The computation of the triply periodic I-WP minimal surface

12 August1994

CHEMICAL

PHYSICS LETTERS ELS E V I E R

ChemicalPhysicsLetters 226 (1994) 93-99

The computation of the triply periodic I-WP minimal surface Djurdje Cvijovir, Jacek Klinowski Departmentof Chemistry, Universityof Cambridge,LensfieldRoad, CambridgeCB2 1El'E,UK Received 17 May 1994;in final form 14 June 1994

Abstract

We give parametric equations for Schoen's I-WP triply periodic embedded minimal surface in terms of the Gauss hypergeometric function. The equations lead to new exact formulae for the normalization factor, surface area and the normalized surfaceto-volume ratio, and enable a straightforward low-cost computation of the surface.

1. Introduction

Triply periodic embedded minimal surfaces provide a useful description of many structures, from cell membranes to liquid and solid crystals [1-3]. Previously thought of as purely mathematical objects [4,5 ], in the last 20 years minimal surfaces have become of considerable interest to physicists, chemists, biologists and materials scientists [6]. A general technique for their systematic mathematical derivation and characterization has recently been developed [ 7-9 ]. This work, as well as our earlier papers [ 10-13 ], stresses the computational aspects necessary for a quantitative comparison of periodic minimal surfaces with actual physical systems.

to be matched to actual structures. Since a triply periodic embedded minimal surface divides three-dimensional space into two disjunct regions (labyrinths) in such way that each region is multiply connected so as to result in a bicontinous structure, structural descriptions of crystal structures with large unit cells and complicated networks of cages and channels are possible. In general, the local Enneper-Weierstrass representation [4,5 ]

x=Re i xexp(i0a) (1 - - z 2 ) R ( ' c ) dz, to o

y = R e i ixexp(i0a) (l + z 2 ) R ( T ) tiT, to o

2. Periodic minimal surfaces

Minimal surfaces are surfaces of zero mean curvature. A triply periodic embedded minimal surface is a surface in three-dimensional space which is infinite, minimal, periodic in three independent directions (has space group symmetry) and free of selfintersections (embedded). Such surfaces possess translation symmetry which in principle enables them

z = R e i 2~cexp(iOB)zR(r) dz,

where the normalization factor r is real, i 2 = -- 1, 0 a is the Bonnet angle and z is complex, enables us to associate with every 'Weierstrass' function R (r), analytical in some simple connected domain in C except at isolated points, a unique (disregarding trans-

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SSD10009-2614 (94) 00708-X

(1)

too

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D. Cvijovi6, 3". Klinowsld /ChemicalPhysics Letters 226 (1994) 93-99

lation) parametrized surface r(za, %) which is guaranteed to be a minimal surface. However, such a surface is not necessarily free of self-intersections and the question of its possible periodicity must be examined separately. The Cartesian coordinates of any point are expressed as the real (Re) part of contour integrals evaluated in the complex plane from some fixed point too to a variable point to. 0B is the parameter describing the Bonnet transformation, which consists of bending the surface without stretching. Different values of 0B lead to different associate surfaces; the associate surface with 0n=~/2 is the conjugate surface.

M 5 a

a

a

M6

M8

Q"

M7

M1

(a) M5

M2

M3 MS

3. Enneper-Weierstrass representation of the I-WP surface 2a/3

The I-WP surface was first described by Schoen [ 14 ], who built its models, demonstrated its cubic symmetry and identified its space group as Im3m. It is body-centred and the labyrinth system seemed to Schoen to resemble a wrapped package, hence the acronym I-WP. Schoen also established that the Fl~ichenstiick (the fundamental piece) of the I-WP surface is inscribed in a quadrirectangular tetrahedron (see Fig. la) which is its kaleidescope (Coxeter) cell. Six such pieces can be assembled to give a larger surface piece contained in a cube (Figs. la and 4a). Following Fogden and Hyde (see Fig. 2b in Ref. [7] ) we refer to this cube, with edge length a, as the bounding cell of the I-WP Fl~ichenstfick. Schoen derived the IWP surface as the conjugate surface of a self-intersecting infinite periodic minimal surface described by Stessmann (surface number VI in Fig. 1 in Ref. [ 15 ] ). The boundary of the Fl[ichenstiick of Stessmann's surface consists of straight lines (Fig. 2). Note that the tetrahedra in Figs. 1 and 2 are of different size (see Eq. (12) below). Karcher [16] gave a rigorous mathematical proof of the existence of the IWP surface by considering the 'conjugate Plateau problem'. Anderson [ 17 ], who was the first to study the surface computationally, showed that it divides space into two sub-spaces of unequal volumes (the volume ratio of the two labyrinths is 0.5361 + 0.0002) and computed its area as well as the surface-to-volume ratio. He also found that the I-WP surface is found in star block co-polymers, and conjuctured that

