982
Langmuir 1992,8, 982-988
Surfaces of Constant Mean Curvature with Prescribed Contact Angle D. S. Bohlen,+H. T. Davis,' and L. E. Scriven Department of Chemical Engineering and Materials Science, Center for Interfacial Engineering, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0132 Received April 12,1991. In Final Form: October 21, 1991
A Galerkin weighted residual formulation of the surface divergence theorem is used with finite element basis functions to compute prototype geometric models of foam and emulsion structure. The models are surfaces of constant mean curvature arranged on a simple cubic lattice so that they meet the boundary planes of the unit cells at prescribed contact angles. Surfaces are computed for a variety of contact angles and mean curvatures. With a contact angle between 5 O and 133O,the structure inverts as the mean curvature decreases, whereas with a contact angle between 134O and 180°,the structure fills the unit cell as the mean curvature decreases, passes through a turning point, and then increases.
Introduction The relationships between the structuresand properties of foams and emulsions are not yet fully understood. An emulsion or a foam is defined in general terms as a reasonably stable dispersion of one fluid phase in another, immiscible, fluid phase. Various additives may prevent the dispersion from separating for a period of time, yet the dispersion is not thermodynamically stable. The relationships between physical structures and physical properties of such dispersions are of practical as well as theoretical interest. These relationships can be computed from three-dimensional models of the structures. At volume fractions below some critical value, there is an "internal" phase that is dispersed in a continuous uexternal" phase and the former's form can be modeled as undistorted spheres. At volume fractions approaching unity, the internal phase forms irregular polyhedral structures. The external phase is limited to thin films between polyhedral faces and channels at the polyhedral edges known as Plateau borders. Three films meet at 120° angles a t the Plateau borders, and four linear Plateau borders meet at tetrahedral vertices with the tetrahedral angle, 109.47', between them. At intermediate volume fractions, there are structures that are not well characterized, much less well understood. Physical properties cannot be accurately simulated without adequate model structures. Lissant and co-workers14 modeled monodispersed emulsions with simple geometric structures. Spheres were packed in unit cells of regular lattices: simple cubic, tetrakaidecahedral, and rhomboidal dodecahedral. The spheres were undistorted until the volume fraction of the internal phase reached the critical volume fraction,namely, that at which the spheres first touched their neighbors. The critical volume fraction, 4, for spheres arranged on a simple cubic lattice is 4 = 52 % ,on a tetrakaidecahedral lattice is 4 = 68%, and on a rhomboidal dodecahedral
* Address correspondenceto Professor H. T. Davis, Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, MN 55455-0132. + Current address: Department of Chemistry, University of Missouri-Rolla, Rolla, MO 65401-0249. (1) Lissant, K. J. J. Colloid Interface Sci. 1966, 22, 462. (2) Liasant, K. J. J . SOC.Cosmet. Chem. 1970, 21, 141. (3) Lissant, K. J.; Mayhan, K. G . J. Colloid Interface Sci. 1973, 42, 201. (4) Lissant, K. J.; Peace, B. W.; Wu, S. H.; Mayhan, K. G . J. Colloid Interface Sci. 1974, 47, 416.
lattice is 4 = 74%. As the volume fraction was raised further, the spheres were either rearranged to form a more stable packing on a different lattice or distorted a t constant volume. Above 74% volume fraction, the spheres must distort. This distortion was characterized by the point of contact between the spheres growing to form a circular disk of thin film. The remaining curved surfaces were assumed to retain spherical shapes as in truncated spheres. Such an assumption required the contact angle between the curved surface and the film to change as the volume fraction of the internal phase was expanded and the film was compressed. Princen5 and co-workers6 recognized that the curved surface would have constant mean curvature, the contact angle between it and the thin film would not change as the volume fraction is raised, and the thin films would not be circular. Princen suggested that advanced numerical techniques might be employed to find constant mean curvature surfaces that meet unit cell boundaries at a specified contact angle. The situation of a constant mean curvature surface separating two immiscible fluids and contacting boundary planes at a contact angle other than zero arises elsewhere than in foam and emulsion structures. Mercury porosimetry is a technique by which a distribution of pore sizes is deduced from the amount of mercury that enters an evacuated porous medium as a function of pressure applied to the mercury. Mercury makes contact angles with solids in the range of 130-140O and is therefore a partially wetting liquid.' A complex three-dimensional pore structure is often modeled as an idealized array of cylindrical capillaries of various diameters. The curvature of the mercury-gas surface, and therefore the diameter of the capillary invaded, is proportional to the injection pressure relative to ambient, AP,according to the YoungLaplace equation, AP = ~ H Qwhere , H is the mean curvature (the reciprocal of capillary diameter in the model) and Q is the surface tension. Real pore space is not a bundle of simple tubes, and the model does not account for such events as mercury being forced into grooves and corners without necessarily invading new pores as the pressure is increased. When a porous medium is occupied (5) Princen, H. M. J. Colloid Interface Sci. 1979, 71, 55. (6) Princen, H. M.; Aronson, M. P.; Moser, J. C. J. Colloid Interface Sci. 1980, 75, 246. (7) Heimenz, P . C. Principles of Colloid and Surface Chemistry;Dekker: New York, 1986; p 337.
