Optical Halo: A Proof of Concept for a New Broadband Microrheology Tool
<p>Two equivalent representations of Jeffreys model of a viscoelastic fluid: (<b>a</b>) is made of a dashpot (of viscosity <math display="inline"><semantics> <msub> <mi>η</mi> <mn>1</mn> </msub> </semantics></math>) connected in series with a Kelvin–Voight element, which is made of a dashpot (of viscosity <math display="inline"><semantics> <msub> <mi>η</mi> <mn>2</mn> </msub> </semantics></math>) and a spring (of modulus <span class="html-italic">G</span>) placed in parallel, whereas (<b>b</b>) is made of a dashpot (of viscosity <math display="inline"><semantics> <msub> <mi>η</mi> <mn>1</mn> </msub> </semantics></math>) connected i parallel with a Maxwell element, which is made of a dashpot (of viscosity <math display="inline"><semantics> <msub> <mi>η</mi> <mn>2</mn> </msub> </semantics></math>) and a spring (of modulus <span class="html-italic">G</span>) placed in series. Both models are connected to a material point of mass <span class="html-italic">m</span>, whose contribution to the dynamics of the system is neglected in this work.</p> "> Figure 2
<p>(<b>a</b>) A proposed experimental configuration of a ring-based optical trap. The setup is based on the configurations reported by Shao et al. [<a href="#B58-micromachines-15-00889" class="html-bibr">58</a>], where an axicon is used to create a smooth optical ring. This can range from a simple fixed design to a design that allows the size of the ring traps to be adjusted [<a href="#B59-micromachines-15-00889" class="html-bibr">59</a>]. (<b>b</b>) Geometrical representation of the torus-shaped optical halo. The torus has major radius <span class="html-italic">R</span> and minor radius <span class="html-italic">b</span> which are related to the stiffness of the halo in the radial direction by <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>r</mi> </msub> <mo>=</mo> <msub> <mi>k</mi> <mi>B</mi> </msub> <mi>T</mi> <mo>/</mo> <msup> <mi>b</mi> <mn>2</mn> </msup> </mrow> </semantics></math>. The stiffness of the optical halo in the <span class="html-italic">z</span> direction is <math display="inline"><semantics> <msub> <mi>κ</mi> <mi>z</mi> </msub> </semantics></math>.</p> "> Figure 3
<p>(<b>a</b>–<b>d</b>) Trajectory of a colloidal particle suspended in a Newtonian fluid and constrained by a toroidal optical trap with major radius <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>R</mi> <mo>ˇ</mo> </mover> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, small radius <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>b</mi> <mo>ˇ</mo> </mover> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </semantics></math>. (<b>e</b>–<b>h</b>) Similar simulation conditions to (<b>a</b>–<b>d</b>), but exploring the effects of varying <math display="inline"><semantics> <mover accent="true"> <mi>R</mi> <mo>ˇ</mo> </mover> </semantics></math> on the MSD in (<b>e</b>), and on the normalised position autocorrelation function in the radial <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> in (<b>g</b>). The inset in (<b>g</b>) shows the steady-state value of <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>r</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> as a function of <math display="inline"><semantics> <mover accent="true"> <mi>R</mi> <mo>ˇ</mo> </mover> </semantics></math>. (<b>f</b>,<b>h</b>) show the effects of varying the ratio <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> </mrow> </semantics></math> on the MSD in (<b>f</b>), and on the normalised position autocorrelation function <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, both only in the axial direction. The insets in (<b>f</b>,<b>h</b>) show the master curves for MSD<sub><span class="html-italic">z</span></sub> and <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> when the same data shown in the main are plotted against <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>z</mi> </msub> <mi>t</mi> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> </mrow> </semantics></math>, respectively, and the MSD<sub><span class="html-italic">z</span></sub> is normalised by the variance of the optical trap in the <span class="html-italic">z</span> direction [<a href="#B26-micromachines-15-00889" class="html-bibr">26</a>].</p> "> Figure 4
