aa r X i v : . [ c ond - m a t . s o f t ] D ec Nonlinear screening of chargedmacromolecules
By Gabriel T´ellez
Departamento de F´ısica, Universidad de los Andes, A.A. 4976, Bogot´a, Colombia
We present several aspects of the screening of charged macromolecules in an elec-trolyte. After a review of the basic mean field approach, based on the linear Debye–H¨uckel theory, we consider the case of highly charged macromolecules, where thelinear approximation breaks down and the system is described by full nonlinearPoisson–Boltzmann equation. Some analytical results for this nonlinear equationgive some interesting insight on physical phenomena like the charge renormaliza-tion and the Manning counterion condensation.
Keywords: Colloids, electrolytes, nonlinear Poisson–Boltzmann equation
1. Introduction and the linear Poisson–Boltzmann equation
A colloidal suspension is a system composed of two substances, one dispersed intothe other. The dispersed phase is composed of macromolecules, with size of theorder 10 − m – 10 − m, while the dispersion medium, or continuous medium, iscomposed of small micromolecules, and/or microions, with size of the order of thenanometer. Of particular interest are the charge-stabilized colloids, where the dis-persed macromolecules have ionizable sites and, when immersed into the dispersionmedium, they acquire a surface electric charge which ensures repulsion betweenthem and thus allows the colloid to stabilize and prevents aggregation.Since the length and time scales of the microcomponents of the dispersionmedium are much smaller than the ones of the dispersed phase, it is convenientto average over the degrees of freedom of the dispersion medium and treat thesystem as a one-component system composed by the macromolecules which inter-act via an effective potential. To understand the physical and thermodynamicalproperties of these systems, it is important to determine this effective interactionbetween the macromolecules, which results not only from the direct interaction be-tween the macromolecules, but also the interaction mediated by the microions ofthe dispersion medium.The basic theory to find this effective interaction for charge-stabilized colloid wasdeveloped independently by [Derjaguin and Landau 1941], and [Overbeek and Verwey 1948],and it is known as the DLVO theory. It is based on the work of [Debye and H¨uckel 1923].Let us consider a spherical charged macromolecule, with radius a , and charge Ze ( e is the elementary charge) immersed in an electrolyte with positive microions ofcharge z + e and average density n + , and negative microions of charge − z − e and aver-age density n − . Without loss of generality we can suppose Z >
0. [Debye and H¨uckel 1923]idea is to treat the microions in a mean field approximation: at temperature T , thelocal density of microions of charge q ± = ± z ± e at a distance r from the macro- Article submitted to Royal Society
TEX Paper
Gabriel T´ellez molecule can be approximated by n ± ( r ) = n ± e − βq ± Ψ( r ) , (1.1)where Ψ( r ) is the electrostatic potential, and β = 1 / ( k B T ), with k B the Boltzmannconstant. Replacing this into Poisson equation of electrostatics yield the Poisson–Boltzmann equation∆Ψ = − πeǫ (cid:0) z + n + e − βez + Ψ − z − n − e βez − Ψ (cid:1) , (1.2)where ǫ is the dielectric constant of the dispersion medium.It is convenient to introduce the following notations: the reduced potential y = βe Ψ, the Bjerrum length l B = βe /ǫ , the Debye length κ − = (4 πl B ( z n + + z − n − )) − / . With these notations, Poisson–Boltzmann equation reads∆ y = κ z + + z − (cid:2) e z − y − e − z + y (cid:3) . (1.3)If the electrostatic coupling between the macromolecule and the microions is small, y ( r ) ≪
