Phase diagram of elastic colloids

Athanasopoulou, Labrini (author), Ziherl, Primož (mentor)

Experiments have shown that at large densities, spherical polymeric nanocolloids self-organize in a number of crystal structures and quasicrystals. Although not identical, the phase sequences of suspensions of dendrimers, star polymers, and block-copolymer micelles are remarkably consistent, often featuring the face- and body-centered cubic lattices (FCC and BCC, respectively), A15, simple hexagonal (H), σ lattice and layered dodecagonal quasicrystals (DQC). The common micromechanical feature of all these nanocolloidal particles, and thus the possible origin of the similarities of their phase diagrams, is softness. So far, the idea of attributing the observed phase behavior to particle softness has been explored explicitly using either simulations at a monomer-resolved level or using effective pair interactions. Although these studies confirm that the non-close-packed structures mentioned above are indeed stable, a different type of argument may be sought within continuum theories, which dispose of most molecular-level details. In this Thesis, we consider the particles as deformable elastic spheres which interact with each other on contact. We describe their repulsive potential using two strain energy density functions from the finite deformation theory: The modified Saint-Venant–Kirchhoff and the Ogden neo–Hookean model. Using the cell approximation, we compute the deformation free energy of a set of Bravais and non-Bravais lattices up to very large deformations and we use these results to construct the phase diagram of elastic spheres with a positive Poisson ratio. We also delimit the small deformation regime, where the Hertz theory of sphere-sphere contact is valid, from the large deformation regime, where non-pairwise interactions are dominant. We show that the validity of either regime depends on the coordination number and on the Poisson ratio, and that the pairwise additivity of contact interaction typically holds only at small indentations just a little beyond the liquid–FCC transition. Furthermore, we study the elastic behavior of a single spherical polymer brush (SPB) upon diametral compression for a set of functionalities and chain lengths. We observe a universal response of the SPBs, which is rationalized using scaling arguments and interpreted in terms of two different coarse-grained models applicable far beyond the small-strain regime. At small and intermediate compressions the deformation can be accurately reproduced by modeling the brush as a liquid drop, whereas at large compressions the brush behaves as a elastic sphere. We also estimate the efective Young modulus and the Poisson ratio of the SPB.

• Language: English
• Keywords: colloids, elasticity, continuum mechanics, finite deformations, Hertz theory, pairwise additivity, many-body interactions, cell approximation, finite-element analysis, Poisson ratio, diametral compression, non-close-packed lattices, phase diagram.
• Work type: Doctoral dissertation
• Typology: 2.08 - Doctoral Dissertation
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2018
• PID: 20.500.12556/RUL-102361
• COBISS.SI-ID: 3213156
• Publication date in RUL: 23.08.2018

Secondary language

• Language: Slovenian
• Title: Fazni diagram elastičnih koloidov

Abstract

Eksperimenti so pokazali, da se pri velikih gostotah sferični polimerni nanokoloidni delci samoorganizirajo v vrsto kristalnih struktur in kvazikristalov. Čeprav ne povsem enaka, so fazna zaporedja suspenzij dendrimerov, zvezdastih polimerov in dvobločnih kopolimerov presenetljivo skladna; pogosto vsebujejo ploskovno in prostorsko centrirano kubično mrežo (znani z angleškima kraticama FCC oziroma BCC), mrežo A15, heksagonalno mrežo (H), σ mrežo in plastovite dodekagonalne kvazikristale (DQC). Skupna mikromehanična lastnost vseh omenjenih nanokoloidnih delcev in s tem možni vzrok za podobnost faznih diagramov je mehkost. Doslej so scenarij, v katerem so opaˇzeno fazno obnašanje pripisali mehkosti delcev, eksplicitno raziskali bodisi s simulacijami z monomerno ločljivostjo bodisi z efektivnimi parskimi interakcijami. Čeprav te raziskave res potrjujejo stabilnost netesnih skladov, omenjenih zgoraj, je mogoče iskati tudi razlage znotraj kontinuumskih teorij, v katerih ni večine moleku- larnih podrobnosti. V doktorskem delu obravnavamo tovrstne delce kot deformabilne elastične sfere, ki druga z drugo interagirajo ob stiku. Odbojni potencial med delci opišemo z dvema modeloma elastične energije iz domene končnih deformacij, in sicer s Saint-Venant–Kirchhoffovim in z Ogdenovim neo–Hookovim modelom. S celičnim približkom izračunamo deformacijske proste energije za več Bravaisovih in ne-Bravaisovih kristalnih skladov in na podlagi slednjih skonstruiramo fazni diagram elastičnih sfer s pozitivnim Poissonovim številom. Obenem določimo meje režima majhih deformacij, kjer velja Hertzova teorija stika dveh sfer, in meje režima velikih deformacij, kjer so dominantne večdelčne interakcije. Pokažemo, da je veljavnost obeh režimov odvisna od koordinacijskega števila in od Poissonovega števila ter da parska aditivnost kontaktnih interakcij tipično velja le pri majhnih indentacijah tik nad prehodom iz tekoče faze v mrežo FCC. Nadalje raziščemo elastično obnašanje ene same sferične polimerneščetke (SPB) ob diametralnem stisku pri vrsti funkcionalnosti in različnih dolžinah verig. Opaženi univerzalni odziv SPB pojasnimo s skalirno teorijo in interpretiramo z dvema različnima kontinuumskima modeloma, ki veljata daleč onstran režima majhnih deformacij. Pri majhnih in zmernih stiskih lahko deformacijo natančno opišemo tako, da SPB obravnavamo kot kapljico, pri velikih stiskih pa se SPB obnaša kot elastična sfera. Ocenimo tudi efektivni prožnostni modul in Poissonovo število SPB.

• Keywords: koloidi, mehanika kontinuov, elastičnost, končne deformacije, Hertzova teorija, parska aditivnost, večdelčne interakcije, celični približek, metoda končnih elementov, Poissonovo število, diametralni stisk, netesni kristalni skladi, fazni diagram.