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and taking into account that, for symmetry considerations coming from the ..... forces and hence independent of the sample thickness ; the second coming from.

N° 4

Tome 51




France 51

Classification Physics Abstracts 4l.lOD- 66.10






15 FÉVRIER 1990,



(*, **)

Laboratoire de



Ion adsorption and of finite thickness G. Barbero


equilibrium distribution of charges in a cell

and G. Durand


le 2 août 1989,

des Solides, Université de Paris-Sud, Bât. 510, 91405


le 26 octobre




En utilisant une méthode self-consistente, on évalue la distribution d’équilibre de dans une cellule d’épaisseur finie, en présence d’adsorption par les parois. L’analyse est charges faite dans les deux cas limites où la densité des neutres est fixée, soit très faible (dissociation totale), soit très grande (dissociation faible). La charge adsorbée en surface est évaluée, en étendant le problème classique d’adsorption de Langmuir aux situations loin du régime de saturation. On discute l’importance du problème considéré sur les propriétés interfaciales des cristaux liquides. On montre en particulier qu’en tenant compte du champ électrique de surface associé aux charges adsorbées, on s’attend à trouver un caractère non local à la partie anisotrope de l’énergie d’ancrage. Ceci est en accord avec des expériences récentes montrant que des échantillons nématiques de même traitement de surface, mais d’épaisseurs différentes, ont des énergies de surface apparente différentes. Résumé.


Abstract. By using a self-consistent method the equilibrium distribution of charges in a liquid cell of finite thickness, when the adsorption phenomenon is present, is evaluated. The analysis is performed for the two limiting cases where the neutral density is fixed : either very weak (total dissociation), or very large (weak dissociation). The surface adsorbed charge is evaluated, by extending the classical Langmuir problem of adsorption to the case far from the saturation regime. The importance of the considered problem on the interfacial properties of liquid is discussed. In particular it is shown that by taking into account the surface electric field associated with the adsorbed charges, a non-local character of the anisotropic part of the anchoring energy of liquid crystals is expected. This agrees with recent experimental observations showing that nematic samples of different thicknesses, but with the same surface treatment, seem to have 2014

different anchoring


(*) Partially supported by

Ministère de l’Education Nationale, de la Recherche et des







di Fisica, Politecnico, C.




Abruzzi 24, 10129 Torino,


Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01990005104028100


1. Introduction.

The surface properties of liquid/solid interfaces depend on the physical chemistry of the two media, and on long range forces like Van der Waals and double layers [1]. In this paper we limit ourselves to consider the effect of the double layer forces, connected to selective surface adsorption of ions dissolved in the liquid. We show that the ion adsorption can be responsible for apparent non locality of interface properties recently observed in nematic liquid crystals

