Long range polarization attraction between two different likely charged macroions

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by  Rui Zhang

Long range polarization attraction between two different likely charged macroions
Rui Zhang and B. I. Shklovskii
Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455
(Dated: February 2, 2008)
It is known that in a water solution with multivalent counterions (Z-ions) two likely charged
macroions can attract each other due to correlations of Z-ions adsorbed on their surfaces. This
5 “correlation” attraction is short-ranged and decays exponentially with increasing distance between
0 macroions at characteristic distance A/2π, where A is the average distance between Z-ions on the
0 surfacesofmacroions. Inthiswork,weshowthatanadditionallongrange“polarization”attraction
2 exists when the bare surface charge densities of the two macroions have the same sign, but are
differentinabsolutevalues. ThekeyideaisthatwithadsorbedZ-ions,twoinsulatingmacroionscan
n
beconsideredasconductorswithfixedbutdifferentelectricpotentials. Eachpotentialisdetermined
a
bythedifferencebetweentheentropicbulkchemicalpotentialofaZ-ionanditscorrelationchemical
J
potential at the surface of the macroion determined by its bare surface charge density. When the
9
two macroions are close enough, they get polarized in such a way that their adjacent spots form a
1
chargedcapacitor,whichleadstoattraction. Inasaltfreesolutionthispolarizationattractiveforce

is long ranged: it decays as a power of thedistance between the surfaces of two macroions, d. The
]
t polarizationforcedecaysslowerthanthevanderWaalsattractionandthereforeismuchlargerthan
f
o itinalargerangeofdistances. Inthepresenceoflargeamountofmonovalentsalt,thepolarization
s attraction decays exponentially at d larger than the Debye-Hu¨ckel screening radius rs. Still, when
. A/2π ≪d≪r , this force is much stronger than the van der Waals attraction and the correlation
t s
a attraction mentioned above. The recent atomic force experiment has shown some evidence for this
m polarization attraction.
-
d PACSnumbers: 61.20.Qg,61.25.Hq,82.45.Gj
n
o
c I. INTRODUCTION (see Fig. 1a) [2]. At small concentrations of Z-ions, the
[ probe is attracted to the surface. With increasing con-

2 Water solutions of strongly charged colloidal parti- centrationofZ-ions,thechargeoftheprobegetsinverted
v cles (macroions) with multivalent (Z valent) counteri- byZ-ions,andthemeasuredforceatlargedbecomesre-
0 pulsive,wheredis the distance ofclosestproximityfrom
ons(Z-ions)areimportantinphysics,chemistry,biology
6 the probe to the surface. The critical concentration of
and chemical engineering. Colloidal particles, charged
7
Z-ions, where this happens is in reasonable agreement
2 lipid membranes,DNA, actin,and evencells andviruses
with the prediction of Ref. [3]. However, an interesting
1 are examples of different macroions. Multivalent metal-
new feature of the repulsive force was observed. With
4 lic ions, dendrimers, charged micelles, short DNA and
0 other relatively short polyelectrolytes like spermine can decreasing d, the repulsive force reaches a maximum at
/ d 100 ˚A (which is roughly equal to the Debye-Hu¨ckel
t play the role of Z-ions. Several interesting, counterintu- ≃
a screening radius of the solution) and start to decrease.
itive phenomena have been discovered in such a system,
m This suggests the existence of a competing attraction.
such as charge inversion, which have attracted signifi-
- cant attention (see review paper Ref. [1] and references Onecanalsoconsideradifferentexperimentalsetup[4]
d
therein). Charge inversion happens in a water solution inwhichtheprobeandthesurfacearelikelychargedand
n
o whenamacroionbindssomanyZ-ionsthatitsnetcharge bothadsorbZ-ions(Fig.1b). Whentheconcentrationof
c changes sign. This phenomenon is especially important Z-ions is high enough so that the charges of the surface
: for gene therapy. To deliver DNA into the cell, charge and the probe are both inverted at large d, it is inter-
v
i of negative DNA (playing the role of macroion) should esting to find out whether the force may be attractive.
X be inverted by positive Z-ions to approach a negatively Preliminaryexperimentsshowedsuchattraction [4]. No-
r chargedcellmembrane. Hereweactuallyassumethatthe ticethatthissetupbringsusbacktotheoriginalquestion
a
membrane is weakly charged and therefore its charge is of attraction of charge-inverted DNA to charge-inverted
notinverted. Thequestioniswhathappensifthecharge cell membrane.
ofthemembraneisalsoinvertedbypositiveZ-ions. Can Motivated by these questions, in this paper we study
DNA still be attracted to the membrane? the interaction between two different macroions in the
A similar question was put forward by the recent presence of a large concentration of Z-ions. In the main
atomic force experiment[2] designed to verify the the- partofthe paper,we focus onthe caseoftwobarenega-
ory of charge inversion based on strong correlations be- tivelychargedmacroionsandpositiveZ-ionscorrespond-
tween Z-ions [3],[1] In the experiment, forces between ing to Fig. 1b, since it is pedagogically simple. Only in
a negatively charged spherical macroion attached to the Sec. V, we discuss the case of oppositely bare charged
cantilever (probe) and a positively charged surface were macroions (Fig. 1a) in the connection with Ref. [2]. We
measured at different concentrations of positive Z-ions assumethatthe valenceofZ-ions,Z 1,butstillmany
≫
2
the macroion. As a result, when the concentration of Z-
ions in the solution is large enough, so many Z-ions are
adsorbedthat the net chargeof the macroionis inverted
and becomes positive [3].
d d As pointed out in Ref. [3], for an already neutral
macroion,adsorptionof additionalZ-ions can be viewed
as charging a conductor of certain capacitance with a
(a) (b)
fixedelectricpotentialφ. Thispotentialisdeterminedby
thedifferencebetweenthechemicalpotentialofZ-ionsin
the solutionandthe correlationchemicalpotentialµ on
c
FIG.1: Schematicillustration oftwoexperimentalsetupsfor
thesurface. Itisaconstantalloverthemacroionsurface
atomic force measurements of likely charged macroions. The
because the net surface charge density of the macroion
forcebetweentheprobe(thebigspherewithnegativecharge)
changes very little and therefore µ is a constant (see
and the surface is measured at different concentrations of Z- c
Sec. II for detail discussion). Actually, this electric po-
ions (small spheres with positive charges). (a) In the setup
tential is similar to the surface potential of a colloidal
of Ref. [2], the bare surface and the bare probe are the op-
positely charged sothat positive Z-ionsare only adsorbed to particle determined by its “potential-determining ions”
the probe and invert its charge. (b) In the setup of Ref. [4], given by the Nernst equation [5]. The only difference is
thebaresurfaceandthebareprobearelikelychargedsothat that in the present case, µ originates from correlations
c
positive Z-ions are adsorbed to both of them, again making of Z-ions instead of chemical bonding.
them likely charged at large d.
