Nanobubbles (ultrafine bubbles)
Nanobubbles are sub-micron gas-containing cavities in aqueous solution.
Interfacial water and water-gas interfaces
Self-generation of colligative properties in water
Evidence for nanobubbles
Nanobubble preparation and use
Rationale for nanobubble stability
Bubbles are gas filled cavities with internal equilibrium pressures at least that of the external environment. Each bubble is surrounded by an interface with different properties than the bulk solution.
Only nanobubbles (see below) are stable for significant periods in suspension, with larger or smaller bubbles disappearing rapidly from aqueous suspensions unless stabilized with surface -active agents.
Solutions containing large numbers of bubbles are made by vigorous mixing of gas and water and are generally produced with a wide range of bubble diameters. When generated, small bubbles can be created at higher concentrations than larger bubbles (see left). The surface area of a volume of bubbles is in inverse proportion to the bubble diameter; thus, one mL of 100 nm diameter bubbles (2x1015 bubbles) has 1000 times more surface (240 m2) than one mL of 0.1 mm bubbles (2x106 bubbles, 0.24 m2). Small bubbles (< 25 µm diameter) have taut inflexible surfaces (like high pressure balloons) that limit distortion whereas large bubbles (~mm) have flexible surfaces (like low pressure balloons) and can divide (break up) relatively easily. The buoyancy of larger bubbles will cause them to rise to the surface of aqueous solutions. From theory, this has been proposed as following Stokes' equation, which is valid for particles at low Reynolds number: l
R = ρgd2/18μ
where R = rise rate (m ˣ s-1), ρ = density (kg ˣ m-3), g = gravity (m ˣ s-2), d = bubble diameter (m) and μ = dynamic viscosity (Pa ˣ s); according to this relationship a 2.5 μm diameter bubble rises 100 times slower (~0.2 mm ˣ min-1) than a 25 μm diameter bubble (~2.3 cm ˣ min-1) . The actual behavior of bubbles is more complex than this, however, with the Stokes equation rarely shown by experiment.
The degree of saturation next to a bubble depends on the gas pressure within the bubble. Smaller bubbles have higher internal pressure (see right) and release gas to dissolves under pressure into under-saturated solution whereas larger bubbles grow by taking up gas from supersaturated solution; thus small bubbles shrink and large bubbles grow. The rates of these processes depend on the circumstances. Also as bubbles rise the pressure on them drops due to the depth of the water and they consequentially enlarge and rise faster. Bubbles less than 1 μm diameter rise so slowly as not to be determinable due to low buoyancy and as they are additionally affected by random Brownian motion. m Accumulation of denser elements at the bubbles' extensive surfaces may contribute significantly to the low buoyancy of such small bubbles. Larger bubbles (25-200 μm diameter) do not obey the above Stokes equation but the Hadamard and Rybczynski equation with terminal velocity 1.5 times the Stokes velocity :
RH-R = 3R/2 = ρgd2/12μ
This is due to interfacial oscillations as the surface becomes greater. Bubbles larger than about 0.2 mm rise at rates proportional to their diameter (Rd=0.2-2 mm = 120 d , m ˣ s-1). Bubbles larger than about 2 mm diameter rise rapidly at almost the same rate irrespective of diameter (Rd>2mm = ~0.25 m ˣ s-1) . Larger bubbles (~> 0.2 mm) undergo significant shape changes on rising through the solution due to the resistance of the liquid. All these rise velocities are relevant to low concentrations of bubbles. Where the solution has high bubble concentrations the bubbles may be physically prevented from rising so fast.
In addition to and in competition to the effect of buoyancy, small bubbles (<25-50 µm diameter) shrink (see below right ), such that the overall behavior of micro-bubbles can be complex.
The electrostatic interactions between nanobubbles can be large enough for little coalescence to occur  but will slow any rise even more. The charge on the bubbles can be determined from their horizontal velocity (v, m2 ˣ s-1 ˣ V-1) in a horizontal electric field, where
v = ζε/μ
where ζ = zeta potential (V), ε = permittivity of water (s2 ˣ C2 ˣ kg-1 ˣ m-3) and μ = dynamic viscosity (Pa ˣ s). The zeta potential is generally negative but mostly independent of the bubble diameter. It depends strongly on the pH (see graph elsewhere) and the dissolved salt concentrations (increased ionic strength reduces zeta potential). As all the bubbles are similarly charged, their coalescence is discouraged. Also their division is not favored unless there is considerable energy input, with smaller bubbles requiring greater energy (see below). Thus, small bubbles can grow or contract, but rarely coalesce or divide.
