Polyelectrolyte brushes in monovalent and multivalent salt solutions

Polymer xxx (2014) 1e13

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Polyelectrolyte brushes in monovalent and multivalent salt solutions Vijeth Sathyanarayana Guptha, Pai-Yi Hsiao* Department of Engineering and System Science, National Tsing Hua University, Hsinchu, Taiwan, ROC

a r t i c l e i n f o

a b s t r a c t

Article history: Received 4 January 2014 Received in revised form 28 March 2014 Accepted 20 April 2014 Available online xxx

Langevin dynamics simulations are performed to study polyelectrolyte brushes by varying the chain grafting density sg, the added salt valence Z, and concentration Cs. The brush thickness H follows a scaling relation in monovalent salt solutions as Cs
0:15 , decreasing more slowly than the scaling prediction. In divalent and trivalent salt solutions, H acquires a minimum value at the stoichiometric point, and then levels off and slightly increases, respectively. The shape and orientation of the chains change with Cs accordingly. An electric double layer is formed around the brush rim, revealed by the calculation of vertical charge distribution. Increasing sg, Z, Cs reduces the width of the layer. Side-view density study further shows how the three parameters transform the brush profile from an individual-chain distribution to a homogeneous one. Finally, about 90% of chain charges are neutralized by the ions trapped in the brushes. It is found that ion condensation on the chains can be described by Manning’s theory in sparse grafting and dilute salt conditions. Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: Polyelectrolyte brushes Density distribution Charge inversion

1. Introduction A polyelectrolyte brush is a system consisting of polyelectrolyte chains grafted on a surface [1,2]. In solutions, the chains are ionized and release counterions. Owing to long-range electrostatic interaction and large degree of freedom of ions and chains, polyelectrolyte brushes exhibit many appealing properties different from neutral polymer brushes and have received considerable attention in the past decades [3]. Technological and scientific importance includes using polyelectrolyte brushes to stabilize colloidal dispersions, to protect vulnerable surfaces, to form lubrication layers, to develop smart nanoactuators, to interact with biomacromolecules, to model biopolymers on membranes, and so on [4e9]. Recently, immense interests have been arisen in design of smart nanochannels using polyelectrolyte brushes, which respond to well-controlled stimulus like temperature, electric field, pH value, and ion concentration [7,10e14]. Polyelectrolyte brush systems are generally classified into two classes: quenched polyelectrolyte brushes and annealed polyelectrolyte brushes [2,15]. In the quenched systems, the positions of ionized monomers are fixed on the chains while in the annealed ones, the ionized monomers dynamically change the positions and

* Corresponding author. E-mail address: [email protected] (P.-Y. Hsiao).

the numbers with the variation of local environmental conditions such as temperature, pH value, and ion concentration [5,7,16,17]. Theoretical study of polyelectrolyte brushes is paved by the pioneering works of Pincus [18] and Borisov et al. [19] For quenched polyelectrolyte brushes, two scaling regimes have been identified when monovalent salt is added in the solutions [20,21]. The first regime is called the osmotic brush regime. It occurs when the number of added salt ions is few and does not alter the distribution of intrinsic counterions in the brush. The equilibrium brush thickness H is determined by the balance of the elastic pressure of brush chains and the osmotic pressure of counterions. Thus, H scales with the degree of polymerization N and the degree of counterion dissociation f, as H w N$f1/2. In this regime, H is independent of chain grafting density sg [18,19]. The second regime is called the salted brush regime, in which the added salt concentration Cs is high enough to modify the brush structure and the ion distributions. The 1=3
1=3 brush thickness thus scales as HwN$f 2=3 $sg $Cs [18]. For annealed brushes, a non-monotonic behavior was predicted [22,23]. When salt concentration is low, the brush thickness increases with Cs but decreases with sg. In high salt conditions, the brushes display a maximal thickness, determined by the pH value, and beyond the maximal point, behave as quenched systems with increasing Cs. Experiments have verified the two scaling regimes for quenched polyelectrolyte brushes [24e26] and the behaviors for annealed brushes [4,27]. A more satisfactory understanding for polyelectrolyte brushes needs to account for certain facts ignored in the scaling theories,

http://dx.doi.org/10.1016/j.polymer.2014.04.035 0032-3861/Ó 2014 Elsevier Ltd. All rights reserved.

