avatars of the gyroid

Physica A 251 (1998) 1–11

Avatars of the gyroidM. Schick ∗

Department of Physics, Box 351560, University of Washington, Seattle WA 98195-1560, USA

Abstract

The unusual gyroid phase is described in three di�erent amphiphilic systems. The way in whichits existence can be understood in terms of surfaces of constant mean curvature is reviewed, as isthe manner in which it can actually be accounted for by standard models of diblock copolymerssolved within self-consistent �eld theory. c© 1998 Elsevier Science B.V. All rights reserved.

PACS: 68.35.-P; 68.35.Md; 68.55.Jk; 68.55.NqKeywords: Gyroid; Amphiphilic system; Lyotropic phase; Diblock copolymer

For Hans van Leeuwen, with gratitude and great pleasure

1. Introduction

The gyroid is the most unusual structure that I know of which nature actually uses.Its appearance was �rst recorded in anhydrous strontium soaps by Spegt and Skoulios[1], and its structure solved by Luzzati and Spegt [2]. It appears in all manners oflipid water systems [3], and was �nally identi�ed a few years ago in melts of diblockcopolymer [4,5]. Magni�cent three-dimensional images of it obtained from electrontomography of a block copolymer system have recently been published [6]. In thisbrief article, I will describe it, and review how one calculates its existence in polymersystems, and understands why it occurs where it does.

2. The structure

The connectivity of the gyroid structure is shown in Fig. 1. It is of space group Ia3d(No. 230). It consists of two interpenetrating networks, one of which is right-handed,the other its mirror image. They are shown in di�erent shades of gray in the �gure.The space group of each network is I4132 (No. 214). One of them is shown in Fig. 2.

∗ Tel.: +1 206 5439948; fax: +1 206 6850635; e-mail: [email protected].

0378-4371/98/$19.00 Copyright c© 1998 Elsevier Science B.V. All rights reservedPII S0378-4371(97)00590 -6

2 M. Schick / Physica A 251 (1998) 1–11

Fig. 1. Two views of the structure of the gyroid: (a) looking down a (1 0 0) axis; (b) looking down a(1 1 1) axis.

Fig. 2. One of the networks comprising the structure of Fig. 1. It can be decomposed into two sublattices.

This network is interesting in itself as it is the three-dimensional analogue of the hon-eycomb lattice. What I mean by this is that, like the honeycomb lattice, the I4132 net-work is three-fold coordinated, and can be decomposed into two sublattices. In Fig. 2,sites of one sublattice are shown in white, while those of the other are shown in black.A site on a given sublattice has three nearest-neighbors, each of which lies on the othersublattice. The two di�erent sublattices have the exact same structure, and one couldbe shifted to coincide with the other. In the honeycomb lattice, the two sublatticesare identical triangular lattices. I was familiar with the three-dimensional analoguefor the following reason. Hans van Leeuwen, Henk Hilhorst, and I had solved the

M. Schick / Physica A 251 (1998) 1–11 3

two-dimensional Ising model exactly by applying a di�erential, real-space, renormal-ization group transformation to the honeycomb lattice [7,8]. The idea was to start with atriangular lattice, employ a star-triangle transformation to generate a honeycomb lattice,then to trace out the sites on the original triangular lattice to leave a second triangularlattice with di�erent couplings. The original triangular lattice is bounded by edges, sothe new triangular lattice is slightly smaller than the original. The �nal step of thetransformation is to expand the second triangular lattice until it is the same size as theoriginal. This transformation can be repeated, its �xed points obtained, and analyzedas usual [ 7–9]. It occurred to me that if one could �nd the three-dimensional analogueof the honeycomb lattice, then one might be able to solve the three-dimensional Isingmodel in a similar manner. I enrolled Eytan Domany and Wolfgang Kinzel in thisproject. We came across the I4132 network of Fig. 2 in the monograph of Wells [10],and later found that series expansions already existed for the Ising model on this lattice[11]. Needless to say, we could not �nd a way to make the renormalization procedurework on it. While we could de�ne an in�nitesimal transformation from one sublatticeto the other, we could not repeat it.

