Preferential interaction coefficient for nucleic acids and other
  cylindrical poly-ions
Preferential interaction coefficient for nucleic acids and other cylindrical poly-ions
Emmanuel Trizac∗
CNRS; Univ. Paris Sud, UMR8626, LPTMS, ORSAY CEDEX, F-91405 and
Center for Theoretical Biological Physics, UC San Diego,
9500 Gilman Drive MC 0374 - La Jolla, CA 92093-0374, USA
Gabriel Téllez†
Departamento de F́ısica, Universidad de Los Andes, Apartado Aéreo 4976, Bogotá, Colombia
The thermodynamics of nucleic acid processes is heavily affected by the electric double-layer
of micro-ions around the polyions. We focus here on the Coulombic contribution to the salt-
polyelectrolyte preferential interaction (Donnan) coefficient and we report extremely accurate ana-
lytical expressions valid in the range of low salt concentration (when polyion radius is smaller than
the Debye length). The analysis is performed at Poisson-Boltzmann level, in cylindrical geometry,
with emphasis on highly charged poly-ions (beyond “counter-ion condensation”). The results hold
for any electrolyte of the form z−:z+. We also obtain a remarkably accurate expression for the
electric potential in the vicinity of the poly-ion.
Coulombic interactions between salt and poly-anions
play a key role in the equilibrium and kinetics of nucleic
acid processes [1]. A convenient quantity quantifying
such interactions and allowing for the analysis and inter-
pretation of their thermodynamics consequences, is the
so called preferential interaction coefficient. Several def-
initions have been proposed and their interrelation stud-
ied, see e.g. [2, 3, 4]. In the present work, they are defined
as the integrated deficit (with respect to bulk conditions)
of co-ions concentration around a rod-like poly-ion. Our
goal is to provide analytical expressions describing the
effect of salt concentration and poly-ion structural pa-
rameters on the preferential interaction coefficient, for a
broad class of asymmetric electrolytes. For symmetric
electrolytes, it will be shown that our formulas improve
upon existing analytical results. For other asymmetries,
they seem to have no counterpart in the literature. Our
analysis holds for highly (i.e. beyond counter-ion conden-
sation [5, 6]) and uniformly charged cylindrical poly-ions,
and is explicitly limited to the low salt regime (i.e. when
the poly-ion radius a is smaller than the Debye length
1/κ). These conditions are most relevant for RNA or
DNA in their single, double, or triple strand forms.
As in several previous approaches [7, 8, 9, 10], we
adopt the mean-field framework of Poisson-Boltzmann
equation, in a homogeneous dielectric background of per-
mittivity ε. The same starting point has proven relevant
for related structural physical chemistry studies of nu-
cleic acids [11]. In a z−:z+ electrolyte, the dimensionless
electrostatic potential φ = eϕ/kT (with e > 0 the ele-
mentary charge and kT thermal energy) then obeys the
equation [12]
z+ + z−
ez−φ − e−z+φ
, (1)
∗Electronic address: trizac@lptms.u-psud.fr
†Electronic address: gtellez@uniandes.edu.co
where r is the radial distance to the rod axis. The va-
lencies z+ and z− of salt ions are both taken positive.
Denoting derivative with a prime, the boundary condi-
tions read rφ′(r) = 2ξ > 0 at the polyion radius (r = a)
and φ → 0 for r → ∞. The latter condition expresses
the infinite dilution of poly-ion limit and ensures that
the whole system is electrically neutral, since it (indi-
rectly) implies that rφ′ → 0 for r → ∞. We consider
a negatively charged poly-anion for which φ < 0 and
the line charge density reads λ = −eξ/ℓB < 0, where
ℓB = e
2/(εkT ) denotes the Bjerrum length (0.71 nm in
water at room temperature). Finally, the Debye length is
defined from the bulk ionic densities n∞+ and n
− through
κ2 = 4πℓB(z
+ + z
The Coulombic contribution to the anionic preferential
interaction coefficient is defined as [7, 8, 9, 10, 13]
Γ = κ2
(ez−φ − 1) rdr, (2)
while its cationic counterpart follows from electro-
neutrality. This quantity –which provides a measure of
the Donnan effect [14]– can be expressed in closed form
as a function of the electrostatic potential, see Appendix
A. As can be seen in (A3) and (A4), Γ depends expo-
nentially on the surface potential φ0, so that deriving a
precise analytical expression is a challenging task. Fur-
thermore, we are interested here in the limit κa < 1
(including the regime κa≪ 1) which is analytically more
difficult than the opposite high salt situation where to
leading order, the charged rod behaves as an infinite
plane, and curvature corrections can be perturbatively
included [15, 16, 17].