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such as excluded volume interaction in brushes [28,29] and nonlinear elasticity of chains [30,31]. Self-consistent-field theory (SCFT) is an alternative to study polyelectrolyte brushes beyond the scaling theories. It is widely used in calculating brush profiles in different ionic strength and solvent conditions [32,33]. Recently, non-local density functional theory (NLDFT) has been applied to investigate polyelectrolyte systems [34,35]. Compared to SCFT, NLDFT includes thoroughly the short-range excluded-volume effect, the long-range electrostatic interactions, and the intra-chain correlations due to chain connectivity [36]. Using the method, Jiang and Wu reported that the brush thickness shows a powere law relationship with the salt concentration in the salted brush regime but the exponent depends on the valence of salt, which are
0.3,
0.19,
0.11 for monovalent, divalent, trivalent salts, respectively [37]. Computer simulations allow us to study the static and dynamic properties of molecular systems in considering the discreteness of constituent particles [38,39]. Using Monte Carlo simulations, profile of polyelectrolyte brushes has been verified with SCFT calculations [40,41]. Dissipative particle dynamics simulations have been used to study electrostatic interactions in charged brushes [42,43], flow controls in nanofluidic channels grafting polyelectrolyte brushes [44], shearing and frictions between two brush systems [45e47]. The most commonly-used method is probably the molecular dynamics simulation incorporating with Langevin thermostat, which is also called Langevin dynamics (LD) simulations. LD simulations have studied polyelectrolyte brushes in different chain flexibilities [48e50], charge fractions and sequences [28,51,52], functional groups [53,54], and solvent qualities [55]. Interactions between polyelectrolyte brushes and nanoparticles have been investigated [49,56e60]. It has also been shown how to control brush thickness through tuning temperature [54,61] and applying external electric fields [62e64] The structure of chains and the distribution of ions have been found to be significantly altered by addition of multivalent salt in the brush systems [1,58,65,66]. However, to simulate polyelectrolyte brushes with added salt remains a non-trivial task because we have to deal with the longrange electrostatic interaction and a large number of ions in the systems, both of which demand large computing resources. Recently, molecular dynamics simulations have revealed that the quenched brush thickness scales in a weaker manner as HwCs
0:15 in the salted brush regime [65] rather than with the wellknown
1/3-exponent. Chen et al. [67] have developed a Florytype theory to explain the phenomena, which accounts for the excluded volume and finite chain extensibility. They pointed out that the brush thickness cannot decrease continuously without restriction and should level off at some moment in the high salt region. As a consequence, H does not scale directly with Cs, but as 1=3 Hwsg ðv0 þ ðAf 2 =4Cs ÞÞ1=3 where v0 is the excluded volume of monomer and A is a fitting parameter to account for the ignored correlation effect. When multivalent salt is added into polyelectrolyte solutions, multivalent counterions interact strongly with the chains. These systems reveal many interesting phenomena such as ion condensation, chain collapsing and charge over-compensation [68e72], which cannot be simply explained by mean-field theory [73e76]. For polyelectrolyte brushes, the chains are end-tethered and the excluded volume effect and the osmotic pressure of counterions in the brushes prevent the chains from complete collapsing, which renders the behavior of the systems quite unpredictable. The grafting density is also a key factor to affect the brush height. Readers can refer to a recent review paper about multivalent ioninduced structural formation of spherical polyelectrolyte brushes for more information [77]. Despite several efforts [43,65,78e80], research reports on systematic study of polyelectrolyte brushes in

different salt concentrations, valences and chain grafting densities are still not many in literature. Molecular simulations provide a rigorous way to verify the scaling theories for brush thickness and to study brush properties properly, using well-defined mathematical models. This motivates us to perform the current study to investigate these important parameters in polyelectrolyte brush systems. The rest of the paper is organized as follows. The model and the setup are described in Section 2. The results and discussions are presented in Section 3. The studied topics include the brush thickness, the chain structure, and the chain orientation in different salt valences, concentrations, and chain grafting densities. The distributions of chains and ions and the vertical charge profile of the brushes are investigated. The side-view density of monomers is also presented. In the last part of the section, ion trapping in the brushes and ion condensation are discussed. The conclusions are given in Section 4. 2. Model and setup A coarse-grained model is used to study the behavior of polyelectrolyte brushes in varying salt concentration and valences. The system comprises M ¼ 16 bead-spring chains, tethered on the bottom surface of a simulation box of dimension Lx  Ly  Lz. Each chain is composed of N ¼ 96 charged monomers plus a neutral end-monomer. Each charged monomer carries a negative unit charge
e and dissociates a monovalent counterion (cation) into the solution. The neutral end-monomer is tethered on the substrate (bottom) surface. The 16 tethered end-monomers are arranged in a 2D hexagonal lattice with lattice spacing pffiffiffiffiffi equal to d. The dimension of the surface is Lx ¼ M d ¼ 4d and pffiffiffi Ly ¼ ð 3=2ÞLx . Periodic boundary condition is applied in the xand y-directions. The pffiffiffi grafting density of chain is given by sg ¼ M=ðLx Ly Þ ¼ 2=ð 3d2 Þ. Salt ions are added in the simulation box and modeled by beads, where a salt cation carries þZe charge and a salt anions
e charge. All the particles, including the monomer and ion beads, interact with each other via the two interactions: the excluded volume interaction and the Coulomb interaction. The adjacent monomers on a chain are, in addition, connected by a bonding potential. The substrate surface is modeled by a mathematical wall, situated at z ¼ 0. A top wall is set at z ¼ Lz to keep the ions inside the simulation box. The excluded volume interaction is modeled by the WeeksChandler-Andersen potential [81],

8       > < 4ε s 12
s 6 þ 1 r r 4 Uex ðrÞ ¼ > : 0

if r 

ffiffiffi p 6 2s

pffiffiffi if r > 6 2s

(1)

where r is the separation distance between two particles, s represents the diameter of particle, and ε is the interaction strength. We assume that all the particles have the same s and ε. The Coulomb interaction is expressed as

