JOURNAL OF CHEMICAL PHYSICSVOLUME 119, NUMBER 2

8 JULY 2003

Phase behavior of charged lipid bilayer membranes with added electrolyte
Shigeyuki Komura,a) Hisashi Shirotori, and Tadashi Kato
Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan

共Received 3 February 2003; accepted 11 April 2003兲
We investigate the phase behavior of the aqueous solution of charged lipid bilayer membranes
forming a lamellar structure in the presence of the added electrolyte. The Goldstein–Leibler model关Phys. Rev. A 40, 1025 共1989兲兴 proposed for neutral lipids is extended by taking into account thescreened electrostatic interaction which belongs to the so called intermediate region of the Poisson–
Boltzmann theory. We mainly focus on the variation of the phase behavior as the added electrolyte
concentration is changed. By decreasing the electrolyte concentration, the electrostatic repulsion
between the neighboring membranes becomes stronger. As a result, the maximum equilibriumswelling composition shifts to larger water content, and the lamellar repeat distance increases
although the membrane thickness remains almost constant. Our results recover not only theexperimentally observed phase diagram for the charged lipid, but also the dependence of thelamellar repeat distance both on the electrolyte concentration and on the lipid composition. Some
discussions are provided for the case when the electrostatic interaction is not screened due to theabsence of any electrolyte. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1579675兴

I. INTRODUCTION

have been also investigated by employing the osmotic stresstechnique.6 In this work, the binding affinity series onto thecharged lipid bilayers of phosphatidylglycerol 共PG兲 andphosphatidylserine 共PS兲 were determined for various cations.
The temperature-composition phase diagram of an anionic dimyristoyl phosphatidylserine 共DMPS兲/water binary
system in the presence of ammonium phosphate has beenstudied by Hauser et al. using x-ray diffraction and differential scanning calorimetry.7,8 In this system, the L ␣ to L ␤ ⬘transition 共the so-called ‘‘main transition’’兲 occurs at progressively lower temperatures as the water content is increased. The lamellar bilayer repeat distance becomes continuously larger with increased water content up to more than
50 wt %. Although the multilayer stacking may be disordered above this water composition, the system does notphase separate up to very high water contents.9 When thesystem is diluted more than 70 wt %, there is a transition
from a single swollen lamellar phase to a two-phase system
consisting of unilamellar vesicles and excess water. Noticethat the corresponding equilibrium swelling composition forneutral PC is at most 40 wt %.3 This difference in the swelling limit implies that the electrostatic repulsion acts to stabilize the swollen lamellar phase.
In the subsequent work by Hauser and Shipley, the effectof monovalent cations on the phase behavior of DMPS as
well as other homologous series was studied.10 According to
the x-ray diffraction data, they showed that the main transition temperature increases only slightly upon the addition of
NaCl up to 1 M concentration. This means that no majorchange in lipid packing within a bilayer occurs for this electrolyte concentration regime. The more significant effect ofadded electrolyte is the pronounced decrease in the lamellar
periodicity due to the salt-induced exclusion of aqueous
buffer from the intermembrane region. In Ref. 11, it was
shown that the addition of higher NaCl concentration up to
6 M increases the main transition temperature significantly.

Lipids are amphiphilic molecules with a hydrophilic
head group and usually two lipophilic hydrocarbon chains.
Phase diagrams for a binary system consisting of lipid andwater have been studied with a variety of experimental
techniques.1 One of the typical structures observed in such abinary system is a lamellar structure which is formed due tothe amphiphilic nature of the lipid molecule.2 The structureof hydrated lipid bilayer membranes are good model systems
for the study of interacting membranes. The lamellar phase is
fluid at high temperature 共the so-called L ␣ phase兲, whereas at
low temperature, the bilayers become a two-dimensional gel(L ␤ ⬘ phase兲. On the other hand, both the L ␣ and L ␤ ⬘ phases
can take up only a restricted amount of water. Adding morewater leads to a phase separation between the lamellar phaseand excess water. As for the electrostatically neutral lipids,
these phase behaviors can typically be observed in thedimyristoyl phosphatidylcholine 共DMPC兲/water binary
system.3 Recently it was reported that the sorption of water
triggers a first-order transition from the L ␤ ⬘ to the L ␣ phase.4
In contrast to the neutral lipids, such as phosphatidylcholine 共PC兲 or phosphatidylethanolamines 共PE兲, much less isknown about the phase behavior of charged lipid bilayer
membranes although several attempts have been made. For
example, force measurement between egg lecitin bilayers
containing various amounts of the charged phosphatidylglycerol 共PG兲 and phosphatidylinositol 共PI兲 was performed using
an osmotic stress technique and x-ray diffraction.5 In thisexperiment, it was claimed that hydration repulsion dominates out to about 20 to 30 Å separation, whereas more
slowly decaying electrostatic repulsion dominates beyond
about 30 Å. Concerning the added electrolyte, effects ofmonovalent ion binding and screening on electrostatic forcesa兲Electronic mail: komura@comp.metro-u.ac.jp

