LIQUID CRYSTALS

Three-dimensional crystals ofadaptive knotsJung-Shen B. Tai1 and Ivan I. Smalyukh1,2,3*

Starting with Gauss and Kelvin, knots in fields were postulated to behave likeparticles, but experimentally they were found only as transient features orrequired complex boundary conditions to exist and could not self-assemble intothree-dimensional crystals. We introduce energetically stable, micrometer-sizedknots in helical fields of chiral liquid crystals. While spatially localized and freelydiffusing in all directions, they resemble colloidal particles and atoms, self-assemblinginto crystalline lattices with open and closed structures. These knots are robustand topologically distinct from the host medium, though they can be morphed andreconfigured by weak stimuli under conditions such as those in displays. A combinationof energy-minimizing numerical modeling and optical imaging uncovers the internalstructure and topology of individual helical field knots and the various hierarchicalcrystalline organizations that they form.

Topological order and phases represent anexciting frontier of modern research (1), buttopology-related ideas have a long history inphysics (2). Gauss postulated that knots infields could behave like particles, whereas

Kelvin, Tait, and Maxwell believed that matter,including crystals, could be made of real-space,free-standing knots of vortices (2–4). These earlyphysics models, introduced long before the veryexistence of atoms was widely accepted, gaveorigins to modern mathematical knot theory(2–4). Expanding this topological paradigm,Skyrme and others modeled subatomic par-ticles with different baryon numbers as non-singular topological solitons and their clusters(3–5). Knotted fields emerged in classical andquantum field theories (3–7) and in scientificbranches ranging from fluid mechanics to par-ticle physics and cosmology (2–11). In condensedmatter, arrays of singular vortex lines and low-dimensional analogs of Skyrme solitons werefound as topologically nontrivial building blocksof exotic thermodynamic phases in superconduc-tors, magnets, and liquid crystals (LCs) (12–14).Could they be knotted, and could these knotsself-organize into three-dimensional (3D) crys-tals? Knotted fields in condensed matter foundmany experimental and theoretical embodiments,including both nonsingular solitons and knottedvortices (7–9, 15–23). However, they were meta-stable and decayed with time (7–9, 15–17) orcould not be stabilized without colloidal inclu-sions (18, 19) or confinement and boundary con-ditions (20–22), and could not self-organize into

3D lattices (22, 23). We introduce energeticallystable, micrometer-sized adaptive knots in chiralLCs that, unexpectedly, materialize the knottedvortices and nonsingular solitonic knots at thesame timeand indeedbehave like particles, under-going 3D Brownian motion and self-assemblinginto 3D crystals.Helical fields, as in the familiar example of

circularly polarized light with electric and mag-netic fields periodically rotating around the Poyn-ting vector, are ubiquitous in chiral materialssuch as magnets and LCs. These helical fieldscomprise a triad of orthonormal fields (Fig. 1A),namely thematerial alignment fieldnðrÞ (of rod-like molecules in LCs or spins in magnets) andthe immaterial line fields along the helical axiscðrÞ, analogous to the Poynting vector, and tðrÞ⊥nðrÞ⊥cðrÞ. For LCs, nðrÞ is nonpolar but can bedecorated with unit vector fields (14, 24). Thedistance over whichnðrÞ and tðrÞ rotate aroundcðrÞ by 2p within the helical structure is thehelical pitchp (Fig. 1A). We demonstrate knottedfields that in nðrÞ are topological solitons withinterlinked, closed-loop preimages resemblingHopf fibration (Fig. 1B). At the same time, thenonpolar nature of cðrÞ and tðrÞ permits thehalf-integer singular vortex lines to form varioustorus knots (Fig. 1C) while retaining the fully non-singular nature ofnðrÞ. Therefore, our topologicalsoliton in the helical field is a hybrid embodimentof both interlinked preimages and knotted vortexlines, which can be realized to have this solitonicnonsingular nature in systems with either polaror nonpolar nðrÞ (12–14). We find these knotsolitons, which we call “heliknotons,” embeddedin a helical background and forming spontane-ously after transition from the isotropic to LCphase when a weak electric fieldE is applied to apositive-dielectric-anisotropy chiral LC along thefar-field helical axis c0 . The materials used areprepared as mixtures LC-1 through LC-3 (24) ofcommercially available, room-temperature ne-matics and chiral dopants. In bulk LC samples oftypical thickness within d = 10 to 100 mm (24),