The model of conductors is convenient to discuss the
interaction and understand the possible attraction be-
tween two different macroions in the presence of Z-ions.
Z-ions are needed to neutralize one macroion. The two
Actually, as we know from electrostatics, if two conduc-
macroions are spherical but different in their bare sur-
tors are charged with different potentials, they may at-
face charge densities. We will see that this is crucial to
tract each other, even though the two potentials have
produce the attraction. Actually this is the reason why
the same sign. For example, consider two conducting
this attraction was not reported before, when the focus
spheres 1 and 2 with same size and potentials φ and φ
was on identical macroions [1]. In calculation, we focus 1 2
respectively and φ > φ > 0. When the two spheres
on the two limiting cases, R =R and R R , where 1 2
1 2 1 ≪ 2 are far away, they repel each other. When we bring
R and R are the radii of the two macroions.
1 2 themcloser,adjacentspotsoftwospheresformacharged
capacitor with potential difference φ φ . In other
1 2
−
r words,spheresgetpolarized. Thepolarizationattraction
A competes with the overallrepulsionbetweentwo spheres
(since q >q >0) and dominates at small distances be-
1 2
tweenspheres. Similarpolarizationleadstoattractionof
2a
a conducting sphere to a close conducting plate holding
at different potentials.
Using the modelofconductors,we findthatas longas
FIG. 2: Z-ions are strongly correlated on the surface of the thebaresurfacechargedensitiesofthetwomacroionsare
macroion. Their short range order is similar to that of a different, they alwaysattracteachother atsmallenough
Wigner crystal with lattice constant A. distances, even when their bare charges or net charges
areofthesamesign. Thispolarizationattractionis even
Before we discuss the mechanism of this attraction, more appreciable in the case that the sizes of the two
let us first briefly review the theory of charge inver- macroions are very different as in Fig. 1. We also find
sion [3]. Let us consider a water solution with one neg- that the attractive polarization force Fp decays with in-
atively charged macroion and many positively charged creasing distance as a power law. In the case of Fig. 1,
Z-ions. Due to the Coulomb interaction, Z-ions are ad- at d rs, we get
≪
sorbed to the surface of the macroion. On the surface,
they strongly repel each other with energy much larger F = πDR1(φ φ )2, (1)
than k T and form a two dimensional strongly corre- p − d 1− 2
B
lated liquid with short range order similar to a Wigner
crystal(WC)(Fig.2). WhenanewZ-ionapproachesthe where rs is the Debye-Hu¨ckel screening radius, D = 80
macroion, it repels already adsorbed Z-ions and creates is the dielectricconstantofwaterandR1 is the radiusof
a negative spot for itself. One can view it as an elec- the spherical probe.
trostatic image of the Z-ion, similar to the image on a We emphasize that the polarization attraction dis-
conventionalconductingsurface. Attractiontotheimage cussed here is different from two standard attractive
leads to an additional negative chemical potential µ (c forces. The first one is the well known van der Waals
c
stands for“correlation”)foreachZ-iononthe surfaceof attractionusedinstandardDLVOtheory[5]. Inthecase
3
of Fig. 1, it is given by and the macroions can be both either undercharged or
overcharged by them. In spite of the same sign of their
R H
1
F = , (2) net charges these macroions attract each other at small
vdW − 6d2
distancesbecause oflocalpolarizationaroundthe points
where H is the Hamaker constant. It is clear that FvdW of closest proximity.
decays with d faster than F . The ratio of these two
p In the presence of large amount of monovalent salt,
forces is
the Coulomb interaction is effectively truncated at the
Fp = 6πDd(φ φ )2. (3) Debye-Hu¨ckel screening radius rs. When φ1 = φ2, the
1 2
F H − two macroions with adsorbed Z-ions repel each other as
vdW
For the typical H =1.0 10−20J =2.4k T and reason- described by the standard DLVO theory [5]. However,
B
× when φ = φ , i.e., when the two macroions are differ-
able φ1 φ2 0.4kBT/e(seeSec.II),thisratioislarger 1 6 2
than|un−ity if|d≃>6 ˚A. ent, the polarization attraction appears. When φ1 and
φ have the same sign, it decays as e−2d/rs at distances
The second competing force is the short range attrac- 2
largerthanr . Still,forA/π d r ,thisattractionis
tionbetweentwomacroionsduetocorrelationsofZ-ions s ≪ ≪ s
muchstrongerthanboththevanderWaalsattraction[5]
on their surfaces [6]. In the spot where two macroions
and the correlation attraction [6], which is proportional
toucheachother,thesurfacedensityofZ-ionsisdoubled
to e−2πd/A.
and the correlation energy is gained. This correlation
attraction is related to Wigner-crystal-like arrangement This paper is organized as follows. In Sec. II, we de-
of Z-ions on the surfaces and therefore decays exponen- scribetheinteractionbetweentwodifferentmacroionsin
tiallywithincreasingdistancease−2πd/A,whereAisthe the presence of Z-ions and show that it is similar to the
“lattice constant”ofthe Wignercrystal(see Fig.2). For interactionbetweentwoconductorswithdifferentpoten-
d A/2π,itbecomesmuchweakerthanthepolarization tials. In Sec. III, we discuss the interaction of two con-
at≫tractiongivenbyEq.(1)(weassumethatr A/2π). ductingspheresandderivethepowerlawoftheattractive
s
In this sense the polarization attraction is a lo≫ng range force. In Sec. IV, we take into account of the effect of
force. screening by monovalent salt. In Sec. V, we generalize
One can understand the polarization attraction from our theory to the case when only one macroion adsorbs
another point of view, i.e., from the concept of contact Z-ions in the connection with the experiment in Ref. [2]
electrification. As well known, when two different solids (see Fig. 1a). We conclude in Sec. VI.
