Institute of High Performance

Transcription

1 Insttute of Hgh Performance Computng (IHPC) Dr. Zshun Lu (Z.S. Lu) Aprl 5, 0 XJTU

2 Sprng Workshop on Nonlnear Mechancs -XJTU Usng Polymerc Gel Theory to Explan Plant Phyllotaxs Z.S. Lu IHPC, Sngapore Natonal Unversty of Sngapore Collaborators: W. Hong: Iowa State t Unversty t Z.G. Suo: Harvard Unversty S. Swaddwudhpong: Natonal Unversty of Sngapore

3 Usng Polymerc Gel Theory to Explan Plant Phyllotaxs OUTLINE Introducton Gel theores Modelng & Smulaton of Gel Deformaton Governng Equatons of Flm Gel Wrnklng Botancal plants modelng Future Study & Dscusson

4 Introducton Cactus Pumpkn Fascnatng complex patterns and shapes & nterestng undulatng surface morphology In plants, nterestng undulatng surface morphology s often observed across a large scale Phyllotaxs: (namely the arrangement of phylla) has ntrgued natural scentsts for over 400 years

5 Introducton Queston: What are the physcal mechansms of ths natural structure deformaton? How to explan the mechansm of flower openng and closure? How to explan the morphogeness of the natural growth of some leaves, flowers and vescles? The challenge: How to provde a ratonal explanaton for the formaton of such patterns, a mechansm or combnaton of mechansms whch capture natural phenomena Many natural tssues of plants and anmals are to some extent polymerc gels. Polymerc gel theory may be used to explan the formaton of phyllotactc patterns at plant Varous phyllotactc lattce and tlng patterns Newell et al, J. Theoy. Bol. 5, 008

6 Introducton What s Gel? Gel= elastomer + solvent Elastcty: long polymers are cross-lnked by strong bonds. Fludty: polymers and solvent molecules aggregate by weak bonds. solvent reversble + Stmul: Physcal --T, electrcal, lght, pressure Chemcal--pH, ons.. polymers gel Jelly ~ 0000% volumetrc change Polyelectrolyte gels

7 Applcatons of Gels Introducton Soft robots; Drug delvery; Tssue engneerng; Water treatment; Sealng n ol wells; Sensor & actuator Repar damaged heart muscle Contact lenses Wchtere, Lm, Nature 85, 7 (960) Tssues, natural or engneered Cartlage & Menscus Nanogel & Drug delvery Uljn et al. Materals Today 0, 40 (007)

8 Introducton Applcatons of Gels Sealng n ol wells Shell, 003 Ion exchange for water treatment Beebe, Moore, Bauer, Yu, Lu, Devadoss, Jo, Nature 404, 588 (000) Valves n fludcs Hydrogel gate valve

9 Gellng System n Ol Well Introducton. Watered-out zone separated from an ol zone by an mpermeable shale barrer-the soluton s to cement n bottom zone.. Shale doesn't reach to producton well. Soluton s nject gel nto the lower zone. 3 Watered-out hgh permeablty 3. Watered out hgh permeablty zone sandwched between two ol zones-the soluton s to solate the zone and nject gel

10 Introducton Challenges of Gel Applcatons. A lack of understandng of the relatonshp between gel composton and response knetcs. The majorty of prevous research efforts of gel are expermental-based 3. More complex shapes are requred and the accurate dmensonal measurements of ther volume transton behavors are not convenent n expermental analyss 4. A lack of completed analytcal theory for Gel 5 Th d f l f b d l d l ll h b 5. The predcton of gel performance by modelng and smulaton wll thus be crtcal for understandng the characterstcs of gel