Ucoul ðrÞ ¼

kB T lB Zi Zj v r

(2)

where kB is the Boltzmann constant, T the temperature, Zi and Zj the corresponding particle valences. The Bjerrum length lB ¼ e2/ (4pε0εrkBT) is the distance between two unit charges, at which the electrostatic energy is equal to the thermal energy kBT, where ε0 and εr are the vacuum permittivity and the relative dielectric constant of the solvent, respectively. The bonding

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potential is described by the finitely extensible nonlinear elastic potential

1 r2 Ubond ðrÞ ¼
kb R20 ln 1
2 2 R0

! (3)

where kb ¼ 5.8333ε/s2 is the spring constant and R0 ¼ 2.0s the maximum extensible bond length. The wall potential is modeled by a 9-3 Lennard-Jones potential

 2 s9 s3 ;
15 [ [

 Uwall ð[Þ ¼ ε

for [  [c

(4)

pffiffiffiffiffiffiffiffi with the cutoff [c ¼ 6 2=5s. A particle is subject to a strong repulsive force from the potential when the particle distance to the wall [ is smaller than [c. Therefore, the substrate has no interaction with the chains except at short distance. The separation between the top and the bottom walls is set to Lz ¼ 2Ns. The space is large enough to allow full extension of chains. We performed Langevin dynamics simulations, using LAMMPS package [82]. The equation of motion reads as

mi

! ! d2 r i dri hi
Vi U þ ! ¼
zi 2 dt dt

(5)

where zi is the friction coefficient, U is the sum of the potentials, h is a stochastic force. The first term on the right hand side and ! i

of the equation accounts for the frictional force when the particle ! i moves through the solvent with the velocity d r i =dt. The second term describes the conservative force acting on the particle. The last term simulates the random collision to the particle by the h i has solvent molecules (which are not modeled explicitly). ! zero mean and satisfies the fluctuation-dissipation theory h i ðtÞ$! h j ðt 0 Þi ¼ 6kB T zi dij dðt
t 0 Þ [83], which is used to control h! the system temperature. We assume that all the particles have identical masses m. The temperature is set to kBp T ffiffiffiffiffiffiffiffiffi ¼ 1.0ε and the friction coefficient is z ¼ 1.0 ms
1 where s ¼ s m=v is the Lennard-Jones time unit. s is 1.5 ps if m ¼ 100 g mol
1 is considered. For many synthetic polyelectrolytes and biomacromolecules, the Bjerrum length in solutions is about 2e5 times of the unit charge distance on the chain backbone. Here, we set a typical value 3s to the Bjerrum length. Notice that the electrostatic interaction in an electrolyte solution is screened due to redistribution of ions, characterized by the Debye screening length. We had verified that no side effect occurred in our model, such as unexpected strong ion associations and ion sticking on the chains, related to the seemingly large setting of the Bjerrum length. Similar setup and models have been widely used in literature [56,60,65,84]. Since lB in water is 7.1 Å at room temperature, the s corresponds to a value, 2.36 Å. Five grafting densities of chains are studied: sg ¼ 9  10
5, 4.5  10
4, 9  10
4, 4.5  10
3, and 9  10
3 s
2. Three valences of the added salt, Z ¼ 1, 2, 3, are investigated and the salt concentration is varied from Cs ¼ 0.0 to 2.025  10
2s
3. The equations of motion were solved using the Verlet algorithm Allen and [38,39] with the integrating time step Dt ¼ 0.005s. The long-range Coulomb interactions were calculated using particleeparticle/particleemesh Ewald algorithm for a slab geometry of system with an error tolerance of 10
3. The system was equilibrated for 107 time steps and followed by a production run of 5  107 time steps used for data analysis. To shorten the notation, the physical quantities will be reported in (s, ε, m, e)-unit system in the rest of the text.

3

3. Results and discussions The behavior of polyelectrolyte brush was studied by varying the salt valence Z, concentration Cs, and the chain grafting density sg. A dimensionless quantity

Ze$C L L L

s x y z b ¼ ð
eÞ$MN

was used to study the effect of salt concentration on the brush chains in view of the ratio of the total charges of the salt cations over the charges of the chain monomers. We also defined S ¼ sg =ðpR2g;0 Þ to describe the grafting density, which primarily factors out the chain size effect in the quantity, where Rg,0 is the radius of gyration of single polyelectrolytes tethered on the substrate surface. S ¼ 1 denotes a condition for brush chains starting to overlap. In the sparse grafting condition S  1, the brush chains are isolated and exhibit mushroom or pancake profiles, depending on the interaction between the chains and the substrate. When the grafting density is dense S [ 1, the occupied space of chains strongly overlaps and the steric effect of monomers stretches up the chains. The system hence exhibits a true brush-like profile. The value of Rg,0 is 18.6 in our system. The five studied grafting densities sg are thus equivalent to S ¼ 0.1, 0.5, 1.0, 5.0, and 10, respectively. 3.1. Brush thickness and chain structure We first studied the thickness H of polyelectrolyte brush as a function of b. Knowing the density profile of monomer fm(z) above the grafting surface, the thickness of brush can be characterized by

