J. CHEM. SOC. FARADAY TRANS., 1994, 90(2), 263-270 New Families of Triply Periodic Minimal Surfaces Andrew Fogden* and Markus Haeberleint Inorganic Chemistry 2,Chemical Center, P.O. Box 124,S-22100 Lund, Sweden The established results regarding balanced minimal surfaces of orthorhombic symmetry are analysed and sup- plemented here. We commence by reviewing the simplest examples, possessing a genus value (per unit cell) of 3. From this foundation we construct a generalised approach, and apply it to the next simplest cases. In particu- lar, we derive two related families of surfaces possessing genera of 5. These two families contain, in total, three previously unrealised examples. For all 11 surfaces considered the exact mathematical equations specifying them are given, along with their space group-subgroup specifications, and illustrations of their elements are provided.Increasing awareness of the geometrical intricacy of atomic or molecular assemblies accessible to nature has elevated the importance of surfaces as a means for describing structures. The description provides an understanding of the way in which the unification of local affinities (on the surface) builds a scheme of global discrimination (either side of the surface). In recent years these developments have been most strikingly exemplified by the observation of bicontinuous phases in a variety of systems, and the modelling of their smooth par- titioning using minimal (zero mean curvature) surfaces. 1-3 For ordered phases the observed space group is often consis- tent with that of known infinite periodic minimal surfaces (IPMS).The interpenetrating pair of tunnel systems defined by such a surface then suggests a possible picture of the topo- logical manner of bicontinuity. Within this known set of IPMS topological options common to the space group, we choose that particular surface which most closely matches any additional experimental information. Typically the infor- mation can distinguish at most the gross topological aspect, namely the value of the genus (per unit cell).4 In the absence of a solid basis for distinction it is usually assumed that the lowest genus is favoured. The restriction to a specific geometrical condition (zero mean curvature) may appear unnecessary since a continuum of bicontinuous partitions fits the same space group. However, the limitation is beneficial for working purposes by the very fact that its admission of only a strictly countable set of pos- sibilities is compatible with the order of typical experimental resolution.Further, the continuum naturally falls into classes sharing the same topology as a member of the IPMS set, rendering this special case an adequate representative of its class and the obvious departure point for perturbation once more information comes to hand. This matching procedure implicitly relies upon a know-ledge of all IPMS, or more specifically, all those without self- intersections. An exhaustive listing is clearly unattainable since there exists a (countable) infinity of examples; instead one should aim to establish a catalogue containing, within all viable space groups (at least of the higher symmetry types), all IPMS possessing relatively simple topologies.To bias the task still further towards the weight of experimental evidence, most attention should be directed at the ‘balanced’ IPMS, for which the two, interpenetrating tunnel systems are sym- metrically interchangeable (by two-fold rotation axes and/or roto-inversion centres lying on the surface). Using real-space methods, based on construction of surfaces spanning crystal- lographic nets, Fischer and Koch have collated a list totalling 43 balanced IPMS of symmetry no lower than orthorhombic 7 Permanent address; Department of Inorganic Chemistry, The Royal Institute of Technology, S-100 44 Stockholm, Sweden. Foror rh~mbohedral.~~~each case the particular space group is provided, together with its (black-white) subgroup derived by abolishing the balancing invariances.Similarly, the genera are identified, displaying a variation through the cases ranging from 3 (the minimum conceivable value) up to 37. Significantly, their list includes three instances of sub- sets of surfaces (R3, MC2, MC3, MC4 and R2, MC6 and HS3, ST1) which possess not only common space groupsub- groups [P6/rncc--Pci/m and 14/mcrn--P4/rnbrn and P6,22-P6,22(2c), respectively] but also identical genera (13 and 9 and 7, respectively). Such coincidences illustrate a general shortcoming of the matching procedure, since the members of the subsets possess different topologies but cannot be differentiated from the gross standpoint of genus.However, in these three instances the genera are sufficiently high to exclude their manifestation in the simpler physical systems. Any such surface derivation relying upon real models in three-dimensional space carries with it a drawback. Although an analysis of space-group scenarios compatible with bal- anced IPMS establishes a rigorous foundation, the options for spanning a given framework of symmetry operations is limited by human imagination, which tends to underestimate the full topological versatility of minimal surfaces. Conse- quently, a number of options may be realised while others closely related to (but topologically distinct from) them may be missed.One of the present authors has recently employed an alter- native approach*-’ ’ based upon the general Weierstrass rep- resentation of a minimal surface in terms of its two-parameter space on the unit sphere, or equivalently, the complex plane. By transfer to its natural mathematical domain the essence of the IPMS problem is reduced to a sequence of necessary conditions, in which space groups are circumvented by spherical groups and topologies by counting schemes imposed on them. This clarified approach facilitates derivation of the Weierstrass function yielding the parametri- sation of the IPMS, and simultaneously embraces all related surface options.In a previous study” this algorithm was used to generate a family of balanced IPMS of trigonal symmetry and genus 7 which includes, in addition to the C(H) and MC1 surfaces listed by Fischer and K~ch,~.