The qualitative method of poking a particle to determine the particle phase (i.e. solid/semisolid vs. liquid) was introduced by . This approach was expanded upon by by quantifying flow rates after poking and determining viscosities from simulations of flow. Figure 1 illustrates schematically the experimental setup for the poke-flow measurements. First supermicron particles of interest (either particles containing sucrose–water or polybutene standards) were deposited on a hydrophobic glass substrate (18 mm diameter) (Hampton Research, Canada). The glass substrate was then mounted onto a custom-made flow cell, similar to those described previously but with a small hole added at the top through which a needle could be inserted . RH within the cell was controlled through the use of humidified ultrapure N2 gas, a stream of which was flowed continuously through the cell. Gas flow was maintained at 1200 sccm for all experiments. The dew point temperature of the humidified N2 gas was measured at the outflow of the cell using a hygrometer with a chilled-mirror sensor (General Eastern, Canada). The hygrometer was calibrated prior to experiments being performed by determining the deliquescence relative humidity (DRH) of ammonium sulfate particles and comparing the observed DRH value to that in the literature (≈ 80.3 % RH at 20 ∘C). The uncertainty (1σ) of the hygrometer was ±0.5 at 80.3 % RH after calibration. The temperature of the sample was measured directly below the flow cell using a thermocouple. Particles were monitored during experiments using a reflectance optical microscope (Zeiss Axio Observer, 40× objective) and recorded using a CCD camera.

Two different types of needles were used for experiments. For particles of low viscosity, a sterilized sharp needle (0.9 mm × 40 mm) (Becton-Dickson, USA) was used. However, at higher viscosity particles stuck to the needle and as a result were removed from the substrate. As such, a second set of needles was used (RS-6063, Roboz Surgical Instrument Co., USA), which were coated with a hydrophobic Dursan coating (SilcoTek, USA) to prevent material sticking to the needles. The needles were mounted to a micromanipulator (Narishige, model MO-202U, Japan) and inserted through the hole at the top of the flow cell. The effect of the presence of the hole at the top of the flow cell upon RH in the cell was determined to be minimal by studying the DRH of ammonium sulfate and potassium carbonate particles with the hole open and the hole closed. The DRH of ammonium sulfate particles differed by < 0.2 at 80 % RH, whilst the DRH of potassium carbonate particles differed by < 0.4 at 43 % RH. The micromanipulator was used to move the needle in the x, y, and z axes.

For a given experiment, the tip of the needle was aligned vertically above the centre of a particle and then moved down in the z direction, first penetrating the particle at the peak of its spherical cap geometry and subsequently coming in contact with the substrate beneath. After the needle was removed, the material of the particle was present in a non-equilibrium half-torus geometry and began to flow in order to minimize the surface energy of the system. Eventually the hole at the centre of the half torus completely closed, and the particle returned to its original spherical cap morphology.

Analysis of optical images captured during each experiment was performed using Zen software (Zeiss). The hole at the centre of the half torus was traced and the area of the hole calculated. An equivalent area diameter of the hole was calculated via the relationship d=(4A/π)1/2, where d is the equivalent area diameter of a hole of area, A . The experimental flow time, termed τexp, flow, was assigned as the time taken for the equivalent area diameter to decrease to 50 % of its original value. This definition of experimental flow time was chosen as it allows experimental flow time of more viscous particles, the holes of which may not fully close on a laboratory timescale, to be measured.

In some of the experiments, a point around the inner edge of the half torus appeared to be pinned to the hydrophobic surface. This behaviour may suggest that in some cases the needle scratched the surface. Particles that exhibited this pinning behaviour were excluded from analysis in order to avoid influencing the results. Pinning affected < 20 % of all the particles that formed a torus geometry after poking.

