document: macromolecules containing ionizable groups when dissolve in a polar solvent such as water , dissociate into charged macromolecules and counterions ( ions of opposite charge ) . depending on acidic or basic property of their monomers , 
   ionizable polymers in solution can be classified into polyelectrolytes and polyampholytes . 
   polyelectrolytes contain a single sign of charged monomers and polyampholytes bear charged monomers of both signs . 
   these macromolecules are often water - soluble and have numerous industrial and medical applications . 
   many biological macromolecules such as dna , rna , and proteins are charged polymers . in polymer science 
   , charged polymers has been an important subject during last several decades @xcite . 
   contrary to a polyelectrolyte chain in which the intra - chain electrostatic interactions are repulsive and tend to swell the chain , in a polyampholyte chain attractive interactions between charged monomers of opposite sign tend to decrease the chain size . 
   oppositely charged monomers can be distributed randomly along a polyampholyte chain or charges of one sign can be arranged in long blocks . with 
   the same ratio of positively and negatively charged monomers ( isoelectric condition ) , behaviors of a single polyampholyte and the solution of polyampholytes depend noticeably on the sequence of charged monomers on the chains . 
   for example , it has been shown that the sequence of charged amino acids ( charge distribution ) along ionically complementary peptides affect the aggregation behavior and self - assembling process in the solution of such peptides @xcite . 
   also , using monte carlo simulations it has been shown that charged monomers sequence of charge - symmetric polyampholytes affect their adsorption properties to a charged surface @xcite . 
   the properties of the system of polymers anchored on a surface are of great interest both in industrial and biological applications and academic research . with a sufficiently strong repulsion between the polymers , 
   the chains become stretched and the structure obtained is known as a polymer brush . 
   planar and curved brushes formed by grafted homopolymers have extensively been investigated by various theoretical methods @xcite . 
   the anchored polymers of a brush may be consisting of charged monomers . in this case 
   , the brush is known as a polyelectrolyte or a polyampholyte brush depending on the charged monomers of the chains being of the same sign or being composed of both signs . in a brush of charged polymers , 
   electrostatic interactions introduce additional length scales such as bjerrum length and debye screening length to the system .    in a polyelectrolyte brush , the repulsion of electrostatic origin between the chains can be sufficiently strong even at low grafting densities , making it easy for the system to access the brush regime . 
   polyelectrolyte brushes have been investigated extensively using both theoretical @xcite and computer simulation methods @xcite . at high enough grafting densities and charge fractions of polyelectrolyte chains , 
   most of counterions are trapped inside the polyelectrolyte brush and competition between osmotic pressure of the counterions and elasticity of the chains determines the brush thickness . 
   this regime of a polyelectrolyte brush is known as the osmotic regime in which some theoretical scaling methods predict no dependence of the brush thickness to the grafting density @xcite . 
   however , other scaling method that takes into account the excluded volume effects and nonlinear elasticity of polyelectrolyte chains and is in agreement with experiment and simulation , predicts a linear dependence of the brush thickness on the grafting density @xcite . 
   also , it has been shown that diffusion of a fraction of counterions outside the polyelectrolyte brush leads to a logarithmic dependence of the average brush thickness on the grafting density @xcite . 
   electrostatic interactions in a polyelectrolyte brush cause most of the counterions to be trapped inside the brush and help the chains to be more stretched and the brush to be more aligned . however , in a brush of overally neutral polyampholyte chains , most of counterions are outside the brush and the electrostatic correlations tend to decrease the chains size and the brush thickness . at a given value of the grafting density , the average thickness and equilibrium properties of such a polyampholyte brush are mainly determined by the chains properties such as fraction and sequence of charged monomers and the bending energy . 
   brushes formed by grafted diblock polyampholytes have been investigated by lattice mean field modeling @xcite and computer simulation @xcite . 
   the effect of chain stiffness , charge density and grafting density on spherical brushes of diblock polyampholytes and interaction between colloids with grafted diblock polyampholytes have been studied using monte carlo simulations @xcite . 
   also , using molecular dynamics ( md ) simulations , the effects of various parameters such as charged monomers sequence , grafting density and salt concentration on the average thickness and equilibrium conformations of planar semiflexible polyampholyte brushes have been investigated @xcite .    in this paper , we study planar brushes of flexible , semiflexible and rodlike diblock polyampholytes using md simulations in a wide range of the grafting density . 