We will proceed in two steps. Focusing first on the
surface potential φ0 = φ(a), we make use of recent re-
sults [18] that have been obtained from a mapping of Eq.
(1) onto a Painlevé type III problem [19, 20, 21]. The
exact expressions thereby derived only hold for 1:1, 1:2
and 2:1 electrolytes, but may be written in a way that is
electrolyte independent. This remarkable feature is spe-
http://arxiv.org/abs/0704.0587v1
mailto:trizac@lptms.u-psud.fr
mailto:gtellez@uniandes.edu.co
z+/z− 1/10 1/3 1/2 1 2 3 10
C -2.51 -1.94 -1.763 -1.502 -1.301 -1.21 -1.06
TABLE I: Values of C appearing in Eq. (4) as a function
of electrolyte asymmetries. For z+/z− = 1, 1/2 and 2, C is
known analytically from the results of [18]. The corresponding
values are recalled in Appendix B. For other values of z+/z−,
C has been determined numerically, see in particular Fig. 6
of Appendix B.
cific to the short distance behaviour of φ and has been
overlooked so far, since not only short distance but also
large distance properties have been studied [18]. We are
then led to conjecture that the corresponding expression
holds for any binary electrolyte z−:z+, and we explicitly
check the relevance of our assumption on several specific
examples.
Technical details are deferred to the appendices. It is
in particular concluded in Appendix B that the surface
potential may be written
e−z+φ0 ≃
2(z+ + z−)
z+(κa)2
(z+ξ − 1)
2 + µ̃2
where
log(κa) + C − (z+ξ − 1)−1
. (4)
Expression (4) is valid for κa < 1 and z+ξ > 1 [in fact
z+ξ > 1 + O(1/| log κa|)]. These conditions are easily
fulfilled for nucleic acids. The “constant” C appearing
in (3) depends smoothly on the ratio z+/z− but is oth-
erwise salt and charge independent. We report in Table
I its values for several electrolyte asymmetries. The de-
crease (in absolute value) of C when z+/z− increases is
a signature of more efficient (non-linear) screening with
counter-ions of higher valencies.
From Eq. (3) and the results of Appendix B, our ap-
proximation for Γ takes a simple form
Γ ≃ −
(1 + µ̃2). (5)
This expression is tested in Figures 1 and 2 against the
“true” numerical results that serve as a benchmark. In
Fig. 1 which corresponds to a monovalent salt (or more
generally a z:z electrolyte), we also show the prediction
of Ref. [9], which is, to our knowledge, the most accurate
existing formula for a 1:1 salt. For the technical reasons
discussed in Appendix B, and that are evidenced in Fig-
ure 6, our expression improves that of Shkel, Tsodikov
and Record [9], particularly at lower salt content. For
1:2 and 2:1 salts, we expect Eq. (5) to be also accurate,
since it is based on exact expansions. The situation of
other salt asymmetries is more conjectural (see Appendix
B), but Eq. (5) is nevertheless in remarkable agreement
with the full solution of Eq. (1), see Fig. 2. To be spe-
cific, in both Figures 1 and 2, the relative accuracy of
our approximation is better than 0.2% for κa = 10−2
(for both ss and ds RNA parameters). At κa = 0.1, the
accuracy is on the order of 1%.
As illustrated in Fig. 3, approximation (4) assumes
that z+ξ > 1. The corresponding expression for Γ there-
fore breaks down when ξ is too low. More general expres-
sions, still for κa < 1, may be found in appendix C. The
inset of Fig. 3 offers an illustration and shows that the
limitations of approximation (4) may be circumvented at
little cost, providing a quasi-exact value for Γ. Moreover,
it is shown in this appendix that for z+ξ = 1, µ̃ reads
log(κa) + C
. (6)
On the other hand, Eq. (3) still holds. The corresponding
Γ is shown in Fig. 4.
