Triply Periodic Bicontinuous Cubic Microdomain Morphologies

by Symmetries

 

Meinhard Wohlgemuth, Nataliya Yufa, James Hoffman,1 and Edwin L. Thomas*

Department of Materials Science and Engineering, MIT, Cambridge, MA 02139

1Mathematical Sciences Research Institute, Berkeley, CA 94720

 

 

 

*To whom correspondence should be addressed

E-mail: elt@mit.edu

 

 

 

 

Abstract

In response to thermodynamic driving forces the domains in microphase-separated block copolymers have distinct intermaterial dividing surfaces (IMDS). Of particular interest are bicontinuous and tricontinuous, triply periodic morphologies and their mathematical representations.

Level surfaces are represented by functions F: R3 ® R of points (x, y, z) Î R3, which satisfy the equation F(x, y, z) = t, where t is a constant. Level surfaces make attractive approximations of certain recently computed triply periodic constant mean curvature (CMC) surfaces and they are good starting surfaces to obtain CMC surfaces by mean curvature flow.

The functions F are the nonzero structure factors F(hkl) of a particular space group, such that the resulting surfaces are triply periodic and maintain the given symmetries. This approach applies to any space group and can, therefore, yield desired candidate morphologies for novel material structures defined by the IMDS. We present a technique for generating such level surfaces, give new examples and discuss certain bicontinuous cubic IMDS in detail.

Introduction

All periodic structures belong to one of the 230 space groups. Usually the primary goal of a structural investigation is to find the specific coordinates within the basic unit cell (a fundamental translational repeat unit) of all the various types of atoms of a material. This knowledge fully describes the structure. There are many structures that are noncrystalline at the atomic level but are crystalline at supermolecular length scales. In these mesoscopic crystalline materials, interest is not in the individual locations of the atoms and molecules but, rather, in the characterization of the interfaces separating adjacent regions of different composition. At high temperatures, entropic forces dominate and the structure is a homogeneous mixture. At lower temperatures, repulsive enthalpic interactions exert themselves, leading to the formation of ordered phases in which the minimization of interfacial area between domains is the dominant term in the free energy balance.1 We therefore focus our attention on the interface between the components. We define this interface as the intermaterial dividing surface (IMDS) and model it as a smooth, mathematical surface since the typical dimensions of the unit cell in block copolymer mesoscopic crystals are 50-100 nm, much larger than the typical width of the composition profile across the interface (3 nm or less).

Motivation and Overview of Triply Periodic Bicontinuous IMDS in Block Copolymer Systems

A block copolymer is a macromolecule comprised of two or more types of monomer units covalently linked in one or more junctions. In 1986, we published the first detailed structural investigation of a periodic bicontinuous structure in a block copolymer.2 The particular block copolymers examined were multiarm star diblock copolymers of polyisoprene and polystyrene. The blocks microphase-separate into two nearly pure types of microdomains, one comprised of the polystyrene (PS) blocks, and the other of the polyisoprene (PI) blocks. The proposed structure was cubic Pnm (space group 224) and called "OBDD," which stands for ordered bicontinuous double diamond, since the proposed structure contained two separate, connected, triply periodic, tetrahedrally coordinated networks comprised of the PS blocks in a matrix of the PI block. Both the PS and PI domains are three-dimensionally continuous. Later we attempted to show3 by comparison of experimental TEM images with two-dimensional simulations of a double diamond structure based on the CMC family of Schwarz’s D surface, that the actual IMDS structure of the star diblock was closely approximated by a recently computed member of the CMC family of D surfaces.4 In 1994, we discovered another bicontinuous cubic structure, the double gyroid, with space groupin a low molecular weight PS/PI diblock.5 Certain TEM images of this double gyroid (DG) structure were strongly reminiscent of the prior OBDD images, suggesting a re-examination of the star diblock structure. Improved X-ray measurements lead us to revise the assignment of the star diblock structure to DG.6

