analysis of the effects of marangoni stresses.pdf

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Analysis of the Effects of Marangoni Stresses on theMicroflow in an Evaporating Sessile Droplet

Hua Hu* and Ronald G. Larson

Department of Chemical Engineering, University of Michigan,Ann Arbor, Michigan 48109-2136

Received October 6, 2004. In Final Form: January 25, 2005

We study the effects of Marangoni stresses on the flow in an evaporating sessile droplet, by extendinga lubrication analysis and a finite element solution of the flow field in a drying droplet, developed earlier.1

The temperature distribution within the droplet is obtained from a solution of Laplaces equation, wherequasi-steadiness and neglect of convection terms in the heat equation can be justified for small, slowlyevaporating droplets. The evaporation flux and temperature profiles along the droplet surface areapproximated by simple analytical forms and used as boundary conditions to obtain an axisymmetricanalytical flow field from the lubrication theory for relatively flat droplets. A finite element algorithm isalso developed to solve simultaneously the vapor concentration, and the thermal and flow fields in thedroplet, which shows that the lubrication solution with the Marangoni stress is accurate for contact anglesas high as 40. From our analysis, we find that surfactant contamination, at a surface concentration assmall as 300 molecules/m2, can almost entirely suppress the Marangoni flow in the evaporating droplet.

1. IntroductionMarangoni effects, manifested as tears of wine, were

observed as early as the 1800s.2 In a wine glass, theevaporation of alcohol generates a surface tension gradi-ent, which produces a traction on thewinesurface causingthe wine to climb up the side of glass where it forms athin film. As the wine accumulates, a bulging rim ofliquid forms along the top of the film, which eventuallypinches into droplets which roll under their own weight,like tears, back into the wine. The Italian physicistMarangoni gave a detailed description of the movementof a liquid surface induced by a surface tension gradient,generated either by a composition or a temperature

variation along the free surface. In 1901, Benard3 dis-coveredthe convection cells in a thin liquid film that were

later named after him. Later, Block4 performed morecareful experiments on a thin liquid film, and Pearson5

gave a detailed theoretical analysis of Benards observa-tion, concluding that a surface tension gradient, i.e., theMarangoni stress, causes theconvection patternsin a thinfilm.

While muchexperimentaland theoretical workhas beenperformed to investigate various surface-tension-driven(i.e., Marangoni) flows in thin films or shallow pools, 6-12

there are only a few papers reporting Marangoni flow inan evaporating sessile droplet. Zhang and Yang13 experi-mentally studied the natural convection in evaporatingdrops, where they observed a Marangoni flow. Davis and

co-workers

14-16

studied both stationary and spreadingdroplets with consideration of evaporation, Marangonistresses, and moving contact lines. They applied alubrication analysis to obtain the evolution equation ofthe droplet free surface profile as a droplet spreads orresides on a heated substrate. The velocity fields in thedroplet were thereafter obtained using the evolutionequation for the droplet profile. Savino and co-workers 17

numerically solved the axisymmetric steady-stateNavier-Stokes equations taking into account the Ma-rangoni stress at the liquid-air interface. They alsoperformed experiments to map the velocity field in thedroplets. Their theoretical resultswere notconsistentwiththe experimental ones due to the errors in using the PIVtechnique to map the velocity field in a spherical cap

droplet, which acts like a lens distorting the real flowfield.Meanwhile, many researchers have taken advantage

of Marangoni flow in drying droplets to affect the pat-tern of deposition from the droplet onto the underlyingsubstrate.18-21 Wang and co-workers,18 for example, pro-duced a patterned porous thin film on a substrate. Stebeand co-workers20,21 used a surfactant that during dryingdevelops a concentration gradient along the droplet sur-face leading to a gradient in surface tension; by varyingthe initial surfactant concentration, they were able to

vary the pattern of particle deposition on the sub-strate. Marangoni flow plays a key role in coating, thinfilm deposition, crystal growth, and production of pho-tonic materials. Therefore, a thorough understanding of

Marangoni flow in an evaporating droplet will be impor-* Corresponding author. E-mail: [email protected].(1) Hu, H.; Larson, R. G. Langmuir 2005, 21, 3963.(2) Scriven, L. E.; Sternling, C. V. Nature 1960, 187, 187.(3) Benard, H. Ann. Chim. Phys., Ser. 7 1901, 23, 62.(4) Block, M. J. Nature 1956, 178, 650.(5) Pearson, J. R. A. J. Fluid Mech. 1958, 4, 489.(6) Scriven, L. E. Chem. Eng. Sci. 1960, 12, 98.(7) Nield, D. A. J. Fluid Mech. 1964 19, 341.(8) Fanton, X.; Cazabat, A. M. Langmuir 1998, 14, 2554-2561.(9) De Gennes, P. G. Eur. Phys. J. E 2001, 6, 421-424.(10) Stroock, A. D.; Ismagilov, R. F.; Stone, H. A.; Whitesides, G. M.

Langmuir 2003, 19, 4358-4362.(11) Mancini, H.; Maza, D. Europhys. Lett. 2004, 66, 812-818.(12) Vogel, M. J.;Miraghaie, R.;Lopez, J. M.;Hirsa, A. H.Langmuir

2004, 20, 5651-5654.(13) Zhang, N. L.; Yang, W. J. Trans. ASME 1992, 104, 656-662.

(14) Ehrhard P.; Davis, S. H. J. Fluid Mech. 1991, 229, 365.(15) Anderson D. M.; Davis, S. H. Phys. Fluids 1995, 7, 248.(16) Oron A.; Davis, S. H.; Bankoff, S. G. Rev. Mod. Phys. 1997, 69,

931.(17) Savino R.; Paterna, D.; Favaloro, N. J. Thermophys. Heat

Transfer 2002, 16, 562-574.(18) Wang, H. T.; Wang, Zh. B.; Huang, L. M.; Mitra, A.; Yan Y. S.

Langmuir 2001, 17, 2572-2574.(19) Maillard,M.; Motte, L.; Pileni, M. P.Adv. Mater. 2001, 13,200-

204.(20) Nguyen, V. X.; Stebe, K. J. Phys. Rev. Lett. 2002, 22, 3282-

3285.(21) Truskett, V.; Stebe, K. J. Langmuir 2003, 19, 8271-8279.

3972 Langmuir 2005, 21, 3972-3980

10.1021/la0475270 CCC: $30.25 2005 American Chemical SocietyPublished on Web 03/26/2005


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tant both in fundamental research and in practicalapplications.

Although Davis and co-workers and Savino and co-workers have developed theories for the Marangoni flowin an evaporating droplet, an analytical theory for thelocally resolved axisymmetricflowin a slowly evaporatingdroplet with a pinned contact line is still lacking. Such atheory is needed, forexample,to predict particle depositionfrom a drying droplet in the presence of the Marangoniflow when, as usual, the particles do not rapidly diffuse

across theheight of thedroplet during flowand deposition.In what follows, in section 2, we develop a lubrication

theory to describe the velocity field in an evaporatingdroplet in the presence of Marangoni stresses. To dem-onstrate the accuracy of the lubrication theory, we alsodevelop a finite element method to solve simultaneouslythe thermal and flow fields in the evaporating droplet. Insection 3, we present the results obtained from thesemethods and discuss the effect of surface-active contami-nants on the Marangoni flow in the evaporating droplet.We finally summarize in section 4.

2. Theory

2.1. Expressions for the Velocity Field with a

Thermal Marangoni Stress Boundary Condition. Inour companion paper,1 we established the governingequations for an evaporating droplet without consideringthe thermal transfer due to latent heat of evaporation.Since in many experiments there is strong evidence of thethermal cooling affecting the flow pattern in the dryingdroplet, we here add the energy equation to the set ofgoverning equations. Therefore, we have the Laplaceequation for the vapor concentration distribution

The flow equations are

where we have neglected inertial terms, since theReynoldsnumber Re FujrR/ is small (0.003) for weak flow in theslowly evaporating dropletconsidered here. Here, we will

alsoneglect thebuoyancy-driven flowbecause of the smallvalue of a dimensionless groupB Fgh02C/7.1375,whichwas introduced by Pearson5 to estimate the relativestrength of the buoyancy-induced flow compared to thatof Marangoni flow. We choose the water density F ) 1 gcm-3; water thermal expansion coefficient22 C ) 2.07 10-4 C-1; the temperature coefficient of surface tensionfor water, ) -0.1657 dyn cm-1 C-1;g ) 980cms-2;andthe droplet height h0 ) 0.04 cm. Thus, we obtain B 3 10-4, which shows that the buoyancy-induced flow is

very weak compared with the Marangoni flow in theevaporating droplet.

We also consider heat transfer in the droplet and theglass coverslip, on which the droplet rests. The energyequation is

where cp is the specific heat and k is the thermalconductivity. Here, we define an inverse Stanton numberSt-1 ) FcpujrR/k, which is a ratio of the convective to thethermal diffusive effects. In typical experiments withwater droplets, such as those described in Hu andLarson,23

parameters in this group have the following approximatevalues: height-averaged radial velocity ujr ) 1 m/s,contact line radius R ) 1 mm, k ) 1.4536 10-3 cal K-1cm-1 s-1 (from Bird et al.24), Cp ) 1 cal g-1 K-1, and waterdensity F ) 1 g cm-3, which gives St-1 about 0.02. Thisimplies that the rate of the convective heat transfer ismuchsmaller thanthe rateof theconductiveheat transfer,and so we can neglect the convection term in the energyequation. In the finite element analysispresentedshortly,we confirm that the heat convection term is negligible.We can also neglect the transient in the energy equation,which we show is valid in Appendix A by estimating theratio of the relative rates of change of droplet height tothat of temperature. Thus, the energy equation (5)simplifies to the Laplace equation

We solve eq 6 for a system containing both the dropletand a glass substrate. So the boundary conditions for eq6 are as follows:

1. On the droplet surface S ) {h(r,t)|r e R}: Jh )Hw(Jn), where Hw is the latent heat of evaporation ofwater and Jh is the heat flux.

2. On the glass surface outside the droplet z ) 0: R 0 ) and completewetting. Hereinbelow is given one of the derivationsof the corresponding equations of the theory, basedon the application of equation (1). For the meniscusin a flat slit H > ho, the curvature of the cylindricalsurface of the meniscus is equal to K =

h"[1 + (h )2]- 312,where h and h" are the first andthe second derivative of the thickness h of a liquidlayer with respect to the coordinate x. Substitutingthis value of K(h ) into equation (1) and using theboundary conditions : h = 0 at h = ho, and h ~ ooat h = H, we obtain the following solution of

equation (1) [13] :

where Ho = H (ho) and denotes the value of theintegral.When 00 > 0, the value of Pc may be expressed

through cos 0 0. In the case of a flat slit, which is hereconsidered, P c = y /r = y . cos 0 OIH, where r isthe radius of curvature of an undisturbed part of

meniscus (Fig. 1). Then, instead of equation (2), weobtain :


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Now we can compare this expression with the known

Young equation

To determine the contact angle 00 fromequation (4) requires knowing the specific interphaseenergies of the solid substrate at the boundary withthe gas phase, ysv, and the liquid, ysL. The Youngequation, in distinction from equation (3), does notallow determination of the contact angle, becausethere are available no methods independent ofequation (4), for determining either each of interph-ase energies or the difference between these. Now,the Young equation is usually employed for solvingan inverse problem- finding a difference ysv - ysLon the basis of the measured values of 00.

