Weighing picogram aerosol droplets with an optical balance

Optical traps are powerful tools to isolate and manipulate microscopic objects in liquid, gas, and vacuum, enabling ultrasensitive force and torque measurements and in situ characterization in fields from biophysics to aerosol science. Despite extensive applications, a broadly applicable method to determine the mass of optically trapped aerosol droplets and particles has been lacking. Electrodynamic balances (EDBs) offer non-destructive, accurate mass measurements but require particle charging, mass calibration, and typically target larger, micrometer-scale particles, missing the submicron range. Photophoretic traps can infer mass but are limited to absorbing, non-volatile particles and larger sizes, with substantial heating and lower accuracy. This study introduces an optical balance based on modulated counter-propagating optical tweezers that drives harmonic oscillations of a trapped particle; analysis of the phase response yields the damping rate and natural frequency, from which the particle mass is determined with high accuracy without requiring prior mass calibration. The approach enables measurements for aqueous droplets down to the lower picogram range and quantifies hygroscopic mass growth across relative humidities.

The authors situate their work within decades of optical trapping research across phases (liquid, gas, vacuum), highlighting applications to biological systems, Brownian motion, particle rotations, and light-matter momentum transfer. In aerosol science, optical traps have been used to study size, shape, refractive index, composition, viscosity, surface tension, and processes including phase transitions, diffusion, evaporation, coagulation, photochemistry, and heterogeneous reactions. For single-particle mass determination in air, EDBs are the reference (percent-level accuracy) but need charged particles and calibration and are biased toward larger sizes. Photophoretic trapping-based mass measurements require absorbing particles, induce heating, and have typical accuracies around 15%, excluding many volatile or aqueous particles. Hygroscopic growth has been extensively probed with HTDMA for submicron particles and with EDBs for supermicron particles, leaving a size gap. The present optical-balance method aims to bridge these gaps by enabling accurate, calibration-free mass determinations for optically trapped aerosols, including aqueous droplets, down to submicron sizes.

Principle: Single aqueous NaCl droplets at ambient pressure are trapped in counter-propagating optical tweezers (CPT) using a 532 nm continuous-wave laser. The power ratio of the two counter-propagating beams is sinusoidally modulated with an electro-optic modulator (EOM), shifting the position of the trap minimum and inducing a driven harmonic oscillation of the particle along the beam axis. Model: The axial motion z(t) obeys a driven, damped harmonic oscillator: z¨ + 2Γ z˙ + (2π f₀)² z = 4π² f₀ Z₀ sin(2π f t). The steady-state response is z(t) = a sin(2π f t − φ). Sweeping the drive frequency f and measuring the phase φ(f) with a lock-in amplifier allows fitting tan φ = (Γ f₀)/(f₀² − f²) to extract the natural frequency f₀ and damping rate Γ₀. Mass retrieval: The droplet mass m is related to Γ₀ via Stokes drag corrected for slip: m = (3 π μ R)/(Γ₀ C_c), where μ is the gas viscosity (nitrogen), R is the droplet radius, and C_c is the Cunningham slip correction factor. The radius R is independently measured via broad-band light scattering (BLS) and Mie-theory fitting. Experimental setup: A 1 W, 532 nm laser passes through an optical isolator and half-wave plate into an EOM controlling polarization and thus the beam-split ratio at a polarization beam splitter, generating two counter-propagating, co-polarized beams focused by aspheric lenses into a trapping cell. A deliberate path length difference (~50 cm) exceeds the laser coherence length to avoid interference. A position sensitive photodiode (PSP) detects scattered light from the droplet to monitor axial position; its output is demodulated by a lock-in amplifier synchronized to the EOM drive. Frequency sweeps span 0–2 kHz (PSP/electronics delay characterized up to 5 kHz and subtracted). Radius measurement (BLS): A broadband Xe lamp illuminates the particle; elastically scattered light (λ = 350–500 nm, collection around θ ≈ 37° over 26.6°) is spectrally recorded. Fits to Mie simulations retrieve R, θ, and the refractive-index dispersion n(λ) via a first-order Cauchy model n(λ) = n_∞ + A/λ². Sample preparation and RH control: Aerosol droplets are generated from 0.5 M NaCl solutions using a TSI 3076 atomizer with humidified nitrogen. In the trap they equilibrate to the cell’s RH, which is set between ~60–90% by mixing dry/wet N₂ and monitored by a nearby RH sensor (SHT31). The trapping cell has fused silica windows for CPT/BLS and ports for gas and sensors. Error handling and calibration: Systematic error is dominated by PSP focusing of scattered light (quantified via repeated measurements at different focus sizes), typically ≤3.5% of mass (2% for reported data). Random error is propagated from uncertainties in Γ₀ (from φ(f) fits at 20–40 mV modulation) and R (from BLS fits). A constant calibration factor C = 0.98 is determined by matching measured droplet densities to bulk solution densities; applying C reduces total error to the random component (2–3%). Harmonicity checks: φ(f), Γ₀, and f₀ are verified to be independent of modulation amplitude Z₀ across EOM voltages 10–40 mV, confirming harmonic behavior. Rare deviations can occur at Mie resonance conditions; slight RH adjustments restore harmonic behavior. Hygroscopic metrics: Mass growth factor m*(RH) = m(RH)/m_NaCl is computed using constant NaCl mass m_NaCl (validated across RH in undersaturated region). Size growth factor R*(RH) = R(RH)/R_dry is likewise derived and extrapolated to 90% RH for comparison with HTDMA data.