According to the widely-accepted Laplace equation, the pressure inside gas bubbles is inversely proportional to their diameter, with excess pressures Pexcess (Pa) given by
Pexcess = 4γ/d
where γ is the surface tension (N m-1) and d is the cavity diameter (m). a Thus 10 µm and 100 µm bubbles contain gas at about 1.3 and 1.03 bar respectively. The pressure within nanobubbles may be affected by other factors and be much lower than expected (see below). a The gas-liquid interface of bubbles may be deliberately (or fortuitously) coated with surface-active materials, such as protein or detergent, that reduce the surface tension and hence the excess pressure in order to stabilize the bubbles. The concentration of surface active agents may also be used to regulate the bubbles size. Such coated microbubbles/nanobubbles may be used as ultrasound contrast agents  or for targeted drug delivery . [Back to Top ]
Evidence for nanobubbles
Stable nanobubbles (diameter 40-300 nm) were first reported (by Nature ) in the white cap of a breaking waves by sonar observation. Nanobubbles i [1632, 1989, 2013, 2108, 2307] are long-lived (» s) gas-containing cavities in aqueous f solution. e Although these bubbles are smaller than the wavelength of light and therefore too small to be visible to the naked eye or standard microscope (giving transparent solutions), they can be visualized by backscattering of the light from a laser pointer. Their concentration is increased by stirring, reduced by filtration and they are not present after degassing  . It is now thought that bulk nanobubbles may be present in most aqueous solutions, possibly being constantly created by cosmic radiation [2108, 2109] and agitation, and that surface nanobubbles are present at most surfaces. Generally as with larger bubbles, they have been found to possess negative zeta potential (~ -25 - -40 mV) in neutral solutions by electrophoretic light scattering. These nanobubbles may have far reaching physical, chemical and biological effects. They are in constant flux with gas molecules both leaving and entering continuously. They are under excess pressure as the surface tension causes a tendency to minimize their surface area, and hence volume. a For the same volume of bubble their surface area (A) increases proportional to the reduction in bubble diameter (D; A = 6/D).
Nanobubbles grow or shrink by diffusion according to whether the surrounding solution is over-saturated or under-saturated with the dissolved gas relative to the raised cavity pressure. The solubility of gas is proportional to the gas pressure and this pressure, exerted by the surface tension in inverse proportion to the diameter of the bubbles, increases greatly at small bubble diameters. a This dissolution process is, therefore, accelerated with an increasing tendency for gases to dissolve as the bubbles reduce in size. Such dissolution is increased by the bubble's movement and contraction during this process, which aids the removal of any gas-saturated solution around the cavities. Early theoretical calculations showed that nanobubbles should only persist for a few microseconds . However, the ease with which water forms larger visible bubbles, under slight tensile pressure well below the tensile strength of water, and the greater difficulty that occurs in this on degassing, both indicate the occurrence of gas-containing nanobubbles (cavities). Clusters of nanobubbles (bubstons) have been proposed that are stabilized by ionic solutes (and magnetic fields ) and containing gas at atmospheric pressure .
A model for the nucleation of stable nanobubbles in water has been developed based on water's cohesive nature . A high density of nanobubbles has recently been created in solution and the heterogeneous mixture lasts for more than two weeks . The total volume of gases in these nanobubble solutions reached about 1% v/v under pressure in 1.9 ˣ 1016 50-nm radius nanobubbles (equivalent to about 600 cm3 when converted to standard temperature and pressure) per 1 dm3 of water, and the liquid density was reduced substantially to about 0.988 g ˣ cm-3 . The surface tension of solutions containing large numbers of nanobubbles seems to be reduced by up to about 15% . Experimental determination, using an atomic force microscope, has shown up to 42% reduction in surface tension in surface nanobubbles, using a number of assumptions .
In contrast to the previous theoretical view, there is now much evidence that sub-micron-sized gas-filled nanobubbles can exist for significant periods of time both in aqueous solution [974, 1172, 1269, 1433, 1532, 1618] and at aqueous submerged hydrophobic surfaces [506, 1270]. d Their presence has been used to explain the behavior  of water bridges . h
Bulk phase nanobubbles can be easily detected by diverse techniques including light scattering j and a resonant mass measurement  technique k (e.g. Archimedes) that can simply distinguish them from solid (or liquid emulsion) nanoparticles. Interestingly, bulk nanobubbles are subject to Brownian motion, m so behave as though they have solid shells similarly to solid nanoparticles. Interestingly, the typical size of detected nanobubbles is about 150 nm diameter, which is the same as that reported for the initiation of bubble nucleation in champagne .
