Brushes of flexible, semiflexible and rodlike diblock polyampholytes: Molecular dynamics simulation and scaling analysis
aa r X i v : . [ c ond - m a t . s o f t ] D ec Brushes of flexible, semiflexible and rodlike diblock polyampholytes: Moleculardynamics simulation and scaling analysis
Majid Baratlo and Hossein Fazli ∗ Institute for Advanced Studies in Basic Sciences, Zanjan 45195-1159, Iran (Dated: September 29, 2018)Planar brushes of flexible, semiflexible and rodlike diblock polyampholytes are studied usingmolecular dynamics simulations in a wide range of the grafting density. Simulations show lineardependence of the average thickness on the grafting density in all cases regardless of different flexi-bility of anchored chains and the brushes different equilibrium conformations. Slopes of fitted linesto the average thickness of the brushes of semiflexible and rodlike polyampholytes versus the graft-ing density are approximately the same and differ considerably from that of the brushes of flexiblechains. The average thickness of the brush of flexible diblock polyampholytes as a function of thegrafting density is also obtained using a simple scaling analysis which is in good agreement with oursimulations.
PACS numbers: 82.35.Rs, 61.41.+e, 87.15.-v
I. INTRODUCTION
Macromolecules containing ionizable groups when dis-solve in a polar solvent such as water, dissociate intocharged macromolecules and counterions (ions of op-posite charge). Depending on acidic or basic prop-erty of their monomers, ionizable polymers in solu-tion can be classified into polyelectrolytes and polyam-pholytes. Polyelectrolytes contain a single sign of chargedmonomers and polyampholytes bear charged monomersof both signs. These macromolecules are often water-soluble and have numerous industrial and medical appli-cations. Many biological macromolecules such as DNA,RNA, and proteins are charged polymers. In polymerscience, charged polymers has been an important sub-ject during last several decades [1, 2, 3, 4]. Contraryto a polyelectrolyte chain in which the intra-chain elec-trostatic interactions are repulsive and tend to swell thechain, in a polyampholyte chain attractive interactionsbetween charged monomers of opposite sign tend to de-crease the chain size. Oppositely charged monomers canbe distributed randomly along a polyampholyte chainor charges of one sign can be arranged in long blocks.With the same ratio of positively and negatively chargedmonomers (isoelectric condition), behaviors of a singlepolyampholyte and the solution of polyampholytes de-pend noticeably on the sequence of charged monomerson the chains. For example, it has been shown thatthe sequence of charged amino acids (charge distribu-tion) along ionically complementary peptides affect theaggregation behavior and self-assembling process in thesolution of such peptides [5, 6]. Also, using Monte Carlosimulations it has been shown that charged monomers se-quence of charge-symmetric polyampholytes affect theiradsorption properties to a charged surface [7].The properties of the system of polymers anchored on ∗ Electronic address: [email protected] a surface are of great interest both in industrial and bio-logical applications and academic research. With a suffi-ciently strong repulsion between the polymers, the chainsbecome stretched and the structure obtained is known asa polymer brush. Planar and curved brushes formed bygrafted homopolymers have extensively been investigatedby various theoretical methods [8, 9, 10, 11]. The an-chored polymers of a brush may be consisting of chargedmonomers. In this case, the brush is known as a poly-electrolyte or a polyampholyte brush depending on thecharged monomers of the chains being of the same signor being composed of both signs. In a brush of chargedpolymers, electrostatic interactions introduce additionallength scales such as Bjerrum length and Debye screeninglength to the system.In a polyelectrolyte brush, the repulsion of electro-static origin between the chains can be sufficiently strongeven at low grafting densities, making it easy for the sys-tem to access the brush regime. Polyelectrolyte brusheshave been investigated extensively using both theoretical[12, 13, 14, 15, 16, 17, 18, 19] and computer simula-tion methods [17, 18, 19, 20, 21, 22]. At high enoughgrafting densities and charge fractions of polyelectrolytechains, most of counterions are trapped inside the poly-electrolyte brush and competition between osmotic pres-sure of the counterions and elasticity of the chains de-termines the brush thickness. This regime of a polyelec-trolyte brush is known as the osmotic regime in whichsome theoretical scaling methods predict no dependenceof the brush thickness to the grafting density [12, 23].However, other scaling method that takes into accountthe excluded volume effects and nonlinear elasticity ofpolyelectrolyte chains and is in agreement with exper-iment and simulation, predicts a linear dependence ofthe brush thickness on the grafting density [17, 18, 19].Also, it has been shown that diffusion of a fraction ofcounterions outside the polyelectrolyte brush leads to alogarithmic dependence of the average brush thicknesson the grafting density [16].Electrostatic interactions in a polyelectrolyte brush
FIG. 1: (Color online) Sample snapshots of brushes of a) flexible ( l p = 5 σ ), b) semiflexible ( l p = 25 σ ) and c) rodlike ( l p = 200 σ )diblock polyampholytes at grafting density ρ a σ = 0 .