3. The phase in three experimental systems

In retrospect, it should not be too surprising that structures of unusual symmetry,such as the Luzzati–Spegt lattice above with symmetry Ia3d, occur in systems of lipidand water. Lipids are amphiphilic, having a polarizable head group, which likes thewater environment, and hydrocarbon tails, which do not. Therefore, the lipids try toarrange themselves into structures in which the tails are separated from water while theheads are in contact with it. In Fig. 1, one should imagine that water �lls both sets oftubes, the lipid head groups line the inner surface of the tubes, and the hydrocarbon tailsof the lipids �ll the space between tubes. It is useful to consider the surface containingthe terminal methyl groups of the tails, those coming from the chains anchored toone set of tubes, and those coming from the other set. This locus forms a surfaceof symmetry I4132. The higher Ia3d symmetry of the total structure of Fig. 1 is dueto the additional re ection symmetry about this central surface. Now if one considersthe free energy of the entire structure to be reducible to an e�ective surface energyof this central surface, then one is led to the construction of in�nite periodic minimalsurfaces; i.e. ones which, for a given symmetry and lattice parameter, minimize thesurface area without any constraint on volume. The lack of volume constraint followsfrom the fact that the very same lipids which are on one side of the surface arealso on the other. Hence, the relative volumes on the two sides is not �xed. Suchsurfaces are characterized by a mean curvature which is zero everywhere. The studyof these mathematical constructs has a long and honorable history, dating back at leastto Schwarz [12] and his student Neovius [13]. They identi�ed �ve such structures,three of which were of cubic symmetry. Much later, Schoen [14] discovered fouradditional surfaces of cubic symmetry. Renewed interest in the physics community in


these surfaces is due to the suggestion by Scriven [15] that such surfaces might befound in amphiphilic systems, and this has led to the discovery of even more suchsurfaces [16,17]. One of the surfaces found by Schoen [14] has the symmetry I4132 ofthe central surface discussed above, and is called the G surface. Originally, the term“gyroid” referred speci�cally to this periodic surface of zero-mean curvature.

As noted above, the actual structure can be thought of as being constructed byattaching the ends of the lipid tails to this central surface, with the water con�ned totubes formed by the lipid head groups. The surfaces separating the lipid head groupfrom the tails form internal interfaces with water and hydrophilic head groups onone side, and hydrophobic tails on the other. There is no reason that these surfacesshould have zero-mean curvature, but it has been suggested [18] that they have a meancurvature which is constant, if non-zero, everywhere [19]. This is not unreasonable,because if one associates the free energy of the total structure with the surface freeenergy of these internal interfaces, then one is led to consider surfaces which minimizethe surface free energy but subject now to the constraint that the volumes on eithersides of the surfaces are �xed. This is due to the fact that the relative amount ofhydrophilic and hydrophobic entities, which are found on either side of the internalinterfaces, are �xed in experiment. The resulting surfaces have constant-mean curvatureeverywhere. The relative volumes of hydrophobic and hydrophilic entities can be variedby altering the architecture of the lipids and changing the water concentration. Werean oil to be added to this system, it would of course enter the hydrophobic region,changing its volume.

A similar situation occurs in a system of AB block copolymers, each of which consistof an A homopolymer chemically joined to a B homopolymer. Here the junctionsbetween A and B blocks form sheets separating the A and B regions. The volume ofeach region can be controlled independently by varying the architecture of the diblock,or by adding A homopolymer and=or B homopolymer. The analogy to a third physicalsystem of amphiphile, water, and oil, which has been the object of much study [20,21]should be obvious.

In all of these systems, one expects that as the relative volumes of the hydrophilic andhydrophobic elements, or A and B monomers, is varied continuously, the constant-meancurvature of the structures formed will vary monotonically. In fact, that is what oneobserves. As the ratio of the volumes departs from unity, one typically (but not invari-ably) observes the following sequence of phases: lamellar, gyroid, hexagonal packingof cylinders, and cubic packing of spheres. The observed phase with the Ia3d symmetryis now generally referred to as “the gyroid”.

I should note that Fig. 1, while displaying the correct topology and space group, ismisleading in that in any of the physical structures, the internal interfaces which formthe tubes are, of course, smooth so that one sees no “ball and stick” geometry as inthe �gure. The junctions of three tubes would look more like that shown in Fig. 3.Furthermore, the diameter of the tubes should be bigger. Typically in experiment, thevolume of the minority component, which is within the tubes, is about 35% of thetotal volume.