[2-5]. These ions can originate from more or less dissociated impurities (the extrinsic contribution), but also from the spontaneous dissociation of the liquid molecules themselves (the intrinsic contribution). The positive and negative ions can have different affinities for the boundary surfaces. Their selective adsorption creates at the liquid interface a surface field which depends on the volume of the sample. This surface field can modify the surface properties of the solid/liquid interface. This idea was already suggested [6], but not quantitatively worked out. The mechanism we imagine is the following : the bulk liquid is neutral but contains positive and negative ions, in fixed concentration p é P e P e in absence of adsorption from the is plate. pc given by the fixed concentration of the totally dissociated impurities, or by the dissociation constant of the liquid. To keep the calculation general, we consider Pe as an independent parameter. We assume a selective adsorption on the solid substrate for, say, positive ions, with an adsorption energy Ea, and for instance, a very large repulsive energy for the negative ions, so that their surface density is always negligible. When putting in contact the liquid and the substrate, positive ions will migrate toward the interface and create a surface electric field Es localized inside a boundary layer of a Debye screening length Ls. Inside this boundary layer, the coupling of Es with the dielectric anisotropy of the liquid gives rise to an additional orientational dielectric free energy. This new contribution can be considered as quasi-local and renormalizes simply the interfacial properties of the liquid/solid system, although Es is, in principle, a non-local quantity. It is indeed sufficient to explain how Es changes with the thickness of the cell d, because for instance of the saturation of adsorption, to explain the apparent long range interaction between the two plates. Let us call cl the adsorbed positive charge density (and the thickness integrated negative volume screening charges). We discuss to simplify a one dimensional model. In the liquid bulk, the new charge densities p ± (z ) depend on the distance z from the solid boundary plates. The aim of the calculation is to estimate the surface field Es (or the absorbed charge u) vs. the initial charge concentration p,,, for samples of various thicknesses d. We use a self-consistent method to solve the problem : we first write that all parts of the liquid, which exchange positive and negative ions, are in equilibrium in an arbitrary potential V (z). This fixes a Boltzmann shape for p± [z, V (z)], which depend only on two parameters, the densities p ± (0) at the center of the cell. We determine V (z ) from the charge densities using Poisson equation, which can be integrated if we know the adsorbed charge density a on the surfaces. cr is defined itself from the equilibrium of positive charges, which are between and the the surface bulk, and depends also on V (z), i.e. on exchanged now the conservation p ± (0). Writting charge equation (or the mass action law), we obtain two the two concentrations p± (0). The to determine finally self-consistency equations one of contains two kind the then problem describing the electric properties, equations : in one section and the the 2, analyzed charge exchange equilibrium, in the bulk describing and the and bulk surface between the (Sect. 2) (Sect. 3). This last equilibrium is very similar to the classical adsorption problem of Langmuir [7]. The electric part has been already partially analyzed in a different context for isotropic liquid [8]. Finally in section 4 we discuss =



the influence of the surface field local effect. 2. Basic

équations of

the electric




energy, to


the cell thickness



Let us consider a liquid containing ions of density p,,, in thermodynamic equilibrium in absence of adsorption. The equilibrium is defined by constant temperature and volume conditions (we neglect the compressibility of the liquid). We analyze a unidimensional problem. In our reference frame the boundary plates are placed at z ± d/2 (see Fig. 1). Furthermore we suppose the initial electroneutrality of the liquid and that the two surfaces, assumed identical, adsorb only positive ions. If a is the surface density of adsorbed ions, p ± (z ) the bulk densities of positive and negative ions, the electroneutrality condition imposes =

1. Expected thickness dependence of the reduced potential u, and the ion charge densities p+ and p_ . The plates are assumed to adsorb only positive ions (surface density o,). pe is the equilibrium concentration of the neutral solution in absence of adsorption. The electric fiels is localized on a Debye screening length close to the plate at ± d/2.



In principle, the ionic dissociation obeys a chemical equilibrium, where the density of the neutral species is important. As we have no information on the practical origin of the ions, we have specialized our analysis for the two limiting cases where the neutral density is fixed : or very weak (total dissociation), or very large (weak dissociation). In the case of totally dissociated impurities, we must impose the conservation of the total number of each (positive or negative) ions, i.e.

In the case of intrinsic action law :

where pci is



constant which



depends only




the temperature.


must use the mass


In the


gas approximation be written as



(for small Pe) the chemical potential

IL:t of the ions in the

where À th is the average thermal de Broglie wave-length [7] of the ions, ka is the Boltzmann constant, T the absolute temperature, q the electrical charge of the ions and V (z ) the macroscopically averaged electrical potential. Since only differences in potential are physically significant, we put V(0) 0, in the center of the cell (where p ± (0 ) = p ô ) and consequently E (o ) 0 because of symmetry. We require at equilibrium that the chemical potentials are uniform across the liquid sample ; from equation (3) we then obtain that the steady-state distributions of mobile charges obey the Boltzmann distribution =


In all considered cases, pô , différent from pe, account (4) Poisson équation reads

where - is


average dielectric constant, and the

and taking into account that, for symmetry hypothesis U(z ) U (- z ) and U’(0) 0, =

from which









It is easy to write :



By taking into

d/dz. By putting :

considerations coming from the electroneutrality we can rewrite equation (5) as :

the electric field at









d/2 gives


U(- d/2). By equation (7)

