Derivation of the Inverse Schulze-Hardy Rule

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by  Gregor Trefalt

Derivation of the Inverse Schulze-Hardy Rule
∗
Gregor Trefalt
Department of Inorganic and Analytical Chemistry, University of Geneva,
Sciences II, 30 Quai Ernest-Ansermet, 1205 Geneva, Switzerland
(Dated: March 21, 2016)
The inverse Schulze-Hardyrule was recently proposed based on experimental observations. This
ruledescribesaninterestingsituationoftheaggregationofchargedcolloidalparticlesinthepresence
of the multivalent coions. Specifically, it can be shown that the critical coagulation concentration
is inversely proportional to the coion valence. Here the derivation of the inverse Schulze-Hardy
6
rule based on purely theoretical grounds is presented. This derivation complements the classical
1
0 Schulze-Hardyrulewhich describes the multivalent counterion systems.
2

r The aggregation of charged colloids is a long-studied Interestingly, in the low particle charge limit, where the
a
M phenomenon. More than 100 years ago Schulze and DL forces can be described by the Debye-Hu¨ckel (DH)
Hardy showed that the aggregation power of salts de- approximation, the same

8 pends strongly on the ion valence [1, 2]. More precisely, 1
1 the critical coagulation concentration (CCC) (i.e., con- CCC∝ (3)
z(z+1)
centration of salt at which particles start to aggregate

] fast) [3] decreases very rapidly by increasing counterion dependenceforbothcounterionsandcoionsisreached[8,
t
f valence. This discovery was later confirmed theoreti- 9]. The latter low charge limit (3) lies between the
o
s callybyDerjaguin,Landau,VerweyandOverbeekandis Schulze-Hardy (1) and inverse Schulze-Hardy (2) depen-
. knownasthe DLVOtheory[4,5]. Theyhaveshownthat dences. This proposed inverse Schulze-Hardy rule there-
t
a by assuming the interaction between particles as a sum fore elegantly completes the understanding the aggrega-
m
of van der Walls (vdW) and double layer forces (DL) in tion in experimentally relevant asymmetric multivalent
- the symmetric z:z electrolyte the CCC is inversely pro- electrolytes.
d
portional to the sixth power of the valence, The Schulze-Hardyrule (1) and the low chargeDH limit
n
(3) were both derived theoretically. On the other hand,
o
1
c CCC∝ (Schulze-Hardy rule). (1) the inverse Schulze-Hardy rule was only given as an
[ z6 empirical dependence based on experimental observa-

tions. [9] Therefore, the aim of this paper is to present
3 The above relation, also named Schulze-Hardy rule, is
the derivation of the inverse Schulze-Hardy rule based
v valid for highly charged particles. This explanation con-
5 solely on theoretical grounds.
firmed the DLVO theory and made it widely accepted.
8 A naive explanation of the inverse Schulze-Hardy rule
Thesymmetricz:z electrolytesareusuallypracticallyin-
2 would come from the original argument of Schulze and
soluble,thereforeinexperimentsonetypicallyusesasym-
3 Hardy. They have explained that the CCC is controlled
0 metric 1:z or z:1 multivalent electrolytes. In the case of
by the counterion concentration. In asymmetric 1:z and
. asymmetric electrolytes,multivalent ions caneither play
1 z:1 electrolytes, where z represents the coions,the coun-
a role of the counterions or the coions, where they have
0 terions are monovalent. In these systems, the concentra-
6 the opposite or the same charge as the colloidalparticle,
tion of monovalent counterions is equal to zc, where c is
1 respectively. It was shownexperimentallyas well asthe-
the salt concentration. If one now assumes that the ag-
: oretically that for highly charged particles the Schulze-
v gregationhappens at constant counterionconcentration,
Hardy rule (1) is a good approximation also for asym-
i thisleadstothe1/zdependenceofthesaltconcentration.
X metric electrolytes where z is the counterion valence [6–
Inthissituation,CCC∝1/zproportionalitystemssolely
r 8]. Recently Cao et al. [9] investigated a complemen-
a from the composition of the 1:z and z:1 salts. However,
tary problem, namely the influence of multivalent coions
as it will be shownbelow, this simple intuitive reasoning
on the aggregation. In this situation experimental data
cannot be justified.
could be reasoned with the inverse Schulze-Hardy rule,
In order to describe the aggregation in the presence of
namely
multivalent coions one can follow the original DLVO ap-
1 proach. This derivation is based on the calculation of
CCC∝ z (inverse Schulze-Hardy rule), (2) the total interaction energy Utotal, between two charged
colloids as a sum of attractive vdW and repulsive DL
wherez is thevalenceofthe coion. Notethatinthecase contributions,
of the coions the dependence on valence is much weaker.