Q743-7463/92/24Q8-Q982$Q3.oO/Q0 1992 American Chemical Society
Langmuir, VoZ. 8, No. 3, 1992 983
Surfaces of Constant Mean Curvature CUBIC UNIT CELL
TETRAHEDRAL CELL
FLATFILMS
v=Q
Figure 2. Reduction of u.nit cell to tetrahedral primitive cell. Figure 1. Single simple cubic unit cell. Flat films are shown on the boundary planes of the unit cell.
by two fluid phases, the grooves and corners, regions of high wall curvature, are the first to fill with wetting liquid when its content is increased. The flow channels these regions form when they are connected are the chief routes by which more of the wetting liquid can enter or, alternatively, can leave as the content of wetting liquid is decreased. Examples that have been examined include wetting fluid being displaced by nonwetting fluid in toroidal pores with grooves in the in square capillaries,+12 and in triangular ~api1laries.l~To a first approximation, the connected channels are covered by menisci of constant mean curvature. To a second approximation, they have small gradients of mean curvature and often can be approximated by the assumption that curvature in the axial direction is s m a l l 4 Anderson and co-workers15J6developed a method to construct smooth constant mean curvature surfaces that are periodic in three dimensions, i.e., that have the symmetries of a space group. The method consists of computing the surface in a single primitive cell of the corresponding point group, the walls of which are symmetry planes and so they are necessarily intersected by the surface at right angles. The full surface can then be assembled from rotated, reflected, and translated replicas of the primitive cell. In the work, Anderson’s method was used to construct constant mean curvature surfaces of simple cubic symmetry that meet the walls of the unit cell, a cube, at a prescribed contact angle. The algorithm was modified, as described below, to compute the surface in the primitive cell, one of the walls of which is not, in general, a symmetry plane.
in a physical system the liquid occupying the internal domain forms a film along the cube wall. The simple cubic unit was subdivided by its planes of reflective symmetry into 48 primitive cells,tetrahedrons, as shown in Figure 2. The portion of the constant mean curvature surface contained within a primitive cell was mapped onto a two-dimensional domain. The desired surface was then computed as a solution to a form of the surface divergence theorem in which the surface itself is unknown. Surface area and volume fractions were computed from the solution. The surface was drawn in threedimensional representation by the reverse process: the twodimensional “computational” domain was mapped into the primitive cell, the primitive cell was reflected in the symmetry planes to recreate the unit cell, and when desired, the corresponding periodic surface was created by translations of the unit cell. The surface patch within a single tetrahedron was mapped onto a two-dimensional domain in the way adopted by Anderson to take advantage of the geometry of the tetrahedron (see Figure 2): two opposed edges, L1 and Lp, of the tetrahedron are orthogonal and never intersect. L1 is a segment of a line connectingthe center of the unit cellto the center of a neighboring cell. Lp is a segment of an edge of the unit cell. The variable u was defined to vary from 0 to 1 on the length of L1; the second variable, u, was defined to vary from 0 to 1 on the length of L2. In Cartesian coordinates, the line segments were Ll(0, u/2,0) and Lp(1/2, 1/2, v/2). The surface is the locus of intersections of the surface and line segments drawn from Ll(u) to L2(u):
m u ,v) = L,(u) + w(u,V)(L,(V) - L,(u))
(1)
The variable w(u,u ) is simply the fractional distance from Ll(u) to the intersection. Thus, each point on the surface maps onto a unique point on the two-dimensional (u, u ) domain provided the line segment Lp(u) - Ll(u) intersects the surface once and only once; otherwise, the mapping fails. With this representation, the unknown surfaceX was computed from the surface divergence theorem, in which it occurs as an unknown:17
Computational Method Within a singleunit cube, the constant mean curvature surface divides space into an internal domain and an external domain (see Figure 1). The internal domain lies between the center of the unit cell and the curved surface; the external domain lies between the surface and the edges and corners of the cube. On each wall of the cube, there is a region contained in the interior of the surface. These regions are referred to as flat films because (8) Roof, J. G. SOC.Pet. Eng. J . 1970, 10, 85. (9) Lenormand, R.; Zarcone, C.;Sarr, A. J. Fluid Mech. 1984,135,337. (10) Arriola, A.; Willhite, G. P.; Green, D. W. SOC.Pet. Eng. J . 1983, 23, 99. (11) Legait, B. J. Colloid Interface Sci. 1983, 96,28. (12) Gauglitz, P. A.; St. Laurent, C. M.; Radke, C. J. SOC.Pet. Eng. J. 1987,27, 753. (13) Singhal, A. K.; Somerton, W. H. J . Can. Pet. Technol. 1970, 9, 197. (14) Ransohoff,T. C.; Radke, C. J. J . Colloid Interface Sci. 1988,121, 392. (15) Anderson, D. M.; Davis, H. T.; Nitsche, J. C. C.; Scriven, L. E. Adu. Chem. Phys. 1991, 77, 337. (16) Anderson, D. M. Ph.D. Thesis, University of Minnesota, Minneapolis, MN, 1986.
Here d S is the curve that bounds the surface S, Vs is the surface divergence operator, F is an arbitrary vector field, V is the unit vector locally normal to d S outwardly pointing, and tangent to the surface, N is the unit vector normal to the surface, and H i s the mean curvature. The mean curvature was used as a constant in this work but can be chosen as an arbitrary function of position. Anderson’s choice of the vector field is the key: F = (dX/dw) &u, u ) = (Lp(u) - Ll(u))&u, u ) , where #(u, u ) is any member of the set of basis functions in which the unknown surface is to be represented. The scalar product V-F is a function of the angle between the surface and, in succession, the boundary planes of the primitive cell. On the symmetry planes it vanishes and on the plane that is a wall of the unit cell it is V-F = v
dX 4’ = I L , ( ~-) L,(u)lbi cos y sin 7 aw
(3)
where y is the specified contact angle and 7 is the computed angle between the surface tangent vector and F. (17) Weatherburn, C. E. Differential Geometry of Three Dimensions; Cambridge University Press: London, 1927; p 239.
Bohlen et al.
984 Langmuir, Vol. 8,No. 3, 1992 Equation 2 was solved by the Galerkin method with finite element basis function. The two-dimensional (u,v) domain was subdivided into 20 X 20 square elements. The unknown variable w was approximated by 441 bilinear basis functions, i.e. w = &aj&u, u). With F chosen as above, the surface divergence theorem then yields 441 weighted residual equations, i = 1-441, for the 441 unknown coefficients, aj:
The finite element representation of the surface enters through the derivatives of X with respect to u (Xu)and v (Xu),and their appearance in Vs, dA,ds, and N:
dA = IX, x X,Jdu dv
(6)
+ [Xu[du
(7)
ds = IX,I du
and
The system of nonlinear algebraic equations, eq 4, was solved for the unknown ais by Newton's method. The first surface computed was the Schwarz surface18of zero mean curvature and 90° contact angle. The Schwarzsurface is continuous and divides space evenly, half of space on one side, half of space on the other. The initial estimate of a, = 0.5 for all j was used for the Schwarz surface. Initial estimates were generated by first-order continuation in mean curvature when families of surfaces with the same contact angle were being computed. The first-order continuation was used as described by Ander~0n.l~ Zeroth-order continuation in contact angle was used when families of surfaces with the same mean curvature but different contact angles were being computed. Newton's method was iterated until the solution, w , was within a specified tolerance: IIwktl - wkll < 10" where k is the number of iterations. The computations were performed on a Cray-2 supercomputer. The total surface area that was computed was the area of the curved constant mean curvature surface plus the area of the six flat films at the walls of the cube (see Figure 1):
The volume fraction was defined as the ratio of internal domain volume to the volume of the unit cube:
2 - d3 ) du du
4 8 s 01 s 01 1 8 ( 2
(10)
Equations 9 and 10 were computed from the solution of eq 4 by Gaussian quadrature with four quadrature points per element. The unit of length throughout the computations was the edge lengtha of the cubic unit cell. Thus, the reported mean curvatures are dimensionless products of mean curvature and edge length. Likewise, surface areas are in units of a2,
Results and Discussion Surfaces of broad ranges of mean curvature and contact angle were computed. The algorithm was first tested by computing simple surfaces for which surface area and volume fraction can be found from mensuration formulas. (18)Schwarz, H.A. Monatsberichte der Koniglichen Akademie der Wissemcaftenzu Berlin; Jahrgang, 1865; p 149.
Table I. Comparison of Surface Areas Computed by the Mensuration Formula of Equation 11 to Those Computed by Solving Equation 4 by Galerkin's Method with 441 Bilinear Basis Functions
H
y
formula area
total area computed area
error, %
2 . m 1.9696 1.8794 1.8126 1.7320 1.6383 1.5321
180.0 170.0 160.0 155.0 150.0 145.0 140.0
3.1416 3.2381 3.5384 3.7744 4.0760 4.4522 4.9140
3.4132 3.3782 3.5371 3.7768 4.0784 4.4543 4.9156
8.65 4.33 0.04 0.06 0.06 0.05 0.03
~
Table 11. Comparison of Volume Fractions Computed by the Mensuration Formula of Equation 12 to Those Computed by Solving Equation 4 by Galerkin's Method with 441 Bilinear Basis Functions
H
y
2.oooO 1.9696 1.8794 1.8126 1.7320 1.6383 1.5321
180.0 170.0 160.0 155.0 150.0 145.0 140.0
volume fraction formula vol fract computed vol fract error, % 0.5236 0.5476 0.6209 0.6764 0.7439 0.8208 0.9002
0.5910 0.5821 0.6201 0.6767 0.7442 0.8211 0.9004
12.87 6.30 0.13 0.04 0.04 0.04 0.02
A sphere centered in a cube intersects the walls if ita diameter exceeds the cube's edge length. The angle at which the portion of spherical surface within the cube contacts the walls is y = c o d (-H/2). The mean curvature, H,of the spherical surface is simply the reciprocal of the sphere radius. The total surface area within the cube is the surface area of the sphere less that of the six caps that lie outside plus that of the six flat circular films on the walls:
The fraction of the cube volume that lies within that surface is
r -(9H l volume fraction = - 8) - 4 31P A comparison of surface areas computed by the mensuration formula of eq 11 to those computed by eq 9 from solutions of eq 4 by Galerkin's method with 441 bilinear basis functions is listed in Table I. A comparison of volume fractions computed by the mensuration formula of eq 12 to those computed by eq 10 from solutions of eq 4 by Galerkin's method with 441 bilinear basis functions is listed in Table 11. The limitingcase is the inscribedsphere, which has mean curvature 2 and makes 180' with the walls at the points of contact; the surface area is u and the volume fraction is d 6 . The surface area computed by eq 9 is in error by 8.65 5% ;the volume fraction computed by eq 10 is in error by 12.95%. The sphere is tangential to the walls of the unit cube; the edge of the surface at the cube wall is a single point. The primitive tetrahedral cell (Figure 2) used in the mapping of surfaces contains a three-sided surface patch. As a consequence, the mapping of eq 1,which maps a four-sided surface patch to a two-dimensional computational region, fails. The computed surface with 170' contact angle and H = 1.9696 has an error in total surface area of 4.33 5% and an error in volume fraction of 6.3 5%. The mapping of this surface is close to failing because the edge of the surface at the cube wall is very small compared to the other three surface edges in the primitive tetrahedron. The radii of
Surfaces of Constant Mean Curvature
Langmuir, Vol. 8, No. 3, 1992 985
Table 111. Comparison of Surface Areas and Volume Fractions Computed by the Mensuration Formula of Equations 11 and 12, Respectively, to Those Computed by Solving Equation 4 by Galerkin's Method with Different Numbers of Bilinear Basis Functions
H
y
2.oooO 180.0'
1.7320 150.0'
basis functions formula 121 256 441 676 961 formula 36 121 256 441 676 961
total area error, % volfract error, % ***** 3.1416 ***** 0.5236 3.3643 7.09 0.5773 10.26 3.3999 8.22 0.5873 12.17 3.4132 8.65 0.5910 12.87 3.4204 8.87 0.5930 13.25 3.4248 9.01 0.5943 13.50 ***** 4.0760 ***** 0.7439 4.1161 0.98 0.7491 0.70 4.0862 0.25 0.7452 0.17 4.0804 0.11 0.7445 0.08 0.7442 4.0784 0.06 0.04 4.0775 0.04 0.7441 0.03 4.0770 0.02 0.7440 0.01 ,
the circular films on the cube walls is 0.09~where a is the edge length of the unit cube. At 180' contact angle, the surface edges on the other three walls of the primitive tetrahedral cell have radii of 0 . 5 ~ . The length of the surface edge at the cube walls increases as contact angle decreases. The radius, r, of the circular films is related to contact angles as r = - 0 . 5 ~tan y. The errors in computed total surface area of surfaces with contact angles between 160' and 140' were less than 0.1 % . The error in computed volume fraction of the surface with a 160' contact angle was 0.13 % while the errors of surfaces with contact angles between 155' and 140' were 0.04 % or less. The accuracy of the computations increased as the contact angle was decreased because the length of the surface edges a t the cube walls increased. Spherical surfaces that make contact angles with the unit cube walls less than 135' are disjoint within the cube; i.e., they are confined to the eight cube corners. Again, the primitive tetrahedral cell contains three-sided surface patches, and the mapping at eq 1fails. Two other surfaces were used for tests. One surface is a polyhedral structure composed of squares at the six walls of the unit cube and a total of eight equilateral triangles connecting the squares (see Figure 5, lower right). This surface has a 125.26' contact angle with the cube walls and H = 0. The surface area is d 3 , and the volume fraction of the internal domain is 5/6. The computed surface area is 1.7245, in error by 0.44 % ,the computed volume fraction is 0.8345, in error by 0.14%. The Schwarz surface has zero mean curvature and makes a 90' contact angle with the wallslS (see Figure 6, upper and lower right). The surface area, 2.3453, and volume fraction, 0.5OO0, computed by this method were within 0.01 % and 0.00 % ,respectively, of the values 2.3451 and 0.5000 reported by Schoen.lg Comparisons of surface areas and volume fractions computed by mensuration formulas to those computed by Galerkin's method with different numbers of bilinear basis functions are listed in Table 111. The error in total surface area of the computed surfaces with 180' contact angles increased from 7.09% for 121 basis functions to 9.01 % for 961basis functions. The error in volume fraction of these same surfaces increased from 10.26 % for 121basis functions to 13.50% for 961 basis functions. The use of more basis functions actually decreased the computational accuracy. The error in total surface area of the computed surfaces with 150' contact angles decreased from 0.25% for 121 basis functions to 0.02% for 961 basis functions. The error in volume fraction of these same surfaces decreased from 0.70% for 121 basis functions to 0.01% (19)Schoen, A. H. NASA Technical Note TND-5541; 1970.
Figure 3. Four surfaces with 180' contact angles: (upper left) H = 2, (upper right) H = 3, (lower left) H = 5,and (lower left) H = 10.
for 961 basis functions. The use of more basis functions increased the computational accuracy. Four hundred forty-one basis functions were used in subsequent calculations. A series of surfaces with 180' contact angles were computed. Four surfaces with 180' contact angles are shown in Figure 3. The surface with H = 2 and a 180' contact angle corresponds to an undistorted sphere. As the curvature is increased, the point of contact between the sphere and the cube wall flattens to form a circular film. However, as the curvature is increased further, the film loses its circular shape and the curved surfaces are no longer spherical. The surface distorts to fill into the edges of the cube. Consequently, the volume fraction of the internal domain approaches 1, and the total surface area approaches 6, the surface area of the cube. During this distortion, the surface remains continuous, which means the external domain also remains continuous as narrow channels along the cube edges. Surfaces of 180' contact angle were computed with mean curvature up to H = 100. The surface areas of the curved constant mean curvature surfaces as functions of mean curvature and contact angle are shown in Figure 9. The area of the films a t the unit cell walls are shown as functions of mean curvature and contact angle in Figure 10. Volume fractions of the internal domain are shown as functions of mean curvature and contact angle in Figure 11. Extensive lists of area and volume fraction for these surfaces are listed by Bohlen.20 Another series of surfaces was computed but with 150' contact angles. Four surfaces with 150' contact angles are shown in Figure 4. Again, the surface distorts to fill the unit cell as the mean curvature is increased from the curvature of a sphere. In this case, the sphere had a mean curvature of 1.732 and extended beyond the walls of the cube. Surfaces of 150' contact angle were computed with mean curvature up to H = 100. As the mean curvature decreases from 1.732, the volume fraction of the internal domain decreases from 0.742 (Figure 11). At H = 1.691, a turning point is encountered and the curvature then increases while the volume fraction continues to decrease (20)Bohlen,D. S. Ph.D. Thesis,University of Minnesota,Minneapolis, MN, 1990.
