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Accurate Modeling of Viruses – Spherical Particle Shape, Size, and Density

Spherical Particle Shape of Viruses and Droplets:

Spherical Particle Shape of Coronavirus Research from CDC
Illustration, created at the Centers for Disease Control and Prevention (CDC), reveals ultrastructural morphology exhibited by coronaviruses.
Photo by CDC on Unsplash

Due to the global pandemic, there has been a big focus on Coronavirus research and a lot of pressure for researchers and scientists to understand how Coronavirus spreads, as well as how we can effectively stop that spread by implementing effective cleaning protocols, prevention of respiratory droplets traveling, disinfecting of surfaces, or incorporating chemical means.

What does coronavirus particle look like?

As the illustration from CDC shows, coronavirus has a spherical particle shape. It approximates a nanosphere of about 100 nanometers (100nm or 0.1um) in diameter. Spikes that adorn the outer surface of the virus impart the look of a corona surrounding the virion when viewed with electron microscope.

We also know that viruses are often transmitted through respiratory droplets produced by coughing and sneezing.

What does a respiratory droplet look like?

Respiratory Droplet Modeling - Spherical Particle Shape
Photo by Nenad Radojčić on Unsplash

The droplets can be larger at greater than 5 μm in diameter and fall rapidly to the ground, or smaller at less than 5 μm in diameter and remain suspended in air for significant periods of time. The characteristic diameter of large droplets produced by sneezing can be as high as 100 μm and sometimes larger.

Respiratory droplets are an order of magnitude larger than the coronavirus particles but can similarly be approximated as having a spherical particle shape of a microsphere.

Virus particles inside of a respiratory droplet

When envisioning the virus particles traveling inside respiratory droplets, we are looking at numerous nanospheres (virus) trapped inside of and being carried by a plurality of microspheres (droplets). This is the system that we need to model in order to best understand the behavior of viruses on the macro scale, the transmission of the virus particles before they get inside a living organism.

Mathematical Modeling of the Spread of the Virus:

Regardless of what Covid-19 mitigation approach is being investigated by the scientists, it almost always starts with designing a computer model with the goal of predictably and reliably simulating the behavior, reactivity, and spread of the virus particles inside of respiratory droplets.

In order to test these mathematical models, scientists need to conduct real-life verification and validation experiments with particles that approximate the physical characteristics of actual viruses and respiratory droplets as much as possible, including particle shape, size, and density.

Modeling with Stoke’s Law

Stoke's Law for Modeling Settling Velocity of Spherical Particle Shape
Terminal fall velocity of a particle. FD is the fluid drag and FG is the submerged weight of the particle.

The investigator needs to understand how spherical particles behave in their particular fluid, at specified environmental conditions. This behavior of spherical particles in another media is best characterized by Stoke’s law.

Stokes’s law is a mathematical equation that calculates the settling velocities of small spherical particles in a fluid medium (fluid may be air). The law, first set forth by the British scientist Sir George G. Stokes in 1851, is derived by consideration of the forces acting on a particular particle as it sinks through a fluid column under the influence of gravity.

Stoke’s law assumes a spherical and rigid particle shape.

In the Stoke’s law, the drag force FD  on a spherical particle of radius r  is given by:

Stoke's Formula

  • FD  is the drag force on the sphere falling through the fluid in newtons (N)
  • η is the viscosity of the fluid in kilograms-per-meter-per-second (kg/m/s)
  • r is the radius of the sphere in meters (m)
  • vT  is the terminal velocity of the sphere in meters-per-second (m/s)

The formula for the settling velocity vT is given by:

vT = gd2p – ρm)/(18μ)
  • vT is the terminal velocity of a spherical particle
  • g is the gravitational acceleration – for Earth, equal to 9.8m/s2
  • d is the particle diameter
  • ρp is the true density of the particle;
  • ρm is the density of the fluid; and
  • μ is the dynamic viscosity of the fluid.

As the formulas show, primary characteristics of importance for calculating fall or settling velocity of particles are spherical particle shape (assumption), particle size (radius), and true particle density.

Spherical Particle Shape

Spherical Particle Shape of a whole influenza (flu) virus.
3D generated rendering of a whole influenza (flu) virus
Photo by CDC on Unsplash

Just like Stoke’s law, most mathematical models are based on the assumption of perfectly spherical particle shape. Conveniently, both the coronavirus particles and the respiratory droplets are close to perfect spheres.

Does the spherical particle shape of the test material really matter in validation of the model? Yes, absolutely.  Particle shape plays a critical role in how the particles travel, behave, and interact with each other and their environment.

Particle shape influences its flow characteristics. Spheres and microspheres are well-known for their ball-bearing effect. They roll and spread out. If you spill spheres on the ground, the floor will turn into a skating rink. Spherical particles glide past each other and surfaces they contact with minimal effort and friction, unlike the non-round granular or flake particles with uneven and sharp edges. This can be part of the reason that coronavirus spreads so easily compared to other viruses that do not have spherical shape.

Particle Size

As you can see from Stoke’s law formula, microsphere particle size is the most critical variable for determining settling velocity. The settling velocity, and, as a result, settling time, are proportional to the diameter of the spherical particle squared. The larger the sphere diameter, the faster the particle will settle. The smaller the particle diameter, the longer it will stay suspended in the fluid.

True Particle Density

The second most critical variable is density delta, or the difference in the true density of the particle and the density of the liquid. Settling time and velocity are proportional to density delta (also known as density mismatch), which means that the closer the particle density is to the density of the medium it is falling through, the longer it will take for it to overcome gravity and fall to the ground.

Keep in mind that even the slightest variation in density matters, and densities of liquids sometimes vary significantly with changes in temperature, pressure, and materials added to it.

Settling velocity is inversely proportional to the viscosity of the fluid. Obviously, the thicker (more viscous) the fluid, the longer the settling time, the thinner (less viscous) the fluid, the faster the settling time.

Validation Particles for Testing a Mathematical Model

A scientist designing a flow visualization experiment that includes spherical tracer particles is designing a complete system that takes into account the exact properties of the fluid medium at specific environmental conditions, matched to the diameter and density of the microspheres, and also ensuring that no external forces are affecting the behavior of the spherical particle.

Microsphere Materials - Solid Polymer Spheres
(Product Code: UVPMS-BY) Fluorescent yellow spheres from Cospheric LLC

Brightly colored highly spherical particles and specifically colored fluorescent polyethylene microspheres are often used by scientists as highly visible tracer particles for the purposes of Particle Image Velocimetry (PIV) or fluid flow visualization.

Saliva density is in the range of 1.002–1.012 g/ml. To model the flow of respiratory droplets, we would want to use microspheres that approximate not only the particle shape, but also the size and true particle density of saliva at around 100um in diameter with a density close to 1.00g/ml (or 1.00g/cc).

It is important for the researches to select the right model test particle to ensure accurate simulation of the target properties, representative performance of the particle in the fluid, the accuracy of data collected, and success of the fluid flow study. By verifying the efficacy of the computer model scientists will ensure confidence in the conclusions drawn from the research study and their transference into real-life scenarios.

For your convenience:

Stokes Law Calculator

Large selection of precision density particles with spherical particle shape