In general, for the block copolymer microdomain structures that had been determined up to 1986, namely spheres packed on a body-centered cubic lattice, cylinders packed on a hexagonal lattice and alternating lamellae, the actual IMDS observed is in all cases very close to a CMC surface. The fundamental reason why the previously identified block copolymer microdomain structures are based on spherical, cylindrical, and planar (lamellar) structures is due to the similarity of the thermodynamic problem of microphase separation to the mathematical problem of area minimization under fixed volume (or volume fraction) constraints, which immediately leads to solutions that are periodic dividing surfaces with constant mean curvature.1 Of course, the polymer physics contains other features in addition to minimization of interfacial area at fixed volume fraction. In particular, both of the component block copolymer chains must uniformly fill the various regions defined by IMDS and these chains would prefer to have random conformations and a uniform environment, requirements which can be frustrated to various extent by an IMDS having everywhere constant mean curvature.7

In recent years, a number of new IMDS microdomain structures have been discovered, particularly in ABC linear and miktoarm star terpolymer systems, where the additional type of block and presence of two junctions per molecule, and/or star architecture, can strongly influence the geometry of the equilibrium microdomain morphology.8-12 Thus, it is important to have the means to generate new models for the host of emerging complex microdomain structures owing to the ongoing synthesis of novel multi-component, variable architecture block copolymers. The additional requirement that the candidate IMDS have precisely constant mean curvature is not of paramount importance, as a CMC surface can often be obtained by curvature flow from a surface which is not CMC,13 and since distinctly non-CMC IMDS microdomain shapes have been found.7, 8, 14

I. Geometry and Symmetries of Level Surfaces

To begin we state some basic facts concerning the geometry of level surfaces. We take on R3 the usual (x, y, z) coordinate system and the Euclidean scalar product < , >R3. Let F: R3 ® R be a smooth function. A point p Î R3 is a point on the surface St if F(p) = t, hence St = F-1 (t) = {p Î R3 : F(p) = t}. If we have two level surfaces F1 and F2 then the level surface given by the sum c1 F1 + c2 F2 has the greatest common symmetries of F1 and F2, unless of course the linear combination is a constant.

An isometry in R3 is a map s : R3® R3 , which preserves distances between points. The symmetries of a surface S are all the isometries which map S onto itself. In R3 the crystallographic symmetries are: translations, rotations, reflections, screws, glide reflections, rotary reflections, and inversions.

We are principally interested in generating surfaces which periodically divide space into 3-D connected labyrinthine regions. Filling the labyrinths with distinct block copolymers is equivalent to coloring the IMDS differently on the inside and the outside. Thus when choosing the appropriate space group for a given structure we do not want to include symmetry operations which would interchange the different polymer regions. For instance, we assign the space group to Schwarz’s P surface instead of a supergroup which would correspond to a single-color P surface with indistinguishable polymer regions on each side.

II. Derivation of Level Set Equations for Space Groups

To find the appropriate surface describing a given microphase-separated morphology we start with its symmetries, which can be determined from transmission electron microscopy (TEM) and small-angle X-ray scattering (SAXS) experiments. Given a set of symmetries of a periodic structure, one can determine from the International Tables for Crystallography (ITC) the space group of that structure.15 To find suitable candidate functions which are invariant under the space group symmetry operations we borrow a tool from X-ray crystallography, the structure factor. The structure factor F(hkl) describes the amplitudes and phases of the three dimensional diffraction pattern due to the scattering of incident radiation off of planes (hkl) of atoms in the crystalline structure. We use the structure factor terms because they have all the symmetries of the structure, however, for special values of h, k, and l the factor F(hkl) can have extra symmetries. For instance, in a non-centrosymmetric space group there can be F(hkl) terms that are centrosymmetric. In group I4132, for example, planes (211) give rise to a structure factor which is invariant under inversion (thus the F(211) level surface is centrosymmetric), while the space group as a whole does not have inversion symmetry.