Surface nanobubbles can be detected by a number of different techniques, prominent amongst which is tapping mode atomic force microscopy (AFM) . Nanobubbles are commonly found on solid hydrophobic surfaces in solutions open to the air, where they appear to be quite stable  and may spread out to form pancake-like structures. Bulk nanobubbles are likely to be repelled from each other, and from negatively charged hydrophilic surfaces, at distance but may attach to such surfaces through water separated films, if they closely approach . Surface and bulk-phase nanobubbles can both give rise to the otherwise difficult to explain long range attraction between hydrophobic surfaces. As the temperature of aqueous solutions rises, the solubility of non-polar gases drops, so increasing the gas released and nanobubble volume and surface coverage  but generally having much lesser effect on nanobubble concentration.
Surface nanobubbles vary considerably in dimensions but typically they might have dimensions of r = 50 nm - 6 µm, rS = 25 - 1000 nm, h = 5 - 20 nm, with contact angles (θ = 135° - 175°) much greater than expected from macroscopic studies. The excess internal pressure is not great when the bubble radius is greater than about a micron. a Surface nanobubbles explain the increased liquid slip at the interface and the resulting lower drag . Extended flat surface nanobubbles are known as micro- or nano-pancakes. Surface nanobubbles and nano-pancakes may be partially stabilized by processes that are not available to the bulk nanobubbles. The hydrophobic wall repels water and raises the solubility of gas molecules as well as increasing the surface radius relative to the bubble size. The lifetime of surface nanobubbles is so long that they may be considered stable , and can withstand near-boiling temperatures .
The likely reason for the long-lived presence of nanobubbles is that the nanobubble gas/liquid interface is charged, introducing an opposing force to the surface tension, so slowing or preventing their dissipation. Curved aqueous surfaces may introduce a surface charge due to water’s molecular structure or its dissociation. It is clear that the presence of like charges at the interface will reduce the internal pressure and the apparent surface tension, with charge repulsion acting in the opposite direction to the surface minimization due to surface tension. Any effect may be increased by the presence of additional charged materials that favor the gas-liquid interface, such as OH- ions at neutral or basic pH or applying negative ions with an anti-static gun that reduces nanobubble diameter (see below) . This charge similarity, together with the lack of van der Waals attraction (the cavities possessing close to zero electron density) tends to prevent nanobubbles from coalescence.
Additional to this is the slow rate of gas diffusion to the bulk liquid surface from both surfaces and bulk-phase nanobubbles [1973, 1987]; in particular, nanobubbles in a cluster of bulk nanobubbles protect each other from diffusive loss by a shielding effect  effectively producing a backpressure of gas from neighboring bubbles that may be separated by about the thickness of the unstirred layer.
This slow dissolution will be even slower than might be expected due to the higher osmotic pressure at the gas liquid interface so both preventing the gas dissolving and driving dissolved gas near the interface back into the nanobubble . Also, it has been proposed that supersaturation around the nanobubbles may reduce the surface tension significantly, so reducing the pressure  and required backpressure. This concept of a thick interfacial layer is supported by the higher forces required to penetrate greater depths of surface nanobubbles [1986, 1987]. There is also evidence of a novel structure of water within this interface by use of synchrotron based scanning transmission soft X-ray microscopy (STXM) . The zeta potential of the nanobubble is shown if you mouse over the figure left. This zeta potential is reduced in the presence of higher ionic strength (positive charges interfering) and lower pH (less OH-) and with greater diameter bubbles (thinner unstirred layer).
Nanobubbles have a tendency towards self-organization  in much the same way as charged oil-water emulsions, colloids  and nanoparticles. This is due to their charge, long range attraction , slow diffusion and interfacial osmotic pressure gradients.
Where there are large numbers of bulk phase nanobubbles, such as in electrolyzed aqueous solutions, there is relatively large amounts of water associated with the surfaces, which can give rise to greater hydration effects due to their greater capacity for forming new hydrogen bonds. Nanobubbles have the effect of increasing the mobility of the water molecules in the bulk as shown by the increase in proton NMR relaxation time T2 .