• The optical balance determines masses of optically trapped aerosol droplets down to at least 4 × 10⁻¹⁵ kg (4 pg) with sensitivity of ~1 × 10⁻¹⁶ kg (~100 fg).
• Accuracy: 5.5–6.5% without calibration (systematic + random), improved to 2–3% with a single constant calibration factor C = 0.98 derived from bulk-solution densities.
• Harmonicity: Phase responses φ(f) overlap for different modulation amplitudes (10–40 mV; Z₀ ~10–40 μm), with fitted Γ₀ and f₀ agreeing within uncertainties (<0.5% at ≥20 mV; up to 2% and 1% at 10 mV due to lower SNR), confirming harmonic oscillator behavior.
• Mass and density vs RH (micron-sized NaCl droplets): At RH 67.5–86.1%, R increases from 1.385 to 1.733 μm; uncalibrated masses m from 14.7 to 25.1 × 10⁻¹⁵ kg; calibrated masses m_c from 14.39 to 24.62 × 10⁻¹⁵ kg; calibrated densities ρ_c from 1294 to 1129 kg m⁻³.
• Hygroscopic growth: Mass growth factors m* range from 3.19 ± 0.02 (67.5% RH) to 5.45 ± 0.03 (86.1% RH) and agree closely with prior EDB measurements. Size growth factor extrapolated to 90% RH gives R*(90%) = 2.40 ± 0.02, consistent with theory and HTDMA observations (2.27–2.46).
• Robustness: Mass extraction from phase φ(f) avoids ambiguity present in amplitude-based fits where parameter correlations can yield indistinguishable a(f) curves for different masses.
• Applicability: Method does not require particle charging or light absorption, enabling measurements for volatile aqueous droplets and extending accessible mass/size ranges beyond those of EDBs and photophoretic traps.

The study addresses the lack of a broadly applicable mass measurement technique for optically trapped aerosol particles by introducing an optical balance that exploits the driven, damped harmonic response of a trapped particle. By fitting the phase response φ(f), the method uniquely retrieves the natural frequency and damping, and thus the mass, avoiding ambiguities inherent to amplitude-based analyses where trap stiffness and resonance parameters can be correlated. This phase-based approach, combined with independent radius determination by BLS and slip-corrected drag, yields accurate, non-destructive mass measurements on second time scales. The results demonstrate high accuracy (≤6.5% without calibration; 2–3% with a simple constant calibration factor), access to lower picogram masses in the submicron to micrometer range, and agreement with established techniques in hygroscopic growth metrics (m* and R*). The method works for uncharged, volatile, non-absorbing aqueous droplets—systems incompatible with photophoretic traps and not readily accessible by EDBs—thus substantially expanding the scope of single-particle mass metrology in aerosols. Rare deviations from harmonic behavior due to Mie resonances are easily mitigated by small RH adjustments. The authors also highlight that the technique can bridge a size gap in hygroscopic growth studies (between typical HTDMA and EDB regimes) and may be transferable to liquid-phase systems, potentially enabling mass measurements of trapped biological objects under overdamped conditions.

An optical balance based on modulated counter-propagating optical tweezers enables accurate, non-destructive mass measurements of single optically trapped aerosol droplets down to at least 4 pg, with sensitivity near 100 fg and accuracy of 2–3% after simple calibration (≤6.5% without). The phase-resolved analysis circumvents ambiguities of amplitude-based methods, and, coupled with BLS radius retrieval, provides absolute masses without requiring particle charging or absorption. The approach reproduces hygroscopic mass and size growth in agreement with EDB and HTDMA benchmarks and opens access to submicron-to-micron particles previously difficult to probe. Future work can exploit this capability to close size gaps in hygroscopicity studies, extend to diverse aerosol compositions and environments, and adapt the method to liquid-phase optical trapping for mass measurements of cells and other biological samples.

Identified limitations include: (1) potential non-harmonic behavior when specific droplet sizes coincide with Mie resonances at the trapping wavelength; this is rare and can be remedied by small RH adjustments. (2) Systematic error in mass primarily arises from focusing of scattered light on the PSP for position tracking (up to ~3.5%); maintaining identical optics across measurements ensures consistent bias, and a single calibration factor (0.98) corrects it. (3) Random error increases at low modulation amplitudes due to reduced signal-to-noise. (4) The method relies on accurate, independent radius retrieval via BLS and known gas properties (viscosity, mean free path); uncertainties here propagate to mass. (5) Demonstrations are for aqueous NaCl droplets in nitrogen at ambient pressure; while broadly applicable in principle, performance for other materials and environments was not experimentally shown here.