We consider first the into account equation

symbol ’


easily :

By using equations (7) and (8) where

are two constants to





impurities. Using equations (4)




that the total system remains neutral since,

using equation (9),

we can


The integrand in equations (10) and (11) present an apparent divergence at z 0; furthermore they are written in absolute units. It is better to rewrite them in a dimensionaless form. By putting =

equations (9), (10)




as :

where :


D is measured in terms of the equilibrium Debye length Le, whereas U e p e Le is a natural surface charge density unit. In the case of intrinsic conductivity (or weakly ionized impurities) equation (16) is changed in =

coming For


from mass action law. fixed cr, the z-dependence of p±

3. Surface




is shown





charge and the adsorption phenomenon.

In order to determine the surface charge density adsorbed by the solid surface let us consider the classical Langmuir problem of adsorption [7]. Let us call 03C3M/q the maximum number of adsorption site per cm2 on the surface (q is the proton charge). The simplest model to explain the selective attraction Ea is to imagine that positive ions are very small compared to negative ions, and that they are attracted by a highly polarizable solid medium. UM can then be estimated as q/m 2, where m is a molecular size. Writing that IL+ is the same on the surface and in the bulk, we obtain the covering ratio 9 = U / UM in the form :





in the considered



adsorption. Using equation (13)

we can


ka T ln (p c À3th) is the chemical potential in the absence of adsorption. By putting equation (20) into equation (19) we obtain

where IL c



equations (13)

have been taken into account, and furthermore


have put


We are interested in A > 1. This leads to


non-saturating regime, i.e.,


follows from

equation (21),

UM/ A. Note that our hypothesis of « infinite » repulsion for negative ions from would not be valid any more if 0-L were comparable to UM, because of the electrostatic attraction between positive and negative ions on the surface, i.e. in the case of a too small A coefficient. Equations (14)-(16) or (16’) and (21) are valid for any d, but are quite complicated in general. We consider first the fully ionized case, i.e. the one where each number of positive and negative charges is separately conserved. We are interested in two limits : 1) the small thickness cells are expected to produce a small u, then a small U ; 2) the large thickness cells are expected to give a saturation of o-, with a large U. In the limit of small U(i.e. q V IkB T « 1 ), which implies ts « 1, we can expand in the usual way equations (14)-(16) to obtain where 0-L the plate










Equations (23)-(25) have been deduced by supposing ts 1; hence they are valid for (u / u c) D 8. By considering that R1 > 0, from equation (25) we also obtain 0’/U,, D/2. Since D is small in the considered limit, from the above conditions we deduce that into equation (21’) and by

equations (23)-(25) hold for D 4. By substituting equation (25) solving with respect to u we obtain :

from which

Equation (27)

shows that in the limit of small thickness, cr is proportional to d. In this regime (Us : 1), the potential drop between the plate and the bulk is small compared to kB T/q. Most of the positive ions are trapped on the surfaces, which are each covered by half of the initial charge number p e d. We point out that equation (26) is not valid in the


limit since it has been deduced equation (26) holds only for D -->


by supposing

that ts

1. From



obtain that

since usually 0-,,/O’L - 1. Previous estimation agrees with the one reported above. In order to have some information at large D let us consider now the case where ts > 1. Since d > Le, the liquid remains practically neutral in the center of the sample, and hence R1 ~ R2 ~ 1. From (21) we obtain for the saturation charge density

and for ts the


can now




critical thickness d * for which the two

condition 1/22 Pc d * = U L’

from which,

by taking

régimes merge. It is given by the equation (28), we obtain

into account

where we use the fact that at room temperature q2/ eka T is an order of magnitude larger than a molecular size m. The maximum surface covering ratio uL/uM’" 1 /A can be taken 0.1, i.e. A - 10. A small ion, of atomic size, could have a dielectric attraction energy with the boundary Ea ’" 1 eV. This results in a practical concentration Pe ’" q .1013 cm- 3, easy to achieve [9]. In this case, we find a maximum value for d* - 50 ilm comparable with typical nematic liquid crystal cell thickness. The fact that d* is macroscopic is obviously related to the amplification factor 40 Le/mA. To conclude, for small cell thickness, (d d* ~ 50 ktm for typical Pe), most positive ions are adsorbed on the two plates with a surface density proportional to d. This is a very reminiscent of the surface purification process already demonstrated [10]. For large thickness, the surface density a is saturated classically by the now fixed p, e of the solution. This thickness dependence of U (d) is sketched in figure 2. =