Z

Lz

2 H ¼

Z

0 Lz

0

zfm ðzÞdz (6)

fm ðzÞdz

which is twice of the “height” of the center of mass of the brush to the surface [65]. The profile fm(z) will be presented in the next section. Fig. 1(aec) show the results of H in monovalent, divalent, and trivalent salt solutions, respectively, at the five grafting densities. In the monovalent salt solutions, the brush thickness gradually decreases upon increasing the charge ratio b (which is proportional to the salt concentration Cs). According to the scaling theory [20,21], a quenched polyelectrolyte brush exhibits two behaviors, which define the so-called osmotic regime and the salted regime. Our result does reveal the two behaviors as the salt concentration is increased: a constant H in the low-salt, osmotic regime, followed by a scaling decrease in the high-salt, salted regime. Nonetheless, the scaling in the latter regime is not consistent with the theoretical prediction H w b
1/3, but follows a weaker scaling with the exponent equal to
0.15, as shown in the logelog plot of the figure. The weaker exponent for the brush thickness has been reported previously by other research groups [65,67]. Russel and coworkers have proposed a modified DaoudeCotton model to describe brushes on curved surfaces [85]. By neglecting electrostatic effect on chain stiffness and taking Odijk’s excluded volume, they argued
1=6 that H should scale with Cs for flat polyelectrolyte brushes. Their prediction has been confirmed by experiments [86,87]. The exponent obtained here supports their result too. Similar to the monovalent salt case, H stays about at a constant value in the divalent and trivalent salt solutions when b is smaller than a critical value b* x 0.1. Further increasing the salt concentration up to b ¼ 1 shrinks the brush thickness drastically. Experiments have confirmed the existence of such critical salt

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Fig. 1. Brush thickness H as a function of charge ratio b in (a) monovalent, (b) divalent, (c) trivalent salt solutions at different grafting densities S. The figures are plotted in logelog scale and the values of S are indicated. (d) Replot of H vs. b in semi-log plot at the three grafting densities S¼0.1, 1.0, 10, focusing on the salt valence (Z) effect.

concentration (CSC) [88]. Below CSC, the added salt basically cannot alter the polyelectrolyte brush. Above it, the brush thickness starts to decrease significantly. Our results show that if b is over the value 1, the brush thickness is leveled-off for the divalent case or becomes slightly increased for the trivalent case. The stoichiometric point b ¼ 1 defines a transition point where the brush exhibits a minimum thickness. Notice that the scaling theory does not predict this kind of swelling behavior at all. The behavior is, in fact, related to a phenomenon called reentrant condensation in polyelectrolyte solutions where condensed polyelectrolytes can re-enter into the solutions by adding excessive amount of multivalent salt [69]. The occurrence of this phenomenon is predominated by charge inversion [72,89]: the brush chains effectively changes the charge sign because of an over-compensated condensation of multivalent cations on the chains and hence, the brush thickness increases due to electrostatic repulsion in the interior of the chains. We will present detailed calculation about charge distributions in the following sections. Experiments have reported the swelling of polyelectrolyte brushes in high salt solutions [90]. The reason was attributed alternatively to an increase of the excluded-volume parameter for monomers, due to the adsorption of salt ions. The effect of valence and ion species of added salts on brush thickness have been investigated by the experiments too [88,90]. It was shown that CSC of brushes depends on the ion species and follows the well-known Hofmeister series. Concerning the influence of grafting density, we found that the denser the grafting, the thinner the brush thickness in the monovalent and divalent salt solutions. The result is opposite to the prediction of the scaling theory which states an increase of thickness as HwS1=3 . In the trivalent solutions, the same behavior is observed for b  1, but reverse variation occurs when b > 1 in which the reentrant brush chains grow thicker in a denser grafting system. Fig. 1(d) re-plots H vs. b, focusing on the effect of salt valence at

a given grafting density. We clearly see that the addition of multivalent salt (Z ¼ 2 and 3) can drastically reduce the brush thickness in comparison with the addition of monovalent salt. We remark that an astonishing “carpet þ brush” structure has been reported below the water surface in the experiments of compressing ionic amphiphilic diblock copolymer systems at air/water interface [91]. In the simulations, we did not observe this kind of “carpet þ brush” structure in varying correspondingly the chain grafting density. The structure of the brush chains was then studied by calculating ! ! the shape factor hR2e i=hR2g i where hR2e i ¼ hð r N
r 1 Þ2 i is the mean P ! ! 2 square of end-to-end distance and hR2g i ¼ hN1 N i ¼ 1 ð r i
r cm Þ i is ! the mean square of radius of gyration of a grafted chain ( r i is the ! position of i-th monomer and r cm the center of mass of the chain). The value of the shape factor is 2, 6.3, and 12, respectively, if the geometry of chain is spherical, coil-like, and rod-like [84]. The variation of hR2e i=hR2g i vs. b at the five studied grafting densities is shown in Fig. 2(a)e(c) for the three valences of added salt. For the monovalent salt case, the shape factor decreases monotonically from a value of about 10.0 to 8.5 with increasing b. Thus, the chains become less expanded in the high concentration solutions, but still more extended than a coiled chain. The denser the grafting, the larger the shape factor, in other words, the more expanded the structure. For the divalent and the trivalent salt cases, the chain shape shows a drastic transition at b ¼ 1. hR2e i=hR2g i decreases when b  1 and turns to become a constant when b > 1. This constant value is not sensitive to S for the divalent salt case, and acquires a value of about 7.3. For the trivalent case, the value depends on S. A smaller shape factor is found for sparser grafting. The smallest value of hR2e i=hR2g i is 5.5 at S ¼ 0.1, showing a structure more compact than an ideal chain. It is anticipated because the trivalent cations have strong binding ability to collapse the brush chains. However, as the chain grafting density increases, the excluded volume effect of monomers becomes important, which