0021-9606/2003/119(2)/1157/8/$20.00

1157© 2003 American Institute of Physics

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1158

J. Chem. Phys., Vol. 119, No. 2, 8 July 2003

From the theoretical point of view, the effects of membrane interactions on the structural phase transition of neutral
lipid bilayers have been studied by Goldstein andLeibler.12,13 In their model, the main transition in an isolated
bilayer is described by a Landau theory in which the membrane thickness is taken as a coarse-grained order parameter.
On the other hand, the dominant molecular intermembrane
forces acting between bilayers are considered within a continuum treatment.14,15 Both of these ‘‘internal’’ and ‘‘external’’ degrees of freedom of the membranes are coupled toeach other when the typical distance between the membranes
is comparable to their thickness. For neutral phospholipid
lamellae, the long-range van der Waals attraction and shortrange hydration repulsion were taken into account. Therefore
it is natural to include the electrostatic repulsion in theGoldstein–Leibler theory to investigate the phase behaviorof charged membranes.
A detailed discussion of the electrostatic properties ofmembranes within the Poisson–Boltzmann theory is given in
Ref. 16. When an electrolyte is added to the solution, theelectrostatic properties 共e.g., electric potential, ionic concentration profiles兲 are strongly screened. The diffusive layers ofions in the solution is characterized by the Debye–Hückel
screening length ⌳ D . In this article, we consider the casewhen the separation between the membrane is large compared to ⌳ D , and the coupling between the two membranes
is weak. This limit is called as the intermediate region of the
Poisson–Boltzmann theory.16 We will show later that this isreally the case in the typical experiments. The main purposeof this article is to calculate the phase diagrams of the aqueous solution of charged lipid bilayer membranes in the presence of the added electrolyte. We pay attention to the variation of the phase behavior as the added electrolyte
concentration is changed. By decreasing the electrolyte concentration, we show that the maximum equilibrium swellingcomposition shifts to larger water content. Then the lamellar
repeat distance increases simultaneously although the membrane thickness remains almost the same.
This article is constructed as follows: In the next section,
we first review the Goldstein–Leibler theory,13 and explain
the newly added electrostatic interaction in the presence ofelectrolyte.16 Based on this generalized model, we then describe how we calculated the phase diagrams. In Sec. III, we
present the calculated results which will be compared withthose from the previous experimental works. Some discussions are provided in Sec. IV.
II. MODELAs a starting point of the present study, we first describe
the Goldstein–Leibler theory briefly.13 Consider a mixture ofN ᐉ lipids and N w water molecules in volume ⍀⫽N ᐉ v ᐉ⫹N wv w , where v ᐉ and v w are molecular volumes of lipid
and water, respectively. The volume fraction of lipid is then␾ ⫽N ᐉ v ᐉ /⍀. As shown in Fig. 1, the multilamellar lattice isassumed to be composed of sharply defined aqueous andmembrane regions of thickness d and ␦, respectively 共Luzzati
approximation兲. For such a stack of lamellae, the incompressibility condition is given by

Komura, Shirotori, and Kato

FIG. 1. The geometry of the lamellar stack. The multilamellar lattice isassumed to be composed of sharply defined aqueous and membrane regionsof thickness d and ␦, respectively.

␦
d

⫽

␾
.
1⫺ ␾

共1兲

In order to describe the main transition in an isolated
bilayer, we introduce an order parameter defined by

␺⬅

␦⫺␦0
,
␦0

共2兲

where ␦ 0 is the thickness in the L ␣ phase coexisting withexcess water. This order parameter is nonzero in the L ␤ ⬘phase, whereas it is small in the L ␣ phase. Since the structural main transition between the L ␣ and L ␤ ⬘ phases is a
first-order transition, the Landau expansion of the stretching
free energy per lipid molecule is expressed as
F L共 ␺ ,T 兲 ⫽ 21 a 2 共 T 兲 ␺ 2 ⫹ 31 a 3 ␺ 3 ⫹ 14 a 4 ␺ 4 ,共3兲where T is the temperature, and a 4 ⬎0 for stability. The inequality a 3 ⬍0 is required in order to ensure the thicker L ␤ ⬘phase. The temperature dependence of a 2 (T) is provided bya 2 (T)⬇a ⬘2 (T⫺T 0 ), where T 0 is the temperature at which
the transition from L ␣ to L ␤ ⬘ is critical if the cubic term isabsent. With these conditions, the main transition temperature T m is given by
T m⫽T 0 ⫹2a 23
9a 2⬘ a 4.

