VOLUME 76, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 1 JANUARY 1996

Self-Consistent-Field Lattice Gas Model for the Surface Ordering Transition ofn-Hexadecane

F. A. M. Leermakers and M. A. Cohen StuartDepartment of Physical and Colloid Chemistry, Wageningen Agricultural University,

P.O. Box 8038, 6700 EK Wageningen, The Netherlands(Received 19 May 1995)

It is known from experiments that the solid-liquid phase transition ofn-alkanes involves surfacefreezing. On cooling, a monomolecular surface layer crystallizes first. Upon further cooling, the bulksolidifies. The orientational ordering aspect ofn-hexadecane freezing is modeled with a self-consistent-field lattice gas theory, showing that both events are first-order transitions. It is assumed that the CH3

segments are more surface active than the CH2 ones. This initiates at highT a slight orientational bias,but at lowT a cooperative alignment of the C16 perpendicular to the interface.

PACS numbers: 68.10.–m, 31.15.Ne, 61.25.Em, 64.70.Dv

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The behavior of surfaces in the vicinity of a solidliquid phase transition has lately attracted interest frthe scientific community. Since molecules situated ingas-liquid or gas-solid interface usually experience lattraction than molecules in the bulk of the dense phaone anticipates that the interface is a region with increaentropy density. One therefore expects the surface tento decrease with temperature, and this is commonly fouIn addition, careful studies of the solid surface hashown that the surface often melts a few degrees bethe bulk. For example, water, a common but not onethe most simple liquids, has a freezing point at 273.15(at p ­ 1 atm). A surface layer of ice remains liquidlikbelow this temperature [1,2], which is (partly) why itpossible to skate on it.

Recently there has been accumulated experimeevidence that the opposite occurs forn-alkanes of highenough molecular weight (range C14 to C50) [3,4]. Here,a surface freezing was found which shows that a liquidlbulk, at well-controlled temperatures near the freezpoint, was capable to support a monomolecular sosurface layer in contact with its vapor. In this Letter wfocus on the question why this phenomenon occursshow that a simple lattice gas model can well accountseveral of the experimental findings.

The theory that is used is based on the inhomogeneself-consistent-field (SCF) approach initiated by Schejens and Fleer for the problem of polymer adsorptionsolid-liquid interfaces. This theory has recently beenviewed in detail [5]. It makes use of a lattice that servas a system of coordinates onto which segments aresitioned. In its simplest form, the theory employs meafield approximations in each lattice layer and first-ordMarkov chain statistics. The theory has been extento study the self-assembly of phospholipids into flatlayer membranes by Leermakers and Scheutjens [6].new element introduced here was the use of bond ortation weighting factors, which could lead to cooperatalignment. The extended theory correctly predicts thafirst-order so-called gel to liquid phase transition exi

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in membranes, where the alkyl chains of the bilayerfrom a high temperature liquidlike disordered state intolow temperature ordered one. Since surface freezing galong with orientational ordering, we decided to use tversion of the SCF theory for the present problem. Belthe term “surface freezing” is used, but it is understothat our model only generates the orientational orderaspects of this phenomenon [7].

It is useful to mention a few characteristics angive the basic equations of this theory. As told abovto reproduce the correct main phase behavior in lipbilayers, it was shown that it is essential, for findinthe probability for unoccupied sites, to go beyond ttraditional mean-field approximation. The probabiliPszd of placing a segment of a molecule in a layerzis commonly taken to bePszd ­ 1 2 wszd where wszdis the fraction of already occupied sites. The next leof approximation is based on the idea that, at hidensities, bonds tend to align parallel, because inconfiguration they do not intersect. It is thus necessto consider pairs of segments; in other words, relatbond directions are also taken into account in calculatP. For lattice models this level of approximation wapioneered by DiMarzio who studied rigid rods on a latti[8,9]. A soon as intermolecular and intramolecular boorientation correlations (excluded volume) are inserinto the weighting factors, one obtains a kind of chastatistics that features cooperative alignment transition

Chain conformations are sampled in a rotational ismeric state (RIS) scheme. This is a third-order Markapproximation where short-range correlations alongchain with gauchesg1, g2d and trans std configurationsare taken into account over a distance of four segme(three bonds). So-calledg1g2 sequences are not excluded, nor do we prevent double occupancies of lattsites by segments that are more than three bonds aTwo (indirect) corrections for the excluded volume prolem are included as discussed below. The energyference between agaucheand atrans configurationUGT

controls the chain stiffness.