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5

Fig. 2. Shape factor hR2e i=hR2g i as a function of b in (a) monovalent, (b) divalent, (c) trivalent salt solutions at different grafting densities S. The values of S are indicated in the figures. (d) Replot of hR2e i=hR2g i vs. b at the three grafting densities S ¼ 0.1, 1.0, 10, focusing on the salt valence (Z) effect.

prevents the chains from further collapsing. As a result, hR2e i=hR2g i takes a value of about 7 at S ¼ 5 and 10. The replot of the data in Fig. 2(d) shows that salt valence has a great effect on the chain structure. Noticeably, the structural difference between the divalent and the trivalent cases diminishes in the dense grafting conditions. In addition to the structure, understanding of the orientation of brush chains is also a relevant subject of study. To achieve this goal, we calculated the mean polar angle hqi of the chain end-to-end vector with respect to the substrate normal axis. The results are presented in Fig. 3. We observe that hqi increases with b in the monovalent salt solutions (Fig. 3(a)). For the divalent and trivalent salt cases (Fig. 3(b) and (c)), the increase slows down when b > 1. The larger angle in high salt conditions results from collapsing of chains into globule structure by the multivalent cations as shown in Fig. 2. Regarding the effect of grafting density, the mean polar angle decreases with increasing S in the monovalent and divalent solutions. The excluded volume interaction raises up the brush and thus, the chains stand more straightly, which reduces hqi. In the trivalent salt solutions, a larger orientation angle, in contrast, was observed when b > 1 at larger S. Since the chain structure is more rod-like than at small S (see Fig. 2(c)), to maintain a small brush thickness as shown in Fig. 1(c) necessitates a larger tilting. Fig. 3(d) replots hqi vs. b, which focuses on the effect of salt valence Z. We can see that Z does not show any appreciable influence on hqi except when the grafting density is high. 3.2. Vertical concentration and charge distribution There are four kinds of particles in the system, which are monomers, counterions (dissociated from monomers), salt cations,

and salt anions. An important subject is to understand the distribution of different sorts of particles present in the systems. Fig. 4 shows the vertical concentrations of monomer fm(z) in monovalent salt solutions at the small (S ¼ 0.1), intermediate (S ¼ 1.0), and large (S ¼ 10) grafting densities. Since the positive ions are attracted by the negative brush chains, we present also the vertical concentrations of counterions and salt cations in the lower panel of the figures for comparison. In this monovalent-salt case, the counterions and the salt cations are identical and carry (þ1)-charge. Therefore, the sum of the two concentrations, the ðsÞ one of the counterions fc(z) and the one of the salt cations fþ1 ðzÞ, is plotted. In the plots, the abscissa is divided by the chain contour length Nb0 to study the degree of chain extension and the ordinate is plotted in units of sg/b0, where b0 ¼ 1.156 is the mean bond length. The unit sg/b0 represents the mean monomer concentration in the condition of a fully stretched brush. In Fig. 4(d), we show the vertical charge density distribution r(z) which was ðsÞ ðsÞ calculated by ð
1Þfm ðzÞ þ ðþ1Þfc ðzÞ þ ðþZÞfþz ðzÞ þ ð
1Þf
1 ðzÞ with Z ¼ 1. fm(z) exhibits a plateau-like profile. In zero-salt solutions, a small hump appears near the chain end at z/(Nb0) x 0.5 before the density drastically decreases to zero for the cases of S ¼ 0.1 and 1.0. Since the number of monomers is conserved in the system, the area under the profile curve should be a constant which is equal to 1 in the unit system. Here we read the plateau value of roughly 2, about twice denser than in a fully stretched brush system. As the salt concentration (i.e. the charge ratio b) increases, the plateau ramps up gradually and the range of plateau (or the brush thickness) ðsÞ decreases. The concentration of cations, fc ðzÞ þ fþ1 ðzÞ, exhibits roughly the same profile as fm(z). At low salt concentration b < 0.5, ðsÞ the value of fc ðzÞ þ fþ1 ðzÞ is smaller than fm(z). But at higher b, it becomes larger than fm(z). Therefore, cations are more numerous

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Fig. 3. Mean polar angle hqi of chain orientation (in degree) as a function of b in (a) monovalent, (b) divalent, (c) trivalent salt solutions at different S. The values of S are indicated in the figures. (d) Replot of hqi vs. b at the three grafting densities S ¼ 0.1, 1.0, 10, focusing on the salt valence (Z) effect.

ðsÞ

Fig. 4. Vertical monomer concentration fm(z) and cation concentration fc ðzÞ þ fþ1 ðzÞ in monovalent salt solutions at the grafting density: (a) S ¼ 0.1, (b) S ¼ 1.0, (c) S ¼ 10. The value b of the added salt is indicated in the figures. The vertical charge density distribution r(z) at the three S is plotted in (d).