1, for any distance r , the nonlinear Poisson–Boltzmann equation (1.3) canbe linearized to obtain ∆ y = κ y . (1.4)For an impenetrable spherical macromolecule with uniform surface charge (totalcharge Ze and radius a ), the solution of this equation is the DLVO potential y ( r ) = Zl B e κa κa e − κr r . (1.5)From this equation, one can see that 1 /κ is the screening length.It is also interesting to consider the case of cylindrical macromolecules, for in-stance stiff polyelectrolytes, ADN, etc. As a first approximation, for an infinitelylong cylinder with radius a and linear charge density e/ℓ , uniformly spread over itssurface, the solution of equation (1.4) gives the electrostatic potential at a radialdistance r from the cylinder y ( r ) = 2 ξ ˆ aK (ˆ a ) K (ˆ r ) , (1.6)where K and K are the modified Bessel functions of order 0 and 1. We have definedthe reduced linear charge density of the cylinder ξ = l B /ℓ , and it is convenient tomeasure the distances in Debye length units: ˆ r = κr and ˆ a = κa . The boundaryconditions that complement the differential equation (1.4) to yield the solution (1.6)are lim r → a r dydr = − ξ , and lim r →∞ ∇ y ( r ) = 0 . (1.7)It should be noticed that at large distances from the cylinder, compared to theDebye length, the potential exhibits again an exponential decay, as for the case ofspherical macromolecules, y ( r ) ∼ ξ ˆ aK (ˆ a ) r π κr e − κr , r ≫ κ − . (1.8) Article submitted to Royal Society onlinear screening of charged macromolecules
2. Charge renormalization
For a highly charged macromolecule, Poisson–Boltzmann equation (1.3) cannot belinearized near the macromolecule surface. However, due to the screening effect,the potential will decay and become small, | y ( r ) | ≪
1, at large distances from themacromolecule surface, r − a ≫ κ − . In that far region, the linear version (1.4)of Poisson–Boltzmann equation holds. Then, for a spherical macromolecule, thepotential will behave as y ( r ) ∼ A e − κr r (2.1)at large distances from the macromolecule. To find the constant of integration A ,one needs to enforce the boundary condition at the surface of the macromoleculethat the normal component of the electric field is proportional to the surface chargedensity. However, the form (2.1) of the potential is not valid in that close region.One needs to find also the form of the potential close to the macromolecule sur-face, and connect it to the large-distance behavior (2.1) to find explicitly the in-tegration constant A . In analogy to the linear solution (1.5), one can write the A = Z ren l B e κa / (1 + κa ), defining a renormalized charge Z ren . The large-distancebehavior of the potential then takes a DLVO familiar form y ( r ) ∼ Z ren l B e κa κa e − κr r , (2.2)but replacing the bare charge Z of the macromolecule by the renormalized one Z ren .In the cylindrical geometry, the renormalized charge concept also applies. In thatcase, the large-distance behavior of the potential is y ( r ) ∼ ξ ren ˆ aK (ˆ a ) K (ˆ r ) , (2.3)with a renormalized linear charge density ξ ren . The determination of the renormal-ized charge requires knowledge of the short-distance behavior of the solution ofthe nonlinear Poisson–Boltzmann equation (1.3). This can be done numerically, asin the original work of [Alexander et al. 1984]. In experimental situations, the un-known renormalized charge is often taken as an adjustable fitting parameter. Thereare also analytical approaches to find the renormalized charge [Trizac et al. 2002,Shkel et al. 2000, T´ellez and Trizac 2004], mostly based on approximations usingthe solution to the nonlinear Poisson–Boltzmann equation in the planar case.Let us illustrate the concept of charge renormalization in the planar geome-try where an explicit solution for the nonlinear Poisson–Boltzmann equation is Article submitted to Royal Society
Gabriel T´ellez known [Gouy 1910, Chapman 1913]. The system is an infinite charged plane, withcharge density σ >
0, immersed in an electrolyte, which, for simplicity, we considersymmetric z + = z − = 1. Let Ox be the axis perpendicular to the plane, which wesuppose located at x = 0, the electrolyte occupies the region x >
0. The nonlinearPoisson–Boltzmann equation in this situation reads d y ( x ) dx = κ sinh y ( x ) . (2.4)It can be integrated once by multiplying by dy/dx , dy ( x ) dx = − κ sinh y ( x )2 . (2.5)where the boundary condition dy/dx → x → ∞ has been used. This lastequation is separable and can be integrated, finally obtaining y ( x ) = 2 ln 1 + Ae − κx − Ae − κx . (2.6)Where A is a constant of integration, which is found using the boundary conditionat the surface of the charged plane dydx (0) = − πl B σ/e . (2.7)Notice that at large distances from the plane, κx ≫