Another approach which can yield simple analytic expressions for periodic bicontinuous partitions of space was developed earlier by von Schnering and Nesper.16,17 They calculated zero-potential (nodal) surfaces based on selected distributions of point charges. This method involves taking the Fourier transform of point charge distributions and leads again to the structure factor of the chosen space group. The principal aim of nodal surface calculations is to obtain simple expressions for the approximation of three-dimensional periodic minimal surfaces, since these surfaces are relevant approximate models for amphiphilic monolayers in bicontinuous mixtures of oil, water and an amphiphile.16 Schwarz and Gompper also have employed Fourier series to approximate nodal surfaces and have computed the variation of mean and Gaussian curvature over the surfaces.18

In order to find level set equations which could correspond to physical intermaterial dividing surfaces we consider the structure factors F(hkl) for small values of h, k, and l. Larger values of h, k and l will, in general, correspond to surfaces with higher genus, which are less favorable due to their larger surface energy. Additionally, we only need to take a single permutation out of all possible ones of ± h, ± k and ± l because for cubic groups these will only differ up to an exchange of axes or a sign. Moreover, we set the numerical coefficient of F(hkl) to unity since scaling does not affect symmetry.

Interesting new level sets can be obtained by taking combinations of several F(hkl) terms. In order to obtain surfaces which belong to a particular group and not its supergroup we need to include at least one term which only has the symmetry of that group.

The general form of F(hkl) is given by:

, (1)

where fj is a factor corresponding to the strength of scattering of the jth type of atoms and (xn, yn, zn) is the nth equivalent position of the jth type of atoms in the unit cell. The symmetry of the space group is conveyed to the structure factor through the set of {xn, yn, zn}, therefore, we can set fj equal to 1. The structure factor then becomes:

(2)

The remaining requirement is to convert the structure factor in equation 1, which is in general complex, into a real level set equation. Let F(hkl) = A + iB. If B is equal to zero, F(hkl) is real and the level set equation is simply F (x, y, z) = A = t. If A is zero, we take F(x, y, z) = B = t. This covers all the cases considered in this paper, however, there are space groups, for instance group No. 220, where low-order structure factor terms have both real and imaginary parts. When this occurs, we consider both the terms of the form FA (x, y, z) = A = t and FB (x, y, z) = B = t.

The ITC tabulate expressions for each space group that can be used to compute the specific form of equation 1. There are 36 groups with cubic symmetry. We illustrate this procedure in detail for seven cubic space groups in the subsequent sections.

III. Level Surfaces with and Symmetry.

The space group Pmm (No. 221) has forty-eight general positions created by its set of symmetries. Since the origin of the unit cell can be chosen on an inversion center, F(hkl) is centrosymmetric, therefore, only the A term from equation 2 contributes to the level set function. The set of 48 equipoints listed for Pmm is:

Equation 2 for the Pmm group has 48 terms:

(3)

which, after dividing through by 8 to make the coefficient equal to 1, becomes:

F(hkl) = cos 2p hx [cos 2p ky cos 2p lz + cos 2p ly cos 2p kz] +

cos 2p hy [cos 2p kz cos 2p lx + cos 2p lz cos 2p kx] + (4)

cos 2p hz [cos 2p kx cos 2p ly + cos 2p lx cos 2p ky]

We arrange the allowed F(hkl) in order of increasing values of h2+k2+l2. For group Pmm the first F(hkl) has h = 1, k = 0, l = 0. Using the simplified expressions in the ITC we find that: F(100) = cos 2p x + cos 2p y + cos 2p z (5)

The second term in the series has h, k, l = (110) and

F(110) = cos 2p x cos 2p y + cos 2p y cos 2p z + cos 2p z cos 2p x (6)

Similarly, the third and fourth terms on the series are given by cos 2p x cos 2p y cos 2p z and cos 4p x + cos 4p y + cos 4p z, respectively.

Level Surfaces of the type F(100) = t

We now examine the level surface family given by the first term: F(100) = t. Figure 1a shows a three-dimensional plot of the surface with t = 0 over a unit cell taken from –1/2 to 1/2. This surface divides space into two continuous regions of equal volume. As previously noted by a number of authors, this level surface is quite similar in appearance to Schwarz’s minimal P surface.19 Of course, the F(100) level surface is not a minimal surface. Anderson et al.4 numerically computed the CMC family based on the P surface. In the CMC family the ratio of the subvolumes varies with the mean curvature. By taking F(100) = t ¹ 0, a family of level surfaces with a strong resemblance to the CMC family of P surface can be constructed.20 The CMC family exists from a volume fraction of 0.5 to 0.25 with f = 0.35399 at pinch-off (the surface no longer subdivides space into two continuous subvolumes), while the level set family exists from 0.5 to the pinch-off at f = 0.21029 with t = 1.

