ARTICLE pubs.acs.org/Langmuir
Effects of the Salt Concentration on Charge Regulation in Tethered Polyacid Monolayers Mark J. Uline,† Yitzhak Rabin,‡ and Igal Szleifer*,† †
Department of Biomedical Engineering, Chemistry of Life Processes Institute, Northwestern University, Evanston, Illinois 60208, United States ‡ Department of Physics and Institute for Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel ABSTRACT: Charge regulation in polyacid monolayers attached at one end to a planar surface is studied theoretically. The polyacid layers are designed to mimic single-stranded DNA monolayers. The effects of the local pH and salt concentration on the protonation states of the polyacid layer are studied using a molecular mean-field theory that includes a microscopic description of the conformations of the polyacid molecule along with electrostatic interactions, acidbase equilibrium, and excluded volume interactions. We predict that, in the case of a monovalent salt, NaCl, the amount of proton binding increases dramatically for high surface coverage of polyacid and low bulk salt concentration. When the polyelectrolyte is almost completely charge neutralized by bound protons, there is an expulsion of sodium from the layer. We show that the degree of protonation can go all the way from 0% to 100% when the bulk pH is kept fixed at 7 by changing the surface coverage of polyacid and the bulk salt concentration. The effects of increasing protonation and the expulsion of the cations from the monolayer are reduced when sodium ions are replaced by divalent magnesium ions. Our theoretical results concur with X-ray photoelectron spectroscopy studies of ssDNA monolayers on gold.
’ INTRODUCTION Most of our knowledge about the electrostatic properties of DNA molecules comes from dilute aqueous solution studies in which the bulk concentrations of mono- and divalent salts and of protons (pH) can be treated as externally controlled parameters that are not affected by the presence of minute amounts of DNA molecules. The extrapolation of these dilute solution concepts to the crowded environment of the cell (e.g., viral capsids, chromatin, and chromosomes) is problematic because both the pH and the local concentration of salt can be strongly affected by the high concentration of the charged biomolecules. In vivo studies of such effects are complicated by the presence of many additional components (e.g., proteins), and to isolate the salient physical/ chemical phenomena that govern the electrostatics of dense DNA assemblies, one has to resort to in vitro experiments. One such method devised by Parsegian and co-workers13 involves enclosing a DNA solution in an osmotic bag and concentrating it by increasing the osmotic pressure in the outside solution (this can be done, e.g., by increasing the concentration of PEG polymers in the outside solution). More recently, Naaman et al.4 used a self-assembled monolayer of thiolated DNA on gold (the DNA molecule is attached to gold via the thiol group at one of its ends). Using X-ray photoelectron spectroscopy, they found that while a sparsely grafted monolayer of relatively short (1526 nucleotides) single-stranded (ss) DNA molecules retains its sodium counterions upon repeated washing with deionized water, a densely grafted monolayer is (a) depleted of sodium r 2011 American Chemical Society
ions and (b) maintains its global electroneutrality. They concluded that the dense DNA monolayer becomes protonated even at physiological pH when the concentration of protons in bulk water is around 107 M. A theoretical study5 based on a combination of the Flory Huggins theory of dense polymer solutions and the Alexanderde Gennes model of polymer brushes found that decreasing the concentration of monovalent salt and/or the pH results in an abrupt protonation transition at which (a) DNA becomes fully neutralized by bound protons and (b) the brush thickness is decreased (the latter prediction has not been tested experimentally so far). To make the model analytically tractable, the authors assumed that DNA is always completely neutralized by counterions or by protons. Changing the bulk salt concentration and pH only affects the relative fraction of the two bound species, and the protonation transition is induced by the larger excluded volume of the bound sodium ions. While the model has the advantage of being analytic, it contains a number of undetermined interaction parameters (note that while thermodynamic data on dense DNA solutions are available for dsDNA, no such data exist for ssDNA) and, more importantly, is based on the simplifying assumption that all the sodium cations and protons in the monolayer are bound to DNA molecules. Received: December 9, 2010 Revised: March 1, 2011 Published: March 22, 2011 4679
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Figure 1. Schematic representation of the end-grafted polyacid in the salt solution environment. The circles on the polyelectrolyte segments represent acid groups; the red segments are negatively charged, and the black segments are protonated and therefore charge neutral. The cations, which are colored blue to denote positive charge, are either monovalent in the case of NaCl or divalent in the case of MgCl2. The magnesium ions are larger than the sodium ions to take into account the larger solvation layer. The negative chlorine ions are shown as small red circles.
To characterize the spatial variation of pH as one moves from the brush interior to the bulk solution (in principle, this can be measured using local pH probes), a more elaborate theory that takes into account sufficient molecular detail is clearly needed. We use a molecular mean-field theory that takes into account the size, shape, electrical properties, and physical configurations of the polyacid along with the size, shape, and physical properties of the chemical species that make up the salt solution. The molecular theory explicitly considers the acidbase equilibrium of the chargeable species on the polyacid. Our model has been shown to properly account for the coupling between acidbase equilibrium, electrostatic, and steric interactions that determines the equilibrium behavior of systems with these intermolecular interactions.6,7 Although all of this work is general to any polyacid with the set of physical properties that is in equilibrium with the surrounding systems used in our model, we are ultimately interested in modeling single-stranded DNA tethered to a planar surface in different bulk salt conditions. For this reason, all of the physical properties that we use for our polyacid are reasonable for ssDNA. A schematic of our system is given in Figure 1. The polyelectrolyte layer is in contact with either a NaCl or a MgCl2 salt solution. There are three major reasons for choosing these two salts: (a) magnesium is divalent, whereas sodium is monovalent, (b) the volume of solvated magnesium is larger than that of sodium, and (c) we can directly compare our results to the experimental observations discussed earlier.4 We also present the effects of the protonation states of the polyacid on the equilibrium density profiles of the ions in the vicinity of the wall. We will start by discussing the model system and the molecular meanfield theory that is used to calculate the equilibrium properties of the system. The results and discussion for high and low surface coverages of the polyelectrolyte in contact with salt solutions with either a high or a low bulk salt concentration are presented in the next section for both NaCl salt and MgCl2 salt. Finally, we present a summary of our main results and conclusions.