1, the potential behaves as y ( x ) ∼ Ae − κx . (2.8)This is the expected behavior for the linear version of Poisson–Boltzmann equa-tion in this geometry y ′′ lin ( x ) − κ y lin ( x ) = 0. The linear solution is y lin ( x ) =4 πl B σe − κx / ( eκ ). Comparing to the large-distance behavior of the nonlinear so-lution (2.8), one can define the renormalized surface charge density σ ren , by writingthe constant of integration A as A = πl B σ ren / ( eκ ) . (2.9)Then, the large-distance behavior of the potential is y ( x ) ∼ πl B σ ren eκ e − κx . (2.10)To find explicitly the renormalized charge, one needs to apply the boundary condi-tion (2.7) at the surface of the charged plane. For this, one needs the short-distancebehavior of the potential. From the explicit solution (2.6), we find y ( x ) = 2 ln 1 + A − A − Ax − A + O ( x ) , (2.11)when x →
0. Using this, we apply the boundary condition (2.7) to find A − A = πσl B eκ . (2.12) Article submitted to Royal Society onlinear screening of charged macromolecules Π l B €€€€€€€€€€Κ Σ Π l B €€€€€€€€€€Κ Σ ren Figure 1. Renormalized charge as a function of the bare charge in the planar case, for asymmetric electrolyte.
Solving, and using (2.9), we find the renormalized surface chargeˆ σ ren = √ ˆ σ + 1 − σ , (2.13)where we have defined reduced charge densities ˆ σ = πl B σ/ ( eκ ) and ˆ σ ren = πl B σ ren / ( eκ ).Figure 1 shows a plot of the renormalized charge as a function of the bare charge.Notice the saturation effect: when σ → ∞ , the renormalized charge approachesa finite value ˆ σ ren →
1. This saturation effect can also appear in other theoriesobtained by modification of Poisson–Boltzmann equation using a density functionalformulation [T´ellez and Trizac 2003]. In that saturation regime, the large-distancebehavior potential becomes independent of the bare charge of the plane y sat ( x ) ∼ y e − κx = 4 e − κx . (2.14)The value y = 4 plays an important role. It can be seen as an effective surface po-tential for the plane, if one wants to match the linear solution of Poisson–Boltzmannequation with the nonlinear one in the close vicinity of the highly charged plane.This is also the starting point to find the renormalized charge at saturation forhighly charged macromolecules of arbitrary shape. For these macromolecules, whenthey are highly charged, the linear Poisson–Boltzmann equation can be solved withan effective boundary condition of constant surface potential y , to find the behav-ior of the potential at large distances and the corresponding renormalized charge,as explained in [Bocquet et al. 2002]. For instance, for a spherical macromolecule,the solution of the linear Poisson–Boltzmann equation with the effective constantpotential boundary condition at the surface of the macromolecule y ( a ) = y = 4, is y sat, sphere ( r ) = y e κa a e − κr r . (2.15)The nonlinear solution, at large distances has the behavior given by equation (2.2).Comparing both equations (2.2) and (2.15), we find an approximate value for the Article submitted to Royal Society
Gabriel T´ellez renormalized charge in the saturation regime Z satren = al B y ( κa + 1) = al B (4 κa + 4) . (2.16)This approximation is based on the planar solution of the nonlinear Poisson–Boltzmann equation, and therefore it is accurate for large macromolecules with κa ≫
1, and only at the first order in κa . One can improve this estimate bydeveloping a planar expansion of the solution of the spherical geometry, as donein [Shkel et al. 2000, Trizac et al. 2003, T´ellez and Trizac 2004]. Up to terms of or-der O (1 / ( κa )), the renormalized charge at saturation for spheres in a 1:1 electrolyteis [Trizac et al. 2003] Z satren = al B (4 κa + 6) . (2.17)The starting point to obtain estimates of the renormalized charge at saturationis the value y of the effective surface potential at saturation in the planar geometry.This depends only on the constitution of the electrolyte. We propose now a simpleformula that gives y in the generic case of