~ three previously unidentified relatives termed the MC(H), H2 and H3 surfaces. Here we apply it in a similar manner to the derivation of two such families possessing orthorhombic symmetry and genus 5. It is found that two members of one of these families possess iden- tical space groupsubgroup specifications, representing another source of ambiguity in the IPMS matching pro- cedure. However, compared with the instances mentioned above, the reduction to orthorhombic symmetry admits ambiguities at this significantly lower genus.Accordingly the situation now borders on relevance to bicontinuous phases in nature, since these two families represent the next-to-simplest IPMS of orthorhombic symmetry. To illustrate the approach we commence with a brief review of the simplest such sur- faces, those of genus 3. Orthorhombic IPMS of Genus 3 Any discussion of IPMS benefits from a familiarity with the situation for genus 3 which, while encapsulating all essential properties of IPMS in general, possesses certain simplifying features. Most importantly, each of the degenerate positions, termed ‘flat’ points, on an IPMS of genus 3 is of first order only, and moreover, necessarily acts as an inversion centre. Consequently, the surfaces are balanced. Further, the collec- tion of surfaces with symmetry no lower than orthorhombic or rhombohedral is free from ambiguities, as each is specified by a distinct space groupsubgroup.This collection is based on five examples: the cubic D, P and G surface^'^,'^ and the tetragonal CLP and hexagonal H surface^.'^-'^ All other members may be obtained by symmetry degradations of these. For the purposes of this study we limit attention to the subtype of these degradations with orthorhombic symmetry containing two-fold axes on the surface in the x, y and z axis directions and/or mirror planes slicing the surface parallel to the yz, zx and xy planes, respectively [i.e. for each of the three coordinate directions either a two-fold axis or a mirror plane (or both) must exist].The members of this subtype are derived from the D, P and CLP surfaces. (The other two examples, H and G, also give rise to degradations in this col- lection, as discovered by recent applications of the Weierstrass parametrisation algorithm. In particular, the H surface was found to admit orthorhombic dist~rtion,~ and distortion of the G surface was exhibited by construction of its rhombohedral variants.’ ’However, no such degradations meet these restrictions on the specific orthorhombic subtype considered here.) The tetragonal versions of the D, P and CLP surfaces give rise to pairs of orthorhombic distortions of this type, denoted oDa, oPa; oDb, oPb and oCLP, oCLP’, re~pectively.~ In Fig. l(a)-(f) we display representatives of the respective surface elements for these six cases, i.e.the smallest subunits bounded entirely by the two-fold axis lines and/or mirror plane curves sketched there. The surface elements in each of Fig. l(c)-(f) possess (first-order) flat points at their_ 2/m vertices, while the elements in Fig. l(a), (b)both exhibit 1 symmetry about their internal flat point. Note that, as opposed to the situation at the mm2 or 222 vertices, the normal-vector directions at each 2/m and i position have one and two degrees of freedom, respectively. Thus all six surface elements carry, in total, two degrees of freedom, naturally giving rise to their two-variable scope for orthorhombic distortion (ignoring the uniform dila- tation factor). The translational units (of genus 3), obtained by repeated two-fold rotation and/or mirror reflection, com- prise 16 such elements for Fig.l(c)-(f) and eight for Fig. l(a), (b).In this way the units for all six cases contain eight flat points. With this information the parametrisations of the six cases may be readily derived uia the abovementioned algorithm.” The method is outlined here in a sufficiently broad context to motivate the generalisations in the subsequent sections. To a minimal surface we apply the stereographically projected Gauss map, taking a point (x, y, z) on it to be the complex number o = u+iu which is the projection into the (complex) plane of the point on the unit sphere marking the direction of the normal vector to the surface there.The properties of this J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1 1 1 Fig. 1 Sketches of the bounding circuits, comprising straight edges along two-fold axes and/or curved edges in mirror planes, of the surface elements for the orthorhombic IPMS, (a) oDa, (b) oPa, (c) oDb, (6)oPb, (e)oCLP, (f)oCLP’. The numerals 1 mark the sites of first-order flat