JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER 15 15 APRIL 1999
Ternary surfactant mixtures in semi-infinite geometryM. Tasinkevych and A. CiachInstitute of Physical Chemistry and College of Science, Polish Academy of Sciences,Kasprzaka 44/52, 01-224 Warsaw, Poland
~Received 14 September 1998; accepted 21 January 1999!
In systems such as surfactant solutions, lipids, or copmers the ordered phases exhibit spatial modulations of dsities, and in the disordered phases correlation functions hdamped oscillatory behavior on the mesoscopic lenscale.1 The effect of the structure on the nanoscale on surfphenomena has not been studied theoretically~with a fewexceptions2–4!. In particular, it is not known whether thgeneral results obtained for phase transitions in the presof surfaces in the case of simple fluids,5,6 including surface-induced ordering or disordering transitions,7–9 apply also tothe systems in which the nanostructure is present. Wedress this question here.
The ordering effects of a hydrophobic or a hydrophisurface in microemulsions and water-surfactant mixtuwere studied by neutron reflectometry.10–13To quantitativelyexplain the measured data, the authors used the LandGinzburg~LG! model of microemulsions introduced first bM. Teubner and R. Strey,14 and then generalized by G. Gompper and M. Schick~GS! model.1,15 The GS model veryaccurately describes the bulk correlation functi^f(0)f(r )& in the microemulsion,
^f~0!f~r !&5exp~2r /jb!sin~2pr /db!
2pr /db, ~1!
where the scalar fieldf is the local water–oil concentratiodifference,db is the characteristic size of the water anddomains, andjb is the correlation length. Near a planar suface the GS model predicts the profile,
^f~z!&;exp~2z/js!sin~2pz/ds1g!, ~2!
whereds5db andjs5jb . g is needed to accommodate diferent surface field strengths.11 Yet, reflectivity R(q) calcu-lated for the structure~2! is essentially lower than the measured values. To fit the data, the authors had to useds
significantly smaller thandb , andjs significantly larger thanjb . Moreover, the difference between the surface and
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bulk parameters increases on the approach to the coexistwith the lamellar phase. The authors concluded:‘‘the inter-facial structure of the microemulsion can not be quantitively inferred from its bulk correlation function.’’11 Thisresult is in evident disagreement with the GS model.
While in reality the microemulsion and the lamellaphase can coexist, in the GS model the corresponding ptransition is only continuous. In simple fluids the surfaphenomena close to continuous or to first-order transitiare qualitatively different; critical adsorption in the first caand wetting in the second case occur. Similarly, in micemulsions, different surface phenomena can occur closthe first- or to the second-order phase transitions. Transibetween microemulsion and lamellar phases is eithercon-tinuous or first order16 in the one-dimensional version of thCiach–Ho”ye–Stell lattice model~CHS!.17 In the experimentsof Refs. 10–12, no ordered phases other than the lamphase, in which the lamellae are oriented parallel tosurface,18 are present. In such a case the direction perpdicular to the surface is distinguished and a one-dimensiodescription is justified. Within the CHS model one can stuthe surface effects in the microemulsion and in the lamephase far off and close to the phase coexistence. In the lcase we show by explicit mean field~MF! calculations that athick lamellar-like film is formed near the hydrophilic extenal wall. The film grows on the approach to the phase trsition as found experimentally.
The results of experiments as well as those we obtaiwithin the CHS model suggest that phenomena similarwetting in simple fluids occur. We thus compare the resuof the explicit MF calculations for the CHS model with thgeneral predictions of the theory for surface phenomenasimple fluids. Perfect agreement is obtained when the lenunit in the complex system is identified with the perioddensity oscillations, rather than with the molecular size. Tsuggests that the general properties of surface phenomecomplex fluids should be described by a Landau–Ginzbfunctional of the same structure as for simple fluids.7,9 In-deed, on the length scale of the size of atoms the simple fl
7549J. Chem. Phys., Vol. 110, No. 15, 15 April 1999 M. Tasinkevych and A. Ciach
exhibits the atomic structure, as described by the pcorrelation function, and only on the larger length scale iuniform. Similarly, on the length scale of;100 Å the com-plex system exhibits the structure as described by dendistributions, and on the length scales larger than the peof density oscillations it is uniform. In the simple fluids, thorder parameter~OP! is the deviation of the local densitfrom the average value in a region of a molecular siBy analogy, the order parameter in the complex systshould be identified with a suitable description of the devtions of the density profile from a constant function inregion of a linear size of the period of modulations. Wexplicitly define such an OP, denote it byl and call ‘‘lamel-lar order-parameter.’’ Next we derive in a systematic wafunctional ofl from the MF description of the CHS modeand we obtain the same form as deduced on gengrounds.19
The rest of this paper is organized as follows: in Secthe lattice model is briefly presented. In Sec. III, MF resufor the near-surface density profiles are described, andthe lamellar OPl and the corresponding surface-excequantity are defined and calculated. In Sec. IV, the newmodel is derived from the CHS model. In the last sectionresults obtained earlier for the functional of density fsimple fluids, being of the same form as our functional ofl,are quoted.
