Garcia-Aguilar et al. [1] have shown that the deforma-
tions of “shape-shifting droplets”, reported in a series of
experimental papers spawned by Refs. [2, 3], are consis-
tent with an elastic model. Here we show that the inter-
play between surface tension and intrinsic curvature in
this model is mathematically equivalent to a physically
very different phase-transition mechanism of the same
process described previously [4, 5]. Hence, the models
cannot distinguish between the two mechanisms, and it
is not possible to claim that one mechanism underlies the
observed phenomena without a more detailed compari-
son of the predictions of both mechanisms with experi-
ments. We suggest that the increasing number of seem-
ingly contradictory experimental results indicates that
the two systems [2, 3] are different. The observed “shape-
shifting” processes are therefore likely to be similar out-
comes of two very different physical mechanisms

Cholakova, Diana

Denkov, Nikolai

Goldstein, Raymond E

Haas, Pierre A

Smoukov, Stoyan K

Physical Review Letters

arXiv

Haas, PA

Goldstein, RE

Cholakova, D

Denkov, N

Smoukov, SK

Queen Mary Research Online

arXiv:2102.03842v1  [cond-mat.soft]  7 Feb 2021Comment on “Faceting and Flattening of Emulsion Droplets: A Mechanical Model”Pierre A. Haas,1, 2, 3, ∗ Raymond E. Goldstein,4, † Diana Cholakova,5, ‡ Nikolai Denkov,5, § and Stoyan K. Smoukov6, ¶1Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany2Max Planck Institute of Molecular Cell Biology and Genetics, Pfotenhauerstraße 108, 01307 Dresden, Germany3Center for Systems Biology Dresden, Pfotenhauerstraße 108, 01307 Dresden, Germany4Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences,University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom5Department of Chemical and Pharmaceutical Engineering, Faculty of Chemistry and Pharmacy,University of Sofia, 1 James Bourchier Avenue, 1164 Sofia, Bulgaria6School of Engineering and Materials Science, Queen Mary University of London, Mile EndRoad, London E14NS, UK(Dated: February 9, 2021)Garćıa-Aguilar et al. [1] have shown that the deforma-tions of “shape-shifting droplets”, reported in a series ofexperimental papers spawned by Refs. [2, 3], are consis-tent with an elastic model. Here we show that the inter-play between surface tension and intrinsic curvature inthis model is mathematically equivalent to a physicallyvery different phase-transition mechanism of the sameprocess described previously [4, 5]. Hence, the modelscannot distinguish between the two mechanisms, and itis not possible to claim that one mechanism underlies theobserved phenomena without a more detailed compari-son of the predictions of both mechanisms with experi-ments. We suggest that the increasing number of seem-ingly contradictory experimental results indicates thatthe two systems [2, 3] are different. The observed “shape-shifting” processes are therefore likely to be similar out-comes of two very different physical mechanisms.Using the notation of Ref. [1], we consider a faceteddroplet deforming under the interplay of surface tensionand bending elasticity, with energyE =∫∫[γ0 + 2κ(H −H0)2]dA; (1)we shall justify neglect of the stretching and gravity termsin Eq. (1) of Ref. [1] presently. If the droplet radius Ris much larger than H−10 , then (H − H0)2 ≈ H20 every-where except in a neighborhood of characteristic extentf(δ)H−10 near the facet edges, in which H ≈ H0, as inFig. 1(d) of Ref. [1]. The dimensionless f(δ) dependson the edge geometry, for example through the dihedralangle δ. With these approximations,E = γ∫∫dA− 2κH0∫f(δ) dℓ, (2a)where γ = γ0+2κH20 , as in Ref. [1], and the line integralis along the facet edges. Rescaling lengths with R, thescaled energy Ê = E/γR2 isÊ =∫∫dÂ− α∫f(δ) dℓ̂, (2b)with dimensionless tension α = 2κH0/γR. In Ref. [5]we obtained the same functional form (2b) for a phasetransition model in which deformations are driven by for-mation of a metastable rotator phase [2, 4, 5] near thedroplet edges. In that case, α = A∆µ/γR, where A isa characteristic cross-sectional area of rotator phase and∆µ is a difference of chemical potentials.To justify neglect of stretching and buoyancy energies,we consider a typical droplet radius R = 10µm [2], soR ≫ H−10 ≈ 60 nm [1] and r ≡ RH0 ≈ 170. The rel-ative importance of stretching, buoyancy, and intrinsiccurvature depends on r, the non-dimensional parametersΥ, Π defined in Ref. [1], and the non-dimensional