Effect of Corrugations of the Three-Phase Line on the Drop Size Dependence of Contact Angles英文版_8页.pdf

Effect of Corrugations of the Three-Phase Line on the Drop Size Dependence of Contact Angles D.LI,F.Y.H.LIN,AND A.W.NEUMANN l Department of Mechanical Engineering,University of Toronto,Toronto,Ontario,Canada M5S IA4 Received April 25,1990;accepted June 19,1990 Conflicting observations of the drop size dependence of contact angles have been reported in literature:while some investigators reported an increase of contact angle with an increase in drop size,others reported a decrease.The former was discussed in terms of negative line tension,and the latter was considered as an effect of positive line tension.While some investigators accept the possibility of positive as well as negative line tension,the present authors suspect that line tension,at least for solid-liquid-fluid systems,is positive.In the present paper,by developing a model to calculate the mean contact angle of a sessile drop with a corrugated three-phase line,we show that the observed increase of contact angles with the increase in drop size does not necessarily imply a negative line tension;it may well be a consequence of the corrugations of the three-phase line caused by heterogeneity of the solid surface.1991 Academic Press,Inc.INTRODUCTION Interest in the drop size dependence of con-tact angles of sessile drops on solid surfaces has increased rapidly in recent years(1-7).The drop size dependence of contact angles can be interpreted in terms of line tensions,and line tension can be determined from such contact angle measurements(8).For a sessile drop on a solid surface,the drop size depen-dence of contact angle can be explained in terms of the modified Young equation(8),a 1 cos0=cos0-39 R Ysv-Ysl cos 0-,1)lv where 3lv,%v,and 7s1 are the surface tensions of the liquid-vapor interface,solid-vapor in-terface and solid-liquid interface,respectively;is the line tension;R is the radius of the three-phase contact circle;0 is the contact an-To whom correspondence should be addressed.0021-9797/91$3.00 Copyright 1991 by Academic Press,lnc All rights of reproduction in any form reserved.gle corresponding to a finite contact radius R,and 0o is the contact angle of an infinitely large drop,i.e.,corresponding to R=.As seen from Eq.1,cos 0 should be a linear function of 1/R,provided that both line ten-sion and liquid surface tension are constant.Since the liquid surface tension 3qv is always positive,then if the line tension is positive,i.e.,r 0,the linear relation cos 0=f(1/R)will have a negative slope,or,in other words,the contact angle 0 will decrease as R increases.If the line tension is negative,i.e.,r 0,then the linear relation cos 0=f(1/R)will have a positive slope,or 0 will increase as R increases.These two cases are illustrated in Fig.1.How-ever,while some observers reported a decrease of contact angle as the three-phase contact ra-dius increases,others have found the opposite dependence;hence,according to the observed drop size dependence of contact angles,both positive and negative line tensions for sessile drops on solid surfaces would seem possible(1-7).For example,Ponter and Boyes(1,2)and Ponter and Yekta-Fard(4,6)showed that,in different environments,there are opposite 224 Journal of Colloid and Interface Science,Vol.142,No.1,March 1,1991 DROP SIZE DEPENDENCE OF CONTACT ANGLES 225 R/t-R slope=-o tv 8 1ope=-v:-S-Ira-I 1/R 1/R(a)(b)FIG.1.The predictions of the modified Young equation with positive and negative line tension.trends of the dependence of the contact angle on drop size:in the pure vapor atmosphere the contact angles decrease with the increase in the contact radius,which will correspond to positive line