heliknotons display 3D particle–like propertiesand form a dilute gas at low number densities(Fig. 1D), with orientations of shape-anisotropicsolitonic structures correlated with their posi-tions along c0 (Fig. 1, D and E). Depending onmaterials and applied voltageU, heliknotons canadopt different shapes (Fig. 1, D to G), which arereproduced by numerical modeling (insets ofFig. 1, F and G) based on minimization of thefree energy (24):

F ¼

∫d3rK

2ð∇nÞ2 þ 2pK

pn � ð∇� nÞ � e0De

2ðn � EÞ2

� �

ð1Þwhere K is the average elastic constant, De isthe LC’s dielectric anisotropy, and e0 is the vac-uum permittivity. The integrand comprises energyterms originating from elastic deformation, chiral-ity, and dielectric coupling, respectively. Minimiza-tion ofF at differentU andDe (table S1) reveals thatheliknotons can be stable, metastable, or unstablewith respect to the helical background (Fig. 1H),comprising localized regions (depicted in gray inFig. 1, B and C) of perturbed helical fields andtwisting rate.Heliknotons undergo Brownian motions (Fig.

1I and movie S1) and exhibit anisotropic inter-actions while moving along c0 and in the lateraldirections (Fig. 1, E and J, and movie S2) (24).The inter-heliknoton pair-interaction potential isanisotropic and highly tunable, from attractive torepulsive and from tens to thousandskBT (wherekB is Boltzmann constant and T is temperature),depending on the choice of LC, U, and samplethickness (Fig. 1J). Similar to nematic colloids(25, 26), interactions between localized helikno-tons arise from sharing long-range perturbationsof the helical fields around them andminimizingthe overall free energy for different relative spa-tial positions of these solitons. These interactionslead to a plethora of crystals, including low-symmetry and open lattices that were recentlyachieved in colloids (26–28) (Fig. 2). In thin cellsof thickness d≲4p , heliknotons localize aroundthe sample’s horizontal midplane, making theiranisotropic interactions quasi-2D. Heliknotonsself-assemble (movie S3) into a 2D rhombic latticeboth when the attractive potential is ~1000 kBT(Fig. 2, A and B) and when it is ~10 kBT (Fig. 2,C and D). From initial positions defined by lasertweezers (24), heliknotons self-assemble into astretched kagome lattice with anisotropic bind-ing energies ~100 kBT (Fig. 2E). Such open lat-tices have interesting topological properties (28),potentially bringing about an interplay betweentopologies of the crystal’s basis and lattice. Crys-tallographic symmetries and lattice parameterscan be controlled through tuning reconfigurableinteractions, such as switching reversibly betweensynclinic and anticlinic tilting of heliknotons viachanging U by 4p,

when anisotropic interactions yield triclinic lat-tices (Fig. 2, H to N). One can watch initially

RESEARCH

Tai et al., Science 365, 1449–1453 (2019) 27 September 2019 1 of 5

1Department of Physics, University of Colorado, Boulder,CO 80309, USA. 2Materials Science and EngineeringProgram, Soft Materials Research Center, and Departmentof Electrical, Computer & Energy Engineering, Universityof Colorado, Boulder, CO 80309, USA. 3Renewable andSustainable Energy Institute, National Renewable EnergyLaboratory and University of Colorado, Boulder, CO80309, USA.*Corresponding author. Email: ivan.smalyukh@colorado.edu