F(110) = t Family

The surface described by

F(110) = cos 2p x cos 2p y + cos 2p y cos 2p z + cos 2p z cos 2p x = t (7)

appears similar to another minimal surface, Schoen’s I-WP surface, for a certain range of t values. Figure 1b shows an I-WP surface with t = -0.25. Anderson et al.4 also numerically computed a CMC family for I-WP.

Combination Level Surfaces

It is interesting to explore two-parameter families, for example, s F(100) + (1-s) F(110) = t. Figure 1c shows an array of images obtained by varying s and t in a region which yields continuous surfaces. By changing s from zero to one we obtain surfaces that change from the I-WP family to the P family. At intermediate values a new family emerges with combined features, as shown in Figure 1c. This level set resembles yet another triply periodic minimal surface, Schoen’s O, CT-O surface.21 The O, CT-O minimal surface can be regarded as a combination of Schwarz’s P minimal surface and Schoen’s I-WP minimal surface.

The O, CT-O surface can also be modeled within the s F(110) + (1-s) F(111) = t family. This family also yielded, for a certain range of s and t values, members of another known constant mean curvature family4 – that based on Neovius’ minimal surface.22 An example of level set approximation of Neovius’ surface, called C(P), is shown in Figure 1e. We also found an expression for the level set approximation to a combination of the P and C(P) surfaces, previously shown by Karcher23 to be another minimal surface with Pmm symmetry. This surface is shown in Figure 1f and is built from a combination of F100, F110, F111, F200, F222 and F300 terms. Another interesting surface from this group resembling a minimal surface, also found by Karcher,23 is the K, shown in Figure 1g.

The cubic space groups and are supergroups of Pmm. They differ only by a body-centering translation in and face-centering translations in The reduced terms of group are non-zero only for h + k + l = 2n and for the group are non-zero only if h, k, l are all odd or all even. Non-zero reduced F(hkl) terms are identical for this subgroup and its supergroups.

A combination of the first two level set terms of the space group gives us an approximation to another triply periodic minimal surface discovered by Schoen,20 the F-RD. Figure 1h shows an example of a member of the F-RD family using the first, third and fifth terms:

FF-RD (x, y, z) = 0.8 F(111) – 0.1F(220) + 0.1F(222) = 0 (8)

Anderson et. al4 also computed the F-RD CMC family.

IV. The Level Surfaces with I4132 and Symmetry.

Level set equations for another pair of cubic space groups, I4132 space group (No. 214) and its supergroup (No.230), are of interest because they relate to the double gyroid microdomain structure found in many block copolymer materials.5,11,12,24 The group contains all the symmetries of I4132 as well as inversion. The first few allowed reflections for I4132 and for are (110), (211), (220) and (211), (220) respectively. Schoen’s infinite periodic minimal G surface, the so-called "gyroid" surface, has the symmetries of space group I4132.21 This cubic space group does not a center of symmetry, so the F(hkl) can have contributions from both A and B terms. The unit cell is body-centered cubic, so in addition to the three translations along the x, y, and z axes there is a centering translation . The set of symmetries for I4132 generates 48 equipoints for the general position. Equation 2 for the I4132 group can be simplified so that its A and B components have six terms each:

(9a)

and

(9b)

The first allowed term is F(110) :

sin 2p y cos 2p z + sin 2p z cos 2p x + sin 2p x cos 2p y (10)

Figure 2a depicts the unit cell of this surface with t = 0. The surface appears quite similar to Schoen’s gyroid minimal surface. A level set family of F(110) exists for the range , which corresponds to values of f from 0.5 to 0.046. Since the gyroid surface does not have inversion symmetry, it cannot be approximated by terms from group .