a multicomponent electrolyte, composedof several species of ions with charges { q α e } and densities { n α } . The nonlinearPoisson–Boltzmann equation in the planar geometry reads now y ′′ ( x ) + 4 πl B X α q α n α e − q α y ( x ) = 0 . (2.18)Multiplying by y ′ ( x ) this equation, it can be integrated once to find( y ′ ( x )) = 8 πl B X α n α ( e − q α y ( x ) − , (2.19)where the boundary condition y ′ ( x ) → x → ∞ has been used. Introducingthe inverse Debye length κ = (4 πl B P α q α n α ) − / , one obtains the formal solution κx = Z y (0) y ( x ) du q P α q α n α P α n α ( e − q α u − . (2.20)Suppose the charged plane is located at x = 0 and positively charged, and we arein the saturation regime, therefore y (0) → + ∞ . At large distances from the plane, κx ≫
1, the potential behaves as y ( x ) ∼ y e − κx , thus κx = ln y − ln y ( x )+ o (ln y ( x ))as y ( x ) →
0. Replacing in (2.20) we findln y = lim y → Z ∞ y du q P α q α n α P α n α ( e − q α u −
1) + ln y (2.21)= Pf. Z ∞ du q P α q α n α P α n α ( e − q α u − . Thus, the value of y is expressed as a Hadamard finite part (Pf.) of the inte-gral (2.21). Alternatively, it can be computed fromln y = Z q P α q α n α P α n α ( e − q α u − − u du + Z ∞ du q P α q α n α P α n α ( e − q α u − . (2.22) Article submitted to Royal Society onlinear screening of charged macromolecules Table 1.
Value of the saturation potential for several electrolytes. z + : z − y z + : z − y In the case of a two-component electrolyte q = z + and q = − z − , the saturationvalue y can be expressed as a function of the ratio r = z + /z − , y = 1 z − exp Pf. Z ∞ r r du q e − ru r + e u r − − . (2.23)In the cases of electrolytes with z + : z − equal to 1:1, 1:2, and 2:1, the integral canbe computed exactly to find the known values [T´ellez and Trizac 2004], y = 4, y = 6(2 − √ ≃ . y = 6. Table 1 gives the value of the saturationpotential for other electrolytes.
3. Cylindrical macromolecules ( a ) Exact solutions for Poisson–Boltzmann equation and the connection problem
In this section, we focus our attention on the study of the screening of athin cylindrical macromolecule, with radius a ≪ κ − . Besides the charge renor-malization effect, another interesting phenomenon that occurs in this geometryis the counterion condensation. This was first realized by Onsager and studiedby [Manning 1969] and [Oosawa 1971]. To understand this phenomenon, considerthe Boltzmann factor between the macromolecule and an ion of opposite charge(counterion): exp( − z − ξ ln r ) = r − z − ξ . It diverges when r →
0, and furthermoreit is not integrable near r → ξ > /z − . This means that, for an infinitely thinmacromolecule, with radius a = 0, the thermodynamics are not properly definedunless ξ < /z − . In real situations a = 0. For κa ≪
1, when ξ > /z − , the densityof counterions will be very large near the surface of the macromolecule. These coun-terions are bounded to the macromolecule: besides the diffuse screening cloud ofions around the macromolecule, there is also a thin layer of condensed counterionsvery near to the surface of the cylinder.This counterion condensation effect can be studied quantitatively in the meanfield approximation, since an analytical solution of the nonlinear Poisson–Boltzmannequation in the cylindrical geometry is available. For a 1:1 electrolyte, this so-lution was found by [McCoy, Tracy and Wu 1977], in a different context, in re-lation to the correlation functions of the two-dimensional Ising model. Later on,[Widom 1997] developed the solution for the asymmetric cases 2:1 and 1:2, and[Tracy and Widom 1998] studied the short-distance asymptotics of the solution andsolved the problem of connecting the large-distance and the short-distance behav-iors of the solution. The consequences of this mathematical work to the screen-ing of cylindrical macromolecules were reported by [McCaskill and Fackerell 1988],[Tracy and Widom 1997] and [Trizac and T´ellez 2006]. A more extensive study is Article submitted to Royal Society
Gabriel T´ellez presented in [T´ellez and Trizac 2006]. We summarize here some of the main findingsof that work.The short-distance behavior of the potential can be obtained by a physical argu-ment. Very close to the charged cylindrical macromolecule, one would expect the po-tential to be the bare Coulomb potential − A ln r + constant, with A some constantrelated to the charge of the macromolecule. By replacing this ansatz into Poisson–Boltzmann equation (1.3) one can compute systematically the following terms of theshort-distance expansion, to find [Tracy and Widom 1998, T´ellez and Trizac 2006,Trizac and T´ellez 2007] y ( r ) = − A ln ˆ r + 2 ln B − (cid:20) − B ˆ r − A − A ) (cid:21) + O (ˆ r A ) . (3.1)We specialize here in the 1:1 electrolyte. For the general case of a z + : z − electrolyte,see [Trizac and T´ellez 2007]. A and B are some constants of integration. The con-stant A can be related to the charge density of the electrolyte by writing the firstboundary condition (1.7) ξ = A − (2 − A )( κa ) − A B − A ) − B ( κa ) − A . (3.2)From the previous section, we already know the large-distance behavior of thepotential, it is the screened potential (2.3) y ( r ) ∼ λK (ˆ r ) , (3.3)where λ is some constant related to the renormalized charge by ξ ren = ˆ aK (ˆ a ) λ/ . (3.4)By using the explicit analytical solution of the nonlinear Poisson–Boltzmann equa-tion from [McCoy, Tracy and Wu 1977] and [Widom 1997], [Tracy and Widom 1998]where able to solve the connection problem of relating the constants of integrationfrom the short-distance behavior A and B , to the one of the large-distance behavior λ . To satisfy the boundary condition y ′ ( r ) → r → ∞ , the constants A and B need to satisfy B = 2 A Γ (cid:0) A (cid:1) Γ (cid:0) − A (cid:1) , (3.5)where Γ is the gamma function, and, A and λ need to satisfy λ = 1 π sin (cid:18) πA (cid:19) . (3.6)The first physical consequence of these relations, is that we can obtain an ana-lytical expression for the renormalized charge, by combining equations (3.2), (3.4)and (3.6). In the simplest situation, when a = 0, this gives, A = ξ and ξ ren = 2 π sin πξ . (3.7) Article submitted to Royal Society onlinear screening of charged macromolecules b ) Counterion condensation
The previous discussion, and in particular equation (3.1) are only valid providedthat
A <
1. Indeed, if A = 1, equation (3.1) becomes singular: the last term becomesof the same order as the second, and besides, the constant B from equation (3.5)becomes undefined. This is the mathematical signature of the counterion conden-sation phenomenon. Notice that for a = 0, the constant A is the linear charge ofcylinder A = ξ , and the value ξ Manning = 1 is precisely the threshold for counterioncondensation discussed earlier.For a cylinder with nonzero radius a = 0, notice that, using equation (3.2), thethreshold A = ξ Manning = 1 corresponds for the linear charge to the threshold ξ c = 1 + 1ln ˆ a + C , (3.8)with C = γ − ≃ − . γ the Euler constant. Note that there is anegative logarithmic correction in the radius a of the cylinder to the Manning value ξ Manning = 1 for the threshold for condensation: ξ c ≤ A as a complex number A = 1 − iµ/
2. Replacing into equa-tion (3.1), the short-distance expansion of the potential now reads y ( r ) = − r − − µ ln ˆ r − µC )4 µ . (3.9)The constant µ can be expressed in terms of the bare linear charge density ξ by usingthe first boundary condition (1.7), and connected to the large-distance expansionof the potential and the renormalized charge by means of equation (3.6), replacing A = 1 − iµ/
2, for details see [T´ellez and Trizac 2006].Notice the first term of expansion (3.9) : − r . It is the bare Coulomb potentialof a charged line with linear charge density ξ Manning = 1. This a characteristic ofthe counterion condensation phenomenon. At intermediate distances of the chargedcylinder, one “sees” a cylinder with an effective charge ξ Manning = 1, if the barecharge exceeds the threshold value ξ c . The second term of equation (3.9) can becomevery large in the close proximity of the cylinder. This second term represents thethin layer of condensed counterions located at the surface of the cylinder. Figure 2,shows a plot of the potential close to the cylinder, and compares it to the bare term − r .