© 1995 The American Physical Society

VOLUME 76, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 1 JANUARY 1996

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We consider a condensed phase of linear alkanCH3-(CH2)N-2-CH3 (chain lengthN , denoted by CN ) inequilibrium with a phase rich in free volume (vaporFree volume is modeled as a monomeric componentV1.There are thus three types of units and consequently thshort-range Flory-Huggins interaction parametersxsx, ydthat characterize the system:x, y ­ CH3, CH2, V1. It isrelevant to mention that these interaction parametersbe measured experimentally.

A segmentx is conjectured to probe a dimensionlessegment potentialuxszd of the type

uxszd ­ u0szd 1X

yxsx, yd fkwyszdl 2 wb

y g , (1)

where u0szd is the so-called excluded-volume potential. It is related to the Lagrange parameter coupledthe density constraint

Px wxszd ­ 1, introduced in opti-

mizing the partition function. The site-average densiis given by kwszdl ­

Pa­21,0,1 lawsz 1 ad ø wszd 1

l1≠2wszdy≠z2. We have usedl21 ­ l0 ­ l1 ­ 1y3, achoice that minimizes so-called lattice artifacts as it alows more easily the placing of a sharp interface onto tlattice than in, e.g., a cubic lattice. In Eq. (1),wb

x is thebulk concentration of segment typex. We take the bulk asthe phase with low alkane density (vapor phase) far awfrom the interface. In the bulk no inhomogeneities onlarger length scale than the lattice sites are present. Fsegment distribution functionsGxszd ­ expf2uxszdg aredefined and generalize such thatGsz, sd ­ Gxszd whensegments is of typex.

The density profiles are computed by an efficiepropagator formalism. This formalism is equivalent witgenerating all allowed conformationsc of a test chain inthe field of all other chains and solvent molecules. Tstatistical weight of each conformation is computed aa summation over configuration space is carried throuA conformationc is fully specified when the coordinateof all segments are known. The numbernc of chains in aconformationc normalized by the number of lattice sitein each layerL is given by

nc

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NYs­1

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(2)where C is a normalization constant. For a conformation c not only the coordinatez of any segments isknown, but also the direction of each bonds. Themultiple product overs gives the overall Boltzmannweights of the segments in conformationc. The sec-ond product runs over all bonds. Below an expressifor the orientation dependent weighting factors for plaing bondss, Gcsz, sd, is given. Finally,lcs accountsfor the gaucheand trans probabilities relevant for con-formationc; lcs ­ lgauche ­ 1yf2 1 expsUGT ykBT dg ifbondss 2 1 ands 1 1 point in different directions andlcs ­ 1 2 2lgauche ­ ltrans otherwise. The energy dif-

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ference,UGT , between a localgaucheand atrans config-uration is experimentally accessible. The two quotienwithin brackets account for placing the firsts ­ 1 andlasts ­ N 2 1 bonds, respectively. Generating all conformationsc and applying Eq. (2) gives access to bothe density distributionwxszd and the density distribu-tions of bonds. These last quantities are collected intwo profiles:w2szd for bonds within a layer andwlszd forbonds between layersz andz 1 1. From these the orien-tation dependent weighting factors are found. If bonds

in conformationc is between layersz andz 1 1, we writeGcsz, sd ­ Glszd and in other casesGcsz, sd ­ G2szdwhere

Glszd ­1 2 wlb

1 2 wlszd, (3a)

G2szd ­1 2 w2by2

1 2 w2szdy2. (3b)

In Eq. (3b) it is assumed that bonds in a layer distribuisotropically between two directions, so that the probabity that a bond in a given direction isw2szdy2. FromEq. (3a) it can be seen that whenwlszd ! 1 the weight-ing factor wlszd rises steeply, which provides the cooperativity needed for the freezing transition. A real divegence does not occur due to finite chain length effects (number of bonds is one less than the number of segmein a chain). The density distribution of the vacanciesfound also by Eq. (2), which reduces for monomerswV szd ­ w

bV GV szd. From the set of conformationshncj it

is possible to generate intramolecular density-density crelation functions, which can be used as a characterisof the average conformation of the molecules. This is ndone in this Letter.

The above set of equations is solved numerically (to 7 significant digits) with reflecting boundaries on bosides of the system. The amount of C16 is fixed [thisspecifies the constantC in Eq. (2)], so that the interfacebetween the condensed alkane and its vapor is displafrom the boundaries.

Results are reported for the orientational orderingC16 mixed with a monomeric componentV1 (vacancies).The Flory-Huggins (FH) interaction parameters are fixratherad hoc to xsCH3, CH2d ­ 0.5, xsCH3, V d ­ 1.0,and xsCH2, V d ­ 1.6. For simplicity it is assumed thatthese parameters are not a function of the temperat[10]. The xsCH2, V d, and to a lesser extendxsCH3, V d,parameter is such that the concentration ofV1 in thehexadecane phase is near 6% in the liquid state. Tlow value is needed to obtain a high density of bondsthat the system is easily triggered to go through the solliquid transition. A much higher vacancy concentratioputs the system too far from this transition point. Threpulsive xsCH3, CH2d interaction gives some internadriving force for orientation of the molecules, significanfor the bulk freezing, but more important is the differenc

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VOLUME 76, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 1 JANUARY 1996

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in repulsion for theV phase between the CH2 and CH3

units. The latter repels theV phase slightly less than thformer. The remaining parameterUGT is used to drive thesystem through its transition. For convenienceUGT ­ 1at T ­ 300 K for any sequence of four segments in thchain. We note that the temperature scaleT ­ 300yUGT

is only qualitatively correct. The system is 60 latticlayers in size and numbered arbitrarily (z ­ 0 is nearthe interface,z , 0 is the vapor, andz . 0 is the alkanephase). It has reflecting boundary conditions and contain all cases 40 equivalent monolayers of hexadecand thus 20 equivalent monolayers of free volume. Tdimension of a lattice cell is equivalent to the size ofC-C bond.

Some typical density profiles are shown for thhexadecane-V interface for three different temperaturecharacteristic for the liquid, liquid plus surface layeand solid in Fig. 1. The interfacial width (the intrinsiwidth) is only a few lattice layers wide (capillary wavexcitations are not included) and does not change mas a function ofT (FH parameters are fixed). For higtemperature only a modest orientation of the C16 at theinterface is noticed. TheV fraction in the fluid is almosthomogeneous throughout this phase. This changesmatically for the intermediate temperature range. Ha monomolecular surface layer has fully oriented. Tcan be seen from the drop in theV fraction by a factorof 10. The CH3 profile shows peaks on both sides

FIG. 1. Segment density profile across aV -C16 interface.Parameters are given in the text. Profiles of theV , CH2, andCH3 units are given for three temperatures as indicated.

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this highly ordered region at the expected distance awfrom the interface, and inspection of the orientationthe molecules shows that indeed the molecules infirst layer are nearly all in an all-trans configuration (notshown). The alignment is not perfect; the probabiliof finding gauchedefects is highest near the ends of thmolecule. The CH3 groups that accumulate on the C16

side of the frozen layer cause a local maximum of the frvolume in the condensed phase. This is the nucleatsite for further ordering of the rest of the hexadecane.the lowest temperature we see that indeed a second lahas solidified. Because of the finite size effects of our bthe third layer could not reach a high level of orderinAt this stage we have no information whether a thirdsubsequent layer order at the same temperature assecond one, or whether these layers “freeze” at slighlower temperatures. Important for a potential cascaof surface ordering phase transitions is the amountsurface-induced orientational bias that is transferred froone (ordered) alkane layer to the next (fluid) one. Tdecay lengthj of this alignment transfer is not expecteto be long for alkanes.

In Fig. 2 the proof is presented that both the surface athe bulk freezes by first-order phase transitions. In thfigure the amount of free volume on variousz coordinatesin the condensed part of the system is given as a functof the temperature. The layersz ­ 8 half-way the firstand z ­ 23 half-way the second C16 layer have beenselected for this. As long as the amount of free volumis as large as 6% the local environment is fluidlikwhereas a solidlike state is characterized by a dramareduction of vacancies (at least a factor of 10 for thfirst layer and slightly less than a factor of 10 for thsecond one). From Fig. 2 it can be seen that the surflayer freezes atT ­ 260 K and the bulk atT ­ 255.5 K.Consistent with experimental findings, it was possibledetect hysteresis regions for both transitions ranging oa few tenths of a degree [11]. It is noteworthy that athe first layer solidifies the density in the liquid buladjoining the surface layer is affected. Perhaps mo

FIG. 2. The amount of free volume [wV ­ 1 2 w(alkane)] atthe midpoint of the first monomolecular layer next to theVphase,z ­ 8, and atz ­ 23 (second layer) as a function of thetemperatureT . The arrows indicate the main transition pointsOnly part of the van der Waals loops can be calculated by onumerical procedure.

