Ordering of paraffin-like molecules at the solid—fluid interface

25 March 1994

CHEMICAL

PHYSICS LETTERS ELSEVIER

Chemical PhysicsLetters 220 (1994) 53-58

Ordering of paraffin-like molecules at the solid-fluid interface X i a n g - Y a n g L i u a, p . B e n n e m a a, L . A . M e i j e r b, M . S . C o u t o a "RIM, Laboratory of Solid State Chemistry, Faculty of Science, University of Nijmegen, Toernooiveld, 6525 EL) Nijmegen, The Netherlands b Department of Physicul and Colloid Chemistry, Wageningen, Agricultural University, Dreijenplein 6, 6703 ItB Wageningen, The Netherlands

Received 2 August 1993; in final form 26 January 1994

Abstract

Structures of the fiquid molecules at the solid-fluid interface for paraffin-like molecule solution systems are studied via calculations of self-consistent field lattice models. Periodic oscillations of the segmental density are directly associated with the length of molecular chains. Molecules in the first and sometimes the second liquid layers lie preferentially parallel to the surface, due to interfacial effects. Comparisons between the calculated data and the results obtained from surface roughening experiments are made for several n-paraffin systems.

The structure and physical properties of solid-fluid interfaces are of fundamental and technological interest. This subject has been studied from both experimental and theoretical points of view [ 1-8 ]. Early theoretical studies of interracial systems were mainly concentrated on solid substrates in contact with simple fluids [ 1-4 ]. Recently attention has focused on solid-fluid systems of complex molecules [5-8]. In this study, we use self-consistent field lattice model calculations [ 9-13 ] to investigate ordering trends of fluid units at the solid-fluid interface. In addition, the influence on some interfacial properties of chain-like molecule systems is investigated. We mainly focus on paraffin molecule systems. The results obtained are compared with experimental data of the roughening phase transition. Within the framework of lattice models, the space is partitioned into ceils with the same size and shape. Each cell should be •led with either a solute unit or a solvent unit. It has to be noted that unlike tradi-

tional homogeneous lattice models, the lattice model used in this work is a so-called inhomogeneous lattice model developed recently [ 14 ]. This model implies that physical properties of a structural unit in the solid or in the fluid phase are a function of the distance (z) from the unit to the dividing surface between the solid and the fluid phase. Using this model, the nature of bond energies and the ordering of fluid units at the solid-fluid interface can be examined [ 14 ]. In our calculations, paraffin molecules are described as chain-like molecules formed by segments of the same size, which are connected to each other. (Each segment represents a CH2 or a terminal CH3 group and occupies one cell.) The system consists of paraffin solutions in contact with paraffin solid substrates. I f not specified, the orientations of static substrates are set in the orientation of (001 ) of the hexagonal lattice. (This is because n-paraffin crystals which we investigated have a pseudo-hexagonal structure in the orientations of { 110}. ) Since in the system van der Waals interactions are dominant, only

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54

X.-Y. Liu et el./ChemicelPhysicsLetters 220 (1994) 53-58

the nearest neighbor interactions need to be considered. For solutions consisting of two different n-paraffins, those with a longer carbon chain arc regarded as the solute (denoted by A), and those with a shorter carbon chain as the solvent (denoted by B). The exchange energy per segment between n-paraffin A and n-paraffin B is supposed to be zero. (This is the socalled good solution approximation.) The interaction energy between a segment of a molecule i ( i = A or B) and a surface site is expressed by the FloryHuggins parameter Z~(i = A or B). This corresponds to the energy change (per segment) (in units of k T : k is the Boltzmann constant, T is the temperature) due to bringing a segment of molecule i from the pure liquid state i into an environment of the pure solid state s. Note that any change of surface structure may cause a change in the number of solid-fluid bonds crossing the dividing surface between the solid and the fluid phase. For a given ;t~, this will lead to different interaction energies between solid and fluid units at the surface. This is the reason that we have to fix the surface structure in the beginning. In the case of crystal growth, the substrates are crystal surfaces. Then Z,~ is directly related to the enthalpy of melting or dissolution. For orthorhombic n-paraffin crystals, X u ~ - 1 . 5 4 + 6 . 0 / n (n>~5) [15]. Here n is the carbon number of a paraffin molecule, and X,~is referred to To= 298.15 K. (Note that differences between CH2 and CH3 groups result in different X,~values. For simplicity, Z,~ used in our calculations is statistically averaged over the value of CH2 and of CH3 in a paraffin chain. Therefore, its value depends on the length of molecular chains.) The calculations were carded out on the basis of formalisms first developed by Scheutjens and Fleer et al. (the so-called SF theory [9-12] ). This theory is based on the same principle as the inhomogeneous lattice model, and the Bragg-Winiams (or mean field) approximation is applied in a segmental scale to calculate the density of segments. This implies that regularly spaced osciUations (or fluctuations) in the dimension of one segment [ 7 ] were neglected. It follows that the average of oscillatory profiles is obtained in our calculations. Generally speaking, the interfacial structure and profiles of the density in the direction perpendicular to the solid surface can in

principle be calculated on the basis of the SF theory. For further details, see refs. [9-12] #1 Segmental density profiles (expressed by the volume fraction ~A(Z)) of n-C21 in n-C6 solutions, for different torsional energies of molecular chains d °~in a constant temperature ( T = To= 298.15 K), plotted versus distance z (z is normalized by the interplanar distance dhk2) from the first fluid layer ( z = 0 ) adjacent to the solid surface, are displayed in Fig. la. (The torsional energy is due to the energy difference be~zThe calculations oftbe SF theory can be carried out using the computer program called Goliath, which has recently been developedby the Department of Physics and Colloid Chemistryof Wageningen University. 0.70

0.00

0.50

vO.40

¢

-

-- --..... .......

0.30

0.20

eta'= ctor= ~t~r= et°r= etoT__ etch=

lOOkTo 4kTo 3kTo 2kTo lkTo 0kTo

o.10

(a) o.o

5.o

10,o

10.o

z (,~)

20.0

25.0

0.125

o.12o

0.1.15 _ _ _ __ _ _ ___

~0.110

0,105

T T T T

-= = --

300 350 400 450

K K K K

0.100 (b) O.0Ofi

0.090 0.0

0.0

10.0

10.0

20.0

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~lO.O

z (n) Fig. 1. (a) Segmentdensity profiles ~^(z) of n-C21moleculesin n-C~solutions, for differenttorsional energiesof molecularchains eTM,plotted versus distance z from the solid surface.The segment density in the bulk ~^=0.1, the temperature is constant (TfT0=298.15 K). (b) Segmentdensity profilesof n-C2sin nC,2 solutions for different temperatures in the case that e~ remains unchanged, et°rffi2kTo, ~A = O. 1.

X.-Y. Liu et aL / Chemical PhysicsLetters 220 (1994) 53-58

tween the trans conformation and the gauche conformation.) Two different profiles of the segmental density can easily be distinguished, according to two torsional energies et°'=0 and et°r=100kTo (To= 298.15 K). Those two energies correspond to two extreme cases: molecules with completely flexible chains and molecules with completely rigid chains. It can be seen that for completely flexible chains the segmental density profile of the solute shows an exponential decay with distance z away from the solid surface. However, for molecules with completely rigid chains, an oscillatory decreasing behavior is shown (c.f. Fig. 2). The density of segments of solute molecules drops drastically after the first fluid layer. It follows that a depletion of solute occurs between roughly the second fluid layer and the in'st maximum of the profile. When the torsional energy etor is reduced from etor= 100kTo to et°~=0, the oscillatory profile of the segmental density is graduatly replaced by the exponential profile. In case e t o ~ lkTo, the oscillation can hardly be identified from the profiles. In the case of normal alkanes, eto, is about 2kTo [ 16 ]. Therefore the oscillation proirfle could be recognized according to Fig. la. On the other hand, for a system having all energy parameters fLxed, the oscillatory profiles will 0.I2

O.lO

.__ ,L~........... t "~

.~ ~ o.o0

~///

0,06

/I " xJi /

---.....

! 2

--

--

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0.03

--

6

)~min

,

0.0

. . . . . . . .

r , , , , , , , , , i

10.0

. . . . . . . . .

20,0

z(,~)

, , , , , , , , , ,

30.0

40,0

Fig. 2. Segmentdensityprofilesin dependenceof the structuroof chain molecules.The structureof substrates:hexagonal;~x=0. I. Curves (1) and (2): n-C21+ n-Ce;curves (3) and (4): n-Cao+nC12;curves (5) and (6): n-C2t+n-Cn; curves (1), (3) and (5): e~flOOkTo; curves (2), (4) and (6): e~=ffi2kTo. For completelyrigidchainmoleculesystems (e~-- 100kTo)the positions of the fast minimum and the first maximum of the profilescorrespond to half of the chain length of solvent moleculesand of solute molecules,respectively.

55

change into exponential profiles if the temperature increases (see Fig. Ib). Nevertheless, the profiles show a characteristic behavior. For a system consisting of completely rigid chain molecules, if the solute or the solvent is given, the position of the first minimum (Z=2mio) or the first maximum ( z = , ~ ) of curve CA(Z) remains almost constant. When the chain length of solute (or solvent) molecules is changed, 2ma~ (or 2rain) will be changed correspondingly. This is explicitly shown in Fig. 2. In the case of normal alkane molecules, the molecular chains are not completely rigid (et°r~.2kT0). Then the maximum and the minimum at curve ~A(Z) are smoothed. However, eminent change in the segment density can still be recognized at z~,~m~ and Z~2mi~ although some lag occurs. We can see from both Fig. 1 and Fig. 2 that 2m~ corresponds to half of the chain length of solvent molecules, and 2 ~ to half of the chain length of solute molecules. This implies that the profiles depend on the structure of the chain-like molecules at the solid-fluid interface. The reason for the oscillation of segmental density can be interpreted as follows. In the solid-fluid interface, rigid molecules show two opposite trends which are competing with each other. To achieve the maximum adsorption energy, chain molecules show a strong tendency to be adsorbed and ordered on the solid surface. (We call this the energy effect. ) On the other hand, to gain a maximum entropy, chain molecules tend to be oriented randomly. In the regions near the surface (Z<2h 2t is half of the length of chain molecule i; i = A or B ), the number of orientations is restricted. This will cause the loss of entropies of interfacial fluid molecules (such as the conformational, the rotational, the translational entropy etc.). Henceforth, from the entropy point of view, chainlike molecules would rather avoid the solid surface. (We call this the entropy effect, and this effect vanishes at z=2~. ) Obviously, these two effects become more pronounced with increasing the length of molecules. In the first fluid layer the adsorption is the dominant effect, and solute (or longer chain) molecules adsorb strongly on the solid surface for energy reasons. In order to release the maximum adsorption energy per molecule, most fluid molecules are oriented in the directions parallel to the surface. (We will in later discussions come back to this issue.) In the successive liquid layers, the entropy effect be-

56

X.-Y. Liu et al. /Chemical Physics Letters 220 (1994) 53-58

comes more important. Consequently, solute (or longer chain) molecules are repelled from these regions. Instead, those layers are preferentially filled with solvent (or shorter chain) molecules whose entropy effect is less pronounced. At Z=Xmm=2e, the entropy effect for solvent molecules vanishes, and the space will be filled with solvent molecules. Then the density of solute molecules is substantially decreased. Similarly, at Z='lm~=2A due to the vanishing of the entropy effect of the solute molecules, ~A(Z) increases remarkably after the first fluid layer. As is known [6], the order parameter, s = ½( 3 ( c o s 2 a ) - 1 ), can be used to characterize the orientation of polymers or chain-like molecules with respect to a given direction; t~ is the angle between a bond connecting two segmental units and a given direction. In our cases, the orientation of a bond occurs with respect to the normal to the solid surface. Here the order parameter s ( z ) is a function o f z for a certain species. Obviously, in case the molecules are completely parallel to the surface, s ( z ) = - ½. A random bond distribution will result for an order parameter s ( z ) = 0 [ 12]. To compare the ordering occurling in different interfacial layers, we use the value of £(z) to characterize the molecule in the layer z. g(z) is the ensemble average o f s ( z ) over the bonds of molecule i ( i = A or B). Now, the order parameter of solnte and solvent molecules for a system of n-Ca3n-C6 solutions is calculated (see footnote 1) and plotted as a function of z, in Fig. 3a. As was expected, it can be seen that first, both solute and solvent molecules are adsorbed parallel to the surface. After the first interracial fluid layer, the value of Y(z) increases considerably, indicating that the degree of ordering decreases drastically. This result can also be seen in both Fig. 1 and Fig. 2 where ~0A(z) drops sharply after the first fluid layer. The implication is that in spite of the parallel "compact" structure of the first interracial layer, molecules tend to be obliquely packed in other interfacial layers. A similar result was obtained by molecular dynamics simulations [ 7]. In general, the solute molecules are more ordered than the solvent molecules. We note that this ordered structure of the first interfacial layer appears to be a general phenomenon of chain-like molecules, and is justified both from computer simulations [ 7 ] and experiments [ 17,18 ]. According to the results of scanning tunneling micros-

0.10

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i,

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,

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z (n) Fig. 3. The averageorder parameter g(z) of moleculesas a function of z for crystalsin contactwith paraffinsolutions. ~ = 2kTo, g^ = 0.1. (a) The order parameter ~(z) of solute and solventmolecules for a n-C23-n-C~solution. (b) The order parameter g(z) of solute molecules for paraffins n-C10,n-C23and n-C~6in n-C6 solutions. Moleculesin the first fluid layer (z=O) are highly ordered. The degreeof the ordering drops drasticallyin the successive layers after the in'st fluid layers. In case z~A~ (i=A or B), g(z) ~.O,correspondingto the vanishingof the ordering. copy experiments of ours and of others [ 17,18 ], when a paraffin solution was applied to a graphite substrate, highly ordered mono-molecular layers of paraffin molecules parallel to the surface were observed. These ordered adsorbed paraffin molecules are limited to the first and sometimes the second mono-molecular layers. This is consistent with the results given in Fig. 3. (Note that for the time being, the ordering of n-paraffins in the plane parallel to the surface cannot be taken into account in our calculations.)