Utotal =UvdW+UDL. (4)
At low salt concentrations the DL interactions are dom-
inant and an energetic barrier develops. With increas-
∗ E-mail: [email protected] ing concentration the barrier diminishes and when it is
2
can be approximated as [26–28]
Πnear(h)= 2π2εε0 · 1 , (5)
DL β2e2 h2
0
where εε0 is the dielectric permittivity, β = 1/kT is the
inversethermalenergy,e0istheelementarycharge,andh
is the separation distance. The above expressionis valid
forthedistanceslargerthantheGouy-Champmanlength
λ = 2εε0 with σ being the surface charge density. The
βe0σ
counteriononlycasesetstheconcentrationofsaltoutside
FIG.1. (a)Schematicpresentationofinteractionenergyevo-
the gap to zero. However, in the case of the added 1:z
lutionwithincreasingsaltconcentration. (b)Disjoiningpres-
surebetweentwonegativelychargedparticlesinthepresence salt, as in Fig. 1b, the concentration outside the gap is
of0.1mM1:4electrolyte, experimentaldatatakenfrom [26]. finite. The near-field pressurecan therefore be corrected
Full Poisson-Boltzmann, near-field, corrected near-field, and for the osmotic pressure outside the gap
far-fieldcurvesarealsopresented. Notethatinthepresented
case thevan derWaals interactions are negligible. Πnear(h)= 2π2εε0 · 1 − (z+1)c, (6)
DL β2e2 h2 β
0
close to zero the particles aggregate, see Fig. 1a. The where c is the bulk concentration of 1:z salt. Note
salt concentration at which the energy barrier vanishes that this simple correction substantially improves the
is a goodapproximationfor the CCC.One cantherefore accuracy of the near-field approximation at larger dis-
understand the effect of multivalent ions on aggregation tances, see Fig 1b. The far-field pressure ΠfDaLr, is of DH
through the effect of such ions on the interactions. The type [26, 27]
interactions in the presence of multivalent counterions
were studied extensively by both experimentalists [10– ΠfDaLr(h)=2εε0κ2ψe2ffe−κh, (7)
17] and theoreticians [18–24]. In particular the effect of
ion correlations and validity of the mean-field Poisson- where ψeff is the effective surface potential, κ= q2βεεe020I
Boltzmann (PB) treatment was addressedin these stud- is the inverse Debye length, and I is the ionic strength
ies. Ontheotherhand,theinteractionsbetweencharged calculatedasI = 1z(z+1)cfor1:z electrolyte. Notethat
2
particles in the presence of multivalent coions,whichare the effective potential for asymmetric electrolytes is not
ofinteresthere,receivedmuchlessattention[19,25,26]. knownanalytically. Wecannowdefineatransitionpoint
Inthesesituationsthedouble-layerinteractionsaremuch ht between the the two limits. Below and above the ht,
softer andlonger-rangedascomparedto the interactions near-fieldandfar-fieldlimitsarevalid,respectively. Such
in the presence ofmonovalentelectrolytes or multivalent treatment successfully describes the experimental force-
counterions. Themultivalentcoionsalsohaveaprofound curves [26]. By increasing the concentration of salt the
influence on the shape of the force-curves. While in the near-field limit is unaffected, what changes are the far-
presenceofmonovalentelectrolyteandmultivalentcoun- field limit and the transition region with the transition
terions the profiles are exponential down to small sep- point,seeFig2a. Thefactthatthenear-field(5)doesnot
arations, the shape of the curves for multivalent coions dependonmonovalentcounterionconcentrationrulesout
is exponential only at large separations. In the latter the simple intuitive explanation of the inverse Schulze-
case the interaction can be decomposed into near-field Hardy rule which is described above. The CCC cannot
algebraic and far-field exponential parts [26]. Fig. 1b be controlled solely by counterion concentration as the
shows such interaction between two negatively charged near-fieldinteractiondoesnotdependonitinthecaseof
particles in the presence of 1:4 electrolyte. Note that multivalent coions. To get the complete picture one has
both the particle and the multivalent ion are negatively to rather look at how the transition point and far-field
charged. An oppositely charged system with positively behavior are affected by the addition of salt.
chargedparticles and multivalent cations can be also re- The transition point ht, between near-field and far-field
alized [26]. The experimental profile can be accurately approximations represents a separation distance, which
described by mean-field PB theory. In this case, no ion- marks the start of expulsion of the multivalent co-ions
correlations effects are expected [19]. The PB equation from the region between the two charged plates upon
is solvednumerically for two chargedplates immersed in closer approach. The exclusion happens due to electro-
anasymmetricelectrolytesolution,fordetailssee[8,26]. static repulsion between charged plates and the coion.