a13 M1

M4

Fig. 1. (a) The Fl~ichenstfick of the I-WP embedded periodic minimal surface and its bounding cell (the cube with edge length a). The Fl~ichenstiick is confined by the quadrirectangular tetrahedron and meets its faces orthogonally. The coordinate system induced by the parametrization has the origin at point O' and the orientation shown in the inset. The coordinates of all points and normal vectors are given in the text. Flat points O' and Q' divide the edges of the cube in the ratio 2: 1. (b) The projection of the the Fliichenstiick onto the O'xy plane and the relationship between the x and y coordinates of the point Q'.

it describes the cubic phase of ternary surfactant mixtures. The I-WP Fl~ichenstiick has a twofold rotation axis perpendicular to it, but there are no straight lines on the surface. The faces of the Coxeter cell are mirror planes for constructing the minimal surface. Thus the infinite I-WP surface can be generated by successive reflections of the Fl~ichenstiick in these faces and in

95

D. Cvijovi~,J. Klinowski / Chemical PhysicsLetters 226 (1994) 93-99 Q,,

~

R"

ac

Fig. 2. The Fliichenstiick of Stessmann's self-intersecting (nonembedded) periodic minimal surface (conjugate to I-WP) is bounded by the edges of the O"P"Q"R" tetrahedron. The origin of the coordinate systemdictated by the parametrization is at 0" and the orientation of the axes is shown in the inset. all new faces. Unlimited continuation of this operation gives a minimal surface without self-intersections which extends to infinity in all directions. The primitive cell ('rhombohedral body-centered cubic, bcc) is composed of 24 Fliichenstiicke, and there are 48 of them in the bcc unit cell. The Weierstrass function of the I-WP surface is [ 8 ]

R ( Q = T-2/3 (T4-[- 1 )-2/3,

(1,0,0),

T imoo 0

P

(2)

where z is complex. However, the integration domain shown in Fig. 2a of Ref. [8] is incorrect. This can be shown by calculating integrals ( 1 ) with Wcierstrass function (2) and On= x/2 along the dotted lines and along the boundary of the shaded area in our Fig. 3a. Straight lines which bound Stessmann's surface (Fig. 2) cannot be obtained in the former case. The shaded region in our Fig. 3a is the correct Gauss map image of the I-WP Fliichenstiick. The standard stereographic projection of the normal vectors (see Fig. la), (0,0,--1),

(b)

(½X/~,--½x/~,O),

Reo)

Q

Fig. 3. (a) Imegration domain of the I-WP Fliichenstiick incorrectly shownin Fig. 2a in Ref. [7 ] (dotted lines) and the correct integration domain (shaded). (b) The mesh used for computing the I-WP surface. The complex-planecoordinates of points O, P, Q and R are O(0, 0), P(1, 0), Q(½x/~, -½x/~) and R( 1/x/~- 1, 1/v/2- 1). The circular arcs which bound the domain are those of the unit circle and the circle of radius v/2 centered at (½x/~, ½x/~).

(-½, -½, - ½4 ), 4. Parametrization of the I - W P surface

onto the complex plane then results in 0,

1,

(1--i)½x/~,

(-- l - - i ) ( 1 - - 1/x/~) i.e. gives the complex plane coordinates of the points O, P, Q and R, respectively (see Fig. 3b).

In what follows F ( z ) and 2F1 (a, fl; 7; z) are the gamma and the Gauss hypergeometric functions, respectively. Their properties are described in standard texts on special functions (for example in Ref. [ 18 ] ). We will show that the Enneper-Weierstrass representation of the I-WP surface involves integrals which

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D. Cvijovid, J. Klinowski / Chemical Physics Letters 226 (1994) 93-99

can be analytically evaluated. The resulting parametric equations are expressible in terms of the Gauss hypergeometric function only. We will also derive a simple closed-form analytical expression for x. First, by combining (1) and (2) and by setting 0B=0 and o90=0, the Cartesian coordinates (x, y, z) of any point on the I-WP Fl~ichenstiick are given by

(x,y,z)=(Re(x*),Re(y*),Re(z*)),

(3a)

where

x*(to)=Io--I2, y*(to)=i(Io +I2) , z*(to) =211,

(3b)

and the Ip are the complex integrals, zj' dr 1)2/3' Ip(o9) =Ij,(0) + x i z2/3(z4+

x(Q) -2. Y(Q)

0

p = 0 , 1 and 2.

(3c)

where the variable limit to is any complex point either inside or on the boundary of the domain shown in Fig. 3a. With z = t t/4 the integrals (3c) become (4a)

0

p=0,1and2.

(4b)

This makes it possible to evaluate them analytically in terms of 2F~ (entry 3.194 in Ref. [ 19] ), and since 2F~ (or, fl; ?; 0) = 1 for any value of or, fl and ~, ( ? # 0 , - 1 , - 2 .... ) w e h a v e to4/t

2

Ip(og) = x - ~ - 2F l ( 3, ]2; fl+ 1; --O94) .

(4C)

Note that 2El (og, fl; ~1;_7) exists for all z, real or complex, such that Iz l ~<1 for all values of the parameters (y # 0, - 1, - 2, ... ) provided that Re ( y - ot - fl) > 0. The well-known formula then holds for z = 1 [ 18 ] 2Fl (Or, fl; ?; 1 ) = r ( y ) r ( 7 - ~ - p )

r(r-a)r(r-p)

Note that when co= ( 1 - i ) ½x/~we have co4= _ 1 and o91/3=sin(5n/12)-isin(~/12). By making use of (5) and of the basic property of the gamma function [181 (7)

and in view of (3a), (3b), (4b) and (4c), we have

sin(5n/12)A+sin(n/12)B sin(5n/12)B+sin(g/12)A where A=F(~2)/F(~) and B=F(7)/F(~2). Fur-

x(Q) y(Q)

where /t=~2(3p+l),

(6)

F(z+ 1) =zF(z) ,

tO4

Ip(o9)=Ip(O)+IX f tu-l(l+t)-2/3dt,

and are finite, since 094= - 1. These values are given by (5). Another singular point, to = 0, is mapped into (0, 0, 0). Thus the Weierstrass function (2) and parametrization derived from it lead to the coordinate system shown in Figs. 1 and 2. In this system the coordinates of the vertices of the bounding cell (Fig. la) are M~(0, 0, ~a), M2(0, 0, -~a), M3(a½v/~_2, 2 1 1 1 a½x/~,-~a), M4(a:,¢;2, a2xf2, ~a), Ms(a½~2, -a½x/2, ~a), M6(a~v/2, -a½x/2 , - 2 a ) , M7(ax/2 , 0, -ia) and Ms(ax/2, 0, ~a). In order to determine x we use, for the first time, the property of the I-WP Fl~ichenstiick which we have discovered: that the point Q' divides the edge of the bounding cell in the 2: 1 ratio. This can be proved by verifying the relationship (see Fig. 1b)

(5)

In this way, despite the fact that the point to= ( 1 - i ) ½ x / 2 (point Q in Fig. 3b) is a singular point of the Weierstrass function (2), in the view of (4e) and (5) the values x(Q), y ( Q ) and z(Q) exist

ther, since [ 18 ]

F(z)F(l-z)= sin(nz-------~' Izl < 1

(8)

and

,¢5-1 2x/~ ,

sin(n/12)-

sin(5n/12)= 2 @ 2 1 , we readily obtain (6). It then follows (Fig. I b) that

x(Q)=2al~/2, y ( Q ) - - a l ~ / 2 .

