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

Significance

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.

Abstract

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 ($L_p$) 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) Phys Rev E Stat Nonlin Soft Matter Phys 67(5):051108]. Using this approach, we report measures of $L_p$ up to 0.5 km. We obtained these measurements on recently designed superparamagnetic $\sim$40-$\mu$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 $L_p$ from 5.1 m to 129 m, corresponding to a bending modulus ranging from 2.2 to 54 (×${10}^{-20}$ 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.

Initiated with the famous estimation of the Avogadro number by Jean Perrin (1), the method that consists of measuring the dispersion of a thermostated system to assess its properties has brought a plethora of scientific successes. For instance, the tracking of Brownian or more complex diffusive particles immersed in a fluid opened up a new era in microrheology (2, 3). The calibration of numerous microtools such as optical tweezers (4), glass fibers (5), or atomic force microscope (AFM) cantilevers (6) also relies on the assessment of thermal motions. The principle of the method was also adapted to measure the intrinsic mechanical properties of nano- and microrods.

This type of stiffness measurement, which is based on the monitoring of the rod-shape deformations caused by the thermal forces, was initially performed on biological polymers (7) and biological filaments of the cytoskeleton including actin (8–10), microtubules (9, 11–13), intermediate (14), and other filaments (15–17). It allowed in vivo measurements (13), although most studies were performed in vitro, in various boundary conditions (14). The method is also used in nanosciences, essentially to probe carbon nanotubes (18–20) that, despite their extremely large elastic modulus, have a sufficiently small radius to exhibit detectable fluctuations. Numerous other methods exist to measure the rod stiffness: Nanoindentation (21), AFM measurements (21, 22), and other techniques based on an external solicitation (21, 23) are commonly performed. However, they require relatively complex equipment. In contrast, the thermal method is neither invasive nor disruptive: It consists of acquiring a set of pictures of the rod, an observation that most often can be performed with an optical microscope only. For this purpose, superresolutive techniques have recently emerged and currently allow the optical observation of nanometric fluctuations, beyond the optical diffraction limit (24, 25).

If it is possible to take the anisotropic internal structure (such as the spontaneous curvature) of the rods into account (26), most analyses rely on the simple isotropic worm-like chain (WLC) model (27): For more than a decade, according to this model, the persistence length $L_p=\frac{C}{k_{B}T}$ (where $C$ is the bending modulus and $k_{B}T$ the thermal energy) was deduced from the experimental data after a decomposition of the filament shape in bending eigenmodes (Fig. S1), each of them being an independent probe of the filament flexural properties (10, 11, 13). However, this technique rapidly reaches its limit for stiff rods because the mode amplitude decays like $n^{-2}$ ($n$ being the mode number) and higher modes become quickly indistinguishable from the noise of the detection system (SI Text, Eigenmode Analysis of the Rod Fluctuations). To circumvent this difficulty, the bending modulus of very rigid objects such as carbon nanotubes (18) or hemoglobin fibers (15) was derived from the fluctuations at a single rod location (tip or middle point), whose variance in real space was estimated from a summation over all modes. Later, Benetatos and Frey (28) renewed the theory by establishing the analogy between the WLC model and the quantum mechanics description of a rigid rotator. Doing so, they determined the probability distribution of the transverse displacement $y_f(l)$ at any curvilinear abscissa $l$ along the rod:

$$p[(y_f(l)]\propto \exp [-\frac{3L_p}{2l^3}{y}_f^2(l)].$$ [1]

Fig. S1.

First eigenmodes of a cantilevered rod fluctuating in a thermal bath. Shown are functions $W_n(l/L)$ described in SI Text, Eigenmode Analysis of the Rod Fluctuations, where $l$ is the curvilinear abscissa and $L$ the contour length of the rod.

This theory simplified the analysis that can thus be performed in direct space at any location along the filament (12).

In this work, we demonstrate the advantage of their theoretical model: We provide a statistical analysis showing that the extensive use of data collected on the entire rod length improves both the accuracy and the precision (by a factor of $\sim$10) of the technique, thus increasing by the same factor the range of measurable stiffness. However, the work presented here is primarily experimental: Using an optical microscope only, we successfully measured persistence lengths typically 1,000 times larger than previously reported (16). Our study was performed on synthetic microrods made of iron-oxide nanoparticles (with a mean diameter of 8 nm or 13 nm) interconnected by a polymer (29, 30). These superparamagnetic rods with a radius $r\sim \mathrm{\hspace{0.17em}350}$ nm and a length $L\sim \mathrm{\hspace{0.17em}20}-70$ $\mu$m were recently used to illustrate the magnetoelastic instability (31, 32) and served as a passive (3) or magnetically driven (33–35) rheological probe to study Newtonian (33) and complex fluids (3, 34, 35), including the cytoplasm of living cells (36). We reached the required precision to detect fluctuations as small as 4 nm by using a standard superresolutive method to locate the rod centerline (11, 13) and by developing an algorithm to track noisy motions of the microscope stage (25). We thus report measures of $L_p$ ranging from 5.1 m to 631 m, corresponding to a bending modulus $C=k_{B}T{L}_p=\frac{\pi r^{4}Y}{4}$ of $(2.2-261)\times {10}^{-20}$ Jm and a Young modulus Y of 1.9–198 MPa. These measurements also provide the most complete experimental check of the theory: We find the expected power law ${\sigma }_f^2(l)\propto l^3$, where ${\sigma }_f^2(l)$ designates the variance of the $y_f$ (Eq. 1). Moreover, we confirm our result by a second type of measure of C that consists of studying the deformation of the rods submitted to a uniform magnetic field. Altogether, we demonstrate the possibility of characterizing rods with an extremely large $L_p/L$ ratio up to ${10}^7$. Our method could henceforth be applied to numerous stiffer items such as bundles of biological filaments (37) or microwires (38).

Notations and Error Calculations.

The raw data collected in our experiments are $S$ individual images of a rod of contour length $L$ set in a cantilevered configuration in an aqueous thermal bath at temperature $T$. Two consecutive pictures are separated in time by an interval $\mathrm{\Delta }t$. These $S$ configurations (or microstates) constitute a particular sampling of the canonic Gibbs ensemble. As described in Materials and Methods, each picture $i$ is independently acquired with a CCD camera, and for each of them, the rod shape is also obtained through $N\sim 1,000$ independent measures performed at the curvilinear abscissa $l$ ($l=\mathrm{\hspace{0.17em}0}$ at the rod anchorage site) regularly chosen along the rod ($l=nL/N$, with $1\le n\le N$). For $1\le i\le S$ and $0\le l\le L$, the rod shape of each microstate is thus described by the transverse measure $y_m^i(l)=y_p(l)+y_f^i(l)+y_e^i(l),$ where $y_p(l)$ describes the profile of the rod at $T$ = 0 K, $y_f^i(l)$ designates the gap produced by the fluctuation, and $y_e^i(l)$ is the error committed during the optical measure of the rod shape. The rod profile $y_p(l)$, independent of $i$, may also account for a systematic bias in the detection of the rod deflection (for instance, caused by fluctuations of the nanoparticle concentration) and thus be indistinguishable from the actual rod shape. According to Eq. 1 the $y_f^i$ follow a centered statistical Gaussian distribution with variance ${\sigma }_f^2(l)=\frac{l^3}{3L_p}$. We also assume that $y_e^i(l)$ is normally distributed with a variance ${\sigma }_e^2$ independent of $l$ because the measurement is identical whatever its position on the rod. With this condition we may write $\overline{y_e}=0$ and $\overline{y_m}(l)=y_p(l)$, where $\overline{x}$ designates the expectation of $x$ that must be distinguished from its experimental estimation, given by the mean of the measurement sample $<x>=\frac{1}{S}{\sum }_i{x}^i$. Under the above conditions the set ${\{y_m^i(l)\}}_{1\le i\le S}$ is a realization of a normal distribution $\mathcal{N}[y_p(l),{\sigma }_m^2(l)]$ with variance

$${\sigma }_m^2(l)={\sigma }_f^2(l)+{\sigma }_e^2=\frac{l^3}{3L_p}+{\sigma }_e^2,$$ [2]

which is evaluated by the unbiased estimator $\widehat{{\sigma }_m^2(l)}=\frac{1}{S-1}\sum _{i=1}^S{\left[y_m^i(l)-<y_m(l)>\right]}^2.$

The difference between the estimator $\widehat{{\sigma }_m^2(l)}$ and the actual variance ${\sigma }_m^2(l)$ yields the error (or uncertainty) on the measure of $L_p$. It is estimated by calculating $Var\left[\widehat{{\sigma }_m^2(l)}\right]$ as detailed in SI Text, Uncertainty on the Experimental Measure of the Persistence Length.

Taking into account that the rod configurations are possibly correlated, which happens if the interval $\mathrm{\Delta }t$ between two consecutive images is shorter than the rod relaxation time $\tau$, the sampling number $S$ is reduced to an effective set of $S\prime \simeq S/f$ independent microstates [where $f=\frac{\tau }{\mathrm{\Delta }t}$ (39)]. With these considerations, the relative uncertainty of a single measure performed at the rod tip is

$$u_L=\frac{\delta L_p(L)}{L_p}=\sqrt{\frac{2}{S\prime }}\left[1+\frac{1}{\sqrt{f}}\frac{3{\sigma }_e^2{L}_p}{L^3}\right],$$ [3]

where $\sqrt{(2/\text{𝑆}\prime )}$ is the “entropic relative error,” owing to the finite size of the sample set, and the second term is the optical relative error. In addition to the uncertainty, such a single-point analysis also introduces an inaccuracy in the measure. With no knowledge of ${\sigma }_e^2$, $L_p$ is computed from ${\sigma }_m^2(L)$ instead of ${\sigma }_f^2(L)$. This yields a systematic relative error $\frac{{\sigma }_e^2}{{\sigma }_f^2}$ (Eq. 2). Hence, the optical error always tends to overestimate the flexibility of the rod.

In our experiments, we take advantage of the $N$ measures along the rod and retrieve the value of $L_p$ by performing a linear regression: Following Eq. 2, $\widehat{{\sigma }_m^2(l)}$ is expected to be linear with $l^3$, thus allowing the determination of the slope $\frac{1}{3L_p}$ and the intersect ${\sigma }_e^2$ by a least-squares fit of the data. With this treatment, a conservative estimation of the relative uncertainty becomes

$$u_{rod}=\sqrt{\frac{2}{S\prime }}\left[1+\frac{5}{\sqrt{2Nf}}\frac{3{\sigma }_e^2{L}_p}{L^3}\right],$$ [4]

which we confirmed by bootstrap calculations (40) (Materials and Methods and Fig. S2).

Fig. S2.

Bootstrap calculation of the persistence length uncertainty. (A) Histogram of $L_p$ computed from ${10}^4$ replicate samples drawn (by a block of seven successive pictures accounting for the time correlation) from the original rod configuration set (original measure and theoretical uncertainty $L_p=38.2\pm 7.1$ m). Mean of the distribution ${<L_p>}_{Bootstrap}$ = 38.3 m; median = 38.1 m; and $L_p$ for the 2.5 and 97.5 percentiles is, respectively, 32.9 m and 44.3 m, corresponding to the 95% confidence interval. (B) Quantile plot of the distribution shown in A (x axis) against the corresponding Gaussian distribution (y axis) of equal mean and variance. (C) Comparison between the error bars of the persistence length theoretically derived from Eq. S27 (blue bars) and the error bars computed by the bootstrap calculations from ${10}^4$ replicas of the original experimental dataset (red bars). Original data are the same as for Fig. 4 of the main text.

In practice, Eq. 3 shows that for a measure at a single location, the minimum fluctuation signal should be at least ${\sigma }_f\sim \mathrm{\hspace{0.17em}3}{\sigma }_e$, so that the optical error does not add up by more than 10% to the entropic error ($f\lesssim \mathrm{\hspace{0.17em}10}$). Considering that ${\sigma }_e\sim \mathrm{\hspace{0.17em}10}$ nm in our experiments, this corresponds to ${\sigma }_f\sim \mathrm{\hspace{0.17em}30}$ nm, yielding a maximum stiffness of $L_p\sim \mathrm{\hspace{0.17em}10}$ m for a typical rod ($L$ = 30 $\mu$m and $r$ = 300 nm). This is also the typical limit achieved by the flexural eigenmode analysis (SI Text, Eigenmode Analysis of the Rod Fluctuations). By contrast, the analysis of the fluctuations over the entire rod improves by a factor of $\sim 10$ the upper limit of the maximum measurable stiffness (up to $L_p$ = 100 m): It allows the detection of fluctuations of amplitude ${\sigma }_f$ comparable to the noise ${\sigma }_e$. In a case where ${\sigma }_e\simeq \mathrm{\hspace{0.17em}4}$ nm (due to the well-contrasted quality-reproducible images) we indeed report a measure of $L_p$ = 631 $\pm$ 118 m. Here, the error, the same as all of the errors in Experimental Results, is computed from Eq. 4 with a prefactor $\alpha =\mathrm{\hspace{0.17em}1.96}$ to obtain a 95% confidence interval.

Experimental Results

We have studied about 120 rods prepared from the “13-nm” nanoparticle stock solutions and 80 rods made of “8-nm” particles (Materials and Methods). After dialysis (Fig. 1A), microrods were found with a very broad length distribution of 1–100 $\mu$m. Only few rods in the suitable configuration were observed in each observation chamber (Fig. 1B), i.e., with a small part stuck on the inner coverslip and perpendicular to it. We retained only the longest rods (above 20 $\mu$m) which undergo the largest fluctuations. To avoid a possible damage of the rod structure, the anchorage to the coverslip was checked after the experiments were performed: The test consisted of strongly bending the rod by submitting it to a magnetic field (Fig. 1C). This procedure filtered out about half of the rods. Other causes of rejection were insufficiently straight rods, failure of the noise rejection procedure (see below), or most often, too stiff rods exhibiting too small fluctuations (${\sigma }_m<\mathrm{\hspace{0.17em}4}$ nm for $l>L/2$) to be analyzed. This last reason happened most for the rods made from 8-nm nanoparticles that appeared to be significantly more rigid than the 13-nm counterpart. At the end, we retained only 2 and, respectively, 10 rods made of 8-nm and 13-nm particles.

Fig. 1.

Rod design process and observations. (A) The rods are made by slow completion of charged polymers with opposite-charged nanoparticles during dialysis in a 1-mL jar held in a large pure water tank and submitted to a 50- to 250-mT uniform field. (A, Inset) Rods grow aligned with the field while salt removal allows nanoparticle aggregation. (B) Sample rods are flowed into a sample cell that contains an $\sim$100-$\mu$m-thick coverslip. Rods suitable for the study are those found across its edge with a small part stuck onto it and the longest part free from any surface interaction. (C) Superposed reflection microscopy images of a rod freely fluctuating (1) and magnetically deflected (2) to test the anchorage (after the measures are performed). (Scale bar, 5 $\mu$m.) The 20-mT magnetic field applied at a 35° angle with respect to the rod axis induces a deflection $\sim 50$ times larger than the fluctuation amplitude.

Analysis of the Thermal Fluctuations.

We usually took $S\simeq \mathrm{\hspace{0.17em}900}$ snapshots with an exposure time of 100–1,000 ms for a total time of $\sim$20 min. Because the optical focus was usually lost after a longer observation time, the number of acquired images $S$ and thus the measurement precision were limited (Eq. 4). We then proceeded to the two-step image recognition procedure. It first consisted of measuring the undesired motion of the sample: In addition to vibrations, the main source of mechanical noise was the drift of the sample that could reach up to $\sim$0.5 $\mu$m at the end of the recording time. Using our home-made software (Materials and Methods) we tracked an object such as a rod fragment stuck on the inner coverslip onto which the studied rod was anchored (Fig. S3 and Movie S1). The second image-recognition step consisted of digitizing the rod centerline for every image (Materials and Methods, Fig. S3, and Movie S2). The sampling was performed every pixel (46.5 nm) with an estimated resolution of $\sim$2–10 nm. At the end of the numerical treatment the resolution that depended mostly on the image quality (contrast and reproducibility over time) was in the range of 2–20 nm.

Fig. S3.

Numerical treatment of the images. (A) Example of a 53-$\mu$m rod seen by reflection microscopy. (Scale bar, 10 $\mu$m.) Red dashed square shows a typical fixed mark on the coverslip on which the rod sticks, used to track the noisy motion of the sample cell. (B) Histogram of the errors committed by the tracking software recognizing 200 piezoelectric steps of 40 nm during a calibration procedure. SD = $9.1\times {10}^{-3}$ pixels, corresponding to 0.42 nm with an 100× magnification. (Movie S1). The calibration was done with a 1.5× and 20× magnification because the software reached the piezo-stage precision ($\sim$5 nm) when using 100×. A comparable result has also been reported (25). (C) Pixel intensity profile taken along the yellow line of A fitted by a Gaussian with the rod center as a free parameter. This analysis is computed at every pixel along the rod axis and for each image (Movie S2).

For each rod, we then computed $\widehat{{\sigma }_m^2(l)}$ (Fig. 2A) as well as the normalized autocorrelation function of the tip fluctuation $\zeta (\mathrm{\Delta }i)\propto <y_m^i(L).y_m^{i+\mathrm{\Delta }i}(L)>$ (Fig. 2B). The fluctuations $y_f(l)$ always appeared smaller than the profile variation $y_p(l)$ that could extend up to $100$ nm (Fig. 2A). It is undecidable whether this profile represents the actual rod shape or a bias in the optical detection. Whatsoever, this measure being reproducible, the calculation of the SD enabled a significant measure of the fluctuations, even if the signal/noise ratio does not permit us to distinguish the rod shape on a single image. Convincingly, the data $\widehat{{\sigma }_m(l)}$ were well fitted by $\sqrt{{\sigma }_e^2+\frac{l^3}{3L_p}}$ with ${\sigma }_e$ and $L_p$ being the free parameters (Fig. 2A).

Fig. 3.

(A) Schema and notations used in the equations (main text and SI Text, Analysis of the Magnetic Experiments) to describe the magnetoelastic experiments. The rod shape is parameterized by the orthonormal coordinates $x(l)$ and $y(l)$ (at the curvilinear abscissa $l$), so that the y axis is aligned with the external induction field $\overrightarrow{B_0}$; the angle of the rod tangent $\theta (l)$ is measured with respect to the x axis and ${\theta }_0=\theta (0)$ is also the field incidence in the clamped boundary condition. (B) Example of a 56.7-$\mu$m-long rod bent by a 3.9 $\pm$ 0.2-mT uniform field ${\overrightarrow{B}}_0$ applied at a ${\theta }_0$ = 35° incidence. At the bottom, the microscope reflection images of the unmagnetized (reference) and magnetized rod are superposed and shown on an orthonormal scale (refers to the x′ axis). Note the reference frame is rotated by comparison with (A). The rod centerlines recognized by our software are drawn on the image (small black lines) and plotted on the graph. When the profile of the undeformed rod (green circles) is subtracted from the magnetically deflected rod (blue circles), the corrected profile (red circles) is well fitted by the theoretical curve provided in the main text (black solid line, maximum difference: 40 nm). (C) Typical automatic sequence of measures of the rod tip deflection $\delta$, from which $C$ is deduced. The measure is performed for three field intensities (3.26 mT, 5.59 mT, and 7.61 mT) to check the independence of $C$ with $B_0$. For each intensity, thermal fluctuations were averaged over five measures. The field was zeroed between each measure to control that the deformation remained elastic.

Fig. 2.

Rod fluctuation analysis. (A) Analysis of the fluctuations of a single rod made of 13-nm particles ($T$ = 301.5 K, $L$ = 42.3 $\mu$m, $r$ = 301 nm, $S$ = 900, $\mathrm{\Delta }$t = 100 ms). Shown is a plot along the rod length of the mean transverse displacement $<y_m(l)>$ (dashed line), the difference $y_m^1(l)-<y_m(l)>$ between one configuration and the mean (thin solid line), and the estimator of the SD $\widehat{{\sigma }_m(l)}$ computed from the $S$ images (thick solid line) fitted by $\sqrt{{\sigma }_e^2+\frac{l^3}{3L_p}}$, yielding $L_p=(7.3\pm \mathrm{\hspace{0.17em}1.4})$ m and ${\sigma }_e=(6.5\pm 0.8)$ nm. A, Inset shows a reflection microscopy image (on the plot scale) of the rod. (B) Autocorrelation function of the tip transverse displacement (same rod as in A). The exponential decay fit gives a correlation time of $1.6$ s corresponding to 16 images ($S\prime =\mathrm{\hspace{0.17em}56}$). (C) Analysis of five different rods including the stiffest and the smoothest. Variance estimators $\widehat{{\sigma }_m^2(l)}$ are plotted as a function of $l^3$. Data are scaled to be shown on the same graph and fitted by a linear regression. C, Inset shows distribution histogram of the entire set of data $\frac{y_m-<y_m(l)>}{l^{3/2}}$ ($\sim 8\cdot {\mathrm{\hspace{0.17em}10}}^5$ values) collected along the rod c. The deflections $y_m$ are scaled by $l^{-3/2}$ to be compared with a Gaussian (main text) fitted to the distribution (shaded line). (It yields an underestimated value for $L_p$ of 5.4 m). (D) The table shows for each rod of C its radius $r$, its length $L$, and the results of the fit: the persistence length $L_p$, the measurement error ${\sigma }_e$, and the linear regression coefficient R. The bending modulus and Young modulus $C$ and $Y$ are computed from $L_p$, $r$, and $T$ ($\simeq$300 K).

Consequently, we calculated $L_p$ for each analyzed rod by a linear regression of $\widehat{{\sigma }_m^2(l)}$ plotted as a function of $l^3$ (Fig. 2 C and D). We also checked that $y_m^i(l)$ follows a Gaussian distribution for every $l$, as well as the entire set of combined data scaled by $l^{-3/2}$: According to Eq. 1, $\frac{y_m(l)-<y_m>}{l^{3/2}}$ is expected to be Gaussian when $y_e(l)$ is negligible (Fig. 2C). The measure of the variance of this Gaussian yields indeed slightly smaller values for $L_p$ than the linear fit, the contribution of the errors $y_e(l)$ accounting for the difference. Fig. 2C shows five examples of rods including the most extremes, with $Lp$ extending from 5.1 m to 631 m, thus indicating that this flexural measurement may be performed for a large range of stiffness.

The autocorrelation function $\zeta (\mathrm{\Delta }i)$ (Fig. 2B) was computed for two purposes: First, the decay to zero indicates that the drift noise had been efficiently removed after our tracking procedure. Second, it provides the correlation time and thus the oversampling rate required to compute the uncertainty on $L_p$ provided by Eq. 4. To check that oversampling did not affect the measure of $L_p$ (besides the uncertainty), we performed three measurements on a same rod with time lapses $\mathrm{\Delta }t$ of 100 ms, 500 ms, and 1,000 ms ($S$ = 900 in each case), which yielded the same value for $L_p$. In most cases, the autocorrelation function exhibited multiple time-constant decays, including an initial drop faster than our sampling time and due to the relaxation of higher modes. The mode entanglement was therefore not suitable to derive $C$ by analysis of this function (39). However, for the most flexible rods, we could extract a dominant single relaxation time [$\tau =\gamma \frac{L^4}{C{q}_0^4}$, where $\gamma$ is the friction coefficient and $q_0=\mathrm{\hspace{0.17em}1.875}$ for the first mode (11, 13)]. We found that $\tau \sim 1$ s was compatible with the water viscosity $\eta \sim {10}^{-3}$ Pa $\cdot$ s [assuming $\gamma =\frac{4\pi \eta }{ln(\frac{L}{2r})+2ln(2)-1/2)}$ (41)], indicating that the outer medium might be the principal source of dissipation of the first mode, rather than an internal dissipation process within the rod.

Magnetoelastic Measurements.

For each rod, immediately after their thermal fluctuations were recorded, we proceeded to the magnetoelastic measure of their bending modulus $C=L_p{k}_{B}T$ by submitting them to a uniform induction field (Fig. S4 and Materials and Methods). At equilibrium, the local elastic torque due to the elastic deformation is balanced by the magnetic torque. We tuned the field intensity ($\sim$2–8 mT) so that the deformation of the rod (Fig. 3) was sufficiently small (the deflection $\delta$ at the tip of the rod was $\sim$1 $\mu$m) to remain in the elastic regime, as the thermal fluctuations. Small deformations, analyzed by our rod recognition software, also avoided the complexity of large deflections for which the internal field has no analytic solution.

Fig. S4.

Magnetoelastic measure of the rod bending modulus $C=L_p{k}_{B}T$. (A) Magnetic setup: The device hangs on a ring fixed on the condenser of the inverted microscope. It is free to rotate in the xy plane perpendicular to the microscope light path. It consists of two 7-cm soft iron bars, planed at one end so they can be brought in the vicinity of the sample, mounted on microstages to vary the distance $d$ between their tips (precision $\sim$10 $\mu$m). The magnetic field is induced by permanent magnets or by induction coils at the remote end of these bars. The axial symmetry of the system ensures that the field in the midperpendicular plane is horizontal. The magnets were used to demagnetize the bar before each experiment, and the coils (powered by a computer-controlled DC generator) were used to avoid mechanical disturbances during the experiments. (B) Calibration of the x component of the field at the expected position of the rod, for (i) a set of three magnets as a function of $d$ (axis in black) and (ii) as a function of the current in the coils (axis in green), for $d=\mathrm{\hspace{0.17em}2.5}$ mm and without magnet. The $\sim 2\%$ error bars are mostly due to the uncertainty on the sample position along $z$ ($\sim$10 $\mu$m) and the weak vertical field gradient ($\partial B_x/\partial z\sim$ 0.6 T/m). (C) Three-dimensional representation of the spatial variation of the x component of the field between the tip iron bars, measured at the expected height of the rod. A–C are reproduced from ref. 32.

The analysis of these experiments requires the knowledge of the magnetic susceptibility $\chi$, which we determined by reproducing experiments (29) that consisted of analyzing the rotational dynamics of freely suspended rods in a viscous fluid while they orient in a magnetic field (Fig. S5 and SI Text, Analysis of the Magnetic Experiments). Because such experiments could not be performed on the cantilevered rods on which we performed the magnetoelastic assays, we measured $\chi$ on 10 other rods, but concomitantly produced during the same synthesis from the nanoparticles. The results showed that the values of $\chi$ for the 10 13-nm rods at $B_0$ = 4 mT [the dependence of $\chi$ on the internal field, proved by the magnetization curve of the nanoparticles (Fig. S5), had to be taken into account] were extremely scattered from 1.3 to 6.1 (mean value $\chi$ = 4.6 $\pm$ 1.2). This large dispersion introduced an important uncertainty of $\sim$70% on the magnetoelastic measures of $C$. For the 8-nm rods we found $\chi \sim 1$. These rods also appeared to be typically 10 times more rigid than the 13-nm counterpart (see below), so that their ratio $C/\chi$ appeared to be irrelevant to perform magnetoelastic measures.

Fig. S5.

Measure of the magnetic susceptibility by analysis of the rod rotational dynamics. Inset shows plot of the angle of a free rotating rod submitted to a uniform field in a viscous Newtonian medium (88% glycerol). Here, only small angles (below the horizontal dashed line) are taken into account to neglect the dependence of $\chi$ with the internal field. Main plot: The rotational experiments are repeated for various magnetic strengths $H_{\parallel }$ and demonstrate the agreement between the rod susceptibility $\chi$ (red crosses) obtained by the rotational experiments and the susceptibility ${\chi }^v$ (open circles) of the ferrofluid made of 1% vol fraction of the nanoparticles (from which the rods are made). The values of ${\chi }^v$ are derived from the magnetization curve of the ferrofluid obtained by VSM (32).

To analyze the magnetoelastic experiments, we relied on a previously published theoretical model (32) (SI Text, Analysis of the Magnetic Experiments and Fig. S6), which shows that the shape of the deformed rod is determined by $x(l)=2\lambda \{\sqrt{1-\frac{\sin ({\theta }_0)}{\sin ({\theta }_L)})}-\sqrt{1-\frac{\sin \left[\theta (l)\right]}{\sin ({\theta }_L)}}\}$ and $y(l)=\lambda {\int }_{{\theta }_0}^{\theta (l)}\frac{\sin (\theta )d\theta }{\sqrt{1-\frac{\sin (\theta )}{\sin ({\theta }_L)}}}$, where the deformation lengthscale $\lambda =\sqrt{\frac{{\mu }_{0}C}{2\pi r^2\mathrm{\Delta }\chi B_0^2}}$ depends on the magnetic susceptibility of the rod $\chi$ through $\mathrm{\Delta }\chi =\chi (H_{\parallel })-\frac{\chi (H_{\perp })}{1+\chi (H_{\perp })/2}$. Here, ${\mu }_0$ is the vacuum magnetic permeability, and $H_{\parallel }$ (resp. $H_{\perp }$) is the longitudinal (transverse) component of the field in the rod (Fig. 3A for schema and notations). We verified that this model matched the shapes of the rod deformed by the field at all applied intensities within an error of $\sim$40 nm, smaller than the rod radius $r$ (Fig. 3B). The values of $\lambda$, from which $C$ followed, were deduced from the measures of the deflections $\delta =-\sin ({\theta }_0)x(L)+\cos ({\theta }_0)y(L)$ obtained for three field intensities (Fig. 3C) and applied at a ${\theta }_0={35}^{\circ }$ incidence with respect to the rod transverse axis, an angle at which the sensitivity of the method is best (Fig. S6).

Fig. S6.

Magnetoelastic deformation of a rod by a uniform induction field. (A) Notations used in SI Text, Analysis of the Magnetic Experiments and main text to describe the magnetoelastic bending of a straight rod by a uniform field. The rod shape is parameterized by the orthonormal coordinates $x(l)$ and $y(l)$ (at the curvilinear abscissa $l$), so that the y axis is aligned with the external induction field $\overrightarrow{B_0}$. The angle of the rod tangent $\theta (l)$ is measured with respect to the x axis, and ${\theta }_0=\theta (0)$ thus designates the tangent at the anchored point. In the clamped boundary condition ${\theta }_0$ is also the field incidence with respect to the transverse direction of the rod. The angle at the tip of the rod is ${\theta }_L=\theta (L)$ and the deflection relative to the straight rod is $\delta =-\sin ({\theta }_0)x(L)+\cos ({\theta }_0)y(L)$ (Eq. S35). (B) Deflection $\delta$ of the tip of the rod plotted as a function of the incidence ${\theta }_0$ of the induction field $B_0$. Each curve represents a different field intensity relative to the critical buckling field intensity $B_c$ given by Eq. S36. Around ${\theta }_0={35}^{\circ }$, most of the curves for $B_0<B_c$ reach approximately a maximum plateau, suggesting this angle is appropriate to perform the magnetoelastic experiments.

Fig. 4.

Persistence length of 12 analyzed rods. $L_p$ is derived from the thermal fluctuation analysis for 2 rods made of 8-nm particles (solid squares) and 10 rods made of 13-nm particles (solid circles) plotted as a function of their length $L$. For each 13-nm rod, the dashed lines associate the thermal measure with the result of the magnetic deflection measurement (open circles) performed on the same specific rod. The important error bars of $\sim 70$% for the magnetic measurements (not plotted for clarity but indicated in Fig. S7) are mostly due to the large variation of $\chi =4.6\pm 1.2$ from one rod to another and measured on different samples by the magnetic rotational experiments (Fig. S5 and SI Text, Analysis of the Magnetic Experiments). All $L_p$ measures overlap but one (indicated by *), which differs by 9%.

Results for 12 Rods.

Fig. 4 shows the results for 12 successfully analyzed rods. For both rod types, the thermal measures of $L_p$ spanned over two orders of magnitude from 5.1 m to 631 m, corresponding to $C$ = (2.2–261)$\times {10}^{-20}$ Jm ($L_p$ = 5.1–129 m and $C$ = (2.2–54)$\times {10}^{-20}$ Jm for the 13-nm rod). For each rod, we verified the expected law ${\sigma }_f^2(l)\propto l^3$. The radius $r$ of each studied rod was measured by analyzing the intensity profile of the rod back-scattered optical image over many cross-sections, from which we computed the rod Young modulus $Y=\frac{4}{\pi }\frac{C}{r^4}$ found to range from 1.9 MPa to 198 MPa (1.9–58 MPa for the 13-nm rod). This broad dispersion was confirmed by the magnetic measurements: A good agreement was found for the two independent measures for 9 of the 10 studied rods, and one differed by 9% only. On average, for the 13-nm rods, the magnetic measures yielded $<L_p>$ = 42 m and the thermal measures $<L_p>$ = 46 m with standard errors of 12 m and 15 m, respectively.

Fig. S7.

Parameters and values with uncertainties of $L_p$ measured by the thermal and magnetic experiments of the rods shown on Fig. 4 of the main text. The symbol (*) designates the rod for which the two measures of $L_p$ do not overlap (9% difference).

Discussion

We have successfully measured the bending modulus of nanostructured microrods by analyzing their thermal fluctuations. To our knowledge, they are the stiffest objects to be characterized by this approach: Our measure of a persistence length over 0.5 km is five orders of magnitude greater than the common measures achieved on microtubules (11) or on single-wall carbon nanotubes (18) and three orders higher than the measures on fibrin fibers (16), the stiffest biomaterial assessed by this technique. This demonstrates that for such measurements, the optical microscope may compete with the electronic microscope, thus improving the “in situ” possibilities such as a combination of thermal and magnetic experiments.

In this study, we found that the stiffness of two 8-nm rods was up to two orders of magnitude larger than in a previous report (42) concerning comparable rods made of nanoparticles of the same size but associated with different polymers. This difference may explain the disagreement between the measures: Given the low volume fraction of particles in the rod ($\sim$20%) (29), we indeed expect its stiffness to be mostly determined by the density and the nature of the polymer mesh. The thermal and magnetic experiments also proved independently that these objects exhibit very dispersed mechanical and magnetic properties: The stiffness distribution extends over more than one order of magnitude for the 13-nm rod and the magnetic susceptibility was found to vary by a factor of 5 (Fig. S5 and SI Text, Analysis of the Magnetic Experiments). We interpret these large dispersions as a consequence of the design process: It relies on a nonlinear instability (29) (the “desalting transition”), certainly very sensitive to local inhomogeneities such as some variations of the magnetic field or of the particle concentration. This dispersion strongly argues for the necessity to characterize individually each rod, until new methods are developed to produce them more homogeneously.

However, for the present study, this broad dispersion first appeared as an advantage: We could experimentally confirm the theory of Benetatos and Frey’s model (28) on an extended range of stiffness. We indeed verified that ${\sigma }_f^2(l)\propto l^3$ for $L_p$ running over two orders of magnitude. This observation, shown in Fig. 2, constitutes the main result of this paper. On the contrary, the dispersion of the rod magnetic susceptibility allowed us to perform the magnetoelastic measurements with a relatively low precision of $\sim 70\%$ only. If this was not sufficient to challenge in detail the thermal method, these independent experiments confirmed the thermal measures and proved the large dispersion of the rod rigidity. In future works, rods synthesized with more monodisperse magnetic properties would advantageously permit an in-depth comparison between both methods. Alternatively, its combination with the thermal measures may provide a direct measure of $\chi$ on individual rods (32).

Most importantly, this work improves the method to measure the persistence length of fluctuating rods: By gathering data along the entire rod we demonstrate that we could reduce the uncertainty (by a factor of $\sim$10). Previous methods that relied either on the analysis of the bending modes or on a measure performed at a single location on the rod would have failed to derive the bending modulus of the stiffest rods we have studied. Our approach can certainly be applied to other rod types. For those with width smaller than the pixel size $p$ ($\sim 50$ nm), the observation of their entire length may be realized by fluorescent or dark-field microscopy as it has been done for actin (8). For such very thin rods, their geometry ensures sufficiently large fluctuations ($C\propto r^4$) to be easily detected even for extremely hard material such as carbon nanotubes ($Y\sim$ TPa). For rods with radius $r$ larger than $p$, the accuracy on its position scales like $\omega \propto p\sqrt{\frac{p}{r}}$. To detect their fluctuations, it must verify $\omega <{\sigma }_f\propto \frac{L}{r^2}\sqrt{\frac{L{k}_{B}T}{Y}}$, thus defining an upper limit for the Young modulus $Y_{max}\lesssim k{(\frac{A}{p})}^3{k}_{B}T$, where $A=L/r$ is the rod aspect ratio, and $k$ is a prefactor mostly determined by the signal/noise ratio of the rod signature (k = 0.02 for our rods observed with our camera). With our equipment, the stiffness measurement of ZnO or plain metal microwires would thus be challenging (21): Their 10- to 100-GPa modulus requires $A\sim$ 1,000. Presently, our thermal method could apply well to rods with $A\simeq \mathrm{\hspace{0.17em}100}$ and $Y<$ GPa such as hydroxyapatite nanowires (38) and many biological filaments.

To conclude, the eigenmode analysis applies well to measure the rigidity of relatively soft semiflexible rods: Each mode provides an independent probe of its stiffness, at a different lengthscale. But this method reaches its limit for rods of larger $L_p/L$ ratios, of which the fluctuations are too small to be detected. For such rods, the present work offers a renewed method to analyze their fluctuations. It enables the possibility to probe numerous stiffer synthetic or biological one-dimensional objects.

Materials and Methods

Microrod Synthesis.

Microrods were prepared by slow completion of opposite-charged polymers with iron-oxyde nanoparticles during a dialysis performed under a permanent magnetic field of $\sim 200$ mT (Fig. 1) according to a protocol already described (30, 32). Here, we used two stock solutions of citrated particles ($1\%$ vol fraction $\simeq$52 g/L) of which the diameter follows a log-normal distribution around 8 $\pm$ 2 nm and 13 $\pm$ 3 nm, respectively. Observation of the rods by electron microscopy has suggested that the internal structure is isotropic (30). The magnetization curve of the ferrofluid made of each particle type was obtained by vibrating sample magnetometry (32) (Fig. S5).

Microscope Sample Preparation.

Rod bulk samples were diluted in water $\sim$100× before being flowed to the observation chamber. It contained an inner coverslip onto which we targeted the rods with a magnet so that they would sediment perpendicularly to its edge (Fig. 1B). The sample was sealed using UV-cured polymer solution (Vitralit) and left for $\sim 30$ min to allow the rods to sediment and to stick to the glass. The temperature was monitored with a precision of 0.1 K by use of two probes hold against each side of the sample cell.

Optical Microscopy and Magnetic Setup.

Rods were observed with a DMIRB inverted Leica microscope equipped with an apoplan 100× NA 1.3 objective and still images were obtained with a Photometrics fx-Coolsnap camera [pixel size: (46.5 nm)²]. The rods were observed either by bright-field or by reflection microscopy by inserting a 50% beam splitter (Melles Griot) into the conventional fluorescent light pathway (Fig. 1C). The system was piloted by ImageJ and Micro-Manager softwares. The magnetic setup, described previously (32), is shown in Fig. S4. The 3D calibration of the field (Fig. S4) using a Hall effect Gaussmeter (LakeShore 410; Cryotronics) proved that the field gradient was below 0.25 T/m in the observation plane (and 0.6 T/m in the vertical direction) at the rod position. This is sufficiently weak not to disturb the uniform field experiments (SI Text, Comparison Between the Deflection Caused by a Magnetic Gradient and a Uniform Field).

Softwares.

A first ImageJ plugin was designed to track the undesired motion of the sample stage (noise). Its principle is to fit the image of a reference object on slice $n$ to its translated image on slice $n+1$ (and proceed for all images of the movie), the displacements $dx$ and $dy$ being the fit parameters of the least-squares minimization Levenberg–Marquardt algorithm (25). We calibrated our software with a P-780 piezoelectric stage (Physike Instrumente) and found it can track displacement as low as 0.4 nm. A second plugin was developed to digitize the rod centerline by fitting a Gaussian to the transverse gray intensity profile for any curvilinear position along the rod, as previously done for microtubules (11, 13) (Fig. S3).

Bootstrap Calculations.

Numerical estimations of the uncertainty of $L_p$ were performed by bootstrap calculations. Using the same software as for computing $L_p$, ${10}^4$ bootstrap samples drawn randomly with replacements from the original set of rod configurations were analyzed. The time correlation between successive pictures was taken into account by the block method (40).

SI Text

Eigenmode Analysis of the Rod Fluctuations.

We consider an elastic rod of length $L$ and bending modulus $C$, straight in its minimum energy state, set in a cantilevered configuration in a thermal bath at temperature $T$. We note the transverse fluctuations $y_f(l)$ at curvilinear abscissa $l$ so that $l=0$ designates the clamped end and $l=L$ the free end of the rod. As previously described by Wiggins et al. (10), each element of the rod follows a force balance equation between the friction force (exerted by the bath fluid or possibly by some internal dissipation) and the elastic restoring force,

$$C\frac{{\partial }^4{y}_f(l)}{\partial l^4}+\gamma \frac{\partial y_f(l)}{\partial t}=0,$$ [S1]

where $\gamma$ is a viscous coefficient. The cantilever configuration and the choice of the coordinate system yield the boundary conditions: ${y_f|}_0={\frac{\partial y_f}{\partial l}|}_0={\frac{{\partial }^2{y}_f}{\partial l^2}|}_L={\frac{{\partial }^3{y}_f}{\partial l^3}|}_L=0$. The solutions of Eq. S1 may thus be written as a series expansion of the fluctuation modes

$$y_f(l)=\sqrt{\frac{1}{L}}{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{a}_n{W}_n\left(\frac{l}{L}\right),$$ [S2]

with the eigenmodes plotted in Fig. S1,

$$W_n(l/L)=\frac{cosh(q_n)+\cos (q_n)}{sinh(q_n)+\sin (q_n)}\left[sinh(q_{n}l/L)-\sin (q_{n}l/L)\right]+\cos (q_{n}l/L)-cosh(q_{n}l/L)$$ [S3]

and where the wave numbers $q_n$ verify the equations $cosh(q_n)\cos (q_n)+1=\mathrm{\hspace{0.17em}0}$ easily computed from the boundary conditions. Its solutions are $q_1\simeq \mathrm{\hspace{0.17em}1.875}$, $q_2\simeq \mathrm{\hspace{0.17em}4.695}$, and $q_n\simeq (2n-1)\frac{\pi }{2}$, for $n>\mathrm{\hspace{0.17em}2}$, with an approximation better than ${10}^{-3}$. The mode amplitudes can be computed by use of the canonical inner product of the eigenmode basis:

$$a_n=\frac{1}{\sqrt{L}}{\int }_0^L{y}_f(l)W_n\left(\frac{l}{L}\right)dl.$$ [S4]

Taking into account the boundary conditions, the expression of the elastic energy may be reformulated after an integration by parts:

$$E=\frac{C}{2}{\int }_0^L{(\frac{{\partial }^2{y}_f(l)}{\partial l^2})}^{2}dl=\frac{C}{2}{\int }_0^L(\frac{{\partial }^4{y}_f(l)}{\partial l^4})y_f(l)dl.$$ [S5]

Using $\frac{{\partial }^4{W}_n(\frac{l}{L})}{\partial l^4}={(\frac{q_n}{L})}^4{W}_n(l)$, Eq. S5 yields a decomposition of the energy $E={\sum }_{n=1}^{\mathrm{\infty }}{e}_n$, the energy in each eigenmode being

$$e_n=\frac{C}{2}{a}_n^2{\left(\frac{q_n}{L}\right)}^4.$$ [S6]

The quadratic form of the energy is a sufficient condition to apply the equipartition theorem and thus, averaging over a Gibbs ensemble of rod configurations,

$$<a_n^2\ge \frac{k_{B}T}{C}{\left(\frac{L}{q_n}\right)}^4.$$ [S7]

From Eqs. S2 and S7, considering the values taken by the $W_n$ (Fig. S1), an order of magnitude of the deflection at the tip of the rod is thus for a given mode:

$${\alpha }_n=\sqrt{\frac{<a_n^2>}{L}{W}_n(1)}\simeq {(\frac{L}{(n-1/2)\pi })}^2\frac{1}{\sqrt{L.L_p}}.$$ [S8]

With a typical measurement error of ${\sigma }_e=10$ nm, the maximum persistence length possibly measured should therefore not exceed $Lp\simeq 10$ m for a rod of length $L=\mathrm{\hspace{0.17em}30}$ $\mu$m: Only some of mode 1 would be detectable from about half of the rod to its tip where ${\alpha }_1$ reaches on average 26 nm. Higher modes could not be seen: The amplitude (SD) of the second mode of such a rod is only 4.7 nm at its tip.

Uncertainty on the Experimental Measure of the Persistence Length.

Uncertainty on L_𝒑 deduced from an analysis performed at a single point on the rod.

As described in the main text, the measures of the rod transverse positions are written as

$$y_m^i(l)=y_p(l)+y_f^i(l)+y_e^i(l),$$ [S9]

where $i$ ($1\le i\le S$) is the index for each rod configuration; $l$ ($0\le l\le L$) is the curvilinear abscissa at which the measure is performed; $y_p(l)$ describes the measure of the rod profile, taking into account the actual rod shape and a systematic bias in the optical measure (note it is independent of $i$), $y_f^i(l)$ being the gap produced by the fluctuations; and $y_e^i(l)$ is the nonsystematic error performed during the optical recognition of the rod position.

From a statistical viewpoint, according to Benetatos and Frey’s model (28) (Eq. 1 of the main text), for any abscissa $l$ ($0\le l\le L$) the set ${\{y_f^i(l)\}}_{1\le i\le S}$ is drawn from a centered Gaussian distribution $\mathcal{N}[0,{\sigma }_f^2(l)=\frac{l^3}{3L_p}]$. The set ${\{y_e^i(l)\}}_{1\le i\le S}$ is also assumed to follow a normal distribution $\mathcal{N}[0,{\sigma }_e^2]$, where the variance ${\sigma }_e^2$ does not depend on $l$ because all of the optical measures are performed identically and independently at any $l$. Because the variables $y_e(l)$ and $y_f(l)$ are uncorrelated, the set ${\{y_m^i(l)\}}_{1\le i\le S}$ is also normally distributed according to $\mathcal{N}[y_p(l),{\sigma }_m^2(l)]$, where its variance is

$${\sigma }_m^2(l)={\sigma }_f^2(l)+{\sigma }_e^2=\frac{l^3}{3L_p}+{\sigma }_e^2.$$ [S10]

The measure of $L_p$ is then performed by computing the estimator of ${\sigma }_m^2(l)$,

$$\widehat{{\sigma }_m^2(l)}=\frac{1}{S-1}{\displaystyle \sum _{i=1}^S}{\left[y_m^i(l)-<y_m(l)>\right]}^2,$$ [S11]

where$<y_m(l)>=\frac{1}{S}{\displaystyle \sum _{1\le i\le S}}{y}_m^i(l)$ is the sample mean of our experimental data at a given location $l$. To estimate the uncertainty $\delta L_p$ we compute the variance of $\widehat{{\sigma }_m^2(l)}$. Assuming at first that the rod configurations are uncorrelated, $\widehat{{\sigma }_m^2(l)}$ follows a $\chi$² distribution (43), and therefore its variance is

$$Var\left[\widehat{{\sigma }_m^2(l)}\right]=\frac{2}{S-1}{\sigma }_m^4(l)$$ [S12]

of which the square root provides the error on the estimation of ${\sigma }_m^2(l)$:

$$\delta {\sigma }_m^2(l)\simeq \sqrt{\frac{2}{S}}\left[{\sigma }_f^2(l)+{\sigma }_e^2\right].$$ [S13]

Neglecting the errors on $l$ and ${\sigma }_e$, and using Eq. S10, the error on ${\sigma }_m^2(l)$ is linked to the error on $L_p$ through $\delta {\sigma }_m^2(l)=\frac{l^3}{3L_p^2}\delta L_p(l)$, which yields (using ${\sigma }_f^2(l)=\frac{l^3}{3L_p}$ and Eq. S13):

$$\frac{\delta {\sigma }_m^2(l)}{{\sigma }_f^2(l)}=\frac{\delta L_p(l)}{L_p}=\sqrt{\frac{2}{S}}\left[1+\frac{3{\sigma }_e^2{L}_p}{l^3}\right].$$ [S14]

Anticipating the next calculations below, each term on the right-hand side of Eq. S13 must be interpreted. We designate the first term ${\delta }_S=\sqrt{\frac{2}{S}}{\sigma }_f^2(l)$ as the entropic error. Scaled by the variance of the fluctuations as it appears in Eq. S14, it yields a relative uncertainty $u_S=\frac{{\delta }_S}{{\sigma }_f^2(l)}=\sqrt{\frac{2}{S}}$ that depends only on $S$, independently of $l$ and of the rod physical property $L_p$. It is a direct consequence of the finite sampling of the Gibbs ensemble (and thus of its “imperfect knowledge”) from which the rod configurations are drawn. The second term in Eq. S13, ${\delta }_e(l)=\sqrt{\frac{2}{S}}{\sigma }_e^2(l)$, is the “optical error.” It originates from the error in the optical measure and is averaged out by the repetition of the $S$ optical independent measures. The corresponding relative uncertainty $u_e(l)=\frac{{\delta }_e}{{\sigma }_f^2(l)}=\sqrt{\frac{2}{S}}\frac{3{\sigma }_e^2{L}_p}{l^3}$ indicates that the most resolutive measurement is achieved when it is performed at the tip of the rod where $l=L$. This term is also relatively smaller for softer rods (with smaller $L_p$) that undergo larger fluctuations.

In the experiments, we most often acquired the rod pictures at a time interval shorter than the typical relaxation time of the rods. We indeed verified that the autocorrelation function of the deflections at the rod tip decays exponentially (Fig. 2B of the main text); i.e., $<y_m^i(L),y_m^{i+i\prime }(L){>}_i\propto \exp (-i\prime \mathrm{\Delta }t/\tau )$, where $\tau$ is the correlation time and $\mathrm{\Delta }t$ is the elapsed time between the acquisition of two consecutive pictures. In the case where $\mathrm{\Delta }t<\tau$, the rod configurations described by $y_m^i(l)$ are thus correlated. As a result, the expression of the uncertainty of Eq. S14 is still approximately valid, up to a rescaling of the sample size. Specifically, defining $f=\frac{\tau }{\mathrm{\Delta }t}$, the error scales as if $S\prime =S/f$ samples were independently obtained from the Gibbs ensemble. This modifies accordingly the statistical uncertainty $u_S$ to $u_{S\prime }\simeq \sqrt{\frac{2}{S\prime }}$. In contrast, the contribution from the measurement error remains identical because the optical measures are still performed independently and are uncorrelated. Therefore, the consideration of the correlation between the rod images leads to a modified version of Eq. S13,

$$\delta \widehat{{\sigma }_m^2(l)}\simeq \sqrt{\frac{2}{S\prime }}{\sigma }_f^2(l)+\sqrt{\frac{2}{S}}{\sigma }_e^2=\sqrt{\frac{2}{S\prime }}\left[{\sigma }_f^2(l)+\frac{{\sigma }_e^2}{\sqrt{f}}\right],$$ [S15]

and of Eq. S14 to

$$u_l=u_S+u_e(l)=\sqrt{\frac{2}{S\prime }}\left[1+\frac{1}{\sqrt{f}}{\left[\frac{{\sigma }_e}{{\sigma }_f(l)}\right]}^2\right]=\sqrt{\frac{2}{S\prime }}\left[1+\frac{1}{\sqrt{f}}\frac{3{\sigma }_e^2{L}_p}{l^3}\right].$$ [S16]

Finally, the smallest relative uncertainty on $Lp$ obtained from a single-point measurement performed at the rod tip is

$$u_L=\frac{\delta L_p(L)}{L_p}=\sqrt{\frac{2}{S\prime }}\left[1+\frac{1}{\sqrt{f}}\frac{3{\sigma }_e^2{L}_p}{L^3}\right].$$ [S17]

Uncertainty on ${\text{𝑳}}_{\mathrm{𝒑}}$ deduced from an analysis performed on N locations along the rod.

In the experiments described in the main text, the analysis is performed over the $N\sim 1,000$ measures of ${\sigma }_m^2(l)$ acquired over the entire rod length. Because the sampling occurs at a regular interval (every pixel), we set $l(n)=nL/N$, with $1\le n\le N$. Our analysis is then achieved by performing a linear regression of the entire data collected over the entire rod length. Reformulating Eq. S10 we write

$$Y(n)=aX(n)+b+r(n)$$ [S18]

with $Y(n)=\widehat{{\sigma }_m^2\left[l(n)\right]}$, $X(n)=l^3(n)={\left(\frac{nL}{N}\right)}^3$, $a=\frac{1}{3L_p}$, and $b={\sigma }_e^2$. Using Eqs. S10 and S18 thus defines the residues $r(n)=\widehat{{\sigma }_m^2(l)}-{\sigma }_m^2(l)$, which designate the gaps between the measures and the expected theoretical values. The coefficients $a$ and $b$ are then obtained from a linear fit of the experimental data (Fig. 2C of the main text), using the classical expressions of the linear regression coefficients (44)

$$a=\frac{{\displaystyle \sum _{1\le n\le N}}[X(n).Y(n)]}{{\displaystyle \sum _{1\le n\le N}}\left[X^2(n)\right]}$$ [S19]

$$b=<Y>-a<X>,$$ [S20]

where $<.>$ designates the sample mean.

The estimation of the errors on $a$ and $b$ is complicated by the entanglement of the entropic and optical errors, both of which contribute to the residues $r(n)$. According to Eqs. S12 and S15, their SD is

$${\sigma }_r=\sqrt{Var\left[\widehat{{\sigma }_m^2(l)}-{\sigma }_m^2(l)\right]}=\delta \widehat{{\sigma }_m^2(l)}\simeq {\delta }_{S\prime }+{\delta }_e=\sqrt{\frac{2}{S\prime }}\left[{\sigma }_f^2(l)+\frac{{\sigma }_e^2}{\sqrt{f}}\right].$$ [S21]

Although difficult to compute, we expect the contribution from the entropic error ${\delta }_{S\prime }$ to be strongly correlated through $n$, essentially because this study concerns the case where $L_p>>L$, which is by definition (27) a condition of strong correlation of the rod shape: $<\overrightarrow{t(l)}.\overrightarrow{t(0)}>=\exp (-l/L_p)$, where $\overrightarrow{t(l)}$ is the tangent vector to the rod at $l$. To avoid the complete mathematical statistical treatment of the correlations, we provide a conservative estimate of the error on $L_p$ after this full rod analysis. According to the interpretation given above to the error terms of Eq. S13, multiplying the measures on each rod configuration does not improve our knowledge of the Gibbs ensemble. Consequently, the entropic error ${\delta }_{S\prime }$ remains unchanged by the full rod analysis. In contrast, the contribution of the optical error ${\delta }_e$ to $r(n)$ is uncorrelated (the measurements are independent from each other and, furthermore, do not depend on the rod shape), so that we expect an efficient reduction of the optical error by the linear regression. To assess how the linear regression modifies the optical error we consider its exclusive contribution ${\delta }_e$ to $r(n)$. In such a case, the entropic term drops from Eq. S13 and according to the independence of the measurement $y_e$, the $r(n)$ do follow a centered Gaussian distribution with ${\sigma }_r=\delta {\sigma }_m^2(l)=\sqrt{\frac{2}{S}}{\sigma }_e^2$. This is a sufficient condition to apply the formulas of the SE on the regression coefficients (44):

$$\delta a=\sqrt{\frac{{\displaystyle \sum _{1\le n\le N}}\left[r^2(n)\right]}{(N-2){\displaystyle \sum _{1\le n\le N}}{\left[X(n)-<X>\right]}^2}}$$ [S22]

$$\delta b=\sqrt{\frac{{\displaystyle \sum _{1\le n\le N}}\left[r^2(n)\right]}{N-2}}\sqrt{\frac{1}{N}+\frac{(N-1){(<X>)}^2}{N{\displaystyle \sum _{1\le n\le N}}{\left[X(n)-<X>\right]}^2}}.$$ [S23]

The calculation of ${\sum }_{n=1}^N{\left[X(n)-<X(n)>\right]}^2={\left(\frac{L}{N}\right)}^{6}N\left[<n^6>-{<n^3>}^2\right]={\left(\frac{L}{N}\right)}^6(\frac{1}{7}-\frac{1}{16})N^7=\frac{9N{L}^6}{112}$ and recognizing ${\sigma }_r$ in Eqs. S22 and S23 finally provide the error estimates

$$\delta a\simeq \frac{{\sigma }_r}{\sqrt{{\displaystyle \sum _{1\le n\le N}}{\left[X(n)-<X(n)>\right]}^2}}\simeq \sqrt{\frac{224}{9NS}}\frac{{\sigma }_e^2}{L^3}\simeq \frac{5{\sigma }_e^2}{\sqrt{NS}{L}^3}$$ [S24]

and

$$\delta b\simeq 2{\sigma }_e\delta {\sigma }_e\simeq {\sigma }_r\sqrt{\frac{16}{9N}}=\sqrt{\frac{32}{NS}}\frac{{\sigma }_e^2}{3}\simeq \frac{2{\sigma }_e^2}{\sqrt{NS}}.$$ [S25]

With this full rod analysis, the uncertainty from the optical error is thus reduced down to

$$u_e^{\prime }(L)=\frac{\delta L_p}{L_p}=\frac{\delta a}{a}=\mathrm{\hspace{0.17em}3}{L}_p\delta a=\frac{15}{\sqrt{NS}}\frac{{\sigma }_e^2{L}_p}{L^3}=\frac{5}{\sqrt{NS}}{\left[\frac{{\sigma }_e}{{\sigma }_f(L)}\right]}^2$$ [S26]

and the final uncertainty on $L_p$, after reintroducing the unchanged entropic uncertainty, is

$$u_{rod}(L)=u_{S\prime }+u_e^{\prime }(L)=\frac{\delta L_p}{L_p}=\sqrt{\frac{2}{S\prime }}\left[1+\frac{5}{\sqrt{2Nf}}{\left[\frac{{\sigma }_e}{{\sigma }_f(L)}\right]}^2\right],$$ [S27]

which is the expression used in Experimental Results to compute the error bars, after a multiplication by $\alpha =1.96$ to obtain a 95% confidence interval. Whereas this statistical treatment in which the two types of errors are considered separately is not rigorously exact in the general case, it is valid in the case where ${\delta }_e>>{\delta }_f$, i.e., ${\sigma }_e>>{\sigma }_f$, and remains a conservative estimate of the uncertainty in the general case.

Bootstrap check of the estimation on the uncertainty of ${\text{𝑳}}_{\mathrm{𝒑}}$.

An independent estimation of the error on the estimation of $L_p$ was obtained using the bootstrap method. First, replicates of the original data set were drawn randomly, following the “block bootstrap” classical method (40). Specifically, a replicate set was constituted of $S$ random configurations chosen among the $S$ original samples, themselves consisting of nonoverlapping blocks of $f$ successive rod configurations, where $f$ equals the correlation length determined from the autocorrelation function. A set of $S/f$ block samples was thus uniformly randomly drawn with replacement from the initial distribution. For each new bootstrap replicate, the value of $L_p$ was computed using the same algorithm as that applied to the original samples (Materials and Methods), yielding a bootstrap estimate of the sampling distribution of our estimator of $L_p$. For each rod, a total of $\sim {10}^4$ bootstrap replications were conducted, yielding computation times of $\sim$24 h per rod (on an eight-core Intel Xeon 3.5 GHz Apple Mac pro computer). The resulting bootstrap distribution of $L_p$ is shown in Fig. S2. The procedure was performed for each rod shown in Fig. 4 of the main text.

For each rod, the bootstrap distribution of $L_p$ was well approximated by a Gaussian distribution, as demonstrated by the quantile plot of Fig. S2B, except for rare cases of long tail deviations, in particular for rod sets with a large $f$. Detailed observations of these rare events show that they correspond to highly degenerate bootstrap resampling configurations, in which the same block was repeatedly drawn, thus yielding an extremely large persistence length (these repeated configurations tend to mimic the type of patterns produced by nonfluctuating filaments). This effect was responsible for a slight positive bias in the calculation of the mean value of $L_p$ from the bootstrap sample (from 0.2% to 3.8%). This bias remained small, however, by comparison with the error we seek to estimate. The 95% confidence intervals were directly obtained from the bootstrap distribution of each rod, by retrieving the 2.5 and 97.5 quantiles. These confidence intervals are compared in Fig. S2C to those computed theoretically, based on Eq. S27. Altogether, the error bars obtained by the two independent methods overlap extremely well, with a mean absolute difference of 4.4% (range: 0.08–14.6%).

Analysis of the Magnetic Experiments.

Notations shared by the rotational and bending experiments.

We consider a rod of length $L$ and radius $r$ submitted to a uniform induction field $\overrightarrow{B_0}={\mu }_0\overrightarrow{H_0}$, where ${\mu }_0$ is the vacuum permeability. We call $\overrightarrow{M}$ the local magnetization and $\overrightarrow{H}$ the field in the rod. The components along the axial and transverse directions of the rods of the various field vectors $\overrightarrow{H_0}$, $\overrightarrow{B_0}$, $\overrightarrow{H}$, and $\overrightarrow{M}$ are respectively indexed by ($\parallel$) and ($\perp$). The magnetic susceptibility, which depends on the local field, is defined as $\chi (H)=\frac{M}{H}$. We also note

$$\mathrm{\Delta }\chi =\chi (H_{\parallel })-\frac{\chi (H_{\perp })}{1+\chi (H_{\perp })/2},$$ [S28]

where each term of the subtraction corresponds to the effective susceptibility along each direction (32).

To obtain the dependence of $\chi$ with $H$, we first measured by vibrating sample magnetometry (VSM) the magnetization curve $M^v(H)$ of a ferrofluid solution containing a 1% vol fraction of the nanoparticles from which the rods are made. We then simply computed ${\chi }^v(H)=M^v/H$ (Fig. S5). As long as the volume fraction of nanoparticles in the rod is low, we expect $\chi (H)=\varphi {\chi }^v(H)$, where $\varphi$ is the volume fraction of the particles in the rod (45).

Rotational experiments: magnetic vs. viscous torque.

To measure the magnetic susceptibility $\chi$ of the rods, we reproduced experiments (29) that consist of analyzing the rotational dynamics of a freely suspended rod in a viscous fluid while it orients in a uniform magnetic fluid. In these experiments, at time $t=\mathrm{\hspace{0.17em}0}$, a uniform field ${\overrightarrow{B}}_0$ is applied at the initial incidence $\alpha (0)$ with respect to the rod axis. The rotation of the rod $\alpha (t)$ in the field is recorded until its axis aligns with the field at ${\alpha }_{\mathrm{\infty }}$ (Fig. S5).

The magnetic induction field induces a magnetic torque per unit volume $\overrightarrow{M}\wedge \overrightarrow{B_0}$ that yields the algebraic expression of the entire magnetic torque on the rod: ${\gamma }_m=\frac{\pi r^{2}L}{2}\mathrm{\Delta }\chi \frac{B_0^2}{{\mu }_0}\sin \left[2\alpha (t)\right]$. At low Reynolds numbers, it counterbalances at each moment the viscous torque ${\gamma }_v\propto \frac{d\alpha }{dt}$. When $\mathrm{\Delta }\chi$ is assumed to be constant, the integration of ${\gamma }_m+{\gamma }_v=0$ is at hand and reads

$$\tan \left[\alpha (t)\right]=\tan ({\alpha }_0)\cdot exp(-kt),$$ [S29]

where $k=\frac{2g(r,L)}{\eta }{(\frac{r}{L})}^2\mathrm{\Delta }\chi \frac{B_0^2}{{\mu }_0}$, ${\alpha }_0=\alpha (t=\mathrm{\hspace{0.17em}0})$, $\eta =\mathrm{\hspace{0.17em}0.16}\pm 0.01$ Pa s is the viscosity of the medium (12% water and 88% glycerol, measured with a Brookfield rheometer), and $g(r,L)=\ln (\frac{L}{2r})+\ln 2-\frac{1}{2}$. In contrast to the experiments of ref. 29, to verify the constraint that $\chi (H_{\parallel })$, $\chi (H_{\perp })$, and thus $\mathrm{\Delta }\chi$ are held constant despite their dependence with $H$, we monitored only small angles with $\alpha (0)<{\mathrm{\hspace{0.17em}20}}^{\circ }$, and the experiments were performed several times on the same rod by varying the intensity of the field. The time constant $k^{-1}$ was deduced from the experimental data, which then yielded $\chi (H)$ and $\varphi$ by comparison with ${\chi }^v$. We checked that the deduced values for $\chi (H)$ were indeed proportional to ${\chi }^v(H)$ (Fig. S5). The experiments were performed on 10 different rods made of 13-nm nanoparticles and we found that although the rods were similar in size, $\chi$ varied considerably from 1.3 to 6.1 (calculated for ${\mu }_{0}H$ = 4 mT), which yielded a mean value of $\chi$ = 4.6 $\pm$ 1.7. On seven other rods of the same type but synthesized in a different batch, we previously found (32) $\chi =\mathrm{\hspace{0.17em}2.9}\pm 0.9$, deduced by the combination of the thermal and magnetoelastic measures, aiming in this case at finding $\chi$. Comparable rotational experiments on the 8-nm rods showed that their susceptibility was too weak ($\chi \sim 1$ at 4 mT) to consistently perform the magnetoelastic experiments, given also their high level of stiffness (Experimental Results in main text).

Bending experiments: magnetic vs. elastic torque.

The bending experiments are performed on the same rod and right after the thermal fluctuation measurements. A uniform induction field ${\overrightarrow{B}}_0$ is applied at a typical ${\theta }_0={35}^{\circ }$ angle with respect to the transverse axis of the cantilevered undeformed rod. The tangent vector of the bent rod is then described relative to the induction field as depicted on Fig. S6A.

We demonstrated (32) that in the rod, the axial magnetization $M_{\parallel }$ is approximatively constant (the magnetization component along this axis is governed by a mean field) whereas the other component $M_{\perp }$ is much smaller (due to the demagnetizing field) and depends only on the local orientation $\theta (l)$, where $l$ is the rod curvilinear abscissa measured from the rod-anchored point.

At each $l$, the bending induces an elastic torque per unit length proportional to the bending modulus $C$ (46),

$${\gamma }_b=C\frac{d^2\theta (l)}{d{l}^2},$$ [S30]

which opposes the magnetic torque (32)

$${\gamma }_m^{\prime }=\left[\chi (H_{\parallel })\sin ({\theta }_L)-\frac{\chi (H_{\perp })}{1+\chi (H_{\perp })/2}\sin \left[\theta (l)\right]\right]\cos \left[\theta (l)\right]\times \frac{\pi r^2{B}_0^2}{{\mu }_0}.$$ [S31]

The deformations being small ($\theta (L)-\theta (0)<3^{\circ }$), we can use the approximation $\sin \left[\theta (l)\right]\simeq \sin \left[\theta (L)\right]$ to allow the integration of ${\gamma }_b+{\gamma }_m^{\prime }=0$. It yields the parameterized shape of the bent rod,

$$x(l)=2\lambda \{\sqrt{1-\frac{\sin ({\theta }_0)}{\sin ({\theta }_L)})}-\sqrt{1-\frac{\sin \left[\theta (l)\right]}{\sin ({\theta }_L)}}\},$$ [S32]

$$y(l)=\lambda {\int }_{{\theta }_0}^{\theta (l)}\frac{\sin (\theta )d\theta }{\sqrt{1-\frac{\sin (\theta )}{\sin ({\theta }_L)}}}$$ [S33]

with $\lambda =\sqrt{\frac{{\mu }_{0}C}{2\pi r^2\mathrm{\Delta }\chi B_0^2}}$. Eqs. S32 and S33 yield the constraint on the curvilinear length $L$,

$$L=\lambda {\int }_{{\theta }_0}^{{\theta }_L}\frac{d\theta }{\sqrt{1-\frac{\sin (\theta )}{\sin ({\theta }_L)}}},$$ [S34]

and the deflection at the tip,

$$\delta =-\sin ({\theta }_0)x(L)+\cos ({\theta }_0)y(L).$$ [S35]

In the experiments, we measured $\delta$ and L. Eqs. S34 and S35 were then numerically solved for ${\theta }_L$ and $\lambda$ from which $C$ can be retrieved, $\mathrm{\Delta }\chi$ being known from the rotational experiments.

For ${\theta }_0=0$, Eq. S34 has no solution for $L<\lambda$, which yields the expression of the magnetoelastic critical buckling field (32)

$$B_c=\frac{1}{rL}\sqrt{\frac{2{\mu }_{0}C}{\pi \mathrm{\Delta }\chi }}.$$ [S36]

The dependence of $\mathrm{\Delta }\chi$ on both components of $H$ (Eq. S28) is taken into account from the rotational experiments (which yield $<\varphi >$) and the variation of $\chi$ with $H$ from the VSM data (Fig. S5). The magnetoelastic experiments were performed for each rod five times at typically three field intensities (from $\sim 2$ mT to $\sim$8 mT) to verify the independence of the measured value of $C$ with the applied field strength.

Comparison Between the Deflection Caused by a Magnetic Gradient and a Uniform Field.

Let us consider the case of an external field $B_0$ with an incidence ${\theta }_0=\pi /4$ deflecting the rod so that ${\theta }_L={\theta }_0+\omega$. In the small deflection limit where $\omega \ll {\theta }_0$, linearizing Eq. S34 of SI Text, Analysis of the Magnetic Experiments yields $\omega ={(\frac{L}{\lambda })}^2\frac{1}{\tan ({\theta }_0)}$, from which we deduce the deflection $\delta =\omega L=\frac{2\pi r^2\mathrm{\Delta }\chi B_0^2{L}^3}{{\mu }_{0}C\tan ({\theta }_0)}$. By comparison, a field gradient induces a force field on the rod $\overrightarrow{q}=(\overrightarrow{m}.\overrightarrow{\nabla })\overrightarrow{B}$. Assuming for simplicity’s sake that the gradient is constant, and considering that only the gradient along the transverse direction of the rod (the y axis) contributes to its deflection, the transverse force is $q_y=\pi r^2\frac{B_0}{{\mu }_0}{\partial }_y{B}_0\frac{\chi }{1+\chi /2}{\cos }^2({\theta }_0)$, where ${\partial }_y{B}_0$ is the notation for $\frac{\partial B_{oy}}{\partial y}$. Using the theory of the weak deformations of elastic rods (46), the shape $y(x)$ verifies the equation $C{y}^{(4)}=q_y$, which yields a deflection $d=\frac{q{L}^4}{8C}=\frac{\chi }{2+\chi }\frac{\pi r^2{L}^4{\cos }^2({\theta }_0)}{4C}\frac{B_0}{{\mu }_0}{\partial }_y{B}_0$. The influence of the gradient is therefore negligible if $d\ll \delta$ or ${\partial }_y{B}_0\ll \frac{16\chi }{\sin (2{\theta }_0)}\frac{B_0}{L}$. In our experiments, ${\partial }_y{B}_0\le$ 0.25 T/m (Materials and Methods), which ensures the above condition was fulfilled: ${\partial }_y{B}_0\frac{L}{B_0}\simeq \mathrm{\hspace{0.17em}0.0025}<<16\chi$ with $\chi \sim 4.6$ and $\sin (2{\theta }_0)=\mathrm{\hspace{0.17em}1}$.