II. LATTICE MODEL
Various systems such as different surfactant mixturcopolymers, lipids, etc., with various details of interparticinteractions, exhibit very similar properties on the nanoscA model in which the interactions have the properties typifor all such systems should correctly describe the phenomcommon for these systems. In the CHS model it is assumthat such crucial properties are:~i! oil and water do not formsolutions unless the temperature is very high, and~ii ! thisend of the amphiphile which attracts water repulses oil avice versa.
If no ordered phases other than the lamellar phasepresent and the lamellae are oriented parallel to the sur~as found experimentally18!, the direction perpendicular tothe surface is distinguished and a one-dimensional destion can be applied. In the one-dimensional problem, torientations of amphiphiles,← and→, are distinguishedwith the tip of the arrow representing the ‘‘head’’ of thamphiphile. In the CHS model,← and→ are treated asdifferent components. In the case of close packing at evlattice site, there are thus four states labeledi 51,2,3,4, cor-responding to water, oil,← and→ respectively. The micro-scopic density distributions of the system are describedr i(z), and assume values 1 or 0 depending on whethersitez is occupied by a particle of speciesi or not. In the case
r-s
ityd
.m-
a
ral
Issos
e
s,
e.lnaed
d
rece
ip-o
ry
yhe
of oil–water symmetry the only relevant chemical potentvariable is the differencem5m12m3 , sincem15m2 . In realsystems, the energies of configurations with the head ortail of the amphiphile oriented toward water differ significantly. To directly account for this amphiphilic propertythe model, we assume that the energies of the configuratid← and←d, are of a different sign. Byd we depicted thewater particle. The same property is assumed foramphiphile–oil interactions. We assume that alike ordinparticles attract each other, and that the different ones dointeract, so that the binary oil–water mixture phase serates.
The Hamiltonian of the one-dimensional systemwidth L in a contact with identical external walls has a for
H5(z51
L S 1
2 (i , j 51
4
~ r i~z!Ui j ~21!r j~z11!1 r i~z!Ui j ~11!
3 r j~z21!!2m~r1~z!1 r2~z!!D1(
j 51
4
~hj~1!r j~1!1hj~L !r j~L !!. ~3!
Herehi is the surface contact field acting on thei-th state,
Ui j ~21!5S 2b 0 2c c
0 2b c 2c
c 2c 0 0
2c c 0 0
D ,
and Ui j (21)5Uj i (11). Within the MF approximation thegrand thermodynamic potential of our system can be writas
V~t,m,L !5(z51
L
(i 51
4
r i~z!S t ln~r i~z!!11
2f i~z!
2m~d1i1d2i ! D1(j 51
4
~hj~1!r j~1!
1hj~L !r j~L !!, ~4!
where t5kT (k being the Boltzmann constant anT—temperature!, f i(z)5( j 51
4 (Ui j (21)r j (z11)1Ui j
(11)r j (z21)) is the mean field, andd i j is the Kroneckersymbol. The distance in Eq.~4! is measured in units of thelattice constant,a;25 Å, comparable to the size of amphiphiles.
The equilibrium densitiesr i(z) correspond to the globaminimum ofV. In practice, however, we can only determinlocal minima ofV by solving the set of the self-consisteequations,16
r i~z!5exp~2 ~1/t! ~f i~z!1hi~z!~d1z1dLz!2m~d1i1d2i !!!
7550 J. Chem. Phys., Vol. 110, No. 15, 15 April 1999 M. Tasinkevych and A. Ciach
The equilibrium densities are then identified with the sotion of Eq. ~5!, which gives the lowest value ofV @Eq. ~4!#.The simplest method of finding the solution of Eq.~5! is bymeans of numerical iterations starting from reasonable inconfigurations. We consider initial densities of three typThe first two correspond to the water-rich phase~in the caseof hydrophilic walls! and to microemulsion. The third typconsists of one-dimensional oscillations. We assume indensitiesr i(z)50 or r i(z)51, as in the ground state.20 Vari-ous periods of the initial densities are chosen.