The Frumkin-Derjaguins theory enables one todetermine the value of ysv for the solid surface

coated by a wetting film. Its value is equal to a sumof the interphase energies of two surfaces of a liquidfilm : one, contacting gas, y, and another, contactingthe solid substrate, ysL, and variations in the free

energy of the film,AG = Jn h . dII = ..1 + no ho,when it is thinning out from ao to the equilibriumthickness ho.As a result, the difference between

and ysL proves to be equal simply to y + Ho ho + d,which just contains equation (3).Equation (3) allows a theoretical determination of

the value of 00 in accordance with the known

H(h) isotherm of the wetting films of a given liquidon a given substrate. The methods for the exper-imental and the theoretical determination of H(h)isotherms are presented in reference [8].Equation (3) has a similar form for cylindrical

drops on a flat substrate [13]. This is not surprising,since at H > ho (in this case H denotes the drop

height) the equilibrium conditions should not dependon the surface curvature sign beyond the radius ofaction of surface forces.Adifference resides only inthat the values of Ho are negative, because the

capillary pressure of the drop having a convexsurface, has another sign Ho = Pc = - y /r, where ris the radius of curvature of an undisturbed portionof the drop. Equation (3) is also applicable atPc = 0, when the surface of the bulk liquid is plane,and its profile is wedge-shaped. In such a case, thesecond term disappears from the right-hand side of

equation (3), since Ho = Pc = 0.For

drops havingthe

sphericalsurface the calcu-

lations are complicated in connection with a necessityof taking into acount the secondary curvature of the

drop surface. If we limit ourselves to the case ofrelatively small contact angles (when it is possible to

assume ah/ax 1), equation (1) may be written inthe following manner [14] :

where h = dh /dp , h" = d2h/dp 2, r = Const. is thedrop surface curvature radius, and p is the radialcoordinate in the substrate plane. In the generalcase, nonlinear differential equation (5) may besolved only numerically. It can be linearized by usinga simplified form of the disjoining pressure iso-therm :

where a and t are the parameters of the isotherm,

which are characteristic of the slope of its stablepart, where 8Hl8h 0, and the radius of action ofsurface forces, accordingly. The form of simplifiedisotherm (curve 1, Fig. 2) is similar to the real one(curve 2), when only thin wetting films are stable.The solution of equation (5) gives the following

expression for the equilibrium contact angle [14] :

where to is the film thickness at II = 0.

a b

Fig. 2. - Simplified isotherm of disjoining pressure (a)and its real analogue (b).

As appears from this expression, the drop contactangle 00 decreases with its dimensions, which is

accompanied by an increase in the negative capillarypressure P c. Such an effect was, in particular,detected experimentally for water drops less than3 mm in diameter [15]. Under otherwise equalconditions, an increase in the radius of action ofsurface forces, t, or a decrease in to causes anincrease in the contact

anglevalues. This is con-

nected with a displacement of the isotherm (6) intothe region of negative values of disjoining pressure.The range of the applicability of equation (3) is

restricted by the values of 00 > 0, when Ho ho +


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A -- 0 and cos 0 0 -- 1. In the case where the continua-tion of the meniscus in a flat slit does not intersect

substrates (Fig. 1, curve 3) equation (3) permits de-termination of the capillary pressure of the meniscusin equilibrium with the film having the thickness

ho. Taking into account that in the equilibrium state,Pc = Ho, we obtain from equation (2) :

This equation correlates the capillary pressure ofmeniscus, Pc, and hence, its curvature radius,r = y /Pc, with the slit half-width H and through 4with the disjoining pressure isotherm. It should betaken into account, however, that ho is a function ofPc. The value of ho can be determined from theisotherm equation, 03A0(h), at Ho = Zizi) = Po. -

Equation (8)enables one to determine, instead of

the contact angle, another parameter, which may beused to characterize the complete wetting con-ditions ; namely, - the diffrence h. = H - r(Fig. 1). The larger the value of h* the smaller themeniscus curvature radius and the better the liquidwets the solid surface. Transforming equation (8),we obtain the following expression for h. :

The values of h* are equal to 0 (curve 4, Fig. 1),when Pc = y/H and Ho ho + 4 = 0. This, according

to equation (3), corresponds to cos 90 = 1 and90=0.At = 0, h * = ho. If the positive values of 4

increase, the values of h * increase, too, exceedingthe equilibrium film thickness. In the case ofH > ho we may consider that y ~ 0394, and Pc = y /H.Then equation (9) transforms into a simpler form :

Then the ratio, S = h*/ho, may be used as aunified characteristic of wetting, which is suitable inthe cases of complete and partial wetting :

The value of S = 0, corresponding to 90 = 0,separates the region of complete wetting (S > 0 )from that of partial wetting (S 0). The higher thepositive values of S the better will be the wetting.Thus, in particular, for the isotherms H(h) =

A/hn>0, taking into account that Pc=J7o=A/hn0, instead of equation (11) we obtain the follow-ing expression :

At n = 3, which corresponds to the wetting filmsthat are stable due to Lifshitz dispersion forces,S = 1.5, and h * = 1.5 ho.At n = 2, which corres-

ponds to the wetting films that are stable due to theelectrostatic repulsive forces, S = 2, and h * = 2 ho.In general, the relationship between h. and hoproves to be a function of the slit width H and theform of the H(h) isotherm, including the different

components of disjoining pressure.In the case of partial wetting (03B80>0), it ispossible to establish, by using equations (3) and(11), a relatively simple functional relationship be-tween the parameter S and the equilibrium contactangle, as formed by the meniscus in a flat slit :

Only the conditions of partial wetting can berealized for equilibrium drops on a solid substrate.The parameter S is here unacceptable, and thecontact angle value remains the only characteristic of

wetting. For drops having a small-radius base, theequilibrium contact angle is also influenced by theline tension of the wetting perimeter, which willfurther be considered (Sect. 8). For the films oncurved surfaces, as for example, for those on fila-ments or in capillaries, a variation in the pressure ina film due to the curvature of its surface adjacent tothe gas phase must be taken into account [8].

3. Isotherms of disjoining pressure of wetting films.

The macroscopic approach allows one to ascertain

what physical phenomenon controls the wetting orthe non-wetting of the surface. For this purpose, we

consider, as an example the isotherms of disjoiningpressure of aqueous wetting films (Fig. 3). Here,curves 1-4 represent the dependences of film thick-

Fig. 3. - Isotherms of the disjoining pressure H(h) ofwater films on the solid surface.


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ness h on disjoining pressure, or, which is the same,on the capillary pressure of a liquid in equilibriumwith the film. Curve 1 relates to water films on a

quartz surface. The known experimental data [8] areindicated by points, while a solid line represents acomputed isotherm [16], accounting for the effect ofthree components of disjoining pressure in the film-molecular 03A0m, electrostatic ne, and structuralIIS. The isotherm branches, where an/ah 0, re-spond to the stable states of the film. The water filmson quartz within the range of thicknesses of 60 to10 nm (curve 1) are unstable, and are not realized.

Ametastable state of thick 3-films (h > 100 nm )forms as a bulk water layer is thinning out. The timeof their transition into a thermodynamically stablestate of thin a-films (h 10 nm ) depends on thecloseness of the capillary pressure to the critical one,Pcr (when ahlah

=

0), and on the surface area of {3-films. The larger the surface area the higher theprobability of the formation of nuclei of the stable a-

phase in metastable /3-films. In the framework of themicroscopic approach to the wetting theory, thep - a transition is denominated as a pre-wettingtransition [5].The existence of thick {3-films of water is con-

ditioned by the electrostatic repulsion of chargedsurfaces of the films (ne:> 0 ).As in this case(Ho ho + ..d) > 0, the 03B2 films are completely wettedby water. Hereinbelow (Sect. 9) this case will be

illustrated by comparison of the results of theexperimental determination of the values of h * withthe theoretical ones, calculated with equation (10).When the meniscus of bulk water is in contact with

a-films, the values of 4 in equation (3) may benegative in view of the H(h) isotherm partly enteringin the region of II 0. The change in the sign of theelectrostatic component of disjoining pressure(curve 1) is connected with the different values ofelectrical potentiels 1/11 and .p2 of the film surfaces.The known tabulated data of electrostatic forces [17]are used to calculate the lle (h) isotherms. The

calculation gives that at h 600, the repulsiveelectrostatic forces transform into attraction forces :lle 0. The repulsive forces do appear again as thefilm thickness decreases further, but these are al-

ready associated with the effect of molecular

(H. > 0 ) and structural (77g > 0 ) forces. Calculationswith equation (3) by using the theoretical isotherm 1(Fig. 3) lead to the contact angle of water on quartz90 = 5 [16], which is close to the experimental data.When the electrostatic repulsive forces are sup-

pressed or the film surfaces have different signs ofelectrical potentials w, the /3-branch of the isothermcannot

be realized. In this case, the isotherm shiftsinto the range of II 0 (curve 2, Fig. 3), whichcauses, in accordance with equation (3), the worsen-ing of wetting.On the contrary, when potentials wi and .p2 (of

the same sign) increase or approach each other intheir values, this causes an increase in the electro-

static repulsive forces.As a result, the whole iso-therm may be found within the H > 0 region, whichmust lead to the complete wetting (curves 3 and 4,Fig. 3).

Thus, the conditions controlling the wetting ofsolid surfaces by water may be formulated as follows.Two effects - those of electrostatic (03A0e) andstructural (77g) forces - can influence the shape ofthe isotherms H(h) of the wetting films of water.The dispersion forces depending on the spectralcharacteristics of water and the solid substrate, are

less sensitive to the composition of an aqueoussolution, temperature, and the surface charge.Three factors can influence the structural forces ;

namely - increasing the electrolyte concentration

and raising temperature, which leads to decreasingthe structural repulsion ; as well as through adsorp-tion of molecules, which changes the character of theinteraction of water molecules with the solid surface.