08. Neutral , positively and negatively charged monomers are shown bywhite, red (dark) and blue (black) spheres respectively. Counterions are not shown and periodic boundary condition is removedfor clarity. cause most of the counterions to be trapped inside thebrush and help the chains to be more stretched and thebrush to be more aligned. However, in a brush of over-ally neutral polyampholyte chains, most of counterionsare outside the brush and the electrostatic correlationstend to decrease the chains size and the brush thickness.At a given value of the grafting density, the average thick-ness and equilibrium properties of such a polyampholytebrush are mainly determined by the chains propertiessuch as fraction and sequence of charged monomers andthe bending energy. Brushes formed by grafted diblockpolyampholytes have been investigated by lattice meanfield modeling [24, 25] and computer simulation [26]. Theeffect of chain stiffness, charge density and grafting den-sity on spherical brushes of diblock polyampholytes andinteraction between colloids with grafted diblock polyam-pholytes have been studied using Monte Carlo simula-tions [27, 28]. Also, using molecular dynamics (MD) sim-ulations, the effects of various parameters such as chargedmonomers sequence, grafting density and salt concentra-tion on the average thickness and equilibrium conforma-tions of planar semiflexible polyampholyte brushes havebeen investigated [29].In this paper, we study planar brushes of flexible, semi-flexible and rodlike diblock polyampholytes using MDsimulations in a wide range of the grafting density. Wefind that in all cases the average brush thickness lin-early depends on the grafting density regardless of thechains different flexibility. Our results also show that thestrength of this dependence is considerably weaker in thecase of the brush of flexible polyampholytes than twoother cases. Despite mentioned same functionality ob-tained for the average thickness versus the grafting den-sity for brushes of different polyampholytes we find that histograms of their equilibrium conformations are notice-ably different. Inter-chain correlations are too weak inthe brush of flexible polyampholytes and the brush prop-erties are dominantly determined by single chain behav-ior. In this case, dependence of the equilibrium confor-mation of the brush on the grafting density is very weak.In the cases of the brushes of semiflexible and rodlikepolyampholytes however, because of the combination ofelectrostatic correlations and strong excluded volume ef-fects, collective behavior of the chains is dominant anddependence of equilibrium conformations on the graftingdensity is strong. In these cases, we also observe separa-tion of the anchored chains into two coexisting fractions.Using a simple scaling method which is consistently ap-plicable for the brush of flexible chains, we describe theo-retically the linear dependence of the brush thickness onthe grafting density.The rest of the paper is organized as follows. In SectionII we describe our model and simulation method in detailand present the results of MD simulations. Our scalinganalysis to describe dependence of the average thicknessof flexible diblock polyampholyte brushes on the graft-ing density is presented in section III. In section IV weconclude the paper and present a short discussion.