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V.S. Guptha, P.-Y. Hsiao / Polymer xxx (2014) 1e13

than needed to neutralize the chains in the brush region. Owing to this “charge” over-compensation, the salt anions diffuse into the brush region and redistribute around the chains. As a result, the charge density (see the S ¼ 0.1 panel in Fig. 4(d)) becomes negative in the brush, mostly near the top, and positive outside the brush, but close to the rim. An electric double layer structure is thus formed near the brush-solution boundary. As b increases, the double layer moves downward, toward the substrate, or moves leftward in the presentation of the figure. The charge density r(z) is flattened and becomes less structured when the grafting density increases. At S ¼ 10, the brush is nearly charge-neutralized in the space. In Figs. 5 and 6, we present the vertical monomer concentration

fm(z) in divalent (Z ¼ 2) and trivalent (Z ¼ 3) salt solutions, respectively, together with the vertical concentrations of chain ðsÞ counterions fc(z) and salt cations fþZ ðzÞ. The charge density distribution r(z) is also plotted for comparison. A remarkable variation is that fm(z) transforms from a plateau profile (as in the monovalent salt case) to a single-peak profile with increasing the salt concentration. The peak is more pronounced for the (3:1)-salt case than for the (2:1)-salt one. Concerning the chain counterion distribution, the dominated profile of fc(z) is plateau in both the salt solutions. As b increases, a reduction in the plateau range takes place first by keeping the plateau value constant, followed by a decrease of the plateau value toward a homogeneous ðsÞ

distribution over the whole space. For the salt cations, fþZ ðzÞ becomes more peaked with increasing the salt concentration and follows basically the profile of fm(z). Notice that the multivalent cations are mostly trapped in the brush region when b  1. Only beyond the stoichiometric point b ¼ 1, some of the cations escape

7

from the brush and the concentration profile exhibits a long tail. Similar trend of variations was observed when the grafting density increases. The plot of charge density distribution r(z) clearly shows the formation of an electric double layer. Inside the top of the brush is a wide, negatively-charged region, while outside the brush becomes a broad, positively-charged space. Increasing b shifts the double layer inward. Associated to this particular layer is a layer of electric dipole near the brush rim. The strength of the dipole is proportional to the thickness of the double layer multiplying the carried charges. It decreases with the parameters b, S and Z, because both the thickness and the charges decrease with increasing the grafting density and the salt valence. 3.3. Side-view density distributions In order to understand the occupation of chains in the space, we studied the side-view surface density distribution of monomer fm(x,z), viewed through the xz-plane, in addition to the vertical concentration fm(z) presented in the previous subsection. Since the brush chains are tethered on the hexagonal lattice points on the xyplane, the surface density was calculated by mapping the volume density along the aligning line of the tethering points, 30 -deviated from the y-axis. Therefore, 4 chain silhouettes are expected to appear in fm(x,z). Fig. 7 shows the results in the monovalent salt solutions. From the series of plots, we are able to see how the monomer density varies with the parameters b and S. At low grafting density S ¼ 0.1 and zero salt, the density plot shows mushroom-like silhouettes. The grafting density is very low such that the chains do not touch with other. Therefore, the sideview density shows individual chain profiles. Increasing the

ðsÞ

Fig. 5. Vertical concentration of monomers fm(z), chain counterions fc(z), and salt cations fþ2 ðzÞ in (2:1)-salt solutions at the grafting density: (a) S ¼ 0.1, (b) S ¼ 1.0, and (c) S ¼ 10. The value b of the added salt is indicated in the figures. The vertical charge density distribution r(z) at the three S is plotted in (d).

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ðsÞ

Fig. 6. Vertical concentration of monomers fm(z), chain counterions fc(z), and salt cations fþ3 ðzÞ in (3:1)-salt solutions at the grafting density: (a) S ¼ 0.1, (b) S ¼ 1.0, and (c) S ¼ 10. The value b of the added salt is indicated in the figures. The vertical charge density distribution r(z) at the three S is plotted in (d).

grafting density brings the chains closer. At S ¼ 1.0, the chains start to touch each other. When the grafting density is high, S ¼ 10, the brush chains are strongly stretched. The density profile is approximately homogeneous in the brush region except near the tethering points and the brush rim. Increasing b decreases monotonically the height of the chain profiles due to ionic screening. Please notice that different scales in the x axis have been used for different grafting densities in the plots. Figs. 8 and 9 present a series of density plots in the divalent and the trivalent salt solutions, respectively. We can see that the divalent and the trivalent cations are very strong condensing agents. They collapse the brush chains into globules, but still maintain individual structures at b ¼ 1.0 when S  1. The effect is more pronounced for the trivalent salt case than the divalent one. The rim of the brush slightly raises up at b ¼ 4 in the trivalent solutions. Similar to the monovalent case, the density in the brush region becomes homogeneous when the grafting density is high, i.e., at S ¼ 10. Entangled mixing between neighboring chains occurs because the brush thickness remains at a small value (Fig. 1(c)) while the chain structure is rod-like (Fig. 2(c)) and also the angle of chain orientation increases (Fig. 3(c)). From the analysis of the results in Figs. 7e9, it is clear that in order to achieve a homogeneous monomer density in a polyelectrolyte brush, the addition of strong condensing agents is not enough; the brush system needs to possess a high grafting density too. 3.4. Ion trapping in brushes and ion condensation The behavior of polyelectrolyte brushes depends very much on the distribution of ions in the systems, as we have seen in the