contactto eachotherin vacuum,due to the difference in
their work functions, certain amount of electrons move
fromonematerialtotheother,developingaelectricvolt-
II. THE MODEL OF CONDUCTORS FOR TWO
age which stops further charging. This contact electrifi-
MACROIONS IN THE PRESENCE OF Z-IONS
cation leads to the well known Coulomb attraction be-
tween them [7]. Here a contact between two objects is
necessary to eliminate the kinetic barrier for electrons In this section we show that the interaction between
and makes electrification possible during time of exper- two macroions in the presence of Z-ions can be consid-
iments. If we wait long enough, the electrification can eredastheinteractionbetweenconductorswithdifferent
happen even without a close contact. In our case of electric potentials. Let us first consider a single spheri-
macroionsinwater,Z-ionsplay the role ofelectrons and calmacroionwithradiusR andbare charge Q<0in a
−
the absolute value ofchemical potentialµ of a Z-ionon watersolutionwithZ-ionconcentrationn. Thebaresur-
c
the surface of the macroion plays the role of the work face charge density of the macroion is σ = Q/4πR2.
− −
function. To keep the same electro-chemicalpotential of As we discussed in Sec. I, Z-ions on the surface of the
Z-ions,sinceµ isdifferentfortwomacroions,theelectric macroionstrongly repel each other to form a strong cor-
c
potentials must also be different, which leads to attrac- related liquid, similar to a structure of Wigner crystal
tion. The difference in our system is that the kinetic in the short range (see Fig. 2). The chemical potential
barrier for Z-ions is relatively small and the equilibrium related to this correlation is dominated by its low tem-
of Z-ions is easily achieved through the solution. perature expression, which can be estimated from the
Similar electrification phenomenon has been studied Coulomb interaction inside each WC cell (see Eq. (9) in
in Ref. [8] in a toy model of two negative spheri- Ref. [3] for the full finite temperature expression of µc)
cal macroions exactly neutralized by Z-ions. The two
spheres had the same radius, but substantially different 1.65Z2e2 1.65√πsZ2e2
µ (s) = . (4)
barecharges. Itwasshownthatunderconditionsoftotal c
≃− Dr − D
neutrality one sphere becomes undercharged (negative)
and the other is overcharged (positive) and, therefore, Here s is the surface density of Z-ions on the macroion,
they attract each other.
and r = A √3/2π is the radius of the WC cell (see
In the present paper, we discuss a more generic and
realistic situation: there is certain concentration of Z- Fig. 2). It sqatisfies πr2s=1.
ions inthe solution(a fixedchemicalpotentialofZ-ions) The equilibrium condition of a Z-ion in the solution
4
then reads n for both macroions, the analogy to conductors holds.
0
The two macroions have electric potentials φ and φ .
ZeQ∗ s3/2 1 2
+µ (s)= k T ln , (5) When d decreases, due to the interaction between two
c B
DR − n macroions, s changes and becomes nonuniform on their
(cid:18) (cid:19)
surfaces. When d is very small, s may become substan-
where Q∗ is the net charge of the macroion combined
tiallydifferentfroms atclosestspotsofthetwosurfaces.
with adsorbed Z-ions. The left hand side is the electro- 0
Then Eq. (8) is not valid and φ start to change with
static energy of a Z-ion at the surface of the macroion 1,2
d [10]. In this paper, we stay in the limit of large s so
plus the correlation chemical potential. The right hand 0
that s s s . Accordingly, Eq. (8) is always valid
side is the difference between the entropic parts of the | − 0| ≪ 0
and our approximation of fixed potentials is fine.
chemical potentials of a Z-ion in the solution and at the
Rewriting Eq. (10) for two macroions, we have
surface. As we will see below, this equilibrium condition
is the key for the analogy with conductors. µ (s ) k T ln(s3/2/n)
When n increases, the number of Z-ions adsorbed to φ = | c 1,2 |− B 1,2 . (11)
1,2
∗ Ze
the macroionalso increases. At some critical n=n , Q
0
becomes zero. According to Eq. (5), we have Herethesubindexes 1,2representtwomacroionsrespec-
tively. s arethenumberdensitiesofZ-ionswhichneu-
1,2
n =s3/2exp −|µc(s0)| . (6) tralize two macroions, i.e., s1,2 = σ1,2/Ze. Notice that
0 0 k T φ are completely determined by n and s , i.e., the
(cid:18) B (cid:19) 1,2 1,2
concentrationof Z-ionsandthe bare surfacechargeden-
Here s0 = σ/Ze is the surface density of Z-ions to sitiesofthe twomacroions. ForagivensolutionofZ-ion
neutralize the macroion. The corresponding radius of concentration n, if σ = σ , φ = φ . For example, for
1 2 1 2
ZiWo2nCes2)/.ceDTllrh0iesr≫erf0o.rkeBInnT0t(ihstehainscaedsxeepfiwonneesean“rtesiatrlilnoytnegsrmleysatlcelhdca,ornµgec(desn0”)trZa≃-- atφyn2pdi=c−alσ0v2.a3l7=ukeBs−TZ0/.e=45a+ne3d/,n6 nmφ21=, 1wφem2Mh6 a,=v−eσ0φ1.14=0=kB−T00../77e75.keB/HTnme/re2e,
tmioonrewZh-iicohnsiscoeamseyttootrheeacmhaicnroeixopnearnimdeQnt∗sb.eFcoormnes>pons0i-, µusci(nsg1)th=e−fu7l.l6ekxBpTresasniodnµ|ocf(sµ−2)g=iv|−en5b.7ykEBqT.a(9re)icnalRcuefla.t[3e]d.
c
tive. AccordingtoEq.(5),whennissolargethattheen- Inorderto calculate the forcebetweentwo macroions,
tropy term is completely negligible, the maximum value wecanimaginethattheyareconductorsinion-freewater
∗ ∗
of inverted Q , Qmax, is achieved [3], with potentials φ1 and φ2 supported by two batteries.