II. MOLECULAR DYNAMICS SIMULATION OFDIBLOCK POLYAMPHOLYTE BRUSHESA. The model and the simulation details
In our simulations which are performed with the MDsimulation package ESPResSo [30], each brush is modeledby M = 25 diblock polyampholyte bead-spring chains oflength N = 24 (24 spherical monomers) which are end-grafted onto an uncharged surface at z = 0. The posi-tions of anchored monomers which are fixed during thesimulation, form an square lattice on the grafting surface( x − y plane) with lattice spacing d = ρ − / a in which ρ a is the grafting density of the chains. The fraction f = of the monomers of each chain are charged andthe chains consist of an alternating sequence of chargedand neutral monomers. Each chain contains the samenumber of positively and negatively charged monomerswith charges e and − e respectively (see Fig. 1). Ex-cluded volume interaction between particles is modeledby a shifted Lennard-Jones potential, u LJ ( r ) = (cid:26) ε { ( σr ) − ( σr ) + } if r < r c , r ≥ r c , (1)in which ǫ and σ are the usual Lennard-Jones parametersand the cutoff radius is r c = 2 / σ . Successive monomersof each chain are bonded to each other by a FENE (finiteextensible nonlinear elastic) potential [31], u bond ( r ) = (cid:26) − k bond R ln(1 − ( rR ) ) if r < R , r ≥ R , (2)with bond strength k bond = 30 ε/σ and maximum bondlength R = 1 . σ . Bending elasticity of the chains ismodeled by a bond angle potential, u bend ( r ) = k bend (1 − cos θ ) , (3)in which θ is the angle between two successive bond vec-tors and k bend is the bending energy of the chains. Thevalue of the persistence length, l p , of the chains dependson the value of k bend as l p = k bend k B T σ . To model brushesof flexible, semiflexible and rodlike chains, we use fourdifferent values of k bend namely 0, 5 k B T , 25 k B T and200 k B T respectively. The simulation box is of volume L × L × L z in which L is the box width in x and y direc-tions and L z is its height in z direction and the graftingdensity is given by ρ a = M/L . We consider M × N × f monovalent counterions to neutralize the chains charge.Positive and negative monovalent counterions are mod-eled by equal number of spherical Lennard-Jones parti-cles of diameter σ with charges e and − e respectively. Allthe particles interact repulsively with the grafting surfaceat short distances with the shifted Lennard-Jones poten-tial introduced in Eq. 1. In addition, a similar repulsivepotential is applied at the top boundary of the simulationbox and in our simulations L z = 2 N σ . All the chargedparticles interact with each other with the Coulomb in-teraction u C ( r ) = k B T q i q j l B r (4)in which q i and q j are charges of particles i and j in unitsof elementary charge e and r is separation between them.The Bjerrum length, l B , which determines the strength ofthe Coulomb interaction relative to the thermal energy, < z m > / a 2 k bend =200 k B T k bend =25 k B T k bend =5 k B T k bend =0 FIG. 2: (Color online) The average thickness of the brushesformed by flexible ( k bend = 0 and 5 k B T ), semiflexible( k bend = 25 k B T ) and rodlike ( k bend = 200 k B T ) diblockpolyampholytes versus dimensionless grafting density, ρ a σ .The lines are linear fits to our simulation data. The slopesof solid, dashed, dotted and dash-dotted lines are 41.6, 40.2and 22.7 and 20.0 respectively. The size of the symbols cor-responds to the size of error bars. k B T , is given by l B = e /εk B T , where ε is the dielec-tric constant of the solvent and we set l B = 2 σ in oursimulations. Periodic boundary conditions are appliedonly in two dimensions ( x and y ). To calculate Coulombforces and energies, we use the so-called M M M tech-nique introduced by Strebel and Sperb [32] and modifiedfor laterally periodic systems (
M M M D ) by Arnold andHolm [33]. The temperature in our simulations is keptfixed at k B T = 1 . ε using a Langevin thermostat.For each value of the bending energy, k bend , we dosimulations of the brush at dimensionless grafting den-sities ρ a σ = 0 . , . , . , . , .
10. In the begin-ning of each simulation, all of the chains are straightand perpendicular to the grafting surface and all theions are randomly distributed inside the simulation box.We equilibrate the system for 1 . × MD time stepswhich is enough for all values of the grafting density men-tioned above and then calculate thermal averages over1500 independent configurations of the system selectedfrom 2 . × additional MD steps after equilibration.MD time step in our simulations is τ = 0 . τ in which τ = q mσ ε is the MD time scale and m is the mass ofthe particles.We calculate the average brush thickness which canbe measured by taking the first moment of the monomerdensity profile h z m i = R ∞ zρ m ( z ) dz R ∞ ρ m ( z ) dz , (5)in which ρ m ( z ) is the number density of monomers as afunction of the distance from the grafting surface. For abetter monitoring of the statistics of the chains conforma-tions, we calculate the histogram of the mean end-to-enddistance of the chains, P ( R ), in which R = M P Mi =1 | R i | and R i is the end-to-end vector of chain i . We alsocalculate the histogram of the average distance of theend monomers of the chains from the grafting surface, P ( z end ), in which z end = M P Mi =1 z i and z i is the z com-ponent of the end monomer of chain i . With the samemethod discussed in Ref. [29], it has been checked thatour results are not affected by finite-size effects (see Sec.IV). B. Results
The average thickness versus the grafting densityfor brushes of flexible, semiflexible and rodlike diblockpolyampholytes are shown in Fig. 2. Dependence of theaverage thickness on the grafting density can be describedwell by a linear function for all values of the bending en-ergy, k bend , that we use in our simulations. Also, it canbe seen that at all values of the grafting density the av-erage brush thickness decreases with increasing the flexi-bility of the chains. Linear fits to the average thickness ofthe brushes of semiflexible and rodlike chains for whichthe persistence length, l p , exceeds their contour length, L c , are of approximately the same slope (see solid anddashed lines in Fig. 2). The slopes of the linear depen-dence in cases of two brushes of flexible chains are alsoapproximately the same and differ by a factor of ∼ fromthose of two other cases (dotted and dash-dotted lines inFig. 2). To analyze such dependencies of the brushesthickness on the grafting density, we look at the equilib-rium conformations statistics of the chains at differentvalues of ρ a . In Figs. 3 and 4 the histograms P ( R ) and P ( z end ) at three different grafting densities are shownfor brushes of flexible, semiflexible and rodlike chains.Because of very similar behaviors of the histograms in k bend = 0 and k bend = 5 k B T cases, the histograms ofthe brush with k bend = 0 are not shown for clarity of thefigures. As it can be seen in Fig. 3, in the case of thebrush of flexible diblock polyampholytes ( k bend = 5 k B T ),dependence of the histogram profile on the grafting den-sity is very weak. Also, in this case, the contribution oflarge values of r ( r ≃ L c ) in the histogram is negligiblewhich shows that the chains are mostly coiled at all graft-ing densities (see a sample configuration of the brush at ρ a σ = 0 .
08 in Fig. 1a). In the case of the brush ofsemiflexible chains ( k bend = 25 k B T ), with increasing thegrafting density, the values of the histogram correspond-ing to smaller values of r become nonzero showing thatpolyampholyte chains take buckled conformations at highgrafting densities [29]. As it is expected, the histogram P ( R ) of the brush of rodlike chains ( k bend = 200 k B T )exhibit no noticeable buckling of the chains. The his-tograms P ( z end ) in Fig. 4 show that in the brush of flex-ible chains at all grafting densities the end monomers are mostly distributed near the grafting surface showing thatthe positive blocks of the chains are mostly turned backtowards the anchored negative blocks. The profile of thishistogram also doesn’t depend noticeably on the graftingdensity. The histograms P ( z end ) for brushes of semiflexi-ble and rodlike chains show that two maxima appear andtheir height increase with increasing the grafting density.By combining the information obtained from histograms P ( R ) and P ( z end ) for the brush of rodlike chains