Sucrose–water particles were produced on siliconized hydrophobic glass substrates using a nebulizer (Meinhard; product no. TR-50-A1). First sucrose (Sigma-Aldrich, purity > 99.5 %) was combined with high-purity water (Millipore, 18.2 MΩ cm) to produce an aqueous solution. The solution was then nebulized to produce submicron-sized particles, which subsequently coagulated on the substrate to form supermicron-sized particles, 25–60 µm in diameter. In the sucrose–water experiments, particles were allowed to equilibrate at a given RH for 30–60 min prior to being poked, which, given their size and approximate viscosity, was sufficient time for the particles to have a water content in equilibrium with that of the gas flowing through the cell (Shiraiwa, 2011; Price et al., 2014).

Particles consisting of polybutene standards (N450000 and N2700000; Cannon Instrument Company, USA) were prepared on hydrophobic glass substrates using a pipette. The samples were heated to 60–70 ∘C over ≈ 30 min to aid the production of particles by reducing the viscosity of the material, with material being picked up on a pipette tip and the pipette being “flicked” towards the substrate, resulting in particles being formed on the substrate. After particle production, the particles were allowed to equilibrate at dry (< 0.5 % RH) conditions at room temperature for 60 min. Particles consisting of polybutene standards that were poked ranged in diameter from 40 to 70 µm.

Flow in the poke-flow experiments was simulated using the laminar two-phase flow, moving mesh mode within the micro-fluidics module of COMSOL (version 4.3a), a finite-element analysis software package. The Navier–Stokes equation, including surface tension, is used to describe the transport of mass and momentum, whilst the arbitrary Lagrangian–Eularian (ALE) method is used to track the evolution of the fluid over time as it flows to attain a spherical cap geometry and thus minimize the surface energy of the system. Simulations were performed with a mesh that consisted of ≈ 5800 elements and had a mesh spacing of 3.92–337 nm.

A top view of the half-torus geometry used to simulate the experimental observations is shown in Fig. 2a, where the dotted line is at the midpoint between the inner and outer edges of the ring of material forming the torus. The initial radius of the hole at the centre of the half torus is denoted as R0–r0, where R0 represents the distance from the centre of the hole to the midpoint of the ring of material that creates the torus, whilst r0 represents the radius of the ring of material. The half torus had two distinct surfaces (Fig. 2b). Surface 1 represents the air–fluid interface, which was allowed to undergo free deformation in all dimensions. Surface 2 is the fluid–substrate interface, which was allowed to undergo free deformation in the horizontal x–y plane, but not in the vertical, z, direction. In the simulations, the size of the hole in the half-torus geometry decreases in an axi-symmetric manner. The time taken for R0–r0 to decrease to 50 % of the initial value was assigned τmodel, flow.

The values of certain physical properties could be defined for a given simulation. The dimensions R0 and r0, and the equilibrium contact angle were determined from measurements, and the values of surface tension (Surface 1, Fig. 2b), slip length (which describes the interaction at Surface 2 in Fig. 2b), and the density of the material were determined based on literature values. These values are detailed in Tables 1 and 2. For each particle that was poked and formed a half-torus geometry, and was not significantly influenced by scratches (determined visually, as discussed above), lower and upper limits of viscosity were determined via simulations. The dimensions of the particle (R0 and r0) were used, as were the relevant parameters in Table 1 (for particles of sucrose–water) or Table 2 (for particles of polybutene standards). Values from row 2 of Tables 1 and 2 were used for simulations of the lower limit of viscosity for a particle, whilst values from row 3 of Tables 1 and 2 were used for simulations of its upper limit of viscosity. The viscosity used in the simulations of a particle was varied until τmodel, flow agreed with the particles τexp, flow (to within 1 %).

A proportion (≈ 30 %) of the sucrose–water particles that were poked had dimensions where r0/R0 > ≈ 0.4. Simulations of the lower limit of viscosity for many of these particles gave rise to a non-physical geometry whereby the inner edge of the half torus geometry appeared jagged and wavy – a phenomenon not observed during experiments. Further study of this phenomenon revealed that this was the result of the stretching of the mesh elements at the moving front of the particle, and use of a finer mesh did not prevent this from occurring. As such, all sucrose–water particles of dimensions r0/R0 > ≈ 0.4 were removed from the study. No particles of the polybutene standard fell into this size range.