   we find that in all cases the average brush thickness linearly depends on the grafting density regardless of the chains different flexibility . 
   our results also show that the strength of this dependence is considerably weaker in the case of the brush of flexible polyampholytes than two other cases . despite mentioned same functionality obtained for the average thickness versus the grafting density for brushes of different polyampholytes 
   we find that histograms of their equilibrium conformations are noticeably different . 
   inter - chain correlations are too weak in the brush of flexible polyampholytes and the brush properties are dominantly determined by single chain behavior . in this case , 
   dependence of the equilibrium conformation of the brush on the grafting density is very weak . in the cases of the brushes of semiflexible and rodlike polyampholytes however , because of the combination of electrostatic correlations and strong excluded volume effects , collective behavior of the chains is dominant and dependence of equilibrium conformations on the grafting density is strong . in these cases , 
   we also observe separation of the anchored chains into two coexisting fractions . using a simple scaling method which is consistently applicable for the brush of flexible chains , 
   we describe theoretically the linear dependence of the brush thickness on the grafting density . 
   the rest of the paper is organized as follows . in section [ simulation ] 
   we describe our model and simulation method in detail and present the results of md simulations . 
   our scaling analysis to describe dependence of the average thickness of flexible diblock polyampholyte brushes on the grafting density is presented in section [ theory ] . in section [ discussion ] 
   we conclude the paper and present a short discussion . 
   in our simulations which are performed with the md simulation package espresso @xcite , each brush is modeled by @xmath0 diblock polyampholyte bead - spring chains of length @xmath1 ( 24 spherical monomers ) which are end - grafted onto an uncharged surface at @xmath2 . 
   the positions of anchored monomers which are fixed during the simulation , form an square lattice on the grafting surface ( @xmath3 plane ) with lattice spacing @xmath4 in which @xmath5 is the grafting density of the chains . 
   the fraction @xmath6 of the monomers of each chain are charged and the chains consist of an alternating sequence of charged and neutral monomers . 
   each chain contains the same number of positively and negatively charged monomers with charges @xmath7 and @xmath8 respectively ( see fig . 
   [ fig1 ] ) . excluded volume interaction between particles 
   is modeled by a shifted lennard - jones potential , @xmath9 in which @xmath10 and @xmath11 are the usual lennard - jones parameters and the cutoff radius is @xmath12 . 
   successive monomers of each chain are bonded to each other by a fene ( finite extensible nonlinear elastic ) potential @xcite , @xmath13 with bond strength @xmath14 and maximum bond length @xmath15 . bending elasticity of the chains 
   is modeled by a bond angle potential , @xmath16 in which @xmath17 is the angle between two successive bond vectors and @xmath18 is the bending energy of the chains . 
   the value of the persistence length , @xmath19 , of the chains depends on the value of @xmath18 as @xmath20 . to model brushes of flexible , semiflexible and rodlike chains , we use four different values of @xmath18 namely @xmath21 , @xmath22 , @xmath23 and @xmath24 respectively . 
   the simulation box is of volume @xmath25 in which @xmath26 is the box width in @xmath27 and @xmath28 directions and @xmath29 is its height in @xmath30 direction and the grafting density is given by @xmath31 . 
   we consider @xmath32 monovalent counterions to neutralize the chains charge . 
   positive and negative monovalent counterions are modeled by equal number of spherical lennard - jones particles of diameter @xmath11 with charges @xmath7 and @xmath8 respectively . 
   all the particles interact repulsively with the grafting surface at short distances with the shifted lennard - jones potential introduced in eq . 
   in addition , a similar repulsive potential is applied at the top boundary of the simulation box and in our simulations @xmath33 . 
   all the charged particles interact with each other with the coulomb interaction @xmath34 in which @xmath35 and @xmath36 are charges of particles @xmath37 and @xmath38 in units of elementary charge @xmath7 and @xmath39 is separation between them . 
   the bjerrum length , @xmath40 , which determines the strength of the coulomb interaction relative to the thermal energy , @xmath41 , is given by @xmath42 , where @xmath43 is the dielectric constant of the solvent and we set @xmath44 in our simulations . 