共4兲

It is believed that the lipid molecules are tilted in the L ␤ ⬘phase.1,2 Since ␺ does not describe any intramembrane symmetry differences, however, we do not distinguish between
the tilted L ␤ ⬘ phase and the untilted L ␤ phase hereafter. In

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J. Chem. Phys., Vol. 119, No. 2, 8 July 2003

Phase behavior of charged lipid bilayer membranes

order to describe such an intramembrane symmetry difference, one needs to include additional order parameters as
mentioned in Ref. 13.
For electrostatically neutral bilayers, there are two contributions to the interaction energy per unit area of themembrane.14,15 The first one is the long-range van der Waals
attraction V d( ␦ ,d):V d共 ␦ ,d 兲 ⫽⫺W

冋

1

2

d

共 d⫹ ␦ 兲

⫺
2

⫹
2

1
共 d⫹2 ␦ 兲

册

,
2

共5兲

where W is the Hamaker constant. The second one is theshort-range hydration repulsion V h(d):V h共 d 兲 ⫽H exp共 ⫺d/⌳ h兲 ,共6兲where H is the amplitude and ⌳ h is the decay length. To
investigate the phase behavior of the aqueous solution of theneutral bilayer membranes consisting of DMPC, Goldsteinand Leibler analyzed the total free energy13
F n⫽N ᐉ F L共 ␺ ,T 兲 ⫹N ᐉv ᐉ

␦关 V d共 ␦ ,d 兲 ⫹V h共 d 兲兴 .

共7兲

Notice that 2 v ᐉ / ␦ is the area per lipid head group. The internal and the external degrees of freedom of the membranes
are coupled through the incompressibility condition 共1兲. Using the above free energy, Goldstein and Leibler were able tocalculate the temperature-composition phase diagram whichis in semiquantitative agreement with the experimental results for the neutral DMPC/water binary system.3 An evaluation of relative importance of the various interactions for
DMPC bilayers was later discussed in Ref. 17.
Next we consider the interacting charged lipid bilayer
membranes in aqueous solution. In the presence of added
electrolyte, it is useful to separate the general solution of theappropriate Poisson–Boltzmann equation into several limits
in which approximate potential energies can be calculated
analytically. In this problem, three important length scales
exist: 共i兲 the spacing between the two membranes d; 共ii兲 the
Debye–Hückel screening length for 1:1 electrolyte 共in SI
base unit兲
⌳ D⫽冉␧ 0 ␧ w k BT
2n 0 q2

冊

2␧ 0 ␧ wk BT,q兩␴兩V e共 d 兲 ⫽32␧ 0 ␧ w共 k BT 兲 2

共9兲

where ␴ is the surface charge distribution assumed to be
negative. As argued in Refs. 16, 18, and 19, the dimensionless parameter space (⌳ D /d,b/d) can be divided into four
asymptotic regions. Among these, we consider the so-calledintermediate region in which the relation bⰆ⌳ DⰆd is satisfied. This is the case where the separation between the membranes d is large compared to the Debye–Hückel screening
length ⌳ D , whereas the coupling between the two mem-

exp共 ⫺d/⌳ D兲 .

q 2⌳ D

共10兲

By assuming complete dissociation, the area perphospholipid head group is about 66⫻10⫺20 m2 for PS.6
Then the Gouy–Chapman length 共9兲 is calculated to beb⬇1.5⫻10⫺10 m with the use of the other numerical values
listed below. For a typical electrolyte concentration, this isrelatively small compared to the values of d or ⌳ D in theexperiment. After the calculation, we will also see that ⌳ D isindeed smaller than d, which is reasonably satisfied in thereal system. These considerations justify our assumption of
the intermediate region.
Using the above asymptotic form of the electrostatic freeenergy 共10兲, we consider the following total free energy forthe charged bilayer membranes
F c⫽N ᐉ F L共 ␺ ,T 兲 ⫹N ᐉv ᐉ

␦关 V d共 ␦ ,d 兲 ⫹V h共 d 兲 ⫹V e共 d 兲兴 ,共11兲

which is a straightforward generalization of 共7兲. Here we
define the reduced free-energy density f c( ␾ ,T; ␺ )⬅F cv ᐉ /⍀ and the reduced parameters by
␭ h⬅

w⬅

⌳h

␦0

Wvᐉ

␦ 30

␭ D⬅

,

,

h⬅

⌳D

␦0

,

Hvᐉ

␦0

,

e⬅

32␧ 0 ␧ w共 k BT 兲 2 v ᐉ
q 2 ␭ D␦ 20

共12兲
.

Using the incompressibility condition 共1兲 and the definitionof the order parameter 共2兲, we obtainf c共 ␾ ,T; ␺ 兲 ⫽ ␾

共8兲

where ␧ 0 is the vacuum permittivity, ␧ w is the dielectric constant of the aqueous solution, k B is the Boltzmann constant,n 0 is the number density 共per unit volume兲 of the added
electrolyte, and q⬎0 is the electron unit; and finally 共iii兲 theGouy–Chapman length given byb⫽branes is weak. In such a limit, the electrostatic interactionenergy per unit area of the membrane has the following
asymptotic form16

冋

1/2

,

1159

⫺
⫹

1
1
1
a 2⬘ 共 T⫺T 0 兲 ␺ 2 ⫹ a 3 ␺ 3 ⫹ a 4 ␺ 4234
wC 共 ␾ 兲
共 1⫹ ␺ 兲

⫹
3

册

h␾
exp关 ⫺D 共 ␾ 兲共 1⫹ ␺ 兲兴1⫹ ␺e␾exp关 ⫺E 共 ␾ 兲共 1⫹ ␺ 兲兴 ,
1⫹ ␺

共13兲

where
C共 ␾ 兲⬅

2 ␾ 5 共 3⫺ ␾ 2 兲
共 1⫺ ␾ 兲

2 2,

D共 ␾ 兲⬅

1⫺ ␾
,
␭ h␾
共14兲

1⫺ ␾
.
E共 ␾ 兲⬅
␭ D␾
The reduced chemical potential is defined by␮ 共 ␾ ,T; ␺ 兲 ⫽冋册⳵ f c共 ␾ ,T; ␺ 兲
.
⳵␾
T, ␺

共15兲

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1160

J. Chem. Phys., Vol. 119, No. 2, 8 July 2003

Komura, Shirotori, and KatoTABLE I. Numerical values of model parameters.

␦ 0 ⫽3.5⫻10⫺9 m
a ⬘2 ⫽2.4⫻10⫺21 J K⫺1

a 4 ⫽1.9⫻10⫺18 J
T 0 ⫽2.6⫻102 K
W⫽1.9⫻10⫺22 J
H⫽1.0⫻10⫺1 J m⫺2⌳ h⫽2.5⫻10⫺10 m
␧ 0 ⫽8.9⫻10⫺12 F m⫺1
q⫽1.6⫻10⫺19 C
a 2⬘ /h⫽7.7⫻10⫺2
a 4 /h⫽60
␭ h⫽0.07av ᐉ ⫽1.1⫻10⫺27 m3
a 3 ⫽⫺9.3⫻10⫺19 J

T m⫽3.1⫻102 Kw⫽4.8⫻10⫺24 J
h⫽3.1⫻10⫺20 J
␧ w⫽80k BT⫽4.1⫻10⫺21 K
a 3 /h⫽⫺30
w/h⫽1.6⫻10⫺4a

The number for w/h given in Ref. 13 contains an error.

If we expand F c as a power series in ␺, we see that interaction act as an external field on the order parameter ␺.13 However, we treat the total free energy 共13兲 in the following
calculation.
For the purpose of calculating the phase diagrams of thelamellar system, we first minimize the total free energy density 共13兲 with respect to ␺. Based on the equality of thechemical potentials and the osmotic pressures, the common
tangent construction is employed to calculate the equilibrium
between two phases with different ␾⬘ and ␾⬙. In the coexisting two phases, the minimized order parameter ␺ can take on
different values from which the membrane thickness ␦ is
calculated using 共2兲. The incompressibility condition 共1兲 further determines the thickness of the aqueous region d forgiven ␾. These calculations have been done numerically. In
order to determine the phase equilibria between a lamellar
structure and excess water, the free energy of the latter solution is necessary as a function of the lipid composition. However, since this phase is extremely dilute in lipid, its freeenergy can be identified with that of pure water.13 Hence theapproximate coexistence for the equilibrium with excess water is determined by the condition of vanishing chemical potential, i.e., ␮⫽0. More rigorous treatment of the phase equilibrium should take into account the so-called Donnan
equilibrium20 which will be discussed in Sec. IV.
The numerical values of the model parameters used in
the present calculation are summarized in Table I. For thequantities which are also used in Ref. 13, we choose thesame numerical values in order to see the effect of the added
term 共10兲 clearly. In principle, the internal structural transition of the lipid bilayer may be affected by the electrostaticinteraction through the modification of a 2 , a 3 , or a 4 as well
as T 0 in 共3兲. Furthermore, the added electrostatic interaction
can give rise to the liner term in ␺ in the Landau expansionof the stretching free energy. In such a case, the electrostaticinteraction acts as an external field on the order parameter.
The free energy is then shifted in a way which favors different membrane thickness. However, we do not consider these
possibilities in this article since we want to focus on theeffect of the electrostatic interaction only through the modification of the intermembrane interaction.
Strictly speaking, the temperature dependence of thepresent model appears both in F L 关Eq. 共3兲兴 and V e(d) 关Eq.
共10兲兴 as well as in ⌳ D 关Eq. 共8兲兴. For the simplicity, however,

FIG. 2. Phase diagram of a mixture of neutral lipid and water. 1⫺␾ represents the water volume fraction, and a 2 /h corresponds to the temperature.
Equilibrium with excess water is denoted by ‘‘EW.’’ The critical point isindicated by a filled circle. This phase diagram is the same with Fig. 5 inRef. 13.

we fix the value of k BT in V e(d) and ⌳ D to that given in
Table I, and the temperature dependence is taken into account only through a 2 共or a 2 /h). With these assumptions,
we will see later in Sec. III that the melting transition 共or,
more precisely, the three phase coexistence temperature between the L ␣ , L ␤ , and excess water兲 occurs at a 2 /h⫽a 2⬘ (T m⫺T 0 )/h⬇3.33. According to the experiments on the
DMPC/water binary system,3,13 the main transition temperature is T m⫽310 K and T m⫺T 0 ⫽50 K 共see Table I兲. Hence,from 共4兲, each unit of the dimensionless temperature variablea 2 /h corresponds to 15 K.13
With the above choice of parameters, the reduced 共dimensionless兲 Debye–Hückel screening length 关see 共12兲兴 as afunction of the number density of the electrolyte n 0 in unitsof 关mol m⫺3兴 is given by.
␭ D⫽2.78⫻n ⫺1/2
0

共16兲

Using this relation, the ratio of parameters e/h is calculated
to bee/h⫽0.016⫻n 1/2
0 .

共17兲

III. RESULTS AND COMPARISON TO EXPERIMENTS

In this section, we show the calculated phase diagrams
based on the free energy 共13兲. The phase diagrams are represented in the (1⫺ ␾ ,a 2 /h)-plane, where 1⫺␾ is the water
volume fraction and a 2 /h corresponds to the reduced temperature. We focus on the variation of the phase behavior as
the electrolyte concentration is varied.
As a reference, we reproduce in Fig. 2 the same phase
diagram with that calculated by Goldstein and Leibler in Ref.
13 for the neutral lipid plus water binary system. This casecorresponds to the limit of the charged lipids when V e(d)→0 for all d. A line of first-order transition between the L ␣and L ␤ phases terminates at a triple point where these twolamellar phases coexist with excess water. The narrowness of

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J. Chem. Phys., Vol. 119, No. 2, 8 July 2003

Phase behavior of charged lipid bilayer membranes

1161

FIG. 3. Phase diagram of a mixture of charged lipid and water. The concentration of the added electrolyte is n 0 ⫽5⫻102 mol m⫺3 (0.5 M). 1⫺␾ represents the water volume fraction, and a 2 /h corresponds to the temperature.
The notations of different phases are the same with Fig. 2. The critical point
is indicated by a filled circle.

FIG. 4. Phase diagram of a mixture of charged lipid and water. The concentration of the added electrolyte is n 0 ⫽1⫻102 mol m⫺3 (0.1 M). 1⫺␾ represents the water volume fraction, and a 2 /h corresponds to the temperature.
The notations of different phases are the same with Fig. 2. The critical point
is indicated by a filled circle.

the tie lines persists to temperatures quite far from the criticalpoint. Due to the skewing of this tilted two-phase region, atemperature scan at fixed composition crosses the two binodal lines. At higher temperature, the first-order line ends atthe critical point 共marked with a filled circle兲 beyond which
the bilayer thickness changes continuously analogous toliquid/vapor type transition. Notice that the equilibriumswelling composition for the L ␣ phase is 1⫺␾⫽0.4, whereas
it varies with temperature for the L ␤ phase.
Next we consider multilamellar dispersions consisting of
charged lipid molecules and water. Figures 3 and 4 are thecalculated phase diagrams for the electrolyte concentrationn 0 ⫽5⫻102 mol m⫺3 (0.5 M) and 1⫻102 mol m⫺3 (0.1 M),
respectively. It is remarkable that the equilibrium swellingcomposition shifts to larger 1⫺␾ 共or smaller ␾兲 by decreasing the electrolyte concentration. For n 0 ⫽1⫻102 mol m⫺3
共Fig. 4兲, the L ␣ phase can be swelled as large as 1⫺␾⬇0.77
water composition. As a result, the region of the single
lamellar phases extends to the lower lipid composition regime. While the locus of the critical point does not change
appreciably for different electrolyte concentrations, the compositions of the L ␣ and L ␤ phases at the triple point temperature shift to larger 1⫺␾. Hence, for small electrolyte concentration n 0 , the two-phase region between the L ␣ and L ␤phases is rather elongated, and the tie lines become narrower.
As has been stated before, the temperature dependence of thepresent model is taken into account only through a 2 (T).
Hence, compared to the neutral lipid case, the addition of theelectrostatic interaction does not modify the temperature at
the critical point or the triple point.
Our results reproduce several features of the experimentally studied phase diagram reported in Ref. 8 for charged
lipid bilayer membranes. In Fig. 6 of Ref. 8, the temperaturecomposition phase diagram of DMPS/water binary systemwith added 0.025 M ammonium phosphate is determined byusing differential scanning calorimetry. In the absence of wa-

ter, pure DMPS undergoes a crystal to liquid crystal transition at 340 K. With increasing the water content the transition between the L ␣ and L ␤ phases occurs at progressively
lower temperature. It finally reaches a limiting value of
310 K at water content of 1⫺␾⬇0.45. Even though themultilayer stacking may be disordered at higher water content, the DMPS/water binary system does not phase separate
up to around 70 wt % water amount.8,9 Further dilution leads
to a transition from a single lamellar phase to a two-phaseregion where vesicles coexists with excess water.9 We see aquite similar behavior of the main transition temperature in
Fig. 4 except that the critical point is not observed in theexperiment. As mentioned in the previous section, this isbecause of the shortcoming of the scalar order parameter 共2兲
which does not distinguish any intramembrane symmetry
differences. The difficulty in observing the coexistence between the L ␣ and L ␤ phases in the experiment may be due tothe narrowness of the binodal region as shown in Fig. 4.
In Fig. 5, we have plotted various lengths, namely, thelamellar repeat distance d⫹ ␦ , the thickness of the water
region d, the thickness of the lipid bilayer ␦, and the Debye–Hückel screening length ⌳ D 关see 共8兲兴 as a function of theelectrolyte concentration n 0 when the L ␣ phase coexists withthe excess water. In other words, we follow the shift of thecoexistence line between the L ␣ phase and the excess water
as the electrolyte concentration is varied. The thickness of
the lipid bilayer ␦ is calculated from the minimized orderparameter ␺ 关see 共2兲兴, and the thickness of the water region dshould satisfy the incompressibility condition 共1兲. For the L ␣phase, we remind that the swelling composition does notdepend on temperature a 2 /h 共see Figs. 3 and 4兲. Figure 6 isthe corresponding plot for the L ␤ phase coexisting with excess water at fixed a 2 /h⫽2.8. We remark again that the coexistence line between the L ␤ phase and the excess water
depends on the temperature a 2 /h. For both the L ␣ and L ␤phases, the thickness of the charged lipid bilayer ␦ 共dotted

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1162

J. Chem. Phys., Vol. 119, No. 2, 8 July 2003

FIG. 5. Various lengths as a function of the electrolyte concentration n 0
when the L ␣ phase coexists with excess water. The solid line is the lamellar
repeat distance d⫹ ␦ , the dashed line is the thickness of the water region d,
the dotted line is the thickness of the lipid bilayer ␦, and the dot-dashed lineis the Debye–Hückel screening length ⌳ D .

line兲 does not change even if n 0 is varied. On the other hand,
the thickness of the aqueous region d 共dashed line兲 decreases
as n 0 is increased. This tendency can be easily understoodby noticing that the electrostatic repulsion between themembranes becomes more screened for larger n 0 , andthe intermembrane distance takes smaller value. For theL ␣ phase in Fig. 5, the lamellar repeat distance d⫹ ␦ deand starts to level off around n 0 ⬇0.5
creases as n ⫺0.52
0⫻103 mol m⫺3 (0.5 M). The level off of the lamellar repeat
distance can be explained as follows: Although the screened
electrostatic repulsion becomes very weak for such a highelectrolyte concentration, the hydration repulsion starts todominate for small intermembrane distances.
The effect of monovalent cations on the phase behaviorof DMPS/water binary system was investigated in Ref. 10. It
is reported that the main transition temperature as well as thetransition enthalpy remained almost unchanged as long as
the added NaCl concentration does not exceed 1 M. Analysisof the x-ray diffraction data confirmed that, even in the presence of added NaCl, no significant change in the hexagonalhydrocarbon chain packing occurs in the L ␤ phase. The major effect of added salt is the electrolyte-concentrationdependent reduction in the lamellar periodicity as the wateris excluded from the intermembrane space. For 50 wt %aqueous dispersion of DMPS, the lamellar repeat distance of
the L ␤ phase at 293 K decreased linearly with increasing
NaCl concentration up to 0.5 M and then leveled off 共seeFig. 5 in Ref. 10兲. Compared to the electrolyte-free case, thelamellar periodicity reduced from 107 to 62 Å upon the addition of 0.5 M NaCl. The corresponding reduction in thelamellar periodicity of the L ␣ phase was from 75 to 52 Å. We
consider that the above experiment has been done under thesituation where the lamellar phase coexists with excess wa-

Komura, Shirotori, and Kato

FIG. 6. Various lengths as a function of the electrolyte concentration n 0
when the L ␤ phase coexists with excess water at a 2 /h⫽2.8. The notationsof different lines are the same with Fig. 5.

ter. Otherwise the lamellar repeat distance is solely determined by the relative composition between lipid and water
since the lipid bilayer thickness remains almost constanteven if the electrolyte is added 关see 共1兲兴. Interestingly, theelectrolyte concentration dependence of the repeat distanced⫹ ␦ in Figs. 5 and 6 shows a similar behavior with theabove mentioned experiment.
In Figs. 5 and 6, we see that the thickness of the aqueous
region d is larger than the Debye–Hückel screening length⌳ D . Moreover both of these lengths are much larger than the
Gouy–Chapman length b⬇1.5⫻10⫺10 m as discussed in theprevious section. These have been a necessary condition to
use the asymptotic form of the electrostatic interaction given
by 共10兲. Hence our calculated results are consistent with therequirement for the intermediate region. However, for n 0
smaller than 1 mol m⫺3 in the L ␤ phase 共not shown in thegraph兲, d is calculated to be smaller than ⌳ D , and it is no
more appropriate to use 共10兲. For such a small electrolyte
concentration case, we are already either in the Gouy–
Chapman region or the ideal-gas region, which will be discussed in the next section.
Fixing the electrolyte concentration to n 0 ⫽1⫻102
mol m⫺3 (0.1 M), we have plotted in Fig. 7 the lamellar bilayer repeat distance d⫹ ␦ as a function of the water content
1⫺␾ at two different temperatures, i.e., a 2 /h⫽2.8 and 4.4.
This result is even in quantitative agreement with Fig. 7 inRef. 8. For both the L ␣ and L ␤ phases, the bilayer repeat
distance expands continuously as the water amount is increased. Compared to the L ␣ phase, the L ␤ phase takes larger
repeat distance d⫹ ␦ as well as the bilayer thickness ␦. Since
we have the constraint of the incompressibility condition 共1兲,larger ␦ in the L ␤ phase means a denser lateral packing of thehead groups in spite of the expected electrostatic repulsion
between the charged groups. Such a reduction of the area per

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J. Chem. Phys., Vol. 119, No. 2, 8 July 2003

Phase behavior of charged lipid bilayer membranes

1163

There are three other asymptotic regimes of the
Poisson–Boltzmann
theoryrepresented
inthe(⌳ D /d,b/d)-parameter space:16 共i兲 Debye–Hückel region关 ⌳ DⰆb and ⌳ D /dⰆ1, or (⌳ D /d) 2 Ⰶb/d and ⌳ D /dⰇ1],
共ii兲 ideal-gas region 关 b/dⰇ1 and (⌳ D /d) 2 Ⰷb/d], and共iii兲 Gouy–Chapman region 关 b/dⰆ1 and ⌳ D /dⰇ1]. In the
Debye–Hückel region, the potential value on the membranesurface is small, and the Poisson–Boltzmann equation can belinearized. The resulting asymptotic form of the energy perunit area is thenV e共 d 兲 ⫽FIG. 7. The bilayer repeat distance d⫹ ␦ as a function of the water content
1⫺␾ when n 0 ⫽1⫻102 mol m⫺3 (0.1 M). The solid line is the repeat distance of L ␣ phase at a 2 /h⫽4.4, and the dashed line is that of L ␤ phase at
a 2 /h⫽2.8.

molecule in the mixture of egg lecitin and charged lipid wasobserved in Ref. 5.
IV. DISCUSSION

In this article, we have investigated the phase behaviorof the aqueous solution of charged lipid bilayer membranes
forming a lamellar structure. We have focused on the effectof the added electrolyte. The Goldstein–Leibler model13 proposed for the neutral lipids has been extended by taking into
account the screened electrostatic interaction. Among various
asymptotic regions of the Poisson–Boltzmann theory, we
considered the so-called intermediate region. We paid attention to the variation of the phase behavior as the added electrolyte concentration is changed. With decreasing the electrolyte concentration 共weak screening兲, the electrostatic
repulsion between the neighboring membranes becomesstronger. Accordingly, the maximum equilibrium swellingcomposition shifts to larger water content, and the lamellar
repeat distance increases although the membrane thickness
remains almost constant. Our results recover not only theexperimentally observed phase diagram for the charged lipid,
but also the dependence of the lamellar repeat distance on theelectrolyte concentration as well as on the lipid composition.
In our calculation, we have used the asymptotic form of
the electrostatic interaction in the intermediate region as
given in 共10兲. It is valid when bⰆ⌳ DⰆd, where ⌳ D is the
Debye–Hückel screening length 共8兲, b is the Gouy–
Chapman length 共9兲, and d is the intermembrane distance.
This condition is satisfied when the separation between themembrane is large compared to ⌳ D although the Poisson–
Boltzmann equation itself cannot be linearized since the surface potential is large. Notice that the asymptotic form of
free energy does not depend on the surface charge density inthis limit.

冋 冉 冊 册␴ 2⌳ Ddcoth⫺1 ,␧ 0␧ w
2⌳ D

共18兲

where ␴ is the surface charge density as before. By replacing
共10兲 with 共18兲 and looking at the total interaction energy as afunction of d 共with an appropriate choice of ␴兲, we haveconfirmed that the phase behavior is not modified at least
qualitatively. This is a natural consequence since both 共10兲and 共18兲 are exponentially decaying functions.
For the ideal-gas and the Gouy–Chapman regions, on
the other hand, the repulsive interaction energy decays algebraically rather than exponentially. In contrast to the intermediate region, it was impossible to construct a globallyconsistent phase diagram due to the long-ranged nature of
the interaction as long as we use reasonable numerical valuesof model parameters. When the repulsive interaction is toostrong, the typical intermembrane distance becomes muchlarger than the membrane thickness. In such a highly swollencase, the internal and the external degrees of freedom of themembranes do not couple each other. It is then reasonable toassume that membranes are infinitesimally thin homogeneous layers characterized by coarse-grained elastic properties. We encounter a similar situation when one takes into
account the long-range Helfrich steric interaction which decays also algebraically.21
When we calculated the phase equilibria between one of
the lamellar phases and the excess water, only the conditionof vanishing chemical potential 共␮⫽0兲 was used. Although
this is justified for neutral lipids, further complication could
arise for charged membranes in the presence of the electrolyte. For example, due to the constraint of electroneutrality inthe membrane system, there is a tendency for the added electrolyte to go into the less crowded excess water region. Thisphenomenon is known as the Donnan effect20 which induces
a difference in the average electrolyte concentration between
the membrane system and the excess water.22 As a result, an
excess osmotic pressure difference should arise between thetwo phases,23 and the location of the phase boundary would
possibly be modified. In order to consider such a problem,
however, one should treat the electrolyte concentration as anadditional degree of freedom, and the chemical potential of
the electrolyte should be taken into account. However, this isout of the scope of the present model. Similar argument
holds also for the phase equilibrium between the L ␣ and L ␤ ⬘phases since the intermembrane distance is different for thesephases. We note that the Donnan effect becomes more important for highly charged membranes.
In the present study we have neglected not only the difference between the L ␤ and L ␤ ⬘ phases, but also the presence

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1164

J. Chem. Phys., Vol. 119, No. 2, 8 July 2003of the so called rippled ( P ␤ ⬘ ) phase where the membrane
structure is modulated. The rippled phase, when it exists, lies
typically between the L ␣ and L ␤ ⬘ phases. Recent experimentshows that the P ␤ ⬘ phase exhibits anomalous dielectric properties due to the strong dipole–dipole correlation.24 In order
to consider such a modulated phase, one needs to use aspace-dependent order parameter. Moreover, it is more appropriate to use a vector order parameter rather than the scalar order parameter.25 The study to obtain the full phase behavior of the aqueous solution of lipid bilayer membranes isunder progress.
ACKNOWLEDGMENTS

We thank D. Andelman and K. Fukada for useful discussions. This work is supported by the Ministry of Education,
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