986 Langmuir, Vol. 8, No. 3, 1992
Figure 4. Four surfaces with 150' contact angles: (upper left) H = 2, (upper right) H = 1.7, (lower left) H = 2, and (lower left) H = 5.
Figure 5. Three surfaces with 120' contact angles: (upper left) H = 2, (upper right) H = 0, and (lower left) H = -0.4. Shown in lower right is the surface with contact of 125.26' and H = 0, which correspondsto portions of octahedra centered a t the comers of the unit cell.
to a minimum of 0.436, H = 2. The volume fraction increases as H increases until H = 2.035 where the surface begins to self-intersect. That is to say, for the surface to satisfy eq 4, it passes through or intersects itself. At the point of self-intersection, the mapping of eq 1fails. The surface area of the curved constant mean curvature surface increases from 2.77 at H = 1.732, reaches a local maximum of 2.78 at H = 1.75, decreases to 2.74 at H = 1.94, and then increases to 3.04 a t H = 2.04 (Figure 9). Surfaceswithcontact angles between 180" to 134" distort to fill the cube as mean curvature is increased. Surfaces with contact angles less than or equal to 133" do not distort to fill the unit cube. Three surfaces with 120" contact angles are shown in Figure 5. The surface with a 120" contact angle and H = 2.04 self-intersects. The internal domain volume fraction decreases as the mean curvature decreases from H = 2.04 until a minimum volume fraction
Bohlen et al.
Figure 6. Four surfaces with 90' contact angles: (upper left) H = 2, (upper right) H = 0, the Schwarz surface, and (lower left) H = -2. Shown in lower right are four unit cells connected to illustrate the periodicity of the three-dimensional structure.
is reached a t H = 1.9. The volume fraction increases as mean curvature decreasesfurther until the surface becomes disjoint at H = -0.415; the surface is confined to the corners of the unit cube and is no longer continuous along the cube edges. When the surface becomes disjoint, the mapping of eq 1fails. As mean curvature decreases from H = 2.04 to H = -0.415, the surface area of the curved surface decreases from 3.0 to a local minimum of 2.34 at H = 1.72, increases to a local maximum of 2.45 at H = 1, decreases to a local minimum of 1.66 at H = -0.25, and increases to 1.92 at H = -0.415. Anderson15 computed a series of surfaces with 90" contact angles. Four surfaces with 90" contact angles are shown in Figure 6. At H = 2 and H = -2, the surfaces self-intersect and the mapping of eq 1 fails. Anderson reported that surfaces with 90" contact angles invert as the mean curvature is changed from positive to negative. As mean curvature decreases from H = 2 to H = -2, the volume fraction decreases from 0.465 to a minimum of 0.25 at H = 1.8, increases to a maximum of 0.75 at H = -1.8, and decreases to 0.535 at H = -2. The area of the curved surface decreases from 2.9 a t H = 2 to a minimum of 2.0 at H = 2 (belowa turning point at H = 2.13), increases to a local maximum of 2.34 a t H = 0, decreases to 2.0 at H = 2, and increases to 2.9 at H = 2 (above a turning point at H = -2.13). A series of surfaces of 60" contact angle were computed between the limits of H = 2.06 and H = -3.09. Four surfaces of 60" contact angle are shown in Figure 7. The surface with a 60" contact angle and H = 2.06 self-intersects. The internal domain volume fraction decreases from 0.453 as the mean curvature increases to a turning point at H = 2.21 and then decreases to H = 1.85 where the volume fraction is a minimum of 0.166. The volume fraction increases to 0.686 as the mean curvature decreases further until H = -2.85. As the mean curvature decreases further, the volume fraction decreases to 0.420 at H = -3.09 where the surface self-intersects. As the mean curvature decreases from H = 2.06 to H = -2.85, the area of the curved surface decreases from 2.86 to a minimum of 1.73 at H = 2, increases to a local maximum of 2.46 at H = -1.1, decreases to 2.30 at H = 2.7, and increases to 3.55 at H = -3.09.
Surfaces of Constant Mean Curvature
Langmuir, Vol. 8, No. 3, 1992 987 I
6.0
5.0
g 2 p:
3
3.0 2.0
v)
1.0
"."
-5.0
-2.5
0.0
2.5
5.0
MEAN CURVATURE Figure 9. Area of curved constant mean curvature surface as a function of mean curvature for contact angles of M O O , EO0, 120°,90°, 60°, 30°,and .'5
4.0
Figure 7. Four surfaces with 60' contact angles: (upper left) H = 2.2, (upper right) H = 0, (lower left) H = -2, and (lower right) H = -3.2.
W p:
2
3.0 20 1.0 0.0
-5.0
-2.5
0.0
25
5.0
MEAN CURVATURE Figure 10. Area of flat filmsat unit cell boundaries as a function of mean curvature for contact angles of 180°,150°, 120°, 90°, 60°, 30°, and .'5 1.0
W
E
3 0
>
0.4
OS
0.0
I
-5.0
Figure 8. Two surfaces with 30' contact angles: (upper left) H = 0 and (upper right) H = -3. Two surfaces with '5 contact angles: (lower Ieft) H = -0.4 and (lower right) H = -3.
A series of surfaces of 30' contact angle were computed between the limits of H = 0.70 and H = -4.02. Two surfaces of 30' contact angle are shown in Figure 8, upper left and right. The surface with a 30' contact angle and H = 0.70 self-intersects. The internal domain volume fraction decreases from 0.46 at H = 0.7 to 0.24 a t H = 0.40, increases to a maximum of 0.65 at H = -3.4, and decreases to 0.26 at H = -4 where the surface self-intersects. As H decreases from 0.7 to -4, the area of the curved surface decreases from 5.68 to 2.77 a t H = -3.2, and increases to 4.8 at H = -4 where the surface self-intersects. A series of surfaces of '5 contact angle were computed between the limits of H = -0.49 and -3.8. Surfaces with H = -0.49 and H = -3.8 self-intersect. Two surfaces of '5 contact angle are shown in Figure 8, lower left and right. The internal domain volume fraction increases from 0.26 at H = -0.49 to 0.64 at H = -3.6, and decreases to 0.46 at
I
I
-2.5
0.0
I
25
I 5.0
MEAN CURVATURE Figure 11. Volume fraction of internal domain as a function of mean curvature for contact angles of 180°,150°,120°,90°, 60°, 30°, and 5 O .
H = -3.8. As H decreases from -0.49 to -3.8, the area of the curved surface decreases from 5.7 to a minimum of 3.4 at H = -3.5, and increases to 4.4 a t H = -3.8. Lissantl defined a dimensionless geometric factor as the total surface area of the internal domain divided by the two-thirds power of the internal domain volume. This geometric factor removes particle size dependence from the surface area. For a given volume fraction, the surface with the lowest geometric factor has the lowest surface area and is the most stable with respect to rearrangement. A filled cube has a geometric factor of A/ W 3= 6. Lissant calculated geometric factors of truncated spheres in rhomboidal dodecahedrons (RDH)and tetrakaidecahedrons (TKDH).These geometric factors as functions of volume fraction are shown in Figure 12 along with the geometricfactors from truncated, simple cubic spheres as
Bohlen et al.
988 Langmuir, Vol. 8,No. 3, 1992
1
6.60
4.60
I
0.60
I
I
0.66
0.06
0.00
1.00
VOLUME FRACMON Figure 12. Geometric factor (A/W3) as function of volume fraction of distorted spheres in simple cubic (SC),rhomboidal packing6 dodecahedral (RDH), and tetrakaidecahedral (TKDH) as well as constant mean curvature surfaces of 180' contact angle.
sh
20.0
t
0.0
1
0.0
I
0.0
0.4
0.8
0.8
1 1.0
VOLUME FRACI'ION
Figure 13. Geometric factor ( A l p 4 as function of volume fraction for constant mean curvature surfaces of 180°,150°, 120°, 90°, 60°, 30°, and .'5
well as surfaces of constant mean curvature with 180' contact angles. The geometric factor of RDH spheres are lowest, that of TKDH spheres are next, followed by that of simple cubic spheres. Packed simple cubic spheres occupy 52% of volume, packed TKDH spheres occupy 68 % of volume, and packed RDH spheres occupy 74% of volume. As the volume contained within the spheres is increased, the distortion of the spheres was modeled as truncated spheres.' The ranking of the geometric factors remains the same until volume fraction of 94 96 when the geometric factors of RDH and TKDH switch ranks. The constant mean curvature surfaces of 180' contact angle have lower geometric factors than truncated simple cubic spheres of the same volume fraction; Le., the 180' surfaces have less surface area than distorted simple cubic spheres of the same volume fraction. However,constant mean curvature surfaces of 180' contact angle have higher geometric factors and surface areas than distorted RDH and TKDH spheres of the same volume fraction. Geometric factors of constant mean curvature surfaces as functions of contact angle and mean curvature are shown in Figure 13. Surfaces with contact angles of 180' have the lowest geometric factor for a specified volume fraction. Surfaces of contact angle less than 180' have higher surface areas and, therefore, higher geometric factors. Clearly, the simple cubic surfaces are not the best choice for models of foam and emulsion structure. As indicated by the geometric factor of A/ VI3, constant mean curvature surfaces on a simple cubic lattice and with 180' contact
angles are more densely packed than truncated spheres of the same curvature. These surfaces are not as densely packed as truncated spheres in RDH and TKDH packings. In addition, surfaces with contact angles less than 90' are physically unrealistic models. However, these surfaces are potentially useful as models of immiscible fluids contained in cubes or pores of noncircular cross-section. Regions of high curvature, namely, grooves and corners, are the first to fill with wetting liquid. The fluid interface is a surface of constant mean curvature and can be modeled with this approach. The method presented here can be readily extended to compute surfaces of constant mean curvature that meet the boundary planes of unit cells at prescribed contact angles arranged on other regular lattices.
Summary A new method to compute constant mean curvature surfaces of simple cubic symmetry that meet the walls of the unit cell a t prescribed contact angle is described and illustrated. The method is an extension of the algorithm developed by Anderson and co-workers15which is a Galerkin weighted residual formulation of the surface divergence theorem used with finite element basis functions. A patch of the surface was mapped onto a two-dimensional computational grid which was subdivided into 20 X 20 square elements. The unknown coefficient was approximated by 441 bilinear basis functions, and the system of nonlinear algebraic equations was solved by Newton's method. Continuation methods in mean curvature and contact angle were used to compute families of surfaces. The computations were performed on a Cray-2 supercomputer. Surfaces were computed for a variety of contact angles and mean curvatures. For each surface, the volume contained (V), the area of the curved surface of constant mean curvature ( A ) , the area of the flat films at the boundary planes, and the geometric factor ( A l W 3 ) are presented. With a contact angle between 5' and 133', the surface inverts as the mean curvature decreases, whereas with a contact angle between 134' and 180°,the surface fills the unit cell as the mean curvature decreases, passes through a turning point, and then increases. These simple cubic structures do not model the generally random character of foam and emulsion structure as well as other regular structures arranged in rhomboidal dodecahedral packing or tetrakaidecahedral packing. Truncated RDH and TKDH spheres are more stable with respect to rearrangement than simple cubic surfaces of the same volume fraction. Also, the surfaces computed with contact angles less than 90° are physically unrealistic as models of foam and emulsion structure. In the cases of two immiscible fluids contained in pores or structures of noncircular cross section, this method of computing surfaces is potentially useful. The family of surfaces illustrates the use of a breakthrough method to compute complicated surfaces, and the family serves as prototypes for surfaces that better approximate true foam and emulsion structure. The method can be extended readily to the computation of surfaces in other regular polyhedra that meet the boundary planes of unit cells at prescribed contact angles. Acknowledgment. Support of this research came from the Minnesota Supercomputer Institute and the NSF Center for Interfacial Engineering.