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’ THEORETICAL APPROACH As seen in Figure 1, the coordinate system is defined so that the z axis is normal to the planar surface, with z = 0 corresponding to the position of the grafting plane. We explicitly consider the inhomogeneity of the system in the direction perpendicular to the surface, z. At each z the system is assumed homogeneous in the x,y plane. Our model polyacid is 12 monomers (i.e., 12 bases) long with a volume of 0.4 nm3 per monomer unit. The monomer unit is designed to mimic the size of the nucleotide unit in ssDNA that consists of the nucleobase along with a five-carbon sugar and phosphate group that comprise the backbone. The polyacid is grafted with a two-dimensional density, σp, in units of molecules per area, and the segment length of the chain is 0.362 nm. Note that if the monomer unit were a sphere, then the diameter would be larger than the segment length. We therefore take the polyacid monomer units to have a shape that allows the volume of adjacent units not to overlap. The grafted polyelectrolyte layer is in contact with a salt solution of either NaCl or MgCl2 in water. The volumes of the solvated Naþaq and Claq are both 0.12 nm3. We take the volume of the solvated divalent magnesium ion to be 0.18 nm3. The water is treated as dissociable from the neutral H2O into H3Oþaq and OHaq, each of the three with a volume of 0.03 nm3. It is convenient to define a local volume fraction, φi(z), which is a function of z and is related to the local number density, Fi(z), by φi(z) = Fi(z)vi, where vi is the molecular volume of that chemical species. We write the Helmholtz free energy of our system as Z βF ¼ σ p ð PðRÞ ln PðRÞÞ þ ÆFp ðzÞæ½fH ðzÞðln fH ðzÞ A R
∑
þ βμ0AH Þ þ ð1 fH ðzÞÞðlnð1 fH ðzÞÞ þ βμ0A Þ dz # Z " 1 dψðzÞ 2 þβ ÆFq ðzÞæψðzÞ ε dz 2 dz Z Z FHþ ðzÞðlnðFHþ ðzÞvw Þ þ Fw ðzÞðlnðFw ðzÞvw Þ 1Þ dz þ Z FOH ðzÞðlnðFOH ðzÞvw Þ 1 þ βμ0Hþ Þ dz þ Z Fþ ðzÞðlnðFþ ðzÞvw Þ 1 βμþ Þ 1 þ βμ0OH Þ dz þ Z ð1Þ dz þ F ðzÞðlnðF ðzÞvw Þ 1 βμ Þ dz The first term in eq 1 is the conformational entropy of the polyacid, where P(R) is the probability of finding a chain in a particular conformation R. A conformation denotes the position of each segment along the chain. The second term describes the free energy arising from the acidbase equilibrium68 on the polyacid chain. Each of the monomer units may exist in two states (protonated or charged), and the equilibrium is given by the following chemical reaction: AH a A þ Hþ
ð2Þ
where A is the charged monomer, Hþ are protons, and AH corresponds to the protonated species. The acidbase term includes the entropy of mixing of the charged and uncharged groups as well as the standard chemical potential of the protonated and unprotonated groups. It is important to note that we neglect any volume change to the polyacid segments due to protonation; i.e., we assume that the segments have the same 4680
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volume whether they are protonated or not. While we do not need to make this assumption in theory, there is not enough experimental quantification of the volume change, and thus, we assume it to be very small. The third term is the contribution of electrostatics,6,911 where ψ(z) is the electrostatic potential, ε is the dielectric coefficient, which is taken to be that of water (78.5 times the value of the dielectric value of a vacuum). Throughout this work, we take the dielectric coefficient to not be a function of position. We performed calculations with a varying dielectric coefficient following the procedure outlined in ref 6 for high and low surface coverage of polyacid for several salt concentrations. Since the dielectric constant in the highly concentrated ssDNA solution is not known, we selected a value of 2 times the dielectric constant of a vacuum to see if this limiting low value would influence the results. We found that there is no qualitative difference and very little quantitative difference in the results, in agreement with previous calculations on related systems.6,10 Therefore, we selected to set the dielectric coefficient to be constant, and its value is that for water. It should be noted that we did not include the self-energy of the mobile ions. We expect that properly including the Born energy12 of all of the ionic species may change some quantitative aspect of the results when one has a varying dielectric coefficient; however, we do not believe that it will have qualitative consequences. ÆFq(z)æ is the ensemble average number density of charges at z given by ÆFq ðzÞæ ¼ ð1 fH ðzÞÞqp ÆFp ðzÞæ þ
∑i qi ÆFi ðzÞæ
∑R PðRÞ vp ðz, RÞ þ Fþ ðzÞvþ þ FðzÞv
þ FHþ ðzÞvHþ þ FOH ðzÞvOH þ Fw ðzÞvw
ð5Þ where Q is a normalization constant ensuring that ∑RP(R) = 1. np(R,z) dz is the number of monomer units located at z when the chain is in conformation R. Minimizing eq 1 yields the following equilibrium expressions for the density profiles of the free species: ð6Þ Fþ ðzÞvw ¼ exp βμþ βπðzÞvþ qþ βψðzÞe F ðzÞvw ¼ exp βμ βπðzÞv q βψðzÞe FHþ ðzÞvw ¼ exp βμ0Hþ βπðzÞvw qHþ βψðzÞe
ð8Þ
ð9Þ Fw ðzÞvw ¼ exp βπðzÞvw
ð10Þ
The functional extreminization of the free energy functional with respect to the electrostatic potential gives a Poisson equation of the form ÆFq ðzÞæ d2 ψðzÞ ¼ dz2 ε
ð11Þ
which is subject to two boundary conditions: dψð0Þ σs ¼ dz ε
ð12Þ
lim ψðzÞ ¼ 0
ð13Þ
zf¥
where σs is a fixed surface charge density which we will always take to be zero in this work. The fraction of protonation, fH(z), is obtained from the minimization of the free energy to be fH ðzÞ φ þ ðzÞ ¼ 0H 1 fH ðzÞ Ka φw ðzÞ