Such conductors are not in equilibrium with each other
∗
Q =0.8 QZe. (7)
max while our macroions are in equilibrium. This however
It is calculated by assuming p does not matter for the calculation of the force which is
the same in both cases. Indeed, the force depends only
s s , µ (s) µ (s ). (8) on potentials and capacitance matrix of the system.
0 c c 0
≃ ≃
This assumption is self-consistent, since for Q Ze we
have (s s )/s =Q∗ /Q=0.8 Ze/Q ≫1. III. ATTRACTION OF TWO CONDUCTING
Nowmwaexc−an0clea0rly seemathxe similarity betwe≪en a neu- SPHERES WITH DIFFERENT POTENTIALS
p
tralized macroion and a neutral conductor. For a given
n>n , the chargingprocess from Q∗ =0 to Q∗ >0 can In this section, let us focus on the interaction of two
0
be viewed as charging a conductor with fixed potential. conducting spheres 1 and 2 with radii R and R and
1 2
Indeed, the equilibrium condition Eq. (5) can be written potentials φ and φ in ion-free water (see Fig. 3). As
1 2
as we explained in the previous section, this interaction is
equivalent to the interaction between macroions covered
∗
Q =DRφ, (9) byZ-ions. Wedefinethethedistanceofclosestproximity
between the two spheres as d and the distance between
with
the centers of the two spheres as L = d+R +R (see
1 2
µ (s ) k T ln(s3/2/n) Fig. 3). Below we use L or d alternatively for conve-
φ= | c 0 |− B 0 . (10) nience. We focus on two limiting cases when R = R
1 2
Ze
andR R ,butourapproachisapplicableforanytwo
1 2
≪
Notice that φ can be expressed through s because of spheres.
0
Eq. (8). Eq. (9) is exactly the same as the expression We start from the total free energy of two conductors
of charge for a conductor with capacitance C =DR and with fixed potentials [7]
charged at fixed potential φ. For n<n , by similar dis-
0 1 1
cussion,wehaveaconductorchargedatφ<0,providing U(L)= C φ2 C φ2 C φ φ . (12)
∗ −2 11 1− 2 22 2− 12 1 2
that Q (n) Q so that Eq. (8) is still satisfied.
| |≪
Now let us consider two macroions in a water solution HereC ,C ,C aretheselfandmutualcapacitancesof
11 22 12
withZ-ions. Clearly,whendisverylarge,ifn iscloseto the two spheres depending on L. Notice that in the case
5
of images induced in the two spheres [9]. We have
R d R
∞
(a) 1 2 1
C = DRsinht , (15a)
11
sinh(2n 1)t
ϕ ϕ n=1 −
X
∞
1 L 2 1
C = DRsinht , (15b)
12
− sinh2nt
n=1
X
where t=arccosh(L/2R). Consequently,
R1 d d R1 ∞
(b) ∂C11 D a2n−1(t)
= , (16a)
1 2 ∂L −2sinht sinh2(2n 1)t
nX=1 −
∞
ϕ ϕ ∂C D a (t)
12 2n
= . (16b)
1 2 ∂L 2sinht sinh22nt
n=1
X
Here a (t) is defined as
n
FIG. 3: The interaction between two spherical macroions in a (t)=nsinhtcoshnt coshtsinhnt. (17)
n
−
the presence of Z-ions can be considered as the interaction
betweentwoconductingsphereswithfixedbutdifferentelec- Since a (t) > 0 for t > 0, we have ∂C /∂L <
n 11
tric potentials. We focus on two cases: (a) The size of the 0, ∂C /∂L > 0. Also one can easily prove that
12
twospheresareequal. (b)Thesizeofsphere2ismuchlarger
∂C /∂L < ∂C /∂L. These inequalities lead to some
11 12
than sphere1 (theradius of sphere2, R2, isnot shown). For |simple res|ults|. Firstly,|when the signs of φ and φ are
d ≪ R2, the interaction between two spheres can be con- 1 2
opposite, F < 0, i.e., the force is attractive, as it should
sidered as the interaction between sphere 1 with its “image
be. Secondly, when φ = 0 or φ = 0, we have F < 0.
sphere” inside sphere 2. 1 2
Physically,thespherewithnonzeropotentialinducesop-
posite chargein the sphere withzeropotential, therefore
they attract each other. Finally, when φ = φ , F > 0,
1 2
of fixed potentials, the workdone by the environment to i.e., the force is repulsive. Indeed, the two spheres with
chargethetwoconductorsshouldbeincludedinthetotal same potential can be considered as a capacitor as a
energyandthereforeeachterminEq.(12)hasanegative whole. Obviously, the closer the two spheres are, the
sign. The generic formula of the force between the two smaller the capacitance is, and the higher the energy is
conductors is given by (remember the negative sign in Eq. (12)).
We are specifically interested in the case where the
F(L)= ∂U = ∂C11φ21 + ∂C22φ22 + ∂C12φ1φ2. (13) dsiigffnesreonft.φ1Wainthdoφu2tlaorseintghegesnaemraelibtyu,twtheecomnasigdneirtutdheescaarsee
−∂L ∂L 2 ∂L 2 ∂L
of φ >φ >0. Introducing α=φ /φ , we have
1 2 2 1
As will be shown, the force is always attractive when
(α2+1) ∂C /∂L ∂C
φ = φ and the two spheres are close enough. And it F(L)= α 11 12 φ2. (18)
1 6 2 − 2 ∂C /∂L ∂L 1
decays as a power law with increasing distance between (cid:20) (cid:12) 12 (cid:12)(cid:21)(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
spheres. (cid:12) (cid:12) (cid:12) (cid:12)
The critical value of α at w(cid:12) hich F =(cid:12)0(cid:12)is ther(cid:12)efore
∂C /∂L ∂C /∂L 2
12 12
α = 1. (19)
c
A. Two spheres of the same size (cid:12)∂C11/∂L(cid:12)−s(cid:12)∂C11/∂L(cid:12) −
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
When α < αc,(cid:12)F < 0, an(cid:12)d vice(cid:12)versa. In(cid:12)Fig. 4, we plot
α as a function of L/2R by taking the first 1000 terms
Let us start from a simple case when the radii of the c
two spheres are equal, R = R = R (Fig. 3a). In this in the series of Eq. (16). We see that the attraction is
1 2
possible at any distance as far as the ratio of the two
case, C =C and Eq. (13) is simplified to
11 22
potentials is smaller than α . The closerthe two spheres
c
are,the largerα is, and the easierthe attractioncanbe
∂C φ2+φ2 ∂C c
F(L)= 11 1 2 + 12φ φ . (14) developed.
