Client Citation Analysis
Primary surface measurement reported
Σ = 3 mN/m for the TPM oil–water interface, by pendant drop. The paper reports it as a single value and notes that it agrees with prior literature: "We measure the surface tension to be Σ = 3 mN/m, which is consistent with previous work" (ref. 35, Kraft et al., J. Phys. Chem. B 2011, 115, 7175).
How the paper credits the instrument
Verbatim, Section 3 (Results): "In our experiment, we measure the TPM surface tension using the pendant drop method (Dropometer, made by Dropletlab). We measure the surface tension to be Σ = 3 mN/m, which is consistent with previous work."
How the surface-tension data were used in the study
Σ is used in two places. First, the authors assume a deformation energy ε = Σd² — "which should be the correct order of magnitude" — and compare it against thermal energy to get an effective temperature Teff = kBT/ε. Second, Σ nondimensionalises the yield-stress axis in Fig. 5(b), as σ₀ = σ_y d / Σ. Everything else in the paper — flow curves, yield stress, Herschel–Bulkley and Three Component fits, droplet sizing, volume fraction — came from the rheometer, the SEM and a weighing method.
What the instrument did, and what it did not do
The Dropometer measured one quantity: the oil–water surface tension of the TPM/water system, Σ = 3 mN/m, by pendant drop. It was not the primary measurement instrument: the rheology data came from an Anton Paar MC302 rheometer, and droplet sizing came from a Topcon DS-150F field-emission SEM. The paper has five figures and the Dropometer produced none of them. The paper uses that single value to convert a droplet diameter into an estimated deformation energy, which places its samples in the crossover regime predicted by simulation, and as the scale for one of the two yield stress normalisations used to compare its data with prior studies that used oils with different surface tensions (Fig. 5b).
Primary surface / interfacial measurement
The TPM oil–water surface tension, measured by pendant drop and reported as Σ = 3 mN/m. It is reported as a single value; replicate count, drop volume, needle geometry and measurement temperature are not stated in the paper, which notes that the value is consistent with previous work (ref. 35). It is referenced as the oil–water surface tension in the paper's yield-stress normalisation (Fig. 5b) and in the assumed deformation energy ε = Σd².
Supporting measurements
The study measures steady-shear rheology (shear stress versus strain rate, γ̇ from 10² to 10⁻³ s⁻¹) across volume fraction φ and uses model fits (Herschel–Bulkley and a Three Component model) to describe yield-stress behavior. Droplet sizes are determined from SEM imaging of AIBN-polymerized droplets, with differential dynamic microscopy as a cross-check agreeing to within 5%; the authors note polymerisation shrinks the droplets, so "it is likely that the emulsion droplets are 2% larger than the numbers we report." Volume fractions are determined gravimetrically by weighing before and after water evaporation, then corrected for a 17.5 nm interdroplet water film taken from prior literature — the authors state "we do not have an independent measurement of h." Volume fractions are correct relative to one another to ±0.003 but carry an additional ±0.02 systematic uncertainty.
Surface tension
Pendant drop tensiometer — "(Dropometer, made by Dropletlab)" — single value, Σ = 3 mN/m; produced no figure of its own
Rheology
Anton Paar MC302 rheometer
Droplet size (SEM)
Topcon DS-150F Field Emission SEM
Droplet size cross-check
Differential dynamic microscopy (DDM); no instrument model stated
Volume fraction
Gravimetric — samples weighed before and after water evaporation; no balance model stated
The Dropometer measured the surface tension of the TPM oil–water system by the pendant drop method. The paper states, in Section 3: "In our experiment, we measure the TPM surface tension using the pendant drop method (Dropometer, made by Dropletlab). We measure the surface tension to be Σ = 3 mN/m, which is consistent with previous work." The paper reports it as a single value and notes that it is consistent with the previous work it cites (ref. 35).
That value is then used in the paper's analysis. The authors assume a deformation energy ε = Σd², in their words: "We assume the deformation energy ε = Σd², which should be the correct order of magnitude", and compare it against thermal energy to obtain an effective temperature Teff = kBT/ε, the parameter Ikeda et al. used to characterise the difference between large athermal and small thermal particles. The authors use it to place their samples in the crossover regime predicted by the simulation. Σ also normalises the yield-stress axis in Fig. 5(b) as σ₀ = σ_y d / Σ. Note that ε is a modelled quantity, not a measured one: its measured inputs are Σ and the droplet diameter d from SEM.
The authors give two estimates of the effective temperature. Their first, using ε = Σd², gives kBT/ε = (5.2–20.0) × 10⁻⁶ for their largest to smallest droplets, which they describe as lying in the crossover regime predicted by the simulation. They then argue that thermal fluctuations need only change a droplet's surface area slightly rather than create the whole surface from nothing, which gives ε = 0.01πΣd², "a factor of 30 smaller than our earlier estimate", and a range of kBT/ε = (1.7–6.4) × 10⁻⁴. The caption to Fig. 4(a) lists Teff = 5 × 10⁻⁶ for the 1.03 µm sample to 1.3 × 10⁻⁶ for the 2.03 µm sample. The paper treats the effective temperature as an order-of-magnitude estimate.