F(211) = t Family and Combination Level Surfaces

The next nonzero structure factor for group I4132 and the first allowed term for group is F(211), given by:

F(211) = A = sin 4p x cos 2p y sin 2p z + sin 4p y cos 2p z sin 2p x + sin 4p z cos 2p x sin 2p y = t (11)

Setting F(211) equal to zero does not yield a smooth surface. However, by varying t we can obtain a continuous surface, G¢ . The surface (shown with t = -0.5 in Figure 2b) is a member of the C(I2-Y**) family discovered by von Schnering. This surface is similar to the gyroid but with extra tunnels inserted along <111> directions to create inversion symmetry, so its space group is

Exploration of the linear combination s F(110) + (1-s) F(211) = t does not lead to a topologically new intermediate surface. The next term in the series, F(220), is simply an I-WP approximation with twice the frequency of equation 7. The combination sF(211) + (1-s) F(220) = t yields two interesting surfaces – an approximation of the double gyroid microdomain structure, denoted DG (called I2-Y** by von Schnering), and another interesting surface named the Lidinoid after S. Lidin.25 This L surface has handles to the face centers, vertices and midpoints of all edges. The s, t array of combination images including DG and L is shown in Figure 2c. The double gyroid has the symmetries of ; it consists of right- and left-handed networks of a single non-minimal gyroid, as shown in Figure 2d. The Lidinoid surface is shown in Figure 2e.

V. Level Surfaces with and Symmetry.

Schwarz’s D surface18 is a triply periodic minimal surface with the symmetries of space group (No. 227). The related family of constant mean curvature surfaces was computed by Anderson et al.4 The set of symmetries for the group generates 192 equipoints for the general position. Equation 2 for the becomes:

(12)

The allowed low-order terms are (111), (220), (311), (222), etc. The first term is given by:

F(111) = cos 2p x cos 2p y cos 2p z + sin 2p x sin 2p y cos 2p z +

sin 2p x cos 2p y sin 2p z + cos 2p x sin 2p y sin 2p z (13) Figure 3a shows the unit cell of this surface for t equal to 0, which is quite similar to the D minimal surface. A family of continuous level surfaces exists for f values from 0.50 to 0.14, corresponding to t values between 0 and ± 1.

The second allowed term in this group is:

F(220) = sin 4p x sin 4p y + sin 4p y sin 4p z + sin 4p x sin 4p z (14)

This term generates a surface which is similar to the I-WP surface of equation (7) but with double the frequency and a phase shift of . The combination of the first two terms, sF(111) + (1-s)F(220), yields a new bicontinuous surface, D¢ , which is shown in Figure 3b. This surface has some features of the P, I-WP and (C)P surfaces.

is a supergroup of . Its low-order nonzero terms are (110), (111), (200), (211), etc. The first term is given by:

F(110) = sin 2p x sin 2p y + sin 2p y sin 2p z + sin 2p x sin 2p z (15)

The above equation also yields an I-WP-like surface with a phase shift. The second term for group is F(111):

F(111) = cos 2p x cos 2p y cos 2p z (16)

This surface is a periodic set of three orthogonal planes. The combination of these two terms yields a structure consisting of two offset surfaces, each one having diamond symmetry. This structure corresponds to the so-called double diamond structure, the OBDD microdomain morphology.2, 26, 27

VI. Combining Level Sets from Different Space Groups

Our library of level set surfaces may also be used intuitively to generate surfaces exhibiting the greatest common symmetries of two existing level set surfaces by simply adding the respective terms and adjusting the parameters to obtain continuous surfaces. As an example of generating such new surface we consider combining the term for the P surface with double the normal frequency () with the double gyroid terms (). By analyzing the generators for the starting space groups, we find the new P2-DG level surface belongs to space group P213, No.198. A plot of the surface, is shown in Figure 4.

Conclusion

We have conducted a systematic search for the cubic space groups 214, 221, 227 and their supergroups – 225, 224, 229 and 230 seeking allowed embedded, connected surfaces with the given symmetries. Searching group 221 yielded level set approximations to all the known triply periodic low genus surfaces: the P surface (see Figure 1a), I-WP (Figure 1b), their combination, the O, CT-O (Figure 1d), Neovius’ surface C(P) (Figure 1e), the intermediate combination of P and C(P) (Figure 1f), and a P-like surface K (Figure 1g). Figure 1c shows an example of how we obtained these surfaces and their level set equations by varying the coefficients s and t in the basic 2-term equation, sF1 + (1-s) F2 = t.

