Overcharging, charge inversion and reentrant condensation: Using highly-charged polyelectrolytes in tetravalent salt solutions as an example of study
aa r X i v : . [ c ond - m a t . s o f t ] M a y Overcharging, charge inversion and reentrant condensation:Using highly-charged polyelectrolytes in tetravalent salt solutionsas an example of study
Pai-Yi Hsiao ∗ Department of Engineering and System Science,National Tsing Hua University, Hsinchu, Taiwan 30013, R.O.C. (Dated: November 2, 2018)
Abstract
We study salt-induced charge overcompensation and charge inversion of flexible polyelectrolytesvia computer simulations and demonstrate the importance of ion excluded volume. Reentrantcondensation takes place when the ion size is comparable to monomer size, and happens in amiddle region of salt concentration. In a high-salt region, ions can overcharge a chain near itssurface and charge distribution around a chain displays an oscillatory behavior. Unambiguousevidence obtained by electrophoresis shows that charge inversion does not necessarily appear withovercharging and occurs when the ion size is not big. These findings suggest a disconnection ofresolubilization of polyelectrolyte condensates at high salt concentration with charge inversion.
PACS numbers: 82.35.Rs, 36.20.Ey, 87.15.Aa, 87.15.Tt et al. recently explained these phenomena, called reentrant conden-sation, using the idea of overcharging [7]. When the surface charge of PE is reduced to nearlyzero by condensed multivalent counterions, the correlation induced short range attractiondominates the Coulomb repulsion, leading to PE condensation. At a more elevated salt con-centration, PE can get overcharged and the Coulomb repulsion dominates; the condensedPE thus reenters into the solution. In their theory, overcharging implies the sign inversion ofthe effective charge, and consequently, the reversal of the moving direction of PE in an elec-tric field was expected. Reversal of electrophoretic mobility has been observed since decadesago [3] and firstly explained by theorists in 1980’s [8]. Nonetheless, the connection with theredissolution of PE is still not well understood. Recent study showed that the resolubiliza-tion of fd virus bundles happened at high salt concentration without mobility reversal [9].Controversy hence arises. Integral equation theories [8, 10, 11] and simulations [12, 13, 14]2ave both confirmed the possibility of charge overcompensation next to a PE surface. Theyhave also demonstrated that the ion size is an important factor to affect the properties ofa macromolecule. Moreover, an oscillatory behavior in the integrated charge distributionaround a molecule has been found in these studies, which indicates the formation of ioniclayers, alternating the sign of charge. It therefore becomes unclear whether the net charge ofthe whole PE-ion complex inverts its sign or not, even if there is overcharging in the vicinityof the surface. Noticeably, theorists also predicted the importance of Bjerrum associationand the dependence of charge inversion on the ion size [15].The main difficulty in calculation of the net charge comes from no precise informationabout the border of a macroion-ion complex. A simple but accurate way to circumvent thisdifficulty is to do electrophoresis in a weak electric field. In this Letter, we use multiple PEs intetravalent salt solutions as an example and computer simulation as a tool to investigate therelationship between overcharging, charge inversion, and redissolution of PE condensation.Our system contains four bead–spring chains and many charged spheres, which model anionicpolyions and monovalent cations, dissociated from the PEs, and tetravalent cations andmonovalent anions, dissociated from the salts. Each chain consists of 48 monomers and eachmonomer carries a negative unit charge − e . The excluded volume of monomers and ions ismodeled by a shifted Lennard-Jones potential, U LJ ( r ) = ε LJ [2( σ/r ) − , truncated at theminimum, with coupling strength ε LJ = k B T / .