tensions;in air or nitrogen sat-urated water vapor,the contact angles increase with the increase in the contact radius,which would correspond to negative line tensions.The line tension values calculated from their data are in the range from 1 to 10 gJ/m.Also,Good and Koo(3)found that the contact an-gle decreased as the contact radius decreased;their results would yield negative line tensions in the range from-6 to-l 7#J/m.However,they suggested that the observed effects were due to the heterogeneity of the solid surface,and not necessarily to line tension.In our own experiments on carefully prepared solid sur-faces(5,7),we found that the contact angle decreased by 3 5 o as the contact radius in-creased from approximately 1 to 5 mm.By Eq.1,our results correspond to positive line tensions of approximately 2 uJ/m.The three-phase line for a solid-liquid-va-por system is formed by the intersection of the three interfaces,liquid-vapor,solid-vapor,and solid-liquid.In analogy to the surface tension defined for a two-dimensional surface,line tension is defined as the force operating in the one-dimensional three-phase line,or al-ternatively,as the free energy per unit length of the three-phase contact line(8,9).Since all the surface tensions are positive,it is dif-ficult to understand that the corresponding line tension could be negative;furthermore,al-though there is no a priori reason to doubt any of these reports,it seems unlikely to us that line tensions could be negative as well as pos-itive for chemically and energetically similar systems.We suspect that,in addition to the influence of different environments as sug-gested by Yekta-Fard and Ponter(6),at least some of the existing problems have to do with difficulties of interpretation and measurement of contact angles on surfaces which are rough and heterogeneous,possibly even to a very small extent.It should be pointed out that the Young equation is derived by assuming that the solid surface is rigid,smooth,and homogeneous.Under this condition,the three-phase contact line of a sessile drop on the solid surface is a smooth circle,with a radius of curvature R.Only for this situation will Eq.1 hold.Oth-erwise,if the solid surface does not satisfy the above condition,then the three-phase line may not be a smooth circle,but rather a corrugated line.Hence 1/R in Eq.1 should be replaced by Kgs,which is the local curvature of the three-phase line in the plane of the solid phase.Therefore,the more general form of the mod-ified Young equation is(8)O cos 0=cos 0oo-Kgs.2 Fly Obviously,as stated above,for a corrugated three-phase contact circle with the radius of curvature varying from point to point,one cannot expect to obtain a meaningful value for the line tension by interpreting the ob-served dependence of the contact angle on the apparent three-phase contact radius in terms of Eq.1.Therefore,we are inclined to attri-bute negative line tension results to the cor-Journal of Colloid and Interface Science,Vol.142,No.1,March 1,1991 226 LI,LIN,AND NEUMANN rugations of the three-phase contact circle which may be due to the heterogeneity of the solid surfaces;in other words,we suspect that negative values of line tension may be an ar-tifact.To investigate the possibility,we present,in this paper,a model investigation into the ef-fects of corrugations of the three-phase contact circle on the drop size dependence of the con-tact angle.CORRUGATED THREE-PHASE LINE MODEL Consider a sessile drop resting on a solid surface.The model surface is smooth but het-erogeneous,consisting of two types of patches.The three-phase contact line of a sessile drop on such a heterogeneous solid surface is a cor-rugated circle.The intensive parameters for the liquid on type 1 and