This investigation has shown that it is possible to find new triply periodic embedded surfaces with cubic symmetries by a simple algorithmic procedure using the structure factor equations from the ITC. We have found two such new surfaces, in groups 198 and 227, the P2-DG and the D¢ surfaces respectively. It is likely that minimal surface equivalents of these surfaces exist and thus the CMC surface families also exist. These bi- and tricontinuous structures may be useful candidate models for novel microdomain structures in block copolymers. Given the increasing interest in synthesis of well-defined, multicomponent, complex architecture macromolecules, these models may also be inspirational to both polymer chemists and also to polymer physicists due to their possible unusual transport and optical (e.g., photonic crystal) properties.28,29,30

Acknowledgements

The authors would like to thank Dr. Karsten Grosse-Brauckmann (University of Bonn) for many useful discussions and insights. We would also like to thank our respective financial sources: J. Hoffman – MSRI, E.L. Thomas – NSF DMR 98-01759, M. Wohlgemuth – Linnen Fellowship of Alexander von Humboldt Society and N. Yufa – MIT UROP and NSF DMR- 9801759.

References

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Figure Captions

Figure 1a. A level set approximation for the P surface:

F(x, y, z) = F(100) = cos 2p x + cos 2p y + cos 2p z = 0.

This surface has genus 3.

Figure 1b. A level set approximation to the I-WP surface:

F(x, y, z) = F(110) = cos 2p x cos 2p y + cos 2p y cos 2p z + cos 2p z cos 2p x = - 0.25.

The I-WP surface has genus 7.

Figure 1c. Two-parameter experiment for Group No. 221. The basic equation is:

s (cos x + cos y + cos z) + (1 - s) (cos x cos y + cos y cos z + cos z cos x) = t,

where s varies from zero to one. The values of t are changing along the horizontal axis and s is varied along the vertical axis. There are distinct regions on the graph where P, I-WP, and the combination surface O, CT-O are located.

Figure 1d. An approximation of the O, CT-O surface. In this figure we use the second and the third terms in the series, but it is also possible to obtain similar, though less minimal-like plots by using the first and the second terms, as shown in Figure 1c.

F (x, y, z) = 0.6 F(110) – 0.4 F(111) =

= 0.6 (cos 2p x cos 2p y + cos 2p y cos 2p z + cos 2p z cos 2p x) +

- 0.4 cos 2p x cos 2p y cos 2p z = 1.

The genus of the O, CT-O is 10.

Figure 1e. An approximation to the Neovius’ surface C(P). The equation is:

F (x, y, z) = 0.6 F(100) - 0.4 F(111)

= 0.6 (cos 2p x + cos 2p y + cos 2p z) - 0.4 cos 2p x cos 2p y cos 2p z = - 0.3.

The genus of this surface is 9.

Figure 1f. P+C(P) surface approximation. The equation is given by:

F (x, y, z) = 0.35 F(111) + 0.2 F(100) + 0.2 F(222) + 0.1 F(200) + 0.05 F(300) +0.1 F(110)

= 0.35 cos 2p x cos 2p y cos2p z +

0.2(cos 2p x+ cos 2p y + cos 2p z) + 0.2(cos 4p x cos 4p y cos 4p z)+

0.1(cos 4p x + cos 4p y + cos 4p z) + 0.05(cos 6p x + cos 6p y + cos 6p z) +

0.1(cos 2p x cos 2p y + cos2p y cos 2p z + cos 2p z cos 2p x) = 0.

A minimal surface with this topology was discovered by Karcher.22 The genus is 12.

Figure 1g. K surface approximation. This surface was also discovered by Karcher.23 The equation is given by:

F(x, y, z) = 0.3 F(100) + 0.3 F(110) – 0.4 F(200) =

= 0.3 (cos 2p x + cos 2p y + cos 2p z) +

0.3 (cos 2p x cos 2p y + cos 2p y cos 2p z + cos 2p z cos 2p x) +

The genus of this surface is 12.