on April 3, 2021

http://science.sciencem

ag.org/D

ownloaded from

http://science.sciencemag.org/

quasi-2D pre-self-assembled crystallites interact-ing with each other while moving in lateral andaxial directions (Fig. 2, J to M, and movie S5),forming different crystallographic planes of the3D triclinic lattice. The helical background LCs,individual heliknotons, and the ensuing latticesare all chiral. The lowest-symmetry triclinic pediallattices can have primitive cells comprising two(Fig. 2, H and J toM, andmovie S5) or three (Fig.2, I andN, andmovie S6) crystallographic planes,depending on relative orientations of helikno-tons within these planes. The two lattices withparallel (Fig. 2M) and orthogonal (Fig. 2N) rela-tive orientations of heliknotons in consecutiveheliknoton layers are just examples as the anglebetween heliknotons within crystallographic pla-nes along c0 can be tuned (Fig. 1E) byU, material,and geometric parameters. Because the helikno-tons have anisometric shape (like LC molecules)and can exhibit spatial twists, hierarchical topo-logical solitons comprising heliknotons couldpotentially emerge. Heliknoton crystals exhibitgiant anisotropic electrostriction (Fig. 2O andmovies S7 and S8). For example, upon changingfrom U = 3.0 to 4.4 V, one lattice parameter inthe insets of Fig. 2O extends by ∼44%, whereasthe other only by ∼4%. This electrostriction isconsistent with the free-energy minimization(Fig. 2P) for 3D crystals of heliknotons withtunable lattice parameters at different U. The ex-perimentally observed soliton crystals correspondto minima of free energy within a broad range

of applied voltages (24), consistent with theirfacile self-organization into triclinic pedial crys-tals and other reconfigurable 3D and 2D lattices(Fig. 2). As the applied field is increased evenfurther, the heliknoton crystals become meta-stable and then unstable with respect to theunwound state with nðrÞ||E (24).Numerical modeling and experiments reveal

detailed structures of the fields within a heli-knoton (24) (Fig. 3, A to I, and fig. S1). InnðrÞ, thecontinuous localized configuration of a heliknotonis embedded in a helical background (Fig. 3, Ato C) and has all closed-loop preimages linkedwith each other once, with the linking number +1(Fig. 3J and fig. S1). Up to numerical precision,this matches the Hopf index calculated throughnumerical integration (22):

Q ¼ 164p2

∫d3rDijkAiBjk ð2Þ

where Bij ¼ Dabcna@inb@jnc , D is the totally anti-symmetric tensor, Ai is defined as Bij ¼ ð@iAj�@jAiÞ=2, and the summation convention isassumed. Spatial structures of cðrÞ and tðrÞ arederived from the energy-minimizing nðrÞ usingthe eigenvector of the chirality tensor (24, 29, 30)(Fig. 3, D to I). They exhibit torus knots of spa-tially colocated singular half-integer vortices,withinwhichcðrÞandtðrÞnonpolar fields rotateby 180° around the vortex line in the plane lo-cally orthogonal to it (Fig. 3, D to I). The closedloop of the vortex line is the righthanded T(2,3)

trefoil torus knot, also labeled as the 31 knot inthe Alexander–Briggs notation (Fig. 3, K and L).The singular vortex knots in cðrÞ and tðrÞ alsocorrespond to a colocated knot of a meron (topo-logically nontrivial structure of a fractional 2Dskyrmion tube) in nðrÞ (fig. S2). Handedness ofthe knots and links matches that of chiral nðrÞ,implying that the sign of Hopf indices of suchenergy-minimizing solitons is dictated by LC’schirality. Simulated and experimental depth-resolved nonlinear optical images of helikno-tons for different polarizations of excitationlight closely agree (Fig. 3, M to O), confirmingexperimental reconstruction of the field (24).Unlike the Shankar solitons (11), which exem-plify condensed matter models with topologyof a triad of orthonormal fields similar to thatof Skyrme solitons in nuclear physics, helikno-tons exhibit nonsingular structure only in oneof the three fields, though they are still overallnonsingular in the material field. Differing fromtransient textures of linked loops of nonsingulardisclinations (15, 16) and metastable loops ofsingular vortices (17) in cholesteric LCs, ourheliknotons are stable torus knots of colocatedmerons innðrÞand vortices incðrÞandtðrÞ thatenable ground-state 3D crystals of knots (Fig. 2and fig S2).In addition to the Q ¼ 1 heliknotons with

equilibrium dimensions between p and 2p,we also find larger Q ¼ 2 topological solitons(Fig. 4A), for which experimental polarizing