Shown in Fig. 3 are examples of optical images of sucrose–water particles at 48.8, 52.7, and 58.8 % RH recorded during typical poke-flow experiments. Prior to being poked the particles may be described geometrically as a spherical cap (Fig. 3a1, b1 and c1). Just after being poked the geometry of the particles can be described as a half torus – a ring of material with a hole at its centre (Fig. 3a2, b2, and c2), which is energetically unfavourable compared to that of a spherical cap. For the particles in Fig. 3, τexp, flow was determined to be 11.25, 3.75, and 1.25 s at 48.8, 52.7, and 58.8 % RH, respectively (Fig. 3a3, b3, and c3). Following τexp, flow the material continued to flow, and eventually re-attained its initial, energetically favourable, spherical cap geometry (Fig. 3a4, b4, and c4).

The τexp, flow values of each of the individual sucrose–water particles poked and analysed is shown in Fig. 4a. Experimental flow times increased from ≈ 150 ms at 59 % RH to ≈ 40 min at 37 % RH. The millisecond time resolution of the camera precluded experiments being performed at RH > 60 % as the closure time for the sucrose–water particles was too fast to measure. Lower and upper limits of viscosity were determined for each individual particle using their dimensions and τexp, flow (Fig. 4b). Between 59 and 36 % RH, the viscosities for individual sucrose–water particles range from 1.0 × 101 to 6.6 × 106 Pa s, with the upper limit of viscosity for a given particle being a factor of 16 to a factor of 140 larger than the corresponding lower limit of viscosity, with the uncertainty mainly due to the value of the physical properties used during simulations.

In Fig. 4c, the viscosities of individual particles are grouped by RH, and previously reported values of the viscosity of sucrose–water particles are included for comparison . Viscosities have been determined by grouping particles based upon RH, with lower and upper limits of viscosity from particles in the group reported. The lower and upper limits of viscosity of the group of particles poked at ≈ 59 % RH are 1.0 × 101 and 1.6 × 104 Pa s, whilst at 37 % RH the corresponding values are 7.2 × 104 and 4.7 × 106 Pa s, respectively. As shown in Fig. 4c, the results are in good agreement with , who measured a viscosity of 103 Pa s at 54 % RH using a rotational controlled stress rheometer, and with , who recently reported mean measured viscosities of ≈ 5 × 102 to ≈ 3 × 106 Pa s between 60 and 37 % RH using holographic optical tweezers.

Figure 5 shows examples of optical images of particles of the polybutene standards recorded during poke-flow experiments at 0 % RH. As for sucrose–water particles, the geometry of the particles of standard solution could be described as a spherical cap prior to being poked (Fig. 5a1 and b1) and a half-torus after being poked (Fig. 5a2 and b2). Upon removal of the needle the material flowed, with the size of the hole at the centre of the half-torus geometry decreasing over time. For the particles in Fig. 5, τexp, flow was determined to be 2.50 and 6.00 s, respectively (Fig. 5a3 and b3). The particle continued to flow after τexp, flow and eventually re-attained its initial, energetically favourable, spherical cap geometry (Fig. 5a4 and b4). The mean experimental flow times, τexp, flow, were 2.79 s (σ=0.39) for the lower viscosity standard #1 and 7.86 s (σ=1.65) for the higher viscosity standard #2.

As done in the sucrose–water experiments, τexp, flow values from individual particles were converted into viscosities using simulations. To determine upper and lower limits for the viscosities of the standards, we took the upper and lower limits of the viscosities determined from the individual particles. Using this approach and the parameters listed in Table 2, the simulated lower and upper limits of viscosity for the particles poked were 2.0 × 102 and 1.2 × 104 Pa s for standard #1 (N450000), and 3.1 × 102 and 2.4 × 104 Pa s for standard #2 (N2700000). These values are in good agreement with value reported by the manufacturer, Cannon Instrument Company (see Fig. 6).