The rest of the paper is not this instrument's. Every flow curve, every yield stress, every Herschel–Bulkley and Three Component fit came from an Anton Paar MC302 rheometer running a 50 mm cone-plate geometry at 25 °C; droplet diameters and polydispersity came from a Topcon DS-150F field-emission SEM; volume fractions came from a balance. The Dropometer contributed one number to a paper with five figures and twenty-five panels. What that number contributes is specific: it sets the scale of the mechanical yield stress normalisation in Fig. 5(b), which puts this study's data on the same axes as Mason et al. (Σ = 9.8 mN/m) and Dinkgreve et al. (Σ = 1.5 mN/m), each scaled by its own surface tension. Fig. 5(a) makes the same comparison with a thermal scaling that does not use Σ.
If you need surface or interfacial tension values for emulsion work, we can show you the same pendant drop workflow on your samples.
Pendant-drop surface tension reported for TPM
The authors report TPM surface tension as Σ = 3 mN/m, measured using "the pendant drop method (Dropometer, made by Dropletlab)," and state that the result "is consistent with previous work" — the comparison is to Kraft et al., J. Phys. Chem. B 2011, 115, 7175 (the paper's ref. 35). The value is reported as a single number.
Σ feeds an assumed deformation-energy scale for Teff
Using ε = Σd², the paper reports kBT/ε = (5.2 − 20.0) × 10−6 for the largest to smallest droplets and describes this as lying in a crossover regime discussed in connection with simulation results.
Alternative ε estimate also depends on Σ
The authors then give a second estimate. Opening "A more cautious approach suggests that the effective temperature may be larger than the above considerations," they argue that thermal fluctuations need only change a droplet's surface area slightly rather than create it from nothing ("If a diameter fluctuation Δd/d = 0.1 is sufficient to allow a droplet to move past another"), giving ε = 0.01πΣd², "a factor of 30 smaller than our earlier estimate," and kBT/ε = (1.7 − 6.4) × 10⁻⁴. The paper therefore presents the effective temperature as an order-of-magnitude range under two assumptions.
Yield-stress scaling uses Σ directly
The paper defines a nondimensional mechanical yield stress σ₀ = σ_y d / Σ and presents yield stress versus φ nondimensionalised by the oil–water surface tension Σ in Fig. 5(b). The authors state: "Neither the thermal yield stress [Fig. 5(a)] nor the mechanical yield stress [Fig 5(b)] collapse the data perfectly." The Σ-normalised plot collapses their own samples fairly well for φ ≳ 0.72, where droplets are deformed, and collapses the combined dataset "fairly well even for lower volume fractions, with the sole exception being our large droplet sample with d = 2.03 µm."
Two distinct transitions in the same material, set by droplet size (Anton Paar MC302 rheometer)
The paper's headline result. Emulsions with droplets near 1 µm show a yield stress from φ ≈ 0.575, consistent with a glass transition at φ_g ≈ 0.58, and a further dramatic rise in yield stress at a jamming transition near φ_J ≈ 0.64. The 2.03 µm sample shows only jamming, with φ_c = 0.635 ± 0.008. The bidisperse sample behaves like the small-droplet samples, suggesting the small droplets dominate. The stated conclusion: "Our results show that liquid-solid transitions in dispersions are not universal, but depend on particle size." All of this was measured on the rheometer.
The raw rheology does not collapse onto a master curve (Anton Paar MC302 rheometer)
The Herschel–Bulkley and Three Component models fit the data within 10% of each other in least-squares error, so the yield stress is robust to model choice. But the HB flow index n varies strongly with volume fraction — near 1 at low φ, falling to 0.4–0.5 at the highest φ — and the authors note "We are unaware of any other data set with n varying so strongly with φ." Because the fitting parameters depend on φ, "our raw rheological data would not collapse onto a master curve," in contrast with prior hydrogel-particle work where n ≈ 0.5 was essentially constant.
What it shows
Yield stress versus volume fraction, nondimensionalised as σ₀ = σ_y d / Σ. This is the only panel in the paper whose axis carries the measured surface tension. Each series is normalised by its own oil's tension (this study's measured 3 mN/m, Mason et al.'s 9.8 mN/m and Dinkgreve et al.'s 1.5 mN/m), which is how this panel compares the datasets; Fig. 5(a) compares them with a thermal scaling instead. The authors report that the collapse is fairly good for their samples at φ ≳ 0.72 and reasonable for the combined data at lower φ, "with the sole exception being our large droplet sample with d = 2.03 µm."