Thecharacteristicshapeofthepressurecurveisthecon- The interaction between a coion and the two plates can
sequence of expulsionof multivalent coions from the gap be estimatedby calculatingthe electrostaticpotentialat
between the surfaces, which in the presented case hap- the mid-plane between two plates. When the plates are
pensatabout60nm,seeFig.1b. Thenear-fieldpressure closer than ht only counterions are present in the slit.
Πnear, therefore corresponds to the monovalent counte- Here we assume positive plates with monovalent anions
DL
rion only case (i.e., multivalent coions are expelled) and as counterions. Note that the analogous situation with
3
The variationof the transitionpoint with z:1 electrolyte
concentration is shown in Fig. 2b. One can observe that
the results from the full PB treatment can be well ap-
proximated with relation (12).
By knowing the position of the transition point, we can
approximate the electrostatic interaction between the
particles by using the near-field limit (5) when h ≤ ht
and the far-field limit (7) for h > ht. The interaction
force between two particles with radius R, can then be
obtained by integration of the pressure and by applica-
tion of the Derjaguin approximation
FIG. 2. (a) Evolution of disjoijning pressures between two
negatively charged plates with increasing 1:4 salt concentra- ht ∞
tion. Full PB solution is presented by full lines, corrected Fnear =πR Πnear(h′)dh′+πR Πfar(h′)dh′,
DL DL DL
Z Z
near-fieldlimitdashedlines,andfar-fieldlimitdashed-dotted h ht
lines. (b) Transition point as a function of salt concentra- (13)
tion for multivalent coions of valence z. Points present the which yields near-field force
full PB calculations, lines are the approximation from the
to−hf5e0trhmmeaClin/ctmoenr2adciitstiiouonnses(d1i.2s).prNesoetnettehda.t oSnulryfatcheecehleacrtgreosdteantiscitpyaortf FπDnReLar = 2βπ22eε20ε0 (cid:18)h1 − h1t(cid:19)+ βεε20eκ20z2 (h−ht)
+2εε0κψe2ffe−κht. (14)
By analogy the far-field force is
negative plates and monovalent cations is possible and
wouldyield the same result. For the former case of posi-
Ffar
tively charged plates the following form of the PB equa- πDRL =2εε0κψe2ffe−κh. (15)
tion has to be satisfied [26, 27, 29]
Integrationoftheforceyieldsthepotentialenergyprofile
d2ψ = e0c−eβe0ψ, (8) in the near-field limit
dx2 εε0 ht ∞
Unear = Fnear(h′)dh′+ Ffar(h′)dh′. (16)
where ψ is the electric potential and c− is the number DL Z DL Z DL
concentration of the counterions. The prefactor c− fixes h ht
the potential at the surface of the plate to the value of The two integralsabovecanbe solvedanalyticallyyield-
surface potential in z:1 electrolyte in the limit of high ing the expression
surface chargedensities.The solutionof Eq.(8) gives the
electric potential at the mid-plane UDneLar =Bln ht
πR (cid:18)h (cid:19)
2 αh
ψM(h)=−βe0 ln(cid:18)2π(cid:19), (9) +B 1− h 1 −1− κht − κ2h2t
(cid:18) ht(cid:19)(cid:18)κht 2π2z 2π2z(cid:19)
Now the multivalent coion with valence z wants to enter h2 κ2h2 B B
the slit and is affected by the electrostatic repulsion ex- +B(cid:18)1− ht(cid:19)(cid:18)4π2zt(cid:19)+ κ2h2t − 2π2z, (17)
ertedby the plates. One canapproximatethat the coion
weleilcltreonstteartitcheenreerggiyonatbtehtewemeind-tphleantewios epqlautaelstowh2eknTthe where constant B = 2βπ22eε20ε0 and the equality ΠnDeLar(ht)=
ΠfDaLr(ht)wereusedtowritetheequationinthecondensed
zβe0ψM =2. (10) form.
The total interaction energy (4) can be now calculated
At this point the separationof the two plates is equal to by summing the vdW and DL energies. The van der
ht. Combining Eq. (9) and Eq. (10) yields the position Waalscontributionscanbeapproximatedbysimplenon-
of the transition point retarded expresions
ht = 2απe−z1. (11) FvdW =−H12R · h12, (18)
HR 1
In the case of asymmetric z:1 electrolyte, where c− =zc UvdW =− · , (19)
and I = 1z(z+1)c we finally arrive at 12 h
2
whereH istheHamakerconstant. Followingtheoriginal
z+1 −1 DLVOcondition[4,5]the CCC canbe estimatedby set-
htκ=2πr 2 e z. (12) ting the energy barrier of the total interaction energy to
4
CCC. Firstwe canget the following expressionfor ht by
equating Eq. (21) to zero
−1
ht =Ce z, (23)
where C = hmaxexp 12πBHhmax + 32 is a constant. By
(cid:16) (cid:17)
using Eq. (12) we arrive at
2π z+1e−z1
2
κ= q . (24)
Ce−1z
FIG. 3. (a) Value of the energy maximum in the interac-
tion total energy potential between two charged particles as For a 1:1 electrolyte (z =1) Eq. (24) yields
a function of relative concentrations of multivalent coions.