Fig. 3. Form of a region of three-fold symmetry serving as a junction of three tubes.

In spite of its reasonableness, there is something unsatisfying to me about this pointof view which considers only the internal interfaces and their properties. It fails toexplain why one observes only a few phases, even though there are other periodicsurfaces of constant-mean curvature with values of curvature intermediate betweenthose of the observed phases. It also fails to predict which phase will appear given thearchitecture of the diblock copolymer, and the volumes of homopolymer. Perhaps thereare satisfactory resolutions to the shortcomings I perceive, and I hasten to add that theapproach utilizing surfaces of constant-mean curvature has many excellent practitioners[ 22–24]. My own penchant is for a more microscopic approach, which is describedin the following section.

4. Microscopic modeling

There are numerous models of small amphiphilic systems [21], and there can beconcern that something special is put into the model that results in it producing exoticphases. Such is not the case with diblock copolymers. There are one or two “standard”models which are transparent, and reproduce the properties of polymers very well. Thesimplest of them is the Gaussian chain [25]. If such models produce exotic phases, thenone believes that they are there for good reason. An additional advantage of studyingpolymers is that mean-�eld theory is extraordinarily good because the objects are solarge.

One begins with some description of the polymer chain. Whatever model one uses,an important ingredient is the location of the A monomers and of the B monomers.A con�guration of the system containing n polymers is speci�ed by the collection ofposition vectors {r�; i} where � = 1; 2; : : : ; n denotes the chain on which the monomer is


located, and i denotes its position on the chain. For a simple AB diblock copolymer, theA monomers are located between 16i6f, and the B monomers between f+16i6N .From these con�gurations one de�nes the dimensionless A monomer density

�̂A(r) =1�

n∑�=1

f∑i=1

�(r− r�; i) ; (1)

and the dimensionless B monomer density

�̂B(r) =1�

n∑�=1

N∑i=f+1

�(r− r�; i) ; (2)

where � is the monomer density, � = nN=V , with V the system volume.The statistical weight of the con�guration depends on two factors; the number

of such con�gurations, P{r�; i}, which follows from the de�nition of the particularmodel, and the Boltzmann weight which depends upon the interaction between poly-mers. This is usually taken to be a repulsive contact interaction of some strength, �in units of kBT , between A and B monomers. The partition function of the system,Z ≡ exp(−Fexact=kBT ), is given by a functional integral over all polymer con�gurations:

exp(−Fexact=kBT ) =∫ n∏

�=1

Dr�; iP{r�; i}exp{−��

∫dr�̂A(r)�̂B(r)

}: (3)

It is the interaction term, which depends explicitly on the polymer coordinates, whichprevents the functional integral from being carried out. What can be done is to introduceauxillary �elds in order to rewrite the partition function exactly as

exp(−Fexact=kBT ) = N

∫D�ADWAD�BDWB exp{−F=kBT} ; (4)

where N is a normalization constant, and

F[�A;WA; �B;WB]nkBT

≡ −lnQ + V−1∫dr[�N�A�B −WA�A −WB�B] ; (5)

with Q the partition function of a single polymer in external �elds WA and WB;

Q[WA;WB] =∫

DriP({ri}) exp

−

f∑i=1

WA(ri) −N∑

i=f+1

WB(ri)

: (6)

What one has done is to rewrite the interaction, which was previously between poly-mers directly, as one intermediated by uctuating �elds WA and WB which interact withthe individual polymers. The mean-�eld approximation consists of replacing the exactfree energy by the minimum of F[�A;WA; �B;WB] and neglecting all uctuations aboutit. The minimum is usually, but not always, sought under the constraint of incompress-ibility, a constraint enforced pointwise by a Lagrange multiplier �(r). The functions for


which the functional F[�A;WA; �B;WB] attains its minimum value, F[�A; wA; �B; wB],subject to this constraint are determined by the set of self-consistent equations

wA(r) = �N�B(r) + �(r) ; (7)

wB(r) = �N�A(r) + �(r) ; (8)

�A(r) + �B(r) = 1 ; (9)

�A = −VQ

DQ

DwA; (10)

�B = −VQ

DQ

DwB: (11)