points. The coordinate axes x, y and z are oriented to the right, into the page and vertically upwards, respectively. (These two conventions will be maintained throughout all subsequent surface-element depictions.) map dictate that its inverse, parametrising the surface, may be expressed in the form’* (x, y, z) = Re [1 -d2,i( 1 + o”),2w’]R(co’)do’ (1)r for some complex-analytic function, R(o).The algorithm thus relies upon interpretation of properties of the IPMS through this Weierstrass representation as necessary conditions on its particular R(w), the so-called Weierstrass function that we seek.In the six cases the Gauss maps of their generic surface elements are pairwise identical. The stereographic projections of the maps of Fig. l(a), (b)and (c), (d)and (e),(f)are imaged in Fig. 2(a),(b)and (c), respectively. The bounding segments of the surface elements, Gauss-mapped to arcs of three mutually perpendicular great circles, appear in the projection as follows. A two-fold line in the y axis direction and/or a mirror curve parallel to the zx plane is imaged along the real axis; similarly, those related to the x axis md/or yz plane and to the z axis and/or xy plane lie along tb imaginary axis and along the unit circle, respectively.The boundary image thus defines a single circuit delimiting one octant in Fig. 2(b)and (c) and a (continuous) double circuit delimiting two octants in Fig. 2(a). Note that the dihedral angles at the (first-order) flat points are increased by a factor of two at their image sites, i.e. Fig. 2 Projected Gauss-map image of the surface elements, (a) oDa and oPa, (b) oDb and oPb, (c) oCLP and oCLP’, in the complex o-plane (Re and Im denote the real and imaginary axes). The double- hatching in (a), and the single-hatching in (b) and (c), represent double and single coverings of the octant, respectively. The numerals 1 indicate the image sites of the first-order flat points, i.e.sites of first-order branch points of the corresponding Weierstrass function (again a convention maintained throughout all subsequent image depictions). J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 from n/2 to 7t in Fig. 2(b),(c) and from 2n to 4n in Fig. 2(a)(in which the site pins the two distinct sheets). Extension of the surface element over a bounding segment via its two-fold rotation or mirror-reflection operation induces a reflection of its Gauss map (on the unit sphere) across this boundary arc. This corresponds, in projection, to application of the operations w -+ 0,co -+ -6 and w -P l/G for the three types of boundary images listed above. Contin- uing this process, the attainment of the translational unit amounts to a uniform covering of two copies of the unit sphere, or complex plane.In this way the sites of the eight flat-point images are symmetric with respect to all reflections in the octant tiling, i.e. the two ‘edge’ sites in Fig 2(b),(c) both possess four equivalents and the ‘face’ site in Fig. 2(a) possesses eight equivalents. Further, each of these sites now pins the two sheets (so the angle of 4n is fulfilled there). These double-sheeted coverings thus define the complete domains of the Weierstrass functions in eqn. (1) The fact that the translational units of genus g = 3 correspond to domains comprising two sheets (s = 2) is consistent with the general IPMS relation’ g=s+l (2) Each of the sites mentioned above are first-order branch points of their Weierstrass function.The existence of eight such sites in each case, so their total branch-point order W = 8, is again consistent with the general IPMS rule’ w= 4s (3) Establishment of the Weierstrass functional forms match- ing the domain structures is a straightforward matter. These forms are composed from polynomials (in w), the root sets of which must all be symmetric with respect to the reflection scheme of the octant tiling underlying Fig. 2. Consider the single octant in Fig. 3. Any polynomial with simple roots sta- tioned at the four reflected equivalents of a single edge site on this octant may be expressed as p,(o) = (aZ+ 1)2 -yew2 (4) for some real ye value.The three kinds of edge are distin- guished by the subdivided ye ranges borne by them in Fig. 3. Similarly, any polynomial with simple roots at the eight I 0 -co >Re w Fig. 3 The labelling of the three kinds of octant edges indicates the (monotonic) ranges of the parameter ye corresponding to them in eqn. (4) equivalents of a single face site on the octant may be written as Pf(4 = C(W2 + -yfo2][(oZ + -YfW2] (5) for some complex number yf in (say) the upper half-plane. The functional forms, thus comprised, pertaining to the cases here are then fixed by the requirement that the Weierstrass function be non-zero throughout (since the Gaussian curvature is finite everywhere on the surface), and that its pair of values (on the two sheets) for each w be equal and opposite (since the corresponding pair of surface points are related by inversion).In particular, the form relevant to Fig. 2(b) and (c)is then ‘(a) = +exp(i8)Cpe,(w)pe,(w)I-lj2 (6) where pe,(w)and pe,(o)are given by eqn. (4) with ye = ye, and ye = ye,, respectively, and that for Fig. 2(a)is ‘(a)= +exp(i8)pf(w)-1/2 (7) The zeros of the eighth-degree polynomials in these forms are the eight branch points. The situation in Fig. 2(b) and (c), differing in the kinds of edges bearing the two branch points, are thus distinguished