Tai et al., Science 365, 1449–1453 (2019) 27 September 2019 2 of 5

Fig. 1. Knots in helices. (A) Helical field compris-ing a triad of orthonormal fields nðrÞ, cðrÞ,and tðrÞ, with nðrÞ being either polar (left) ornonpolar (right). (B) Preimages in nðrÞ of aheliknoton colored according to their orientations

on S2 (top right inset). (C) Knotted colocatedhalf-integer vortex lines in cðrÞ and tðrÞ. In (B)and (C), the gray isosurfaces show regionsof distorted helical background. (D) A gas ofheliknotons in LC-1 sample of thickness 30 mmat U = 4.5 V. (E) Two heliknotons interact in 3Dwhile forming a dimer in LC-2 sample of thickness30 mm at U = 11.0 V. (F and G) Polarizing opticalmicrographs of metastable and stable heliknotonsat U = 4.3 and 4.5 V, respectively, in a samplewith d ¼ 10 mm, with computer-simulated counter-parts shown in the bottom right insets. (H) Freeenergy of individual heliknotons versus E, whereenergy of the helical state equals zero; the helical

state and heliknotons are unstable atffiffiffiffiffiffiDe

pE ≳ 1V/mm

when the field tends to align nðrÞ||E (24).(I) Displacement histograms Dx and Dy showingdiffusion of the heliknoton in (F) in orthogonallateral directions perpendicular to c0. Experimentaland numerical data were obtained for LC-1 in (D)and LC-2 in (E) to (I). (J) Pair interaction ofheliknotons. Data shown in red (at voltages ○,4.2 V; △, 2.9 V) were obtained for LC-2 andthose in blue (○, 1.4 V; △, 1.0 V; ◇, 1.7 V)for LC-1 at d ¼ 10 mm (24). Scale bars indicate10 mm and p ¼ 5 mm.

RESEARCH | REPORTon A

pril 3, 2021

http://science.sciencemag.org/

Dow

nloaded from

http://science.sciencemag.org/

micrographs also match their numerical coun-terparts. A Q ¼ 2 heliknoton contains a largerregion of distorted helical background in both thelateral and axial directions (Fig. 4, A to D, andfigs. S3 and S4). Preimages for two antiparal-lel vertical orientations of nðrÞ form a pair ofHopf links (Fig. 4H), linked twice, like all otherpreimage pairs. Singular vortex lines incðrÞ andtðrÞ form closed cinquefoil T(2,5) torus knots(also labeled as 51 knots) colocated with a sim-ilar knot of a meron tube in nðrÞ. A Q = 3heliknoton contains three Hopf links of preim-ages with a net linking number of 3 for eachpreimage pair (Fig. 4I and figs. S5 and S6). Thesingular vortex lines incðrÞandtðrÞ form a T(2,7)

torus knot (the 71 knot), colocated with the sameknot of a meron tube in nðrÞ (Fig. 4, E, F, andI). Figure 4, G to I, shows both preimages of theantiparallel vertical orientations of nðrÞ andvortex lines in cðrÞ and tðrÞ , as well as theReidemeister moves simplifying their struc-tures. Topologically disinct heliknotons havedifferent numbers of crossings in the free-standing knots of vortex lines and differentlinking of preimages, which were key topo-logical invariants in early models of atoms andsubatomic particles (2–6). For different heli-knotons, Q is related to crossing number N ofthe vortex knots: N ¼ 2Q þ 1. The closed loopof preimages in nðrÞ are interlinked with the

torus knots of vortices incðrÞandtðrÞ, as shownin Fig. 4, G to I.Differing from transient vortex lines, which

shrink with time owing to energetically costlycores and distorted order around them, vortex-meron knots in heliknotons are energeticallyfavorable because they are nonsingular in thematerial nðrÞ field and comprise twisted struc-tures with handedness matching that of the LC.The stability of our 3D topological solitons asspatially localized structures is assisted by thechiral term in Eq. 1, which introduces their fi-nite dimensions and plays a role analogous tothat of high-order nonlinear terms in solitonicmodels of subatomic particles (3–6) and the