type 2 surfaces are 71v,svl,Yll,al,0t,010o lv,sv2,s12,02,02,020o.where 3qv refers to the liquid surface tension;Ysv and 3 refer to the solid-vapor and the solid-liquid interfacial tensions,respectively;and 0 represent the line tension and contact angle,respectively;the subscripts 1 and 2 in-dicate the patches of type 1 and type 2,re-spectively.We consider that the two type patches are uniformly distributed over the whole surface area,and type 1 patches represent the main material of the surface and type 2 patches rep-resent the impurity on the surface.We as-sume that all the parts of the corrugated three-phase line in contact with type 1 patches have the same shape,and all the parts of the cor-rugated three-phase line in contact with type 2 patches have the same shape.For simplicity,we further assume that the whole corrugated three-phase contact line consists of n type 1 circular arcs and n type 2 circular arcs,and they are smoothly linked to each other so that the three-phase contact line has continuous first-order and second-order derivatives.An illustrative configuration of such a corrugated three-phase contact circle is given in Fig.2.R 2-/(27r.Let 6 2=Kffl,where K is a constant for a given system.Thus,Eq.8 can be rewritten as cos 0 R(R+R2+R3)cos 0+R2R3cos 02=(R1+R2)(R1+R3)Vlv k(R1+R2)(R1+R3)91 Equation 9 implies that the mean contact angle on such a heterogeneous surface is a function of 3iv,01,02,el,K(or 62),R,R2,and R3.For a given model system,cos 0j,cos 02,3qv,a and K(or 62)are constant.Therefore,as seen from Eq.9,the value of the mean contact angle of a sessile drop with the corrugated three-phase line will depend not only on the apparent radius of the three-phase contact circle,R3,but also on the local radii of curvature,R1 and R2.MODEL CALCULATIONS AND DISCUSSION Generally,the local radii of curvature R and R2 will change as the apparent three-phase contact radius R3 increases.How R and R2 vary with R3 will depend on the geometry of the heterogeneous patches.However,these changes can be divided into two general pat-terns:either R and R2 increase as R3 increases,Rl=al+-blR3 Re=a2+-b2R3.However,this will not restrict our final con-clusions,since the purpose of this paper is only to explore the possibilities of different patterns of the drop size dependence of the contact an-gle due to the corrugated three-phase line.In our model calculations,the parameters in Eq.9 were chosen as follows:3qv=72.8 mJ/m2;01o=70;02o=80;guided by our experimental results(5,7),we took or1=2 gJ/m;with respect to the size of the hetero-geneous patches,the local radii of curvature R1 and R2 were chosen to be of the order 10 tm.For the above parameters,the calculated apparent contact angles as a function of the apparent contact radius R3 of the corrugated three-phase line are plotted in Figs.4 and 5,with K varying from 0.5 to 1.5.As shown by these curves,although the line tensions are positive,the apparent contact an-gle may increase as well as decrease with in-creasing drop size,due to the local curvature effects of the corrugated three-phase line;and,for all cases,the contact angles change up to approximately 10 as the apparent contact ra-dius increases approximately from 1 to 10 mm;this order of magnitude compares well with actual observations(1-7).Also,for the case where RI and R2 increase with R3,as shown in Fig.4,when K 1,contact angles increase with increase in drop size,which corresponds to the so-called negative line tension effect observed in some experiments.For the case where R1 and R2 decrease with increasing R3,as shown in Fig.5,the two different patterns Journal of Colloid and lnterfaceSeience,Vol.142,No.1,March 1,t991 DROP SIZE DEPENDENCE OF CONTACT ANGLES 229 0(degrees)90-86-82-78-74 70 K-0.5 K-0.6 K-0.7 K=0.8 K=0.9.66-62 0 KI.I K=1.2 K=I.3 K 1.4 K=l.5 I I I I 2 4 6 8 10 R 3(mm)cosO 0.448-0.408-0.368-0.328 0.288-0.248 0.208-0.168-0.128-0.088-0.048-0.008 0/J .K=1.5.