11 Theores of Gel Deformaton THB Theory: Tanaka, Hocker, and Benedek, J. Chem. Phys. (973) Multphasc theory Bphasc theory: Bowen, Int. J. Eng. Sc. (980) Trphasc model: La, Hou, Mow, J. Bomech. Eng. Trans. (99) Mxture theory: Sh, Rajagopal, Wneman, Int. J. Eng. Sc. (98) Lmtatons No nstant response to loads Lnear theory, small deformaton Basc prncples are unclear, hard to extend Monophasc theory: Regard a gel as a sngle phase Start wth thermodynamcs Analogy to sold mechancs Hong, Zhao, Zhou, Suo, JMPS (007) Vew solvent and solute as two or three phases Unclear physcal pcture. Unmeasurable quanttes. X K W F B δxdv W T N F K δxda W μ δcdv 0 C Gbbs, The Scentfc Papers of J. Wllard Gbbs, 84, 0, 5 (878): Derve equlbrum theory from thermodynamcs. Bot, JAP, 55 (94): Use Darcy s law to model mgraton. Hong, Zhao, Zhou, Suo, JMPS (008) Hong, Lu, Suo, IJSS (009) Lu, Hong, Suo, Somsak, Zhang, COMMAT, (00) Lu, Somsak, Cu, Hong, Suo, IJAM (0)

12 3D nhomogeneous felds: x X F X C X,t W F,C Theory of Gel Deformaton Deformaton gradent Concentraton of solvent Chemcal potental In equlbrum, the change n the free energy of the gel, assocated wth arbtrary change n dsplacement feld and concentraton feld, equals the work done by the mechancal loads and the envronment A gel s n contact wth a solvent of a fxed chemcal potental, and subject to a mechancal load and geometrc constrant Free-energy of system changes by G WdV B x dv Free-energy densty changes Tx da CdV geometrc constrant Gel gel solvent μ δw F, C W F, C δf δ C W, F C W F,CC mechancal P load System = gel + load+ solvent Hong, Zhao, Zhou, Suo, JMPS (008)

13 The chemcal potental defnton: S, V, N FT, V, N GT, p, N U N N N Introducng the energy, entropy and volume per molecular u, s, and v u Ts pv st vp Chemcal Potental The chemcal potental of the solvent ( ) n general depends on the temperature (T) and pressure (p) μ μ( T, p) μ In equlbrum: the chemcal potental nsde the gel be a constant and equal to the chemcal potental of the external solvent For constant Temperature ˆ( p, T ) v( T )( p ) ˆ 0 p0 ˆ ˆ( p0, T ) kb T log( p / p0 ) μ μˆ for ncompressble lqud phase for deal gas v( p p0) ˆ kbt log( p / p 0 ) f p p0 f p p 0 where p 0 s the equlbrum vapor pressure and depends on the temperature, At the equlbrum vapor pressure ( p p 0 ), the external chemcal potental: In a vacuum (p = 0): μˆ μˆ 0

14 Theory of Gel Deformaton In equlbrum condton 0 CdV da x T dv x B WdV G C δ C C W δf F C W W δ F, F, Free-energy densty changes da x N F W T x dv B F W X K Applyng the dvergence theorem A G G d dv V A gel s n contact wth a solvent of a fxed chemcal potental, and subject to a mechancal load and geometrc constrant 0 CdV C W F F X K K geometrc constrant solvent μ 0, K B F C W X F K T N F C W, F gel C W F Gel C C W F, P mechancal load System = gel + load+ solvent W,C F Hong, Zhao, Zhou, Suo, JMPS (008)

15 Theory of Gel Deformaton Introduce a new free-energy functon Ŵ by a Legendre transform Wˆ F, μ W F, C μc New Equlbrum condton: δwdv ˆ B δx dv T δx da Defne nomnal stress as the work conjugate to the deformaton gradent The change n free energy δw δw W F s s W F, C F F, C W F, C δf δc δf μδc C δw ˆ s δf Cδμ W ˆ F, Wˆ F, μ s F C μ The true stress σ j F jk s det F F jk det F W F F, C F jk Wˆ F, μ det F F

16 Molecular Incompressblty n Gels + = V dry + V sol = V gel vc detf v volume per solvent molecule Assumptons: Indvdual solvent molecule and polymer are ncompressble. Gel has no vods. Ths molecular ncompressblty condton can be enforced as a constrant by ntroducng a Lagrange multpler Π W F, C W F, C vc det F

17 Theory of Gel Deformaton Flory-Rehner Free Energy n Gels Swellng ncreases entropy by mxng solvent and polymers, but decreases entropy by straghtenng the polymers. Followng Flory and Rehner, the free energy densty can be wrtten as Free-energy functon W F, C W FW C s m s W F Free energy of stretchng W C W s F NkT F F 3 logdet F kt χ Free energy of mxng W m C vc log v vc vc Flory, Rehner, J. Chem. Phys.,, 5 (943) Flory-Huggns polymer theory