4. Concluding remarks
To summarize and conclude the review presented here, we would like to stressthe differences between the different linear charges densities that we presented forcylindrical macromolecules: renormalized charge, Manning charge, threshold chargefor counterion condensation.The renormalized charge characterizes the behavior of the potential far from thecharged macromolecule. At those large distances, r ≫ κ − , the potential exhibitsan exponential decay. The prefactor of this exponential decay is proportional to therenormalized charge, as shown in equation (2.3). Article submitted to Royal Society Gabriel T´ellez κ r y (r) Figure 2. Short-distance expansion of the potential (3.9) (thick line) compared to − r/ (4 µ )) (dashed line). Parameter µ = 0 . At short distances, there are two possible behaviors, depending on the value ofthe bare linear charge density ξ compared to the threshold value ξ c given by equa-tion (3.8). If ξ < ξ c , the potential behaves as given by equation (3.1). It is a bareCoulomb potential with a prefactor given in terms the bare charge of the macro-molecule. If ξ > ξ c , the counterion condensation takes place. The short-distancebehavior of the potential is given by equation (3.9). A thin layer of counterionsis bound to the surface of the cylinder, which makes the potential very large inthat region. Beyond this layer, the potential behaves as − ξ Manning ln r : the bareCoulomb potential for a charged cylinder but with an effective charge ξ Manning = 1,the Manning charge for counterion condensation, see figure 2. Notice that if theradius of the cylinder a = 0, the Manning value differs from the threshold value: ξ Manning > ξ c .The counterion condensation phenomenon can be only noticed at close proximityof the charged cylinder, by the change of the short-distance behavior of the poten-tial from (3.1) to (3.9). At large distances, the potential is always given by (2.3),regardless if the counterion condensation has taken place or not. When ξ = ξ c , nosingularity appears in equation (2.3), nor in the renormalized charge ξ ren whichcharacterizes only the large-distance behavior of the potential. Also, notice that ξ ren = ξ Manning and ξ ren = ξ c . This has caused some confusion in the past, since inthe original work of [Manning 1969], in the condensed phase, the diffuse screeningcloud of the remaining uncondensed counterions around the charged cylinder wastreated using the linear Poisson–Boltzmann equation and using an effective chargeof the cylinder given by ξ Manning = 1 to account for the counterion condensation.It turns out that this picture is not completely correct, since besides the counte-rion condensation, there are also additional nonlinear effects in the uncondensedcloud that are responsible for an additional charge renormalization. As a result
Article submitted to Royal Society onlinear screening of charged macromolecules ξ ren = ξ Manning . For example, in the limiting case a = 0, from equation (3.7), weobtain ξ ren = 2 /π ≃ . < ξ Manning = 1.
The author acknowledges partial financial support from Comit´e de Investigaciones yPosgrados, Facultad de Ciencias, Universidad de los Andes.
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