Embed Size (px)
Citation preview
PHYSICAL REVIEW E FEBRUARY 1999VOLUME 59, NUMBER 2
Phase diagram of charge-stabilized colloidal suspensions:van der Waals instability without attractive forces
Renevan Roij and Marjolein DijkstraH. H. Wills Physics Laboratory, Royal Fort, University of Bristol, Bristol BS8 1TL, United Kingdom
Jean-Pierre HansenDepartment of Chemistry, University of Cambridge, Cambridge CB2 1EW, United Kingdom
~Received 6 October 1998!
A careful analysis of the classic Derjaguin-Landau-Verwey-Overbeek theory of the interaction energy in asuspension of charge-stabilized, spherical colloidal particles~polyions! in the presence of salt shows that inaddition to the usual screened-Coulomb effective pair interaction between polyions, there exists a structure-independent but state-dependent contribution~the ‘‘volume’’ term!, which has almost invariably been over-looked. A variational procedure based on the Gibbs-Bogoliubov inequality is used to calculate the contributionof the polyion pair interactions to the free energy of the suspension. The latter is then combined with the‘‘volume’’ term to derive the phase diagram of the colloidal suspension. Although the effective pair interactionbetween polyions is purely repulsive, it is shown that the volume term may drive a van der Waals–likeinstability in highly deionized suspensions~salt concentrations less than 20mM ) for experimentally relevantchoices of the polyion radius and charge. If the latter are sufficiently large, the fluid-fluid phase separation ispreempted by the fluid-solid freezing transition which broadens considerably. Reentrant behavior is predictedon the solid side of the phase diagram. The predicted phase diagrams may provide an explanation of somesurprising recent experimental results. They also show that the observation of a fluid-fluid phase separation ina charge-stabilized colloidal dispersion does not necessarily imply the existence of an attractive component inthe effective pair interaction between highly charged polyions.@S1063-651X~99!04002-7#
PACS number~s!: 82.70.Dd, 64.10.1h, 83.20.Di, 64.60.Cn
nt-etim
o-namet
tiv
ao
ale-pela
enc
ote-tio-pa
theotand
eeidsder
the
r
asedi-
ing
lloi-oly-
inthetedenre
es,rge
I. INTRODUCTION
Phase separation of an initially homogeneous fluid idense~or concentrated! and dilute fluid phases is a very common phenomenon in molecular systems. In one-componsystems, involving a single molecular species, the separainto liquid and gas phases observed below the critical teperatureTc is attributed, since van der Waals, to intermlecular attractions which balance the loss of configuratioentropy upon condensation. In mixtures of two or more coponents, the mechanism for demixing into phases of differconcentrations of the various species is less clear-cut. Incase of molecules of comparable size, the role of attracinteractions is again believed to be preeminent@1#, sincepurely repulsive interactions do not appear to lead to phseparation, as long as they are additive. The situation is mcomplicated for colloidal suspensions, which are essentimulticomponent in nature and involve large sizasymmetries between the individual species. Such sussions consist of mesoscopic colloidal particles, a molecusolvent, and most frequently at least one third componsuch as polymer coils or microscopic ions, which introduan intermediate length scale~the radius of gyrationRg forpolymers or the Debye screening lengthlD for ions!. Thisthird component plays a crucial role, since it induceseffec-tive interactions between the colloidal particles, which arelargely entropic origin. Effective interactions result quinaturally from a formal contraction of the initial multicomponent system into an effective one-component descripinvolving only the colloidal particles. An important difference with simple molecular systems is that the control
PRE 591063-651X/99/59~2!/2010~16!/$15.00
o
nton-
l-nthee
serely
n-rt,
e
f
n
-
rameter is not the temperature but the concentration ofthird component. It is worth noting that the solvent is ndirectly involved in the phase separation mechanism,plays the role of a passive ‘‘spectator’’ phase.
In the case of sterically stabilized colloidal particles, frpolymer coils induce a depletion interaction between collowhich is predominantly attractive and of a range of the orof Rg beyond the colloid particle diameter@2,3#. There isample experimental and theoretical evidence@4–6# that thedepletion attraction can induce a liquid-gas separation ofsuspension into a high colloid concentration~‘‘liquid’’ !phase and a dilute~‘‘gas’’ ! phase, at least if the polymecoils are assumed to be ideal~nonadditive!. This gas-liquidtransition is thus reminiscent of the van der Waals phtransition in ordinary molecular fluids. Recent results incate, however, that additivity of the pair interactions~such asin an asymmetric binary hard-sphere mixture! tends to drivethe gas-liquid transition metastable with respect to freez@7#.
In the present paper, we focus on charge-stabilized codal suspensions, made up of highly charged, spherical pions and microscopic coions and counterions dissolvedwater. The microions form electric double layers aroundcharged surface of the polyions, and it is generally accepthat, at least in the bulk, the effective interactions betweelectric double layers surrounding different polyions apurely repulsive. Direct measurements@8#, as well as first-principles computer simulations@9#, point to the quantitativevalidity of the Derjaguin-Landau-Verwey-Overbeek~DLVO!@10# potential between charge-stabilized colloidal particlprovided an adequate choice is made for the effective cha
2010 ©1999 The American Physical Society
heacneetiv
abulcos,ultex
doenu
thth
es
d
n
otelsihiry
ivnsata
heao
nds
ailurveohe
e
geof
ticalan-
il-
e
onn-
il-dgeus-id-ac-he
nons.se
es.mut
geis
u-lyiontral.ngoneent-a-
,
PRE 59 2011PHASE DIAGRAM OF CHARGE-STABILIZED . . .
of the polyions, which is generally significantly less than tstructural charge related to the number of ionizable surfradicals. We shall not consider here the case of conficolloids, where the presence of charged walls has bshown to induce an attractive component to the effecinteraction between like-charged polyions@11–13#.
According to the classical van der Waals picture, thesence of attractive interactions between polyions in bseems to preclude any phase separation into dilute andcentrated~fluid or solid! colloid phases. This picture seemhowever, to be contradicted by some experimental resThe relevant experiments are invariably performed attremely low salt concentration, of the order of a fewmM ,such that the influence of attractive van der Waals–Londispersion forces may be safely ruled out. In one experimthe measured lattice spacings in crystalline phases of squasideionized suspensions were found to be smallerexpected on the basis of a space-filling structure ofknown colloid concentration@14#. A natural explanation ofthis observation would be that the dense crystalline phasnot pure, but coexists with a much more dilute gas phaMore evidence of phase coexistence has been providethe observation of extremely dilute voids~a gas phase! in thebulk of an apparently homogeneous deionized suspensio~aliquid @15# or a solid@12#!. In fact, even a fully equilibratedgas-liquid coexistence has been reported@16#, although thisobservation aroused some controversy@17#.
There thus appears to be a contradiction between theservation of the coexistence of dilute and concentraphases and the generally accepted view of a purely repueffective interaction between like-charged polyions. In tpaper we carefully reexamine the traditional DLVO theoWe recover an effective screened-Coulomb~or Yukawa! pairinteraction, as expected, but in addition the total effectpotential energy for any given configuration of the polyioturns out to contain a structure-independent but stdependent volume term, the existence of which had alrebeen pointed out by Silbert and Grimson@18#. It will beshown that this volume term contributes significantly to tfree energy of the polyions, and varies sufficiently rapidlyvery low salt concentrations to have a very marked effectthe phase diagram. Under experimentally achievable cotions it leads to complex fluid-fluid-solid phase diagramwhen the colloid and salt concentrations are varied. In pticular, a van der Waals–like phase separation between dand concentrated fluid phases is predicted, despite the purepulsive nature of the effective pair potential. We beliethat the calculation to be presented provides a plausibleplanation of experimental observations. The present wsupplements and corrects a preliminary version publiselsewhere@19#.
II. THE MODEL
The colloidal dispersion under consideration is assumto contain~i! Np identical spherical polyions of radiusR andcharge2Ze ~uniformly distributed over the particle surface!;~ii ! Nc5ZNp identical counterions of charge1e; and ~iii !Ns fully dissociated pairs of monovalent salt ions of char6e. The solvent~water! is assumed to be a continuumdielectric constante ~‘‘primitive model’’ !, and the total vol-
ed
ene
-kn-
s.-
nt,chane
ise.by
b-dves.
e
e-dy
tni-
r-te
elyex-rkd
d
ume is denoted byV. All microions are assumed to be poinparticles, which cannot penetrate the interior of the spherpolyions. The total numbers of microscopic cations andions in the suspension areN15Nc1Ns and N25Ns . Thecorresponding mean~macroscopic! number density will bedenoted byn65N6 /V, the total microion density byn5n11n2 , and the polyion density bynp5Np /V.