ð14Þ
where K0a = exp[β(μA0 þ μHþ0 μ0AH)]. In the bulk solution the acidbase equilibrium is determined by the equilibrium constant, Ka, given by
ð4Þ
where the first term represents the volume fraction of the polymer at z, with vp(z,R) dz being the volume that the polymer chain in conformation R occupies at z. The next five terms correspond to the volume fraction at z of the cations, anions, protons, hydroxide ions, and water, respectively. To determine the expressions for the equilibrium values of P(R), Fi(z), fH(z), and ψ(z), we find the stationary point of the
ð7Þ
FOH ðzÞvw ¼ exp βμ0OH βπðzÞvw qOH βψðzÞe
ð3Þ
where i runs over each of the components in the fluid that are not the polyacid segments and qi corresponds to the charge of the molecule of type i, i.e., qp = qOH = e, where e is the elementary charge. The angular brackets represent an ensemble average that includes a sum over all of the conformations of the polyacid molecule weighted by the equilibrium probability for that conformation. The fourth term in eq 1 represents the translational entropy of the water molecules. The fifth and sixth terms in eq 1 represent the translational entropy of the dissociated H3Oþaq and OHaq species along with the respective standard chemical potentials of each group. The water molecules are treated as being dissociable into H3Oþaq and OHaq, with the water dissociation constant in the bulk equal to Kbw = [H3Oþaq][OHaq], where pKbw = 14. The seventh and eighth terms in eq 1 represent the translational free energy of the cations and anions of the salt as well as the bulk chemical potential of both species (in the bulk solution in contact with the monolayer). The hard-core repulsive interactions are included through packing constraints, which are 1 ¼ σp
free energy (eq 1) subject to the imcompressibility constraint (eq 4) by introducing Lagrange multipliers, βπ(z). βπ(z) enforce the sum of all of the volume fractions to be equal to unity at each z. From functional minimization, we get the following expression for the probability distribution function of the polyacid chains as a function of configuration R: Z 1 βπðzÞ vp ðR, zÞ dz PðRÞ ¼ exp Q Z Z np ðR, zÞ ln½1 fH ðzÞ dz þ βψðzÞ enp ðR, zÞ dz
Ka ¼
½A ½Hþ ½AH
ð15Þ
The equilibrium constant is determined by the free energy change of the reaction, ΔG0 = μA0 þ μHþ0 μ0AH, where μ0i represent the standard chemical potentials of the molecules involved. The free energy and the acid equilibrium constant are related by Ka = C exp(βΔG0) = CK0a , where C is a dimensional constant. It is important to note that in the above discussion of the acid 4681
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Figure 2. Average number density profile of the polyacid and Naþaq ions as a function of the distance from the wall for the polyacid layer in contact with a NaCl bulk solution. The bulk salt concentration is 0.1 M. The insets show the degree of protonation as a function of the distance from the wall. (A) The surface coverage of polyacid is 0.05 molecule/nm2. (B) The surface coverage of polyacid is 0.45 molecule/nm2. Note the very different scales for graphs A and B and the insets.
equilibrium constant (eq 15) we assumed that the activity for each of the species is given by its molar concentration. We take the pKa of the polyacid groups to be 3 to try to capture the electrostatic properties of DNA. This value of the pKa is used to mimic a conservative estimate based on literature values for the nucleobases adenine (A), guanine (G), and cytosine (C), which are in the range of 2.54.5.1315 The measured pKa of the phosphate backbone is about 01,13 so that neutralization of a nucleotide takes place through the binding of a proton by the nucleobase (rather than by the negatively charged phosphate). Our results are applicable to an ssDNA chain that does not include a substantial amount of thymine (T), since the pKa value for T to gain a proton is less than 2. For reference, we note that when the acid groups are monomers in the bulk solution, the fraction of protonated groups is related to the bulk pH and pKa by the following expression: fH;bulk ¼
1 1 þ 10pH pKa
ð16Þ
where one can easily see that when the bulk pH equals pKa, the fraction of protonated sites is 1/2. Due to the connectivity of the monomers in the polyacid and the highly inhomogeneous environment surrounding the chains, the fraction of protonated monomers is a function of the position away from the grafting surface, fH(z) (in principle, this allows us to define a local pKa(z)). We solve the minimization equations by discretizing space into 0.3 nm increments in the z coordinate, inserting eqs 510 into the incompressibility constraint, eq 4, and solving this set of equations together with the one obtained from the discretized Poisson equation (eq 11) to get the equilibrium values of βπ(z) and βψ(z). We determine the conformations of the polymers by Flory’s rotational isomeric state (RIS) model16 to generate the chain conformations. While the RIS model was originally devised to model alkane chains, this model captures the considerable flexibility of the polymer chains, and the model is easily adaptable to mimic ssDNA. We are then able to determine the equilibrium density profiles of each of the components in the system, the degree of protonation, and the probability distribution function of the polymer chains. From their knowledge all of the thermodynamic and structural properties as a function of the bulk solution condition can be determined. For more details about the numerical methodology see refs 6 and 17.