Figure 1h. F-RD surface approximation. The equation is:

F(x, y, z) = 0.8 F(111) + 0.1F(222) - 0.1F(220)

= 0.8 cos 2p x cos 2p y cos 2p z + 0.1(cos 4p x cos 4p y cos 4p z) +

The genus of this surface is 6.

Figure 2a. G (gyroid) level surface:

F(x, y, z) = F(110) = sin 2p y cos 2p z + sin 2p z cos 2p x + sin 2p x cos 2p y = 0.

The genus of this surface is 3.

Figure 2b. G¢ surface level set:

F(x, y, z) = F(211) =

= sin 4p x cos 2p y sin 2p z + sin 4p y cos 2p z sin 2p x + sin 4p z cos 2p x sin 2p y = -0.32. G¢ has several new features as compared to the gyroid, e.g. extra holes and tunnels. Unlike the gyroid, G¢ belongs to space group 230. (A surface from this family which uses F(211) +0.5 F(220) has been named C(I2-Y**) by von Schnering and Nesper.17) The G¢ surface has genus 12.

Figure 2c. Two-parameter experiment for space group No. 230. The basic equation is

F(x, y, z) = s F211 + (1-s) F220 = t.

Figure 2d. Double gyroid surface approximation. The equation is:

F(x, y, z) = 0.8 F(211) – 0.2 F(220) =

0.8 (sin 4p x sin 2p z cos 2p y + sin 4p y sin 2p x cos 2p z + sin 4p z sin 2p y cos 2p x)

- 0.2 (cos 4p x cos 4p y + cos 4p y cos 4p z + cos 4p z cos 4p x) = 0.

Figure 2e. An approximation to the L surface. The equation is:

F(x, y, z) = 0.5 F(211) - F(220) =

0.5(sin 4p x cos 2p y sin 2p z + sin 4p y cos 2p z sin 2p x + sin 4p z cos 2p x sin 2p y)

- 0.5(cos 4p x cos 4p y + cos 4p y cos 4p z + cos 4p z cos 4p x = - 0.15.

Figure 3a. D surface level set approximation, shifted here by p /4 for a more familiar representation:

F(x, y, z) = F(111) = cos 2p x cos 2p y cos 2p z + sin 2p x sin 2p y cos 2p z +

sin 2p x cos 2p y sin 2p z + cos 2p x sin 2p y sin 2p z = 0.

The genus of the D surface os 3.

 

Figure 3b. D¢ surface level set:

F(x, y, z) = 0.5 F(111) - 0.5 F(220)

= 0.5 (cos 2p x cos 2p y cos 2p z + cos 2p x sin 2p y sin 2p z + sin 2p x cos 2p y sin 2p z +

+sin 2p x sin 2p y cos 2p z) +

This is a new triply periodic surface with symmetry and genus 9.

Figure 3c. Double diamond surface, shown here in a shifted unit cell for easier visualization.

F(x, y, z) = 0.5 F(110) + 0.5 F(111) =

= 0.5 (sin 2p x sin 2p y + sin 2p y sin 2p z + sin 2p x sin 2p z)

+ 0.5 (cos 2p x cos 2p y cos 2p z) = 0.

Figure 4. The P2-DG surface. This new surface belongs to space group No. 198. It arises as a combination of terms of the double gyroid and P-surface level sets. The equation for this surface is given by:

F(x, y, z) = 0.7 FA(211) - 0.1 F(220) - 0.2 F(200) =

= 0.7 (sin 4p x cos 2p y sin 2p z + sin 4p y cos 2p z sin 2p x + sin 4p z cos 2p x sin 2p y)

- 0.1 (cos 4p x cos 4p y + cos 4p y cos 4p z + cos 4p z cos 4p x)

- 0.2 (cos 4p y + cos 4p z + cos 4p x) = 0.

The first two terms are from space group No. 230 and the third term is from space group No. 221.

 

 

Figure 1a

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Figure 1g

 

Figure 1h

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Figure 2c

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Figure 2e

 

Figure 3a

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Figure 3c

Figure 4