The worsening of wetting, which is required, forexample, for enhancing the flotation effectiveness, is

usually attained through adsorption of ionic surfac-tants. In this case, it would be of importance that asurfactant would be selectively adsorbed on one ofthe film surfaces, imparting to it a charge, whose

sign is reverse with regard to the charge of anothersurface. Thus the forces of electrostatic attraction

arise (lle 0 ), which shifts the isotherm into theII 0 region. Adsorption of surfactants may alsosimultaneously lead to hydrophobization of the solid

substrate, which reduces the repulsive structuralforces.Ahigh degree of hydrophobization may alsoreverse the sign of structural forces, too (77g : 0)[18, 19].At 03A0e 0 and IIS 0, still higher contactangle values may be attained [16].On the contrary, all the measures causing an

increase in the forces of electrostatic and structural

repulsion, improve the wetting. This aim is attainedeither by imparting a high potential of the same sign

to the film surfaces and/or through hydrophilizationof the substrate, as for example, by increasing thenumber of centers that are able to form hydrogenbonds with water molecules.Adsorption of nonionicsurfactants or polymers leads to an additional effectof the sterical repulsion of adsorption layers. Thus,in each specific case, one can choose the optimummethods for controlling the wetting.Asimilar program of the theoretical calculation of

contact angles on the basis of equation (3) wasperformed for alkanes on the Teflon surface [20]. Inthis simpler case, one could restrict oneself to the

takinginto account of

onlyone molecular

componentof disjoining pressure llm 0.Agood agreement ofthe calculation results with the experimental valuesof e o has thus been obtained.The region of the applicability of equation (2) is


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limited by such thicknesses of wetting films, whenthese can yet be considered as a liquid phase. Withpoor wetting ( 00 > 90 ), a two-dimensional adsorp-tion phase is formed on the solid surface, and thethickness of films does not exceed a monolayer. In

this case, another expression following from Gibbsequation, correlating the adsorption T with a changein the interphase tension ysv depending on theadsorbate vapour pressure, p, is applicable [21, 22] :

where ps is the pressure of the saturated vapour.Equation (14) transforms into equation (2) with a

formal replacement r = holv., where vm is themolar volume of liquid, and with the use of a knownthermodynamic relationship between the equilibriumvapour pressure over the film p and its disjoiningpressure II [1, 8] :

where R is the gas constant, and T is temperature.

4. Effect of concentration of electrolyte and surfac-

tants.

The main difficulty involved in calculating the con-tact angle values 00 from equations (2), (3), and (14)resides in that at 90 > 0 the isotherms of disjoiningpressure 77 (A) or the isotherms of adsorptionr (p / p s) are partly disposed in the supersaturationregion (H -- 0, p/ps > 1). The values of cos 0 0 are

proportional to the value of d - that is, to thedifference between the areas a and b in figure 3(curve 1).As it is difficult to determine experimen-tally the isotherms in the supersaturation region,these parts of the isotherms can be determined onlytheoretically. Such calculations were made on thebasis on the theory of surface forces for different

aqueous solutions [16, 23, 24].In figure 4 are represented the results of calcu-

lation with equation (3) of contact angles 00 for theaqueous KCI solutions of different concentrations

(curve 1) and different pH values (curve 2), whilepreserving in the latter case a constant ionic strengthof the solution, 3 = 10- 2 mole/1 [23, 24]. In carryingout calculation of 03A0e(A), the known dependences onthe concentration of quartz-solution potentials t/11and the aqueous solution-air interface t/12 were used.For the dispersion forces nm =A/6 7Th3,the valueof constantA= 7.2 x 10-13 erg was adopted.Aknown exponential dependence [16] was adopted for

the isotherm of structural forces Hs(h), whoseparameters were used as adjusting ones in makingthe theoretical calculation of 00 agree with the

experimental data [25].An increase in the values of

Fig. 4. - The results of calculation of contact angles

00 for aqueous KCI solutions of different concentrations(curve 1) and at different pH-values (curve2)..

9o with the electrolyte concentration (curve 1) maybe explained by two causes : i) a reduction in thethickness of the boundary layers of water, whichleads to a reduction in the structural repulsiveforces ; ii) a decrease in the values of potentials"11 and 03C82, which decreases the electrostatic repul-sive forces.

Adecrease in the contact angles as the pH valueincreases

(curve 2),was caused

mainly byan increase

in the potentials 03C81and "2 of the film surfaces as aresult of adsorption of the potential-determiningOH-ions. We note that at 3 of 10- 2 mole/1 (curve 2),the dependences of contact angles are described byusing the H(h) isotherm, including only two compo-nents of disjoining pressure ; namely- an electros-tatical, and a molecular one.At such an electrolyteconcentration the structural forces are small

(03A0s ~ 0), and these could not be taken into account.In figure 5 are presented the results of calculation

of the isotherms of disjoining pressure of the wettingfilms of 10- 3 mole/1 KCI aqueous solution withadditions of ionic surfactants. The isotherm indicated

by curve 6, is the initial one, for KCI solutionwithout addition of surfactants. In this case, the

potentials of the quartz and film surfaces wereassumed to be equal to : 03C81 = - 100 mV, and03C82 = - 25 mV, respectively. The value of 80 = 8(as is also shown in Fig. 4) was obtained by calculat-ing with equation (3). In the calculations done, theinfluence of surfactants was expressed as a variationin the values of potential 03C82 due to the adsorption ofsurfactant on the film-gas interface.Adsorption ofan anionic surfactant increases the negative values of

t/12 and leads to an improvement in wetting. Thus,for example, at 03C82 = - 35 mV the calculated valueof 00 reduces to 7, while at 03C82 = - 45 mV it reducesto 5.Afurther increase in the absolute values of


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Fig. 5. - 11(h) isotherms calculated for a 10- 3 mole/1 KCIaqueous solution with additions of surfactants at 1/1 =

- 100 mV = Const. ; lP2 = -100 (curve 1) ; 03C82 = - 75

(curve 2) ; 03C82 = - 65 (curve 3) ; 03C82 = - 45 (curve 4) ;03C82 = - 35 (curve 5) ; 112 = - 25 (curve 6) and 03C82 =+ 100 mV (curve7).,

03C82 (curves 1-3) corresponds to the complete wettingof the quartz surface.

Adsorption of a cationic surfactant charged posi-tively the film-gas interface (03C82 = + 100 mV). Whenthe substrate remains negatively charged (curve 7),the contact angle increases up to 28 owing to theelectrostatic attraction of the film surfaces (1-I,, 0).

The calculations are in good agreement with theresults of direct measurements of the contact anglesof KCI solutions with additions of anionic sodium-

dodecylsulfate and cationic cetyltrimethylammoniumbromide

[26].5. Contact angle hysteresis.

In a number of cases to form equilibrium contact

angle requires a long time. This is connected withthe retarded mass exchange between the bulk liquidand the thin wetting film [27]. The kinetics oftransition into the equilibrium state is controlled bythe viscous resistance of films and the diffusion of

the , dissolved components, whose equilibrium con-centrations in the bulk phase and in a thin film maybe different. In view of this, at first a partial

mechanical equilibrium can rapidly be established inthe bulk part of a drop or meniscus in the absence ofboth the mechanical and the thermodynamic equilib-rium with the film.Apossibility of realization of anumber of states of the mechanical equilibriumresults in the phenomenon of the static hysteresis ofthe contact angle [13]. In this case the transitionzone between the meniscus and the film plays asubstantial role (Fig. 6). The state of its mechanicalequilibrium, which is determined by fulfillment ofthe condition (1) breaks up at two values of thecapillary pressure PC PA and PC PR, and ac-

cordinglyat two values of the contact

angle

-

an

advancing 8 A > 8 0, and a receding 9 R e o.Aplurality of the mechanical equilibrium states can beestablished in the whole interval between 8 A and6 R. This interferes with determination of the equilib-rium value of the contact angle [28]. The better thewetting - that is, the larger the film thicknessho, the quicker occurs the transition into the equilib-rium state, and the smaller the difference between

the values of 8 A and 9 R.

Fig. 6. - Influence of the transition zone on the static hysteresis of the contact angle in a flat slit : (a) advancingmeniscus ; (b) receding meniscus.


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It was considered earlier that the contact anglehysteresis is due either to the surface roughness or toits chemical heterogeneity - the presence of theareas that differ from one another in the values of

equilibrium contact angles. Examination of the stabi-

lity of the transition zone has demonstrated thathysteresis is possible also on a smooth, uniform

surface, too. In this case, the values of 6A and

B R can also be determined on the basis of theisotherms of disjoining pressure, H(h) [13]. For theS-shaped isotherms (curves 1, Fig. 3) it was shownthat the values of 8 A are within the range between

00 and 90 (depending in the n (h) Eq.), whilst thevalues of 9 R are close to zero, because a thickmetastable a-film remains behind the retreatingmeniscus.

6. Wetting transition.

The macroscopic approach enables one also toexamine the known phenomenon of the wettingtransition, which is at present widely discussedwithin the framework of the microscopic approach[5, 29, 30].An increase in temperature T influences the

H(h) isotherms, the consequence of which is tran-sition from the complete wetting to the partial one.For quartz and glass substrates a worsening of thewetting by water is experimentally detected, as T is

raised [31-33]. In view of the H(h) isotherms con-sidered in figure 3, the transition of wetting, i.e.,transition from isotherms 4 and 3 to isotherms 1 and

2, can result from a number of causes. The mostprobable is the influence of temperature on thestructure of the boundary layers of water. It hasexperimentally been shown that the raising of tem-perature from 20 C to 70 C causes gradual disinte-

gration of the special structure of the boundarylayers of water and its approach to the bulk structure[34].Adecrease in the structural repulsive forces asT is raised, can enhance the extent of isotherm 3

(Fig. 3) enteringthe

rangeof II 0. This

maycause

transition from the complete wetting to the partialone. The structural mechanism of transition is con-firmed by a thermal reduction in the thickness of a-films [35], as well as by the fact that the higher thesubstrate hydrophilicity the higher the sensitivity ofcontact angles to variation in temperature [31].The cause of wetting transition is quite different

for the films of nonpolar liquids on dielectric subs-trates, when only dispersion forces are acting.As isknown [36], the disjoining pressure of such thin films(h 10-20 nm) is determined by a difference in thedielectric function of the substrate 1 and the

liquid 3 :

Here Il is the Planck constant, subdivided by2 tut ; 8 (w ) are the frequency dependences of dielec-tric permeability, where co = i 03BE is the circular

frequency along the imaginary frequency axis.The wetting transition corresponds to the change

in the sign of the integral in equation (16). Atsl > 83 in the frequencies range, making the maincontribution to the integral (1016_16 17 rad/s), thevalues of 03A0m > 0. This is the case of completewetting. At 03B51 E3, the transition to the partialwetting can occur. For the same liquid and substrate,this is possible when its temperature dependencesE (T) are different. The values of e depend ondensity (the number of molecules per unit volume).Then, at a higher coefficient of thermal expansion ofthe liquid, than that of substrate, the values ofE3 (larger than El at a low temperature) may become

less than el as temperature T is raised. In accordancewith equation (16), in this case an increase inT should lead to a reversal of the disjoining pressuresign and to the wetting transition. Since in this casethe film thickness changes jump-wise, the wettingtransition is of the first order.

Apossibility of the wetting transition ciue to thetemperature dependence of the electric potentials ofthe film surfaces is also not excluded. However, thisis possible at the adsorption energy values of poten-tial - determining ions comparable to kT. In thiscase, as in that of structural forces, the film thicknesscan

change gradually,and the transition

maybe of

the second order.

7. Dynamic contact angles.

Up to now, we have considered the state of ther-modynamic or mechanical equilibrium of the menis-cus-film system. When drops or menisci move, thedistribution of pressures throughout the transitionzone and the film changes, which cause also thecurvature of the meniscus to change. If we continuean undisturbed by surface forces meniscus profileuntil it intersects the substrate, then the contact

angle values determined by that formal method willindicate the dependence on the flow rate v. Thedynamic contact angles 8a begin to differ from thestatic values of 00 and exceed these at v > 10-3 cm/s

[37-39].The theory of dynamic contact angles has so far

been developed only for the case of completewetting, and when the meniscus advances at aconstant velocity onto an equilibrium wetting film.The numerical calculations were made on the basis

of equation (1) [40].Assuming that the conditiondh/dx 1 holds also for the flowing transition zone,

the following expression for the pressure gradientcan be obtained :


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983

Fig. 7. - The meniscus profiles of completely wetting liquid in a flat capillary. (a) at rest ; (b) and (c) in motion at anincreasing velocity, V = Const.

where an approximation K= d2h/dx2 is used for thesurface curvature of

liquid layer.In equation (17) P = Po - Pc(x) - n(x) is thehydrodynamic pressure in a film. It is equal to thepressure in the gas phase Po minus the local values ofthe capillary and the disjoining pressure [41].

Expression (17) for dP /dx may now be substitutedinto the known equation of hydrodynamics of thinlayers at v = Const. [42] :

where 11 is viscosity, h (x) is the local thickness of

liquid layer, and ho is the thickness of an equilibriumfilm.Asteady-state solution of the differentialequation gives the profile h (x ) of a flowing liquid.

In figure 7 are shown the computed profiles of ameniscus moving at different flow rates v through aflat slit [40]. For determination of ed, we have used apart of the profile of a constant curvature, which isdirectly adjacent to the flow zone, and which istherefore found still within the region of the slopingliquid layer dh/dx 1.

In figure 8 are pregented computed dependencescos 0 d on the capillary number Ca = v 11 / Y. Calcu-

lationswere

made for the isotherms H(h)=

A/h3(withA= 10-14 erg) and different half-widths of aslit H.As appears from figure 8, an increase in theflow rate causes an increase in the values of

03B8d. Differences of Od from the static value of00 = 0, begin to show up at Ca > 10-3. The nar-rower the slit and accordingly the smaller thethickness of an equilibrium wetting film ho - thesmaller the meniscus profile is disturbed.Experimental investigations of dynamic contact

angles for water on the quartz surface had shownthat the values of ad began to exceed the equilibriumvalues

( 6 0 =10 )at V 10-3

cm/s,thus

attaininga

value of 75 to 80 at v > 0.1 cm/s [39].As appearsfrom figure 9, these data only qualitatively agreewith the theoretical ones (full line in Fig. 9). Thequantitative discrepancy is associated with the fact

Fig. 8. - The dependences of the dynamic contact angle9d on the meniscus flow rate. (1) H =10-2 cm,ho =150 ; (2) H = 1.25 x 10- 3 cm, ho = 74 ; (3)H = 1.25 x 10-5 cm, ho =16.

v, culs

Fig. 9. - The experimental dependence 8d (v ) (shown bypoints) obtained for water in quartz capillaries 30 2013 35 03BCmin diameter. The solid line indicates the results of theoreti-

cal simulation [40].

that the theory has been developed only for theconditions of complete wetting and monotonous

H(h) isotherms (curve 4, Fig. 3). For water on


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985

An analytical expression for K was obtained for asimplified isotherm of disjoining pressure (6) [14] :

where a and t are the parameters of a modelisotherm II (h ).At r ~ oo, the values of Ktend to a constant value

of Ka = - 2 y t tg 9 0. For small values of (Jo, when agradual profile of the transition zone is formed, thevalues of Kare negative, and amount to 10- 6 dyne intheir order of magnitude, which is in agreement withthe experimental data for foam films [48].

9. Transition zone.

Quite recently, the profile of the transition zonebetween the meniscus and a film on the solidsubstrate has for the first time been experimentallyinvestigated, and the correctness of equations (8)and (10) of the theory of complete wetting has beenverified [49].The wetting films were formed on a polished

quartz plate 1 (Fig. 11) by making approach to it themeniscus of liquid in a tube 2 about R = 1 mm inradius. The liquid is sucked from the tube throughslots 3. The thickness of a film ho in the equilibriumstate with the meniscus surrounding it was obtainedfrom the intensity of reflected light. Depending on

the capillary pressure of meniscus Pc the radius ofthe film ro amounts to several scores microns.

Simultaneously with measuring thickness, the inter-ference rings from meniscus were photographe.This allowed its profile h (r) to be determined, wherer is the radial coordinate.

An analysis of experimentally obtained profiles ofthe meniscus for water and aqueous KCI solutions of

low concentration has shown that their continuation

does not intersect the substrate plane- that is, the

complete wetting takes place. The undisturbed pro-file was calculated with the Laplace equation for an

axisymmetrical meniscus :

where 0 (r) is the current angle value, for whichtg e = ah/ar (Fig. 11).

Solution of equation (20) allows the coordinatero to be determined, at which the theoretical profileof meniscus passes through a minimum. Its positioncorresponds to the thickness h * (compare Figs. 1 and11).As has been demonstrated above, the value ofS = h */ho is a quantitative characteristic of completewetting.

For the aqueous KCI solutions, Z. M. Zorin hasobtained the following values of ho and h * ; namely- for a 10- 4 molell solution ho =1025, andh.=1450; for a 10-3 mole/l solution, ho =

Fig. 11. - Formation of an equilibrium wetting filmhaving the thickness ho and radius ra on the solid substrate1 in pipe 2, when the liquid is being sucked off throughslit 3 under the effect of the capillary pressure.

590, and h * = 710. This leads to the h */hovalues equal to 1.41 and 1.2, respectively.The experimental values of S may be compared

with theoretical ones. Large values of the equilib-rium thicknesses of films ho and the completewetting indicate that the long-range electrostatic

repulsion forces predominantly act there.Assumingthe potentials of quartz surface t/1 1 and film surface

t/12, it is possible to calculate from Devereux and deBruyn tables [17] the isotherm of electrostatic forces03A0e(h). On the basis of references [50, 51], for10- 4 molell KCI solution, it is possible to assumethat t/11 = - 150 mV, and t/12 = - 45 mV, while for a10- 3 mole/1 solution, t/11 = - 125 mV, and 112 =- 45 mV. Under the condition of t/1 = Const., thecalculated isotherms lle (h ) are linearized in logarith-mic coordinates with the correlation coefficient

being equal to 0.996. This enables one to approxi-mate the /3-part of the isotherm by a power-function,H = Alh. For a 10- 4 mole/1 KCI solution n =

2.87 ;and for a

10- 3mole/1 KCI solution the value of

n was obtained to be equal to 6. Substituting thesevalues of n into equation (12), we obtain thetheoretical values of the parameter S, which are

equal to S = 1.5 for the 10- 4 mole/1 concentration,and to S = 1.2 for the 10- 3 mole/l. These valuessatisfactorily agree with the experimental ones.Similar results were obtained for water and for a

number of aqueous solutions [49].At a low electrolyte concentration, the extensionof the transition zone amounts to about 5-10 03BCm,which allows its investigation by means of opticalmethods. On the basis of a photogramme of theinterference patterns, Z. M. Zorin has calculated

the profile of the meniscus and that of the transitionzone (curve 1, Fig. 12). Curve 2 gives the profile ofthe meniscus that has not been disturbed by surface


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986

Fig. 12. - The profiles of the transition zone between themeniscus and a flat film

havingthe radius r ^ 10

03BCm,both

determined experimentally (curve 1) and calculated theor-etically (curve 3). Curve 2 indicates the profile of anundisturbed meniscus (10-4 mole/1 NaCI).

forces, as plotted with the use of equation (22).Aminimum on this curve determines the layer thick-ness h *.

Then the theoretical profile of the transition zonemay also be attempted to be constructed. However,the theory of the transition zone has so far been

developed only for a meniscus in a flat slit, and forthe isotherms of the type II =A/hn [11].Acorre-sponding equation for the profile of the transitionzone h (x ) has the following form :

where x is the tangential coordinate.The width of the slit H, which is equivalent to the

round cell, was determined by equalizing the capil-lary pressure of the cylindrical meniscus in a flat slit(Fig. 1) and that of the meniscus in the tube(Fig. 11). This gives in both cases the equality of thedisjoining pressure Ho = Pc in the films. The profileof the transition zone calculated with equation (23)(at n = 2.87, and A = 9.54 x 10-12, if77is expressedin dynelcm2,and h in cm) is indicated by curve 3 infigure 12.Agreement between the experimental andthe theoretical profile is a satisfactory one.

10. Conclusion.

Thus, the review of the current state of the macro-

scopic theory of the wetting shows that the disjoiningpressure isotherms can be successfully used fortheoretical evaluation of contact angles. This ap-proach is limited to solid surfaces which are suffi-ciently lyophilic that the contact angles did notexceed a value of about 30 2013 40.

The object of further investigations will consist inthe obtaining of experimental isotherms of disjoiningpressure for wetting films of different liquids ondifferent substrates. This will enable one to rendermore correct the values of the parameters of the

isotherms, which will make the results of the calcu-lation more reliable.

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[40] STAROV, V. M., CHURAEV, N. V.,KHVOROSTJANOV,A. G. , Kolloid. Zh. U.S.S.R.39 (1977) 201.

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Col l o i d & Po l ymer Sc i . 256 , 784-792 (1978) 1978 Dr. Dietr ich Steinkopff Verlag, DarmstadtI S S N 0 3 0 3 - 4 0 2 X / A S T M - C o d e n : C P M S B ( f o rm e r ly K Z ZP A F )

Equipe de Recherche C. N . R .S . associ& a l' Universitd Pa ris V , Pa ris (France)S t r u c t u r a l d i s j o i n i n g p r e s s u r e i n t h i n f i l m o f l i q u i d c r y s ta l s

I . : T h e r m o d y n a m i c s a n d F r e d e r ik s z t r a n s i t io n w i t h s u r fa c e f i e l d s * )E . P e r e z , J . E . P r o u s t , a n d L . T e r - M i n a s s i a n - S a r a g a

W i t h 8 f i g u re s(R ece i v ed J an u a ry 4 , 1 9 77 )

L i s t o f s ym b olsh T : f i l m t r an s i t i o n t h i ck n es shs : c r i t i c a l f i l m s t ab i l i ty t h i ck n es sq0 su pe r-s c r i p t : f i l m p h as eA : a reaf : f o r ce p e r u n i t a r eah : f i l m t h i ck n es sT : t e m p e r a t u r eV : v o l u m eF : f r ee en e rg y, / : d i s j o i n i n g p re s s u red r : w o r k p e r f o r m e d i n v a r y i n g hPc : c ap i l l a ry p re s s u res : s p ec i f i c s u r f ace en t r o p yA : f i l m t en s i o n# : ch em i ca l p o t en t i a la : s u r f ace t en s i o nF ( h ) : n u m b e r o f m o l e c u l e s p e r u n i t s u r fa c eo f f i l m a t t h i ck n es s h: f i l m p e r i m e t e rs u b s c r i p t h : u p p e r su r f a c es u b s c r i p t 0 : l o w er s u r f acew s u r f ace f r ee en t h a l p yG : f r ee en t h a l p yW ' : s p ec i f ic b u l k f r ee en t h a l p y i n f i l m o f L .C .W b : s p ec i f ic b u l k f r ee en t h a l p y o f L .C .W(h ,z ) : s p ec i f ic ex ces s f r ee en t h a l p y i n t h i n f i l mo f L . C .: l i q u i d c ry s t a l d i r ec t o r0 : a n g l e b e t w e e n d i r e c t o r a n d x , y p l a n~b : az i mu ta l ang leW a : e l a s t i c en e rg y d en s i t yW u : s p ec i f i c b u l k f r ee en t h a l p y o f t h eu n d i s t o r t e d f i lm

* ) T h i s p a p e r h a s b een p re s e n t ed a t t h e 5 0 t hE U C H E M C o n f e r e n c e C h e m i s t r y o f I n t e r f a c e s ,C o l l i o u r e ( F r a n c e ) , A p r i l 2 8 -3 0 , 1 97 6. a n d V i e I n t e r -n a t i o n a l L i q u i d C r y st a l C o n f er e n ce , K E N T , U . S . A .2 3 - 2 7 A u g u s t 1 9 7 6 .P a r t I I o f t h i s p a p e r h a s b e e n p u b l i s h e d i n t h e i s s u e :No 255 , 1133--1135 o f th i s journal .

w', Aw' : i s o t r o p i c a n d a n i s o t r o p i c c o n t r i b u t i o nt o t h e i n t e rf a c i a l t e n s i o n o f L . C .K : av e rag e e l a s t i c co n s t an t o f n em a t i cl i q u i d c ry s t a lAu : n o n e l a s t i c co n t r i b u t i o n t o th e f il mt e n s i o nAa : e l a s t i c co n t r i b u t i o n t o t h e f i l m t en s i o nF e ] : e l a s t i c t o rq u eFs : s u r f ace t o rq u er )a : c l a s s i ca l d i s j o i n i n g p re s s u re s : s t ru c t u ra l d i s j o i n i n g p re s s u reB ( h) : V a n d e r W a a l s p a r a m e t e rAn : o p t i c a l a n i s o t r o p y o f t h e l iq u i d c r y s ta l6 : o p t i c a l r e t a rd a t i o nI : l i g h t i n t en s i t yh * : r ed u ced t h i ck n es s

I n t r o d u c t i o nT h e p r o p e r t i e s o f d i s p e r s e sy s t e m s i n l i q u i d s

d e p e n d o n t h e n a t u r e a n d s t a t e o f t h e i r in t e r -f a c es a n d o f t h e t h i n f i lm s s e p a r a t i n g t h ep a r t i c le s o f t h e d i s p e r s i o n .

A l t h o u g h i n t e r m o l e c u l a r f o r ce s a re s h o r t -r a n g e , t h e r e i s i n c r e a s i n g b e l i e f t h a t n e x t t ot h e s e i n t e r f a c e s , t h e i n i t i a l p e r t u r b a t i o n o f th et w o p h a s e s i n c o n t a c t t a k e s p l a c e o v e r a f i n i t et h i c k n e s s o f t h e l i q u i d i n t e r m e d i a t e f i l m . I t h a sb e e n a s s u m e d t h a t t h e p e r t u r b a t i o n m a y s t r u c -t u r e ( 1 , 2 ) o r d e s t r u c t u r e ( 3, 4) t h e l i q u i d . T h ee f fe c t o f t h e d i s p e r s i o n c o n s t i t u e n t s p o l a r i t i e sh a s b e e n e m p h a s i z e d , a l t h o u g h r e c e n t e x -p e r i m e n t s ( 4 , 5 ) h a v e d e m o n s t r a t e d t h a t th i sp a r t i c u l a r p r o p e r t y i s n o t e s s e n t ia l fo r i n t e r -f a c ia l s t r u c t u r i n g .

F o r t h i n f i l m s , Derjaguin ( 1, 2) h a s p o s t u l a t e dt h e o c c u r r e n c e o f a s t r u c t u r a l d i s j o i n i n g p r e s -s u r e r e l a t e d t o t h e i n t e f f a c i a l s t r u c t u r i n g .

T h e p o l a r t h e r m o t r o p i c l i q u id c r y s t a l s o rL . C . , w h i c h h a v e a s t r o n g m o l e c u l a r f i e ld ,

W 704


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Per ez et aL, Structur al disjoiningpressure in thin film of liquid crystals, I. 785a r e s y s t e ms a p p r o p r i a t e f o r t h e s t u d y o f i n t e r -fac ia l s t ruc tu r ing . E las t i c fo rces r e s i s t anys t r u c t u r e p e r t u r b a t i o n o r d i s t o r t i o n i n b u l k s ot h a t a n i n t e r r a c i a l s t r u c t u r i n g wi l l p r o p a g a t ea w a y f r o m t h e i n t e r f a c e .

T h e p r o p e r t i e s o f L C i n b u l k h a v e b e e ne x t e n s i v e l y s t u d i e d ( 6 , 7 ) . A r e v i e w o f t h er e c e n t r e s u l t s o n t h e e f f e c t o f s u r f a c e f o r c e so n L C o r i e n t a t io n s p e ci fi c it y m a y b e f o u n di n r e f . ( 7 ) . I n t h e p r e s e n t p a p e r we d e a l o n l ywi t h t h e s u r f a c e f o r c e s o f p h y s i c a l c h e mi c a ln a t u r e .A d i r e c t a p p r o a c h t o t h e s t u d y o f t h e s ef o r c e s i s t h e s t u d y o f we t t a b i l i t y o r a d h e s i o nof the sys tem LC-subs t ra te so l id (8 , 9 , 10 , 11)o r l i q u i d ( 1 2 ) w h i c h p r o v i d e i n f o r m a t i o n o nL C s u b s t r a t e i n t e r r a c i a l t e n s i o n v a r i a t i o n wi t hchanges in in te r rac ia l s t ruc tu re .F i n a l l y , s u r f a c e s t r u c t u r i n g a n d o r i e n t a t i o ni n t h i n f i l ms o n s o l i d s u b s t r a t e s h a s b e e n c o n -s i d e r e d a s a n e p i t a x i a l g r o wt h ( 2 ) .

I n t h e p r e s e n t p a p e r , th e t h e r m o d y n a m i c s o fa s y m m e t r i c n e m a t i c l i q u i d c r y s t a l , N L C , t h i nf i l ms o n wa t e r a r e d i s c u s s e d , u s i n g t h e c o n -c e p t o f s u r f a c e t e n s i o n a n i s o t r o p y ( 1 3 , 1 4 ) i . e .v a r i a ti o n o f N L C i n te r ra c ia l t e n s i o n w i t hmo l e c u l a r o r i e n t a t i o n a t t h e i n t e r f a c e .

Derjagu in (1), Sheludko (15) and E v e r e t t (16,1 7 ) t h e r m o d y n a m i c a p p r o a c h e s a r e u s e d t od e d u c e t h e a p p r o p r i a t e f i l m t h e r m o d y n a m i cp o t e n t i a l a n d d i s j o i n i n g p r e s s u r e . Us i n g t h ef o r m a l i s m o f J e n k i n s a n d B a r r a t t (13) and theresu l t s o f Parsons (14) , we d i scuss the pa r t i cu la re q u i l i b r i u m d i s j o i n i n g p r e s s u r e f o r N L C f i l m so f v a r i o u s t h i c k n e s s e s f o r v a r i o u s s u r f a c eo r i e n t a t i o n s u n d e r t w o a l t e r n a t i v e a s s u m p -t i o n s :

A) t h i c k n e s s i n d e p e n d e n t s u r f a c e o r i e n t a -t i o nB ) t h i c k n e s s d e p e n d e n t s u r f a c e o r i e n t a t i o n .I n b o t h c a s e s , i t i s p r e d i c t e d t h a t v e r y t h i nf i l ms a d o p t a n o n - d i s t o r t e d , e p i t a x i a l , s t r u c -t u r e wh e r e a s t h i c k f i l ms a r e d i s t o r t e d .H o w e v e r , t h e t r a n s i t i o n t h i c k n e s s h T f r o mt h e u n d i s t o r t e d t o t h e d i s t o r te d N L C f il m isd i f fe ren t in cases (A) and (B) .

Th i s d i f fe rence o r ig ina tes spec i f ica l ly in thed i f f e re n t b e h a v i o u r o f t h e d i s j o i n i n g p r e s s u r ei n t h e t wo c a s e s ( A) a n d ( B ) . T h i s t r a n s i t i o ni n s t r u c t u r e , a n a l o g o u s t o t h e w e l l k n o w nF r e d e r i k s z t r a n s i t i o n ( 18 ) o b s e r v e d w h e ne x t e r n a l e l e c t r i c o r ma g n e t i c f i e l d s a r e a p p l i e dt o NL C s l a b s , i s d i r e c t e d b y s u r f a c e f o r c e s i n

t h e c a se o f a s y m m e t r ic t h i n f i lm s o f N L C o ns u b s t r a t e s .I n t h e c o n c l u s i o n , w e c o m p a r e o u r r e s u l t sw i t h t h o s e o b t a i n e d b y d i f f e r e n t m e t h o d s f o rd i f fe ren t sys tems .

T h e r m o d y n a m i c p o t e n t ia l t h in f il m s a n ds t r u c tu r a l d i s j o i n i n g p r e s s u r e o f l i q u i dcrys t a l sT h e s y s t e m o f fi g u r e 1 i s a n a l o g o u s t o t h a t o f

A s h , E v e r e t t a n d R a d k e (16, 17).T h e t h i ck n e s s h o f f lu i d c o r r e s p o n d s t o a

f o r c e A f e x e r t e d o n t h e p l a te s . W o r k i s p e r -f o r m e d t o v a r y h : d T = - - A f d h . D e r ja g ui n' sd is jo in ing p ressu re (1 , 2 ) i s by de f in i t ion :~ 7 = - - ( 1 / A ) ( 8 F I S h ) A , T , v . A s d F = d z , i tf o l l o w s t h a t ~ = f a n d i s m e a s u r e d b y t h ecap i l l a ry p ressu re (1 , 2 ) Pc .

T h e v a r i a ti o n o f t h e f re e e n e r g y F ~ o f th ef i lm phase , no ted ~ , i s equa l tod F ~ = S ~ d T q - A d A - f A d h q - # dn ~ - - p d V ~

[ 1 ]w h e r e A i s t h e f i lm t e n s i o n d e f i n e d b y Sheludko(1 5) . I n t e g r a t i o n a t c o n s t a n t T , h , # , A l eads toF ~ = A A q - # n ~ - - p V ~ . [2 ]

A r e l e v a n t t h e r m o d y n a m i c p o t e n t ia l ~ f o rf i lm phases i s the re fo re :q ) = F ~ q - p V ~ - - n ~# = A A = 2 ~ A

= g * - - n # [3 ]wh e r e ~ i s t h e s u r f a c e t e n s i o n o f t h e f i l m ( 1 6 ,1 7) a n d ~ c o r r e s p o n d s t o d e f i n e d v a l u e s o fT , P , # . F r o m [2 ] a n d [3 ] t h e v a r i a t i o n o f is :d ~ = -- S ~ d T q - A d A - - f A d h - - n ~d #

+ [4]~ P

Tp VlJ- ~ h - I

f l u i dA f .

Fig. 1. Film of fluid between tw o surfaces. Uniformtemperature T, pressure p, che m ical potential #;volume V, n mo le number, A surface area, h separa-tion, f force on plates


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7 8 6 Colloid and Polymer Sdenee, Vol. 256 No. 8 (197 8)

a n d t h e d e f i n i t i o n o f t h e e q u i l i b r i u m d i s -j o i n i n g p r e s s u r e i s d e d u c e d :1 [ O q }] = f = ~ . [5]A - ~ - T , A ,~ ,~

A c o r r e s p o n d i n g G i b b s - D u h e m e q u a t i o n f o rf i lms i s ob ta ined f rom [3] and [4 ]- - s ~ d r - - d a - - f d h - V ( h ) d e + h dp = 0 [6]w h ich l ead s to a sec o n d ex p ress io n fo r ~7[ ] [ d l

_ dd = - - 2 L - N -J T , ~ , . [ 7 ]= - ~ - T , ~ , ew h e n e i t h e r q~ o r A a r e k n o w n , t h e d i s j o in i n gp ressu re ~ may b e d ed u ce d u s in g [5 ] o r [7] .N e m a t i c l i q u i d c r y s ta l ( N L C ) t h i n f il m sa) T hermod ynamic pote nt ia l and equi l ibrium ofstructure

J e n k i n s a n d B a r r a t t (13 ) fo rm al i sm i s u sedt o f i n d f o r a d o m a i n o f N L C o f t h i c k n e s s h ,c ro ss sec t io n A an d p e r im e te r 65 as sh o w n inf igure 2 .T h e f i l m d o m a i n a r e a i s l a r g e c o m p a r e d t oi ts t h i c k n e ss . E x c e p t o n t h e p e r i m e t e r , w h e r e ad e f e c t o c c u r s , t h e m o l e c u l e s a r e u n i f o r m l yo r i e n t e d o n e a c h m o l e c u l a r l a y e r o f t h e N L Cf il m . I t i s a s s u m e d t h a t t h e c o n t r i b u t i o n o f t h ed e f e c t fr e e e n t h a l p y t o t h e f r e e e n t h a l p y o f t h ed o m a i n i s n e g li g ib l e . T h e f il m is a s y m m e t r i ca n d h a s a n u p p e r s u r f a c e f r e e e n t h a l p y w ~ ( h )a n d a l o w e r s u r f a c e f r e e e n t h a l p y wo(h) d e-p e n d e n t o n t h e l o c a l m o l e c u l a r o r i e n t a t i o n sw h i c h m a y v a r y w i t h h.T h e f i lm f r e e e n t h a l p y is e q u a l t o :

G * = y W ' d v + y [~,o(h) + w ~(h )]dA . [8]V A

, "

W o Asubs t ra te z< 0F i g . 2 . D o m a i n o f t h i n f i lm o n a s u b s t r a t e : A I = a r e a ;V I = v o l u m e . W ' = b u l k s p e c i fi c f r e e e n t h a l p y ; w /~ ,Wo = s u r f a c e s p e c i f i c f r e e e n t h a l p y

L e t t h e s p ec if ic b u l k f r e e e n t h a l p y i n t h ef i lm W ' = W ' (h, z ) b e :W ' (h , z) = W ( h , z ) q - W b [9]w h e re W~ = 9 # i s t h e sp ec if ic b u lk f r ee en -t h a l py o f t h e v e r y t h ic k a n d e x t e n d e d d o m a i na n d W ( h , z) i s t h e ex cess f r ee en th a lp y d en s i tyin th e th in f i lm .F ro m [8 ] an d [9 ] w e o b ta in fo r G th e ex -p r e s s i o n :

hu s = A ~ { e~ h + Y g ( h , z ) d z + w o ( h )0+ w ~ ( h ) ) . [ 10 ]

T h e sp ec i f ic f i lm p o ten t i a l # i s o b ta in edf ro m [9 ] an d [1 0 ] :(q~/A) = A ----[ G e ( h ) I A ] - - q l* h [11]

A t h e s p ec if ic fi lm t h e r m o d y n a m i c p o t e n ti a lfo r u n i t su r face A is o b ta in ed f ro m [1 0] an d [11 ]hA = f [ W ( h , z ) ] d z J r w o (h ) q - w h ( h ). [12]

oW h e n h - ~ oo , w e h a v e A m = (W o q - W h ) ~ .T h e d i s j o i n i n g p r e s s u r e i s o b t a i n e d f r o m[5] [7] and [12]:

h

0O(w~ + wo) = Pc. [13]Oh ~,p,~W h e n P c = O , t h e e q u i l i b r i u m t h i c k n e s s o ft h e f i l m d o m a i n i s g i v e n b y t h e s o l u t i o n

o f t h e e q u a t i o n :h

0- - [O(w~+w)]T,e,v=O'oh [14]b) tT,x p r es s io n f o r A a n d ,1 o f N L C

N L C a r e ch a r a c te r i ze d b y th e s t r o n g m u t u a la l i g n m e n t o f t h e i r m o l e c u l a r a x i s ~ a l o n g a nax is o f u n iax ia l sy m me t ry . In f ig u re 3, a s imp l i-f ied case (6 , 7 ) i s sho w n.F i g u r e 4 a s h o w s a n u n d i s t o r te d d o m a i n .F i g u r e 4 b s h o w s a d is t o r te d d o m a i n s u c h th a t


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Per ez e t aL, Structura l disjoiningpressure in th& fi& of liquid crystals, I. 78 7

~ I I / / / /~z = s in O(z)/ ! /

/ / /

F i g . 3 . O r i e n t a t i o n o f t h e u n i a x ia l s y m m e t r y a x i s ~ ; nx ,n a = c o m p o n e n t s o f ~ ; 0: a z i m u t a l a n g l e

t h e a n g l e 0 b e t w e e n h a n d t h e x d i r e c t i o n v a ri e sf r o m t h e b a s e t o t h e t o p o f t h e f i l m . T h i sd i s t o r t i o n a d d s a n e l as t ic p o s i ti v e c o n t r i b u t i o nt o t h e f r e e e n t h a l p y d e n s i t y o f t h e u n d i s t o r t e df i l m o f t h i c k n e s s h . T h i s c o n t r i b u t i o n i s v e r yr o u g h l y ( 6 , 7 ) :1w e = g K [ 15 ]

W h e r e K i s t h e a v e r a g e e l a st ic m o d u l u s o f th eN L C . T h e n , i f t h e s p e c if ic b u l k f r e e e n t h a l p yo f t h e u n d i s t o r t e d f i lm i s W n ( h, z )W ( h , z ) = W ~ ( h , z ) - - 9/* q - W e ( h , z ) . [16]E l i m i n a t i n g W ( h , z ) f r o m [ 1 2 ] a n d [ 1 6 ]

h0 h4 - f g e ( h , z ) d h + w o (a ) w h ( h ) . [ 1 7 ]

o

A c c o r d i n g t o r e f e r e n c e s [ 1 3 , 1 4 ] , t h e a n i -s o t r o p y o f t h e i n t e r r a c i a l t e n s i o n i s e x p r e s s e da s f o l l o w s :w h = w h + A w h s in 2 0 hWo = Wo + Aw o sin20o [ 1 8 ]w he re wh (z~ /2) < wh (0) an d Wo@ /2 ) > Wo (o).F r o m [ 15 ], [ 1 8 ] a n d [ 1 7 ]:

h

A = A ~ , + A a = f [W ' (h , z ) - - 9 # ] d z + w f ,0h

, x fW o + - ~ \ d z ] d z + A w o s i n 2 0 o0

+ A w h sin20h. [19]A u, A a a r e r e s p e c t iv e l y t h e c o n t r i b u t i o n s

i n d e p e n d e n t o r d e p e n d e n t o n 0 . D i s t o r t i o no r c h a n g e i n Oh o r 0 o m o d i f i e s t h e t h e r m o -

d y n a m i c p o t e n t i a l A a , t h e c o n t r i b u t i o n A ub e i n g i n d e p e n d e n t o f O h o r 0 o.T h e e q u i l i b r i u m v a l u e o f (dO~dr) o r O ( z ) iso b t a i n e d b y m i n i m i z i n g A a as usua l (6 , 7 ) ,u s i n g t h e e q u i l i b r i u m b o u n d a r y c o n d i t i o n sd i s c u s s e d in r e f e r e n c e s (1 3 , 1 4 ) . I t i s o b t a in e d :

Oh -- Oo dO, Oh -- Oo0 ~ - - ~ z O r d z - - - ~ [ 2 0 ]A t e a c h s u r fa c e , t h e e q u i l i b r iu m o r i e n t a t i o no f N L C m o l e c u l e s i s t h e r e s u l t o f b a l a n c e o ft w o t o r q u es :

- - * P e l : e l a s t i c t o r q u e o p p o s i n g d i s a l i g n m e n to f m o l e c u l e s- - / ' 8 : s u r fa c e t o r q u e o p p o s i n g a n i n c r ea s e o fs u r f a c e fr e e e n t h a l p y o r s u r f a c e t e n s i o n .

T h e t o r q u e s a r e e q u a l r e s p e c t iv e l y toa) F e l = K "~zb ) I s , h = A w h s in 2 0 h ; F , , o = A w o s in 200 .