energydifferences ∆ES , ∆EG, ∆EH computed from its Table I.With Υ ≈ 4, Π ≈ 10−8 [1], for the icosahedron-platelettransition,|∆EH | r≈9400, Υ |∆ES | r2≈35, Π |∆EG| r4≈29. (3)Hence |∆EH | r ≫ Υ |∆ES | r2,Π |∆EG| r4; the same sepa-ration holds at the sphere-icosahedron transition. Thus,from Eq. (3) of Ref. [1], intrinsic curvature swampsstretching and buoyancy, justifying (1).Estimating α reinforces the equivalence: for the elas-tic mechanism, using Fig. 2(d) of Ref. [1] to estimateΓ ≈ 0.02 at the icosahedron-platelet transition, we findα = 2(Γr)−1 ≈ 0.6; for the phase transition mechanism,A ≈ 0.3µm2 [6], ∆µ ≈ 6 · 105N/m2 [4], γ ≈ 5mN/m [7],so α ≈ 4.The calculations of Garćıa-Aguilar et al. considerstatic shapes [1], and cannot show, for example, thatan icosahedral droplet would flatten dynamically into ahexagonal platelet rather than a different, lower energyshape. Because of the model equivalence, the results ofRef. [5] showing that an icosahedral droplet can flattendynamically into a hexagonal platelet under the phase-transition mechanism also show it is possible under theelastic mechanism of Ref. [1].Experimental studies of “shape-shifting” droplets haveobtained seemingly contradictory results: surface ten-sion measurements [7, 8] differed by orders of magni-tude; cryoTEM experiments showed monolayers at thedroplet surface [9], while differential scanning calorime-try detected multilayers [6]. However, the cationic sur-factant C18TAB used in Refs. [3, 8, 9] has a relativelyhigh surface freezing temperature, while Refs. [2, 5–7]used different surfactants covering a range of freezingtemperatures. These real differences of the experimental2systems [6, 7, 10] and the corresponding and mathemati-cally equivalent phase-transition and elastic mechanismsare therefore physically different realizations of a moregeneral “shape-shifting” mechanism based on the inter-play of positive surface tension and negative edge tensionin faceted droplets.We thank L. Giomi for discussions. This workwas supported in part by the Max Planck Society(P.A.H.) and Fellowships EP/M017982/1 (R.E.G.) andEP/R028915/1 (S.K.S.) from the Engineering and Phys-ical Sciences Research Council.∗ haas@pks.mpg.de† r.e.goldstein@damtp.cam.ac.uk‡ dc@lcpe.uni-sofia.bg§ nd@lcpe.uni-sofia.bg¶ s.smoukov@qmul.ac.uk[1] I. Garćıa-Aguilar, P. Fonda, E. Sloutskin, and L. Giomi,Faceting and flattening of emulsion droplets: A mechan-ical model, Phys. Rev. Lett. 126, 038001 (2021).[2] N. Denkov, S. Tcholakova, I. Lesov, D. Cholakova, andS. K. Smoukov, Self-shaping of oil droplets via the for-mation of intermediate rotator phases upon cooling,Nature (London) 528, 392 (2015).[3] S. Guttman, Z. Sapir, M. Schultz, A. V.Butenko, B. M. Ocko, M. Deutsch, and E. Slout-skin, How faceted liquid droplets grow tails,Proc. Natl. Acad. Sci. USA 113, 493 (2016).[4] P. A. Haas, R. E. Goldstein, S. K. Smoukov,D. Cholakova, and N. Denkov, Theory of shape-shiftingdroplets, Phys. Rev. Lett. 118, 088001 (2017).[5] P. A. Haas, D. Cholakova, N. Denkov, R. E. Goldstein,and S. K. Smoukov, Shape-shifting polyhedral droplets,Phys. Rev. Research 1, 023017 (2019).[6] D. Cholakova, N. Denkov, S. Tcholakova, Z. Valkova,and S. K. Smoukov, Multilayer formation in self-shapingemulsion droplets, Langmuir 35, 5484 (2019).[7] N. Denkov, D. Cholakova, S. Tcholakova, and S. K.Smoukov, On the mechanism of drop self-shaping incooled emulsions, Langmuir 32, 7985 (2016).[8] S. Guttman, Z. Sapir, B. M. Ocko, M. Deutsch,and E. Sloutskin, Temperature-tuned facetingand shape changes in liquid alkane droplets,Langmuir 33, 1305 (2017).[9] S. Guttman, E. Kesselman, A. Jacob, O. Marin,D. Danino, M. Deutsch, and E. Sloutskin, Nanostruc-tures, faceting, and splitting in nanoliter to yoctoliterliquid droplets, Nano Lett. 19, 3161 (2019).[10] D. Cholakova and N. Denkov, Rotator phases in alkanesystems: In bulk, surface layers and micro/nano-confinements, Adv. Colloid Interface Sci. 269, 7 (2019).

English

Comment on "Faceting and Flattening of Emulsion Droplets: A Mechanical Model"

Haas, P.

Goldstein, R.

Cholakova, D.

Denkov, N.

Smoukov, S.

MPG.PuRe

Comment on “Faceting and Flattening of Emulsion Droplets: A Mechanical Model”

Comment on "Faceting and Flattening of Emulsion Droplets: A Mechanical Model".

Apollo (Cambridge)

http://hdl.handle.net/21.11116/0000-0009-A98D-A

Comment on "Faceting and Flattening of Emulsion Droplets: A Mechanical Model".

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