VOLUME 76, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 1 JANUARY 1996

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surprisingly, when the second layer solidifies, the densin the first layer is affected as well; clearly the first layecompacts even more, and the orientational order ofchains increases. These density changes are importaestimatej and to judge how many subsequent orderin(freezing) transitions might be expected. Furthermothere is a critical value of alkane density at which thordering sets in. The fluid both in the first layer andthe next layer become unstable and then solidify asfraction of V drops belowwV ­ 0.055. Again, this canbe controlled not only by the chain stiffness but also bthe appropriate FH interaction parameter. It is alsoconsiderable interest to mention that the density in the fifrozen layer is significantly higher than in the second onThis is because the CH3 groups cause the alkanes to orieat the interface, and this helps to reduce the numberdefects. In the second layer there are many more defewhich may partly be due to the inability of the third layeto solidify.

The results shown above show good resemblance wexperimental data [3,4]. In experiments the freezing pofor C16 is at higher temperatures. In our model the poiat which surface ordering is found increases by, e.increasing the FH parameter for CH2 and V contacts.More important is the difference between the bulk ansurface freezing temperatures, which is in practice onlythe order of 1 K. The too large gap between the surfaand bulk ordering is introduced in the model by a slightoverestimated difference in surface affinity between tCH3 and CH2 groups.

In this Letter we have concentrated on the C16 sys-tem. To extend our model to cover the full range of chalengthsN it is necessary to choose, in addition to the temperature dependence of the chain stiffness, an approate scaling for the FH parameters with temperature, ex , T21. Then interesting issues like the chain lengdependence of the difference between the bulk and sface freezing can be addressed. Moreover, it is then psible to examine why there is a lower limitN , 14 andan upper limitN , 50 for this peculiar freezing phenom-enon. Work along this line is in progress.

Related to the orientation of the alkane molecules atinterface is that those alkanes have interesting autophowetting behavior. This has been noted before by oth[3], and is convincingly reproduced by our calculations.

In summary, we developed a theory based upon the idthat two effects are important for alkane surface freezinFirst, there must be a tendency for the chains to beco

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rigid and aligned in a dense packing. Furthermore, thmust be a driving force for the enrichment in the surfaof one particular kind of group, which we supposedthe CH3 group. It is shown that a simple SCF lattice-gamodel can capture these aspects, and a proof is given-hexadecane indeed can develop a highly orientedordered surface layer. The surface-induced 2D orderis known to be present for other molecular weigh(according to experiments for alkanes between C14 andC50), and for molecules such as those forming liqucrystals. Our model can be used to study these systas well.

[1] D. Beaglehole and P. Wilson, J. Phys. Chem.97, 11 053–11 055 (1993).

[2] A. Lied, H. Dosch, and J. H. Bilgram, Phys. Rev. Lett.72,3554 (1994).

[3] X. Z. Wu, E. B. Sirota, S. K. Sinha, B. M. Ocko, and MDeutsch, Phys. Rev. Lett.70, 958–961 (1993).

[4] X. Z. Wu, B. M. Ocko, E. B. Sirota, S. K. Sinha, M.Deutsch, B. H. Cao, and M. W. Kim, Science261, 1018–1021 (1993).

[5] G. J. Fleer, J. M. H. M. Scheutjens, M. A. Cohen StuaT. Cosgrove, and B. Vincent,Polymers at Interfaces(Elsevier, London, 1993).

[6] F. A. M. Leermakers and J. M. H. M. Scheutjens, J. ChePhys.89, 6912–6924 (1988).

[7] In a lattice model the in-plane crystalline order is set bthe lattice type used. Noa priori information is generatedon the in-plane crystalline ordering of the solid alkanphase.

[8] E. A. DiMarzio, J. Chem. Phys.35, 658 (1961).[9] E. A. DiMarzio, J. Chem. Phys.66, 1160 (1977).

[10] Any reasonable temperature dependence for theparameters will keep the transitions as discussed intIt will, however, influence details such as the differencbetween the surface and bulk freezing temperaturUsually, however, the correct influence of the temperatuon xsx, yd is not known as it may contain both entropiand enthalpic contributions.

[11] Calculations near the transition zone are met with cosiderable difficulties. The freezing of the bulk was modifficult to calculate as this event causes a shift (for tnumerical procedure catastrophic) of the interface and tha shift of the first layer to slightly higherz values. It wastherefore not possible to exactly “measure” the sizes ofvan der Waals loops of the transitions. A detailed analyof the transitions is not the main goal of this Letter.

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