previous sections. In this section, we go further to study the amount of ions trapped in the brushes and the amount of ions condensing on the individual chains. Here we start from the topic of ion trapping. An ion is said to be trapped in a brush if the distance of the ion to the grafting surface is smaller than the mean chain height in the configuration plus the Bjerrum length. Since there are three sorts of ions in the system, we calculated the three charge trapping ratios: sc, sþZ, and s
1, which are, respectively, the absolute ratios of the charges of the trapped counterions (c) of chains, the trapped salt cations (þZ), and the trapped salt anions (
1) over the total charges of brush monomers. The ratio for the total charges of all the trapped ions, stot, was also computed. The results are presented in the upper and lower panels of Fig. 10 (a)e(c) for the three salt valences by varying the salt concentration b and the grafting densities S. In the monovalent salt solutions, stot is approximately 0.9 over the range of b. It shows that about 90% of brush charges are neutralized by the trapped ions. Even in the salt-free solution, the charges have already been 90%-neutralized by the own counterions of the brush. The calculation in the lower panel of Fig. 10(a) further shows that sc decreases with b, but sþ1 increases. Therefore, the trapped chain counterions are gradually replaced by the salt cations as the salt concentration increases. In this monovalent case, the counterions (charge þe, each) are identical to the added salt cations. Hence, the amount of the trapped cations has contribution from both the ions, portioned according to their populations in the system. Consequently, there is a crossover between the two curves, sc and sþ1, at the stoichiometric point b ¼ 1.0. Concerning about the salt anions, strong Coulomb interactions repel them from the brush monomers, because they all carry negative charges. Nonetheless, there are increasingly salt anions trapped in the brush as b

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Fig. 7. Side-view surface density of monomer fm(x,z) in monovalent salt solutions. The charge ratio b and the grafting density S are indicated on the table. The color bar is in logarithmic scale in the plots. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8. Side-view surface density of monomer fm(x,z) in divalent salt solutions. The charge ratio b and the grafting density S are indicated on the table. The color bar is in logarithmic scale in the plots. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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V.S. Guptha, P.-Y. Hsiao / Polymer xxx (2014) 1e13

Fig. 9. Side-view surface density of monomer fm(x,z) in trivalent salt solutions. The charge ratio b and the grafting density S are indicated on the table. The color bar is in logarithmic scale in the plots. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. Absolute ratio of total trapped charges in the brush (stot), and absolute ratios of the trapped chain counterion charges (sc), of the trapped salt cation charges (sþZ), and of the trapped salt anions charges s
1 as a function of b in (a) monovalent, (b) divalent, and (c) trivalent salt solutions. The grafting density S is indicated in the figures. (d) Replot of the ratios vs. b at S¼0.1, 1.0, 10, focusing on the salt valence (Z) effect.

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V.S. Guptha, P.-Y. Hsiao / Polymer xxx (2014) 1e13

increases, manifested by an increasing s
1. It is obviously due to the trapped monovalent cations which mediate between the monomers and the salt anions. We found that these ratios are not very sensitive to the grafting density S, except at the lowest one S ¼ 0.1 where stot is relatively smaller than the others. In the divalent and trivalent salt solutions, stot also acquires a value of about 0.9 when the salt concentration is smaller than the stoichiometric point. But the value sharply raises to 0.95 and 1.0, respectively, when b  1.0. The resulting polyelectrolyte brush is very close to a neutralized object in the high salt condition. Similar to the monovalent case, the trapped counterions are replaced by the salt cations gradually with increasing the salt concentration, as evidenced by a monotonic decrease of sc and monotonic increase of sþZ. Both curves exhibit a kink at b ¼ 1.0: the value of sþZ is approximately 1, while sc is zero. It marks a brush neutralization solely by the trapping of multivalent salt cations. Beyond this point, excessive amount of salt cations gets trapped in the brush and the coions (i.e. the monovalent anions dissociated from the salt) start to enter into the region. Nearly no chain counterion appears in the brush in this condition. Fig. 10(d) is a replot of the data at the small (S ¼ 0.1), medium (S ¼ 1.0), and large (S ¼ 10) grafting densities, focusing on the effect of salt valence (Z). stot is roughly a constant if b < 1 but increases with Z when b > 1.0. The larger the salt valence, the higher the trapped charges from the salt cations. An interesting finding is that more co ion charges are trapped in the brush region when Z is smaller. It is a counter-intuitive result because the excessively trapped salt cations with larger salt valence are thought to have a greater power to attract more coions. In the second part of this section, we studied the ion condensation on the chain. An ion is said to condense on a chain if the smallest distance of the ion to the chain monomers is smaller than the Bjerrum length. Fig. 11(a)e(c) present the calculation of the