1 2
∂L 2 ∂L In order to gain more intuition about the interaction,
also determine the dependence of the attractive force on
Accordingtothe standardmethodofimagecharges,C the distance analytically,we study asymptotic behaviors
11
andC canbe calculatedbyconsideringainfinite series of the force in the case of φ > φ > 0. In the limit of
12 1 2
6
α Remarkably, in this limit α =1 and the force is always
c c
attractive (the only exception is φ =φ when the lead-
1 2
ingordertermgivenbyEq.(23)iszeroandonehastogo
0.8
to the next order). Therefore we conclude that for any
smallφ φ ,twoconductingspheresalwaysattracteach
0.6 + 1− 2
other atsmallenoughdistance. This originatesfromthe
0.4 factthatinthe limit ofd 0,C11 andC12 both diverge
→
and diverge in the same way (see Eq.(22)).
0.2 - It should be mentioned that for the interaction be-
tween two macroions, the model of conductors is valid
L /2R only for d A. Indeed, instead of smeared charge dis-
2 3 4 5 ≫
tribution on a real conductor surface, on the surface of
the macroion, Z-ions are discrete and form a WC-like
liquid with “lattice constant” A (see Fig. 2). When d
FIG. 4: The critical αc = φ2/φ1, at which the interactive becomescomparablewithA,theelectricpotentialφgets
force between two spheres is zero, is plotted as a function of a periodic component along the surface, which does not
L/2R numerically. The minus (plus) sign under (above) the exist for a real conductor. In this paper, we always fo-
curve means that the interaction between the two spheres is cus on the case when d A. Still, for Ze Q and,
attractive (repulsive). therefore, A R, there i≫s a big window of A≪ d R
≪ ≪ ≪
in which the model of conductors works and Eq. (23) is
applicable.
L 2R,keeping the leading orderterms in Eq.(15), we
≫
get
B. A small sphere close to a big sphere
DR3 DR2
C =DR+ , C = , (20)
11 L2 12 − L
Now let us consider the interaction between a small
and sphere with radius R and a big sphere with radius R .
1 2
We are interested in the limit R R (Fig. 3b). We
1 2
F(L)= DR2φ φ DR3(φ2+φ2). (21) can solve this problem by a similar≪procedure as in the
L2 1 2− L3 1 2 lastsubsection, startingfroma formulalike Eq.(15) but
with both R and R in it [9]. Instead of doing this
Thephysicalmeaningisclear. Whenthetwospheresare 1 2
complicated calculation, let us use physics intuition and
far away from each other, they interact like two point
look at the interesting limit d R . In this limit, the
charges. Inthezeroorder,themagnitudeofthechargeis ≪ 2
size of the large sphere, R , becomes irrelevant to the
determined by their own capacitance and potential, i.e., 2
problem, and the interaction can be considered as the
DRφ and DRφ . This gives the first term in Eq. (21)
1 2 interaction between a sphere with a metallic semi-space
(one D in charges is cancelled out since the interaction
(see Fig. 3b).
itself is proportional to 1/D). In the next order, each
Let us first consider the case when φ = 0. As well
point chargeinduces opposite charge in the other sphere 2
knownin electrostatics,a point chargeinduces an image
with the magnitude DR2φ/L following the standard
− charge in a metallice semi-space which describes the in-
methodofimages[9]. Theinteractionbetweenthecharge
teraction between the charge and the metal. Similarly,
and its image gives the second term in Eq. (21). From
usingaseriesofimagecharges,onecanshowthatthein-
Eq. (21), one can easily get α = R/L. Therefore the
c teraction between a sphere and a metallic semi-sphere is
attraction dominates only if φ is much smaller than φ .
2 1 equivalent to the interaction between the sphere and its
The attractive force is proportional to (R/L)3.
“image sphere” induced in the metal (see Fig. 3b). The
Themoreinterestinglimitiswhenthetwospheresare
image sphere and the originalsphere have the same size,
very close to each other, i.e., d R. In this limit, we
≪ and their positions are symmetric about the boundary
have
of the metal. If the original sphere has fixed potential
DR 16β2R φ1, its imagesphere hasfixedpotential φ1. Since these
−
C11 = ln , (22a) two spheres are oppositely charged, we immediately see
4 d
thataconductingsphereisalwaysattractedtoametallic
DR β2R
C = ln , (22b) semi-space. Thereforespheres1and2attracteachother.
12
− 4 d
Nowweconsiderthecasewhen φ =0. Thispotential
2
6
is equivalent to a charge sitting at the center of sphere
where β = 1.78 (see Appendix for the derivation of
2, with magnitude DR φ . It induces an image charge
Eq. (22)). According to Eq. (14), we have 2 2
in sphere 1. In the limit of R , this image charge
2
→ ∞
DR is located at the center of sphere 1 with the magnitude
F(d)=− 8d (φ1−φ2)2. (23) −DR1φ2 (seediscussionofmethodofimagesinRef.[9]).
7
Itisequivalenttoaddanpotential φ tosphere1. Asa In most cases, the effect of screening by monovalent salt
2
−
result,thepotentialofsphere1isrenormalizedtoφ φ . can be described by the linear Debye-Hu¨ckel screening
1 2
−
Therefore,inthisparticularcase,theinteractionbetween radius, r (we discuss the possibility and effect of non-
s
two spheres with potentials φ and φ is the same as linear screening at the end of this section). What we
1 2
two spheres with potentials φ φ and 0. Clearly, the discussedinthelasttwosectionscorrespondsto thecase
1 2
−
attractivenatureoftheinteractionbetweensphere1and r R ,R ,d. Now we wouldlike to considerthe oppo-
s 1 2
≫
itsimagesphereisnotchangedbythenonzeroφ (except site limit, A r R ,R . We show that even though
2 s 1 2
≪ ≪
the special case φ = φ when there is no interaction attractionissuppressedandlosestorepulsionatd r ,
1 2 s
between the two spheres). the two spheres still attract each other at d<r . ≫
s
Having established the attractive nature of the force, ∼
we now calculate it quantitatively. Since the interaction
isessentiallybetweensphere1anditsimagesphere,C , A. The method of images in the presence of
11
C and C can be expressed through a linear combi- monovalent salt
12 22
nation of C and C , where C and C are self and
aa ab aa ab
mutual capacitances of sphere 1 and its image sphere Let us first discuss how the method of images is mod-
given by Eq. (15). We have ified in the presence of monovalent salt. We start from
the simplest situation in which there is a point charge q
C11(d) = Caa(2d) Cab(2d), (24a) close to a metallic semi-space in a water solution with
−
C (d) = C (2d) C (2d), (24b) monovalent salt described by screening radius r . When
12 ab aa s
−
r , it is well known that the equal potential con-
C (d) = DR +C (2d) C (2d), (24c) s
22 2 aa ab → ∞
− dition at the boundary of the metal can be satisfied by
′
HereweputtheddependenceineachCtoremindusthat putting an image chargeq = q inside the metal at the
−
thedistancebetweenspheres1and2isdbutthedistance position symmetric to q (Here and below, as a premise,
betweensphere1anditsimagesphereis2d(seeFig.3b). water should also be introduced with the image charge
Noticing that C = C and ∂C /∂d = ∂C /∂d, to produce the same dielectric constant). In the pres-
11 12 11 22
−
using Eq. (13), we have ence of monovalent salt, one can easily check that the
sameboundaryconditioncanbe satisfiedby introducing
∂C11(d)(φ1 φ2)2 thesameimagechargeq′ atthesameposition,providing
F(d) = −
∂d 2 thatavirtualcloudofmonovalentionswiththescreening
∂Cab(2d) ∂Caa(2d) (φ1 φ2)2 radius rs is produced together with the image charge.
= − .(25)
− ∂d − ∂d 2
(cid:18) (cid:19)
Since ∂C /∂d< 0 and ∂C /∂d> 0 as we discussed in rs rs
aa ab R R
the last subsection, the force is attractive for any given d
φ andφ (exceptthe specialcase φ =φ whenthere is
no1 intera2ction between the two sphe1res). 2 rs q q q rs
Theexpressionsofforcesareparticularlysimpleincer- ϕ
tainlimits. WhenR d R ,usingEq.(20),wehave
1 2 L
≪ ≪ L
F(d)= DR12(φ φ )2. (26) (a) (b)
− 4d2 1− 2
When d R , using Eq. (22), we have
≪ 1 FIG. 5: Images in a conducting sphere in the presence of
DR monovalent salt with screening radius rs: (a) a point charge
F(d)= 1(φ φ )2. (27) q induces an image charge q′ = −Rq/L with a virtual cloud
1 2
− 4d − ′
of monovalent ions with screening radius r = Rr /L; (b) a
s s
ThisresultcanbecomparedwithEq.(23)ifR =R. We conducting sphere with potential φ is equivalent to a point
1 charge q′′ = DRφeR/rs with a virtual cloud of monovalent
see that the force is stronger by a factor of 2 in the case
ions with screening radius r .
of two spheres with different sizes. This is simply due to s
the fact that the surfaces of the two spheres are “closer”
Asamorecomplicatedsituation,letusconsiderapoint
and the capacitances diverge faster in the present case.
charge q and a grounded conducting sphere in a water
solution with monovalent salt described by r (Fig. 5a).
s
As well known [7], when r , the magnitude and
IV. THE EFFECT OF SCREENING BY s
→ ∞
MONOVALENT SALT position of the image charge are given by (see Fig. 5 for
definition of all lengthes)
In a water solution of macroions and Z-ions, nor- R R2
′ ′
mally there is also certain amount of monovalent salt. q = q , L = . (28)
− L L
8
When r is finite, similar to the case of metallic semi- from the leading order image charges only. To see this,
s
space, a virtual cloud of monovalent ions should be cre- one just remember that the image charges really repre-
ated together with the image charge. Due to the special sentsurfacechargedensities induced onthe twospheres.
geometry of a sphere, as one can check, the screening Eachimageinthe infinite seriesgivesacorrectionto the
radius of this virtual cloud is given by surfacechargedensitycalculatedfromthepreviousorder
image. As we discussed after Eq. (31), the new correc-
rs′ = RLrs, (29) htiiognheinrdourdceedr cgoertrseoctnieonmsoarreefeaxcptoorneen−tdi/arlsly.sSminaclleadn≫d crasn,
different from r in the solution. be completely neglected.
s
Anotherrelevantsituationis aconductingspherewith
fixedpotentialφinawatersolution(seeFig.5b). Inthis
case, the potential in the solution can be solved exactly r
s
using linearized Poisson-Boltzmann equation and inter-
R R
pretedbythemethodofimages. Actuallyitisequivalent
d
′′
to the potential produced by a image charge q at the
center of the sphere. We can create a virtual cloud of q q q q
1 1 2 2
monovalent salt together with the image charge as our
ϕ ϕ
convenience. The magnitude of q′′ is then determined L
1 2
not only by the boundary condition but also the virtual
cloud we choose. As we will see in the next subsection,
it is convenientto introduce r inside the sphere and get
s
q′′ =DRφeR/rs. (30) FIG.6: Thefirsttwoimagechargesinducedineachspherein
consideringtheinteraction between twospheres. Each image
Letuscalculatetheinteractionenergybetweenapoint charge has its own virtual cloud (not shown).
charge and a grounded conducting sphere (Fig. 5a) for
the purpose of the next subsection. Considering the in-
′ In Sec. III, the energyis calculatedusing capacitances
teraction energy between q and q , we have
(Eq. (12)). In the present case, it is more convenient
′
U = qq e−(R−L′)/rs′e−(L−R)/rs to considerinteractionenergybetween imagechargesdi-
2D(L L′) rectly. The two methods are equivalent providing that a
− factor 1/2 is added to the interaction energy between
Rq2
= e−2d/rs, (31) a charge and its image [7] in the later method (see
−2D(L2 R2)
− Eq. (36)). The two image charges in each sphere are
whered=L R. Inthelimitofr ,theexponential given by (see Fig. 6)
s
− →∞
factorgoesto1andtheenergygoesbacktothestandard
expression. The additional factor e−2d/rs can be under-
q = DRφ eR/rs, q =DRφ eR/rs, (32a)
stood in the following way. The charge induced by q on 1 1 2 2
the surface of the sphere is proportional to e−d/rs, the q′ = DR2φ2eR/rs, q′ = DR2φ1eR/rs.(32b)
interaction with it is also proportional to e−d/rs. 1 − L 2 − L
Hereq andq arezeroorderimagechargesineachsphere
1 2
B. Two spheres of the same size whichtakeintoaccountofφ andφ respectively,similar
1 2
′′
to q discussed in the last subsection (see Eq. (30) and
Nowweconsidertheinteractionbetweentwoconduct- Fig.5b). Thescreeningradiusaccompaniedwiththemis
ing spheres of the same size (see Fig. 3a) in the presence just r . The first order image charges q′ and q′ induced
s 1 2
ofmonovalentsalt. Duetothecomplexityofthemethod by q and q are accompanied by r′ = Rr /L, similar
2 1 s s
ofimagesinthepresentcase,insteadofgivingthegeneral to q′ discussed in the last subsection (see Eqs. (28), (29)
expression for the force, we focus on two limits: d r andFig.5a). SimilartoEq.(31),theinteractionenergies
s
≫
and d r . between these images are
s
≪
Let us first discuss the case when d r . As well
s
≫
known,when r , there is aninfinite series ofimage
charges in eachss→ph∞ere [9]. For finite rs, similar to what U(q1,q2) = DR2φ1φ2e−d/rs, (33)
L
we discussed in the last subsection, the magnitudes and
positions of all image charges are the same but a virtual U(q ,q′) = DR3φ21 e−2d/rs, (34)
cloud of monovalent salt is created together with each 1 2 −L2 R2
−
of them. In the particular case of d ≫ rs, we can actu- U(q ,q′) = DR3φ22 e−2d/rs, (35)
allycutofftheinfiniteseriesandincludethecontribution 2 1 −L2 R2
−
9
and
1 R r
U(L) = U(q ,q )+ [U(q ,q′)+U(q ,q′)] s
1 2 2 1 2 2 1
θ d
= φ φ DR2e−d/rs φ21+φ22 DR3 e−2d/rs. 0
1 2 L − 2 L2 R2
−
(36) x
Correspondingly,
DR2 1 1
F(L) = φ φ e−d/rs +
1 2
L r L
(cid:18) s (cid:19) FIG.7: Aschematicillustrationofthecontactregionbetween
(φ2+φ2) DR3 e−2d/rs 1 + L . two spheres. The contact region includes the surfaces of two
− 1 2 L2 R2 r L2 R2 spheres inside acylinder with thecross section shown bythe
− (cid:18) s − (cid:19) rectangular.
(37)
Weseeclearlythatanadditionalfactorofe−d/rs appears
Here DR2/r is the capacitance of a single sphere in the
for each order of interaction. When r , this equa- s
s → ∞ presence of monovalent salt. When the sphere gets close
tion goes back to Eq. (21) as expected.
totheothersphere,itlosesitscapacitanceinthecontact
Atd r ,thefirstterminEq.(37)dominates. When
φ1 =φ2≫, weshave the standard double layer repulsion of regTiohne,nwwheosecoanrseiadeisr πaRno(rthse−rds)p.ecial case when φ =
DLVOtheory[5]. Whenφ andφ havesamesigns,only 1
1 2 φ =φ. Inthiscase,wehaveq =(C C )φ= q .
thesecondtermrepresentsattraction,whichisnegligible − 2 1 11− 12 − 2
Outside the contact region, the capacitance is the same
comparing with repulsion. This is analogous to Eq. (21)
as in the previous case. Inside the contact region, the
butattractionismuchweakerhere. Sointhelimitofd
≫ capacitance can be estimated as
r , the force is attractive only if potentials are opposite
s
in signs or one of potentials is equal to zero.
θ0 dS θ0 2πR2sinθdθ r
Now let us consider the case when d rs. In this C =D =D =πDRln s,
cchasaer,getsheshfoauctldorbee−idn/crlsud≃ed1. aSnindcealleahcihghime≪raogredcehrairmgeagies Z0 x(θ) Z0 d+2R(1−cosθ) (39d)
accompanied by its own virtual cloud, the interaction is where θ0 is the angle for the boundary of the contact
very complicated. Instead of calculating it exactly, we region (see Fig. 7). The absolute value of the charge in-
estimate C and C using a simple method as follows. ducedoneachsphereinthecontactregionis2Cφ. There-
11 12
We divide the surface ofthe two spheres into two pieces. fore
Inthefirstpiece,thedistancebetweenthesurfacesofthe
DR2 DR(r d) r
twospheresissmallerthanrs sothatthetwospheresin- C11 C12 = s− +2πDRln s. (40)
teract with each other in an unscreened Coulomb way. − rs − 4rs d
We call this piece “contact region” (see Fig. 7). In the
second piece, the distance between the two surfaces is Combining Eqs. (38) and (40), we get
largerthanr andthe interactionbetweenthemis expo-
s
nentially small and negligible. Therefore, we can safely DR2 DR(rs d) rs
C = − +πDRln , (41a)
11
assume that the modification to the capacitances of the r − 4r d
s s
two spheres due to their proximity happens only in the rs
C = πDRln . (41b)
contact region. 12 − d
In order to calculate C and C , let us first consider
11 12
a special case when φ1 = φ2 = φ. In this case, the Inthecaseofrs ∼R,theseresultsmatchEq.(22)which
charge induced on two spheres are equal and given by was obtained for rs R.