   periodic boundary conditions are applied only in two dimensions ( @xmath27 and @xmath28 ) . to calculate coulomb forces and energies 
   , we use the so - called @xmath45 technique introduced by strebel and sperb @xcite and modified for laterally periodic systems ( @xmath46 ) by arnold and holm @xcite . the temperature in our simulations 
   is kept fixed at @xmath47 using a langevin thermostat .    for each value of the bending energy , @xmath18 , we do simulations of the brush at dimensionless grafting densities @xmath48 . 
   in the beginning of each simulation , all of the chains are straight and perpendicular to the grafting surface and all the ions are randomly distributed inside the simulation box . 
   we equilibrate the system for @xmath49 md time steps which is enough for all values of the grafting density mentioned above and then calculate thermal averages over @xmath50 independent configurations of the system selected from @xmath51 additional md steps after equilibration . 
   md time step in our simulations is @xmath52 in which @xmath53 is the md time scale and @xmath54 is the mass of the particles . 
   we calculate the average brush thickness which can be measured by taking the first moment of the monomer density profile @xmath55 in which @xmath56 is the number density of monomers as a function of the distance from the grafting surface . for a better monitoring of the statistics of the chains conformations , 
   we calculate the histogram of the mean end - to - end distance of the chains , @xmath57 , in which @xmath58 and @xmath59 is the end - to - end vector of chain @xmath37 . 
   we also calculate the histogram of the average distance of the end monomers of the chains from the grafting surface , @xmath60 , in which @xmath61 and @xmath62 is the @xmath30 component of the end monomer of chain @xmath37 . with the same method discussed in ref . 
   @xcite , it has been checked that our results are not affected by finite - size effects ( see sec . 
   [ discussion ] ) . 
   the average thickness versus the grafting density for brushes of flexible , semiflexible and rodlike diblock polyampholytes are shown in fig . 
   dependence of the average thickness on the grafting density can be described well by a linear function for all values of the bending energy , @xmath18 , that we use in our simulations . 
   also , it can be seen that at all values of the grafting density the average brush thickness decreases with increasing the flexibility of the chains . 
   linear fits to the average thickness of the brushes of semiflexible and rodlike chains for which the persistence length , @xmath19 , exceeds their contour length , @xmath63 , are of approximately the same slope ( see solid and dashed lines in fig . 
   [ fig2 ] ) . 
   the slopes of the linear dependence in cases of two brushes of flexible chains are also approximately the same and differ by a factor of @xmath64 from those of two other cases ( dotted and dash - dotted lines in fig . 
   [ fig2 ] ) . 
   to analyze such dependencies of the brushes thickness on the grafting density , we look at the equilibrium conformations statistics of the chains at different values of @xmath5 . in figs . 
   [ fig3 ] and [ fig4 ] the histograms @xmath57 and @xmath60 at three different grafting densities are shown for brushes of flexible , semiflexible and rodlike chains . because of very similar behaviors of the histograms in @xmath65 and @xmath66 cases , the histograms of the brush with @xmath65 are not shown for clarity of the figures . as it can be seen in fig . 
   [ fig3 ] , in the case of the brush of flexible diblock polyampholytes ( @xmath67 ) , dependence of the histogram profile on the grafting density is very weak . 
   also , in this case , the contribution of large values of @xmath39 ( @xmath68 ) in the histogram is negligible which shows that the chains are mostly coiled at all grafting densities ( see a sample configuration of the brush at @xmath69 in fig . 
   [ fig1]a ) . in the case of the brush of semiflexible chains ( @xmath70 ) , with increasing the grafting density , 
   the values of the histogram corresponding to smaller values of @xmath39 become nonzero showing that polyampholyte chains take buckled conformations at high grafting densities @xcite . 
   as it is expected , the histogram @xmath57 of the brush of rodlike chains ( @xmath71 ) exhibit no noticeable buckling of the chains . 
   the histograms @xmath60 in fig . 
   [ fig4 ] show that in the brush of flexible chains at all grafting densities the end monomers are mostly distributed near the grafting surface showing that the positive blocks of the chains are mostly turned back towards the anchored negative blocks . 
   the profile of this histogram also does nt depend noticeably on the grafting density . 
   the histograms @xmath60 for brushes of semiflexible and rodlike chains show that two maxima appear and their height increase with increasing the grafting density . by combining the information obtained from histograms @xmath57 and @xmath60 for the brush of rodlike chains 
   it can be understood that at high grafting densities a fraction of the chains which are perpendicular to the grafting surface coexist with the remaining fraction which fluctuate in the vicinity of the grafting surface . in the case of the brush of semiflexible chains also , 
   the fraction of perpendicular chains to the brush surface coexist with the chains which are buckled towards the grafting surface @xcite . 
   histograms @xmath57 and @xmath60 show that conformations of the brushes of semiflexible and rodlike chains change noticeably with changing the grafting density despite the case of the brush of flexible chains . 
   as mentioned in the previous section , our md simulations show that the brush thickness is a linear function of the grafting density regardless of different values of grafted diblock polyampholytes bending energy . also , as it is shown in fig . 
   [ fig2 ] , in cases of the brushes of flexible chains the slope of the fitted line to the average thickness versus the grafting density is considerably smaller than that in two other cases . 
   we use here a simple scaling theory similar to that of the solution of charge - symmetric diblock polyampholytes @xcite to describe the linear dependence of the brush thickness on the grafting density . 
   this scaling analysis is applicable to the brushes of flexible diblock polyampholytes . 
   consider @xmath72 flexible diblock polyampholyte chains end - grafted to a flat surface of area @xmath73 . 
   the degree of polymerization of each chain and the fraction of charged monomers are @xmath74 and @xmath75 respectively . 
   electrostatic attractive interactions between oppositely charged blocks of the chains lead them to form a dense layer of positive and negative monomers of average thickness @xmath76 near the grafting surface ( see fig . 
   [ fig5 ] ) . inside the layer , to use the blob concept we define correlation length @xmath77 as a length scale that electrostatic interactions do nt perturb the chains statistics at smaller length scales and are dominant over thermal fluctuations at larger length scales . 
   let suppose that the number of monomers inside the correlation blob is @xmath78 and we have @xmath79 where @xmath80 and @xmath81 are the kuhn length and the flory exponent respectively . 
   accordingly , the layer is a melt of correlation blobs in which positive blobs with a high probability are surrounded by negative blobs . 
   electrostatic interaction between any two neighboring blobs is of the order of the thermal energy , @xmath41 @xmath82 thus we obtain @xmath83 . as a result , 
   the local monomer concentration inside the blob is @xmath84 considering that correlation blobs are space filling , the global monomer concentration , @xmath85 , approximately equals to the local monomer concentration , @xmath86 . thus , dependence of the thickness of a brush of diblock polyampholytes on the grafting density can be obtained as @xmath87 the main prediction of this equation in the range of its validity is as follows . 
   the brush thickness , @xmath76 , is a linear function of the grafting density , @xmath5 , irrespective of the values of the system parameters . 
   only the slope of this linear function , @xmath88 , depends on the values of the system parameters . 
   one should note here that despite the bulk solution of flexible diblock polyampholytes , in a polyampholyte brush the fact that the chains are end grafted to the surface introduces an additional length scale to the system , namely @xmath89 . in the scaling analysis presented here , 
   if the size of the correlation length , @xmath77 , exceeds the separation between the grafting points , @xmath90 , the scaling approach becomes inconsistent . 
   although linear dependence of the brush thickness on the grafting density is observed at all values of the bending energy used in our simulations , the condition @xmath91 is valid only in the case of the brush of flexible chains with @xmath65 . in this case 
   the values of @xmath92 corresponding to lowest and highest values of the grafting density we have used in our simulations are 0.19 and 0.45 respectively . 
   for the set of parameters used in our simulations , in the case of the brush with @xmath65 the upper limit of the validity of the scaling method ( @xmath93 ) corresponds to the grafting density @xmath94 which is quite far from our range of the grafting density . for a brush of flexible chains with @xmath65 using the values 
   @xmath95 and @xmath96 gives the slope @xmath97 for the brush thickness versus the grafting density for the used set of system parameters . as it is shown in fig . 
   [ fig2 ] , the value of @xmath98 obtained from our simulations of flexible chains is @xmath99 which is in reasonable agreement with prediction of the scaling method . 
   our scaling method is not consistently applicable for the brush of flexible chains with @xmath100 because the range of the grafting density used in our simulations is higher than the validity range of the condition @xmath91 in this case . 
   this method is not also applicable for the brushes of semiflexible and rodlike chains because in these cases the kuhn length exceeds the contour length of the chains . 
   the fact that the average thickness linearly depends on the grafting density at all values of @xmath18 used in our simulations shows that although the scaling method presented here can not be used to describe all the simulation results , this linear dependence persists over a wide range of the system parameters . 
   brushes of flexible , semiflexible and rodlike diblock polyampholytes have been studied using md simulations and a scaling analysis has been presented to describe the results of the simulation of flexible chins brush . 
   the average thickness as a function of the grafting density and histograms of equilibrium conformations of the brushes are obtained . 
   strong dependence of the system conformations on the grafting density and separation of the chains into two coexisting fractions at high grafting densities have been observed in cases of the brushes of semiflexible and rodlike chains . in cases of the brushes of flexible chains , 
   single - chain behavior is dominant and dependence of the brush conformations on the grafting density is very weak . in spite of above mentioned differences , it has been observed that dependence of the average brush thickness on the grafting density is linear for brushes of all different chains . 
   this linear dependence resulted from our md simulations has been described well using a simple scaling method in the case of the brush of flexible chains . 
   brushes of polyelectrolytes and polyampholytes are dense assembly of these macromolecules in which the interplay between electrostatic correlations , strong excluded volume effects and bending elasticity of the chains determine equilibrium properties of the system 
   . the main differences between brushes of polyelectrolyte and polyampholyte chains originates from opposite trends of inter- and intra - chain electrostatic interactions and different rules of counterions osmotic pressure in these brushes . in a brush of polyelectrolyte chains 
   most of counterions are contained inside the brush and their osmotic pressure tends to increase the brush thickness . in a brush of overally neutral polyampholytes 
   however counterions are outside the brush and have no effect on the brush thickness . 
   linear dependence of polyelectrolyte brush thickness on the grafting density has theoretically been described @xcite . 
   the results of a recent simulation of semiflexible polyampholytes @xcite and our simulations and theoretical analysis here show that linear dependence of the average thickness on the grafting density is also the case in the brush of polyampholyte chains . 
   amount of the flexibility of the chains which control the strength of excluded volume effects and inter - chain correlations , causes the equilibrium conformations of the brushes of three different polyampholytes to be different . in brushes of semiflexible and rodlike polyampholytes , strong dependence of the equilibrium conformations on the grafting density and separation of the chains into two coexisting fractions at high grafting densities 
   are resulted from strong electrostatic and excluded volume correlations between the chains . 
   similar phenomenon has been observed in the brush of rodlike polyelectrolytes @xcite .    to pay attention to possible finite - size effects in our simulations 
   , we have calculated the average size of the chains lateral fluctuations as well as the histograms @xmath60 and @xmath57 for a larger brush containing @xmath101 semiflexible chains at the grafting density @xmath102 @xcite . 
   the average size of lateral fluctuations is defined as @xmath103 , where @xmath104 and @xmath105 denotes averaging over equilibrium configurations . 
   we have found that the histograms @xmath57 and @xmath60 do not change noticeably with increasing the system size @xcite . 
   also , we have found that the value of @xmath106 is smaller than the lateral size , @xmath26 , of the simulation box of a brush containing @xmath0 chains at the same grafting density . 
   this result shows that in simulation of the brush of @xmath0 polyampholytes , the chains do not overlap with their own images . 
   accordingly , to avoid time consuming simulations of larger brushes , we concentrated on the simulations of the brushes containing @xmath0 chains . 
   j. rhe , m. ballauff , m. biesalski , p. dziezok , f. grhn , d. johannsmann , n. houbenov , n. hugenberg , r. konradi , s. minko , m. motornov , r. r. netz , m. schmidt , c. seidel , m. stamm , t. stephan , d. usov , h. zhang , adv . 
   sci . * 165 * , 79 ( 2004 ) .
summary: planar brushes of flexible , semiflexible and rodlike diblock polyampholytes are studied using molecular dynamics simulations in a wide range of the grafting density . 
   simulations show linear dependence of the average thickness on the grafting density in all cases regardless of different flexibility of anchored chains and the brushes different equilibrium conformations . 
   slopes of fitted lines to the average thickness of the brushes of semiflexible and rodlike polyampholytes versus the grafting density are approximately the same and differ considerably from that of the brushes of flexible chains . 
   the average thickness of the brush of flexible diblock polyampholytes as a function of the grafting density is also obtained using a simple scaling analysis which is in good agreement with our simulations .