We provide in Appendix C a general expression of the
short scale (i.e valid up to κr ∼ 1) radial dependence
of the electric potential, see Eq. (C1). The bare charge
should not be too low [more precisely, one must have
ξ > ξc with ξc given by Eq. (C5)], and µ̃ –which encodes
the dependence on ξ– follows from solving Eq. (C2). In
general, the corresponding solution should be found nu-
merically. However, one can show a) that µ̃ vanishes
for ξ = ξc, b) that µ̃ takes the value (6) when z+ξ = 1
and c) that µ̃ is given by (4) when z+ξ exceeds unity
by a small and salt dependent amount. In practice, for
DNA and RNA, we have ξ > 2 and Eq. (4) provides
0.001 0.01 0.1 1
0.001 0.01 0.1
FIG. 1: Preferential interaction coefficient for a 1:1 salt. The
main graph corresponds to ss-RNA with reduced line charge
ξ = 2.2 while the inset is for ds-RNA (ξ = 5). The circles
correspond to the value of (2) following from the numerical
solution of Eq. (1). The prediction of Eq. (5) with eµ given
by (4) and C ≃ −1.502, shown with the continuous curve, is
compared to that of Ref. [9], shown with the dashed line. As
in all other figures, the opposite of Γ is displayed, to consider
a positive quantity.
0.001 0.01 0.1
0.001 0.01 0.1
2.5 dsRNA
ssRNA
FIG. 2: Same as Figure 1 for a 1:3 and a 3:1 electrolyte. From
Table I, we have C ≃ −1.21 in the 1:3 case and conversely
C ≃ −1.94 in the 3:1 case. The symbols correspond to the
numerical solution of Eq. (1) and the continuous curves show
the results of Eq. (5) with again eµ given by (4).
1 1.5 2 2.5 3
1 1.5 2 2.5
FIG. 3: Preferential interaction coefficient for a 1:1 salt (hence
C ≃ −1.502) and κa = 10−2. The circles show the numerical
solution of PB theory (1), the continuous curve is for (5) with
(4) and the dashed line is the prediction of Ref. [9]. Although
approximation (4) breaks down at low ξ, the inset shows that
eµ following from the solution of Eq. (C2) gives through (5) a
Γ (continuous curve), that is in excellent agreement with the
“exact one”, shown with circles as in the main graph.
excellent results whenever κa < 0.1. To illustrate this,
we compare in Figure 5 the potential following from the
analytical expression (C1) to its numerical counterpart.
We do not display 1:1, 1:2 and 2:1 results since in these
cases, Eq. (C1) is obtained from an exact expansion and
fully captures the r-dependence of the potential. For the
asymmetry 1:3, Fig. 5 shows that the relatively simple
form (C1) is very reliable. A similar agreement has been
found for all couples z−:z+ sampled, with the trend that
0.001 0.01 0.1
0.001 0.01 0.1
FIG. 4: Same as Fig. 1 for ξ = 1 and z+/z− = 1. The same
quantities are shown: our prediction for Γ [Eqs. (5) and (6)
with C ≃ −1.502] is compared to that of Ref. [9]. The inset
shows −z+Γ/z− for a 1:2 salt such as MgCl2 where C takes the
value -1.301. Circles : numerical data; curve : our prediction.
the validity of (C1) extends to larger distances as z+/z−
is decreased. In this respect, the agreement shown in Fig.
5 for which z+/z− is quite high (3), is one of the “worst”
observed.
0.01 0.1 1
0 0.5 1
0.01 0.1 1
FIG. 5: Opposite of the electric potential versus radial dis-
tance in a 1:3 electrolyte with κa = 10−2. The continuous
curve shows the prediction of Eq. (C1) with eµ given by (4) ;
the circles show the numerical solution of Eq. (1). The po-
tential for ξ = 2.2 is shown in the main graph on a log-linear
scale, and on a linear scale in the lower inset. The upper inset
is for ξ = 5.
