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10 2 Probing elastic anisotropy from defect dynamics in Langmuir Monolayers region of these singularities, the molecular tilt becomes normal to the interface (Tabe et al., 1999)), and a pure bend texture around the core (see Fig. 2.8). Since the system is not chiral, equal amount of droplets with clockwise and counter-clockwise bending textures are present in the monolayer. (a) (b) θ x z φ n y Fig. 2.6. (a) Constant tilt of the amphiphilic molecules with respect to the air-water interface normal (z). The orientational order of the domains is described by the director field n, defined as the projection of the molecule on the monolayer plane (xy). (b) Director field on the monolayer plane (red) of a droplet with a positive +1 defect. Coalescence of droplets with the same chirality is hindered and seldom observed. On the other hand, droplets with opposed chirality fuse abruptly (Fig. 2.8), creating a pair of new point defects upon intersection. Fusion of droplets with the same chirality is energetically unfavorable since involves the creation of a line defect consisting of antiparallel arrangements of molecules in the contact line between droplets. The energy barrier to overcome scales then with the size of the droplet, the proportionality constant being the elastic splay constant of the material. For droplets with opposite chirality, the contact line between droplets have parallel director fields, Fig. 2.7. (a) (b) Fig. 2.7. Coalescence of two droplets of (a) opposite chirality and (b) same chirality.
2.3 Results 11 Upon droplet fusion, a pair of defects appear at the surface of the newly formed droplet. Since the total charge must be preserved inside the closed domain, we consistently assign a −1/2 charge to each boundary defect. These semi-integer defects are necessarily confined to the droplet boundary since they are not allowed inside polar nematic domains. In the simplest route to collapse, typical in the fusion of droplets of different size, one of the originally central s =+1 defects is annihilated in a two-step mechanism, Fig. 2.8A. First it migrates to merge at the boundary with one of the newly created singularities, resulting in a +1/2 boundary defect. The latter, and the remaining −1/2 boundary defect approach following an asymmetric dynamics along the contour of the fused droplet, with +1/2 defects always moving faster (see Fig. 2.8B for a representation of defect trajectories). Relaxation of the droplet shape following coalescence is much faster than defect dynamics. As a result, droplet circularity does not change measurably during the final stages of defect collapse, where a roughly constant linear velocity along the curved boundary is observed. The ratio of linear velocities in this regime measured for different surface pressures is used below to quantify the asymmetry in defect mobilities. -30 -20 -10 0 time (s) ! (C) ! ! " ! ! Fig. 2.8. (A) Experimental coalescence process of a clockwise and an anti-clockwise " bend domain at T = 32◦C and Π = 0.3mN m−1 leading to a droplet with a single s =+1 defect. BAM analyzer is set at 60◦ counterclockwise from the plane of incidence, which includes the vertical axis in the images. A sketch of the molecular field is shown below each experimental image. Merging of one of the inner s =+1 defects and one of the two −1/2 boundary defects created upon fusion (b) results in a ±1/2 pair of boundary defects that attract and annihilate (d) following the arclengths towards their meeting point shown in (B). The straight lines yield the average linear velocity in the pseudo-constant velocity regime prior to collapse. Elapsed times from panel (b) are 90 s (c) and 124.6 s (d). The ruler is 100µm long. Video image digitization and processing (Rasband, 2006) is used to set the quasi-circular droplet contour as fixed in space. L ,L + -(µm) 20 0 -20 -40 -60 -80 (B)
Page 1: Studies of dynamical phenomena in s
Page 5 and 6: Acknowledgements This thesis has be
Page 7 and 8: Contents 1 General introduction . .
Page 9: Contents v 4.6 Conclusions . . . .
Page 12 and 13: 2 1 General introduction gies. In t
Page 14 and 15: 4 1 General introduction Finally, i
Page 16 and 17: 6 2 Probing elastic anisotropy from
Page 18 and 19: 8 2 Probing elastic anisotropy from
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Page 33 and 34: 3 Self-organization and cooperativi
Page 35 and 36: 3.1 Introduction 25 Fluctuating for
Page 37 and 38: Transition rates 3.1 Introduction 2
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60 4 Membrane-cortex interactions:
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5 General conclusions In this work
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5 General conclusions 115 number of
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A Resum en català El present treba
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A.1 Auto-organització i cooperativ
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A.2 Mesura de l’anisotropia elàs
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A.3 Interaccions de membrana i còr
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A.3 Interaccions de membrana i còr
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B Résumé en Français Ce travail
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B.1 Auto-organisation et coopérati
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B.2 Mesure de anisotropie élastiqu
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B.3 Interactions de membrane et cor
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B.4 Conclusions 135 une perturbatio
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References Alberts, B., Johnson, A.
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REFERENCES 139 Evans, E. (2001). Pr
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REFERENCES 141 Kruse, K., Joanny, J
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REFERENCES 143 Sit, P., Spector, A.
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