X.- Y. Liu et al. / Chemical Physics Letters 220 (1994) 53- 58

In addition, it is shown in Fig. 3b that in case the chain length of solute molecules increases from n-C~o to n-C36, the distance from the solid surface to the fluid layer when paraffin chains distribute randomly (g(z) ~, 0) increases correspondingly. Compared with Fig. 3a, it is explicit that at z ~ 2 i ( i = A or B) paraffin molecules show a random packing. Obviously, this is due to the vanishing of the entropy effect. These resuits are consistent with Fig. 2. In the context of interfacial properties, it is of interest to see how bond energies are influenced by the interracial structure. This can be characterized by a so-called surface scaling factor C? [ 8,14 ]. In the language of Ising models [8,17], the interfacial exchange bond energy ¢}(0) (per molecule), which corresponds to the energy change due to bringing a molecule from the interracial solid phase to the interfacial fluid phase, is expressed in direction j for a twocomponent system, as

• j(o) ~ ½[ ~j(o) ' ~ - ~j(o) ss] + [1 --XA(0) ]2¢}j(0)¢

(1)

and ~ j ( O ) O = ~ j ( O ) ~ m - ½ [ ~ j ( O ) A - ~ + f P j ( O ) nB ] ,

(la)

where superscripts AA, BB, AB and SS denote solute-solute, solvent-solvent, solute-solvent and solidsolid interactions, respectively, XA(0) denotes the mole fraction of solute molecules in the first fluid layer. The value of ~j(0) can be determined by measuring the roughening temperature T ~ of a crystal surface [ 14,19 ], according to the relation 2kT ~ 0r= - -

~A0)

(2)

(0 ' is the dimensionless roughening temperature, and can be calculated for a given crystal surface [ 14,19 ].) Alternatively, the corresponding exchange bond energy in the bulk phase (denoted by q~j) can be evaluated from the data of the dissolution enthalpy A/-7~,~ (=Z]'=I g~j, m is the coordinate number) [17,19]. Then the ratio of C7 = q~j(0)/q~j can be determined from experiments [20-22]. For n-C2~, n-C23 and nC25 paraffin crystals grown from n-hexane solutions, the results of CT(exp.) are given in Table 1. On the other hand, this ratio is directly related to the segment density proffies of the solute [ 14].

57

Table 1 Calculatedand experimentalvaluesof the surfacesealingfactor C'7 for various paraffinsolution systems Solutions

~A"

¢}A(0)

C?(calc.)

C?(exp.) b

n-C21/n-C6 n-C23/n-C6 n-C2s/n-C6

0.100 0A00 0.100

0.519 0.543 0.579

0.406 0.392 0.382

0.414 0.401 0.372

"The valueofoAis chosento be closeto experimentalconditions [21-23]. b See refs. [8,21-231. Therefore, it can also be obtained from the SF theory calculations, based on the following relation [ 14 ]:

• j(o) C?=

~j

ln[X,,(o) ] ~

ln(XA)

(3)

Here XA is the corresponding value in the bulk. The corresponding calculated values of C T ( c a l c . ) f o r those three paraffin systems are also listed in Table 1. It can be seen that the experimental and calculated results agree in a quantitative way with each other. We notice from the aforementioned discussions that the density of paraffin molecules in the first fluid layer is associated with the interracial structure of systems (see Fig. I ). Therefore, it can be concluded from Table 1 that our calculations are quite convincing. The results presented above reveal the ordered structure of alkanes at the solid-fluid interface [ 2325 ]. This gives physical insight to the molecular behavior at the interface, and offers a better understanding of the influence of solvents on the morphology of crystals. Some progress has been made in this respect, and the results will be published elsewhere. It is noted that the molecule-dependent oscillatory profdes for chain-like molecular systems is, to the best of our knowledge, identified for the first time by our calculations. Without doubt, further investigations are needed. We hope that our results will attract more attention to this issue both from the experimental and theoretical points of view.

Acknowledgement

Helpful conversations with Dr. F.A.M. Leermakers and Professor Dr. G.J. Fleer are gratefully acknowledged. We are much indebted to Drs. R. Geertman

58

X.-Y. Liu et al. / Chemical Physics Letters 220 (1994) 53-58

for t e c h n i c a l support, Dr. C.S. S t r o m a n d Dr. H. Meekes for a critical r e a d i n g o f the m a n u s c r i p t . T h i s research was s u p p o r t e d b y Shell N e t h e r l a n d s B.V.

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