Conclusion. The poly-ion ion preferential interaction
coefficient Γ describes the exclusion of co-ions in the
vicinity of a polyelectrolyte in an aqueous solution. We
have obtained an accurate expression for Γ in the regime
of low salt (κa < 1). The present results are particu-
larly relevant for highly charged poly-ions (z+ξ > 1, that
is beyond the classical Manning threshold [22]), but are
somewhat more general and hold in the range ξc < ξ < 1,
where ξ stands for the line charge per Bjerrum length and
ξc is a salt dependent threshold, given by Eq. (C5). Our
formulae have been shown to hold for arbitrary mixed
salts of the form z−:z+ (magnesium chloride, cobalt hex-
amine etc). They have been derived from exact expan-
sions valid in 1:1,1:2 and 2:1 cases, from which a more
general conjecture has been inferred. The validity of this
conjecture, backed up by analytical arguments, has been
extensively tested for various values of z+/z−, poly-ion
charge and salt content. These tests have provided the
numerical value of the constant C reported in Table I,
which only depends of the ratio z+/z−. As a byprod-
uct of our analysis, we have obtained a very accurate
expression for the electric potential in the vicinity of the
charged rod (r < κ−1).
It should be emphasized that the validity of our
mean-field description relying on the non-linear Poisson-
Boltzmann equation depends on the valency of counter-
ions (z+), and to a lesser extent to the value of z− [12, 23].
For the 1:1 case in a solvent like water at room temper-
ature, micro-ionic correlations can be neglected up to a
salt concentration of 0.1M [8]. For z+ ≥ 2 or in sol-
vents of lower dielectric permittivity, they play a more
important role. Our results however provide mean-field
benchmarks from analytical expressions, from which the
effects of correlations may be assessed in cases where they
cannot be ignored (see e.g. [8] for a detailed discussion).
Acknowledgments
This work was supported by a ECOS
Nord/COLCIENCIAS action of French and Colom-
bian cooperation. G. T. acknowledge partial financial
support from Comité de Investigaciones, Facultad de
Ciencias, Universidad de los Andes. This work has
been supported in part by the NSF PFC-sponsored
Center for Theoretical Biological Physics (Grants No.
PHY-0216576 and PHY-0225630).
APPENDIX A
In order to explicitly relate the preferential coefficient
Γ in (2) to the electric potential, we follow a procedure
similar to that which leads to an analytical solution in
the cell model, without added salt [24]. Implicit use will
be made of the boundary conditions associated to (1).
First, integrating Eq. (1), one gets
[r′φ′(r′)]ra =
z+ + z−
e−z+φ − ez−φ
r′dr′, (A1)
where the notation [F (r′)]ra = F (r) − F (a) has been in-
troduced. Then, multiplying Eq. (1) by r2φ′ and inte-
grating, we obtain
z+ + z−
(r′φ′)2
e−z+φ
+ r′2
e−z+φ
dr′.(A2)
Combining both relations with adequate weights, in order
to suppress the integral over counter-ion (+) density, we
r′(ez−φ − 1)dr′ =
2(z+ + z−)
ez−φ0 − 1
e−z+φ0 − 1
where φ0 = φ(a) is the surface potential. Equation (A3)
will turn useful in the formulation of a general conjec-
ture concerning the surface potential φ0, see Appendix
B. We also note that for the systems under investigation
here, the surface potential is quite high, and a very good
approximation to (A3) is
r′(ez−φ − 1)dr′ ≃
a2z−e
−z+φ0
2(z+ + z−)
APPENDIX B
We start by analyzing a 1:1 electrolyte, for which it
has been shown [19, 20] that the short distance behaviour
reads
eφ/2 =
2µ log
− 2Ψ(µ)
+ O (κr)
where Ψ denotes the argument of the Euler Gamma func-
tion Ψ(x) = arg[Γ(ix)] [19, 20]. In (B1), µ denotes the
smallest positive root of
tan [2µ log(κa/8)− 2Ψ(µ)] =
ξ − 1
. (B2)
Expressions (B1) and (B2) require that ξ exceeds a salt
dependent threshold [denoted ξc below and given by Eq.
(C5)] that is always smaller than 1 [18]. They thus always
hold for ξ ≥ 1 and in particular encompass the interesting
limiting case ξ = 1, which is sufficient for our purposes.