18 Theory of Gel Deformaton Flory-Rehner Free Energy n Gels Consder the molecular ncompressblty condton as a constrant by ntroducng a Lagrange multpler Π Followng Flory and Rehner, the free energy densty can be wrtten as Free-energy functon W F, C W F W C Π vc detf s m (F, C) s W F W F s H det F W Wm v C C Consder other energy terms Open Space. s W W s H det F F F F W W m v C C C

19 Theory of Gel Deformaton Free energy W F, C W F W C Π vc detf s m kt W F NkTF F 3 logdet F χ C vc log s W m v vc vc s W F s ΠH detf Assume the prncpal nomnal stresses are s s NkT( NkT ( 3 ) 3 ) 3 Wm μ Πv C s s s s vc kt log vc vc ( vc ) s3 s 33 v s3 NkT( 3 ) Consttutve equatons: vs Nv 3 log 3 kt 3 3 kt vs kt vs3 kt Nv 3 log kt Nv 3 3 log kt 3 F 3

20 Theory of Gel Deformaton General Form of Equatons of State n Gels Enforce molecular ncompressblty as a constrant by ntroducng a Lagrange multpler P s W F, C H detf W F, C F Use the Flory-Rehner free energy C v s NkT F H H det F vc kt log vc vc vc v Consttutve Equatons vs kt Nv det F det F F H det F log det F H kt

21 FEM Implementaton for Deformaton n Gels Consttutve equatons (Equaton of State) vs kt Nv det F det F F H det F log det F H We select new reference state of the gel by free swellng λ λ λ 3 λ 0 λ0 F FF 0 F 0 λ0 I ' F ' F ' J' det F' λ 0 kt 3 J χ 3 log 6log 0 0 log μ NkT λ I J λ λ J λ J v J λ 0 λ0j v, μ ˆ 3 0 W F Consttutve equatons (Equaton of state) vs kt Nv 3 χ μ λ F H λ F log det F H 0 0 kt det g 3 λ 3 6 det F λ λ det F kt ABAQUS UMAT ABAQUS UHYPER Hong, Lu, Suo: IJSS (009) Lu, Hong, Suo, Somsak, Zhang, Com. Mat. Sc. (00)

22 Neutral Gel Theory: xx F Deformaton gradent X X Concentraton of solvent Chemcal potental molecule C,t The change of free-energy of system G X K WdV W F B B x dv x dv W T F Theory of Gel Tx da N K vc detf ncompressble constrant v volume per solvent x da CdV W C CdV 0 geometrc constrant W F,C gel Gel P solvent μ mechancal load System = gel + load+ solvent Hong, Zhao, Zhou, Suo, JMPS (007) X K W F, C F W F, C F N K B T 0 W F, C C Defne nomnal stress s W F, C F δw W F F, C W F, C δf δc C Flory, Rehner, J. Chem. Phys., (943) W vs kt F, C W FW C vc det F net sol Consttutve Equatons Nv det F det F kt F H detflog det F H W net F NkTF F 3 logdet F kt W C vc log sol v vc vc vc detf Hong, Lu, Suo, IJSS, (009)

23 Analytcal soluton of -D beam gel bucklng Incremental Modulus of the Gel For column or bean case vs kt Nv log kt ~ E s / s / v kt log Nv kt ~ E Nv 0 0. / kt 0.00 Incremental modulus of gel varyng wth stretch for a) varous ntal chemcal potentals b) varous dmensonless measures of the enthalpy of mxng Lu, Somsak, Cu, Hong, Suo, Zhang, IJAM, (0)

24 Analytcal soluton of -D beam gel bucklng Crtcal values of beam gel under bucklng the crtcal stress of a column wth hnged ends (Tmoshenko and Gere) ( ) s cr ~ E ( l / r) kt Nv v NkT log kt ( l / r) Dfferent stress curves varyng wth stretch Stablty dagram showng normalzed crtcal stress of gel beam varyng wth ts slenderness rato of beam for dfferent chemcal potentals Lu, Somsak, Cu, Hong, Suo, Zhang, IJAM, (0)