The last two equations identify �A(r) and �B(r) as the average densities of A andB monomers at r as calculated in an ensemble of non-interacting polymers subject tothe �elds wA(r) and wB(r), which act on A and B monomers, respectively. As in anymean-�eld approximation, the problem of calculating the partition function of inter-acting objects has now been reduced to the calculation of the partition function of asingle object in an external �eld. This is trivial when the object is an Ising spin withfew, discrete, degrees of freedom, less so for model polymers with their enormousnumber of continuous degrees of freedom. None the less, within the standard polymermodels, the con�guration of the chain is described by one kind of random walk oranother, so that the partition function, Q[wA; wB], is obtained by solving some sort ofmodi�ed di�usion equation containing a potential to be determined self-consistently[25]. In this way, the problem is much like a quantum–mechanical Hartree calculationin which one solves the Schrodinger equation, a di�usion equation in imaginary time.Lipids, however, are not so long so as to obey the assumptions on exibility inherent inthe standard models, so their partition functions cannot be obtained so simply. Insteadone determines it by Monte Carlo sampling, generating representative single-lipid con-�gurations which are then weighted appropriately [26,27]. With the partition functionin hand, the mean-�eld equations can be solved, in principle. But it proved di�cultto solve these equations in practice for the ordered phases of interest. The lamellarphase is simple to calculate, as it varies only in one direction. The hexagonal arrayof cylinders and the cubic array of spheres could be handled approximately [28] byreplacing their exact Wigner–Seitz cells by cylinders and spheres, respectively, leavinga one-dimensional problem. But solving the above real-space equations for a structureas complicated as the gyroid appeared hopeless.

This problem was solved a few years ago by Matsen and me [29]. One expandsall functions of position in a complete set of functions, fj(r), which possess a par-ticular space-group symmetry, such as Ia3d. Such complete sets are readily availablein tables [30]. The self-consistent equations then become equations for the expan-sion coe�cients, which one solves numerically to a desired accuracy. Whatever thelevel of numerical precision, the free energy evaluated with this solution is guaran-teed to have the desired symmetry. One repeats the calculation with basis sets of


Fig. 4. Phase diagram of diblock copolymer as a function of the interaction parameter � and fraction f ofA monomer in the diblock. The labels are as follows: L (lamellar), G (gyroid), C (hexagonal packing ofcylinders), S (body-centered cubic packing of spheres), and Scp (close-packing of spheres). From Ref. [31].

Fig. 5. Portion of the phase diagram of the diblock polyisoprene–polystyrene, after Ref. [34].

other symmetries, and �nds the phase with the lowest free energy for a value ofthe interaction constant, �, and architecture, f. Proceeding in this way, Matsen andI calculated the phase diagram of the AB copolymer system within mean-�eld the-ory [29]. The most recent version of this phase diagram utilizing the standard Gaus-sian polymer model is given by Matsen and Bates [31], and shown in Fig. 4. Thegratifying results, in complete agreement with experiment, is that the gyroid phaseappears between the lamellar and hexagonal phases, just as in experiment, and isthe only one of the “non-standard” phases to appear. In fact the space group ofthe experimentally observed non-standard phase was previously identi�ed as being“double diamond”, of space group Pn3m [32]. After the prediction that this phasewas not stable [29], but that the Ia3d phase was, reexamination of experiment showedthat the Pn3m identi�cation was in error [33], and the phase was, in fact, gyroid.A portion of the phase diagram of polyisoprene–polystyrene [34] is shown in Fig.5. The di�erences between the experimental phase diagram and that predicted bymean-�eld theory are due to the e�ect of uctuations which have opened up direct


transitions between the disordered phase and the gyroid, the hexagonal packing ofcylinders, and the cubic packing of spheres. In particular, the single triple point pre-dicted by mean-�eld theory between gyroid, lamellar, and hexagonal phases has beenreplaced by a pair of such points between the disordered, gyroid, and hexagonal, anddisordered, gyroid, and lamellar phases, respectively. This e�ect is understood theoret-ically [ 35–37].