2π
Resultsforthevalenceofthecoionsbetween1and5arepre- κ≈ (25)
sented. The concentrations are normalized by concentration C
of 1:1 electrolyte where the energy barrier is 0 kT. Curves
for the full PB solution and the near-field solution (17) are and for a z:1 electrolyte (z >1) with
presented. (b) Relative CCC as a function of coion valence
calculated with the full PB, the near field solution and the 2π z+1
κ≈ (26)
analytically derived inverse Schulze-Hardy 1/z dependence. C r 2
Low-potential Debye-Hu¨ckel and Schulze-Hardy 1/z6 depen-
dences are also shown. Surface charge density of 0.4 C/m2 AttheCCCtheDebyelengthforz:1electrolyteisdefined
wasusedforPBcalculations. Hamakerconstantof1·10−18 J as
and R=150 nm were used throughout.
2βe2 z(z+1)
κ2 = 0 · ·CCC (27)
εε0 2
zero. This condition can be mathematically written as
Combining Eqns. (25)–(27) we arrive at the expressions
dUtotal =−Ftotal(hmax)=0 and Utotal(hmax)=0 for CCC in 1:1 electrolyte
dh (cid:12)
(cid:12)(cid:12)hmax (20) CCC(z =1)= 2π2εε0, (28)
(cid:12) C2βe2
The full PB solution for the interaction energies be- 0
tweentwochargedparticlesinthepresenceofmultivalent
and in z:1 electrolyte for z >1
coions shows that the position of the maximum in the
epnaerregdytoprtohfielepiossgiteinoneraolflythaettsrmanasllietirosneppaoriantti,ohnmsaaxs<cohmt-. CCC(z >1)= 2Cπ22βεeε20 · 1z. (29)
Therefore, Eqs. (12), (14–15), (17–19) can be used with 0
condition (20) to numerically calculate the evolution of From Eq. (28) and Eq. (29) the inverse Schulze-Hardy
thevalueoftheenergymaximumwiththeconcentration rule immediately follows
within the near-field approximation, see Fig. 3a. At the
concentration when the energy maximum reaches zero CCC 1
= . (30)
(i.e., the barrier is 0 kT) the particles aggregate. The CCC(z =1) z
numerical near-field solution confirms that ht ≫ hmax
and that hmax at CCC is practically independent of the Finally, in Fig. 3b the relative CCC as a function of
coion valence. We can use these criteria to further ap- ion valence is plotted. One can observe that the full
proximate the near-field energy limit and calculate the PB and near-field numerical solutions match closely the
total energy at maximum using Eq. (17) and Eq. (19) inverse Schulze-Hardy 1/z dependence. These curves
represent the high-charge limit and their dependence
Utotal(hmax) =Bln ht +Bf(z)− H , (21) is weaker as compared to the low-charge Debye-Hu¨ckel
πR (cid:18)hmax(cid:19) 12πhmax limit. The case of multivalent counterions, which yields
much stronger Schulze-Hardy 1/z6 dependence in the
where f(z) is a function of coion valence high-charge limit, is also shown.
Inconclusion,thederivationoftheinverseSchulze-Hardy
f(z)= 1 −1− κht − κ2h2t + 1 − 1 , rule is shown. The simple inverse proportionality of the
(cid:18)κht 2π2z 4π2z κ2h2t 2π2z(cid:19) CCC on the coion valence is not caused by monovalent
(22) ionconcentrationasonewouldnaivelyexpectbutrather
which can be well approximated with f(z)≈ 3 − 1. Let by the interplay between counterion only and Debye-
2 z
us now calculate the concentration at which the energy Hu¨ckeltypeofinteractions. Thetransitionpointbetween
barrier vanishes, this concentration corresponds to the near-field and far-field regime, at which coions begin to
5
be expelled from the slit, turns out to be critical for un- tinguish between counterions and coions. I hope that
derstanding the aggregation process. This work com- these resultswill stimulatefurther researchonthe use of
plements the classical Schulze-Hardy rule and extends multivalent coions in tuning the stability of colloids and
our understanding of aggregation in multivalent asym- could be possibly used for colloidal self-assembly.
metric electrolytes. Now both, counterion and coion, I acknowledgefruitful discussions with MichalBorkovec,
high-charge limits are explained. Furthermore, the low- SvilenKozhuharov,andDuˇskoCˇakaraandfinancialsup-
charge Debye-Hu¨ckel limit is the same for counterions port from the University of Geneva and the Swiss Na-
and coions, as the weakly charged surface does not dis- tional Science Foundation.
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