it canbe understood that at high grafting densities a fractionof the chains which are perpendicular to the grafting sur-face coexist with the remaining fraction which fluctuatein the vicinity of the grafting surface. In the case of thebrush of semiflexible chains also, the fraction of perpen-dicular chains to the brush surface coexist with the chainswhich are buckled towards the grafting surface [29]. His-tograms P ( R ) and P ( z end ) show that conformations ofthe brushes of semiflexible and rodlike chains change no-ticeably with changing the grafting density despite thecase of the brush of flexible chains. III. LINEAR DEPENDENCE OF THE BRUSHTHICKNESS ON THE GRAFTING DENSITY:SCALING APPROACH
As mentioned in the previous section, our MD simula-tions show that the brush thickness is a linear functionof the grafting density regardless of different values ofgrafted diblock polyampholytes bending energy. Also, asit is shown in Fig. 2, in cases of the brushes of flexiblechains the slope of the fitted line to the average thicknessversus the grafting density is considerably smaller thanthat in two other cases. We use here a simple scalingtheory similar to that of the solution of charge-symmetricdiblock polyampholytes [34] to describe the linear depen-dence of the brush thickness on the grafting density. Thisscaling analysis is applicable to the brushes of flexible di-block polyampholytes.Consider M flexible diblock polyampholyte chains end-grafted to a flat surface of area A . The degree of poly-merization of each chain and the fraction of chargedmonomers are N and f respectively. Electrostatic at-tractive interactions between oppositely charged blocksof the chains lead them to form a dense layer of posi-tive and negative monomers of average thickness h nearthe grafting surface (see Fig. 5). Inside the layer, touse the blob concept we define correlation length ξ asa length scale that electrostatic interactions don’t per-turb the chains statistics at smaller length scales andare dominant over thermal fluctuations at larger lengthscales. Let suppose that the number of monomers insidethe correlation blob is g and we have ξ ≈ bg ν where b and ν are the Kuhn length and the Flory exponent re-spectively. Accordingly, the layer is a melt of correlationblobs in which positive blobs with a high probability aresurrounded by negative blobs. Electrostatic interactionbetween any two neighboring blobs is of the order of thethermal energy, k B Tk B T | l B f g ξ |≈ k B T. (6)Thus we obtain ξ ≈ l B f g ≈ b ( l B f b ) νν − . As a result,the local monomer concentration inside the blob is ρ local ≈ gξ ≈ ξ − νν b ν ≈ b ( l B f b ) − νν − . (7)Considering that correlation blobs are space filling, theglobal monomer concentration, ρ global = MNAh = ρ a Nh , ap-proximately equals to the local monomer concentration, ρ global ≈ ρ local . Thus, dependence of the thickness of abrush of diblock polyampholytes on the grafting densitycan be obtained as h ≈ ρ a N b ( l B b f ) ν − ν − . (8)The main prediction of this equation in the range of itsvalidity is as follows. The brush thickness, h , is a lin-ear function of the grafting density, ρ a , irrespective ofthe values of the system parameters. Only the slope ofthis linear function, α = N b ( l B b f ) ν − ν − , depends on thevalues of the system parameters.One should note here that despite the bulk solutionof flexible diblock polyampholytes, in a polyampholytebrush the fact that the chains are end grafted to the sur-face introduces an additional length scale to the system,namely d = ρ − / a . In the scaling analysis presented here,if the size of the correlation length, ξ , exceeds the separa-tion between the grafting points, d , the scaling approachbecomes inconsistent. Although linear dependence of thebrush thickness on the grafting density is observed at allvalues of the bending energy used in our simulations, thecondition ξ < d is valid only in the case of the brush offlexible chains with k bend = 0. In this case the values of ξd corresponding to lowest and highest values of the graft-ing density we have used in our simulations are 0.19 and0.45 respectively. For the set of parameters used in oursimulations, in the case of the brush with k bend = 0 theupper limit of the validity of the scaling method ( ξ ≃ d )corresponds to the grafting density ρ a σ ≃ .