With the mean-�eld theory calculation in hand, one can reexamine the arguments asto why particular phases occur. This has recently been considered by Matsen and Bates[38]. They �nd that it is indeed correct that the average mean curvature of the surfaceseparating A and B monomers in the Ia3d phase has a value which is intermediatebetween that in the lamellar phase (which is clearly zero), and that in the hexagonalphase. It is also correct that there are other phases with intermediate curvature whichare not observed to be stable. However, they emphasize that an important piece ofphysics is ignored if one considers only the minimization of the amount of internalinterface; that physics is the constraint that the polymers must pack to �ll all space.This consideration favors domains which are relatively uniform in thickness so thatthe entropic stretching penalty is distributed uniformly among the molecules. It isthis consideration which drives the surfaces of A–B junctions to di�er from surfacesof constant mean curvature. The free energy of the actual structure is a compromisebetween the cost of the internal interfaces and the cost of the chain stretching to �ll thevolume between the interfaces. In this interplay, the double diamond, and other unusualphases, lose out to the gyroid. Matsen and Bates speculate that as the temperature islowered and the chains tend to become ever more strongly stretched, the energy costof �lling the volumes of the gyroid grows, and causes this phase to become unstablewith respect to the lamellar and cylindrical phases, in which the volumes are moreeasily �lled.

The latest wrinkle in this story is an interesting one. In a series of papers [ 39–41],Noolandi and colleagues worked out the contribution of uctuations in the Gaussianapproximation to the extremum calculated within mean-�eld theory. The calculationis amusing. As noted above, in the standard polymer models, the partition functionof the single-polymer problem is found by solving a di�usion equation containing apotential, much like the Schrodinger equation. To calculate the Gaussian uctuationsabout the self-consistent solution, one solves a similar equation in which the potentialis periodic, re ecting the space-group of the mean-�eld solution. Thus this problem issimilar to a band-theory calculation. One obtains a set of excitations, the least energeticof which, dominates the uctuations. There are many interesting results produced bythese calculations. The relevant one here is the following. The excitations of the gyroidphase were examined at the highest temperature at which it exists in mean-�eld theory,at its triple point with the lamellar and hexagonal phases.

Surprisingly, it was found that some excitation had a negative energy, with theconsequence that the gyroid phase is unstable at this point [40,41]. In other words,the extremum found by mean-�eld theory is not a minimum, but a saddle point! Ihave a great deal of di�culty with this result for it seems clear experimentally that


the gyroid phase is very stable indeed. On further re ection, I decided that the resultneed not be wrong. What is being said is that the gyroid is not stable at its triplepoint with lamellar and hexagonal phases, and this is certainly correct as seen in thephase diagram of Fig. 5. What one would like to see in this calculation is for thegyroid phase to become stable at a lower temperature, so as to agree with experiment.However as the temperature is decreased, the number of basis functions which mustbe kept to ensure a given level of accuracy rapidly increases to a point at which thecalculation becomes no longer feasible. There is another check of this result whichcould be performed, however. If the gyroid phase is truly a saddle-point solution ofthe mean-�eld equations at the value of � and f corresponding its triple point, thenthere must be another solution of the self-consistent equations with a symmetry di�erentfrom those of the other phases and with a lower free energy. It should be possible to�nd this solution. But if it does exist, one would like to know why this phase is notexperimentally observed somewhere in the phase diagram. Clari�cation of this pointwould be welcome, but would hardly make the gyroid phase less strange!

Acknowledgements

I thank Anatoly Frenkel and Anna Poiarkova for preparing Figs. 1 and 2. Thiswork was supported in part by the National Science Foundation under Grant No.DMR9531161.

References

[1] P.A. Spegt, A.E. Skoulious, Acta Crystallogr. 21 (1966) 892.[2] V. Luzzati, P.A. Spegt, Nature 215 (1967) 701.[3] V. Luzzati, A. Tardieu, T. Gulik-Kryzwicki, E. Rivas, F. Reiss-Husson, Nature 22 (1968) 485.[4] D.A. Hajduk, P.E. Harper, S.M. Gruner, C.C. Honeker, G. Kim, E.L. Thomas, L.J. Fetters,

Macromolecules 27 (1994) 4063.[5] M.F. Schulz, F.S. Bates, K. Almdal, K. Mortensen, Phys. Rev. Lett. 73 (1994) 86.[6] R.J. Spontak, J.C. Fung, M.B. Braunfeld, J.W. Sedat, D.A. Agard, L. Kane, S.D. Smith,

M.M. Satkowski, A. Ashraf, D.A. Hajduk, S.M. Gruner, Macromolecules 29, 4494.[7] H.J. Hilhorst, M. Schick, J.M.J. van Leeuwen, Phys. Rev. Lett. 40 (1978) 1605.[8] H.J. Hilhorst, M. Schick, J.M.J. van Leeuwen, Phys. Rev. B 19 (1979) 2749.[9] H.J.F. Knops, H.J. Hilhorst, Phys. Rev. B 19 (1979) 3689.

[10] A.F. Wells, Three-Dimensional Nets and Polyhedra, Wiley, New York, 1977.[11] J.A. Leu, D.D. Betts, C.J. Elliot, Can. J. Phys. 47 (1969) 1671.[12] H.A. Schwarz, Gesammelte Mathematische Abhandlung, vol. 1, Springer, Berlin, 1890, p. 6.[13] E.R. Neovius, Bestimmung zweir speziellen periodischen Minimal �achen, J.C. Frenkel, Helsingfors,

1883.[14] A.H. Schoen, NASA Report No. TN D-5541, 1970, unpublished.[15] L.E. Scriven, Nature 263 (1976) 123.[16] W. G�o�zd�z, R. Ho lyst, Phys. Rev. Lett. 15 (1996) 2726.[17] W. G�o�zd�z, R. Ho lyst, Phys. Rev. E 54 (1996) 5012.[18] E.L. Thomas, D.M. Anderson, C.S. Henkee, D. Ho�man, Nature 334 (1988) 598.[19] D.M. Anderson, H.T. Davis, L.E. Scriven, J.C.C. Nitsche, Adv. Chem. Phys. 37 (1990) 337.[20] G.J.T. Tiddy, Phys. Rep. 57 (1980) 1.


[21] G. Gompper, M. Schick, Self-Assembling Amphiphilic Systems, Academic Press, San Diego, 1994.[22] See International Workshop on Geometry and Interfaces, Aussois, France, Colloque Phys. C-7 23

(Suppl.) 1990.[23] S.T. Hyde, S. Andersson, B. Ericsson, K. Larsson, Z. Kristallogr. 168 (1984) 213.[24] S.T. Hyde, A. Fogden, B.W. Ninham, Macromolecules 26 (1993) 6782.[25] M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, Oxford, 1986.[26] I. Szleifer, A. Ben-Shaul, W.M. Gelbart, J. Chem. Phys. 85 (1986) 5345.[27] M. Schick, M. M�uller, Bull. Am. Phys. Soc. 42 (1997) 145.[28] J.D. Vavasour, M.D. Whitmore, Macromolecules 25 (1992) 5477.[29] M.W. Matsen, M. Schick, Phys. Rev. Lett. 72 (1994) 2660.[30] N.F.M. Henry, K. Lonsdale, International Tables or X-Ray Crystallography, Kynoch, Birmingham, 1969.[31] M.W. Matsen, F.S. Bates, Macromolecules 29 (1996) 1091.[32] D.B. Alward, D.J. Kinning, E.L. Thomas, L.J. Fetters, Macromolecules 19 (1986) 215.[33] D.A. Hajduk, P.E. Harper, S.M. Gruner, C.C. Honeker, E.L. Thomas, L.J. Fetters, Macromolecules

28 (1995) 2570.[34] F.S. Bates, M.F. Schulz, A.K. Khandpur, S. F�orster, J.H. Rosedale, K. Almdal, K. Mortensen, Faraday

Discussions 98 (1994) 7.[35] G.H. Fredrickson, E. Helfand, J. Chem. Phys. 87 (1987) 697.[36] G. Floudas, S. Pispas, N. Hadjichristidis, T. Pakula, I. Erukhimovich, Macromolecules 29 (1996) 4142.[37] V.E. Podneks, I.W. Hamley, JETP Lett. 64 (1996) 617.[38] M.W. Matsen, F.S. Bates, J. Chem. Phys. 106 (1997) 2436.[39] A.-C. Shi, J. Noolandi, R.C. Desai, Macromolecules 29 (1996) 6487.[40] M. Laradji, A.-C. Shi, R.C. Desai, J. Noolandi, Phys. Rev. Lett. 78 (1977) 2577.[41] M. Laradji, A.-C. Shi, R.C. Desai, J. Noolandi, R.C. Desai, Macromolecules 30 (1997) 3242.

avatars of the gyroid

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