(9)

The expression for x, 8~

1

x= 3x/~F3(I ) a ,

(10)

is found starting from the first equation in (9) in the

D. Cvijovi~,J. Klinowstd/ Chemical PhysicsLetters 226 (1994) 93-99 way already used for deriving (6). Thus, x is a positive constant which depends on the size of the bounding cell. Finally, Eqs. (3a), (3b), (4b), ( 4 c ) a n d (10)are the parametric equations of the I-WP Fliichenstiick with a parameter domain shown in Fig. 3a. Note that in view of the use of the Gauss hypergeometric function they differ from the parametric equations of the other triply periodic minimal surfaces [ 10-13 ] which use incomplete elliptic integrals of the first kind.

5. Properties of the I-WP surface

Equations derived above enable us to deduce certain geometric properties of the I-WP surface and its conjugate surface. First, using (5), (7) and (8) we easily verify that z(Q) = z ( P ) = ½a.

( 11 )

Thus, the singular points of the Weierstrass function (2), i.e. the points O and Q, are mapped into points O'(0, 0, 0) and Q'(2a½v/2, a~x/~, -~a) on the I-WP surface. Geometric analysis shows that the points O' and Q' divide a (the edge of the cube) in the 2: 1 ratio. Furthermore, the length O'Q' with value a~x/ql is an invariant of the surface. There seems to be no simple similar relationship for points P' and R'. However, we conjecture that O'P'=R'Q' and O'R'= P'Q'. There is strong numerical evidence for this, but we have been unable to prove it analytically. For 0B=n/2 the conjugate Fl~ichenstiick is obtained by introducing the multiplicative factor i (the imaginary constant) into expressions (4c). It is found that O"(0, 0, 0),

a"(O, -av/~3, 0 ) ,

Q"(0,-av/~L a/x/~),

R"Calx/~, - a i r ~ 6 , 0 ) .

Since P"Q" is the length of the edge of the cube, ac (see Figs. 1 and 2) a¢=a/ qr3 .

(12)

This unexpectedly simple relationship allows us to derive analytical formulae for the perimeter ~ , the surface area ~¢ and the normalized surface-to-volume ratio ~¢/~2/3 (since the ratio must be dimensionless we take ~2/3 instead of ~ ) of the I-WP

97

Fl~ichenstiick. Since the Bonnet transformation preserves the metric, the perimeter and the area of the conjugate Fl~ichenstiick and of any associate surface remain unchanged. The value of ~ is calculated by computing the perimeter of the conjugate surface of the I-WP. According to Fig. 2 and in view of ( 12) this is given by = 2ac(x/~+ 1 ) = 2a(x/~-23+ 1/x/~) . Further, if the area of the minimal surface bounded by the tetrahedron is ~¢, r is the inradius of the tetrahedron (the radius of a sphere which touches each face of the tetrahedron at one point) and the length of the perimeter is ~, then [20] ~¢=½~r. Hence a2 I 2 d = ½a ac= 2x/~ - ~acx/~ , since the inradius is r= ½a(x/~-1 ) for the tetrahedron shown in Fig. I a. Finally, by observing that ~a 3 is the volume of the quadrirectangular tetrahedron which encloses the IWP Fllichenstiick, we have ~ / ~ 2 / 3 = ( 4t) I/6 (for the Fl~ichenstiick),

d/~ V'2/3--22/3V/3 (for the bcc primitive cell), ~¢/~2/3 = 2x/~

(for the bcc unit cell).

The ratio d/'/r2/3 for the bcc unit cell of the I-WP surface was computed earlier by different methods: Anderson [ 17] obtained the numerical value of 3.467 _+0.003, Lidin et al. 3.46410161 [ 22 ] and Terrones 3.44436 [22 ]. Our result is the exact value. We have thus proved the conjuncture in Ref. [21 ] that 2x/~=3.464101615 is the normalized surface-tovolume ratio for the bcc unit cell of the I-WP surface.