The question arises as to why these surface charge effects are not seen to affect the determination of the surface tension when different conditions such as pH and solute are used. The answer may be partly that small nanobubbles are constantly moving such that they lose counter ions beyond their slip planes, and partly that the effect of the charged surface is stronger through the low-dielectric gas phase formed by the tightly curved surfaces.
Under 260 nm excitation wavelength, nanobubbles seem to give two wide structure-less photoluminescence bands
at 345 nm and at 425 nm that may be due to the electronic charge density induced by the concentration of hydrated ionic compounds at their interface [800b]. It has been shown that nanobubbles in (still) mineral water can be magnetized and retain this magnetization for more than a day . [Back to Top ]
Nanobubble preparation and use.
Nanobubbles may be initiated in under-saturated water due to temperature fluctuations with raised temperatures reducing solubility and causing saturation fluctuations . Such bubbles may then take considerable time to relax or attach themselves to surfaces. Nanobubbles can be made by electrolysis , by the introduction of gas into water at a high shear rate [1618, 2306], from smaller bubbles containing volatile liquids  , by saturation at higher pressures followed by pressure drop or by saturation at low temperatures followed by fast temperature increase jump. The stability of nanobubbles, together with their high surface area per volume, endows them with important and useful properties. Water containing nanobubbles can be used in water treatment  and as a surface cleaning material  (for example in toilets) without the need for additives or detergents and with low water usage. Their large gas-liquid interfaces pick up insoluble, but surface-active, fragrance and flavor molecules into aqueous solution for use in the food and fragrance industries . Recently, plasmonic nanobubbles are showing great promise in cancer treatment . Unfortunately, a continuing and general lack of awareness concerning nanobubbles in both ordinary and complex aqueous solutions has meant that many scientists have been uninformed of the effects that their presence and concentration may make on their studies.
The effect of surface charge on surface tension and nanobubble stability.
In the analysis that follows it is shown that surface charge can counter the surface tension (see also ), so preventing high pressures within the nanobubbles. Clearly the final net charge density at the surface is that required for stability. It may be expected that as the nanobubble shrinks, the charge density will increase. During this process, some charge density will be expelled to the bulk but it is not clear to what extent this will occur as the energy required for expulsion b must be less than the increase in energy due to the approach of the charges. In any case, surface charge density will always slow down the process of nanobubble collapse. Even at the equilibrium charge density, contained gas will dissolve if the solution interface is under-saturated (but see above), although this is unlikely if the exposed liquid water surface is also in contact with similar gas at similar pressure.
The effect of charges at the water/gas interface is shown opposite, with the surface negative charges repelling each other and so stretching out the surface. The effect of the charges is to reduce the effect of the surface tension. As the repulsive force between like charges increases inversely as the square of their distances apart the charges cause strongly increasing outwards pressure as bubble diameter lessens. As well as tending to increase the nanobubble diameters, surface charge will clearly also tend to increase the contact angles. The greater van der Waals attraction across the gas bubble also assists in flattening surface nanobubbles .
The surface tension tends to reduce the surface whereas the surface charge tends to expand it. Equilibrium will be reached when these opposing forces are equal. The expected increase in surface charge density as bubbles reduce in volume has been confirmed .
Assume the surface charge density on the inner surface of the bubble (radius r) is Φ (C ˣ m-2). The outwards pressure (Pout, Pa) can be found to give , c where D is the relative dielectric constant of the gas bubble (assumed unity), ε0 is the permittivity of a vacuum (= 8.854 pF ˣ m-1). The inwards pressure (Pin, Pa) due to the surface tension on the gas is , where γ is the surface tension (0.07198 N ˣ m-1, 25 °C). Therefore if these pressures are equal (Pout = Pin), rΦ2 = 4Dε0γ = 2.55 ˣ 10-12 ˣ C2 ˣ m-3 = ~ 0.1 nm ˣ (e- ˣ nm-2)2. For nanobubble diameters of 10 nm, 20nm, 50 nm, 100 nm and 200 nm the calculated charge density for zero excess internal pressure is 0.14, 0.10, 0.06, 0.04 e- nm-2 and 0.03 e- ˣ nm-2 bubble surface area respectively. a
Such charge densities are achievable; e.g. one surface anion to every about 250 surface water molecules would stabilize a 100 nm diameter nanobubble. The nanobubble radius increases as the total charge on the bubble increases to the power 2/3. Under these circumstances at equilibrium, the ‘effective’ surface tension of the water at the nanobubble surface is zero and the excess pressure is also zero. The presence of charged gas in the bubble clearly increases the size of the stable nanobubble. Further reduction in the bubble size would not be indicated as it would cause the reduction of the internal pressure to below atmospheric pressure and consequent inward gas flow into the bubble. The surface charge is discouraged from dissipating when the bubbles shrink due to the interfacial osmotic pressure gradient giving rise to the metastable nanobubble structures.
