Force-Strain Curves Of Microcapsules With Atomic Force Microscopy Eli Lansey
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Force-Strain Curves Of Microcapsules With Atomic Force Microscopy Eli Lansey, Fredy R. Zypman Yeshiva University Gerofsky Physics Center 2495 Amsterdam Avenue New York, NY 10033 USA ABSTRACT We develop an algorithm to measure elastic properties of microcapsules with Atomic Force Microscopy (AFM). The AFM is used as an indenter and presses down on a spherical microcapsule. We study the system from an atomic point of view (considering interactions between the atoms in the system via Equivalent Crystal Theory) and calculate the force produced by the system to balance the external AFM force. We plot this force as a function of the indentation depth, and from that curve we extract the interatomic parameters of ECT that are related with elastic constants. Our calculations model measurements of force-strain curves including non-linear effects. This is relevant as classical elasticity theory breaks down in the AFM indentation regime, when atomic interactions must be considered explicitly. INTRODUCTION Mechanical properties of microcapsules have been studied extensively in the last few years. First, by immersing the capsules in solution and studying the osmotically-driven swelling [1]. Second, by measuring the enlargement when filled with liquid [2]. Finally, the deformation of the microcapsules has been recorded as a function of the applied load with an Atomic Force Microscope (AFM) [3]. In that work, they obtained force-strain curves experimentally and compare them with theoretical expressions. By fitting to theoretical expressions they extract estimates for the Young’s modulus. In this paper, we develop a theoretical algorithm that includes nonlinearity, to calculate Force-Strain curves. In the next section we briefly review Equivalent Crystal Theory (ECT), which will serve as the framework in which we develop the subsequent energy calculations. There, we also describe the geometric details of the microcapsule and its possible deformations. Then, we apply it to a model system. Finally, we present conclusions.
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THEORY We consider an initially spherical shape that, upon application of external stresses deforms into an oblate spheroid (Figure 1).
Figure 1. Cross-section of a sphere of radius R as it is compressed along the z direction into a spheroid. The axes are normalized to the radius of the sphere
Points in the undeformed sphere ⎧ x = R cos θ cos ϕ ⎪ ⎨ y = R cos θ sin ϕ ⎪ z = R sin θ ⎩
(1)
are mapped into ⎧ x = a cosh u cos θ cos ϕ ⎪ ⎨ y = a cosh u cos θ sin ϕ ⎪ z = a sinh u sin θ ⎩
(2)
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π
π
, 0 ≤ ϕ ≤ 2π , where u0 determines the outer surface 2 2 of the spheroid. The distance 2a is the width of a very flat spheroid. Since the radius of the original sphere R is known, we obtain the parameter a by imposing
with 0 ≤ u ≤ u0 , −
≤θ ≤
R = 1 lim (aeu0 ) 2
(3)
a →0 u0 →∞
e u0 when u0 is large. 2 We create a mesh inside the spheroid and, at each node, locate an atom. The total energy of the system will be calculated by following the Equivalent Crystal Theory (ECT) [4]. In ECT the tot
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