6. Computation of the I-WP surface

Schoen's report contains photographs of plastic models of the I-WP surface (Fig. 9 in Ref. [ 14] ). The surface was first computed by Anderson [ 17 ] and 15/ 8 unit cells were shown in Fig. 5a in Ref. [23]. The drawings of the surface in Ref. [ 16 ] and in Fig. 27 in Ref. [ 5 ] and the colour photograph of its unit cell

D. Cvijovi~, J. Klinowski / Chemical Physics Letters 226 (1994) 93-99

98

given as plate 7(a) in Ref. [5] and Plate 13 in Ref. [24] are obtained by Polthier [25 ]. Our computation of the I-WP surface is as follows. The evaluation of functions F and 2F~ which appear in Section 4 is computationally straightforward, and appropriate programs are easily available either as user-callable routines in general purpose subroutine libraries or built-in functions in various software packages. For instance, Mathematica (Wolfram Research) has F and 2F~ as built-in functions Gamma and Hypergeometric2F1. The same software contains powerful 3D graphics packages and we find that ParametricPlot3D and ListSurfacePlot3D are particularly useful. The Cartesian coordinates of the points of the IWP Fliichenstiick were computed by the Eqs. (3a), (3b), (4b), (4c) and (10) and 10×10 mesh (Fig. 3b). In order to calculate larger pieces of the surface it is convenient to rotate the coordinate system shown in Fig. la by - l t / 4 about the z axis and translate its origin along the z axis by ]. This results in a new coordinate system with origin at point M2 and edges M2M6, M2M3 and M2M~ determining the x, determining the x, y and z axes, respectively. Hence the entire bounding cell is in the first octant and three of its faces are coordinate planes. The new coordinates of the I-WP Fl~ichenstiick are those of Fl (see Table 1 ). F2 and F 3 a r e obtained by reflection of F~ through planes determined by the points listed in Table 1. Similarly, F2 and F3 give F5 and F4 respectively, while F 6 c a n be obtained either from Fs or F4. By reflecting a piece of the I-WP surface obtained in this way (Fig. 4a) through the coordinate planes (simple change of Table 1 Cartesian coordinates x', y' and z' of the six Fl~ichensfiicke the assembly of which gives a piece of the I-WP surface inscribed in a cube. The coordinates x, y and z are given by Eqs. (3a), (3b), (4b), (4c) and (10) while ~= ½x/~ and ~/= ]. The points Mr, M2 etc. (see Fig. la) determine the mirror planes Fliichenstiick

x'

y'

z'

Points

F,

~x-~

¢x+ (:v

z+,7

-

F2 F3 F4 F5 F6

~c+(y ~X--~F ~¢.q-(~., Z-l-t/ 2"-I-~/

~x-~V 2"-I-q z-l-r/ ~X--~{3.' ~¢-}-~

z+~/ ~jt'+ ~).' ~f -- ~,Y ~'-I- ~[1,' ~ x - ~.F

M1, M2, Ms M2, M,, Ms M2, M3, Ms M2, Ms, Ms M2, M6, M8

M5

us

M7

(a)

ua

(b) Fig. 4. The transformed coordinate system used in this figure is explained in the text. (a) A piece of the I-WP surface composed of six Fliiehenstiicke is inscribed in the bounding cell. Six edges of cube are divided in the ratio 2:1. (b) Unit cell of the I-WP surface made by combining eight pieces shown in (a). The exact surface area of this piece is 24a2/v/3. The origin of the coordinate system is at the centre of the cube, and the orientation of the axes is as shown.

the sign of the coordinates) we arrive at the complete I-WP unit cell (Fig. 4b).

7. Concluding remarks

A particular minimal surface is mathematically well-described by its Enneper-Weierstrass represen-

D. Cvijovi~, J. Klinowski / Chemical Physics Letters 226 (1994) 93-99

tation. The Weierstrass function then completely determines the differential geometry of the surface (its first and second fundamental forms). The EnneperWeierstrass representation in principle enables the computation of the surface, since the integrals involved can be evaluated numerically. However, this work illustrates the importance of closed-form expressions for the parametric equations. By giving them for the I-WP surface we have been able to obtain analytical expressions for the normalization factor, perimeter length, surface area and the surface-tovolume ratio, something which cannot be obtained by numerical integration. It is therefore important to ensure that the relevant integrals cannot be analytically evaluated before resorting to numerical methods. We note that the analytical formulae for the perimeter and the surface area of the I-WP Fl~ichenstiick are the first such results. Our analytical expression for the surface-to-volume ratio is the first such result since the results of Schwarz (for the tetragonal saddle surface) and Schoen (for the D, G, P and the Neovius surface, see Table 1 in Ref. [2] ). Further, the parametric equations derived in this paper enable straightforward computation of the I-WP surface. Its first calculation [ 17,23] was performed with a specially designed numerical method for solving the minimal surface equation (a second-order partial differential equation) in tetrahedral coordinates, using Cyber 124 and Cray computers. Our calculations were done with a simple program [26 ] t written in the Mathematica language (a commercially available general-purpose mathematical software) on a PC 386 computer. This software allows us to visualise the surface.

Acknowledgement We are grateful to Mr. M.J. Springett for invaluable help and to Mr. Clemens WSgerbauer for assistance with computer graphics. I An electronic PostScript version of this thesis is available through an anonymous ftp account on model.ch.cam.ac.uk (131.111.112.8).

99

References [ 1 ] A.L. Mackay, Nature 314 (1985) 604. [2] A.L. Mackay and J. Klinowski, Comput. Math. Appl. 12B (1986) 803. [3] S. Andersson, S.T. Hyde, K. Larsson and S. Lidin, Chem. Rev. 88 ( 1988 ) 221, and references therein. [4]J.C.C. Nitsche, Lectures on minimal surfaces, Vol. 1 (Cambridge Univ. Press, Cambridge 1989). [ 5 ] U. Dierkes, S. Hilderbrandt, A. Kiister and O. Wohlrab, Minimal surfaces (2 Vols. ) (Springer, Berlin, 1992 ). [6]E. Dubois-Violette and B. Pansu, eds., International workshop on geometry and interfaces, Colloque de physique, J. Phys. (Paris) Suppl. to Vol. 51 (1990) C7. [ 7 ] A. Fogden and S.T. Hyde, Acta Cryst. A 48 (1992) 442. [8] A. Fogden and S.T. Hyde, Acta Cryst. A 48 (1992) 575. [9] A. Fogden, Acta Cryst. A 49 (1993) 409. [10] D. Cvijovi6 and J. Klinowski, J. Phys. I (Paris) 2 (1992) 137. [ 11 ] D. Cvijovi6 and J. Klinowski, J. Phys. I (Paris) 2 ( 1992 ) 2191. [ 12 ] D. Cvijovi6 and J. Klinowski, J. Phys. I (Paris) 2 (1992) 2207. [13] D. Cvijovi~ and J. Klinowski, J. Phys. I (Paris) 3 (1993) 909. [ 14 ] A.H. Schoen, Infinite periodic minimal surfaces without selfintersections, NASA Technical Report No. TN D-05541 (1970). [ 15 ] B. Stessmann, Math. Z. 38 (1934) 417. [ 16] H. Karcher, Manuscr. Math. 64 (1989) 291. [ 17 ] D.M. Anderson, Ph.D. Thesis, University of Minnesota (1986). [18] M. Abramowitz and I.A. Stegun, eds., Handbook of mathematical functions ( Dover, New York, 1980). [ 19] I.S. Gradshteyn and I.M. Ryznik, Table of integrals, series, and products (Academic Press, New York, 1980 ). [20] B. Smyth, Invent. Math. 76 (1984) 411. [21 ] S. Lidin, S.T. Hyde and B.W. Ninham, J. Phys. (Paris) 51 (1990) 801. [ 22 ] H. Terrones, Ph.D. Thesis, University of London ( 1992 ). [23] D.M. Anderson, H.T. Davis, L.E. Striven and J.C.C. Nitsche, Advan. Chem. Phys. 77 (1990) 337. [24] K. Polthier, in: Geometric analysis and computer graphics, eds. P. Concus, R. Finn and D. Hoffman (Springer, Berlin 1991 ) pp. 141-145. [ 251 K. Polthier, Diplomarbeit, Bonn (1989). [26] D. Cvijovi6, Ph.D. Thesis, University of Cambridge (1994).