Morphological Response of a Spherical Polyelectrolyte Brush to

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Morphological Response of a Spherical Polyelectrolyte Brush to Solvent Quality and Electrostatic Interaction Strength Qing-Hai Hao,*,† Gang Xia,† Bing Miao,*,‡ Hong-Ge Tan,† Xiao-Hui Niu,† and Li-Yan Liu† †

College of Science, Civil Aviation University of China, Tianjin 300300, China College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China

‡

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S Supporting Information *

ABSTRACT: We study surface morphologies of a spherical polyelectrolyte brush in the presence of trivalent counterions using molecular dynamics simulations. Solvent quality and electrostatic interaction strength are varied to generate a series of structures. Through a careful analysis on snapshots of morphologies, shape factor of tethered chains, and monomer−monomer pair correlation function we find a nonmonotonic dependence of surface morphology on electrostatic strength, which represents clearly the electrostatic correlation effect mediated by the multivalent counterions. Due to the very importance of counterions, we further study the correlation effect by classifying counterions into four states, calculating the monomer− counterion pair correlation function and diffusion coefficient of counterions. Our simulation results clearly demonstrate that ordered patterns can be induced by the electrostatic correlation effect in the presence of trivalent counterions, which is absent in the system with monovalent ions. Also, our results can serve as a guide for rational materials design in nanoscale based on spherical polyelectrolyte brushes.

I. INTRODUCTION Polyelectrolyte (PE) brushes are formed by end-tethered PE chains on a planar or curved surface. In water or other polar solvents, the ionic groups on the PE chains can dissociate counterions into solution and leave a charged polymer backbone. The electrostatic interactions between charged units (counterions and ionic groups) play an important role in determining the morphology of the PE brush, which is crucial for its potential applications into a wide range such as colloidal stabilization, biological lubrication, advanced drug delivery, and surface modification protection.1−5 On the other hand, it is known that the structure of the PE brush is strongly influenced by solvent quality, dictating the excluded volume interactions between polymer segments,6,7 which can be tuned through adding organic solvents into aqueous solutions8 or varying temperature. 9 Elucidating how the PE brush morphology is adjusted by the combining effects between electrostatic interaction and solvent quality is an essential goal for the rational design of novel surfaces in nanoscale for widespread applications. There were only a few attempts to investigate the effect of solvent quality (hydrophilic or hydrophobic properties as well as dielectric constant) on the structures and properties of the PE brush.8,10−16 For example, in poor solvent condition for the polymer backbone, the overall swelling of the PE brush is determined by a balance between the osmotic pressure of counterions and the hydrophobic interactions between polymer segments. Theoretical studies have predicted the stretch−collapse transition of the planar PE brush induced by the decrease of solvent quality.10,11 Using X-ray reflectivity and grazing incidence diffraction, Günther et al.12 studied the bundle formation of the vertically oriented chains within planar © XXXX American Chemical Society

PE brushes under poor solvent conditions. By means of a quartz crystal microbalance with dissipation (QCM-D), Wang et al.8 investigated changes in the ion-specific conformational behavior of planar poly(sodium styrenesulfonate) (PSS) brushes as the solvent was changed from water to methanol. With increasing methanol content, the swelling-to-collapse transition of the brush induced by the increase of monovalent salt concentration became less obvious. However, for the multivalent cations, the brush exhibited swelling-to-collapse-toreswelling transitions, which was influenced only weakly by the methanol content. Yamada and co-workers13 reported the preparation of a lipophilic PE brush and its morphology variation induced by immersing in various organic solvents. A series of fibrous structures was identified by the atomic force microscope (AFM) images in the solvents with different dielectric constants. Molecular dynamics (MD) simulations conducted by Dobrynin’s group14,15 considered the formation of lateral morphologies of the PE brush induced by the electrostatic interaction strength under poor solvent conditions, where only the case of monovalent ions was considered. Employing MD simulations, He et al.16 investigated the morphological changes of the planar PE brush in the presence of monovalent counterions. The homogeneous brush, the vertical bundle of chains, and the collapsed brush were observed successively with decreasing temperature (or solvent quality). Quite recently, Tergolina et al.17 reported the effect of dielectric discontinuity on the morphology of a spherical PE brush in a salt-free medium and obtained the Received: July 10, 2018 Revised: September 29, 2018

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II. MODEL AND SIMULATION METHOD A well-tested coarse-grained model14−16,23,27 is used to study the strong spherical PE brush. A group of atoms is modeled as a unit with a diameter σ. The flexible PE chains are created as a bead−spring model, and the degree of polymerization (number of monomers) for each chain is N = 36. Each monomer carries an elementary charge −e, corresponding to the charge fraction f = 1.0. PE chains, M = 23, are uniformly tethered at one end to the surface of a spherical colloid with radius of R = 5.0σ. To achieve electroneutrality of the brush, the number Nc = MNf/3 of trivalent counterions is added to the simulation box. The simulation box has dimensions of Lx × Ly × Lz = 90.0σ × 90.0σ × 90.0σ and periodic boundaries, which is large enough to avoid interactions between the brush and its periodic image. The excluded volume interactions between any two beads are modeled through the truncated-shifted Lennard−Jones (LJ) potential ÅÄÅ ÑÉ l o ÅÅij yz12 ij yz6 i y12 i y6ÑÑÑ o o σ σ σ σ Å j z j z o j z j z Å j z j z ÑÑ o o o 4εLJÅÅÅÅjjjj r zzzz − jjjj r zzzz − jjj r zzz + jjj r zzz ÑÑÑÑ rij ≤ rc ULJ(rij) = m ÅÅk ij { k c{ k c { ÑÑÑÖ o k ij { o ÅÇ o o o o o 0 rij > rc o n

density profiles of monovalent counterions through MD simulations. A different picture arises when replacing monovalent counterions by a concomitant number of multivalent ions.18 Multivalent ions induce a marked collapse of the PE brush layer. On one side, this is due to the depression of osmotic pressure of counterions with three monovalent counterions replaced by one trivalent ion. On the other side, it is due to the strong electrostatic absorption of multivalent counterions to the charged residues of the PE chains, which has been identified by experiments.19,20 Furthermore, the electrostatic correlation effects, which were important for systems with multivalent ions, were shown to be of importance for the observed abrupt collapse.21,22 By means of experiments and coarse-grained MD simulations, the lateral structure inhomogeneities of the planar PE brush induced by multivalent ions were studied.23,24 Brettmann et al.25 conducted a theoretical analysis on an analogous collapse of the PE brush, providing an interpretation based on energy balance for the formation of pinned micelles and cylindrical bundles in the presence of multivalent ions. Very recently, Jackson et al.26 revealed that solvophobic collapsed brushes and multivalent-ion collapsed brushes exhibited a distinct mechanism, demonstrating the distinguishing feature of multivalent “bridging”. Considering the common existence of multivalent ions in industrial formulations and biological systems, the effects of the solvent quality and the strength of electrostatic interaction on the conformational behavior of PE brushes in the presence of multivalent counterions are still far from being fully understood. In this paper, we carry out MD simulations to explore the surface morphologies of spherical PE brushes with trivalent counterions under good, theta, and poor solvent conditions. Our focus is to elucidate the effect of electrostatic interaction strength, in the presence of high valent counterions, on the structuring of the PE brush, which can provide important insights into the understanding of responsive behavior of the brush morphology to the environmental conditions. Through a careful analysis on the snapshots of brush morphologies, the shape factor of tethered chains, and the monomer−monomer pair correlation function, we clearly demonstrate that the electrostatic correlation effect results in a nonmonotonic dependence of the brush morphology on the electrostatic interaction strength, which is absent in the case of monovalent counterion system. While through lateral phase separation ordered lateral patterns of the brush can be induced by the excluded volume interaction with decreasing solvent quality, we demonstrate that, as the manifestation of electrostatic correlation effect, the driving force for the pattern formation can also be the attractive interactions between charged polymer segments mediated by the trivalent counterions which are condensed onto polyion backbones and play the roles of intrachain/interchain bridges. Due to the very importance of high-valent counterions, we carefully study the trivalent counterions in the system by classifying them into four states, calculating the monomer−counterion pair correlation function, and studying their dynamic behavior through calculating the diffusion coefficient. This paper is organized as follows: section II presents the model and simulation method used in this study. In section III, we discuss the main results through a careful analysis on morphological snapshots and different quantities calculated in the simulation. In section IV, we end with a conclusion.

(1)

where rij is the distance between any two particles, εLJ defines the strength of the pair interaction and the depth of the attractive well of the potential, σ is the bead diameter selected to be the same regardless of the bead type, the minimum of the potential is achieved at rij = 21/6σ, and rc is the cutoff distance. Note that the choices of εLJ and rc for the monomer−monomer interactions are used to mimic the solvent quality for the polymer backbone within the implicit solvent,14−16 that is, εLJ = 1.0kBT (kB is the Boltzmann constant and T is the absolute temperature) and rc = 21/6σ, by which the attractive part of the potential is neglected, is for good solvent condition; εLJ = 0.3kBT and rc = 2.5σ is for theta solvent condition; εLJ = 1.0kBT, and rc = 2.5σ is for poor solvent condition. For other pairwise interactions, the εLJ and rc are set as εLJ = 1.0kBT and rc = 21/6σ. The bond distance between the neighboring beads is constrained by the finite extensible nonlinear elastic (FENE) potential developed by Kremer and co-workers28 É ÅÄÅ 2Ñ Å ij rij yz ÑÑÑÑ 1 2 Å Å j z UFENE(rij) = − KR 0 lnÅÅÅ1 − jj zz ÑÑÑ j R 0 z ÑÑ ÅÅ 2 k { ÑÖ ÅÇ (2) where K = 30kBT/σ is the spring constant and R0 = 1.5σ is the maximum bond length. These parameters prevent the bonds from crossing each other and generate a mean bond length of 0.97σ.29 Electrostatic interactions between any two charged beads with charge valences Zi and Zj, separated by a distance rij, are mimicked by the Coulomb potential UCoul(rij) = kBTZiZj

λB rij

(3)

where λB = e /εkBT is the Bjerrum length. In our simulations, the effect of the strength of electrostatic interaction is controlled by varying the value of λB, specifically, λB is changed in a wide range between 0.001 and 10.0σ. The larger λB, the 2

B

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Figure 1. Snapshots of the spherical PE brush in good, theta, and poor solvents with trivalent counterions as a function of the Bjerrum length.

III. RESULTS AND DISCUSSION Figure 1 illustrates snapshots of morphologies of the spherical PE brush at varying solvent conditions and electrostatic strengths, representing the excluded volume effect and the electrostatic effect, respectively. As shown, we observe the morphological evolutions induced by the solvent-mediated segment−segment excluded volume interaction at given electrostatic strengths along the columns. In the weak electrostatic regime of λB < 0.1, phase separation on the spherical surface, resulting in pattern formation, induced by the excluded volume effect is observed from top to bottom along the column. Because the concentration field of PE is defined on a spherical surface, patterns can be classified by the spherical harmonics Yml (θ,ϕ)describing the polar and azimuthal ordering. It is found that with increasing the electrostatic strength l increases, indicating a phase separation at a smaller length scale; in particular, one observes that the morphology transforms from l = 1 to l = 2 at around λB = 0.01 and from l = 2 to l = 3 at around λB = 0.05. The snapshot at the boundary value of λB = 0.1 appears as an intermediate crossover morphology between ordered pattern and homogeneous micelle type. In the regime of 0.1 < λB ≤ 0.5, phase separation induced by the excluded volume effect almost disappears along the column at given λB. Instead, the PE brush appears as a micelle-type morphology irrespective of the solvent condition. The reason underlying the observation that excluded-volume-effect-induced phase separation occurs in a smaller length scale, then disappears, with the increase of electrostatic strength is that, being repulsive interaction for λB < 1.0, the electrostatic interaction counterbalances the attractive interaction between polymer segments induced by the poor solvent condition, eventually leading to the remelting of the poor-solvent-induced collapsed morphology when the electrostatic repulsion gets large enough to render a net repulsive segment−segment interaction. In the regime of λB > 1.0, PE brushes collapse in all solvent conditions. This is the representation of electrostaticcorrelation-induced phase separation of PE, which is beyond the description of mean field theory.32 Specifically, an attractive interaction between polymer segments is mediated by the trivalent counterions under a large λB, which results in the collapse of PE brush in all solvent conditions. Furthermore, it is worthwhile to stress that the surface morphology of the spherical PE brush depends on the electrostatic effect nonmonotonically at given solvent conditions in the presence of trivalent counterions, as observed in

stronger the electrostatic interaction. The calculations of the Coulomb interaction are divided into two parts by introducing the cutoff distance, which is specified as re = 10.0σ, that is, the pairwise electrostatic interactions within re are computed directly. Also, electrostatic interactions outside re are calculated in the reciprocal space, namely, the particle−particle particle− mesh (PPPM) algorithm30 with a sixth-order charge interpolation scheme, and an estimated accuracy of 10−3 is adopted for calculations of the long-range part of Coulomb potential. The open source software LAMMPS31 is used to carry out the MD simulations. Simulations are performed in a canonical ensemble. The constant temperature is maintained by coupling the system to a Langevin thermostat.30,31 The Langevin equation for the motion of bead i holds m

d2ri(t ) dt

2

= −∇Ui − ξ

dri(t ) + FiR (t ) dt

(4)

where m is the mass of the bead i, and it is the same regardless of the bead type. Ui denotes the total potential of the system, that is, Ui = ULJ + UFENE + UCoul. ξ and FiR(t) are friction coefficient and stochastic force of the solvent, respectively. The friction coefficient is selected as ξ = 0.143τ−1, where τ is the standard LJ time τ = σ(m/εLJ)1/2. The stochastic force FiR(t) is assumed to have a zero average value and satisfies the fluctuation−dissipation theorem ⟨FiR(t)·FjR(t′)⟩ = 6mξkBTδijδ(t − t′). The velocity−Verlet algorithm with time step Δt = 0.005τ is applied for integrating the equation of motion. The temperature in our simulations is kept as T = 1.0εLJ/kB. The LJ units are used throughout this paper, that is, all quantities are unitless based on the fundamental quantities, mass (m), distance (σ), and energy (εLJ). Simulations are carried out in the following procedure. Initially, the grafted colloid is fixed at the center of the simulation box. Also, the tethered chains in the fully extended conformation stretch along the radial direction. The trivalent counterions are randomly inserted into the simulation box. After the initial configurations are prepared, each system is preequilibrated for 2 × 106 steps. Then the production run lasts 3.0 × 106 steps. During the production run, one configuration is sampled every 2.0 × 103 steps and 1501 configurations are saved to collect data for analyzing the spherical PE brush properties in our program. C

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length at different solvent conditions. The nonmonotonic dependence is clearly illustrated in this plot. Specifically, we can identify four regimes of λB as indicated by the vertical dashed lines in Figure 2. In the first regime of λB ≤ 0.01, the electrostatic strength is weak and the excluded volume effect dominates, leading to more extended chain conformation with increasing solvent quality. Note that compared to a clear increasing of shape factor for good and theta solvent conditions, shape factor is almost not responsive to the increase of λB in the poor solvent case, which is due to, as shown in Figure 1, the fact that while the morphology of the PE brush in good and theta solvents is micelle-like with coil conformations of chains whose sizes are responsive to Coulomb repulsions, the corresponding morphology in poor solvents is phase-separation-induced pinned patch with globule-like conformations of chains whose global sizes are weakly responsive to Coulomb repulsions. In the second regime of 0.01 < λB ≤ 0.25, a distinguished increase of shape factor, indicating expansion of chains, occurs for all solvent conditions. The responsive behavior for the poor solvent case is attributed to the fact that a sequence of transitions from low l to high l spherical pattern is induced by Coulomb repulsions, as demonstrated in Figure 1 in the third row, leading to the increase of shape factor. In the third regime of 0.25 < λB ≤ 2.5, shape factor decreases with increasing λB. This clearly represents the electrostatic correlation effect mediated by trivalent counterions, as clarified above, namely, the counterion condensation and the intrachain/interchain bridging by condensed ions induce the collapse of the brush. On further increasing the Bjerrum length to the fourth regime of λB ≥ 2.5, a slight fluctuation of shape factor with increasing λB is observed for good and theta solvent conditions, while the shape factor increases for poor solvent condition. The corresponding morphologies in this regime are demonstrated in Figure 1. The increasing trend of shape factor for λB ≥ 2.5 is attributed to the variation of the states of counterions in the system, specifically, more condensed counterions play the role of interchain bridging leading to an expansion of the collapsed chains. This will be further clarified by studying the states of counterions in the following. In Figure S2 of the Supporting Information, the corresponding shape factor plot in the case of divalent counterions is presented. It is observed that as found from comparing Figures 1 and S1 for snapshots of morphologies, different regimes of shape factor responsive to the electrostatic strength shift to higher values of λB for the divalent case compared to that of the trivalent case due to weaker electrostatic coupling strength in the presence of divalent counterions. Since the electrostatic correlation effect is mediated by counterions of high valence in the system, we further study the states of trivalent counterions by classifying them into four states, characterized by f free, f NNC, f IAB, and f IEB, corresponding to the fractions of free, nearest neighbor condensation, intrachain bridging, and interchain bridging counterions, respectively. This has been demonstrated schematically in Figure 3. The fractions are defined as ratios of the number of trivalent counterions that are in the corresponding state to the total number of counterions, which are quantified in terms of a topological distance parameter α. The topological distance parameter α is defined as23

our simulation. This is the manifestation of the strong electrostatic correlation effect existing in an electrostatic system with multivalent counterions. In the cases of good and theta solvent conditions, increasing the electrostatic strength, the spherical PE brush is swollen up under the action of repulsive Coulomb interactions between polymer segments for small λB < 0.5; it is collapsed for large λB > 0.5 due to the counterion condensation and then the trivalentcounterion-mediated attractive segment−segment interaction. In the case of poor solvent condition, at weak electrostatic strength, the surface morphology starts with the pinned patches resulted from poor-solvent-induced spherical-surface lateral phase separation with a small l, transforms to the pinned bundles corresponding to phase separation with larger l driven by electrostatic repulsions; when λB > 0.5, electrostatic correlations due to the trivalent counterions result in lateral phase separation, leading to the reformation of pinned patches of the surface morphology. We note that this observed nonmonotonic dependence of surface PE morphology on the electrostatic strength is absent in the system with monovalent counterions,15 where electrostatic coupling, which is determined by both the Bjerrum length and the valence of ions, is not large enough to manifest the electrostatic correlation effect. In Figure S1 of the Supporting Information, we also present the snapshots of the spherical PE brush at different solvent conditions and electrostatic strengths in the presence of divalent counterions. Similar trends for the morphological response to solvent conditions and electrostatic interaction strengths are observed as compared to the case of trivalent counterions. In particular, the electrostatic correlation mediated by the divalent counterions in the strong electrostatic coupling of λB > 1.0 induces the lateral inhomogeneous morphologies (patches) in all solvent conditions. The difference from the trivalent case is that, at a given solvent condition, with increasing electrostatic strength the morphology transition points shift to larger values of λB, which is due to the relatively weaker electrostatic coupling strength for the divalent counterions compared to the trivalent case. The shape factor, defined as ⟨Re2⟩/⟨Rg2⟩, is a quantity to capture the global structure of polymer chains. Here, Re and Rg are the end-to-end distance and the radius of gyration of a grafted chain, respectively. It is known that the larger the shape factor is, the more extended the chains are. In Figure 2, to further demonstrate the electrostatic effect on the spherical PE structures, we plot the shape factor with respect to the Bjerrum

Figure 2. Shape factor of tethered PE chains versus the Bjerrum length for different solvent conditions in the presence of trivalent counterions. Filled square, filled circle, and filled triangle indicate good, theta, poor solvent conditions, respectively. Vertical dashed lines are used to discriminate the different regimes of the profiles. D

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Figure 3. Schematic for classifying trivalent counterions into four states, that is, the free state, the nearest neighbor condensation state, the intrachain bridging state, and the interchain bridging state. Green bead represents the grafting colloid. Lavender beads correspond to the monomers. Red beads represent the trivalent counterions.

α=

1 len({NL})

Figure 4. Four fractions of trivalent counterions as a function of the Bjerrum length in good (filled square), theta (filled circle), and poor (filled triangle) solvents. (a) Fractions of free f free, (b) fractions of nearest neighbor condensation f NNC, (c) fractions of intrachain bridging f IAB, and (d) fractions of interchain bridging f IEB.

{NL}

∑ (i − i

condensation. On further increasing λB, condensed counterions appear as three states as shown in Figure 4 b−d. While the role of nearest neighbor condensed counterions is mainly to renormalize the polyion charge to a smaller effective value, the intrachain and interchain bridging counterions induce an effective attractive interaction between polymer segments, manifesting the electrostatic correlation effect and resulting in the lateral phase separation of the PE brush. To better characterize the variation of counterions’ state with increasing λB at different solvent qualities, we list data of fractions of counterions in Table 1. It is shown that at given λB, f free,good > f free,theta > f free,poor, indicating that the poorer the solvent the more condensed counterions. Because chain conformation is more compact with decreasing solvent quality, this observation is consistent with the previous result that a more compact conformation of charged chain leads to more counterions condensed.35 For λB > 5.0, especially at λB = 10.0, with f free ≈ 0, almost all counterions are condensed. Also, at λB > 5.0, for the condensed counterions, with increasing λB, f NNC and f IAB decreases while f IEB increases. This is due to the fact that under the action of condensed counterions chains are collapsed compared to the case of weak electrostatic strength, as can be observed in Figure 2 for the shape factor. The collapsing leads to an effective higher grafting density away from the tethered spherical surface, resulting in a shorter interchain distance, which then provides a favorable environment for multivalent counterions to be interchain bridging. Since the total condensed counterions are fixed to be almost 1.0 at this strong electrostatic coupling regime, the increasing f IEB leads to the decreasing of f NNC and f IAB. Furthermore, in combination with Figure 4, it can be found that a different state of condensed counterions leads to a different level of chain collapse, represented by the decrease of shape factor. Specifically, counterions of the nearest neighbor condensation and the intrachain bridging induce more chain collapse, while counterions of the interchain bridging induce less. Thus, with a larger value of f IEB, the chain tends to be more expanded relatively resulting in a larger value of shape factor. This explains the observation that the shape factor rises up at λB > 2.5 because f IEB begins to have a larger value indicating the more important role played by the interchain

i ̅ )2 (5)

where len({NL}) denotes the length of the neighbor list of charged monomers for one trivalent counterion, i is the topological index of the monomer, and i̅ is the average topological index of the neighbor list. The topological index i of the monomer is assigned as follows: because the brush consists of M tethered chains and each chain contains N monomers, the integer i is assigned for each monomer along one chain as (1, 2, ..., N), then the neighboring chain is specified as (N + 1, N + 2, ..., 2N), and the process is continued until all monomers are uniquely enumerated. For each MD trajectory snapshot, we search for all monomers within a cutoff distance (Rc) of the position of the counterion and form a neighbor list ({NL}) (noted as topological indices) of all monomers within Rc. For calculating the parameter α, a cutoff distance Rc = 21/6 is taken for constructing the neighbor list of each trivalent counterion, which corresponds to the cutoff distance of the pairwise interaction between segments and counterions. Clearly, the sum of all fractions is 1.0. Note that the length of the neighbor list equals 0, 0 < α ≤ 0.816, 0.816 < α ≤ 5.0, and α > 5.0 corresponding to free, nearest neighbor condensation, intrachain bridging, and interchain bridging counterions, respectively.23 The dependences of fractions of counterion states on the electrostatic interaction strength, at different solvent conditions, are plotted in Figure 4. Due to the competition between the translational entropy of counterions and the polyion−counterion electrostatic attraction, a fraction of counterions is condensed onto the polyions and leads to a renormalized linear charge density of polyions, which is called the counterion condensation. Although it is only well defined for a rod PE solution in the infinitely dilute limit, we can calculate a rough value for the onset of counterion condensation for the present system according to Oosawa− Manning theory33−35 by which the onset λB* = 1/Z = 0.33 for the counterion valence Z = 3. This value is approximately consistent with that observed from Figure 4a for the fraction of free counterions, where f free begins to decrease from 1.0 at around λB = 0.3, indicating the setting in of counterion E

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Macromolecules Table 1. Fractions of Trivalent Counterions in the Four States for λB ≥ 1.0 f free

f NNC

f IAB

f IEB

λB

good

theta

poor

good

theta

poor

good

theta

poor

good

theta

poor

1.0 2.5 5.0 10.0

0.637 0.181 0.016 0.001

0.606 0.143 0.009 0.000

0.456 0.078 0.005 0.000

0.310 0.450 0.170 0.012

0.323 0.399 0.119 0.008

0.361 0.269 0.075 0.010

0.048 0.233 0.356 0.265

0.062 0.243 0.300 0.205

0.102 0.211 0.227 0.179

0.005 0.137 0.458 0.721

0.010 0.215 0.572 0.787

0.081 0.442 0.693 0.811

monomers with increasing λB decrease the ordering of the brush layer. For λB = 10.0 in Figure 5d, the long-range structural order in the PCF becomes pronounced, which corresponds to the electrostatic correlation induced pinned patch morphologies. Note that the PCF characterizing the electrostatic-correlation-induced order in Figure 5d resembles that characterizing the excluded-volume-interaction-induced order in poor solvent condition in Figure 5a; however, different from Figure 5a, the electrostatic-correlation-induced order in Figure 5d exists for all three solvent conditions. It is worthwhile to stress that ordered structures could be induced by the electrostatic correlation effect even in the good solvent condition through increasing the electrostatic strength with the presence of multivalent counterions. Furthermore, we find that the structure induced by the electrostatic correlation (see open square in Figure 5d) is more ordered than that by the solvophobic interaction (see open triangle in Figure 5a), which is demonstrated as the higher intrachain correlation peak and the more pronounced subpeaks. The comparison is more clearly demonstrated in the inset of Figure 5d, where the monomer−monomer PCF for the poor solvent case in Figure 5a (open triangle) and that for multivalent case in Figure 5d (open square) are plotted together. One can find that (1) the amplitude of the primary peak, at r = 1.0, is larger for the multivalent case (red curve) than the poor solvent case (blue curve), (2) the widths of the peaks are narrower in the multivalent case than the poor solvent case, indicating less fluctuations and more orders in the multivalent case, and (3) the secondary and higher order peaks, as the green arrows show, are more distinct in the case of the multivalent counterions, indicating that the multivalent counterionsinduced order is longer ranged than that of the poor solvent induced order. These observations point to our claim that the multivalent case induces more ordered structures. Meanwhile, one observes that the global amplitude of the correlation profile is larger in the poor solvent case than in the multivalent case. This indicates that the lateral heterogeneity occurs in a larger length scale for the poor solvent case, leading to a larger global scattering. Combining together, compared to the poor solvent case, the order induced by the multivalent counterions are (1) more heterogeneous laterally with smaller characteristic length scale and (2) longer ranged. The similar trends for solvophobic- and multivalent-counterions-induced order have also been reported26 for the planar PE brushes. The monomer-trivalent counterion PCFs under a variety of λB in good, theta and poor solvent conditions are presented in Figure 6. It is expected that there exists no correlation between charged monomers and trivalent counterions in the case of λB = 0.001, which is in fact demonstrated in Figure 6a. With increasing the electrostatic interaction strength, a pronounced peak is developed in the gmc profile shown in Figure 6b and 6c. Moreover, the peak is more pronounced with the decrease of solvent quality for a fixed λB. The width of the peak gets narrower as λB changes from 0.05 to 1.0, which means a

bridging of counterions. In particular, for the poor solvent case, at λB = 2.5, already f IEB is the largest fraction; with increasing, λB > 2.5, chains are more expanded due to the increase of f IEB. The reason that counterions of interchain bridging render relatively less chain collapse requires further careful investigations. We hypothesize that this may be due to the fact that lateral heterogeneities of surface morphology are mostly induced by the interchain bridging counterions which collapse chains in a binding-together fashion. Then the lateral heterogeneity leads to a more expanded chain and larger value of shape factor. To investigate the local structuring of the PE brush, we calculate the pair correlation function (PCF). The monomer− monomer PCF with different Bjerrum lengths in good, theta, and poor solvent solutions are illustrated in Figure 5. For

Figure 5. Monomer−monomer pair correlation functions for different electrostatic strengths in good (open square), theta (open circle), and poor (open triangle) solvents: (a) λB = 0.001, (b) λB = 0.05, (c) λB = 1.0, and (d) λB = 10.0. Inset of d replots monomer−monomer pair correlation functions in poor (λB = 0.001) and good (λB = 10.0) solvents for a direct comparison. Inset blue curve and red curve correspond to the open triangle in a and the open square in d, respectively. Long-range order peaks of the inset profiles are marked with green arrows.

clarity, only four different Bjerrum lengths are provided, corresponding to the four regimes defined in Figure 2. For λB = 0.001 (Figure 5a), solvent quality is the dominant factor determining the phase behavior of the brush. Representing the pinned patch formed in the poor solvent condition, the PCF has both the intrachain and the interchain pronounced correlation peaks as shown in Figure 5a (open triangle). On increasing solvent quality, the chain adopts a relatively extended conformation in theta and good solvents, leading to only one intrachain correlation peak in Figure 5a (open circle and square). With the increase of electrostatic interaction strength, both the intrachain and the interchain correlations are weakened regardless of solvent quality in Figure 5b and 5c. The enhanced electrostatic repulsions between charged F

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ies.19,37,38 In refs 19 and 37, through experiments and simulations for a spherical PE brush in the presence of multivalent counterions, the mobility of trivalent counterions has been demonstrated to be significantly reduced with increasing the electrostatic strength due to the enhancing binding to the PE backbone. In ref 38, a similar dependence of counterion diffusional motion on the electrostatic strength for the dendrimer PE solution has also been observed. Overall, the electrostatic strength plays a dominant role in the dynamic properties of counterions and the effect of solvent quality is weak. The result for the dependence of diffusion behavior of counterions on the electrostatic strength is consistent with that for the monomer−counterion correlation behavior demonstrated in Figure 6, namely, with stronger electrostatic strength, counterions are driven to a more ordered state with smaller fluctuation and diffusion.

Figure 6. Monomer−counterion pair correlation functions for different electrostatic strengths in good (open square), theta (open circle), and poor (open triangle) solvents: (a) λB = 0.001, (b) λB = 0.05, (c) λB = 1.0, and (d) λB = 10.0.

IV. CONCLUSIONS On the basis of a coarse-grained model, the electrostatic effect on the surface morphologies of a spherical PE brush in the presence of trivalent counterions under good, theta, and poor solvent conditions is investigated systematically by means of MD simulations. Varying the Bjerrum length, the spherical PE brush can form a variety of surface morphologies, including pinned patches, pinned bundles, and micelle-like brushes. The electrostatic correlation effect, which sets in for strong electrostatic strengths in the presence of multivalent counterions, renders the PE brush to demonstrate a nonmonotonic dependence on the electrostatic strength, which is absent for the system with monovalent counterions. The shape factor, which captures the global structure of the PE brush, demonstrates four different regimes in terms of the Bjerrum length, corresponding to different responsive behaviors to the electrostatic strength. For the local structuring, the monomer− monomer PCF is calculated to reveal the electrostaticcorrelation-induced order in the presence of multivalent counterions, even in the good solvent condition. It is worthwhile to discuss the difference between the poorsolvent-induced and the multivalent counterions-mediated electrostatic correlation-induced order. On one hand, based on the idea that both the solvophobic- and the multivalent counterions-induced formation of lateral structures rely on an effective attraction between polymer monomers, following the treatment for the case of poor solvent, a theoretical model25 has been developed to study the multivalent counterionsinduced lateral order for the planar PE brush by introducing an effective, multivalent counterions-induced negative second viral coefficient of the monomer−monomer interaction. On the other hand, the above treatment may need to be improved, because the physical origins of the solvophobic- and the multivalent counterions-induced lateral structure formation are different subtly, as recently pointed out.26 From the mechanism point of view, from previous studies it has been shown that the multivalent counterions-induced lateral structure is due not only to the effective monomer−monomer attraction as in the case of poor solvent but also to the “bridging” effect, where the multivalent counterions act as a nucleating site to bridge intra- and interchain monomers, collectively leading to structure formation. From the responsive property point of view, the lateral structures induced by the multivalent counterions are more ordered than those induced by the poor solvents, as found previously26 and in the present work, which is attributed to the role of

stronger order is created by increasing the electrostatic strength. For the largest electrostatic strength λB = 10.0, both the nearest neighbor correlation and the long-range correlation become pronounced regardless of solvent quality, which is due to the strong electrostatic correlation effect in this parameter. Besides the equilibrium quantities, we further study the electrostatic effect on the dynamic behavior of counterions through calculating the diffusion coefficient D of counterions, which is defined by the Einstein expression36 D = lim

t →∞

1 6Nct

Nc

∑ [ri(t ) − ri(0)]2 i=1

(6)

where ri(t) is the position vector of the i-th counterion at time t. The diffusion coefficients D of the counterions as a function of λB in good, theta, and poor solvents are plotted in Figure 7.

Figure 7. Diffusion coefficient of counterions for different electrostatic interaction strengths in good (filled square), theta (filled circle), and poor (filled triangle) solvents.

It is observed that the diffusion coefficient of trivalent counterions remains a large value regardless of solvent quality for λB ≤ 0.01, which is attributed to the weak electrostatic interaction between counterions and polyion monomers. With increasing electrostatic strength, the diffusion coefficient decreases sharply because of the counterion condensation. Further increasing to λB > 1.0, almost all of the trivalent counterions are confined around the charged monomers and the diffusion coefficient approaches zero. The observed dependence of the diffusion coefficient of counterions on the electrostatic strength is consistent with the previous studG