25 Thn Flm Gel wth Wrnkle Mode In-plane Incremental Modulus of thn flm gel v kt ~ E 3 Nv v kt ~ E Nv In plane ncremental modulus of thn flm gel varyng wth chemcal potental In-plane ncremental modulus of thn flm gel varyng wth chemcal potental for a) dfferent ntal chemcal potentals, b) dfferent dmensonless measures of the enthalpy of mxng

26 Thn Flm Gel wth Wrnkle Mode The bucklng stress and wave length of thn flm gel The bucklng stress and wave length of thn flm gel Schematcs of thn gel flm before and after buckng Wrnklng of a solgel-derved thn flm bq w N w D 4 ) ( l x w w m sn l x w l A q m sn A bh D h l A l h bh D bh h l bh 0 To mnmze 3 / ) ( 4 A NkT h l cr cr / / ) ( 4 3 ) ( 4 3 ) ( A NkT A A NkT NkT cr Lu, Somsak, Cu, Hong, Suo, Zhang, IJAM, (0)

27 Thn Flm Gel wth Wrnkle Mode The bucklng stress and wave length of thn flm gel Normalzed stresses varyng wth chemcal potental for dfferent gel flm and substrate stffness factor of NkT/Es Normalzed crtcal stress, normalzed wavelength and crtcal chemcal potental of thn flm gel varyng wth gel flm and substrate stffness factor of NkT/Es Lu, Somsak, Cu, Hong, Suo, Zhang, IJAM, (0)

28 PDMS membrane gel on substrate n swellng Swellng-nduced bfurcaton Zhang Y et. al. Nano Letter 008 The pattern of PDMS membrane gel wth a square lattce of holes before and after swellng A square lattce of cylndrcal holes bfurcates nto a perodc structure of ellpses wth alternatng drectons Hong, Lu, Suo, IJSS, (009)

29 Rectangular Strp Membrane Gel Mora & Boudaoud (006) Fx one edge Fx two edges 30 λ /H w/h The nstablty wavelength λ as a functon of the wdth w of the swollen rectangular membrane gel strp for constranng one edge and two edges Lu, Hong, Suo, Somsak, Zhang, Com. Mat. Sc. (00)

30 How To Explan Varous Phyllotactc Patterns Al leaf growng and ddryng smulaton The deformaton pattern of a leaf when dryng by usng membrane gel

31 Swellng of Corona membrane Gel n Equlbrum R / H 4 R / H 6 R / H 8 The bucklng shapes of corona membrane gel for dfferent nner radus (the rato of outer radus 0 and thckness R / H 0, ntal chemcal potental, ) 0 kt

32 A leaf growng and dryng smulaton The deformaton pattern of a leaf s dryng by usng membrane gel deswellng The deformaton pattern of a leaf s growng by usng membrane gel swellng

33 How To Explan Varous Phyllotactc Patterns Al leaf growng and ddryng smulaton Epsca The deformaton pattern of a leaf when dryng by usng membrane gel

34 A leaf growng and dryng smulaton The deformaton pattern of a leaf when dryng by usng membrane gel ph change

35 Phyllotaxs: A Swellng of Sphercal Shape Shell Gel How to provde a ratonal explanaton for the formaton of frut patterns, a mechansm or combnaton of mechansms whch capture natural phenomena? A swellng of spherodal thn layer of gel to represent the growng process of plant (ph effect)

36 How To Explan Varous Phyllotactc Patterns A Swellng of Sphercal Annulus Gel Shell z Shell (stffer gel) Corpus (softer gel) x Schematc of the geometry of spherod model Phyllotactc pattern lattce and tlng pattern of plant A swellng of spherodal thn flm/substrate system of gel to represent the growng process of plants

37 How To explan Varous Phyllotactc Patterns Cucumber growng can be smulated by gel swellng wth changng the chemcal potental

38 How To explan Varous Phyllotactc Patterns Apple growng can be smulated by gel swellng wth changng the chemcal potental

39 Concludng remarks & Future Work Modelng and smulaton results are nsprng Some natural phenomena can be explaned by gel theory Acdc Leaves (actual dmenson and components) Acdc fruts (actual dmenson and components) Natural or engneered tssues; bo-materal Varous phyllotactc lattce and tlng patterns Newell et al, J. Theoy. Bol. 5, 008

40 Thank You