For large ξ, we have proposed in [18] an approximation
which amounts to linearizing the argument of the tangent
in (B1) in the vicinity of −π, and similarly linearizing Ψ
to first order: Ψ(x) ≃ −π/2 − γx + O(x3) where γ is
the Euler constant, close to 0.577. It turns out however
that finding accurate expressions for exp(−z+φ0), which
is useful for the computation of the preferential interac-
tion coefficient, requires to include the first non-linear
correction in the expansion of the tangent. After some
algebra, we find :
log(κa) + C − (ξ − 1)−1
6(log(κa) + C − (ξ − 1)−1)4
(ξ − 1)3
ψ(2)(1)
where the constant C = C1:1 reads C1:1 = γ − log 8 ≃
−1.502 and ψ(2)(1) = d3 ln Γ(x)/dx3|x=1. From (B3) and
(B1) where the sinus is expanded to third order, we ob-
(κa)2e−φ0 ≃ 4[(ξ − 1)2 + µ̃2] (B4)
where µ̃ is given by
log(κa) + C − (z+ξ − 1)−1
. (B5)
In writing (B5), we have introduced the change of vari-
able µ̃ = 2µ [25]. The reason is that similar changes for
other electrolyte asymmetries allows to put the final re-
sult in a “universal” (electrolyte independent) form, see
below. A similar reason holds for introducing z+, here
equal to 1, in the denominator of (B5).
The functional proximity between our expressions and
those reported in [9] in the very same context is striking.
We note however that our µ̃ (denoted β in [9]) involves
a different constant C. More importantly, the functional
form of (B1) differs from that given in [9]. The compari-
son of the performances of our results with those of [9] is
addressed below, and is also discussed in the main text.
Performing a similar analysis as above in the 1:2 case
where z+ = 2 and z− = 1, we obtain from the expressions
derived in [18]:
(κa)2e−z+φ0 ≃ 3[(ξ − 1)2 + µ̃2] (B6)
and similarly, in the 2:1 case (z+ = 1, z− = 2):
(κa)2e−z+φ0 ≃ 6[(ξ − 1)2 + µ̃2]. (B7)
In both cases, provided again that ξ is not too low (see
below) µ̃ is given by (B5) [26], with however a different
numerical value for C [C1:2 = γ− (3 log 3)/2− (log 2)/3 ≃
−1.301 and C2:1 = γ − (3 log 3)/2− log 2 ≃ −1.763].
The similarity of expressions (B4), (B6) and (B7) leads
to conjecture that this form holds for any z−:z+ elec-
trolyte :
(κa)2e−z+φ0 ≃ A[(z+ξ − 1)
2 + µ̃2]. (B8)
We then have to determine the prefactor A as a function
of z+ and z−. To this end, we make use of the exact
relation (A3) [or equivalently (A4)], where in the limit of
large ξ, the lhs is finite while the two terms on the rhs
diverge. This yields the leading order behaviour :
(κa)2 exp(−z+φ0)
z+ + z−
(z+ξ − 1)
2. (B9)
It then follows that A = 2(z+ + z−)/z+ so that our gen-
eral expression (B8) takes the form:
(κa)2e−z+φ0 ≃ 2
z+ + z−
(z+ξ − 1)
2 + µ̃2
. (B10)
This expression holds regardless of the approximation
used for µ̃. If Eq. (B5) is used, then z+ξ should not
be too close to unity (see appendix C for more general
results including the case z+ξ = 1).
In order to test the accuracy of (B10) in conjunction
with (B5), we have solved numerically Eq. (1) for several
values of κa < 1 and electrolyte asymmetry and checked
that for several different values of z+ξ > 1, the quantity
Q = −π
(κa)2e−z+φ0
2(z+ + z−)
− (z+ξ − 1)
]−1/2
− log(κa) + (z+ξ − 1)
−1 (B11)
is a constant C, which only depends on z+/z− but not on
salt and ξ [it should be borne in mind that Eq. (B5) is a
small κa and large ξ expansion, which becomes increas-
ingly incorrect as κa is increased and/or ξ lowered]. This
is quite a stringent test (since the two terms on the rhs of
(B11) are large and close] which requires high numerical
accuracy. This is achieved following the procedure out-
lined in [27]. In doing so, we confirm the validity of (B10)
and collect the values of C given in Table I. In the 1:1
case, we predict that C = γ− log 8 ≃ −1.507, in excellent
agreement with the numerical data of Figure 6. On the
other hand, the prediction of Ref. [9] that Q reaches a
constant close to -1.90 (shown by the horizontal dashed
line in Fig 6) is incorrect. Figure 6 shows that the quality
of expression (B10) deteriorates when κa increases, as ex-
pected. It is noteworthy however that for κa = 10−1, its
accuracy is excellent whenever ξ > 2. The inset of Fig.