2, where k B denotes Boltzmann’s constantand T the absolute temperature. We assume that the monomers, monovalent cations andanions have equal effective diameter σ = σ m and vary the effective diameter of tetravalentcation σ t from 0 to 4 . σ m . This size variation covers a broad range of interest for theoristsand experimentalists, discussing from point charges to large charged colloids. The virtualsprings jointing monomers on a chain are described by the FENE potential [16], U FENE ( b ) = − . kb max ln(1 − b /b max ), with maximum extension b max = 2 σ m and spring constant k =7 ε LJ /σ m . Solvent is simulated as a medium of uniform dielectric constant ε , and hence, theCoulomb interaction between two particles of valences z i and z j is equal to k B T λ B z i z j /r where λ B ≡ e / (4 πεε k B T ) is the Bjerrum length and ε the vacuum permittivity. We set λ B = 3 σ m . The model is representative of a prototypical PE such as sodium polystyrenesulfonate in an aqueous solution at room temperature. The system is placed in a periodiccubic box and subject to a uniform external electric field pointing to +ˆ x direction. Langevindynamics simulations are employed in this study [17] and the equation of motion reads as3 ¨ ~r i = − mγ i ˙ ~r i + ~F ( c ) i + z i eE ˆ x + ~η i ( t ) where the particles are assumed to have identical mass m and mγ i is the friction coefficient. According to the Stokes law, γ i is linearly proportionalto the particle size σ i ; we thus choose γ i = ( σ i /σ m ) τ − with τ = σ m q m/ ( k B T ). ~F ( c ) i is theconservative force acting on i , and ~η i is the white noise satisfying the fluctuation-dissipationtheorem. E is the strength of the electric field and we set E = 0 . k B T / ( eσ m ). The monomerconcentration is fixed at C m = 0 . σ − m . This C m is well below the overlap threshold andthe results have been shown representative for the range 0 . σ − m ≤ C m ≤ . σ − m [14].In the rest of the text, we use σ m , τ , e , and k B T as the units of length, time, charge, andenergy, respectively. Therefore, concentration will be measured in unit σ − m , strength ofelectric field in unit k B T / ( eσ m ), and so on.We first verified that E = 0 .
05 is a weak field for our system so that the conformation ofthe chains and the structure of the solution are almost not modified by the electric field. Itis justified by comparing both the mean square radius of gyration R g and the mean center-of-mass distance between chains d cm in the presence and in the absence of the electric field,shown in Figs. 1(a) and 1(b). The difference between the two set of data is very small. Wehave verified that the electric field applied here is much weaker than needed to deform thechains.In Fig. 1(a), we observed that the chain size for σ t < . C s up to C ∗ s (= 0 . C s is increased beyond C ∗ s : R g increasesor in other words, the chains swell. This chain collapsing and swelling can be regarded as asingle-chain version of the macroscopic phase separation and redissolution of PEs happeningin multivalent salt solution [6], and have been reported in our previous study [14]. For thecases with larger σ t , the collapsing and swelling basically do not take place. It demonstratesa strong influence of ion excluded volume on the chain size. In addition to the variationof single-chain size, multivalent salt can also provoke subsequently multi-chain aggregationand segregation. When the ion size is intermediate ( σ t = 1 . . d cm curve in Fig. 1(b)shows a deep valley near C ∗ s , with its value smaller than 2 R g , which strongly indicateschain aggregation. At high C s , d cm increases to the value for neutral polymers; PEs hencesegregate. It is worth noticing that even there occurs single-chain collapsing in the region C s ≤ C ∗ s , multichain aggregation may not take place, for example, when σ t = 0 . Q ( r ) around a PE. Q ( r ) denotes the4 -4 -3 -2 d cm R C s (a) -4 -3 -2 C s (b) neutral 0.0 0.5 1.0 2.0 3.0 4.0 FIG. 1: (a) R g and (b) d cm as a function C s for different values of σ t indicated in the figures.The symbols denote the data obtained in the presence of the electric field, whereas the solid curvesnearby are the ones obtained in the absence of it. The dotted curve is the data for neutral polymers. total charge inside a wormlike tube which is the union of the spheres of radius r centeredat each monomer center of a PE. For C s < C ∗ s , Q ( r ) increases monotonically from −
48 to 0with r , due to electroneutrality [14]. If C s > C ∗ s , a positive peak appears in Q ( r ) next to thePE surface. This peak results mainly from an excess number of the tetravalent counterionsappearing near the surface; the chains are thus overcharged. The degree of overchargingincreases with C s . At high C s , Q ( r ) eventually shows an oscillatory behavior around zero,which suggests a repetitive overcompensation of charge inside a wormlike tube. Fig. 2 showsan example of Q ( r ) at high C s = 0 .
008 for different σ t . We observed that the positive peaknear the surface and the oscillatory behavior of Q ( r ) are both enhanced as σ t is increased.These phenomena could be explained from the point of view of entropy [18]. Increasing ionsize decreases the free moving space of ions and thus, decreases the entropy of the solution.The correlation of particles therefore becomes stronger, manifested, for instance, in Q ( r )curve with a more pronounced peak and oscillatory behavior. Notice that for large σ t ,monovalent cations can become energetically competitive with tetravalent ones, to condense5 Q(r) r
FIG. 2: Q ( r ) at C s = 0 .
008 for different values of σ t indicated in the figure onto a PE. It is the condensation of monovalent cations which leads to the small bumps at r = 1 . σ t = 3 . Q eff = v ( c ) d / ( µ ( c ) E ), providedthat the drift velocity v ( c ) d in the electric field E and the kinetic mobility µ ( c ) of the complexare known. Since PE chains move with complexes, we approximated µ ( c ) by chain mobility µ ( p ) , calculated using Einstein relation: µ ( p ) = D ( p ) / ( k B T ). D ( p ) is the diffusion coefficientobtained by taking time derivative of the mean square displacement of the center of massof a PE chain in ˆ y - and ˆ z -directions and the results are shown in Fig. 3. We found that thereferenced D ( p ) for neutral polymers decreases with increasing C s due to the jamming effectof the salt ions presented in solutions. For PEs, D ( p ) is smaller than the referenced D ( p ) while C s < . C ∗ s , the main condensed ions are tetravalent; D ( p ) depends sensitively on σ t . For σ t = 0 .
0, there is no dragging force acting on the tetravalentions; D ( p ) is hence large and the curve shows a hump structure. For σ t = 1 .
0, the dragging6 -4 -3 -2 D (p) C s FIG. 3: D ( p ) as a function of C s for σ t = 0 . force takes effect and in addition, chains aggregate and form big complexes, which leads toa drastic decrease of the diffusivity. The importance of the latter effect can be seen fromthe absence of the drastic decrease in D ( p ) for σ t = 3 . σ t . The decrease of D ( p ) is aresult of the combination of the two effects: ionic jamming and PE complexation with largenumber of ions.We also approximate v ( c ) d by the chain drift velocity v ( p ) d . The computed Q eff are presentedin Fig. 4. In the low-salt region C s < . Q eff weakly depends on the ion size andattains a value roughly equal to −
20. This value is smaller than the prediction of Manningcondensation theory [19], which states a reduction of the effective line charge density on aninfinite rodlike chain to − e/λ B and yields an effective chain charge − . Q eff turns to become sensitive to σ t while C s is larger than 0.001. For σ t < . Q eff increases and surpasses zero around C ∗ s . Itclearly demonstrates the occurrence of charge inversion. Q eff attains a value of roughly 10at high C s for point tetravalent ions and the value gradually decreases as σ t is increased. For7 -4 -3 -2 -40-30-20-10010 Q eff C s FIG. 4: Q eff as a function of C s for different σ t . The value of σ t is given near the associated curve.The dotted curve denotes the data for neutral polymers. σ t = 1 . .
4, a hump appears in Q eff curve and crosses zero line twice, indicating thatcharge inversion takes place only in a window of salt concentration. For σ t ≥ .
0, althoughshowing a hump, the curve stays completely in the negative region — no charge inversiontakes place. Q eff at high C s can be even more negative than in a salt-free solution for largeion size such as σ t = 3 . Q eff obtained here stands for the mean netcharge of a PE complex which may contain multiple chains, and thus, should not be simplyinterpreted as the net charge of a single PE, specially in the mid-salt region where chainsaggregate. A general behavior of decreasing of Q eff in the high-salt region was found. Theonset of the associated phenomenon, the decrease of the electrophoretic mobility at high C s ,has been observed in experiments [11, 21] but never been emphasized before. Both meanfield theory and integral equation theory fail to predict this decrease. Noticeably, from acomparison between Fig. 2 and Fig. 4, we can see that the location of the slipping plane ofa PE in the notion of electrophoresis depends on the ion size. The smaller the ion size, thecloser the plane to a PE. The location is roughly at r = ( σ m + σ t ) / C s for σ t = 2 . σ t = 2 . E = 0 .
005 to 0 .
1, andfound consistent electrophoretic mobility and ion distributions around a PE. It confirmsthat the applied field is weak enough so that the following scenario does not happen: PEsare stripped of their condensed counterions by the electric field, which leads to the negative Q eff . Even though there is a strong segregation for the case σ t = 1 .
0, the absolute value of Q eff is small. These findings suggest that chain redissolution does not come from Coulombrepulsion owing to the inversion of the net charge to a non-negligible value. Moreover,the reversal of electrophoretic mobility can take place only when the size of multivalentcounterions is small, complying with the theory of Nguyen et al. [7] in which they assumedpoint ions. Solis and Olvera de la Cruz [15] predicted a different result: no charge inversionfor small ions. The difference comes from their assumption that the ions have identical size.It artificially aggrandizes the effect of ion association when the ions are small. Simulationsusing a model similar to theirs (cf. the second paper in Ref. [14]) gave consistent results withtheir prediction.We propose the following mechanism for chain decondensation. If C s is high, the increaseof energy related to the segregation of chains will be small because tetravalent counteri-ons present abundantly in the bulk solution and interact with the segregated chains, whichreduces the energy. Therefore, the entropy increase related to segregation can become adominated term, which lowers the free energy. Consequently, the system favors chain segre-gation. The effective charge is another issue. If σ t is large, the electrostatic correlation willbe small. The entropy effect related to the ion excluded volume does not produce sufficientlystrong complexation to lead to charge inversion, although the local overcharging can still bepossible.In summary, we have studied “overcharging” and “charge inversion” of PEs in tetravalent9alt solutions. The crucial role of the ion excluded volume on the PE properties has beendemonstrated. Overcharging does not necessarily bring in charge inversion and the effectivecharge is a nonmonotonic function of salt concentration. Unambiguous evidence has beenpresented that chain redissolution induced by tetravalent salt can take place without chargeinversion.This material is based upon work supported by the National Science Council, the Re-public of China, under Grant No. NSC 95-2112-M-007-025-MY2. Computing resources aresupported by the National Center for High-performance Computing. ∗ E-mail: [email protected][1] A. Y. Grosberg, T. T. Nguyen, and B. I. Shklovskii, Rev. Mod. Phys. , 329 (2002);M. Quesada-P´erez, E. Gonz´alez-Tovar, A. Mart´ın-Molina, M. Lozada-Cassou, and R. Hidalgo-´Alvarez, Chem. Phys. Chem. , 234 (2003); J. Lyklema, Colloids Surf. A , 3 (2006).[2] V. I. Perel and B. I. Shklovskii, Physica A , 446 (1999); B. I. Shklovskii, Phys. Rev. E ,5802 (1999).[3] H. R. Kruyt, ed., Colloid Science , vol. II (Elsevier Publishing Company, New York, 1949);M. Elimelech and C. R. O’Melia, Colloids Surf. , 165 (1990); M. de Frutos, E. Raspaud,A. Leforestier, and F. Livolant, Biophys. J. , 1127 (2001).[4] The key words, overcharging and charge inversion , have been widely used in literatures withdifferent meanings. Refere to, for example, the references in Ref. [1] and F. Jim´enez- ´Angelesand M. Lozada-Cassou, J. Phys. Chem. B , 7286 (2004), for different definitions.[5] V. A. Bloomfield, Curr. Opin. Struct. Biol. , 334 (1996).[6] M. Olvera de la Cruz, L. Belloni, M. Delsanti, J. P. Dalbiez, O. Spalla, and M. Drifford,J. Chem. Phys. , 5781 (1995); E. Raspaud, M. Olvera de la Cruz, J.-L. Sikorav, andF. Livolant, Biophys. J. , 381 (1998).[7] T. T. Nguyen, I. Rouzina, and B. I. Shklovskii, J. Chem. Phys. , 2562 (2000).[8] E. Gonzales-Tovar, M. Lozada-Cassou, and D. Henderson, J. Chem. Phys. , 361 (1985).[9] Q. Wen and J. X. Tang, J. Chem. Phys. , 12666 (2004).[10] M. Lozada-Cassou, E. Gonz´alez-Tovar, and W. Olivares, Phys. Rev. E , R17 (1999);M. Lozada-Cassou and E. Gonz´alez-Tovar, J. Colloid Interface Sci. , 285 (2001).
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