[211F r o m [2 0] a n d [ 2 1 ] t h e e q u i l i b r iu m m o l e c -

u l a r o r i e n t a t i o n a t t h e f i l m s u r f a c e a r e :K ( O h - - 0 0 )a) h - - A w h s in 20hK (0 h - - 0 o )b ) h - - A w o s in 200

A w h < 0

A w o > 0[22]

F r o m [ 19 ] a n d [ 20 ] w e d e d u c e :K ( 0 h - 0 0 ) 5A = Au + -~- h ' + Awh s in20h

- k A w o sin20o [23]

0h i i ; ~,Oh /

/So o0a b

Fig. 4. Thin films of N LC . a) undistorted Oh = Oo = 0 ;b) d istorted 0h # 0o = O


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788 Colloid and Polymer Science, Vo l. 256 N o. 8 (1978)an d f ro m [7 ] an d [2 4 ] a l lo w in g fo r [22 a] a n d[2 2 b ] , t h e fo l lo w in g ex p ress io n i s o b ta in edfo r th e d i s jo in in g p ressu re :

~ = ~ q - ~ s = - - ( - ~ b ) T , p , u- t - 2 K (O h --h 0 0 ) q - 2K_h ( O h - - 0 o ) " -~h 'O[24]

F~ is the c lass ical d is jo in ing p ressure (1 , 2 )o f t h e u n d i s t o r t e d N L C o r F a = B ( h ) h - a. T h el a st t e r m s a re s t a n d i n g f o r r /s , t h e s t r u c t u r a ld i s jo in in g p ressu re .

c) Sta bil ity a nd transition critical fil m thicknessT h e b o u n d a r y c o n d i t i o n s [ 22 aJ h a v e b e e nd i scu ssed in r e fe ren ce (14 ).I t i s f o u n d t h a t , w h e n 00 = 0 , a c c o r d i n g t oth e f ig u res 4 a an d 4 b , t h e so lu t io n o f [2 2 a] i sKa) 0 a = 0 h < h T = 2 d w hKb ) Oh 0 h > h T - - 2 d w h " [25]

M a t e r ia l a n d e x p e r ime n t a l me t h o d sT h e N L C : 4 - - 4 ' - p e n ty l c y a n o b i p h e n y l (5 C B)h a s b e e n k i n d l y o f f e re d b y B D H . i t s p u r i t y i sb e t t e r t h a n 9 9 % .T h e su b s t r a t e : d i s t i l l ed w a te r 3 X i s sw e p tb e f o r e d e p o s i t i n g t h e N L C . T h e f i l m s a r ef o r m e d b y s p r e a d in g 2 - - 6 /~ l o f N L C o nv ar io u s a reas : 4 , 1 0 , 4 0 sq cm, o f su b s t r a t eco n ta in ed in an o p t i ca l ce l l a t 2 3 C . T h e ce l li s e x a m i n e d b e t w e e n c r o s s e d N i c o l s , u s i n g ac o m m e r c i a l p o l a r i z i n g m i c r o s c o p e a n d t r a n s -m i t t e d l i gh t .I n g e n e r a l , 2 0 X a n d 1 0 0 X m a g n i f i c a t i o n sw e r e u s e d a n d 2 00 X e x c e p t io n a l ly .T h e o p t i c a l a n i s o t r o p y o f 5 C B i s 3 n ~

+ 0 , 1 5 ( 1 9 ) . T h e o p t i c a l r e t a r d a t i o n f o r a ni n c i d e n t l i g h t c r o s s i n g t h e f i l m o f f i g u r e 4 ao f th i ck n ess h is eq u a l t o d = h An .T h e f i g u r e 5 r e p r e s e n t s t h e e x p e r i m e n t a ls e t up . A c o m p e n s a t i n g p l a te L e q u i v a l e n t t oa r e t a rd a t io n o n d0 is p l aced b e tw een p o la r i ze ra n d s a m p l e . F o r a u n i f o r m f il m , w i t h L a b s e n t ,t h e i n t e n s i t y o f t h e l i g h t c r o s s i n g t h e s a m p l e

B e lo w a t r an s i t i o n th i ck n ess h T th e s t ru c tu reo f f ig u re 4 a i s s t ab le . A b o v e h T t h e s t r u c t u r eo f 4 b i s s t ab le . T h i s t h i ck n ess h T d e p e n d s o nth e r a t io ( K / A w h ) i . e . o n t h e m a g n i t u d e s o ft h e t o r q u e s F e l a n d F s .F r o m [24 ] an d [2 5 ] i t fo l lo w s th a t t h e s t ru c -tu ra l t r an s i t i o n w i l l mo d i fy th e d i s jo in in gp ressu re ex p ress io n as fo l lo w s :B (h) K O~a) ~ - - h a q - ~ - h---g h > h TB (h) h < aT [26]b) ~7 - - ha

T h e s t ru c tu ra l d i s jo in in g p ressu re is p o s i t i v ean d s t ab i l i zes th e f i lm sh o w n in f ig u re 4 bo n ly .F r o m [2 2 a], [2 5 a an d 2 5 b ] a n d [2 6] a fin a lex p ress io n i s o b ta in ed fo r [ 2 6 ] .B (h) 1 Aw ~ sin~ 2 Oh [27]

T h e c o n d i t i o n f o r N L C t h i n f i l m s t a b i l i t yis d iscussed below. I t leads to a cr i t ica l s ta-b i l it y th i ck n e ss hs ~ hT.

' g i x . . . . . . . .

Fig. 5. Experim ental device for observation of liquidcrystal film ; P i polariser; A l l or Aa_: analyser parallelor perpendicular to the polariser; h: f ilm thickness;~: liquid cry stal director defining the optica l axis;/ /2 0 = polar angle; : azimu tal angle


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Pe rez e t a l . , S tru c tura l d is jo in ing pressure in th in f i lm of f iquid crysta ls, I . 789

p ara l l e l o r p e rp en d icu la r t o th e p o la r i ze r Presp ec t iv e ly i s eq u a l t oa) I / / = I o l l - ( s i n 2 4)~ s in ( ~ - ~ ) ,b ) I z = Io ( s in z 2 ) 2 s in ( - ~ - ) [28 ]w h e r e Io i s the inciden t l igh t in tensi ty ( f ig . 5 ) .W h e n w h i t e l i g h t i s u s e d , a c o l o u r e d i m a g ei s o b ta in ed .W h e n p o l a r i z e r a n d a n a l y z e r a r e p a r a l l e l ,t h e z e r o t h o r d e r o f t h e c o l o u r s e q u e n c e i sw h i t e ( s i l v e r ) , I t i s b l ack fo r p e rp en d icu la rp o l a r i z e r a n d a n a l y z e r . T h e o t h e r c o l o u r s a r ed e t e r m i n e d b y (d /k ) , t h e r e s p e c t i v e i n t e n s i t yd ep en d in g o n . Wh en L i s i n se r t ed ( f ig . 5 ) ,c o r r e s p o n d s t o t h e f o l l o w i n g v a l u e s o f :~ ' = d + d o fo r = - - ~ - : k =

7~d " = ~ - - d 0 fo r - - 4 ~ n "F o r d i s to r t e d f i lms (f ig . 4 b ) , t h e av e rag eo p t i ea l r e t a rd a t io n i s ca l cu la t ed u s in g th e

v a r i a t i o n o f t h e p o l a r a n g l e o f m o l e c u l a r a x i s0 (z) in t he f i lm g ive n b y [20].F o r u n i fo rm samp les , d o = d co s~ 0 (20 ). W ea s s u m e t h a t f o r a d i s t o r t e d s a m p l e t h e a v e r a g ere t a rd a t io n g i s eq u a l t o :

h= An j" co s ~ O(z) dz [29]0w h ere h i s t h e r ea l f i lm th i ck n ess .

R e s u l t sa) Or ien ta t ion o f N L C mo/ecu /es a t the f i lmi n t e r f a c e s

T h e t h i n f il m s s t u d ie d d i s p l a y m a n y d e f e c t so f o r i e n t a t io n . T h e s e a r e d e s c r i b e d i n r e f e r e n c e( 12 ). H o w e v e r , t h e e x t e n s i o n o f t h e u n i f o r md o m a i n s o f t h e o r d e r o f 1 m m i s l a r g e c o m -p a r e d t o t h e w i d t h o f t h e d e f e c t w a ll s o r l in e sw h i c h a r e o f t h e o r d e r o f 1 0 # m . W e c a na s s u m e t h a t t h e i r c o n t r i b u t i o n t o t h e f r e ee n t h a l p y o f t h e u n i f o r m d o m a i n s i s n e gl i g ib l e .T o f i n d t h e v a l u e o f Oh, w e m e a s u r e d ~ f o rv a r i o u s k n o w n v a l u e s o f h a n d u s e d e q u a t i o n s[20] and [29].

0, 70,60 ,5Off I E

0,2O ,l

1 2 3 ~ 5 6 7 8 9h . m i c r o n s

F i g . 6 . V a r i a t i o n o f t h e a v e r a g e r e t a r d a t i o n ~ w i t h t h ef i lm t h i c k n e s s h . I ) u n d i s t o r t e d . I I ) d i s t o r t e d f i l m s

T h e v a l u e s o f h w e r e d e d u c e d f r o m t h ek n o w n v o l u m e s v o f s pr e a d N L C a n d t h ea rea a o f t h e su b s t r a t e . T h e e r ro r o n h i ss m a l l er th a n 1 5 % . T h e r a n g e o f v a l u e s o f hi s 0 . 5 - - 1 0 # m .T h e o p t ic a l r e t a r d a t i o n 8 m e a s u r e d iss m a l l e r t h a n 1 # m . T h e o b s e r v e d c o l o u r sco r resp o n d to th e 1 s t , 2 n d , 3 rd o rd e r s i nN e w t o n c o l o u r s e q u e n c e . T h e y a l l o w f o r t h ee v a l u a t i o n o f 8 w i t h a s a t i s f a c t or y a c c u r a c y

~oas ~ ___ N - ~ - w he re ,to ~ 0 ,56 # m for w hi tel ig h t . T h e f ig u re 6 r ep resen t s t h e r e su l t s .T h e s lo p e o f I i n f ig u re 6 i s eq u a l t oA n = 0 . 1 5 . T h e s l o p e o f H i n t h e s a m e fi g u r eis equal to 0 .07 and c lose to ( A n ~ 2 ) = 0 . 0 7 5 .T h e i n t e r c e p t o f H w i t h t h e o r d i n a t e a i r s i seq u a l t o 0 .0 8 . T h e t r an s i t i o n I - ~ I I o c c u r s a th =2 .2 - t -0 ,1 m .I n t e r p r e t a ti o n o f t h e r e s u l t s

F r o m [29 ], [2 0 ] an d [25 ], w e o b ta in th ea v e r a g e r e t a r d a t i o n

1 1 A n h= ~ zln h q- ~ 0---7-sin 2Oh. [30]T w o c a s e s a r e c o n s i d e r e d f o r t h e d i s t o r t e df i lms sh o w n in f ig u re 4 b .