6 shows the validity of (B10) for a 3:1 electrolyte. When
z+ξ is close to 1, Eq. (B5) becomes an irrelevant approx-
imation to the solution of (B2), and can therefore not be
inserted into the general formula (B10). This explains
the large deviations between Q and the asymptotic value
C observed in Fig. 6 for the lower values of ξ reported.
We come back to this point in Appendix C.
The present results hold for z+ξ > 1 +O(1/| log κa|).
In this regime, our analysis shows that Eq. (B10) [with µ̃
given by (B5)] is correct up to order 1/ log4(κa) for any
(z−, z+). On the other hand the results of [9] ,valid in
the 1:1 case, appear to be correct to order 1/ log2(κa).
In addition, our expression for the surface potential may
be generalized to a broader range of ξ values and an ex-
pression for the short distance dependence of the electric
potential may also be provided. This is the purpose of
appendix C.
1 2 3 4 5
1 2 3 4
Q = -1.90
FIG. 6: Plot of the quantity Q defined in (B11) versus line
charge ξ for a 1:1 electrolyte at κa = 10−3 (continuous curve)
and κa = 10−1 (dashed curve). The value reached at large ξ is
compared to the prediction of [9] Q → eγ + log 2− γ ≃ −1.90
(horizontal dashed-dotted line) whereas Eqs. (B10) and (B5)
imply Q → γ− log 8 ≃ −1.50, shown by the horizontal dotted
line. The inset shows the same quantity for a 3:1 electrolyte
at κa = 10−5 [such a very low value is required to determine
precisely the value of the asymptotic constant C, that can
subsequently be used at experimentally relevant (higher) salt
concentrations]. Here, we obtain Q → −1.94 (dotted line)
which is the value reported for C in Table I.
APPENDIX C
In Appendix B, the “universal” results valid for all
(z+, z−) have been unveiled partly by a change of variable
µ → µ̃ from existing expressions [18]. In light of these
results, and of their accuracy (assessed in particular by
the precision reached for the preferential interaction co-
efficient), it is tempting to go further without invoking
approximations of (B2), or related expressions for other
asymmetries than 1:1. Inspection of the results given in
[18] for the 1:1, 1:2 and 2:1 cases lead, with again the
help of (A4), to the conjecture that
ez+φ/2 ≃
2(z+ + z−)
sin [µ̃ log(κr) + µ̃ C] (C1)
tan [ µ̃ log(κa) + µ̃ C ] =
z+ξ − 1
. (C2)
We emphasize that (C1), much as (B1), is a short dis-
tance expansion and typically holds for κr < 1 (hence
the requirement that κa < 1). In appendix D we give
further analytical support for conjecture (C1). A typical
plot showing the accuracy of (C1) is provided in the main
text (Fig. 5). For κr < 0.1, the agreement with the ex-
act result is better than 0.1%, and becomes progressively
worse at higher distances (20% disagreement at κr = 1).
From (C1), it follows that the integrated charge q(r)
in a cylinder of radius r [that is q(r) = −rφ′(r)/2] reads
z+q(r) = −1 + µ̃ tan
µ̃ log
where the so-called Manning radius [18, 28, 29] is given
κRM = exp
. (C4)
The Manning radius is a convenient measure of the coun-
terion condensate thickness. It is the point r where not
only z+q(r) = 1 but also where q(r) versus log r exhibits
an inflection point [30]. For high enough ξ, the logarith-
mic dependence of 1/µ̃ with salt [see (B5)] is such that
RM ∝ κ
−1/2.
The two relations (C1) and (C2) encompass those given
in Appendix B and allow to investigate the regime z+ξc <
z+ξ, and in particular the case z+ξ = 1, the so-called
Manning threshold [5]. However, (C1) and (C2) are not
valid for ξ < ξc, with
z+ξc = 1 +
log κa+ C
. (C5)
Note that ξc < 1, since the constant C is negative and
that salt should fulfill κa < 1. For κa = 10−2 and
z+/z− = 1, we obtain ξc ≃ 0.836. This is precisely the
point where −Γ = 1 in the inset of Fig. 3. This inset
also shows that the value of Γ resulting from the use of
the solution of (C2) is remarkably accurate.
At this point, it seems useful to investigate the Man-
ning threshold case z+ξ = 1 (which corresponds to
the onset of counterion condensation when κa → 0
[5, 18, 30]). It is readily seen that the solution of (C2)
reads
z+ξ=1
log(κa) + C
, (C6)
which should be inserted in (C1) to obtain the potential
profile, or in (5) to get the interaction coefficient.