Solutions of Eq.~5! can be found only for a finite systemof a sizeL. To find the value ofL at which we may neglecthe finite size effects, we calculate the excess quantitys,
s~t,m,L !5 12~V~t,m,L !2Lvb~t,m!!, ~6!
as a function ofL. The grand potential of the bulk systeper lattice site,vb , is calculated by the same method asRef. 16, that is by using periodic boundary conditionssystems of various sizes. The thermodynamic potentialsite of such a system is equal to the thermodynamic poteper site of infinite periodic structure. The lowest thermodnamic potential per lattice site corresponds to the staphase and determines the period of the structure. In Eq~6!the values ofL commensurate with the period of the budensity oscillations are chosen, to avoid contributions toVcoming from frustrations or deformations of the structuwhich are not present in the semi-infinite case. We verifithat s achieves a constant value for sufficiently largeL5Ls(t,m). Hence forL>Ls(t,m) the finite size effects arenegligible.
III. MF RESULTS
We use the water–water interaction energy as the enunit, and setb51. The (t,m) phase diagram of the bulsystem calculated forc54 ~strong surfactant! is presented inFig. 1. In the insets, the first-order transition lines in a sthe analog of a capillary condensation, are shown. For str
FIG. 1. Bulk phase diagram inT,m plane forc54 ~in units of water–waterpair potential!. In the regions O/W, L, M, the coexisting oil and water-riclamellar, and microemulsion phases are stable, respectively. Soliddashed lines indicate first- and second-order transitions, respectively; dline is a Lifshitz line. The solid dots indicate the location of the tricriticpoints. In the insets the lines that are analogs of the capillary condensaare presented. The distance between walls isL5199.
-
l.
al
erial-le
,d
gy
,g
surfactants there are two tricritical points separating the ctinuous and the first-order transitions between the micemulsion and the lamellar phase. Surfactant concentratioone of them is low,rs'10%, like in the experiments oRefs. 10–13.
We first study the near-surface structure in the micemulsion and in the lamellar phase far off the coexistenFor microemulsion the model predicts the exponentiallycaying lamellar order near the surface, as shown in Fig. 2~a!,in accordance with the GS model. In order to study the effof the surface on the period of the lamellar phase, we choparameters corresponding to a swollen lamellar phase, sfor large periods the discreteness of the model plays aimportant role. We first find the density profile in the buphase using the method described above. The equilibrprofile is shown in Fig. 3~a!. Next we take boundary condi
ndted
ns
FIG. 2. The thermodynamic variablest, m correspond to the stability of themicroemulsion (t52.8,m54.4214,c54). The distance from the first-ordetransition between the microemulsion and the lamellar phase isuDmu50.1. Walls are covered by water.~a! The density of surfactant as a function of a distance from the wall in units of the lattice constant.~b! LamellarOP l as a function of a distance from the wall measured in units ofperiod of the lamellar structure. Dashed lines are to guide the eye.
FIG. 3. Thermodynamic variablest,m,c correspond to the stability of aswollen lamellar phase (t50.84,m50.7739,c51). ~a!: The density of wa-ter in the bulk. The period of the density modulation isp513. ~b!: Thedensity of water near the hydrophilic wall. The period of the density molation is p512. Dashed lines are to guide the eye.
hawa
eeic
nto3
ona
aasbthaarfa
rmrene
eac
llaW
een
heden-
i-of
en-ad-OP,
on-
he
s
to
om
lla
to
ticell
7551J. Chem. Phys., Vol. 110, No. 15, 15 April 1999 M. Tasinkevych and A. Ciach
tions corresponding to water-covered walls. It turns out tin such conditions the period of the structure giving the loest thermodynamic potential per lattice site is smaller,shown in Fig. 3~b!. In a semi-infinite geometry the interfacbetween these two types of profiles will appear. To undstand this effect let us adopt a microscopic picture in wha probability of a microscopic state$r i(z)% is given by theBoltzmann factor exp(2bHMF@ r i(z)#), whereHMF@ r i(z)# isthe MF Hamiltonian. Deviations from the average positioof the monolayers occur with finite probability and leaddensity profiles slowly varying in space, as shown in Fig.The solid wall covered by water suppresses these deviatiand the average distance between the monolayers is smthan in the bulk.
We turn now to the surface phenomena near the phcoexistence between microemulsion and the lamellar phIn this model the order of the transition can be changedvarying the thermodynamic variables and the strength ofsurfactant. We focus our attention on the region of the phspace (t,m,c) where the microemulsion and the lamellphase coexist. Close to the phase transition a thick film owell-defined lamellar structure is formed near the wall,shown in Fig. 4~a!. Hence, close to the coexistence, the foof rs(z) is in apparent disagreement with the GS model pdiction, and with the shape of the bulk correlation functioexhibiting damped oscillatory behavior in the samconditions.16 The shape of the density profile@Fig. 4~a!# in-dicates that the reflectivityR(q) should be substantiallylarger than that found within the GS model. Moreover,R(q)should be larger the closer to the coexistence, as in theperiments. When the surface is neutral and the lamellar phis stable in the bulk, a surface film of a microemulsion ocurs near the phase coexistence@see Fig. 5~a!#.
In the context of wetting we are interested in the lameorder as a function of the distance from the surface.introduce lamellar order-parameter in the following way:
FIG. 4. The vicinity of the first-order transition from the microemulsionthe lamellar phase (t52.8,m54.3214,c54,uDmu50.0003). Walls are cov-ered by water.~a!: The density of surfactant as a function of a distance frthe wall in units of the lattice constant.~b!: Lamellar OPl as a function ofa distance from the wall measured in units of the period of the lamestructure. Dashed lines are to guide the eye.
t-s
r-h
s
.s,
ller
see.ye
se
as
-,
x-se-
re
l~z!5 (z85z
z1p A(i 51
4
~r i~z8!2 r i !2, ~7!
where r i5(z85zz1p r i(z8), and p is the period of the density
oscillations in units of the lattice constanta. l(z) is definedon a coarse-grained lattice with the lattice constantp•a.r i(z) is periodic in the bulk lamellar phase, andl(z) is justa constant. In the uniform phasel50 by definition. Hence,l is discontinuous at the first-order phase transition betwthe two phases. In Figs. 2~b! and 4~b! we presentl(z) cal-culated far off and close to the transition, respectively.
In the case of simple fluids the ordering effect of tsurface is quantitatively described by the surface excesssity ~also called adsorption or coverage!, G5*0
`dz(r(z)2rb), wherer(z) andrb are the local and the bulk densties, respectively. However, this quantity is not capabledescribing the ordering effect in the case of oscillating dsities. To quantitatively describe this effect we define ansorption parameter as a surface excess of the relevantwhich in this case should be identified withl(z),
G l 5(z>1
l~z!. ~8!
We verified thatG l ;2 log(Dm), whereDm is the differencebetweenm and its value at the phase-coexistence~see Fig.6!, and that at the transitions corresponding to capillary cdensationsG l ; logL ~see Fig. 7!.
IV. LANDAU–GINZBURG MODEL
Close to the continuous transition the amplitude of tdensity oscillations in the lamellar phase behaves as;u(t2tT)/tTu1/2 in the MF approximation, wheretT is the tran-sition temperature.21 Hence, by definition~7!, l also van-ishes as ;u(t2tT)/tTu1/2 at this transition. At themicroemulsion-lamellar phase coexistence,l has a disconti-nuity. Close to the coexistence@see Fig. 4~b!# the shape ofl(z) strongly resembles the density profile in simple fluid7
r
FIG. 5. The vicinity of the first-order transition from the lamellar phasethe microemulsion (t52.8,m54.3203,c54,uDmu50.0008). Walls are notpreferential for any statei , i.e., hi521 for i 51,2,3,4.~a!: The density ofthe surfactant as a function of a distance from the wall in units of the latconstant.~b!: Lamellar OPl as a function of a distance from the wameasured in units of the period of the lamellar structure.
in-g
tnd
-dtin
llanmin
ic
r--in
er--
-
allsn
e-
il–the
ery
7552 J. Chem. Phys., Vol. 110, No. 15, 15 April 1999 M. Tasinkevych and A. Ciach
in the vicinity of the phase-coexistence above the wetttransition. The shape ofG l (Dm) is the same as the corresponding shape ofG(Dm) in the case of continuous wettinin simple fluids~see Fig. 6!, and the dependences ofG l andG on the size of the system at capillary condensations aresame too~see Fig. 7!. The analogy between our system asimple fluids in the semi-infinite geometry,l(z)↔r(z),G l ↔G, suggests thatl is a proper OP for the lamellar ordering. Since we study the phase transition between theordered and ordered phases, which in general can be conous or first order, the Landau–Ginzburg~LG! functional oflin the semi-infinite geometry should have the form7
V@l~z!#5E0
`
dzF1
2S dl
dzD2
1Al21Bl41Cl6G2H1l~0!1H2l~0!2. ~9!
The length unit is chosen equal to the period of the oscitions. In this length scale the system is already uniform awe are able to reduce the problem of wetting in the systewith the mesoscopic structure to the problem of wettingsimple fluids. On the other hand, the functional~9! does notdescribe the mesoscopic structure of microemulsion, whis already ‘‘integrated out.’’
FIG. 6. Excess lamellar OPG l vs log(Dm) calculated along the patha ~see.Fig. 1! in the case of water-covered walls.Dm is the difference betweenmand its value at the bulk phase transition.
FIG. 7. Excess lamellar OPG l at the transitions corresponding to capillacondensations as a function of log(L). L is a distance between the walls.
g
he
is-u-
-ds
h
To relate the phenomenological parametersA,B,C to thethermodynamic variables of the CHS lattice modelt,m,c,we first introduce continuous approximation for the MF themodynamic potential~4!. A procedure of deriving the continuous approximation for the CHS model is describedRef. 22 forD53 (D is the lattice dimension!. For the lamel-lar order the resulting functional depends on the three ordparameter fields:f(z)—the concentration difference between oil and water,r(z)—the deviation of the local densityof surfactant,r3(z)1r4(z), from the average valuers andthe fieldu(z)5r3(z)2r4(z) describing the orientational ordering of the surfactants.u(z).0 if at the pointz the headsof the amphiphiles are mainly oriented toward the left wandu(z),0 if the direction of the majority of amphiphiles ireversed.u(z)50 in the case of no orientational order. Thethe functional assumes the form
V@f,r,u#5V2@f,r,u#1V int@f,r,u#. ~10!
Here, forD51, the Gaussian part22 is
V2@f,r,u#5E S 1
4~~]zf!21~]zr!2!1a1f21a2r2
1a3~~]zu!21u2!22cu]zf Ddz. ~11!
The length unit in Eq.~10! is by construction equal to thelattice constanta. All the coupling constants are given in thAppendix @Eq. ~A1!#. Amphiphilic interactions are represented by the term;2u]zf, which favors the configura-tions with properly oriented amphiphiles located at the owater interface. The effect of this term is the stronger,stronger the amphiphilic interactionsc.
V int can be written in the compact form
V int@fa#5E S (a<b
(n1m53
6
aabn1mfa
nfbmD dz, ~12!
where we introduced the notationf5f1 ,r5f2 ,u5f3 .Close to the bifurcation linet5tT(rs),f1 ,f35O(e), andf25O(e2),22 wheree;u(t2tT)/tTu1/2. Since for arbitraryt,a1,a2 @see Appendix, Eq.~A1!#, the instability of theuniform phase is induced byf1 andf3 . Using the Fourierrepresentation forfa , and minimizingV with respect tof2(k) andf3(k), we can expressf2(k) andf3(k) in termsof f1(k) and we obtain
V5E ~e2A2~ k!1e4B2~ k!!uf1~k!u2dk1E dS (n51
4
knD3~A4~$kn%!1e2B4~$kn%!!)
n51
4
f1~kn!dkn
1E dS (n51
6
knDA6~$kn%!)n51
6
f1~kn!dkn1O~e8!,
~13!
whereuku252Aa1 is the wave number corresponding to thbifurcation. Expressions forAn ,Bn are given in the Appen-
frit
s
ite
p-b
ar
t
fo
olfb
al
-s
on
le
sit
iodcanthedel
theg in
R.spe-das.
22,-dif-
7553J. Chem. Phys., Vol. 110, No. 15, 15 April 1999 M. Tasinkevych and A. Ciach
dix @Eqs.~A3!, ~A4!, ~A5!, ~A7!#. The Fourier amplitudes othe lamellar structure just below the bifurcation can be wten in the form
f~k!5F~d~k2 k!1d~k1 k!!. ~14!
Inserting the above into Eq.~13! gives
V/V5w2F21w4F41w6F61O~F8!. ~15!