in eqn. (6) by the ranges of their real parameters ye, and ye, as per Fig. 3. Hence we take ye, > 4 and 4 > ye, > 0 for Fig. 2(b)and 4 > ye,, ye, > 0 for Fig. 2(c).In eqn. (7) the parameter yf, specifying the site of the face branch point in Fig. 2(a),is complex as mentioned above. The Weierstrass functional forms in eqn. (6)and (7) contain three degrees of freedom [ignoring a constant, real factor which merely effects uniform dilatation of the surfaces uia eqn. (l)]. In the context of the surface cases considered here, the real variable B(mod n) is restricted to particular values (particular members of the Bonnet associates it defines), thus reducing the forms to two-variable freedom, as required. Spe- cifically, surfaces containing two-fold axes and/or mirror planes correspond to the value 8 = 0 or 8 = 42. The two choices, termed the surface-adjoint surface pair, are related by interchanging the identification of two-fold axes and mirror planes on the same Gauss map.It is precisely these two choices which restore the six surface cases in Fig. 1 from their pairwise identical images in Fig. 2. The pair oDb, oPb and the pair oCLP, oCLP’ correspond to eqn. (6) (for the abovementioned variable ranges pertinent to Fig. 2(b)and (c), respectively) with 8 = n/2, 8 = 0 for both. Similarly, the pair oDa, oPa [imaged to Fig. 2(a)] is in turn given by eqn. (7) with f3 = 42, 8 = 0. The totality of surface information regarding these six cases is summarised, for reference, in the first six rows of Table 1. Generalisationto a Family of Genus 5 The basis for all IPMS of genus 3, namely their exhibition of flat points of lowest (i.e. first) order, also suffices for ortho- rhornbic IPMS of any genus.By this we mean that non-degenerate or simple degenerate positions (flat points of order 0 or 1, respectively) on a minimal surface can exhibit all point-group symmetries admissible to an orthorhombic space group. In particular, non-degenerate positions can access the symmetries mm2, 222, m, 2 (either lying on the surface or per- pendicular to it) and 1, while simple degeneracits can display symmetries 2/m, m, 2 (necessarily on surface), 1 and 1. Flat points of higher orders may possibly occur as a coalescence of simple degeneracies, for example of second order at mm2 or 222, and of third order at 2/m, positions due to merging of a pair of first-order positions on an m or 2. However, these are merely special occurrences; any analysis of orthorhombic IPMS must originate from the assumption that flat points of, at most, first order are involved.J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 1 Summary of all information relating to the 11 orthorhombic, balanced IPMS studied here (listed in column 1); the information relating to their real-space manifestation is listed in columns 2-4, while that specifying their parametrisation is listed in columns 5-8 minimal space group surface subgroup genus oPa Immm-Pmmm 3 oDa Pnnn-Fddd 3 oPb Fmmm-Cmmm 3 oDb Cmma-Imma 3 oCLP‘ Cmmm-Pmmm 3 OCL P Pccm-Cccm 3 PT Fmmm-Cmmm 5 lll Fmmm-Cmmm 5 Cmmm-Pmmm 5 VAL Cmma-Cmmaa 5 PV Pccm-Pccmb 5 ~~ a (c’ = 2c). (a’ = 2aj or (b‘ = 2b). surface element variable ranges element image Weierstrass function (Fig.) (Fig.) (eqn.1 Bonnet angle, 0 ye, Ye2 l(b) l(4l(d) 1(c)Uf1 l(4 6(4 2(4 2(42(b) 2(b) 2(c)2(c) 5(4 (7) (7)(6) (6) (6)(6) (9) subject to (10) (minus sign) and (1 1) 0 4 0 42 0 XI2 0 if Yel + ye2 -~ee> 0 XI’ if Ye, + Ye2 -Yee 0 (4, a) (4, a) (0, 4) (0, 4) (-a,Oj (094) (0, 4) (0,4) (0, 4) (4, a) 6(c) 5(c) (9) subject to (10) (plus or minus sign) and (12) 42 (4, a) (YeeT 4) 6(b) 5(b) (9) subject to (10) (minus sign) and (12) 42 (4, a) (4, Yet) 8(b) 7(b) (1 3) subject to (14) (plus or minus sign) and (15) 0 (4, a) (Ye,, 4) 8(4 7(4 (minus sign) and (15) (13) subject to (14) 0 (4, a) (4, Yel) In this section we again address only that subtype of orthorhombic IPMS possessing surface elements bounded by (on-surface) two-fold axes in the x, y and z axis directions and/or mirror planes parallel to the yz, zx and xy planes, respectively.To generalise the results of the preceding section we now reverse the sequence traced there, establishing our primary considerations within the parametric space on the unit sphere, or complex plane. The projected image of any such surface element must define some region over the complex plane bounded by segments lying along the real axis, the imaginary axis and the unit circle (all three kinds of edge of the underlying octant tiling). The image of a (first-order) flat point at a 2/m vertex lies on an edge of the region, and hence an edge site ‘e’ on an octant.The image of a flat point on an rn or 2 boundary again resides on an octant-edge site, which we now denote ‘ee’ since it is a double-edge site of the region (i.e. the boundary jmage wraps one full turn around it). For a flat point at a 1 or 1 position in the interior, the region winds twice around the image, which lies on an octant-face site ‘f’. By (projected) reflection over its boundary segments, the region must cover some number, s, of copies of the plane such that the superposed, complete distribution of flat-point image sites (totalling W)is symmetric with respect to the reflection scheme of the octant tiling, and the full angle of 4n is subtended at each of the flat-point images. Whatever the values of W and s may be, we require that their ratio matches eqn.