The above model is characterized by the total Hamtonian
H5Hp1Hm1Hmp , ~1!
whereHp andHm are the Hamiltonians associated with thpolyions and microions, andHmp is the microion-polyioninteraction term. For later reference we write the polyiHamiltonian explicitly as a sum of kinetic and potential eergy,
Hp5Kp1Vp~$Ri j %!5Kp1(i , j
Np Fv0~Ri j !1Z2e2
eRi jG , ~2!
whereKp is the kinetic energy andVp the pairwise potentialenergy consisting of a hard-sphere contributionv0(Ri j ) anda Coulombic contribution. HereRi j 5uRi2Rj u denotes theseparation between polyioni and j, Ri , j being their positionvectors, respectively.
Several remarks are in order concerning the model Hamtonian ~1!. ~a! It contains exclusively short-range excludevolume and long-range Coulombic interactions. Long-randispersion interactions are omitted. This turns out to be jtified, since the low salt concentration regime to be consered later gives rise to weakly screened Coulomb intertions that completely mask the dispersion interactions. Tstrong screening~high salt concentration! regime was explic-itly considered in Ref.@20#, where it was shown that the vader Waals–London-Hamaker attraction between the polyimay indeed lead to a van der Waals phase separation~b!Stability of the system against classical Coulomb collaprequires, strictly speaking, all ions to have finite hard corHowever, the mean-field density-functional formalisadopted later on allows microions to be point ions withocausing any divergences.~c! The total Hamiltonian~1! ad-mits a proper thermodynamic limit, despite the infinite ranof the Coulomb interactions, because the colloidal systemglobally charge-neutral. This is no longer true when maniplating each of the three terms separately because the poand microion components are not separately charge-neuThis difficulty may be overcome by adding compensatiuniform backgrounds of opposite charge density, as is din the theory of liquid metals, considered as two-componion-electron fluids@21#. An alternative, but equivalent, procedure which will be adopted implicitly throughout this pper is to replace the bare Coulomb interactione2/er by ascreened Coulomb interaction
e2
er→
e2
ere2lr ~3!
and going to the limitl→0 at the end of the calculationafter the thermodynamic limit has been taken.
ly
cem
rg
sena
mth
is
nte
orx
di
uliza
ee
as
he
ely
er,
ity-.
in a
nof
en-les
2012 PRE 59van ROIJ, DIJKSTRA, AND HANSEN
At fixed inverse temperatureb51/kBT, the Helmholtzfree energyF of the colloidal suspension may be formalexpressed as
exp@2bF#5TrpTrmexp@2bH#[Trpexp@2bHpeff#, ~4!
where the traces Trp and Trm denote canonical phase-spaintegrals over the polyion and microion degrees of freedoThe effective polyion HamiltonianHp
eff is defined as
Hpeff5Hp1F8[Kp1Vp
eff~$Ri%!, ~5!
whereF8 is given by
exp@2bF8#5Trmexp@2b~Hm1Hmp!#. ~6!
Clearly,F8 may be interpreted as the Helmholtz free eneof an inhomogeneousfluid of microions in theexternalfieldof polyions at fixed positionsRi . Consequently, it dependparametrically on the polyion positions, and the potentialergy of the effective one-component system of polyions,ready introduced in Eq.~5!, is given by
Vpeff~$Ri%!5Vp~$Ri%!1F8~$Ri%!. ~7!
The above reduction of the initial polyion-microion systeto an effective one-component polyion system shows thatlatter interact via an effective potential energy that consof a contributionVp , given by Eq.~2! as a sum of pairinteractions, and the microion-induced contributionF8,which is state-dependent and not,a priori, pairwise additive.
III. DENSITY-FUNCTIONAL THEORY
A. Formulation
The inhomogeneous distribution of microscopic catioand anions in the external field of the polyions is characized by equilibrium density profilesr1(r ) andr2(r ), whichare the fundamental quantities in density-functional the~DFT! @22#. This theoretical framework is based on the eistence of a free-energy functionalF @r1
(1) ,r2(1)# of varia-
tional density profilesr6(1)(r ). The equilibrium profiles
r6(r ) must satisfy the Euler-Lagrange or stationarity contion
S dF @r1~1! ,r2
~1!#
dr6~1!~r !
D Ur
6~1!~r !5r6~r !
5m6 , ~8!
where, within the canonical formulation, the Lagrange mtipliers m6 must be chosen such as to satisfy the normaltion conditions
EVdrr6~r !5N6 . ~9!
The corresponding equilibrium value of the Helmholtz frenergyF8 is then given by
F85F @r1 ,r2#. ~10!
It is customary to split the functionalF into ideal gas, exter-nal field, Coulomb interaction, and correlation terms@22#,
.
y
-l-
ets
sr-
y-
-
--
F5Fid1Fext1FCoul1Fcorr. ~11!
The ideal gas contributions can be written down explicitly
Fid5kBT (a56
E drra~1!~r !@ ln„ra
~1!~r !La3…21#, ~12!
whereLa is the thermal wavelength of microion speciesa56. The external field contribution may be cast into tform
Fext5 (a56
E drra~1!~r !Ua~r !, ~13!
where the external potentialsUa(r ), due to the polyions, aremulticentered sums
U6~r !5(j 51
Np
u6~r2Rj ! ~14!
over the polyion-microion pair interactionsu6(r ), given by
u6~r !55 7Ze2
e
1
r, r .R
7Ze2
e
17g6
R, r ,R.
~15!
These polyion-microion pair interactions are thus purCoulombic beyond the polyion core radiusR, and chosen tobe constant inside the core region. Within anexactdensityfunctionalF, the most relevant choice for the constantsg6
would be the hard-core limitg6→`, ensuring that the mi-croions are excluded from the polyion cores. Howevwithin the framework of theapproximatefunctional adoptedbelow, which allows an analytic solution of the stationarconditions~8!, the excluded volume condition will be satisfied for finite values ofg1 andg2 to be determined later onThe choice ofu6(r ) for r ,R is reminiscent of ion-electron‘‘pseudopotentials’’ widely used in the theory of metals@21#.A polyion-microion pseudopotential has also been usedfirst-principles simulation of charge-stabilized colloids@9#,but for different technical reasons.
The mean-field contribution to the microion-microioCoulomb interaction term in the free-energy functional isthe familiar Hartree form
FCoul5e2
2eE drE dr 8@r1~1!~r !2r2
~1!~r !#
31
ur2r 8u@r1
~1!~r 8!2r2~1!~r 8!#. ~16!
To conclude the specification of the functionalF, the follow-ing two approximations are made:~a! The correlation contri-bution to the interaction term is neglected, i.e.,
Fcorr50. ~17!
This mean-field approximation corresponds to a multictered Poisson-Boltzmann theory for the density profi
ona
th
o
l-s
ts
nthf
l
s
e
-
w.
PRE 59 2013PHASE DIAGRAM OF CHARGE-STABILIZED . . .
r6(r ) which can only be handled numerically.~b! In orderto obtain an analytically tractable theory, the ideal gas ctributions ~12! are expanded to quadratic order in the locinhomogeneities,
Dr6~1!~r !5r6
~1!~r !2n6 . ~18!
In view of the constraints~9!, this leads to the followingapproximate expression for the ideal gas contribution tofunctional:
Fid. (a56
S F id~V,T,na!1kBT
2naE dr „Dra
~1!~r !…2D ,
~19!
where F id(V,T,na)5VkBTna@ ln(naLa3)21# is the homoge-
neous ideal gas contribution from ionic speciesa56.
B. Diagonalization
The free-energy functional to be used in this paper is ndefined by Eqs.~11!, ~13!, ~16!, ~17!, and ~19!; it is qua-dratic in the density profilesr6
(1)(r ). This suggests diagonaizing the functionalF by changing from the variational fieldr6
(1)(r ) to their linear combinations
r~1!~r !5r1~1!~r !2r2
~1!~r !,~20!
s~1!~r !5n2r1
~1!~r !1n1r2~1!~r !
n11n2.
Clearly,r (1)(r ) is the local microion charge density in uniof e. In the limit of high salt concentration~such thatn1
.n2),s (1)(r ) reduces to one-half of the local total microionumber density. At lower salt concentrations, however,physical meaning ofs (1)(r ) is less obvious; it vanishes ins50. The quadratic functionalF is diagonal in the newfields defined in Eq.~20!, and can be written as
F @r~1!,s~1!#5F id~V,T,n1!1F id~V,T,n2!
1Fel@r~1!#1Fhc@s~1!#, ~21!
where the ‘‘electrostatic’’ functional is of the form
Fel@r~1!#5kBT
2~n11n2!E dr @r~1!~r !2 r #2
1E drr~1!~r !U~r !