V is the volume of the system. For the sinusoidal shaper i(z), that is just below the second-order transition line,ltakes the form
l~z!5Ez
z1pS f2
21
u2
21r2D 1/2
dz. ~16!
The relation betweenf and u just below the bifurcation~obtained by minimizingV with respect tou) is given byu25f2k2c2/a3
2(11 k2)21O(e3). Thus l2.F2(11 k2c2/a3
2(11 k2)2), and neglecting higher order terms we can wrA[w2(11 k2c2/a3
2(11 k2)2)21, B[w4(11 k2c2/a32(1
1 k2)2)22, C[w6(11 k2c2/a32(11 k2)2)23. With the above
equalities the functional~9! describes the near-surface proerties of ternary surfactant mixtures in terms of measuraparameters, a single phenomenological parameterc related tothe amphiphilicity of surfactant, and two parameters charterizing the surface (H1 and H2). The expressions fown ,n52,4,6 @Appendix, Eqs.~A9!, ~A10!, ~A11!# allow usto express the results obtained for the simple functional~9!in terms of t,rs , and c. For example, the tricritical poingiven byw25w450 can be found in the (rs ,t) phase spacefor various surfactant strengthsc.
V. SUMMARY AND CONCLUSIONS
The functional~9! was extensively studied7,9 in the con-text of simple fluids. Therefore, all the results obtainedsimple fluids in the semi-infinite geometry7,9 also hold in thecomplex systems such as microemulsions, lipids, or copmers, but with themesoscopiclength unit. The results othese studies, relevant for the structured fluids, are listedlow.
~1! Surface-induced ordering~SIO! and disordering~SID!transitions were found in the phase space (H1 ,H2).9 Thetransitions can be either continuous or first order.
~2! The thickness of the ordered~or disordered! film on theapproach to the phase-coexistence grows logarithmicwith the inverse distance from the transition.
~3! For the system of the widthL at the capillary condensation, the thickness of the ordered surface film behave; logL.
~4! The analytical expressions forl(z) are given in Ref. 7for various situations.
~5! Near the continuous transition the critical adsorptitakes place,6 and G diverges as;tb2n with b and nbeing the standard critical exponents.
The predictions of the LG model~9! listed above are alin very good agreement with the results of the CHS modOn the other hand, the LG functional ofl is not capable ofproviding the detailed information about the shape of den
-
of
le
c-
r
y-
e-
ly
as
l.
y
profiles. In particular, the effect of the surface on the perof density modulations in lamellar or hexagonal phasesonly be studied within models that are able to describemesoscopic structure. By considering the lattice CHS moand the new LG functional ofl, we provide both the de-tailed information about the near-surface order as well asgeneral conclusions concerning surface-induced orderinstructured fluids.
ACKNOWLEDGMENTS
We would like to thank Professor G. Findenegg, Dr.Steitz, and Professor S. H. Chen for discussions, and ecially Dr. A. Maciołek for many valuable discussions ancomments at the early stage of this work. This work wpartially supported by the KBN Grant No. 3 T09A 073 16
APPENDIX
Because the calculations proceed exactly like in Ref.where expressions forw2 ,w4 were obtained for threedimensional systems, we present only the results of theferent stages of calculation.
~i! Coupling constants ofV2:
a15c1t21
2, a25
~c11c2!t21
2, a35
c2t
2, ~A1!
wherec151/(12rs) andc251/rs .~ii ! Coupling constants ofV int :
H a223 5
~c122c2
2!t
6, a12
2115c1
2t
2,
a2311252
c22t
2 J ;H a114 5
c13t
12, a22
4 5~c1
31c23!t
12,
a334 5
c23t
12, a12
2125c1
3t
2, a23
2125c2
3t
2 J ;
H a225 5
~c142c2
4!t
20, a12
4115c1
4t
4,
a2311452
c24t
4, a12
2135c1
4t
2, a23
31252c2
4t
2 J ;
H a116 5
c15t
30, a22
6 5~c1
51c25!t
30, a33
6 5c2
5t
30,
a124125
c15t
2, a23
2145c2
5t
2, a12
2145c1
5t
2,
a234125
c25t
2 J . ~A2!
e25(a3/41a1a32c2)/Aa31Aa1a3 and e250 givesthe bifurcation line.
~iii ! Coupling constants in Eq.~13!:
A2~k!5k2Aa3
a3~k!, B2~k!5
1
a3~k!S 2k2a3
a3~k!21D . ~A3!