A ) Oh = ~ /2 in d e p en d en t o f h . T h en1= 7 d n h .

B) 0 < Oh < ~ / 2 w h e n h v a r ie s . [ 3 1 ]T h e s e c a se s c o r r e s p o n d r e s p e c t i v e l y t ow e a k s u rf a c e fo r c es o r " a n c h o r i n g " o f m o l e -


29/31

790 C o l lo id a n d P o l y m e r S c ie n c e, V o L 2 5 6 N o . 8 ( 1 9 7 8 )cu les a t t h e f i lm f r ee su r face . T h e su b s t r a t ea n c h o r s s t r o n g l y t h e m o l e c u l e s . W e e l i m i n a t es in 2 O h f ro m [3 0 ] u s in g [22 ] an d [2 5 ] an do b t a i n :

A nh ( K ) A n-- 2 1 2b AW h = ~ (h + hT)[32]

i n w h i c h t h e d e f i n it i o n [ 2 5 ] o f hT h a s b e e nu s e d .T h e p l o t o f 8 v s . h a c c o r d i n g t o [ 3 1 ] o ra s s u m p t i o n 1 s h o u l d p r o v i d e a l in e w i t h z e r oi n t e r c e p t a t t h e o r i g i n a n d s l o p e e q u a l t o(An~2). T h e a n a l o g o u s p l o t a c c o r d i n g t o [ 3 2 ]o r a s s u m p t i o n 2 s h o u l d p r o v i d e a l i n e w i t h ap o s i t i v e in t e rcep t eq u a l t o (An/2)hT.F o r u n d i s t o r t e d f il m s o r Oh = 0 o = 0 ( s eef ig . 4 a) t h e r e t a r d a t i o n b e c o m e s :

= Anh . [33]T h e s lo p e o f t h e p lo t g v s h i s eq u a l t o An.In th e f ig u re 6 w e sh o w th i s p lo t t i n g . I t i ss e e n t h a t :a ) F o r h < 2 .2 # m , th e f i lms v e r i fy th e eq u a-t ion [33] ( l ine I ) .b ) F o r h > 2 . 2 # m t h e r e s u l ts v e r i f y t h ee q u a t i o n [3 2] c o r r e s p o n d i n g to t h e d i s t o r te df il m s w i t h Oh v a r y i n g w i t h h . F r o m t h e i n t e r -cep t i t i s o b ta in ed fo r t h e s t ru c tu ra l t r an s i t i o nt h i c k n e s s : hT = 1 . 1 # m . F r o m t h e d e f in i ti o n[25] of hT, w e f i n d f o r t h e a n i s o t r o p y f a c t o ro f s u r f a c e t e n s i o n o f 5 C B , Awh = 4 . 5 10 - 6

J m - e , f o r K = 1 0 - ' 1 N .T h e r e f o r e th e e x p r e s s i o n o f t h e s u r fa c et e n s i o n o f 5 C B m a y b e w r i t t e n :[ lS b is ) w h = w ~ - 4 ,5 10-6 s in 2 0~ [Jm -2] .

T h e a n i s o t r o p i c c o n t r i b u t i o n i s n e g l i g i b l ec o m p a r e d t o t h e f i r s t i s o t r o p i c c o n t r i b u t i o no f t h e o r d e r o f 3 8, 1 0 - a J m - z .H o w e v e r , t h e o b s e r v e d s t r u c t u r a l t ~ a n s i -t i o n th i ck n ess hT i s sma l l e r t h an th e d i sco n -t in u i ty in a o b se rv e d a t h = 2 .2 # m ( fig. 6) .T h i s d i s a g r e e m e n t m a y b e d u e t o o u r m e t h o do av e ra g in g th e o r i en ta t io n s in e q u a t io n [29 ].T h e s t ru c tu ra l d i s jo in in g p ressu re ex i s t so n ly fo r h > 2 .2 # m.I t h a s a m a x i m u m v a l u e f o r O h = ~ / 4, a c -c o r d i n g t o i t s e x p r e s s io n [ 2 7 ] a n d a s s u m i n g

that ~/a < < ~]s.

A t a th i ck n ess ca l l ed h , o f mech an ica l s t a -b i l i ty l imi t , Oh = x /4 , the s t ru ctu r a l d is jo in ingp r e s s u r e is m a x i m u m a n d e q u a l t o F ].Aw~ K (~ /4 ) 2

8 = _ [ 3 4 ]~s 2K 2 (hs)2

F r o m [ 34 ] a n d A w = 4 . 5 1 0 - 6 J m - 2 w ef ind hs = 1 ,78 # m an d r /] = 1 Jm -a .F o r 0 < O h < ~ / 4 a n d h < h , , a c c o r d i n g t oth e mech an ica l s t ab i l i t y co n d i t io n an d to [3 4 ] ,th e s t ab i l i t y c r i t e r iu m.& 2 Aw~d~ - -- K cos 2 0 h

d0hsin 2 0h d~ > 0 . [35]

T h e d i s to r t ed f i lm o f f ig u re 4 b v e r i f i e s [3 5 ]:g 9~fo r -~- < Oh < -~- i .e. fo r h > h8 = 1 .8 # m w hil e

a c c o r d i n g t o [ 2 5 ] d i s t o r t e d s t r u c tu r e s m a yp er s i s t i n th e r an g e h > h T = l . 1 # m . T h el o w e r l i m i t o f d i s t o r te d f i lm s is d e t e r m i n e d b yth e i r mech an ica l s t ab i l i t y , r a th e r t h an b y th es t ab i l it y o f t h e i r s t ru c tu re ( co n d i t io n s [26 ]) .

e r q s 5 . /c r n32341" / / /

t-~ / 4 z r / 2O h

Fig. 7. Variation of the structural disjoining pressure~s with film molecular orientation 0h at the freesurface of film. I) Stable distorted film; unstablemechanically. II) Mechanically stable distorted films

J , , i , , , , L2 3 1 , 5 6 7 8h IJmFig. 8. Variation of structural disjoining pressure ~ T sof 5CB with the reduced thickness h* = ( h / h e )


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Per ez e t aL, Stru ctura l disjoining pressure in thin film of liquid crystals, L 7 9 1

T h e " F r e d e r i k s z " t r a n s i t i o n i n a s y m m e t r i ct h i n f il m s m a y h a v e t o c o n f o r m t w o s e ts o fc o n d i t i o n s: m e c h a n i c a l [ 3 5 ] a n d t h e r m o d y -n a m i c a l o r p h y s i c o c h e m i c a l : [ 2 2 ] a n d [ 25 ].In f igure 8 , the resu l ts ~7 shown in f igure 7h a v e b e e n p l o t t e d h * = h / h r . F o r 1 < h * < 1 .6 4.T h e f i l m o f f i g u r e 4 b a n d c o r r e s p o n d i n g t o a0 h c h a n g i n g w i t h h w o u l d b e u n s t a b l e a c c o r d -ing to [34] un less th e f i rs t in [24] has a s ign i f i -can t co n t r ib u t io n . H o w ev e r , i f Oh =~ z /2 , ac -c o r d i n g t o [ 2 6 a ] i n t h i s r e g i o n t h e p o s i t i v eco n t r ib u t io n o f ~]s en h an ces th e f i lm s t ab il it y .T h e r e f o r e t h e v a l i d i t y o f a s s u m p t i o n 1 a n d 2m a y b e c h e c k e d b y o p e r a t i n g i n t h e r a n g e1 < h * < 1 . 6 4 .I n t h i s r a n g e w h i c h i s o f t h e o r d e r o f 1 # mfo r 5C B, let t ing [B(h)] __ 10-z0 J. T he dis-p e r s i v e c o n t r i b u t i o n w o u l d b e o f t h e o r d e ro f 1 0- 2 J m - 2 a n d m u c h s m a l l e r t h a n t h e v a lu e ss h o w n i n t h e f i g u r es 7 a n d 8 ,I f the resu l ts d isp lay a d ecre ase in Ws fo r1 < h * < 1 , 6 4 , t h e o r i e n t a ti o n o f m o l e cu l e sc h a n g e s w i t h f i l m t h i c k n e s s a n d t h e i n t e r -f ac i a l t en s io n o r t h e en th a lp y v e r i f i e s eq u a t io n(18b is) . I f the l imi t o f f i lm s tab i l i ty and f i lmd i s to r t io n s t ru c tu re d i sap p ear a t t h e same f i lmth ick n ess h * , t h e o r i en ta t io n o f t h e su r facemo lec u les Oh i s f i x ed . T h ese tw o cases co r re -s p o n d i n g t o a s s u m p t i o n s A a n d B a b o v e a r en a m e d w e a k a n d s t r o n g a n c h o r a g e s r e s p e c -t ively .D i s c u s s i o n a n d c o n c l u s i o n

A F red er ik sz t r an s i t i o n i s fo u n d in th ea b s e n c e o f e x t e r n a l e l e c t r i c a l o r m a g n e t i cm a c r o s c o p i c f i e l d s w h i c h i n o u r c a s e a r e r e -p l a c e d b y s u r f a c e s h o r t r a n g e f o r c e s a c t i n ga t t h e b o u n d a r y o f a s y m m e t r i c t h i n f il m s o f5 C B f o r m e d a t t h e s u r f a c e o f w a t e r .E v i d e n c e is f o u n d t h a t t h e i n te r r a c ia l t e n-s i o n a n i s o t r o p y a t t h e f i l m - w a t e r i n t e r f a c eis s t r o n g e r t h a n a t t h e f i lm v a p o u r i n t e r f a c e .A t t h e l a s t i n t e rf a c e , th e w e a k e r a n i s o t r o p yi s b a l an ced b y th e b u lk e l a s t i c an i so t ro p y .T h e n t h e m o l e c u l a r o r i e n t a t i o n a t t h i s i n t e r -f a c e i s d e p e n d e n t o n f i l m t h i c k n e s s b e y o n d ap r e d i c t e d t r a ns i ti o n t h ic k n e s s h T = l , 1 # ma n d o b s e r v e d o n e a t h T = 2 , 2 # m . T h i s v a r i a -t i o n o f m o l e c u l a r o r i e n t a t i o n a l l o w s a s y m -m e t r i c e q u i l i b r i u m d i s t o r t e d N L C f i l m s t o b ef o r m e d o n w a t e r . I t c o r r e s p o n d s t o a n a n i s o -t r o p i c f a c t o r c o n t r i b u t i o n t o t h e f r e e s u r f a c e

t e n s io n o f N L C e q u a l t o A w h = 4 , 5 1 0 -6J m - 2 independent of film thickness and structure.T h e r e f o r e , w e d e d u c e t h a t Aw h i s r e l ev an t t olocal , short range surface orces only. A s t w o o fu s ( 2 5 ) h a v e f o u n d t h a t 5 C B h a s n o p o l a rc o n t r i b u t i o n t o

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