What it shows
Yield stress versus volume fraction for all four samples, each labelled by mean droplet diameter — the figure that shows the small-droplet samples retaining a yield stress down to φ ≈ 0.58 while the 2.03 µm sample loses it below 0.643. Its caption reports the effective temperature, which uses the measured surface tension: "The effective temperature ranges from Teff = 5 × 10⁻⁶ for the d = 1.03 µm sample to 1.3 × 10⁻⁶ for the d = 2.03 µm sample." Panel (b) of this figure is a simulation result reprinted from Ikeda, Berthier and Sollich (Phys. Rev. Lett. 109, 018301) and contains no experimental data from this study.
A central goal of the paper is to compare glass-like and jamming-like rheological transitions across emulsions with different droplet sizes while keeping the emulsion formulation consistent (same oil, continuous phase fluid, and surfactant across samples). Within this framework, the surface tension Σ provides a single interfacial-material parameter used in the paper’s scaling arguments.
The reason a single interfacial number matters here is comparability. The authors point out that in earlier work "the large and small emulsion samples used different oils (thus with different surface tensions), so it was difficult to directly compare the rheological data between the samples." This study addresses that by using the same oil, continuous phase fluid and surfactant for all samples. The measured Σ then fixes the deformation-energy scale ε against which thermal energy is judged, and puts this study's yield stresses on the same axes as datasets measured on oils at 9.8 mN/m and 1.5 mN/m (Fig. 5b). Whether a dense emulsion stops flowing at φ ≈ 0.58 or φ ≈ 0.64 turns out to depend on droplet size, and the paper uses the effective temperature, which includes Σ, to describe how thermal the droplets are.
It is worth being precise about the size of that contribution. The Dropometer produced one scalar in a paper with five figures; the rheometer produced the rheology data and the SEM produced the droplet sizes. The surface tension is identical across all four of this study's samples, so within this dataset it separates nothing — the authors note that "for our own data at least, kBT and Σ are constant." Its role is at the boundaries of the study: setting the absolute energy scale, and providing one of the two scalings used to compare these results with the wider literature. That is a small share of the measurements, used in specific parts of the interpretation.
Pendant-drop Σ can be a key scaling input
Here, a single TPM surface-tension value (Σ = 3 mN/m) measured by "the pendant drop method (Dropometer, made by Dropletlab)" is used in two places: the assumed deformation energy ε = Σd² behind the effective temperature, and the mechanical normalisation σ₀ = σ_y d / Σ in Fig. 5(b). If you are running the same comparison, measure Σ on your own system rather than taking a literature value — the three emulsion studies compared in Fig. 5(b) span 1.5 to 9.8 mN/m for three different oil-in-water systems.
Use Σ with droplet size to estimate kBT/ε
The study uses ε = Σd² (and also ε = 0.01πΣd² under a stated Δd/d = 0.1 assumption) to compute kBT/ε ranges for the droplet sizes studied. The two choices differ by a factor of 30 — (5.2 − 20.0) × 10⁻⁶ versus (1.7 − 6.4) × 10⁻⁴ — and the authors describe the second as the "more cautious" one. Treat the effective temperature as an order-of-magnitude bracket, which is how the paper treats it: ε = Σd² is assumed and "should be the correct order of magnitude."
Normalize yield stress with Σ for mechanical scaling
The paper defines σ₀ = σ_y d / Σ and shows yield stress nondimensionalised by Σ in Fig. 5(b) to compare behaviour across droplet sizes and datasets. Expect partial success: the paper states that neither this normalisation nor the thermal one "collapse the data perfectly," with the authors' own 2.03 µm sample the one exception they note at lower volume fractions. The paper expects the mechanical (Σ) scaling to set the yield stress once droplets are deformed (above about φ ≈ 0.66 in the simulations), and its own samples collapse fairly well for φ ≳ 0.72; near φ_g ≈ 0.58 the paper argues the surface tension should be irrelevant and thermal motion dominates.
Keep formulation constant when comparing size effects
The experiments are framed around using the same oil for the droplets, the same continuous phase fluid, and the same surfactant for all samples while varying droplet diameter.
The authors used the pendant drop method on a Dropometer to obtain a single surface tension value for the TPM oil, Σ = 3 mN/m, which they note is consistent with previous work. This is the only Dropometer measurement in the paper; the flow curves and yield stresses came from an Anton Paar MC302 rheometer, droplet sizes from SEM and volume fractions from weighing.
Σ sets the energy scale in the authors' estimate of an effective temperature, using an assumed deformation energy ε = Σd², which gives kBT/ε of (5.2 to 20.0) × 10⁻⁶ from the largest to the smallest droplets. A more cautious second estimate, ε = 0.01πΣd², gives (1.7 to 6.4) × 10⁻⁴, and Σ is also used to nondimensionalize the yield stress in Fig. 5(b) as σ₀ = σ_y d / Σ.
From rheology, the authors report that emulsions with droplets near 1 µm show both a glass transition near φ ≈ 0.58 and a jamming transition near φ ≈ 0.64, while the 2.03 µm sample shows only jamming, so the liquid to solid transitions depend on droplet size. The paper reports Σ as one value without stating replicate count, drop volume or measurement temperature, and treats the deformation energy built on it as an order of magnitude estimate.
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