The theory above, would predict that greater surface charge density would allow decreased nanobubble diameter, as found in dilute salt solutions . However, nanobubble size also depends on the bulk properties and increased pH leads to increased nanobubble diameter together with the increase in OH- concentration . It has recently been shown how the stability of nanobubbles in the presence of an amphiphile varies with pH and ionic strength .
It is possible that the bubble could divide to give smaller bubbles due to the surface charge. Assuming that a bubble of radius r and total charge q divides to give two bubbles of shared volume and charge (radius r½= r/21/3, charge q½=0.5q), and ignoring the Coulomb interaction between the bubbles, calculation of the change in energy due to surface tension (ΔEST) and surface charge (ΔEq) gives:
ΔEST = +2 ˣ 4πγr½2 - 4πγr2 = 4πγr2(21/3 – 1)
The bubble is metastable if the overall energy change is negative which occurs when ΔEST + ΔEq is negative,
which gives the relationship between the radius and the charge density (Φ):
For nanobubble diameters of 5 nm, 10 nm, 20nm, 50 nm and 100 nm the calculated charge density for bubble splitting is 0.12, 0.08, 0.06, 0.04 and 0.03 e- nm-2 bubble surface area respectively. For the same surface charge density the bubble diameter is always about three times larger for reducing the apparent surface tension to zero than for splitting the bubble in two. Thus, bubbles will never divide unless there is a further energy input.
The presence of salt ions adversely affects nanobubble stability causing aggregation followed by coalescence at higher salt concentrations . The aggregation behavior appears similar to that of the salting out of colloidal particles due to the screening of the particle charge by the ionic strength of the solution. Coalescence is due to changes at the gas-water interface. [Back to Top ]
a The pressure inside gas cavities is theoretically given by the Laplace equation, Pin = Pout + 2γ/r,
where Pin and Pout are the cavity internal
and external pressures respectively, γ is the surface tension and r is the cavity radius. This equation is simply derived by equating the free energy change on increasing the surface area of a spherical cavity (= γΔA = 4πγ(r+δr)2 - 4πγr2) to the pressure-volume work (= ΔPΔV = ΔP(4/3)π(r+δr)3 - ΔP(4/3)πr3). Although it is not certain
that the Laplace equation holds at very small radii [1129, 1807] and it has been shown that surface tension may increase almost 20-fold to 1.3 N m-1 for 150 nm diameter droplets  in the absence of other effectors such as surface charge, this equation appears correct down to about a nanometer or so, below which a small correction must be applied . However, there may well be further contributions to the work required , due to the removal of surface-bound material, as the surface area contracts, that would lower the excess pressure. In the absence of any other surface effects such as solutes or charges, the excess pressures expected in a 50 nm radius spherical nanobubbles and a 50 nm diameter surface nanobubble (rS = 50 nm, r = 1000 nm), due to the surface tension minimizing the cavity surface, are 5.8 MPa and 0.14 MPa respectively. It has been proposed that supersaturation around the nanobubbles may reduce the surface tension significantly, so allowing stable nanobubbles . [Back]
b The free energy of surface absorption is expected to vary from about 4-10 kJ mol-1at higher concentrations to 25-40 kJ mol-1 at low surface concentrations (~10-3-10-4 nm-2) . [Back]
c This equation may be simply derived by considering the total surface charge (area ˣ charge density, 4πr2Φ) as equivalently concentrated at the center of the spherical cavity and that this charge then exerts a force on the same charge at the surface. The force would be (4πr2Φ)2/4πDε0r2 from basic electrostatics (Coulomb's law), and therefore the pressure (force per unit area) would be (4πr2Φ)2/(4πDε0r2 x4πr2) = Φ2/Dε0. However for each part of the surface the force has been double counted (towards the surface and towards the center; equivalent to the surface again), therefore the final pressure concerns only half this force (= ). [Back]
d There is some dispute over whether the density depletion often found at hydrophobic surfaces is real in some cases . Some hydrophobic liquid-water interfaces behave differently, with no vapor-like layer being observed . [Back]
e Another interesting phenomenon in aqueous solutions is the antibubble where a water drop is held, surrounded by a gaseous film, within the bulk liquid  [Back]
f Although first reported that nanobubbles seem to be specific to water and aqueous solutions , more recent work indicates that surface nanobubbles can be produced in non aqueous liquids that can form three-dimensional hydrogen-bonding networks, such as formamide, but not hydrogen-bonding liquids that cannot form three-dimensional networks, such as ethanol .. Such bubbles produced in other non-aqueous solutions are not stable and disappear rapidly. Freely existing nanobubbles have so far only been found in water and aqueous solutions. [Back]
g Similar results to these have been published for dilute salt solutions , where the nanobubble size is seen to increase with reduction in the salt concentration. [Back]
h The floating water bridge is a stable nearly cylindrical tube of water of 1-2 mm diameter extending up to 25 mm between two beakers of pure water under the influence of a large (15-25 kV) applied electric potential difference . [Back]
i Nanobubbles are generally recognized in current (2015 and prior) scientific literature as those gaseous cavities with diameters less than a micron. As such cavities (bubbles) are often greater than 100 nm in diameter but the term 'nano' is applied mostly to particles of smaller diameter (< 100 nm, ISO/TS 27687:2008), sub-micron bubbles should probably be known as ultrafine bubbles in the future. [Back]
j Dynamic light scattering (DLS). This determines the fluctuations in the scattering of laser light traveling through the sample solution . The fluctuations are due to the Brownian motion m of the particles with larger bubbles giving greater scattering but slower fluctuations. Analysis of the total signal gives both the concentration and size distributions of the nanobubbles.
Nanoparticle tracking analysis (NTA) ia a related technique (e.g. NanoSight) can track individual bubbles within a small volume (e.g. 100 µm ˣ 80 µm ˣ 10 µm, 80 pL), so ascertaining the x- and y- movement in a given time. The NTA method can be used to analyze lower bubble concentrations than DLS. The calculated diffusion constant can be used to determine the hydrodynamic sphere-equivalent radius, r, of the particles using the Stokes-Einstein equation where Dt is the translational diffusivity (m2 ˣ s-1 ), R is the gas
constant (J K-1 mol-1), T is the temperature (K), N is Avogadro's
number (mol-1) and η is dynamic viscosity (Pa ˣ s). Each bubble is individually sized with the bubble concentration determined from the number of bubbles within the field of view. Some instruments also give the zeta potentials of the bubbles. [Back]
k Resonant mass measurement. Nanobubbles, in about 1 nL solution, passing around the resonator (shown green in (a) right) changes its effective mass and shifts the resonator's resonant frequency (b). Movement of the resonant frequency (Δω shown in (c) right) gives an accurate and precise measure of a particle's buoyant mass (< ~ 1 fg, < ~100 nm diameter cavity), so easily distinguishing bubbles (positive buoyant mass; increasing frequency) from particles (negative buoyant mass; decreasing frequency). [Back]
l Bubble and particle flow and liquid molecular and turbulent motions are often described by means of a number of empirical relationships, involving
dimensionless numbers. The most important of these is the Reynolds number (Re), which relates the inertial force due to the flow of solution to the viscous
force resisting that flow. Low Re indicates streamlined flow whereas higher Re indicates progressively more turbulence, there being a critical value for Re,
dependent on the configuration of the system, at which there is a transition
from streamlined flow to turbulent flow. Re is defined in terms of Lfm/h or Lf/n, where L is the characteristic length of the system (m), fm is the mass
flow rate (kg ˣ m-2 ˣ s-1), f is the fluid velocity (m
ˣ s-1), h is the dynamic
viscosity (kg ˣ m-1 ˣ s-1) and n is the kinematic viscosity (m2 ˣ s-1). Re is higher at high flow rates and low viscosities where turbulent flow occurs and lower at low low rates and high viscosities where laminar flow occurs, characterized by smooth motion. [Back]
m Brownian motion, named after the botanist Robert Brown who noticed the phenomenon in 1827, is the irregular (apparently random) motion of small particles suspended in water (or other fluids) resulting from their collision with the water molecules . This irregular movement is observable down to extremely small timescales. The mean squared displacement is proportional to the time elapsed (t) and the diffusivity (D); the averaged net displacement (Δx) being proportional to the square root of the elapsed time.
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