Reduced surface charge density versus reduced thickness for a fully dissociated impurity. Fig. 2. Below D*, most positive ions are adsorbed on the plate and the surface field increases with D. Above, D*, the surface density saturate, and also the surface field. -


discuss the case of intrinsic conductivity (or weakly ionized impurities). Equations (14)-(15) and (16’) must be solved with equation (21’). Using equations (16’) and (21), from equation (14) we obtain Let



in the limit of we have

uL/2 o-c > 1, usually verified. By substituting equation (24’) into equation (15)

giving D = D (R1 ) and plotted in figure 3. This approximated graph of the previous case (Fig. 2).






figure 2,

As previously, let equations (14’), (25’)

us we

but for





must be


with the

dissociated system.

consider first the limit of small ts, which obtain :


small D. From


Equations (24"), (25") hold for D 4 (2 U e/ UL)I/3, usually very small. Hence these approximate equations are not very useful. In the opposite limit of large D we have 7?i - R2 - 1. Consequently, as in the previous case, tS - U L/2 u c and cr = 0- L, as expected. One can practically define a d * - 5 Le for which the saturation is obtained. For samples thicker than d *, one is in the classical Langmuir limit : the surface charge a is fixed by the ion concentration and not by the volume of the sample. Below d *, U decreases with the thickness d. 4. Influence of the adsorbed substrate surface energy.




anisotropic part of

the nematic

liquid crystal,

In sections 2 and 3 it is shown that the ionic charge that is stuck at the solid-liquid surface a diffuse layer of ions of the opposite sign in the liquid. The adsorbed charge and the



layer of oppositely charged ions that it attracts constitute an intrinsic double layer depending only on the solid and liquid. The thickness of this double layer is of the order of the Debye screening length. It follows that the selective surface adsorption of ions creates an electric surface field Es which extends in the bulk over the Debye screening length Ls. If an isotropic liquid is considered this surface electric field can change the effective surface tension, which is an isotropic quantity. In the opposite case where an anisotropic liquid, as a nematic liquid crystal, is considered, the surface field can also change the anisotropic part of the surface energy. In this section we wish to analyse the influence of the selective adsorption phenomenon on the so-called nematic /substrate anchoring energy. As known a nematic material is characterized by anisotropic physical properties [1] as dielectric, or magnetic, permittivity. The physical properties of a nematic depend on the average molecular orientation, usually indicated by fi. û is known as nematic director. When a nematic is put over a solid substrate, û alignes parallel to a well defined direction à called easy direction [1]. If now an extemal field is applied, whose effect is such to change the surface orientation of n, the surface tends to maintain the surface director ns parallel to Tf, by means of a restoring torque. This restoring torque takes origin by the anisotropic part of diffuse

the surface energy which controls the surface orientation of the nematic. The surface energy results from short range forces, so that it is expected to be a quasi-local property of the nematic/substrate interface. Recently some experimental determination of surface energy have shown an apparent long range dependence of this parameter vs. the macroscopic size of the cell d (d 3 --. 100 ktm ). Since a direct influence between the two surfaces a macroscopic d apart is difficult to imagine we think this strange d-dependence could be explained by considering the influence of the surface electric field E,, coming by the selective adsorption, on the surface energy. Since the nematic presents anisotropic physical properties, the simplest coupling of Es with the director is the one related to the anisotropy of the dielectric constant 8. Calling, as usual, Ea El - E 1. the dielectric anisotropy (where Il and 1 refer to the dielectric of n), coupling Es with n gives a surface free energy density =