(3). Moreover, the accumulated order of the flat points residing on a single element must amount to four times the area fraction of its image region, i.e. one-half of the number, N, of octants that it occupies. Distinguishing the flat-point sites according to the three varieties defined above, and denoting the numbers of each on the image region by n,, neeand n,, its accumulated order must then obey the relation an, + $nee + n, = 3N The first such situation is then a single-octant region, N = 1. In this scenario, image sites of the last two types, ee and f, are inadmissible, so nee = n, = 0 and the only solution forthcoming from eqn. (8) is n, = 2. Since all three octant edges are originally equivalent, there are only two distinct possibilities for placement of the two edge sites, represented by the images displayed above in Fig.2(b)and (c). Extending to the situation of a double-octant region, N = 2, one solution of eqn. (8) is then n, = nee = 0, n, = 1. The solution pertains to a scenario of two overlain octants on distinct sheets pinned at this single face site, represented by the image given above in Fig. 2(a). This scenario is more naturally grouped with the first situation since the site dis- tribution is identical on the two octants (although it cannot be symmetrically subdivided by octant segments). Hence these simplest scenarios correspond to precisely those three images in Fig. 2. The subsequent derivation of the Weierstrass functional forms replicates that in the preceding section, and the consequent reconstruction of the surface- adjoint surface pairs defined by them returns us to the orig- inal point of departure, the six cases of genus 3 in Fig.1. The situation N = 2 also admits a second scenario, that of two adjoining octants comprising, say, the first quadrant of the complex plane. Face sites are thus rendered inadmissible, and the solutions of eqn. (8) (with n, = 0) are then n, = 4 or n, = 2, nee= 1 or nee = 2. We must ignore solutions pos- sessing no bounding segment(s) along the unit circle (since they relate to IPMS of symmetry other than the orthorhom- bic subtype considered here) and those that do, but are inter- nally reflection-related across it (since they reduce to the N = 1 situation).By these criteria the solutions n, = 4 and nee = 2 are both discarded, leaving n, = 2, nee = 1 as the only viable candidate. The basic image region associated with this solution is rep- resented in Fig. 4.The boundary arc following the unit circle wraps around the site ee. Internal symmetry across this arc is broken by the distribution of the two edge sites designated el and e,, which may each be located on any of the six seg- ments a, b, c, a‘, b’, c’. Irrespective of the locations of el and e, ,exhaustion of all boundary reflections generates a set of 16 regions covering four sheets. In this way the four sheets are pinned pairwise by the two flat-point images stationed at each of the four reflec- tion equivalents of the site ee in the octant tiling.Further, two of the four sheets are pinned by the single flat-point image residing at the four reflection equivalents of the sites e, and e2, while the other two sheets are unpinned there by virtue of the absence of internal symmetry (i.e. a, b, c are distinct from a’, b’, c’ in the region). The determination of the Weierstrass functional form matching this four sheeted domain structure is a natural gen- eralisation of the derivation in the preceding section. Again the form is composed from polynomials, the roots of each necessarily displaying reflection equivalence with respect to this same, underlying octant tiling. In particular, the poly- nomials possessing simple zeros at the four equivalents of the J. CHEM.SOC. FARADAY TRANS., 1994, VOL. 90 Im o II Fig. 4 The image template associated with the solution n, = 2, nee = 1, occupying (a single copy of) one quadrant. Its boundary comprises six segments, a and a’ along the real axis, joined by the unit-circle ‘tongue’ formed by b and b’ wrapping around the double- edge site ee, together with c and c‘ joined continuously along the imaginary axis. The two additional edge sites e, and e2 are not indi- cated since their locations are not yet fixed, we may assign them to any of these six segments. edge sites ee, el and e, ,denoted pee,pel and p,, , are given by eqn. (4) with ye = ye,, ye = ye, and ye = ye,, respectively. Recalling Fig.3, since ee lies on the unit circle, we require 4 > ye, > 0, while the distinction of six segments a, b, c, a’, b‘, c’ on which el and e, may reside reduces to the choice of three ranges for ye, and ye,, since the segments a, a’ both correspond to ye > 4, the segments b, b’ both to 4 > ye > ye,, and c, c’ to 0 > ye. The Weierstrass functional form, with its four branches pinned at first-order branch points as described above, and meeting all other requirements, is then given by R(o)= el fexp(i0) Here pe, is an extra polynomial, similarly given by eqn. (4), with ye = ye,. This variable ye, is, in turn, expressed in terms of the three branch point variables ye,, ye, ,ye, by Note that the right-hand side is real, as required, since the product ?,(ye -ye,) is non-negative for all three ranges of ye cited above.Further, it has two alternative values [the fsigns in eqn. (10) are independent of those two defining the four branches of R(o)in eqn. (9)]. These two alternatives resurrect the distinction between the six segments from the three ranges of the edge variable. A choice of values of ye, and ye, (at fixed ye,) each admits two locations, say el on segments q1 or q; and e, on segments q, or q;, and when taken simul- taneously, presents the two possible combinations (el, e,) on (ql, q2) or (ql, 9;) [the other two combinations (qi, 9;) or (qi, q,) are symmetry-related to these by unit-circle reflection]. These two possibilities are precisely those given by the fsigns in eqn.