-K=l.4.-K=1.3-.-K=1.2 K=I.1 K=0.5 I I I I I 0.2 0.4 0.6 0.8 1 1/R 3(mm-1)FIG.4.The contact angle drop size dependence calculated by Eq.9l with RI=1.0 10-5+0.02R3(m)andR2=1.0 10 5+0.02R3(m).of contact angle drop size dependence are re-versed for K 1 and K 1.In all cases,the different patterns become more pronounced as K differs more and more from unity,i.e.,as the two line tensions differ.It should be pointed out that the existence of the two types of patterns of contact angle drop size dependence does not depend on the particular forms of linear relationships among Rl,Rz,and R3,Other type functions have been used without qualitative impact on the two types of patterns.As an example,the sire-Journal of Colloid and Interface Science,VoL 142,No,I,March 1,!991 230 LI,LIN,AND NEUMANN 92,5 87.5-82.5-77.5-0(degrees)72.5-67.5-62.5-57.5-52.5/Jj K=0.5 K=0.6 K=0.7 K=0.8 K=0.9 K=I I 4 R3(mm)K=1.1 K=1.2 K=13 K=l.4 K=15 I I 8 10 0.6 0,5-0.4-0.3-cos0 0.2-0.I-0-0.1 0 f I 0.2 I I I I 0.4 0.6 0.8 1 1/R3(rnm-)-K=l.5 K=IA K=l.3 K=1.2 K=I.1 K=I K=0.9 K=0.8 K=0.7 K=0.6 K=0.5 FIG.5.The contact angle drop size dependence calculated by Eq.9 with R1=6.0 10-s-0.003R3(m)and R2=6.0 10-5-0.003R3(m).ilar curves obtained by using a logarithmic function are plotted in Fig.6.CONCLUSIONS For a sessile drop with a corrugated three-phase line,the observed drop size dependence of contact angle cannot be simply interpreted in terms of the modified Young equation,Eq.1,because the curvature of the three-phase line is not a constant.According to the cor-rugated three-phase line model as developed here,it is possible that,due to the changing patterns of local curvatures with increase in drop size,the contact angle drop size depen-dence of a sessile drop with a corrugated three-phase line may show qualitatively different patterns.Thus,the model developed in this paper indicates that an experimental obser-vation of an increase in the contact angle with increasing drop size does not necessarily imply a negative line tension;instead,it may be the effect of corrugation of the three-phase line caused by the imperfections of the solid sur-face.Journal of Colloid and lnterfizce Science,Vol.142,No.1,March 1,1991 0(degrees)88-84-80-76-72-68-64-60 0 DROP SIZE DEPENDENCE OF CONTACT ANGLES J f I 2 I I I I 4 6 8 10 R3(mm)K=0.5 K=0.6 K=0.7 K=0.8 K=0.9 K=I K=I.I K=1.2 K=1.3 K=1.4 K=1,5 231 cos0 0,495-0,455-0,415-0.375-0,335-0.295-0.255-0.215-0.175-0.135-0.095-0.055-0.015-0.025 I I I I 0.2 0.4 0.6 0,8 1/1t 3(ram-1)/K=l.5-K=1.4 K=l.3 K=1.2 K=I.1 K=I K=0.9 K=0.8 K=0.7 K=0.6 1K=0.5 1 FIG.6.The contact angle drop size dependence calculated by Eq.9 with RI=1.0 10 5+1.0 10-log(5000R3)(m)and R2=1.0 10.5+1.0 X 10-Slog(5000R3)(m).ACKNOWLEDGMENTS This study was supported by the Natural Sciences and Engineering Research Council of Canada(Grant No.A8278)and by a University of Toronto Fellowship.REFERENCES 1.Ponter,A.B.,and Boyes,A.P.,Canad.J.Chem.50,2419(1972).2.Boyes,A.P.,and Ponter,A.B.,J.Chem.Eng.Japan 7,314(1974).3.Good,R.J.,and Koo,M.N.,J.ColloMInterfaceSci.71,283(1979).4.Ponter,A.B.,and Yekta-Fard,M.,Colloid Polym.Sci.263,1(t985).5.Gaydos,J.,and Neumann,A.W.,Z Colloid Interface Sci.120,76(1987).6.Yekta-Fard,M.,and Ponter,A.B.,J.Colloid Interface Sci.126,134(1988).7.Li,D.,and Neumann,A.W.,Colloids Su 43,195(1990).8.Boruvka,L.,and Neumann,A.W.,J.Chem.Phys.66,5464(1977).9.Gibbs,J.W.,The Scientific Papers,Vol.1,p.288.Dover,New York,1961.10.Boruvka,L.,Gaydos,J.,and Neurnann,A.W.,Col-loids Surf 43,307(1990).Journal of Colloid and Interface Science,Vol.142,No.1,March l,1991