1e2
2eE drE dr 8r~1!~r !r~1!~r 8!
ur2r 8u, ~22!
while the ‘‘hard-core’’ functional reads
Fhc@s~1!#5kBT
2 S 1
n11
1
n2D E dr @s~1!~r !2s#2
1E drs~1!~r !W~r !. ~23!
-l
e
w
e
In Eqs. ~22! and ~23!, r and s are the macroscopic spatiaaverages of the variational fields
r5n12n25Znp , ~24!
s52n1n2
n11n25
2n1n2
n, ~25!
while U(r ) andW(r ) are the following linear combinationof the external fields~14!:
U~r !5n1U1~r !2n2U2~r !
n[(
j 51
Np
u~r2Rj !,
~26!
W~r !5U1~r !1U2~r 
reut
te
thore-delintehe
edn-
tr
t
-oing
the
re
on
2014 PRE 59van ROIJ, DIJKSTRA, AND HANSEN
The Lagrange multiplierms follows from the normalizationcondition and reads
ms5hw0 , ~33!
where we defined the colloid packing fraction
h54pR3
3np . ~34!
The constantw0 will now be adjusted to satisfy the hard-cocondition of zero density within the core and constant oside. It follows from Eqs.~25! and ~32! that
s~r !5sF111
2kBTS hw02(j 51
Np
w~r2Rj !D G , ~35!
which reduces to a multicentered sum of Heaviside sfunctionsQ(x),
s~r !5s
12h (j 51
Np
Q~ ur2Rj u2R!, ~36!
provided we set
bw052
12h. ~37!
The resulting hard-core contributionFhc8 to the colloid freeenergy follows from evaluation ofFhc@s# and reads
Fhc8 5kBTVn1n2
n S 221
2bw0~12h! Dhw0
5kBTVhs
12h. ~38!
Note that the equilibrium profile given in Eq.~36! doesnot, strictly speaking, satisfy the hard-core condition, insense that the spherically symmetric profile centered abpolyion i gives rise to a nonvanishing density within the coof another polyionj Þ i . This problem is inherently connected to the expansion of the functional to quadratic orin the profiles: the resulting stationarity equations give aear relation between the profile and the multicentered exnal field, which is unphysical for hard-core interactions. Tspecific choice forw0 given in Eq.~37! is, in that sense, thebest one within a quadratic functional or a linearizPoisson-Boltzmann theory. A similar problem will be ecountered in the minimization ofFel .
D. Minimization of Fel
The Euler-Lagrange equation associated with the elecstatic part~22! of the functionalF reduces to
-
p
eut
r-r-
o-
mr5S dFel@r~1!#
dr~1!~r !D U
r~1!~r !5r~r !
5kBT
nDr~r !1U1DU~r !1
e2
e E dr 8r1Dr~r 8!
ur2r 8u,
~39!
where U denotes the volume average ofU(r )5U1DU(r ), and wheremr is a Lagrange multiplier. Care musbe exercised in handling Eq.~39! in the thermodynamiclimit, since bothU and the Coulomb integral involve divergent volume integrals of the Coulomb potential. The twdivergences cancel, but a proper evaluation of the remainfinite constant is most easily achieved by the use ofscreened Coulomb potential~3! and the subsequent limitingprocedurel→0. Substitution of the screened for the baCoulomb potential transforms Eq.~39! into
mr5kBT
nDr~r !1U ~l!~r !
1e2
e E dr 8exp~2lur2r 8u!
ur2r 8u@ r1Dr~r 8!#, ~40!
where
U ~l!~r !5(j 51
Np
u~l!~r2Rj ! ~41!
is a multicentered sum over the modified polyion-microipseudopotential
u~l!~r !55 2Ze2
e
exp~2lr !
r, r .R
2Ze2
e
exp~2lR!
R~12g!, r ,R.
~42!
We now introduce the Fourier transform
f k5EVdr f ~r !exp~ ik•r ! ~43!
of an arbitrary functionf (r ) defined in a finite volume withperiodic boundary conditions. Fourier transforming Eq.~40!yields
bmr~2p!3d~k!51
nDrk1bUk
~l!
14p l
~k21l2!@~2p!3rd~k!1Drk#,
~44!
wherel 5be2/e is the Bjerrum length, and with the Diracd
d~k!51
~2p!3 EVdr exp~ ik•r !. ~45!
e
tri
anonlti
n
er
ed
or
n
herium
onthendn-qs.
ge-n-
PRE 59 2015PHASE DIAGRAM OF CHARGE-STABILIZED . . .
Now mr is determined by the normalization~9!, which in-volves thek50 Fourier component
Drk5050. ~46!
If we use that (2p)3d(k50)5V, Eq. ~46! implies
bmr51
VbUk50
~l! 14p l r
l2. ~47!
The explicit solution of the stationarity condition~44! nowreads
Drk52nk21l2
k21kl2
bUk~l!1
~2p!3d~k!
Vn
l2
kl2
bUk50~l! ,
~48!
where we defined the Debye screening parameterk5lD21
given by
k254p ln ~49!
and its modified counterpartkl25k21l2. Clearly, kl re-
duces tok when the limitl→0 is taken at the end of thcalculation. It is essential to substitute the solution~48! intothe Coulombic part of the functional to calculate the conbution to the equilibrium free energy before taking thel→0 limit ~cf. the following section and Appendix A!. How-ever, the form of the charge-density profile in that limit cbe directly obtained by Fourier transforming the soluti~48! at l50. This can be seen if one realizes that the mucentered character ofU (l)(r ), given in Eq.~41!, leads to
Uk~l!5uk
~l!(j 51
Np
exp~ ik•Rj !, ~50!
with uk(l) the Fourier transform ofu(l)(r ) defined in Eq.
~42!,
buk~l!524p lZ
exp~2lR!
k21l2
3Fg cos~kR!1~12g!sin~kR!
kR1~lR!
sin~kR!
kR
1~lR!2~12g!sin~kR!2kRcos~kR!
~kR!3 G . ~51!
It is thus easily checked that for any wave vectork,
liml→0
~k21l2!buk~l!524p lZFg cos~kR!1~12g!
sin~kR!
kR G .~52!
The inverse Fourier transformation of Eq.~48! leads, uponinserting Eqs.~50! and ~52! in the limit l→0, to the equi-librium profile r(r )5 r1Dr(r ) of the form
r~r !5(j 51
Np
r0~r2Rj !, ~53!
-
-
with the spherically symmetric ‘‘orbitals’’
r0~r !5Z
~2p!3E dk exp~2 ik•r !k2
k21k2
3Fg cos~kR!1~12g!sin~kR!
kR G
55Z.k2
4p
exp~2kr !
r, r .R
Z,k2
4p
sinh~kr !
r, r ,R.
~54!
The effective chargesZ. andZ, are defined by
Z.5ZS g cosh~kR!1~12g!sinh~kR!
kR D ,
~55!
Z,5ZS exp~2kR!
kR D @12g~kR11!#.
For anyg,k, andR, these profiles satisfy the normalizatio*drr0(r )5Z. To complete the specification ofr0(r ), achoice must be made for the yet undetermined parametgin the pseudopotential~27!. According to Eqs.~54! and~55!the polyion-microion excluded volume condition is satisfiprovided that
g51
11kR. ~56!
Substitution of Eq.~56! into Eq. ~55! leads then directly tothe DLVO value of the effective polyion valence@10#,
Z.5ZS exp~kR!
11kR D . ~57!
The final expression for the microion-charge distribution,double layer, around a polyion is thus
r0~r !5H 0, r ,R
Zk2
4p
exp~kR!
11kR
exp~2kr !
r, r .R.
~58!
The coion and counterion density profilesr6(r ) are obtainedfrom Eq. ~20! by taking the appropriate linear combinatioof s(r ) andr(r ).
We finally note that even the choice forg given in Eq.~56! and the resulting profile of Eqs.~53! and ~58! do nottake a truly proper account of the hard-core conditions. Treason is the same as encountered above for the equilibprofiles of the hard-core functionalFhc: the profile aroundpolyion i actually penetrates the core of any other polyij Þ i . This shortcoming is, as already discussed, due tolinear relation between the inhomogeneity of the profile athe external potential of the polyions in the stationarity codition. Thus, although the key result summarized by E~53! and ~58! is in itself not new ~see, e.g.,@9,10#!, thepresent derivation shows that it is, in fact, the best chardensity profile within the framework of linearized Poisso
e-ia
snV
sotov
nst
ei
c
l
oln
,eh
r
ly-
en-esstri-alof
nlyionItf
ison-ge-old
reeen
ill
n-
y-nce
will
en-
a
s:
nd
2016 PRE 59van ROIJ, DIJKSTRA, AND HANSEN
Boltzmann theory ~corresponding to the quadratic freenergy functional!. Other choices of the pseudopotentparameterg are possible. In particular,g50 leads back tothe effective polyion valenceZ.5Zsinh(kR)/kR advocatedby Sogami and Ise@23#. Clearly, however, this choice leadto microion orbitals that penetrate ‘‘their own’’ polyiocores, and is for that reason less favorable than the DLchoice of Eq.~56!.