<p>The mean squared displacement (<b>a</b>,<b>c</b>) and the normalized position autocorrelation function (<b>b</b>,<b>d</b>), evaluated along the three main directions <span class="html-italic">r</span>, <span class="html-italic">z</span>, and <math display="inline"><semantics> <mi>θ</mi> </semantics></math> of a toroidal optical trap from Brownian dynamics simulations of an ensemble of 1000 particles on a torus with <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>R</mi> <mo>ˇ</mo> </mover> <mo>=</mo> <mn>150</mn> </mrow> </semantics></math>, a ratio of axial to radial stiffness of <math display="inline"><semantics> <mrow> <msub> <mi>κ</mi> <mi>z</mi> </msub> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math>; these are parametric (colour coded) using <math display="inline"><semantics> <mi>De</mi> </semantics></math>, as depicted in the legend. The simulations were performed by assuming two different Jeffreys fluids: (i) a fluid with an elastic component stiffer than that of the optical tweezers, i.e., <math display="inline"><semantics> <mrow> <msup> <mi>κ</mi> <mo>∗</mo> </msup> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>16</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) and (ii) a fluid with an elastic component softer than that of the optical tweezers, i.e., <math display="inline"><semantics> <mrow> <msup> <mi>κ</mi> <mo>∗</mo> </msup> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>). Note that the normalized position autocorrelation function for the azimuthal component is not reported in order to maintain clarity in the diagrams. This is because the data would consistently hover near a value of unity throughout almost the entire time window, falling to zero only at large lag times.</p> "> Figure 5
<p>The mean squared displacement (<b>a</b>,<b>c</b>) and the normalised position autocorrelation function (<b>b</b>,<b>d</b>), evaluated along the three main directions <span class="html-italic">r</span>, <span class="html-italic">z</span> and <math display="inline"><semantics> <mi>θ</mi> </semantics></math> of a toroidal optical trap, from Brownian dynamics simulations of an ensemble of 1000 particles on a torus with <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>R</mi> <mo>ˇ</mo> </mover> <mo>=</mo> <mn>150</mn> </mrow> </semantics></math>. The outcomes are parametric (colour coded) using the ratio of axial to radial stiffness <math display="inline"><semantics> <mrow> <msup> <mi>κ</mi> <mo>∗</mo> </msup> <mo>/</mo> <msub> <mi>κ</mi> <mi>r</mi> </msub> </mrow> </semantics></math> as depicted in the legend. They show two cases: <math display="inline"><semantics> <mrow> <mi>De</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> (<b>a</b>,<b>b</b>) and <math display="inline"><semantics> <mrow> <mi>De</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (<b>c</b>,<b>d</b>). Note that the normalised position autocorrelation function for the azimuthal component is not reported in order to maintain clarity in the diagrams. This is because the data would consistently hover near a value of unity throughout almost the entire time window, falling to zero only at large lag times.</p> "> Figure 6
<p>(<b>a</b>) The MSD along the three main directions <span class="html-italic">r</span>, <span class="html-italic">z</span>, and <math display="inline"><semantics> <mi>θ</mi> </semantics></math>. (<b>b</b>) The viscoelastic moduli <math display="inline"><semantics> <mrow> <msup> <mi>G</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>ω</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msup> <mi>G</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>ω</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> derived from each of the three MSD functions shown on the left. The MSD data have been obtained from Brownian dynamics simulations of an ensemble of 1000 particles on a torus with <math display="inline"><semantics> <mrow> <mover accent="true"> <mi>R</mi> <mo>ˇ</mo> </mover> <mo>=</mo> <mn>150</mn> </mrow> </semantics></math> and by assuming a value of the fluid’s plateau modulus of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> Pa, a relaxation time of <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math> s, and the solvent to have a viscosity of <math display="inline"><semantics> <mrow> <mn>0.001</mn> </mrow> </semantics></math> Pa·s (e.g., water).</p> ">
Abstract
:1. Introduction
2. Theoretical Background
2.1. Linear Rheology
2.2. Passive Microrheology
2.3. Range of Accessible Dynamics and Measurable Viscosities
2.4. The Link between Bulk and Micro-Rheology
2.5. Hybrid Microrheology with Optical Tweezers
2.6. Jeffreys Model
2.7. Generalized Langevin Equation for an Optically Trapped Particle Moving in a Jeffreys Medium
2.8. A Possible Experimental Configuration of an Optical Halo
2.9. Geometrical Representation of an Optical Halo
2.10. Overdamped Particle Moving in a Newtonian Fluid with a Toroidal Optical Trap
3. Results and Discussion
3.1. Overdamped Particle Moving in a Jeffreys Fluid with Toroidal Optical Trap
3.2. Simulation of a Biologically Relevant System with Realistic Numbers
4. Conclusions
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Ramírez, J.; Gibson, G.M.; Tassieri, M. Optical Halo: A Proof of Concept for a New Broadband Microrheology Tool. Micromachines 2024, 15, 889. https://doi.org/10.3390/mi15070889
Ramírez J, Gibson GM, Tassieri M. Optical Halo: A Proof of Concept for a New Broadband Microrheology Tool. Micromachines. 2024; 15(7):889. https://doi.org/10.3390/mi15070889
Chicago/Turabian StyleRamírez, Jorge, Graham M. Gibson, and Manlio Tassieri. 2024. "Optical Halo: A Proof of Concept for a New Broadband Microrheology Tool" Micromachines 15, no. 7: 889. https://doi.org/10.3390/mi15070889