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absolute ratio ctot of total condensed ion charges on the chains, including all sorts of ions, over the chain bare charges, in the monovalent, divalent, and trivalent salt solutions, respectively. According to Manning’s theory [92], the counterions in bulk solutions condense onto polyelectrolyte chains when G < lB, where G (¼b0) is the mean length per unit charge distributed on the chains. The condensed counterions effectively reduce the chain charges and thus, increase G. The chains have a resultant value G* equal to lB, which occurs when the fraction of chain charge, c* ¼ 1
(b0/lB), is neutralized by the condensed ions, assuming that the chains are rigid and of infinite length in a dilute solution. Recall that in our study, b0 ¼ 1.156 and lB ¼ 3. The theory hence predicts a neutralization of a fraction c* ¼ 0.615 of chain charges, disregarding that our system is flexible brush which does not rigorously respect the assumptions. Our simulations show a surprisingly good consistency of ctot with the predicted value at the low grafting density S ¼ 0.1 when the salt concentration is low, for the three salt valences. In the monovalent solutions, ctot increases smoothly with b. But in the divalent and trivalent solutions, a sharp increase occurs near b ¼ 1.0, which is due to the replacement of the condensed chain counterions by the salt cations (Z ¼ 2 or 3), similar to what we have observed in the study of the trapped ions. Moreover, increasing S significantly raises the value of ratio ctot from the Manning’s value. The higher the grafting density, the larger the ctot value and hence the more condensed counterions on the brush chains. For the trivalent case, ctot becomes larger than 1 above the stoichiometric point, which indicates the occurrence of charge overcompensation, or called ‘charge inversion’, in the proximity of the chains. The very high ion condensation has been observed experimentally as the grafting density and the salt concentration increases [88,91]. The replot in Fig. 11(d) reveals that the degree of this charge over-compensation decreases with increasing S. The

Fig. 11. Absolute ratio of the total condensed ion charges over the chain bare charges, ctot, as a function of b in (a) monovalent, (b) divalent, and (c) trivalent salt solutions. The values of S are indicated in the figures. (d) Replot of ctot vs. b at the three grafting densities S ¼ 0.1, 1.0, 10, focusing on the salt valence (Z) effect.

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effect of salt valence on ctot can be seen clearly in the replot too. The difference between salt valences diminishes at S ¼ 10. We remark that the ion trapping and ion condensation take place in a dynamical way. The ions are not permanently trapped in the brush or condensed on the chains, but can escape from the brush region or from the chains due to thermal fluctuations. Other ions come in, take the vacancies, and become the new parts of trapped or condensed ions. An equilibrium is established. These processes were clearly witnessed in the simulations. 4. Conclusions We have performed Langevin dynamics simulations to study the properties of polyelectrolyte brush in salt solutions by varying the chain grafting density, the salt valence and salt concentration. In monovalent salt solutions, the brush thickness H decreases monotonically with the salt concentration (or equivalently, the charge ratio b). In the high salt condition, H scales asymptotically with b
0.15 where the exponent is close to the prediction
1/6 of Russel et al. [85]. For the divalent and trivalent salt cases, H is leveled off or becomes slightly increased as b > 1.0 due to the effect of charge inversion. The study of shape factor hR2e i=hR2g i showed that the brush chains gradually collapse into coil-like structure as the salt concentration increases, accompanied by a sharp structural transition happened near the stoichiometric point b ¼ 1.0 in the divalent and trivalent salt solutions, associated to the drastic decrease of the brush thickness. Noticeably, the brush chains remain expanded at b > 1.0 when the grafting density is high (at S ¼ 5 and 10) for the trivalent case. Consequently, the orientation of chains tilted to maintain, at the same time, a small brush thickness. A larger tilting angle is opposite to the general trend that hqi decreases with increasing S because of the steric exclusion of monomers which stands the chains up when the grafting density increases. The study of mean vertical concentrations revealed that increasing salt concentration decreases the brush thickness and thus, changes the plateau-like distribution of monomers to a peaklike profile. The peak is more pronounced in the solutions with larger salt valence. The profiles of the counterions and the salt cations followed basically the profile of the monomer distribution. An electric double layer was formed near the brush rim. The charge distribution is negative inside the brush, while positive outside the brush. The layer width reduces with increasing the salt concentration as well as with the grafting density. A series of side-view monomer density were presented, sweeping over a wide range of the parameters b and S. It showed how the density profile changes from individual chain silhouettes to a continuous, quasihomogeneous slab distribution. Finally, the ion trapping in the brush was studied. About 90% of brush charges are neutralized by the trapped ions when b < 1.0. Above the stoichiometric point, the brush is nearly neutralized in the trivalent salt solutions. We showed that the trapped chain counterions are gradually replaced by the salt cations. And in the high salt solutions, the chain counterions are completely expelled outside the brushes. Our study further revealed that Manning’s theory prediction, originally derived for bulk polyelectrolytes, also hold for the ion condensation in polyelectrolyte brush systems in dilute salt and small chain grafting density conditions. Increasing grafting density promotes the ion condensations. Increasing salt concentration also increases the charges condensing on the chains. Moreover, the chain charge can be over-compensated by the condensed ions in the trivalent salt solutions, which results in the charge inversion and hence the slight increase in the brush thickness.

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Please cite this article in press as: Guptha VS, Hsiao P-Y, Polyelectrolyte brushes in monovalent and multivalent salt solutions, Polymer (2014), http://dx.doi.org/10.1016/j.polymer.2014.04.035