APPENDIX D
In this appendix we give further support for the con-
jecture (C1) which gives the short-distance expansion
of the electric potential. Let us suppose initially that
the charge is below the Manning threshold ξ < ξc.
It is straightforward to verify that Poisson–Boltzmann
equation (1) admits solutions which behave as φ(r) =
−2A ln(κr) + lnB + o(1) for κr ≪ 1. Injecting this ex-
pansion into equation (1) allows us to compute higher
order terms. To study the regime beyond the Man-
ning threshold, we compute all higher order terms of the
form r2n(1+z+A) (for a negatively charged macroion) and
r2n(1−z−A) (for a positively charged macroion), with n a
positive integer. These terms turn out to present them-
selves as the series expansion of the logarithm, thus re-
summing them we obtain
φ(r) = −2A ln(κr) + lnB (D1)
−z+ (κr)2(1+z+A)
8(z+ + z−)(1 + z+A)2
− (κr)2(1−z−A)
8(z+ + z−)(1 − z−A)2
+ · · ·
The dots represent terms of order r2n(1+z+A)+2m(1−z−A)
with n and m two nonzero positive integers. When the
Manning threshold is approached, z+A + 1 = 0 for neg-
atively charged macroion, the terms r2n(1+z+A) (second
line of Eq. (D1)) become of order one, but the rest of the
terms (third line of Eq. (D1) and dots) remain higher or-
der: a change in the small distance behavior of φ occurs.
A similar situation is reached for 1 − z−A = 0 which is
the Manning threshold for a positively charged macroion.
A and B in the previous equations are constants of in-
tegration, which should be determined with the bound-
ary conditions rφ′(r) = 2ξ at the polyion radius (r = a)
and φ → 0 for r → ∞. Thus to proceed further, we
have to connect the long and the short distance behavior
of φ. This connection problem has been only solved in
the cases 1:1, 1:2 and 2:1 in Refs. [19, 31]. In particular,
once A has been chosen (notice that for a = 0, A = −ξ),
B should be one and only one function of A in order to
satisfy φ→ 0 for r → ∞. The results from [19, 31] show
B = 26Aγ ((1 +A)/2)
(1 : 1) (D2)
B = 33A22Aγ (2(1 +A)/3)γ ((1 +A)/3) (1 : 2)
B = 33A22Aγ ((1 + 2A)/3)γ ((2 +A)/3) (2 : 1)
where γ(x) = Γ(x)/Γ(1 − x). B turns out to have some
interesting properties in the cases 1:1, 1:2 and 2:1, where
its exact expression (D2) is known. Namely, at the Man-
ning threshold 1 + z+A = 0,
A→−1/z+
8(z+ + z−)(1 + z+A)2
= 1 (D3)
Furthermore if we put 1 + z+A = iµ̃, and define
e2iΨ(eµ) =
8(z+ + z−)(1 + z+A)2
then for µ̃ ∈ R, Ψ(µ̃) ∈ R is a real function of µ̃, with
Ψ(0) = 0.
Let us now study the regime beyond the Manning
threshold for a negatively charged macroion. From
Eq. (D1) we can write
ez+φ(r)/2 ∼ (κr)−z+ABz+/2
−z+ (κr)2(1+z+A)
8(z+ + z−)(1 + z+A)2
neglecting terms of higher order when z+A is close to −1.
Let us conjecture that the properties of B as a function
of A presented above hold in the general case z− : z+.
Then using the parameter µ̃ defined above we find after
some simple algebra
ez+φ(r)/2 =
2(z+ + z−)
sin [µ̃ log(κr) + Ψ(µ̃)]
+O(r3+2z−/z+) (D6)
Recalling that |µ̃| ≪ 1 we can approximate Ψ(µ̃) ≃ µ̃C,
where C = Ψ′(0). Replacing this approximation into (D6)
and imposing the boundary condition aφ′(a) = 2ξ leads
to (C1) and (C2). Numerical values obtained for the
constants C are reported in Table I, for different charge
asymmetries z− : z+. The previous analysis shows that
analytical predictions for C could be made if the con-
nection problem is solved and the equivalent of expres-
sions (D2) are found for the general case z− : z+.
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