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(7) Willott, J. D.; Murdoch, T. J.; Webber, G. B.; Wanless, E. J. Physicochemical Behaviour of Cationic Polyelectrolyte Brushes. Prog. Polym. Sci. 2017, 64, 52−75. (8) Wang, T.; Long, Y. C.; Liu, L. D.; Wang, X. W.; Craig, V. S. J.; Zhang, G. Z.; Liu, G. M. Cation-Specific Conformational Behavior of Polyelectrolyte Brushes: From Aqueous to Nonaqueous Solvent. Langmuir 2014, 30, 12850−12859. (9) Albright, P. S.; Gosting, L. J. Dielectric Constants of the Methanol-Water System from 5 to 55 Degrees. J. Am. Chem. Soc. 1946, 68, 1061−1063. (10) Borisov, O. V.; Birshtein, T. M.; Zhulina, E. B. Collapse of Grafted Polyelectrolyte Layer. J. Phys. II 1991, 1, 521−526. (11) von Goeler, F.; Muthukumar, M. Stretch-Collapse Transition of Polyelectrolyte Brushes in a Poor Solvent. J. Chem. Phys. 1996, 105, 11335−11346. (12) Günther, J. U.; Ahrens, H.; Forster, S.; Helm, C. A. Bundle Formation in Polyelectrolyte Brushes. Phys. Rev. Lett. 2008, 101, 258303. (13) Yamada, T.; Kokado, K.; Higaki, Y.; Takahara, A.; Sada, K. Preparation and Morphology Variation of Lipophilic Polyelectrolyte Brush Functioning in Nonpolar Solvents. Chem. Lett. 2014, 43, 1300− 1302. (14) Sandberg, D. J.; Carrillo, J. M. Y.; Dobrynin, A. V. Molecular Dynamics Simulations of Polyelectrolyte Brushes: From Single Chains to Bundles of Chains. Langmuir 2007, 23, 12716−12728. (15) Carrillo, J. M. Y.; Dobrynin, A. V. Morphologies of Planar Polyelectrolyte Brushes in a Poor Solvent: Molecular Dynamics Simulations and Scaling Analysis. Langmuir 2009, 25, 13158. (16) He, G. L.; Merlitz, H.; Sommer, J. U. Molecular Dynamics Simulations of Polyelectrolyte Brushes under Poor Solvent Conditions: Origins of Bundle Formation. J. Chem. Phys. 2014, 140, 104911. (17) Tergolina, V. B.; dos Santos, A. P. Effect of Dielectric Discontinuity on a Spherical Polyelectrolyte Brush. J. Chem. Phys. 2017, 147, 114103. (18) Jusufi, A.; Borisov, O.; Ballauff, M. Structure Formation in Polyelectrolytes Induced by Multivalent Ions. Polymer 2013, 54, 2028−2035. (19) Mei, Y.; Lauterbach, K.; Hoffmann, M.; Borisov, O.; Ballauff, M.; Jusufi, A. Collapse of Spherical Polyelectrolyte Brushes in the Presence of Multivalent Counterions. Phys. Rev. Lett. 2006, 97, 158301. (20) Farina, R.; Laugel, N.; Yu, J.; Tirrell, M. Reversible Adhesion with Polyelectrolyte Brushes Tailored via the Uptake and Release of Trivalent Lanthanum Ions. J. Phys. Chem. C 2015, 119, 14805−14813. (21) Brettmann, B. K.; Laugel, N.; Hoffmann, N.; Pincus, P.; Tirrell, M. Bridging Contributions to Polyelectrolyte Brush Collapse in Multivalent Salt Solutions. J. Polym. Sci., Part A: Polym. Chem. 2016, 54, 284−291. (22) Ezhova, A.; Huber, K. Contraction and Coagulation of Spherical Polyelectrolyte Brushes in the Presence of Ag+, Mg2+, and Ca2+ Cations. Macromolecules 2016, 49, 7460−7468. (23) Yu, J.; Jackson, N. E.; Xu, X.; Brettmann, B. K.; Ruths, M.; de Pablo, J. J.; Tirrell, M. Multivalent Ions Induce Lateral Structural Inhomogeneities in Polyelectrolyte Brushes. Sci. Adv. 2017, 3, 1497. (24) Yu, J.; Jackson, N. E.; Xu, X.; Morgenstern, Y.; Kaufman, Y.; Ruths, M.; de Pablo, J. J.; Tirrell, M. Multivalent counterions diminish the lubricity of polyelectrolyte brushes. Science 2018, 360, 1434− 1438. (25) Brettmann, B.; Pincus, P.; Tirrell, M. Lateral Structure Formation in Polyelectrolyte Brushes Induced by Multivalent Ions. Macromolecules 2017, 50, 1225−1235. (26) Jackson, N. E.; Brettmann, B. K.; Vishwanath, V.; Tirrell, M.; de Pablo, J. J. Comparing Solvophobic and Multivalent Induced Collapse in Polyelectrolyte Brushes. ACS Macro Lett. 2017, 6, 155−160. (27) Liu, L.; Pincus, P. A.; Hyeon, C. Heterogeneous Morphology and Dynamics of Polyelectrolyte Brush Condensates in Trivalent Counterion Solution. Macromolecules 2017, 50, 1579−1588.

nucleating sites played by the bridging multivalent counterions in the brush layer. With increasing electrostatic strength, counterions tend to locate in an ordered state with smaller fluctuations, which is demonstrated by investigating the monomer−counterion PCF and the diffusion coefficient of the counterions. Our systematic analysis in this work provides a clear manifestation of electrostatic correlation effect in a PE-multivalent counterions system, which is beyond the mean field model. On the technological side, our results can serve as a guide for the rational materials design in nanoscale based on the responsive properties of PE systems.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b01466. Snapshots of the spherical PE brush in good, theta, and poor solvents with divalent counterions as a function of the Bjerrum length; shape factor of tethered PE chains versus the Bjerrum length for different solvent conditions in the presence of divalent counterions (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (Q.H.H.). *E-mail: [email protected] (B.M.). ORCID

Qing-Hai Hao: 0000-0001-9506-5883 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful for the financial support provided by the National Natural Science Foundation of China (NSFC) (Grant Nos. 21544007 and 21774131) and the Fundamental Research Funds for the Central Universities (Grant No. 3122018L007).

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REFERENCES

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