IV. EFFECTIVE POLYION INTERACTION ENERGY
The microion-induced contributionF8 to the effectivepolyion interaction energy is obtained by substituting thelutions s(r ) andr(r ) of the Euler-Lagrange equations inthe free-energy functional. This has already been achiefor the hard-core part, resulting in Eq.~38!. As alreadystressed earlier, the explicit calculation of the minimumFel8of Fel is somewhat trickier, due to nontrivial cancellatiobetween Coulomb divergences. This requires a detour viaauxiliary screened Coulomb potential~3!, followed by thelimit l→0 onceFel8 has been evaluated for any finite valuof l. The different steps of this procedure are detailedAppendix A.
Gathering results from Eqs.~7!, ~21!, ~38!, and~A11!, wearrive at the final result for the potential energy in the effetive one-component Hamiltonian,
Vpeff~$Ri%!5F01(
i , j
Np
veff~Ri j !, ~59!
where veff(R) is the usual DLVO effective pair potentiabetween polyions,
veff~r !5H `, r ,2R
Z.2 e2
e
exp~2kr !
r, r .2R,
~60!
and whereF0 is a state-dependent term. This so-called vume term isstructure-independent, i.e., it does not depend othe coordinates$Ri% of the polyions, and is given explicitlyby
F0~V,T,np ,ns!5F id~V,T,n1!1F id~V,T,n2!
2Z2e2
2eR
NpkR
11kR
1kBThs
12hV2
1
2
4pe2
ek2~Znp!2V,
~61!
where s,h, andk are defined in Eqs.~25!, ~34!, and ~49!.The last two terms on the right-hand side of Eq.~61! wereerroneously omitted in our earlier analysis@19#. The impor-tance of these terms, and more particularly of the latterwell illustrated by the significant quantitative differences btween the phase diagrams to be discussed later and tobtained in Ref.@19#.
The existence of a volume term in the total polyion inteaction energy is a consequence of the reduction~mapping! of
l
O
-
ed
he
n
-
-
is-ose
-
the initial multicomponent system~polyions and microions!into an effective one-component system of ‘‘dressed’’ poions. The third term on the right-hand side of Eq.~61! maybe interpreted as the ‘‘self-energy’’ of theNp electric doublelayers associated with the individual polyions, or as theergy due to the potential well in which the colloidal particlreside because of the surrounding oppositely charged dibution of microions. The fourth term accounts for additionexcluded volume for the microions due to the hard corethe polyions; it vanishes for polyion radiusR50 since thenh50. It also vanishes in the salt-free case, i.e., when opolyions and counterions are left. The physical interpretatof the last terms of Eq.~61! is less transparent, however.resembles, but is in generalnot identical to, the negative othe polyion-polyion mean-field contribution,Fp
mf , to the to-tal Helmholtz free energy, which with Eq.~60! can be evalu-ated as
Fpmf5
V
2np
2Er .2R
dr veff~r !
51
2
4pe2
ek2~Znp!2VS 112kR
~11kR!2D . ~62!
The difference betweenFpmf and the last term of Eq.~61! is,
apart from the minus sign, the bracketed factor in Eq.~62!.This factor is unity only for point polyions,R50. We there-fore suggest that the final term of Eq.~61! is a reminder ofthe fact that the free energy of a purely Coulombic systemonly due to correlations and fluctuations, and does not ctain mean-field contributions because of the charneutrality condition. Since such a constraint does not hfor hard-core contributions, the cancellation ofFp
mf and thefinal term of Eq.~61! is not perfect for finite radiusR of thepolyion, due to the coupling of the Coulomb and hard-cocontributions. Remarks along these lines have also bmade by Warren in the appendix of Ref.@24#.
In any given thermodynamic state, the volume term whave no influence on the equilibriumstructureof the poly-ions which is entirely determined by the effective pair potetial ~60!. F0 does, however, contribute to all equilibriumthermodynamicproperties of the system, including the polion free energy, and is hence expected to have an influeon the phase diagram of the colloidal suspension. Thisbe examined in the next two sections.
V. FREE ENERGY OF THE COLLOIDAL SUSPENSION
Given the effective polyion interaction energy~59!, thenext task is to calculate the total free energy of the suspsion, for fixed temperatureT and volumeV, as a function ofthe polyion and salt concentrations, or equivalently asfunction of the colloid packing fractionh and the salt con-centrationns .
Substituting Eq.~59! into expression~4! for the total freeenergy we find that the latter naturally splits into three term
F5F01Fp~id!1Fp
~exc! , ~63!
whereF0 is given explicitly by Eq.~61!, Fp~id! is the trivial
ideal free energy of a system of noninteracting polyions, a
er
u
tepoote
izi
sethe
e-e
es
terloicdu
.th.
at
nchen
cuaenaaketh-p
elor
e
eir-ner-ith
wse itse.y of
in-
rted-
ons
he
ies,
PRE 59 2017PHASE DIAGRAM OF CHARGE-STABILIZED . . .
Fp~exc! is the excess part of the free energy of polyions int
acting via the effective pair potential~60!,
Fp~exc!52kBT lnXE dRNp
VNpexpS 2b(
i , j
Np
veff~Ri j !D C.~64!
This excess free energy may be calculated variationally,ing the Gibbs-Bogoliubov inequality
Fp~exc!<F ref
~exc!1^DV& ref , ~65!
whereF ref~exc! is the excess free energy of a reference sys
under the same thermodynamic conditions but with pairtentialsv ref(Ri j ). The second term on the right-hand sideEq. ~65! is the ensemble average over the reference sysdenoted •••& ref , of the potential-energy difference
DV5(i , j
Np
@veff~Ri j !2v ref~Ri j !#. ~66!
The right-hand side of Eq.~65! may be minimized with re-spect to one or several variational parameters characterthe reference pair potentialv ref(r ). This minimum, then, isthe best estimate for the required quantityFp
~exc! .To calculate the variational free energy of the fluid pha
a hard-sphere~HS! reference system was adopted, andright-hand side of Eq.~65! was minimized with respect to threference hard-sphere diameterd, or equivalently the refer-ence packing fractionh ref5(p/6)npd3. The very accurateCarnahan-Starling equation of state was used forF ref
~exc! ,while ^DV& ref was estimated using the Verlet-Weis improvment of the Percus-Yevick pair distribution function of threference HS fluid@25#. The resulting upper bound estimatof the excess free energy of the Yukawa system,f Y
~exc!
5bFp~exc!/Np , have been tested against ‘‘exact’’ Mon
Carlo evaluations of this quantity. In these Monte Casimulations ofNp5256 polyions we employ thermodynamintegration: for a given thermodynamic state point we graally switch on the full pair interactionveff(r ) starting fromthe hard-core fluid@26#. Typical results are reported in Fig1: the variational estimates are seen to lie very close to‘‘exact’’ results, as will be discussed in more detail below
The excess free energy in the solid phase was calculusing an Einstein solid as a reference system. In this casevariational parameter is the single Einstein phonon frequevE which determines all thermodynamic properties of treference system, as well as the mean-square displacemethe particles around their lattice sites. The variational callations were carried out for fcc and bcc lattices. Underthermodynamic conditions which were explored, the freeergy of the fcc lattice turned out to be slightly lower than thof the bcc crystal, but the differences are extremely smThe accuracy of the variational estimates was checagainst MC calculations of the free energy based onFrenkel-Ladd integration scheme@27#, and against the predictions of a more realistic phonon model, described in Apendix B. The latter reproduce the MC data remarkably wwhile the variational free energies turn out to be once m
-
s-
m-
fm,
ng
,e
-
e
edthey
t of-
ll-
tll.de
-l,e
surprisingly accurate in view of the great simplicity of thcalculation, as illustrated in Fig. 1.
As expected, the variational free energies~full curves! ofboth the fluid and the solid phase are slightly above thsimulated values~symbols!. The differences are small, however. The phonon model gives extremely accurate free egies in the solid phase, as follows from a comparison wthe simulations in Fig. 1. Unfortunately, the model shounphysical characteristics in the low-density regime, wherpredicts pathologically low free energies for the solid phaFor that reason we merely used it as a test for the accuracthe variational calculations, which prove to be reliabledeed. Consequently, the variationalFp
~exc! was used through-out in the construction of the phase diagrams to be repoin the following section. We are thus in a position to com
FIG. 1. Excess Helmholtz free energyf Y(exc) ~see text! of a one-
component system with pairwise hard-core-Yukawa interacti~60! for chargeZ57300 and diameterD5652 nm, as a function ofpacking fractionh, for salt concentrations~a! ns51 mM and ~b!ns52 mM . The variational calculations~full curves! are in allcases above the results from ‘‘exact’’ Monte Carlo simulation. Tharmonic phonon calculations for the solid free energy~dashedcurve! reproduce the simulations perfectly, except at low densitwhere its prediction off Y
(exc) is pathologically low. The inset in~a!shows the fluid free energy in more detail.
icva
v
a-
iahic
ll
at
-mem
ahr
iatu
ste
at
el
sf
-alifofhe
lly
on-ingal
n-rctout-
va-e
-the
f
eral
lac-
ge-romthe
uch
2018 PRE 59van ROIJ, DIJKSTRA, AND HANSEN
pute the total free energy~63! of the colloidal suspensionsystematically and efficiently for any set of thermodynamconditions, as well as all relevant thermodynamic deritives.
VI. PHASE DIAGRAMS
The phase diagram of a system of particles interactinga screened Coulomb~Yukawa! potential of the form~60!~with R50) is well known, thanks to the extensive simultion work of Robbinset al. @28#. As expected from thepurely repulsive nature of the pair potential, the phase dgram exhibits a single high-temperature fluid phase, witfreezing line towards a low-temperature crystal phase, whis of fcc structure for largek, and of bcc symmetry at smak, with a bcc-fcc-fluid triple point aroundknp
21/3.4.9.These calculations do not take into account the stdependence of the effective pair potential~60!, embodied inthe variations ofk andZ. with polyion and salt concentration. Neither do they include the state-dependent voluterm ~61!. In this section we show that these factors havprofound influence on the topology of the phase diagraparticularly at low salt concentrations.
Due to the charge-neutrality constraints, the polyion-ssystem behaves as a binary mixture characterized by tintensive variables, the temperatureT, the polyion densitynp@or, equivalently, the packing fractionh as defined in Eq.~34!#, and the salt~i.e., anion-cation pair! concentrationns .To make contact with experimental conditions, all phase dgrams presented below were calculated at room tempera(T5300 K!, and for a solvent dielectric constante578. Thisamounts to a Bjerrum lengthl 5e2/ekBT57.2 Å. For agiven polyion diameterD52R and valenceZ, the phase dia-grams are represented in the (h,ns) plane. At any giventemperatureT, the conditions of coexistence of two phaselabeled by 1 and 2, are the equalities of the chemical potials of polyions (mp) and salt (ms), and of the osmoticpressuresp,
mp~np~1! ,ns
~1!!5mp~np~2! ,ns
~2!!,
ms~np~1! ,ns
~1!!5ms~np~2! ,ns
~2!!, ~67!
p~np~1! ,ns
~1!!5p~np~2! ,ns
~2!!.
The chemical potentials and pressures may be calculfrom the free energy per unit volume,f5F/V, as obtainedin the preceding section, by the usual thermodynamic rtions
mp5S ]f~np ,ns!
]npD
ns
,
ms5S ]f~np ,ns!
]nsD
np
, ~68!
p5npmp1nsms2f.
In practice, phase coexistence is determined by fixing theconcentration in one phase~e.g.,ns
(1)), and solving the set o
-
ia
-ah
e-
ea,
ltee
-re
,n-
ed
a-
alt
three coupled equations~67! for the three remaining concentrations numerically. A practical problem of the numericroot-finding procedure involved is its poor convergencethe initial guess is far from the actual solution. The choicea good initial guess is greatly facilitated by considering tLegendre transform to a ‘‘semigrand potential’’v5f2nsms , which describes the colloidal species canonica~fixed h) and the added salt grand-canonically~fixed ms).From curvesv(h,ms) as a function ofh at fixed ms , ex-amples of which are shown in Fig. 2, common tangent cstructions can be performed that are equivalent to solvEqs. ~68!. We used a combination of these two numericschemes to map out the full phase diagram in the (h,ns)plane for various values ofZ andD.
In the strong screening~high salt-concentration! limit,such thatkR@1, the two negative van der Waals–like cotributions to the volume term~61! are nearly constant, osmall, or linear innp , so that they are not expected to affethe phase diagram very much. This expectation is borneby explicit calculations which show that for polyion diametersD of the order of a few hundred nanometers andlencesZ of the order of a few thousand, the effect of thvolume termF0 is insignificant whenns>20 mM . For suchsufficiently highns , our calculations predict a simple fluidfcc solid phase coexistence similar to that observed whenvolume term is omitted altogether@28#. The situationchanges dramatically, however, at salt concentrationsns<20 mM , i.e., under highly deionized conditions typical orecent experimental data@8,11,12,14–16#. This is illustratedin Fig. 2, where the Legendre transformv5f2nsms dis-cussed above is shown versus packing fraction at sev
FIG. 2. Thermodynamic potentialv5f2msns in the fluidphase of colloids with chargeZ53650 and diameter 2R5462 nmas a function of colloid packing fractionh at several fixed chemicapotentialsms of the salt. The corresponding reservoirs are charterized by salt concentrationsns
r of ~1! 46.4mM , ~2! 21.5 mM , ~3!10.0 mM , ~4! 4.64 mM , ~5! 2.15 mM , and ~6! 1.00 mM . Thehigh-salt curve~1! is convex, and thus describes a stable homoneous fluid phase; the lower salt curves exhibit concave parts fwhich common tangent constructions can be performed to yieldh ’s of the coexisting gas and liquid phases. Combination of sfluid curves with similar curves for the solid phase~not shown! canbe used in the determination of fluid-solid coexistence.
he
uf
huid
hethe
e
t,
ith
-alt
-of
id
ri t
PRE 59 2019PHASE DIAGRAM OF CHARGE-STABILIZED . . .
fixed values ofms . These curves, which correspond to tfluid phase of a system with particle diameterD5461 nmandZ53650, are seen to develop a concave region at sciently low reservoir salt concentrationns
r ~correspondingwith ms), signaling a van der Waals–like instability whicresults in the separations into low- and high-density flphases of the colloidal particles.
Figures 3–6 show the evolution of the topology of tphase diagram with particle size and valence, assumingZ;R2 ~i.e., the total polyion charge is proportional to thparticle surface!. The case of the largest particles (Z57300and D5652 nm! is plotted in Fig. 3. The fluid-solid phas
FIG. 3. Room-temperature phase diagram of aqueous collosuspension~chargeZ57300 and diameterD5652 nm! as a func-tion of colloid packing fractionh and salt concentrationns (mM).The narrow fluid~F! to fcc-solid transition atns.20 mM broadensand narrows again at lower salt concentrations. The~thinner! tielines connect coexisting state points on the~thicker! phase bound-aries.
FIG. 4. As in Fig. 3, but withZ53650 andD5461 nm. Thedotted curve denotes a metastable gas-liquid binodal with the ccal point indicated by3.
fi-
at
transition, which forns>20 mM is very ‘‘narrow’’ ~i.e.,shows only a small jump in colloid packing fraction!, sud-denly broadens enormously at lowerns , signaling the coex-istence of the fcc solid with a very low-density fluid. In facthe fluid phase packing fraction is so low (h,1024) that thefluid side of the coexistence curve appears to coincide wthe ns axis on the scale of the figure. Note that asns islowered below 20mM , the packing fraction of the coexisting solid first increases, before decreasing at still lower sconcentrations~reentrant behavior!. At the very lowest saltconcentrationsns,1 mM , the dissociation of water becomes significant. Assuming a pH of 7, the concentrations
al
ti-
FIG. 5. As in Fig. 3, but withZ52086 andD5349 nm. Thereis now stable coexistence of a gas~G! and liquid~L! phase betweena critical point (3) and triple points (D). Above the critical pointthe homogeneous fluid~F! phase is stable at lowh and freezes intothe fcc solid at higherh. Below the triple point, there isG-fcccoexistence.
FIG. 6. As in Fig. 3, but withZ51217 andD5266 nm. Herethe van der Waals–like instability only persists at such lowh that itis decoupled from the freezing transition. TheG-L coexistence ex-hibits two critical points (3) while the freezing transition does noshow any broadening at lowns .
nav
thnc
.lid
id
t
f
isdorg
x--
vehe
eid-f
tha
thh
sov
atttheng.a
s-nnltasf a
hes-a-n-
atell
inthe
s aialthectn
viorol-
htio
of
f
2020 PRE 59van ROIJ, DIJKSTRA, AND HANSEN
H1 and OH2 ions have been added to the counterion acoion concentrations in all calculations; this does not hany significant effect as long asns.1 mM . The salt con-centration in the coexisting phases may be read fromintersections of the inclined tie lines and the coexistecurves in Figs. 3–6.
The scenario changes continuously asZ decreases. In Fig4 (Z53650,D5461 nm! the reentrant character on the soside is less pronounced than forZ57300, while a metastablefluid-fluid phase transition is barely preempted by the flusolid transition. For still lower charge~e.g., the caseZ52086 andD5349 nm is shown in Fig. 5! the fluid-fluidtransition becomes stable with a ‘‘gas-liquid’’ critical poinat a salt concentrationns above that of a gas-liquid-solidtriple point. This phase diagram roughly resembles that osimple molecular system, where the temperatureT plays therole of the salt concentrationns in the colloidal dispersion. Adifference is, of course, that the temperature in two coexing phases is the same, while the salt concentrations neenecessarily be identical; the tie lines are therefore not hzontal in Figs. 3–6. At still lower charge the phase topolochanges again, as illustrated in Fig. 6 forZ51217 andD5266 nm. A reentrant ‘‘gas-liquid’’ coexistence curve, ehibiting an upper and a lower critical point, is now completely separated from the fluid-solid coexistence curTherefore there is no longer a triple point in this case. Tfluid-solid-coexistence is now hardly affected by the volumterm F0 in the free energy. For particles and charges a fpercent smaller than those of Fig. 6, the region of the flufluid instability shrinks further and finally disappears completely, leading to a stable homogeneous fluid phasepacking fractions below the freezing line.
Figure 7 shows how the correspondence betweenfreezing transition in the colloidal suspension and that ofunderlying HS transition is completely lost as soon ascontribution of the volume term becomes significant. T
FIG. 7. The reference packing fractionh ref ~minimizing theGibbs-Bogoliubov inequality! of the fluid phase that coexists witthe fcc solid phase as a function of the reservoir salt concentrans
r for Z57300 andD5652 nm. The sharp drop ofh ref below nsr
'25 mM indicates that the rule-of-thumb freezing criterionheff'0.5 fails when the volume termF0 affects freezing.
de
ee
-
a
t-noti-y
.e
w-
or
enee
packing fractionh ref of the HS reference fluid that minimizethe free energy of the fluid phase in the Gibbs-Bogoliubinequality~65! is plotted as a function ofns
r along the freez-ing line. Herens
r is the salt concentration in a reservoirchemical potentialms . As long as ns
r exceeds abou25 mM ,h ref is nearly constant, and reasonably close tovalue;0.5 corresponding to a hard-sphere fluid at freeziBelow that salt concentration,h ref decreases sharply tovalue near zero, indicative of a very dilute gas phase~a‘‘void’’? ! coexisting with the crystal.
The reentrant nature of the melting line is clearly illutrated in Fig. 8, which shows the variation of the Lindemaparameteru in the coexisting solid with the reservoir saconcentrationns
r . The Lindemann parameter is definedthe ratio of the root mean square of the displacement ocolloidal particle~from its equilibrium lattice position! andthe lattice spacing~i.e., the nearest-neighbor distance!. It iseasily calculated from the Einstein solid that minimizes tfree energy of the fcc colloidal crystal in the GibbBogoliubov inequality. While at the higher salt concentrtions u turns out to be fairly constant and close to the geerally accepted ‘‘universal’’ valueu.0.15, it drops sharplyat lowerns
r , goes through a minimum, and then rises againvery low salt concentration. This behavior correlates wwith the melting line of Fig. 3.
The Donnan effect is the reduction of salt concentrationa colloidal dispersion, compared to that of a reservoir atsame chemical potentialms , as the colloid packing fractionincreases. This effect is illustrated in Fig. 9, which showfamily of ns-h curves for 10 values of the chemical potentms ; the salt concentration in the reservoir corresponds tons value ath50. The figure shows that the Donnan effebecomes more pronounced as the reservoir concentrations
r
decreases.
VII. DISCUSSION AND CONCLUSION
The present analysis shows that complex phase behamay be expected in highly de-ionized charge-stabilized c
n
FIG. 8. Lindemann parameteru of the fcc solid along the melt-ing line as a function of the reservoir salt concentrationns
r for thesame system parameters as in Fig. 7. The melting criterion ou'0.15 does hold at highns
r but fails belownsr,25 mM .
actu
loen-a
hed
bemu
teran
p-heataiseyioeasneur
u
tho
tee
useges.e to
na-
the
rk-id-alt
ofea--blehelysce of
ia-ons
cedac-mlly,,elyap-inity
e
uid-
n-ug-
ndated-of-the
withng-to
redeenter-re-
ns
foressRef.
PRE 59 2021PHASE DIAGRAM OF CHARGE-STABILIZED . . .
loidal dispersions. While long-range van der Waals attrtions might induce fluid-fluid phase separation, and evenally irreversible flocculation, at high salt concentrations@20#,the behavior predicted by the present analysis at verysalt concentrations is of purely electrostatic origin. The kfinding is that charge-stabilized colloidal particles may udergo a fluid-fluid~or gas-liquid! phase separation for sufficiently low ionic strengths, although they interact viapurely repulsive effective pair potential. In other words, texperimental observations of a fluid-fluid phase transitionnot necessarily imply the existence of attractive forcestween the charged colloidal particles or polyions. This seeingly surprising result is a direct consequence of the redtion of the initial multicomponent problem, involvingmesoscopic polyions and microscopic coions and counons, to a one-component system of dressed polyions inteing via effective screened forces; the effective interactioare pairwise additive only within the framework of the aproximate, quadratic functional introduced in Sec. III. Tprice to pay for this reduction is the appearance of the stdependent volume termF0 in the free energy, which hasprofound effect on the phase behavior of the colloidal dpersion. The reduction also leads to a decoupling betwdensity fluctuations, characterized by the measurable polstructure factorS(k), and the observed thermodynamic bhavior. In a genuine one-component system, fluid-fluid phseparation is signaled by enhanced density fluctuationsthe critical point, resulting in a sharp increase of the structfactor at smallk ~long wavelengths!. In the effective one-component system of dressed colloidal particles,S(k) is un-affected by the vicinity of the critical point of the fluid-fluidphase transition, since it is determined by the purely repsive effective pair potential~60!, which can only lead toregular behavior in thek→0 limit.
It is worth noting that a volume term similar toF0 alsooccurs naturally in the theory of simple metals, wherefull ion-electron system is mapped onto a system of pseudtoms interacting via an effective state-dependent pair potial @21#. However, in the metal case this volume term do
FIG. 9. Donnan effect forZ57300 andD5652 nm. For eachcurve the salt chemical potentialms is fixed; the correspondingreservoir salt concentrationns
r equals the value ofns at h50.
--
wy-
o--
c-
ri-ct-s
e-
-enn
-eare
l-
ea-n-s
not have a significant incidence on phase behavior becaof the restricted range of electron densities and ionic charIn the colloid case the parameter space is much wider duthe large range of variation ofnp , R, andZ, and due to theexistence of an additional salt component, which has no alog in metals.
The wealth of possible phase diagrams, induced bystructure-independent contributionF0 to the free energy, isillustrated in Figs. 3–6, which summarize the most remaable scenarios. For large, highly charged polyions, the flufcc solid freezing transition dramatically widens below a sconcentration of about 20mM ; the packing fraction of thecoexisting fluid drops to extremely small values while thatthe coexisting fcc crystalline phase exhibits a reentrant fture upon loweringns . The sudden broadening of the fluidsolid density gap is attributed to an underlying metastafluid-fluid phase separation. This is shown explicitly by tresults for smaller polyions in Figs. 4 and 5: the initialmetastable fluid-fluid transition~Fig. 4! becomes stable athe polyion size and charge decreases, with the emergena fluid-fluid critical point above a fluid-fluid-solid triplepoint. This scenario is vaguely reminiscent of the phase dgrams observed in sterically stabilized colloidal suspensiin the presence of free polymer coils@4,6#. However, in thiscase the change in topology of the phase diagram is induby a change in the range of the effective depletion intertion, which is essentially attractive, while a volume tersimilar toF0 has no incidence on the phase diagram. Finafor yet smaller polyions~Fig. 6!, the triple point disappearsand the fluid-fluid part of the phase diagram completseparates from the fluid-solid coexistence line, with thepearance of a lower fluid-fluid critical point. Note thatboth Figs. 5 and 6 the packing fractions of the low-dens‘‘gas’’ phase are always very small@except very close to thecritical point~s!#, which may provide an explanation for thobserved ‘‘voids’’ reported in Ref.@15#. For diameters andcharges a few percent smaller than those of Fig. 6, the flfluid instability shrinks to a smallerh-ns region, and finallydisappears completely.
The possibility of a fluid-fluid phase separation of suspesions or solutions of like-charged polyions was already sgested by Langmuir@29#, who referred to it as ‘‘unipolarcoarcervation.’’ His argument was only qualitative, abased on the behavior of the osmotic pressure, evaluwithin Debye-Huckel theory for simple electrolytes. A similar, more careful argument was put forward in the thesisVoorn @30#, with specific applications to linear polyelectrolytes. He pointed out that the addition of salt decreasestendency for phase separation, in qualitative agreementthe predictions of the present work. However, neither Lamuir’s nor Voorn’s considerations are directly applicablethe case of highly charged colloidal particles considehere. More recent studies of effective interactions betwcharged colloidal particles in suspension have revealed inesting mechanisms for pairwise attraction. Some of thesequire multivalent salt ions, and originate from correlatioand fluctuations in the double layers@31,32#. The resultingattractions are, however, very short-ranged, and can thereprobably not drive a fluid-fluid transition. Moreover, it washown in Ref.@33# that a spherical geometry of the particlesuppresses attractions compared to a plate geometry. In
heecncta
litexilatr
trten
rao
as
.Gd-at
ln
ouosRth
ed
tth
-
ing
l-
2022 PRE 59van ROIJ, DIJKSTRA, AND HANSEN
@34# another mechanism for attraction is put forward. Tattraction there originates from the Coulomb-depletion effcaused by an intricate interplay between Coulombic asteric interactions. It would be interesting to study its effetogether with the volume term introduced here, on the phdiagrams.
The main conclusion of this paper, namely the possibiof a fluid-fluid phase separation due to Coulombic andcluded volume interactions alone, is reminiscent of a simphase separation predicted for very dilute symmetric eleclytes, modeled by the ‘‘restricted primitive model’’@35# ofoppositely charged hard spheres. Both in this symmemodel and in the extremely asymmetric case investigahere, the phase separation is driven by a delicate balabetween like-charge repulsion and opposite-charge atttion. The polyion-microion asymmetry and the presencesalt appear, however, to add further complexity to the phdiagrams.
ACKNOWLEDGMENTS
The authors have benefited from discussions with J.ThOverbeek, H.N.W. Lekkerkerker, T. Biben, A. van Blaaeren, and, particularly, P.B. Warren and R. Evans. Theygrateful to A. Vrij and H.N.W. Lekkerkerker for pointing outhe work by Voorn@30#, and to the ‘‘Ecole Normale Supe´r-ieure de Lyon’’ and CECAM for hospitality during the initiastages of this work. R.v.R. also thanks P.B. Warren for seing notes on the subtleties involved in the divergent Clomb integrals. This work was supported by Grant NERBFMBICT971869 and ERBFMBICT972446 of the TMProgram, by EPSRC Grant No. GR/L89013, and byEPSRC Liquid Matter Network.
APPENDIX A
In this appendix we calculate the minimum valueFel8 ofthe functionalFel defined in Eq.~22! by inserting the mini-mizing profile~53!. It proves essential to use the regularizCoulomb potential~3! throughout, and take the limitl→0 atlate stages of the calculation. It proves convenient to useFourier representation throughout. In this representationfunctional reads
bFel51
2n
1
~2p!3E dkk21kl
2
k21l2Dr2k
~1!Drk~1!
11
~2p!3E dkbU2k~l!Drk
~1!1 rbUk50~l! 1
1
2
4p l
l2r2V,
~A1!
with Uk(l) defined in Eqs.~50! and~51!, k andkl in Eq. ~49!,
n5n11n2 the total microion number density, andl5e2/ekBT the Bjerrum length. Inserting the equilibrium profile of Eq. ~48! into the functional~A1! gives its minimumvalue
td,se
y-r
o-
icdcec-fe
.
re
d--.
e
hee
bFel8 52n
2
1
~2p!3E dkk21l2
k21kl2
bU2k~l!bUk
~l!
1n
2V
l2
kl2 ~bUk50
~l! !21 rbUk50~l! 1
1
2
4p l
l2r2V.
~A2!
We first focus on the evaluation of the second line of~A2!,and then treat the first line.
It follows directly from Eq.~50! thatUk50(l) 5Npuk50
(l) , andhence from Eq.~51! that
rbUk50~l! 524p l rZnpS 1
l22
R2~112g!
6 D V1O~lR!,
~A3!
where theO(lR) are irrelevant in the limit of interestl→0. A similar expansion reveals
n
2V
l2
kl2 ~bUk50
~l! !254p l ~Znp!2S 1
2l21
R2~112g!
62
1
k2D V
1O~lR,l2/k2!. ~A4!
Using the results of Eqs.~A3! and ~A4! in the evaluation ofthe second line of Eq.~A2! yields, neglecting theO(lR,l2/k2) terms,
n
2V
l2
kl2 ~bUk50
~l! !21 rUk50~l! 1
1
2
4p l
l2r2V
51
2
4p l
l2~Znp2 r !2V2
1
2
4p l
k2~Znp!2V
521
2
4p l
k2~Znp!2V, ~A5!
where we used the electroneutrality conditionr5Znp . Theg-dependent contributions cancel identically. The remainterm of Eq.~A5! is independent ofl, and therefore consti-tutes thel→0 limit of the second line of Eq.~A2!.
We now calculate the first line of Eq.~A2!, which can berewritten with Eq.~50! as
2n
2
1
~2p!3E dkk21l2
k21kl2
bu2k~l!buk
~l!(i 51
Np
(j 51
Np
exp~ ik•Ri j !
52n
2
4p
~2p!3(i 51
Np
(j 51
Np E0
`
dksin~kRi j !
kRi j
3k2~k21l2!
k21kl2
bu2k~l!buk
~l! , ~A6!
where we performed the angular integrals explicitly. It folows from Eq.~51! and a straightforwardl expansion that
nh
e
eairs
edcal-
of
e
e
n--
-
seon
assa
ven
nan
rib-
PRE 59 2023PHASE DIAGRAM OF CHARGE-STABILIZED . . .
k2~k21l2!
k21kl2
bu2k~l!buk
~l!
5~4p lZ !2
k21k2 S g cos~kR!1~12g!sin~kR!
kR D 2
1O~lR,l2/k2!, ~A7!
which upon insertion into Eq.~A6! leads to integrals that cabe performed easily with contour integration techniques. Tterms in the double sum in Eq.~A6! must then be distin-guished between those withi 5 j ~for which Ri j 50) andthose withiÞ j ~for which Ri j .2R because of the hard-correpulsion between the polyions!. The termsiÞ j give rise tothe integral
2n
4
4p
~2p!3E2`
`
dksin~kRi j !
kRi j
3~4p lZ !2
k21k2 S g cos~kR!1~12g!sin~kR!
kR D 2
5Z2l
2 S g cosh~kR!1~12g!sinh~kR!
kR D 2
3exp~2kRi j !
Ri j2
Z2l
2
1
Ri j
5Z.
2 l
2
exp~2kRi j !
Ri j2
Z2l
2
1
Ri j, ~A8!
where we used the definition of the effective chargeZ. ofEq. ~55!. The factors 1/2 in Eq.~A8! correct for doublecounting of iÞ j terms in the double sum in Eq.~A6!. Thetermsi 5 j in Eq. ~A6! give rise toNp identical integrals
2n
4
4p
~2p!3E2`
`
dk~4p lZ !2
k21k2 S g cos~kR!1~12g!sin~kR!
kR D 2
52Z2l
2kh~g,kR!, ~A9!
where the dimensionless functionh(g,x) is given by
h~g,x!5~12g!2exp~22x!2112x
2x2
12g~12g!12exp~22x!
2x1g2
exp~22x!11
2.
~A10!
Gathering the results of this appendix reducesFel8 fromEq. ~A2! to
e
Fel8 5(i , j
Np S Z.2 e2
e
exp~2kRi j !
Ri j2
Z2e2
e
1
Ri jD
21
2
4pe2
ek2~Znp!2V2
Z2e2
2ekh~g,kR!Np .
~A11!
The first line in Eq.~A8! is recognized as the differencbetween screened and bare Coulomb interactions of all pof polyions at separationsRi j , with the effective chargenumberZ. determining the strength of the pairwise screenrepulsions. The bare Coulomb interactions cancel, in theculation of the effective polyion potentialVp
(eff) , the barepolyion-polyion contribution fromHp of Eq. ~3!. The secondline of Eq. ~A11! is independent of the coordinatesRi of thepolyions. Its first term, which was missed in the analysisRef. @19#, appears after carefully taking the limitl→0. Itsphysical interpretation is discussed in the main text.
It is easily checked thath„(11x)21,x…5(11x)21, andhence the second term in the second line of Eq.~A11! equals,for each of theNp polyions,2(Ze)2/@2e(k211R)# for ourchoiceg5(11kR)21. This is of the order of the Coulombicenergy of charges2Ze and 1Ze separated by a distanck211R in a medium of dielectric constante. It thus repre-sents the self-energy of anNp dressed polyion, i.e., barpolyions of charge2Ze with double layers of thicknessk21
and net charge1Ze. It can also be interpreted as the potetial energy of theNp polyions due to the potential well created by the double layers. For the Sogami-Ise choiceg50, itis easily checked fromh(0,x)5@exp(22x)2112x#/(2x2) thatthe second term in the second line of Eq.~A11! gives rise toa free-energy contribution (Ze)2/(4eR)@exp(2kR)2112kR#/(kR) per colloidal particle. This contribution is identical to the one in Eq.~44! of Ref. @23#. Although it has thesame characteristics as the one we obtained fromg51/(11kR), it was not considered as a driving term for phaseparation in Ref.@23#. Instead, the focus there has beenthe ~controversial! transformation from the Helmholtz toGibbs effective pair interaction@36#.
APPENDIX B
We calculate the free energyFD of a system ofN5nVindependent three-dimensional harmonic oscillators of mm fixed onN sites in a volume V. Throughout we assumeDebye spectrum
vs~k!5csuku ~B1!
for the longitudinal (s51) and transversal (s52,3) modes.Later on we adjust the sound velocitiescs(s51,2,3) to beconsistent with the screened Coulomb pair potentials giin Eq. ~60!, such thatFD1F0 is an estimate for the freeenergyF of the solid phase of a colloidal crystal. Withiclassical statistical mechanics, each mode contributesamount kBTln@\vs(k)/kBT# to the total free energyFD ,while the ground-state energy, or Madelung energy, contutes an amountum per particle. Hence we have formally
y
heiti
at-
thd
3
i-s
ds intric
di-e.ical
2024 PRE 59van ROIJ, DIJKSTRA, AND HANSEN
FD5Num1kBTVE dvg~v!ln@\v/kBT#, ~B2!
where we introduced the total phonon densityg(v)5(s51
3 gs(v). The phonon densitygs(v) of the branchess51,2,3 is defined by
gs~v
e
-
n-
f Eq.g
en
PRE 59 2025PHASE DIAGRAM OF CHARGE-STABILIZED . . .
M5I (RÞ0
Fv8~R!
R1S v9~R!2
v8~R!
R D ~R• z!2G . ~B14!
Using Eq.~B14! it is straightforward to evaluate the latticsums in Eq.~B12! for a given pair potentialv(r ); the latticeconstanta ~along a principal axis! is fixed by the numberdensity and the symmetry of the lattice. For fcc we havea5(4/n)1/3, for bcca5(2/n)1/3. The resulting numerical values form2 andm4 then provide the left-hand side of Eq.~B6!explicitly, from which numerical values for the dimensio
ecek
v
ce
.
r,
.
J.
e
v.
less sound velocitiesmci2/kBT and mc'
2 /kBT follow. Thesecan, subsequently, be used to evaluate the second line o~B4!. The full evaluation ofFD also requires the MadelunenergyuM , given by the lattice sum
um5 (RÞ0
v~R!, ~B15!
which can also be evaluated directly numerically for a givv(r ).
s. 2
-
-
G.
ys.
Y.
@1# J.S. Rowlinson and F. Swinton,Liquids and Liquid Mixtures,3rd ed.~Butterworths, London, 1982!.
@2# S. Asakura and F. Oosawa, J. Polym. Sci.33, 183 ~1958!; A.Vrij, Pure Appl. Chem.48, 471 ~1976!.
@3# The depletion force has been reassessed in a number of rpapers: see, e.g., Y. Mao, M.E. Cates, and H.N.W. Lkerkerker, Physica A222, 10 ~1995!; T. Biben, P. Bladon, andD. Frenkel, J. Phys.: Condens. Matter8, 10 799 ~1996!; R.Dickman, P. Attard, and V. Simonian, J. Chem. Phys.107, 205~1997!; B. Gotzelmann, R. Evans, and S. Dietrich, Phys. ReE 57, 6785~1998!.
@4# A.P. Gast, C.K. Hall, and W.B. Russel, J. Colloid InterfaSci. 96, 251 ~1983!.
@5# T. Biben and J.P. Hansen, Phys. Rev. Lett.66, 2215~1991!.@6# H.N.W. Lekkerkerker, W.C.K. Poon, P.N. Pusey, A
Stroobants, and P.B. Warren, Europhys. Lett.20, 559 ~1992!.@7# M. Dijkstra, R. van Roij, and R. Evans, Phys. Rev. Lett.81,
2268 ~1998!.@8# J.C. Crocker and D.G. Grier, Phys. Rev. Lett.73, 352 ~1994!;
77, 1897 ~1996!; K. Vondermasson, J. Bongers, A. Muelleand H. Versmold, Langmuir10, 1351~1994!.
@9# H. Lowen, J.P. Hansen, and P.A. Madden, J. Chem. Phys98,3275 ~1993!.
@10# J.W. Verwey and J.Th.G. Overbeek,Theory of the Stability ofLyotropic Colloids~Elsevier, Amsterdam, 1948!.
@11# G.M. Kepler and S. Fraden, Phys. Rev. Lett.73, 356 ~1994!.@12# A.E. Larsen and D.G. Grier, Nature~London! 385, 230~1997!.@13# D. Goulding and J.P. Hansen, Mol. Phys.95, 649 ~1998!.@14# N. Ise, T. Okubo, M. Sugimura, K. Ito, and H.J. Nolte,
Chem. Phys.78, 536 ~1983!.@15# K. Ito, H. Hiroshi, and N. Ise, Science263, 66 ~1994!; B.V.R.
Tata, E. Yamahara, P.V. Rajamani, and N. Ise, Phys. RLett. 78, 2660~1997!.
@16# B.V.R. Tata, M. Rajalakshmi, and A.K. Arora, Phys. ReLett. 69, 3778~1992!.
@17# T. Palberg and M. Wu¨rth, Phys. Rev. Lett.72, 786 ~1994!.
ent-
.
v.
@18# M.J. Grimson and M. Silbert, Mol. Phys.74, 397 ~1991!.@19# R. van Roij and J.P. Hansen, Phys. Rev. Lett.79, 3082~1997!.@20# J.M. Victor and J.P. Hansen, J. Chem. Soc., Faraday Tran
81, 43 ~1985!.@21# See, e.g., N.W. Ashcroft and D. Stroud, Solid State Phys.33, 1
~1978!.@22# For recent reviews, see R. Evans, inFundamentals of Inhomo
geneous Fluids, edited by D. Henderson~Marcel Dekker, NewYork, 1993!; or J.P. Hansen, inPhase Transitions in ComplexFluids, edited by M. Baus, L.F. Rull, and J.P. Rijckaert~Klu-wer, Dordrecht, 1995!.
@23# I. Sogami and N. Ise, J. Chem. Phys.81, 6320~1984!.@24# P.B. Warren, J. Phys. II7, 343 ~1997!.@25# See, e.g., J.P. Hansen and I.R. McDonald,Theory of Simple
Liquids, 2nd ed.~Academic Press, London, 1986!.@26# D. Frenkel and B. Smit,Understanding Molecular Simula
tions: From Algorithms to Applications~Academic Press,Cambridge, 1996!.
@27# D. Frenkel and A.C. Ladd, J. Chem. Phys.81, 3188~1984!.@28# M.O. Robbins, K. Kremer, and G.S. Grest, J. Chem. Phys.88,
3286 ~1988!.@29# J. Langmuir, J. Chem. Phys.6, 873 ~1938!.@30# A summary of this thesis is contained in J. Michaeli, J.Th.
Overbeek, and M.J. Voorn, J. Polym. Sci.23, 443 ~1957!.@31# M.J. Stevens and M.O. Robbins, Europhys. Lett.12, 81
~1990!.@32# J.L. Barrat and J.F. Joanny, Adv. Chem. Phys.XCIV , 1
~1996!.@33# M. Stevens, M.L. Falk, and M.O. Robbins, J. Chem. Ph
104, 5209~1996!.@34# E. Allahyarov, I. D’Amico, and H. Lo¨wen, Phys. Rev. Lett.81,
1334 ~1998!.@35# An excellent account of the recent work is contained in
Levin and M.E. Fisher, Physica A225, 164 ~1996!.@36# C.E. Woodward, J. Chem. Phys.89, 5140~1988!.
