Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods

Abstract

<p id="d6859306e156">Under the influence of thermal forces, microscopic rod-shaped objects immersed in a fluid exhibit fluctuations: They deform mechanically. The energy provided by the thermal forces being well known, the measure of these fluctuations provides a convenient means of probing the rigidity of many biological filaments (DNA, microtubules, etc.) or synthetic microrods such as carbon nanotubes. However, only relatively flexible items could be studied by this technique: If they are too rigid, the fluctuations are too small to be detected. We substantially improved the method and measured the rigidity of rods 1,000 times stiffer than in previous studies. The method could henceforth be applied to numerous harder microscopic objects. </p><p class="first" id="d6859306e159">The rigidity of numerous biological filaments and crafted microrods has been conveniently deduced from the analysis of their thermal fluctuations. However, the difficulty of measuring nanometric displacements with an optical microscope has so far limited such studies to sufficiently flexible rods, of which the persistence length ( <span class="inline-formula"> <math id="i1" overflow="scroll"> <msub> <mi>L</mi> <mi>p</mi> </msub> </math> </span>) rarely exceeds 1 m at room temperature. Here, we demonstrate the possibility to probe 10-fold stiffer rods by a combination of superresolutive optical methods and a statistical analysis of the data based on a recent theoretical model that predicts the amplitude of the fluctuations at any location of the rod [Benetatos P, Frey E (2003) <i>Phys Rev E Stat Nonlin Soft Matter Phys</i> 67(5):051108]. Using this approach, we report measures of <span class="inline-formula"> <math id="i2" overflow="scroll"> <msub> <mi>L</mi> <mi>p</mi> </msub> </math> </span> up to 0.5 km. We obtained these measurements on recently designed superparamagnetic <span class="inline-formula"> <math id="i3" overflow="scroll"> <mo>∼</mo> </math> </span>40- <span class="inline-formula"> <math id="i4" overflow="scroll"> <mi>μ</mi> </math> </span>m-long microrods containing iron-oxide nanoparticles connected by a polymer mesh. Using their magnetic properties, we provide an alternative proof of validity of these thermal measurements: For each individual studied rod, we performed a second measure of its rigidity by deflecting it with a uniform magnetic field. The agreement between the thermal and the magnetoelastic measures was realized with more than a decade of values of <span class="inline-formula"> <math id="i5" overflow="scroll"> <msub> <mi>L</mi> <mi>p</mi> </msub> </math> </span> from 5.1 m to 129 m, corresponding to a bending modulus ranging from 2.2 to 54 (× <span class="inline-formula"> <math id="i6" overflow="scroll"> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>20</mn> </mrow> </msup> </math> </span> Jm). Despite the apparent homogeneity of the analyzed microrods, their Young modulus follows a broad distribution from 1.9 MPa to 59 MPa and up to 200 MPa, depending on the size of the nanoparticles. </p>