Herea3(k)5a3(11k2).
7554 J. Chem. Phys., Vol. 110, No. 15, 15 April 1999 M. Tasinkevych and A. Ciach
A4~$kn%!5V4~$kn%!2V2~k1 ,k2!V2~k3 ,k4!
4a2~k11k2!, ~A4!
B4~$kn%!5S 8Aa3
a3~k!2
2
Aa1a3D S V4~$kn%!2
c13a3
6c2D
2V2~k1 ,k2!V2~k3 ,k4!
8Aa3a22~k11k2!
~k11k2!2
k1k2
2S 4Aa3
a3~k!2
1
Aa1a3D
3V2~k1 ,k2!~V2~k3 ,k4!2c1
2a3 /c2!
2a2~k11k2!, ~A5!
here
a2~k!5a21k2
4,
V2~k1 ,k2!5a3
c2S c1
21c2
2c2k1k2
a3~k1!a3~k2!D , ~A6!
V4~$kn%!5a3
6c2S c1
31c2
3c4k1 • • • k4
a3~k1! • • • a3~k4!D .
Further,
A6~$kn%!5V6~$kn%!2V2~k1 ,k2!
2a2~k11k2!W4~k3 , . . . ,k6!
1V2~k1 ,k2!V2~k3 ,k4!
2a2~k11k2!a2~k31k4!
3W3~k31k4 ,k5 ,k6!2~c1
22c22!a3
24c2
3V2~k1 ,k2!V2~k3 ,k4!V2~k5 ,k6!
a2~k11k2!a2~k31k4!a2~k51k6!, ~A7!
where
W3~$kn%!52c1
3a3
c21S a3
a3~k11k2!21D 2c2
2a3c2k2k3
a3~k2!a3~k3!,
W4~$kn%!5c1
4a3
2c21S 2a3
3a3~k11k21k3!2
1
2D3
c23a3c4k1 . . . k4
a3~k1! . . . ~k4!, ~A8!
V6~$kn%!5c1
5a3
151S a3
9a3~k11k21k3!2
1
15D3
c24a3c6k1 . . . k6
a3~k1! . . . ~k6!.
And finally,
~iv! coefficients in the expansion ofV in powers ofF @Eq.~15!#:
w252e2
a3~ k!H k2Aa31e2S 2a3k
a3~ k!21D J . ~A9!
w456V4($k%)2V2
2~ k,k!
2a2~2k!2
V22~ k,2 k!
a2
1e2H S 8Aa3
a3~ k!2
2
Aa1a3
D S 6V4($k%2c1
3a3
c2D 2
V22~ k,k!
Aa3a2~2k!2
2S 4Aa3
a3~ k!2
1
Aa1a3
D FV2~ k,k!~V2~ k,k!2 c12a3 /c2!
a2~2k!1
2V2~ k,2 k!~V2~ k,2 k!2 c12a3 /c2!
a2G J . ~A10!
w654c1
5a3
3c2
12c2
4a3
3S ck
a3~ k!D 6H 22
a3
3 S 1
a3~3k!1
9
a3~ k!D J 22a3
V2~ k,k!
a2~2k!H c1
4
c2
1c23S ck
a3~ k!D 4
3F12a3
3 S 1
a3~3k!1
3
a3~ k!D G J 2
a3
a2
V2~ k,2 k!H 3c14
c2
2c23S ck
a3~ k!D 4F32
4a3
a3~ k!G J 12a3
V2~ k,k!2
a2~2k!2
3H 2c13
c2
1c22S ck
a3~ k!D 2F22a3S 1
a3~3k!1
1
a3~ k!D G J 18a3
V2~ k,k!V2~ k,2 k!
a2a2~2k!H c1
3
c2
2c22S ck
a3~ k!D 2F12
a3
a3~ k!G J
18a3
a22
V2~ k,2 k!2H c13
c2
1c22S ck
a3~ k!D 2F12
a3
a3~ k!G J 2
~c12c2!2a3
c2a2
V2~ k,2 k!H V2~ k,2 k!2
3a22
2V2~ k,k!2
2a2~2k!2J . ~A11!
In the above expressionsk252Aa1 denotes the wave number corresponding to bifurcation.
a,r,
a B
7555J. Chem. Phys., Vol. 110, No. 15, 15 April 1999 M. Tasinkevych and A. Ciach
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