’ RESULTS AND DISCUSSION The bulk pH in all calculations is 7. We will be varying the bulk salt concentration and surface coverage of the tethered polyacid along with the type of salt comprising the bulk solution. We will show how the optimization of the structure of the polymer layer results from the coupling between the different interactions, the acidbase equilibrium, and the inhomogeneous distribution of the different molecular species and in particular from the need to minimize the strong electrostatic repulsions. There exist three mechanisms to reduce these strong electrostatic repulsions: stretching of the chains, bringing in bulky counterions to help neutralize the charge, and shifting the acidbase equilibrium, increasing protonation of the polyacid, and reducing the polymer charge at the expense of the chemical free energy. Since there are only 12 monomer units comprising the chain, stretching of the chain to reduce electrostatic repulsions is not going to be a major factor. The competition between the other two mechanisms for reducing repulsions changes with the bulk environment, and this is the major focus of the next section. NaCl Salt Solution. Let us begin by looking at the polyacid and sodium ion number density profiles in different bulk salt environments and surface coverages of polyacid. We are taking a grafting density of 0.05 molecule/nm2 to represent the lower surface coverage and 0.45 molecule/nm2 to represent the higher surface coverage. A surface coverage of 0.05 molecule/nm2 corresponds to an average volume fraction of the polyacid in the layer of 0.09, and a surface coverage of 0.45 molecule/nm2 corresponds to an average volume fraction in the layer of 0.60. Figure 2 show the density profiles at a bulk salt concentration of 0.1 M. In Figure 2A the surface coverage of the polyacid is at 0.05 molecule/nm2. The insets in Figure 2 show the local degree of protonation as a function of the distance from the wall. In Figure 2A the equilibrium protonation fraction is around 5 104, which means that the layer is neutralized by the counterions present in solution. This is largely due to the high salt concentration in comparison to that of the free protons in the bulk solution (106 more Naþaq ions relative to H3Oþaq molecules in the bulk). Figure 2B shows the higher surface coverage of the polyacid, and we see that there is a considerably higher fraction of protonated segments (by roughly 3 orders of magnitude), so less sodium cations are needed to help neutralize the brush. Figure 2B clearly shows that the density of sodium ions is consistently below the polyacid density profile. The peak in the sodium ion concentration at the outer boundary of the brush is a consequence of the fact that the polymer concentration in this region 4682
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Langmuir falls rapidly from a large value in the bulk monolayer to zero, and the bulky hydrated counterions which neutralize the charged brush are repelled from the interior of the brush and accumulate at its outer periphery in an attempt to keep the monolayer charge neutral. The first conclusion that we can draw from Figure 2B is that the higher local density of polyacid results in a dramatic shift in the acidbase equilibrium to minimize the electrostatic repulsions. In other words, the system pays in chemical free energy to reduce the electrostatic repulsions and to avoid the localization of a very high concentration of ions, i.e., counterion confinement. Figure 3 shows the local pH(z), defined as pH(z) = log [H3Oþ(z)], where [H3Oþ(z)] is the molar concentration of protons, as a function of the distance from the wall for low and high polyacid surface coverages at a bulk salt concentration of 0.1 M. In Figure 3 the local pH decays to the bulk value within 10 nm of the surface, which is consistent with the Debye screening length being about 1 nm for this bulk salt concentration. As we mentioned earlier, at the lower surface coverage of polyacid, we saw that the sodium ion density profile matched up nicely with the polymer profile, and that indicated that the sodium was neutralizing the layer and thus reducing the electrostatic repulsions. This effect is obvious when one considers that there are 106 more sodium ions relative to H3Oþaq molecules in the bulk, and at this low surface coverage of polymer the excluded volume
Figure 3. Local pH(z) as a function of the distance from the wall for a tethered polyacid layer with surface coverages of 0.05 and 0.45 molecules/nm2. The layer is in contact with a 0.1 M NaCl bath.
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repulsions that prevent the bulky sodium cation from entering the layer are minimal. Figure 3 shows that the local pH(z) does decrease, but only to a value of about 6.25 for the lower surface coverage, and this explains why there is very little protonation of the ionizable sites on the polyacid for this system. As we move to the higher surface coverage, the amount of protonation increases greatly. As seen in Figure 3, the local pH(z) drops to a value of almost 4 inside the layer, and there is an increase of 2 orders of magnitude in the local concentration of protons. The greater reduction in the local pH(z) is a result of the system trying to reduce electrostatic repulsions when faced with much greater steric repulsions. The dense layer makes it more difficult for the bulky sodium ions to enter the layer, so the system decides to reduce electrostatic repulsions by shifting the acidbase equilibrium at the price of chemical free energy. This can be understood by rewriting eq 14 in the form: fH ðzÞ ¼
1 Ka0 φw ðzÞ 1þ φHþ ðzÞ
ð17Þ
According to eq 17, the acidbase equilibrium is shifted by bringing in the smaller protons from the bulk and therefore increasing the local volume fraction of protons, shifting the protonation of the layer, and ultimately decreasing the electrostatic repulsions. This can be thought of as a local Le Chatelier principle, and it explains the much lower local pH(z) in Figure 3 and the much higher protonation fraction seen in Figure 2B for the higher surface coverage system. Due to a combination of excluded volume interactions and the entropy penalty of confining so many counterions in the polyacid layer, the system shifts the mechanism of lowering the electrostatic repulsions from counterions in the layer to charge regulation by shifting the acidbase equilibrium. Figure 4 depicts the polyacid and sodium ion number density profiles for a bulk salt concentration of 1 105 M, at which the bulk concentrations of sodium cations and of protons differ by only 2 orders of magnitude. The polyacid is approximately 56% protonated at the low surface coverage and almost completely protonated in the high surface coverage case. As a result there is almost no need for the sodium cations to penetrate into the layer to neutralize the brush, as can be seen in Figure 4. Figure 4A shows the density distributions and the local protonation fraction at a low surface coverage of the polyacid, and we see that there is still a small enhancement of sodium cations in the layer
Figure 4. Average number density profile of the polyacid and the Naþaq ions as a function of the distance from the wall for the polyacid layer in contact with a NaCl bulk solution. The bulk salt concentration is 1 105 M. The insets show the degree of protonation as a function of the distance from the wall. (A) The surface coverage of polyacid is 0.05 molecule/nm2. (B) The surface coverage of polyacid is 0.45 molecule/nm2. Note the very different scales for graphs A and B and the insets. 4683
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Langmuir compared to the bulk. In Figure 4B we see that the concentration of sodium cations in the dense layer is close to the bulk value, except for a peak localized at the outer boundary of the layer. Figure 5 shows that the local pH decays much slower in the 1 105 M bulk salt than in the higher salt case of Figure 3. The local pH in the low salt case decays to the bulk value after 300 nm, reflecting the 100-fold increase of the Debye screening length as the salt concentration is reduced by 4 orders of magnitude. The values of the local pH inside the layer are also lower in Figure 5 than in Figure 3. In the case of lower surface coverage, the local pH has a minimum value of 2.82, which is a bit lower than the assumed bulk pKa of the polyacid. In the higher surface coverage case, due to a combination of excluded volume and counterion confinement effects, the local pH reaches a value of 2.00, which is a full unit of pH below the bulk pKa value of the polyacid. As the local pH in the layer decreases, the acidbase equilibrium is shifted following Le Chatelier’s principle (see eq 17) to increase
Figure 5. Local pH(z) as a function of the position from the wall for a tethered polyacid layer in contact with a NaCl salt bath with various surface coverages at a bulk salt concentration of 1 105 M. The surface coverage is 0.05 and 0.45 molecule/nm2.