≫
φ(C11+C12). In the contact region, the screening effect Using Eq. (14), and remembering d≪rs, we have
bymonovalentsaltcanbeignoredandtheadjacentspots
of the two spheres form a capacitor with zero voltage, F = πDR(φ φ )2. (42)
1 2
therefore the charge is zero [11]. While for the rest part − 2d −
of the sphere, the interaction between two spheres is ex-
The force is independent of r in this limit because the
ponentiallysmallandthechargeoneachsphereisalmost s
changeofcapacitanceshappensmainlyinthecontactre-
the same as if the other sphere is not there. Therefore
gion(thelogarithmterminEq.(41)). Again,theattrac-
we have
tion shows up at small d and the attractive force decays
DR2 DR(r d) asapowerlaw. Forthe modelofconductorsto work,we
s
C +C = − . (38)
11 12 rs − 4rs require d≫A which is valid since A≪rs.
10
C. A small sphere and a big sphere to φ (see Eq. (44)). This potential is certainly different
c
from the electric potential of the probe determined by
Now we consider the interaction between a small Z-ions (Eq. (11)). We again have two conductors with
sphere and a big sphere (see Fig. 3b) in the presence differentfixedpotentials. Evenwhenboth potentialsare
of monovalent salt. The approach we used is similar to positive(thenetchargesofthesurfaceandtheprobeare
that in the last subsection. positive),theforceisstillattractiveatsmalldistancesas
When d r , calculating the interaction between we discussed in the last section.
s
leading orde≫r image charges similar to Eq. (37), we get The atomic force experiment in Ref. [2] is actually a
realization of the situation discussed above. The maxi-
F(L) = φ φ DR1R2e−d/rs mumoftherepulsiveforcementionedinintroductioncan
1 2 Lrs beunderstoodfollowingEq.(43),extrapolatedtod≃rs.
R φ2 R φ2 DR R When both potentials are positive, at d rs, the first
1 1 + 2 2 1 2e−2d/rs.(43) term(repulsionterm)inEq.(43)dominate≫sandtheforce
− (cid:18)L2−R22 L2−R12(cid:19) rs is repulsive. At d < rs (rs 100 ˚A in the experiment),
thesecond,attract∼ionterm≃inEq.(43)dominatesdueto
Again, when φ and φ have the same signs, attraction
1 2
its largerpre-factorofthe exponential. This leads to the
is exponentially weaker in this limit. When d r , we
s
≪ maximum of the repulsion at d r . When d r , we
can estimate C , C and C similar to Eq. (41). We s s
11 12 22 ≃ ≪
have the polarization attraction given by Eq. (1).
get an attractive force given by Eq. (1).
For d<r , Eq. (3) can be used to compare the polar-
s
izationattractionwiththevanderWaalsone. According
In the end of this section, let us discuss the possi- to discussions in introduction and Sec. II, we conclude
bility of nonlinear screening by monovalent salt in a that for substantial difference between σ1 and σ2, the
polarizationforceF ismuchlargerthanF atd A
specialcasewhenφ >0. Theresultcanbegeneralized p vdW
1,2 ≫
to other cases. The condition for linear screening reads (A=21˚AforZ =3and σ = 0.75e/nm2). Therefore
− −
the van der Waals attraction has nothing to do with the
kBT attraction observed in the experiment at d 100 ˚A.
φ1,2 <φc = e |ln(cv0)|, (44) So far we have considered two cases wh≃en the elec-
tric potentialsofthe twomacroionsaredifferentandthe
where e is the proton charge and c and v are the con-
0 attraction between likely charged macroions is possible.
centration and volume of the monovalent ion. Here φ
c One is that positive Z-ions are adsorbed to both neg-
is determined by entropy of monovalentions in the solu-
ative macroions; the other is that positive Z-ions are
tion. AccordingtoEqs.(4)and(11),φ <Ze/Dr
Ze/Da, where a r is the radiu1s,2o∼f a Z-ion1,2(s≪ee onlyadsorbedtoonenegativemacroion,butmonovalent
≪ 1,2 counterions are adsorbed (nonlinearly screening) to the
Fig. 2). If φ > φ , Ze/Da φ . Therefore monova-
1,2 c ≫ c other positive macroion. It is interesting to relate them
lent counterions are adsorbedto Z-ions. As a result, the
to the more standard case when two likely charged (say,
effective charge of a Z-ion is renormalized to
negative)macroions are nonlinearly screened by positive
∗ monovalent counterions (no Z-ions in the solution). In
Z e=Daφ . (45)
c thiscase,thetwomacroionshavethesameelectricpoten-
tialφ givenbyEq.(44)andneverattracteachother[5].
This effective charge is not changed when a Z-ion is ad- c
∗
sorbedonthesurfaceofthemacroionsinceZ e/Dr
1,2
∗ ≪
Z e/Da. Consequently, φ should be calculated using
1,2
Z∗ and satisfy φ < Z∗e/Da = φ . In summary, the VI. CONCLUSION
1,2 c
nonlinearscreeningofmonovalentsaltisimportantwhen
Z-ions are strongly charged. In this case, our theory is In this paper, we discussed a long range polarization
∗
stillapplicableprovidingthatZ is replacedbyZ every- attraction of two likely charged spherical macroions in
where. the presence of multivalent counterions (Z-ions). We
show that the necessary condition for the attraction is
that the bare charge densities of the two macroions are
V. ATTRACTION OF TWO MACROIONS, different. This polarization attraction is much stronger
WHEN ONLY ONE OF THEM ADSORBS Z-IONS and longer ranged than the van der Waals force [5] and
theshortrangecorrelationattraction[6]. Inthepresence
Inthissection,wegeneralizeourtheorytoalittlemore of large amount monovalent salt, it adds an additional
complicatedcasewhenthetwobaremacroionsareoppo- termtothestandarddoublelayerrepulsionofDLVOthe-
sitely charged. As shown in Fig. 1a, Positive Z-ions are ory when the two macroions are different. We discussed
onlyadsorbedtothenegativeprobeandinvertitscharge. twocaseswhenthepolarizationforcebetweentwodiffer-
Inthepresenceoflargeconcentrationofmonovalentsalt, entmacroionscanbe attractive,eveniftheirnetcharges
monovalentcounterionsareadsorbedtothe positivesur- havethe samesign. Inthe firstcase,both macroionsad-
face (nonlinear screening), reducing its electric potential sorb Z-ions (Fig. 1b). In the second case, one macroion