(10). In total, the three ranges for ye, and ye, together yield six choices (since the designations e, and e, are arbitrary): ye, > 267 Ye, > 4; Ye, > 4 and 4 > ye, > Yee; 0 > ye, and ye2 > 4; 4 > Ye, > Ye, > Yee; 0 > Ye, and 4 > Ye2 > Ye,; 0 > ye, > ye,. This then doubles into the 12 possibilities for (el, e,) on: (a, a), (a, a’); (a, b), (a, b’); (c, a’), (c, a); (b, b), (b, b’); (c, b’), (c, b); (c, c), (c, c’); respectively. In each of these six pairs the first men- tioned possibility is given by the minus sign in eqn. (lo), and the second by the plus sign. The imposition of all necessary conditions pertaining to this scenario in image space thus results in the Weierstrass functional form of eqn.(9) [in conjunction with eqn. (lo)], which embraces this list of 12 possibilities, each having four degrees of freedom 6, ye,, yel, ye,. We must now return to real space via eqn. (1) to assess the materialisation of these possibilities. As stated in the analysis for genus 3, the requirement that the image boundary is manifested as a surface-element boundary comprising two-fold axes and/or mirror planes dic- tates that the variable 6(mod IL)takes either the value 6 = 0 or 0 = 42. So each image possibility gives rise to two such cases, a surface-adjoint surface pair, further doubling the possibilities to a family of 24 surfaces. However, as opposed to the genus 3 cases, each of the present surfaces now con- tains three variables y,,, y,,, ye,, corresponding to the edge sites borne by the image region. These singly variable normal vectors of the flat points on the element boundary provide the surface with three degrees of freedom (in addition to uniform dilatation), i.e.one degree in excess of the scope for orthorhombic distortion. This excess arises naturally in each of the 24 surface ele- ments, as is made apparent by a ‘qualitative’ reconstruction of the cases. For the 12 possibilities listed above, we sketch the boundary circuit of the generic surface element possessing this image circuit, in the two cases related by interchanging the identification of two-fold axes and mirror planes on it (in the same way that Fig. 1 may be reconstructed from Fig.2). Each of these cases is found to require one supplementary (real) constraint forcing the relative arrangement of these point-group symmetries to be commensurate in the y axis direction. Although a (countable) infinity of ratios are acces- sible, we will always address the simplest representative, since all others are guaranteed to generate self-intersections of the surface. Surface elements given by alternatives (ql, b) and (ql, b’) are continuously related (by merely pushing a flat point through a 2/m vertex) thus it serves to unify the pairs of pos- sibilities (a, b), (a, b’); (b, b), (b, b’); (c, b), (c, b’) as (a, bb’); (b, bb’); (c, bb’), respectively, thus spanning both sign alterna- tives in eqn. (10). Further, the minimal surfaces must adopt the highest symmetry compatible with the fixed boundary conditions.Accordingly, for the (ql, 9;) possibilities (a, a’), (b, bb’), (c, c’) the supplementary constraint will lead to halving of the surface elements, via two-fold symmetry in the z-axis direction, or mirror symmetry parallel to the xy plane in the adjoints. So these cases will reduce to the genus 3 surfaces oDb, oCLP, oPb, respectively, and their adjoints to oPb, oCLP’, oDb, respectively. [Equivalently, the image region becomes unit-circle symmetric, thus with ye, = ye,, the plus sign in eqn. (10) yields ye, = ye, = ye,, collapsing the Weierstrass function in eqn. (9) to the genus 3 form in eqn. (6).1This reduction leaves only 12 separate, non-trivial cases, namely the surface-adjoint surface for the six possibilities (a, a), (a, bb’), (c, a), (c, a’), (c, bb’), (c, c).Of these, it is found that nine cases still give rise to self-intersections on rotation and/or reflection of their surface element ;the three remaining cases are, however, true (i.e.self-intersection-free) IPMS. Spe- cifically, these three cases are the surface for (c, a’) and the adjoint surfaces for (a, a) and (a, bb’). The three image regions J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 are represented in Fig. 5(a),(b),(c), respectively, with the per- tinent surface case discerned from its adjoint surface in each by requirement of the affixed labelling scheme p and 1 (indicating boundary sections corresponding to mirror planes and two-fold lines, respectively) rather than its opposite, 1 and p.[In the last case the possibility (a, b) has been dis- played, that for (a, b‘) is obtained effectively by sliding ee down the unit circle to below e,, carrying the labelling along with it]. The boundary circuits delimiting surface elements reconstructed from Fig. 5(a), (b),(c)are sketched in Fig. qa), (b), (c), respectively. [The possibility (a, b‘) is similarly Imo Re o \ I / Im o \ I 1 el 1 el Fig. 6 Sketches of the bounding circuits delimiting the surface ele- ments for the IPMS, (a) PT, (b) ll2, (c) lll, obtained as the simplest commensurate reconstructions of the images in Fig. 5(4, (b), (c), respectively IlTi( 1 +o”)R(o’)do’ I= 1[:i( 1 +d2)R(o’)do’ (11) Fig.5 Particular possibilities of Fig. 4, given by (el, ez) on (a)(c, a’), (b) (a, a), (c) (a, bb’), The IPMS cases corresponding to these three possibilities (and conforming to the labelling scheme of p and I symbols explained in the text) are referred to as the (a) PT, (b) ll2, (c) Ill, surfaces. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 (directed along the y-axis) of its central point. The (projected) image of any such element is bounded by segments lying along only the real axis and the unit circle, i.e. two of the three kinds of octant-tiling edge (over which it is continued uia the edge-reflection operations o -P 0 and o -,l/&, respectively). In addition, the image displays invariance under the internal operation w +P -l/w.Note that the quadrant scenario illustrated above in Fig. 4 gives rise to two families, since there are two means for con- tinuation over the imaginary axis segment (i.e. the third kind of octant-tiling edge). Assignment of the reflection operation o + -6 corresponds to the previous subtype, for which the family was derived in the preceding section. On the other hand, doubling of the region over this segment uia the oper- ation o + -l/wcorresponds to this new subtype, the family of which will be analysed here. Since it is so closely related to its predecessor, much of the analysis is common to both. To evoke the similarity we employ an identical notation. In par- ticular, we denote the double-edge site by ee and the extra two edge sites by el and e2.The location of these latter two edge sites is now restricted to the segments a, b, a’, b’, since the other two segments c, c’ in Fig. 4no longer act as reflec- tion edges. Again 16 regions (eight double regions) cover four copies of the complex plane such that the overall distribution, and occupancy number, of these sites is completely analogous to that of the previous case. The Weierstrass functional form associated with the domain structure is now given by R(w)= & exp(i8) in which As before, the variable ye, carries two independent alterna- tives, which are guaranteed to be real-valued since the quan- tity ye -ye, is non-negative for any edge site e on segments a, b (or a‘, b’). The distinction of the possibilities embraced by eqn.(13) and (14)is identical to that in the preceding section, save for this eradication of the third edge type. To repeat, the two ranges for ye, and ye, provide three simultaneous choices: Ye1 > Ye, > 4; Ye, > 4 and 4> ye2 > Yee; 4> Ye1 > Ye, > Yee, then developing pairwise to the six possibilities (a, a), (a, a’); (a, b), (a, b’); (b, b), (b, b’); the first (second) member of each pair is given by the minus (plus) sign alternative in eqn. (14). Inserting eqn. (13)into eqn. (l),the region boundary abb’a’ of reflection segments corresponds to a surface element bound- ary of two-fold axes and/or mirror planes only for the values 8 = 0 and 8 = n/2 [although the operation co -+ -l/w is manifested as a two-fold axis along the normal (y) direction for any 81.Hence again, each image-space possibility above doubles into two such real-space cases (its surface and adjoint surface) containing the three free variables yee, ye,, ye?, giving a total of 12 of these cases to consider. Qualitative recon-struction of these three-variable surface elements verifies that each is in need on one supplementary constraint to ensure commensurability, now in the z-direction. As previously, we apply the simplest constraint to each case, yielding its two- variable orthorhombic version which minimises any self- intersection. Merging of those possibilities which are continuously related in real space reduces the distinction to (a, a), (a, a’), (a, bb’), (b, bb’).Of these, (a, a’) and (b, bb’) are symmetrically subdivided under the action of their supplementary con-straints (specifically, the surfaces revert to oDb and oCLP and their adjoint surfaces to oPb and oCLP’). Only the sur- faces derived from the two possibilities (a, a) and (a, bb’) are true IPMS (their adjoint surfaces both generate self-intersections). Their corresponding regions are represented in Fig. 7(a) and (b),respectively, retaining the p and 1 labelling convention. In both cases we display the double region (with internal symmetry co c,-l/o)to provide a closed boundary of reflection segments. The boundary circuits of two-fold lines and mirror-plane curves delimiting the surface elements (possessing this internal, rotational invariance) are recon-structed, in their simplest commensurate state, in Fig.8(a) and (b),respectively. The first case constitutes a new IPMS, which we call the PV surface. As in the preceding section, we have displayed only the possibility (a, b) for the second case [its merging partner (a, b’) is readily obtained by sliding ee below e2 in Fig. 7(b),and hence pushing ee through e2 to form an inflection point on the top- and bottom-face curves in Fig. 8(b)].In the limit bridging these two subcases, the flat point ee, and its mirror equivalent, coincide at the flat point Im wIk \1u I Re o Fig. 7 Particular possibilities of the image template associated with the second subtype (in which the two quadrants are symmetry-related about o = i by composition of reflections in the imaginary axis and unit circle), given by (e,, ez) on (a) (a, a), (b) (a, bb’).The corresponding IPMS cases (again complying with the imposed p and 1 schemes) are referred to as the (a) PV, (b) VAL surfaces. fb 1 Fig. 8 Sketches of the bounding circuits of the surface elements for the IPMS, (a) PV, (b) VAL, reconstructed, in the simplest com- mensurate state, from their images in Fig. 7(a), (b), respectively. The dashed line (in the y direction) denotes the internal two-fold axis. e2 to form a third-order flat point there. The IPMS in this special limit was discovered previously and called the VAL surface.lo The surface was correctly parametrised there (although note that its image in displayed erroneously in Fig.5 of that study, the branch point at o = 1/A should instead reside at o = -l/A). However, it was not appreciated that the extra degree of freedom present in the generic case, which we will continue to refer to as the VAL surface here, would be required to fit a supplementary constraint. For both the PV and VAL surfaces this supplementary constraint demands that the plane containing the pair of two- fold lines (and the internal two-fold axis) in the y direction must be parallel to the xy plane. Hence, along the imaginary axis segment of the images we must impose that Re sd2o’R(o’)do’ = 0 (15) where the integration may be taken on any of the four branches of their Weierstrass functions.The abovementioned facts relating to the parametrisation of these two cases are listed in rows 10 and 11 of Table 1, together with the genus value [again 5 from eqn. (2)] and space groupsubgroup of their translational units. Conclusions In this study we have addressed the general algorithm for parametrisation of IPMS in the context of these two ortho- rhombic symmetry subtypes. Any such surface is categorised according to the number, N, of spherical octants occupied by the image of its basic element. This number is directly related to the Euler characteristic per surface element: xE = -N/4. The IPMS translational unit, comprising A4 such surface ele- ments, then possesses Euler characteristic xT = -MN/4 and hence genus g = 1 -xr/2 = 1 + MN/8.For the simplest situ- ations N = 1 and N = 2 we have exhaustively derived all possibilities, resulting in a total of 11 IPMS (all of which are balanced). The information pertaining to each is summarised in Table 1. The study may readily be extended to situations N > 2 of more complicated surface-element topology. In all of these J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 situations the product MN must be at least 32, so the genus can be no lower than 5. As an illustration, for N = 3 one solution of eqn. (8)is n, = 4, nee = 1, n, = 0. This solution, for which M = 16, thus corresponds to non-trivial families of balanced IPMS with genus 7. The families warrant further investigation, since no examples of such surfaces are presently known.Of all the previously established IPMS, there is only one additional example with orthorhombic symmetry and genus 5. This surface, which is likewise balanced and of the first symmetry subtype, was discovered by Koch and Fischers*6*20and named oMC5, since it is an orthorhombic distortion of their tetragonal MC5 surface, It derives from the situation N = 8, and specifically, the solution n, = 0, nee = 8, n, = 0 of eqn. (8), for which M = 4. In summary, the two families with genus 5 analysed here are clearly the simplest orthorhombic IPMS which cannot be obtained via crystallographic degradation of surfaces of higher symmetry (i.e.cubic, tetragonal or hexagonal). For this reason, we would anticipate the two families to represent plausible options for interfacial shapes in self-assembling bicontinuous phases.One of the authors (A.F.) was supported by the Swedish Natural Research Council (NFR). References 1 L. E. Scriven, Nature (London), 1976,263, 123. 2 E. L. Thomas, D. M. Anderson, C. S. Henkee and D. Hoffmann, Nature (London), 1988,334, 598. 3 J. Prost and F. Rondelez, Nature (London) (Suppl.) 1991,350, 11. 4 P. Barois, S. T. Hyde, B. W. Ninham and T. Dowling, Langmuir, 1990,6, 1 136. 5 W. Fischer and E. Koch, Acta Crystallogr., Sect. A, 1989,45, 726. 6 E. Koch and W. Fischer, Acta Crystallogr., Sect. A, 1990,46, 33. 7 E. Koch and W. Fischer, Acta Crystallogr., Sect. A, 1993,49, 209. 8 A. Fogden and S. T. Hyde, Acta Crystallogr., Sect. A, 1992, 48, 442. 9 A. Fogden and S. T. Hyde, Acta Crystallogr., Sect. A, 1992, 48, 575. 10 A. Fogden, Acta Crystallogr., Sect. A, 1993,49, in the press. 11 A. Fogden, J. Phys. I, France, 1992,2,233. 12 A. Fogden, Z. Kristallogr., in the press. 13 H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Vol. I, Springer, Berlin, 1890. 14 A. H. Schoen, Infinite Periodic Minimal Surfaces without Self- intersections, NASA Techn. Rep. D-5541,1970. 15 S. Lidin and S. T. Hyde, J. Phys. France, 1987,48, 1585. 16 S. Lidin, J. Phys. France, 1988,49,421. 17 A. Fogden, M. Haeberlein, S. Lidin, J. Phys. I France 1993, in the press. 18 K. Weierstrass, Untersuchungen iiber die Flachen, deren mittlere Kriimmung iiberall gleich Null ist, Monatsber. d. Berliner Akad. 1866, p. 612. 19 W. Fischer and E. Koch, J. Phys. Fr. C7 Colloq., 1990,51, 131. 20 E. Koch and W. Fischer, Acta Crystallogr., Sect. A, 1989, 45, 169. Paper 3/03579D; Received 22nd June, 1993