Tai et al., Science 365, 1449–1453 (2019) 27 September 2019 3 of 5

Fig. 2. Crystals of heliknotons. (A and B) Snapshots showing self-assembly of a 2D crystal (d ¼ 10 mm, U = 3.5 V). (C to E) 2D closedrhombic [(C) and (D)] and open (E) lattices of heliknotons at U = 1.9 and1.7 V, respectively (d ¼ 15 mm). (F and G) Crystallites with aligned (F) andanticlinically tilted (G) heliknotons at U = 1.8 and 2.3 V, respectively(d ¼ 17:5 mm). (H and I) Primitive cells of 3D heliknoton crystals wherethe solitons in neighboring horizontal layers have relative parallel (H)or perpendicular (I) orientations. Isosurfaces (gray) show the localized

3D regions of heliknotons with distorted helical background whencolocated with both vortex knots (light red) and preimages ofantiparallel vertical orientations in nðrÞ (black and white). (J to L) 3Dinteractions and self-assembly of heliknoton crystallites (d ≈ 30 mm and

U = 2.8 V). (M and N) 3D heliknoton lattices comprising crystalliteswith parallel (M) or perpendicular (N) orientations, where (N) and itsinset are polarizing micrographs obtained when focusing at differentcrystalline planes ∼10 mm apart (d ≈ 30 mm in both cases and U = 2.8 and3.4 V, respectively). (O) Electrostriction of a heliknoton crystal. Insetsshow lattices at different U, with the lattice parameters shown inblue and red (d ¼ 10 mm). (P) Free energy of heliknoton crystals perprimitive cell for two LCs at different E. The heliknotons become

metastable with respect to the unwound state atffiffiffiffiffiffiDe

pE ≳ 0.8 V/mm

for LC-1 and atffiffiffiffiffiffiDe

pE ≳ 1.2 V/mm for LC-2. Data were obtained using LC-2

in (A), (B), and (O) and using LC-1 in (C) to (N) (24). Scale barsindicate 10 mm and p ¼ 5 mm.

RESEARCH | REPORTon A

pril 3, 2021

http://science.sciencemag.org/

Dow

nloaded from

http://science.sciencemag.org/

Tai et al., Science 365, 1449–1453 (2019) 27 September 2019 4 of 5

Fig. 3. Structure of an elementary heliknoton. (A to I) Computer-simulated cross-sections of nðrÞ [(A) to (C)], cðrÞ [(D) to (F)], and tðrÞ [(G)to (I)] of a heliknoton. Vertical cross-sections and the viewing directionsare marked in (A), (D), and (G), respectively. nðrÞ is shown with arrowscolored according to S2 [(A), inset], and cðrÞ and tðrÞ are shown with ellipsoidscolored according to their orientations on the doubly colored S2=ℤ2 [(D) and(G), insets].The vortex lines in cðrÞ and tðrÞ are marked by red circles in (D) to(I). (J) Preimages of vertical orientations of nðrÞ forming a Hopf link (bottom

right inset) and the cross-section of nðrÞ. (K and L) The singular vortex linein cðrÞ and tðrÞ forming a trefoil knot (bottom right insets) visualized bylight-red tubes and the cross-sections of cðrÞ and tðrÞ, respectively.(M to O) Computer-simulated and experimental nonlinear optical images ofnðrÞ in the cross-sections of a heliknoton obtained with marked linearpolarizations [(M) and (N)] and circular polarization [(O)]. Images on the leftare numerical and those on the right are experimental, all obtained for LC-3at d ¼ 10 mm and U = 1.7 V (24). Scale bars indicate 5 mm and p ¼ 5 mm.