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the amount of protons bound to the polyacid, so the total charge in the polyacid layer decreases. To demonstrate the effect of the surface coverage and bulk salt concentration on the degree of protonation, we plot the ensemble average protonation fraction as a function of the surface coverage and bulk salt concentration in Figure 6. The ensemble average protonation fraction is defined as Z ¥ Æφp ðzÞæfH ðzÞ dz ð18Þ ÆfH æ ¼ 0 Z ¥ Æφp ðzÞæ dz 0
Starting in the high bulk salt case, Figure 6 shows that the degree of protonation increases slowly from approximately 0 to 15% protonation at the densest reported layer with increasing surface coverage. In this regime the electrostatic interactions are screened by the high salt concentration, so it takes a high surface coverage of polyacid for the system to start to shift the acidbase equilibrium to reduce the electrostatic repulsions. At low bulk salt the situation is very different. Now the electrostatic repulsions are not screened, so the degree of protonation is already large at low coverages. Increasing the surface coverage increases the shift in chemical equilibrium, but since the degree of protonation is so high to begin with, the amount of protonation saturates for higher polymer density. This reduction in the effect of screening the electrostatic interactions leads to a change in curvature of the plot as the bulk salt concentration decreases. Figure 6 shows that by keeping the surface coverage fixed and changing the bulk salt concentration the degree of protonation experiences a similar change in curvature. The main conclusion that can be drawn from Figure 6 is that the transition from counterions reducing the electrostatic repulsions in the layer to protonation reducing the electrostatic repulsions in the layer is highly nonlinear in the surface coverage and salt concentration. Namely, this is not simply a linear function of the polymer density due to the nonlinear coupling between the physical interactions and the chemical equilibrium. This effect is due to
Figure 6. Average protonation fraction as a function of the surface coverage and bulk salt concentration of a tethered polyacid layer in contact with a NaCl bulk solution. The bulk pH is 7. 4684
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Figure 7. Sodium and proton density profiles for the low bulk salt concentration case of 1 105 M. (A) Sodium cation density profiles for high and low surface coverage. (B) Proton density profile for high and low surface coverage. Note the very different scales for graphs A and B.
Figure 8. Integrated charge density as a function of the position away from the wall. High and low bulk salt concentrations are plotted on the same plot for different surface coverages. (A) The surface coverage is 0.05 molecule/nm2. (B) The surface coverage is 0.45 molecule/nm2. Note the very different scales for graphs A and B.
the complex coupling of interactions in the highly inhomogeneous layer, in particular, the relaxations in counterion confinement interactions by paying in acidbase chemical free energy. Figure 6 shows how the system can go from a degree of protonation equal to 0 all the way to a degree of protonation equal to 1 by changing the surface coverage and bulk salt concentration all with the same bulk pH of 7. To understand the sodium cation and proton distributions for the low bulk salt concentration, we display the profiles for the high and low surface coverage of the tethered polyacid together in Figure 7. Inside the polymer layer (z < 3.6nm), Figure 7A shows that the sodium number density profiles have similar values in the dense and the dilute monolayers, presumably because the opposing effects of the driving force (number of negative charges) and excluded volume on the cation concentration inside the monolayer result in a similar net effect. As has been discussed earlier in this paper, the sharp peak in sodium concentration at the outer boundary of the dense layer where the local polyacid concentration is abruptly reduced is the result of the weakening of excluded volume effects in this region and the need to neutralize the layer. Figure 7B shows that although the concentration of free protons inside the dense brush is much larger than that inside the dilute one, it is still an order of magnitude smaller that that of the sodium cations (even at local pH 2; see Figure 5), and we conclude that electrostatic screening is always controlled by the latter ions. Note that the role of increasing the proton concentration is not to act as counterions but rather to shift the chemical equilibrium and uncharge the polymer segments; see Figure 6. We conclude this section by looking at the integrated charge density as a function of the distance from the wall. The integrated
charge, σint(z), is given by the following expression: Z σint ðzÞ ¼ 0
z
ÆFq ðz0 Þæ dz0
ð19Þ
Figure 8 shows the integrated charge for the high and low bulk salt concentration at the same polyacid surface coverage. Each of the plots shows a minimum value in the integrated charge near the end of the layer. The width of the minimum of the charge profile reflects the width of the interface of the monolayer and is therefore much smaller for the denser brush. In both plots the charge neutrality for the high bulk salt concentration is recovered at distances approximately twice the thickness of the brush. There is a considerably longer tail in the decay of the integrated charge for the low salt concentration cases, which corresponds to a larger Debye screening length. Both the high and low coverages reach zero integrated charge around 200 nm for the 1 105 M bulk salt concentration. It is interesting to note that the integrated charge is independent of the salt concentration up to 2.5 nm, i.e., more than 70% of the brush. This is true for both surface coverages, indicating that it is the polymers that determine the optimal local environment regardless of what the solution screening length is, or in other words, the relevant length scale in the interior of the layer is determined by the brush and not the solution conditions. This is of course not the case for the decay of the free ion profiles to their bulk values. MgCl2 Salt Solution. In this section we look at the effects of changing the salt from NaCl to MgCl2. The differences between the magnesium and the sodium ions are the valency and the size of the ions (Mg2þ is 50% larger in volume). 4685
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Figure 9. Average number density profile of the polyacid and the Mg2þaq ions as a function of the distance from the wall for a tethered polyacid in contact with a MgCl2 bulk solution. The bulk salt concentration is 0.1 M. The insets show the degree of protonation as a function of the distance from the wall. (A) The coverage is 0.05 molecule/nm2. (B) The coverage is 0.45 molecule/nm2. Note the very different scales for graphs A and B and the insets.
Figure 10. Average number density profile of the polyacid and the Mg2þaq ions as a function of the distance from the wall for a tethered polyacid in contact with a MgCl2 bulk solution. The bulk salt concentration is 1 105 M. The insets show the degree of protonation as a function of the distance from the wall. (A) The coverage is 0.05 molecule/nm2. (B) The coverage is 0.45 molecule/nm2. Note the very different scales for graphs A and B and the insets.
Figure 11. Local pH(z) as a function of the distance from the wall for a polyacid layer in contact with a MgCl2 salt bath for various surface coverages and bulk salt concentrations. The surface coverages are 0.05 and 0.45 molecule/nm2. (A) The bulk salt concentration is 0.1 M. (B) The bulk salt concentration is 1 105 M. Note the very different scales for graphs A and B.
Figure 9 shows the polyacid and magnesium number density profiles at a bulk salt concentration of 0.1 M. In Figure 9A the surface coverage is at 0.05 molecule/nm2. The equilibrium protonation fraction remains low (around 2 104), which means that the layer is neutralized by the magnesium cations. We can see in Figure 9A that the density profile of the cation follows the DNA profile (recall that the magnesium cations carry two positive charges, so if the number density is almost half that of the polymer number density, there is an almost equal amount of positive and negative charge in that region). Figure 9B shows the higher surface coverage, and since the fraction of protonated segments is still very small (around 1%), even at high coverage the brush is neutralized mainly by magnesium cations. This means that while there is still an effect of excluded volume and
counterion confinement and some protons enter the layer to shift the acidbase equilibrium and reduce the electrostatic repulsions, this effect is greatly reduced in the MgCl2 salt case compared to the NaCl salt. Namely, the Mg2þ confinement is a better choice for the system to neutralize the charge of the polymer. The free energy cost of confinement, both excluded volume and entropy, is less than the chemical free energy cost of protonation. Figure 9 shows that the magnesium cations have no trouble getting into the monolayer even though they have a larger volume compared to the sodium ions. This effect is easily understood when one considers the fact that one doubly charged magnesium ion does the job of two sodium ions in neutralizing the layer, and therefore, the number of magnesium ions needed 4686
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Langmuir to neutralize the brush is half that of sodium ions. Since we took the volume of the hydrated magnesium ion to be 50% larger than that of the hydrated sodium ion while the number of magnesium ions inside the brush is about 2 times lower than in the case of sodium, the total volume occupied by magnesium in the brush is much lower than the corresponding volume occupied by sodium ions. This reduction in steric repulsive interactions, together with the lower loss of entropy by confinement, for magnesium compared to sodium aids in the enhanced retention of magnesium ions in the dense brush relative to the sodium case (compare Figures 4 and 9). Figure 10 also shows that the system reduces its electrostatic repulsions by enhancing the bulky salt cation penetration into the layer to neutralize the polymer charge even for the very low salt concentration of 1 105 M. This is very different from the monovalent cation case where the system prefers to pay acidbase free energy to shift the chemical equilibrium to reduce the electrostatic repulsions for the low bulk salt case. The degree of protonation seen in the insets of Figures 9 and 10 shows that acidbase equilibrium is not shifted as much with the divalent cation. The maximum degree of protonation is half of what was seen with the sodium salt. Figure 11 shows the local pH(z) as a function of the distance from the wall for several different surface coverages of polyacid and bulk magnesium salt concentrations. In comparing Figures 11 with Figures 3 and 5, the local pH never falls below a value of 3.6, and therefore, we expect the polyacid monolayer to be less protonated for the magnesium compared to the sodium salt. These results demonstrate that the free energy benefit of having a divalent ion in the polyacid layer outweighs the free energy penalty of having the very bulky counterion in the layer
Figure 12. Average protonation fraction as a function of the surface coverage of a tethered polyacid layer in contact with a MgCl2 solution at bulk salt concentrations of 0.1 and 1 105 M.
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even at a very high surface coverage and low bulk salt concentration, where one may expect excluded volume effects to dictate a shift in acidbase equilibrium to reduce the electrostatic repulsions and therefore exclude Mg2þaq from the layer. Figure 12 shows that the ensemble average protonation fraction (eq 18) increases even more slowly with increasing surface coverage in MgCl2 salt compared with NaCl salt. We see the same general trend with increasing surface coverage at constant salt concentration, but the effect is not as great on the absolute magnitude of the protonation fraction. For the same salt concentration of 0.1 M and the same surface coverage of tethered polyacid of 0.45 molecule/nm2, the ensemble average protonation fraction is more than an order of magnitude smaller for the magnesium compared to the sodium cations. We finish this section by looking at the integrated charge profiles for the MgCl2 salt. Figure 13 shows the integrated charge for the high and low bulk salt concentration at the same surface coverage of polyacid. As with Figure 8, the width of the minimum of the charge profile reflects the width of the interface of the monolayer. The systems in contact with the MgCl2 salt solution become charge neutral closer to the surface boundary of the polyacid relative to the sodium case. This is due to the Debye screening length being shorter in the presence of the divalent cation compared to the NaCl case for the same bulk salt concentration. Both Figures 8 and 13 show the tendency for the total charge density to be positive at the wall (z = 0). This is a common effect in systems where there is a polyacid tethered to a planar surface. The reason is that there is always a depletion of polymer density at the repulsive surface, and as a result the concentrations of all of the other species are enhanced at the wall. Since there are considerably more positive than negative mobile ions present to neutralize the layer, the resulting total charge density at the surface of the wall is positive. Effects of Decreasing Salt on the Density Profile of the Polyacid Monolayer. Figure 14 shows the polyacid volume fraction profile for 0.05 molecule/nm2 coverage for a variety of bulk salt concentrations in both NaCl and MgCl2 solutions. Starting with the sodium case in Figure 14A, we can see that the layer stretches upon decreasing the bulk salt concentration from 1 to 0.01 M. This stretching is due to increasing the Debye screening length and therefore increasing the electrostatic repulsions. At 0.1 M the Debye length is essentially 1 nm, and 23 Debye lengths correspond to the length of the entire chain, so the chain is almost completely stretched at 0.1 M. Decreasing the bulk salt concentration from 0.01 to 0.001 M does nothing to the chain since it is fully stretched at this point, and one can see in
Figure 13. Integrated charge density as a function of the distance from the wall in the MgCl2 salt solution. High and low bulk salt concentrations are plotted on the same plot for different surface coverages. (A) The coverage is 0.05 molecule/nm2. (B) The coverage is 0.45 molecule/nm2. Note the very different scales for graphs A and B. 4687
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Figure 14. Volume fraction profiles for a surface coverage of polyacid of 0.05 molecule/nm2. The figure shows the effect of the bulk salt concentration on the polyelectrolyte density profile. (A) The monolayer is in NaCl salt. (B) The monolayer is in MgCl2 salt.
Figure 14A that the polyacid profiles for these two salt concentrations correspond to fully stretched chains and the plots lie completely on top of each other. The fraction of protonation at 0.001 M is still very low (only 4% bound protons), so the chain is overwhelmingly charged at a large Debye screening length. At 1 105 M, the story changes and the chain starts to collapse again. This onset of collapse corresponds to the chain being 57% protonated, and as a result of this increased protonation, there is a decrease in electrostatic repulsions along the chain. This stretching and collapsing effect with decreasing bulk salt concentration was seen recently in experimental and theoretical studies of poly(acrylic acid) brushes,17,18 as well as in previous studies of polyelectrolytes tethered to surfaces.6,1923 The reentrant behavior (stretching followed by collapse) of the brush, as the concentration of monovalent salt is decreased, is the consequence of the fact that electrostatic interactions inside the brush can be reduced both by screening and by protonation. Initially, as the bulk concentration of salt is decreased from 1 to 0.001 M, the screening length increases, resulting in enhanced electrostatic repulsions between the charged chains, and this results in increased chain stretching as the system tries to reduce the repulsions. At some point (0.010.001 M salt) the brush becomes very highly stretched, and further reduction of the salt concentration makes the free energy cost of electrostatic repulsions prohibitively high, so the system minimizes the repulsions by shifting the acidbase equilibrium and neutralizing the negatively charged chains by bound protons. Figure 14B shows the polyacid volume fraction profiles in the MgCl2 salt solution. In this case the chain continues to stretch as the bulk salt concentration is decreased. This result is consistent with the fact that the Debye screening length in this salt solution is about half the length in the NaCl salt solution, so we need to get to lower salt concentrations to see the electrostatic repulsions drive the chain to be fully stretched. As mentioned above, the benefit of having a cation with two charges neutralizing the polyacid compensates for the steric repulsions and the entropic loss of having Mg2þaq in the layer. Therefore, there is not enough of a benefit in the free energy for the protons to enter the layer to shift the acidbase equilibrium. The local pH never gets low enough in the magnesium salt to drive the acidbase equilibrium to very high levels of protonation on the polyacid. As a result we never see a transition back to collapse of the layer with decreasing salt concentration with the magnesium (divalent) salt. We show the results for the lower surface coverage of the polyacid because there is enough volume for the chain to expand and contract. For the higher surface coverages this effect is greatly reduced due to
greater steric repulsions in the monolayer and the chains already being highly stretched.6
’ CONCLUSIONS We have employed a molecular mean-field theory to study the effects of the bulk salt concentration on the charge regulation of polyacid layers, designed to mimic an ssDNA layer, tethered to a planar surface. In general, we find that by lowering the bulk salt concentration we are increasing the amount of protons binding to the polyelectrolytes in the layer. We examined both the low and the high surface coverages of the polyacid to quantify the effect of increasing the total amount of steric repulsions, as well as the total amount of negative charge in the layer. We started with the NaCl solution and observed at the low surface coverage that decreasing the bulk salt concentration increased the protonation, but only to a point where the sodium cations are still needed to neutralize the charge in the layer. At the high surface coverage, decreasing the bulk salt concentration increased the protonation to a point where the sodium is no longer needed to neutralize the layer and nearly all the sodium ions are expelled from the layer. This is in agreement with recent experiments of ssDNA4 that showed that, as the surface coverage increased, repeated washing of the tethered ssDNA layer (reducing the salt concentration) expelled the sodium ions from the layer. We also found that replacing the sodium salt by that of magnesium decreased the protonation of the polyacid and reduced the cation expulsion from the layer (because of the 2-fold higher valency of Mg compared to Na). Even though all the qualitative trends predicted by our model are in agreement with experimental observations on tethered ssDNA on gold surfaces discussed earlier,4 quantitative comparisons cannot be made at this time because the experimental salt concentrations are not well-defined (the salt concentration is affected by the process of washing). While most of the present results agree with the predictions of the simple model of ref 5, contrary to this work in which protonation of DNA resulted in a sharp decrease in the layer thickness, here we find that the density profile of the brush experiences more subtle changes by changing the bulk salt concentration (and therefore by protonation). It is important to mention that the main conclusion from this work is the nontrivial coupling that exists between chemical equilibrium, physical interactions, and molecular organization with inhomogeneous polymer layers. The dual role of the salt to screen and regulate charge is highly nonlinear and clearly depends on the valency of the counterions. The main point is that the free energy treatment should contain all of the contributions 4688
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Langmuir and that the optimization should be done in a coupled way to obtain the proper behavior. Another recent publication from this group by Tagliazucchi et al.24 further strengthens this argument. Previous work in our research group has shown that the theory provides quantitative predictions, as compared to experimental observations,7,17,18,25 for the behavior of polyacids and polybases in a variety of environments all in monovalent salt. The quality of the approximations in the theory for multivalent ions is yet to be tested; however, we believe that the main qualitative effects are captured in this approach. However, the degree that electrostatic coupling of divalent ions is captured needs to be tested.26 Finally, we comment on DNA nucleobase sequence dependence, which was not considered in the present work. Our results are consistent with the experimental observation of similar behavior for ssDNA in which no sequence dependence was reported.4 The agreement is somewhat unexpected in view of the rather large range of pKa values of the different nucleotides. A tentative explanation for the absence of the sequence effect is the possibility of strong attractive interactions between the grafted DNA chains induced, e.g., by the stacking of bases of neighboring chains. Such attractive interactions could lead to the formation of very dense domains of DNA in which packing considerations promote increased protonation, therefore effectively increasing the local pKa of all of base units. Further experimental and theoretical studies are clearly needed to address these issues.
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(14) Verdolino, V.; Cammi, R.; Munk, B. H.; Schlegel, H. B. J. Phys. Chem. B 2008, 112, 16860. (15) Lippert, B. Chem. Biodiversity 2008, 5, 1455. (16) Flory, P. J. Statistical Mechanics of Chain Molecules; WileyInterscience: New York, 1969. (17) Gong, P.; Wu, T.; Genzer, J.; Szleifer, I. Macromolecules 2007, 40, 8765. (18) Wu, T.; Gong, P.; Szleifer, I.; Vlcek, P.; Subr, V.; Genzer, J. Macromolecules 2007, 40, 8756. (19) Pincus, P. Macromolecules 1991, 24, 2912. (20) Israels, R.; Leermakers, F.; Fleer, G. J.; Zhulina, E. B. Macromolecules 1994, 27, 3249. (21) Fleer, G. J. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 936. (22) Bieslaski, M.; Ruhe, J. Macromolecules 2002, 35, 9480. (23) Zhulina, E. B.; Borisov, O. Macromolecules 2005, 38, 6726. (24) Tagliazucchi, M.; Olvera de la Cruz, M.; Szleifer, I. Proc. Natl. Acad. Sci. U.S.A. 2010, 107, 5300. (25) Tagliazucchi, M.; Calvo, E. J.; Szleifer, I. Langmuir 2008, 24, 2869. (26) Ermoshkin, A. V.; Olvera de la Cruz, M. Phys. Rev. Lett. 2003, 90, 125504.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT This work was supported by the National Science Foundation under Grant CBET-0828046. The work of Y.R. was supported by grants from the Israel Science Foundation and the Israel-France Research Network Program in Biophysics. We also thank the Materials Research Science and Engineering Program of the National Science Foundation (Grant DMR-0520513) at Northwestern University for facilitating the visit of Y.R. ’ REFERENCES (1) Rau, D. C.; Lee, B.; Parsegian, V. A. Proc. Natl. Acad. Sci. U.S.A. 1984, 81, 2621. (2) Leikin, S.; Rau, D. C.; Parsegian, V. A. Phys. Rev. A 1991, 44, 5272. (3) Rau, D. C.; Parsegian, V. A. Biophys. J. 1992, 61, 246. (4) Ray, S. G.; Cohen, H.; Naaman, R.; Rabin, Y. J. Am. Chem. Soc. 2005, 127, 17138. (5) Rappaport, S. M.; Medalion, S.; Rabin, Y. Soft Matter 2009, 5, 3010. (6) Nap, R.; Gong, P.; Szleifer, I. J. Polym. Sci., Part B: Polym. Phys. 2006, 44, 2638. (7) Tagliazucchi, M.; Calvo, E. J.; Szleifer, I. J. Phys. Chem. C 2008, 112, 458. (8) Raphael, E.; Joanny, J.-F. Europhys. Lett. 1990, 13, 623. (9) Schwinger, J.; DeRaad, L.; Milton, K.; Tsai, W.-Y. Classical Electodynamics; Perseus Books: New York, 1998. (10) Croze, O. A.; Cates, M. E. Langmuir 2005, 21, 5627. (11) Wang, Z.-G. J. Theor. Comput. Chem. 2008, 7, 397. (12) Wang, Z.-G. Phys. Rev. E 2010, 81, 021501. (13) Bloomfield, V. A.; Crothers, D. M.; Tinoco, I. Nucleic Acids: Structures, Properties, and Functions; University Science Books: Herndon, VA, 2000. 4689
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