in the

simplest exponential approximation for E (z ). In (29), ûs is the surface director and LS penetration length given by : Ls Le[(7?i + R2)/21+ 1/2, not very far from Le. According to the sign of Ea, FE is minimum for ns parallel (Ea:> 0) or perpendicular (Ba : 0) to Es, which is obviously normal to the limiting surface. The dielectric energy FE must be added to the intrinsic surface energy form, which tends to align û along the easy axis Tr (fr2 =1 ), normal or parallel to the plates. We assume that the Rapini-Papoular form [11] is a good approximation for the intrinsic anchoring energy. The total surface free energy density is then : a


where Ws is the anchoring strength and Es is given by equation (8). Since Es depends now on d, a non-local « size effect » is expected. Note that Ls appearing in equations (25-31) also depends on d, but in a weak way. In fact this parameter changes from Le to Le for d ranging from infinite to zero. Equation (30) shows that the anisotropic part of the nematic/substrate surface energy contains two terms : the first connected to short range interaction forces and hence independent of the sample thickness ; the second coming from the dielectric interaction between the nematic and the surface electric field Es. Since



Es depends on the adsorbed charge, which is thickness dependent, we conclude that also the effective anisotropic surface anchoring energy is expected to be thickness dependent. This agrees with recent measurement performed by a Russian group [2-5], showing a long range dependence of the anchoring energy vs. the macroscopic size of the cell d (3 : 100 um). We can now compare these predictions with the experimental data reported in reference [4]. Note first that the various samples at various d are differents, which allows each sample to represent an equilibrium situation at constant p e, probably the same for each sample. The starting geometry is planar and the used nematic liquid crystal is 5 CB, with a positive Ea =13 [12]. This combination is the one of the two which can give rise to a decrease of Ws and even to an instability if Ea > 2 WS E2/ (Ls ul) as follows from equation (30) and equation (8). In reference [4], only a decrease and a saturation is observed, which means that the above mentioned condition is probably not fulfilled. To fit the data of reference [4], we should know the conductivity of used sample, to estimate Pe. The only reported data connected with the conductivity is the 1 kHz AC voltage frequency which suppresses the electrohydrodynamical effects. The dielectric relaxation frequency of the sample O’IE is then lower than 103 Hz. Using for the ions mobility the value of reference [9], we estimate pc~ 10-4 C/cm3, and hence L,, - 0.1 itm. We can now try to fit the reference [4] data to determine Ws and 0-L. Note first that the model of weak dissociation is not very good to explain the observed saturation around d - 20 = 30 lim, since it predicts a d* - 5 Le - 0.5 um. We have tried a fit with the fully dissociated impurities model. The result is shown in figure 4. The best values are Ws - 1 erglcm2, and 0 L - 4.3 x 10 - 7 C/CM2

Predicted dependence of the anchoring energy versus thickness, for the case of nematic liquid 5 CB. One assumes a simple dielectric coupling between the nematic orientation and the surface field. The points are experimental data from reference [4].

Fig. 4. crystal


The value for Ws is a bit larger than the one usually measured [13], but must be accepted since it is a reasonable extrapolation of the data of reference [4]. From OL’ we can estimate the mean distance between ions on the surface is l> ~ (UL/q )-1/2 ’" 2 x 10- b cm, and hence A - 102. These values are at the limit of validity of our model. They just indicate that the general trend is correct. To check the model more carefully, one should know better the experimental parameters, and describe in more details the interaction between neighbouring sites, which has been neglected in our model.


5. Conclusion. In this paper we have extended the classical Langmuir problem of adsorption in order to take into account the electrical properties of a sample selectively adsorbing ions contained in a given liquid. The analysis shows that the surface electric field built by trapped ions depends on the sample thickness, in a macroscopic range. This implies that in ordinary liquids the interface properties strongly depend on the adsorbed charge. In particular we have focused our attention on the anisotropic part of the anchoring energy nematic/solid substrate, showing that apparent long range dependence of this quantity can be easily explained with our model. The solutions derived here for the electrical problem can be useful to test different mechanics proposed in order to study the surface properties of liquids, when a selective adsorption is



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