RESEARCH | REPORTon A

pril 3, 2021

http://science.sciencemag.org/

Dow

nloaded from

http://science.sciencemag.org/

Dzyaloshinskii–Moriya term in models of mag-netic skyrmions (13, 14). Applied field along c0tends to reorient nðrÞ along E as compared tothe helical state withnðrÞ⊥E. Consequently, theknots emerge as local or global energy minimawithin a certain range of voltages (24) by reduc-ing the dielectric term inEq. 1 comparedwith thehelical state (Figs. 1H and 2P). At low E, elasticenergetic costs are high and heliknotons collapseinto the helical background through spontane-ous creation and annihilation of singular defectsin nðrÞ. The strong dielectric coupling betweennðrÞ and E aligns nðrÞ||E at high applied fields,eventually making both the helical structure andsolitons unstable, but heliknotons are the glob-al free-energy minima within broad, material-dependent ranges of E (24).We have demonstrated 3D topological solitons

in helical fields of chiral LCs that can be ground-state and metastable configurations, forming 3Dcrystalline lattices. Unlike the atomic, molecular,and colloidal crystals, heliknoton crystals ex-hibit giant electrostriction andmarked symmetrytransformations under voltage changes

Three-dimensional crystals of adaptive knotsJung-Shen B. Tai and Ivan I. Smalyukh

DOI: 10.1126/science.aay1638 (6460), 1449-1453.365Science

, this issue p. 1449; see also p. 1377Sciencearrangements.topologically distinct from the host medium and diffuse and organize like colloidal particles, forming regular crystallinestructures in cholesteric liquid crystals using electric fields (see the Perspective by Alexander). The knots are polymers, proteins, DNA, and even chemical molecules. Tai and Smalyukh describe the creation of localized knottedsystems spans many disciplines, including fluid and optical vortices, Skyrmion states, liquid crystals, excitable media,

Although for some, knots are merely a frustration caused by poorly tied shoelaces, interest in knots in physicalOptical micrograph of a self-assembled lattice of knots in a chiral liquid crystal

Knot all that it seems

ARTICLE TOOLS http://science.sciencemag.org/content/365/6460/1449

MATERIALSSUPPLEMENTARY http://science.sciencemag.org/content/suppl/2019/09/25/365.6460.1449.DC1

CONTENTRELATED http://science.sciencemag.org/content/sci/365/6460/1377.full

REFERENCES

http://science.sciencemag.org/content/365/6460/1449#BIBLThis article cites 42 articles, 10 of which you can access for free

PERMISSIONS http://www.sciencemag.org/help/reprints-and-permissions

Terms of ServiceUse of this article is subject to the

is a registered trademark of AAAS.ScienceScience, 1200 New York Avenue NW, Washington, DC 20005. The title (print ISSN 0036-8075; online ISSN 1095-9203) is published by the American Association for the Advancement ofScience

Science. No claim to original U.S. Government WorksCopyright © 2019 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of

on April 3, 2021

http://science.sciencem

ag.org/D

ownloaded from

http://science.sciencemag.org/content/365/6460/1449http://science.sciencemag.org/content/suppl/2019/09/25/365.6460.1449.DC1http://science.sciencemag.org/content/sci/365/6460/1377.fullhttp://science.sciencemag.org/content/365/6460/1449#BIBLhttp://www.sciencemag.org/help/reprints-and-permissionshttp://www.sciencemag.org/about/terms-servicehttp://science.sciencemag.org/

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages false /GrayImageMinResolution 150 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages false /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects true /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/CreateJDFFile false /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > > /FormElements true /GenerateStructure false /IncludeBookmarks true /IncludeHyperlinks true /IncludeInteractive false /IncludeLayers false /IncludeProfiles true /MarksOffset 6 /MarksWeight 0.250000 /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PageMarksFile /RomanDefault /PreserveEditing true /UntaggedCMYKHandling /UseDocumentProfile /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice