitative agreement. All qualitative aspects discussed are present in both theoretical approaches, and in most cases also semiquantitative agreements are observed. Nevertheless, a closer inspection reveals that the lattice mean-field theory in general predicts broader distributions as just discussed for a single case. Regarding the distributions of the A segments, at γ ) 1 a tail of the A-segment probability distribution appears beyond the upper nonzero limit of the B-segment probability distribution (see panels b and d of Figure 6 and panels b and d of Figure 7), not appearing in the primitive model. Interestingly, at γ ) 0 there is an enhanced probability of finding A segments very close to the surface (cf. panels a and b of Figure 6 as well as panels a and b of Figure 7). Furthermore, broader distributions of the B segments, that is, the B blocks are more stretched, are predicted by the lattice mean-field theory for both diblock polyampholytes at all charge conditions. That is demonstrated clearest by the AB-junction probability distributions (cf. panels g and h of Figure 6 as well as panels g and h of Figure 7). 3.4. Lateral Spatial Correlations. Both the primitive model solved by MC simulations and the lattice meanfield theory show that the structure in the radial direction of the diblock polyampholyte brushes strongly depends on the charge of the B block. We will now examine the angular correlations in the spherical brush, that is, lateral (34) Linse, P.; Gunnarsson, G.; Jo¨nsson, B. J. Phys. Chem. 1982, 86, 413.

surf Figure 9. Radial distribution function gcom,com (r) for (a) A100B10 and (b) A50B50 diblock polyampholytes at indicated values of the charge compensation parameter γ obtained from MC simulations of the primitive model. Here, r denotes the arclength separation between projected positions of the chain center-of-mass on the surface of the particle; see text for further details. The largest possible value of r is πRsph ) 56.6 Å.

correlations. Of course such correlations are suppressed in the lattice mean-field theory, and we will resort to results of the primitive model only. Only the locations of the center-of-mass (com) of the chains will be used in the analysis of the lateral correlation. Let rcom,i denote the position of the com of chain i with respect to the center of the particle. In the following, we will consider the com-com radial distribution function surf gcom,com (r) using the arc length separation r between rcom,i and rcom,j, projected on the surface of the particle according to r ) Rsph cos-1(rˆ com,i‚rˆ com,j) with rˆ com,i ≡ rcom,i/|rcom,i|. As for three-dimensional radial distribution functions, surf gcom,com (r) is normalized to unity for a uniform distribution. surf (r) for the two diblock polyamFigure 9 shows gcom,com pholytes with uncharged (γ ) 0) and charged (γ ) 1) B blocks. In the case of A100B10, at γ ) 0 we observe that surf gcom,com (r) is zero at short separations and displays a maximum at r ) 20 Å, which is followed by a minimum and another maximum before it settles to approximately unity (Figure 9a), in analogy to radial distribution functions of dense fluids of soft spheres. Hence, the negatively charged chains are laterally strongly separated because of electrostatic repulsion between them (see Figure 2a). At γ ) 1, the long-range oscillations are lost, surf (r) displays a nonzero value at r ) and moreover gcom,com 0. In other words, the volumes of the collapsed chains overlap, confirming the view that the polyelectrolyte complex is composed of a network of entangled chains (see Figure 2c). Regarding the A50B50 chains, at γ ) 0 a weaker lateral ordering is found (Figure 9a), consistent with the smaller total charge as compared to the A100B10 chains, whereas at γ ) 1 the degree of lateral chain intermixing appears to be similar. Finally, the rather large noise at γ ) 1, in particular for the A100B10 chain, is due to the considerable entangling, which makes the generation of new com-com separations less efficient. 3.5. Structure Factors. Structure factors have also been calculated to characterize the structure of the

10360

Langmuir, Vol. 20, No. 23, 2004

Akinchina et al.

tures at q ) q0 ≈ 0.04 Å-1 and q1 ≈ 0.08 Å-1 are discerned. The relative locations of these features are also consistent with the previous conclusions from the radial probability distributions that the A50B50 diblock polyampholyte gives rise to a thinner polyelectrolyte brush [q0(A50B50) > q0(A100B10)] and a thicker polyelectrolyte complex [q1(A50B50) < q1(A100B10)]. 4. Summary

Figure 10. Segment-segment partial structure factor Sseg-seg(q) for (a) A100B10 and (b) A50B50 diblock polyampholytes at the charge compensation parameters γ ) 0, 0.2, 0.4, 0.6, 0.8, and 1.0 obtained from MC simulations of the primitive model.

systems. Below, we will restrict ourselves to partial structure factors involving chain segments. Structure factors are proportional to the scattering intensities, which constitute primary results from scattering experiments. In neutron scattering it would be possible to observe the polyampholyte chains only by using contrast-matched solvent. The segment-segment structure factor Sseg,seg(q) is defined as

Sseg,seg(q) )

〈

1

|∑

Nseg

Nseg j)1

|〉

exp(iq‚rj)

2

(15)

where q is a wave vector with magnitude q, rj the position of segment j, Nseg is the total number of segments in the system [Nseg ) Nchain(NA + NB)], and 〈...〉 denotes an ensemble average. In the present system, where all the chains are grafted to the same particle, Sseg,seg(q) becomes the form factor of the polymer brush with the limits Sseg,seg(q) f Nseg as q f 0 and Sseg,seg(q) f 1 as q f ∞. Figure 10 shows Sseg,seg(q) for the two types of chains at various values of γ. Starting with A100B10, we observe that the partial structure factor indeed fulfills the expected limits (Figure 10a). In addition, the characteristic features appearing at q ) q0 ≈ 0.02 Å-1 for γ ) 0 and at q1 ≈ 0.1 Å-1 for γ ) 1 are consistent with the polyelectrolyte brush at γ ) 0 and the polyelectrolyte complex at γ ) 1 both having fairly well-defined extensions and being sphericallike. The feature occurring at q ≈ 1 Å-1, corresponding to a real-space separation of ≈6 Å, comes from the separation of bonded segments. The corresponding Sseg,seg(q) for A50B50 possesses less sharp features in the range q ≈ 0.01-0.1 Å-1 (Figure 10b), suggesting more diffuse edges or larger shape fluctuations of the polyelectrolyte brush and of the polyelectrolyte complex, as also was supported by the radial probability distributions. Nevertheless, weak fea-

The structure of spherical brushes formed by diblock polyampholytes end-grafted onto spherical particles has been considered. The outer block was always kept charged, whereas the charge of the inner one ranged from zero (γ ) 0) to full charge compensation (γ ) 1). The former case constitutes the polyelectrolyte limit and the latter a polyampholyte with zero net charge. Two different polyampholytes have been considered, one asymmetric, with a long outer and a short inner block, and one symmetric in block lengths. The linear charge densities of the outer blocks of the two polyampholytes were the same. The properties of the model systems were investigated by using two theoretical approaches: the primitive model solved by MC simulation and lattice mean-field theory. Chain end-to-end distance and shape-ratio data have been used to characterize the extension and shape of individual chains, and the structure of the polymer brush has been analyzed in terms of segment probability distributions as well as free-end and block-junction probability distributions as a function of the radial distance from the grafting surface. In addition, lateral correlations in the brush have been analyzed, and the overall brush structure has been examined using partial structure factors and illustrated by representative snapshots. From our analysis we found that the structure of the polyampholyte brush depends strongly on the charge of the inner block. In particular, the following were found: (A) In the polyelectrolyte limit (uncharged inner block), the chains are stretched and form a polyelectrolyte brush with the chains separated from each other. (B) In the limit of a polyampholyte with zero net charge, the chains are collapsed and form a polyelectrolyte complex with entangled chains close to the particles. (C) At intermediate charge compensation, stretched and collapsed chains coexist. At this condition, the polymer brush has two substructures: an outer polyelectrolyte brush and an inner polyelectrolyte complex. (D) As compared to the asymmetric diblock polyampholyte, the symmetric one displays several different features related to the weaker electrostatic interactions: (i) less stretched chains in the polyelectrolyte limit and (ii) less compact polyelectrolyte complex in the limit of a polyampholyte with zero net charge. (E) The structures of the polymer brushes, as calculated by the two theoretical approaches, are qualitatively similar. However, the lattice mean-field approach predicts broader segment distributions and less pronounced bimodality. Nevertheless, we suggest the lattice mean-field theory to be an excellent tool to establish an overview of the characteristics of such polymer brushes, which could be complemented with more computer-demanding simulations to focus on specific effects. Acknowledgment. Financial support form the Swedish Research Council (VR) is gratefully acknowledged. LA0490386