55 which isquite far from our range of the grafting density. For abrush of flexible chains with k bend = 0 using the values b ≃ σ and ν ≃ gives the slope α f ≃
35 for the brushthickness versus the grafting density for the used set ofsystem parameters. As it is shown in Fig. 2, the valueof α f obtained from our simulations of flexible chains is α f ≃ . l p = 5 σ because therange of the grafting density used in our simulations ishigher than the validity range of the condition ξ < d in this case. This method is not also applicable forthe brushes of semiflexible and rodlike chains because in these cases the Kuhn length exceeds the contour length ofthe chains. The fact that the average thickness linearlydepends on the grafting density at all values of k bend used in our simulations shows that although the scalingmethod presented here can not be used to describe all thesimulation results, this linear dependence persists over awide range of the system parameters. IV. CONCLUSIONS AND DISCUSSION
Brushes of flexible, semiflexible and rodlike diblockpolyampholytes have been studied using MD simulationsand a scaling analysis has been presented to describe theresults of the simulation of flexible chins brush. The av-erage thickness as a function of the grafting density andhistograms of equilibrium conformations of the brushesare obtained. Strong dependence of the system confor-mations on the grafting density and separation of thechains into two coexisting fractions at high grafting den-sities have been observed in cases of the brushes of semi-flexible and rodlike chains. In cases of the brushes of flex-ible chains, single-chain behavior is dominant and depen-dence of the brush conformations on the grafting densityis very weak. In spite of above mentioned differences, ithas been observed that dependence of the average brushthickness on the grafting density is linear for brushes ofall different chains. This linear dependence resulted fromour MD simulations has been described well using a sim-ple scaling method in the case of the brush of flexiblechains.Brushes of polyelectrolytes and polyampholytes aredense assembly of these macromolecules in which theinterplay between electrostatic correlations, strong ex-cluded volume effects and bending elasticity of the chainsdetermine equilibrium properties of the system. Themain differences between brushes of polyelectrolyte andpolyampholyte chains originates from opposite trendsof inter- and intra-chain electrostatic interactions anddifferent rules of counterions osmotic pressure in thesebrushes. In a brush of polyelectrolyte chains most ofcounterions are contained inside the brush and their os-motic pressure tends to increase the brush thickness. In abrush of overally neutral polyampholytes however coun-terions are outside the brush and have no effect on thebrush thickness. Linear dependence of polyelectrolytebrush thickness on the grafting density has theoreticallybeen described [17, 18, 19]. The results of a recent sim-ulation of semiflexible polyampholytes [29] and our sim-ulations and theoretical analysis here show that lineardependence of the average thickness on the grafting den-sity is also the case in the brush of polyampholyte chains.Amount of the flexibility of the chains which control thestrength of excluded volume effects and inter-chain cor-relations, causes the equilibrium conformations of thebrushes of three different polyampholytes to be different.In brushes of semiflexible and rodlike polyampholytes,strong dependence of the equilibrium conformations onthe grafting density and separation of the chains into twocoexisting fractions at high grafting densities are resultedfrom strong electrostatic and excluded volume correla-tions between the chains. Similar phenomenon has beenobserved in the brush of rodlike polyelectrolytes [22].To pay attention to possible finite-size effects in oursimulations, we have calculated the average size of thechains lateral fluctuations as well as the histograms P ( z end ) and P ( R ) for a larger brush containing M =8 × ρ a σ = 0 . h l lat i = M P Mi =1 h R i || i , where R i || = | R i − R i . ˆ z | and h ... i denotes averaging over equilibriumconfigurations. We have found that the histograms P ( R )and P ( z end ) do not change noticeably with increasing the system size [29]. Also, we have found that the value of h l lat i is smaller than the lateral size, L , of the simulationbox of a brush containing M = 25 chains at the samegrafting density. This result shows that in simulationof the brush of M = 25 polyampholytes, the chains donot overlap with their own images. Accordingly, to avoidtime consuming simulations of larger brushes, we con-centrated on the simulations of the brushes containing M = 25 chains. Acknowledgments
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02, b) ρ a σ = 0 .
06 and c) ρ a σ = 0 . P ( Z e nd ) Z/ k bend =200 k B T k bend =25 k B T k bend =5 k B T (c) k bend =200 k B T k bend =25 k B T k bend =5 k B T P ( Z e nd ) Z/ (b) k bend =200 k B T k bend =25 k B T k bend =5 k B T P ( Z e nd ) Z/ (a) FIG. 4: (Color online) Histogram of the average distanceof the end monomers of the chains from the grafting sur-face for the brushes of flexible, semiflexible and rodlike di-block polyampholytes at grafting densities a) ρ a σ = 0 .
02, b) ρ a σ = 0 .
06 and c) ρ a σ = 0 . FIG. 5: (Color online) The blob picture of a flexible diblockpolyampholyte brush. The blobs of positive charge are moreprobably surrounded by blobs of negative charge. Positivelyand negatively charged monomers are shown by red (+) andblue ( − ) sphere respectively. The neutral monomers are notshown for clarity. h and ξξ