Size matters. In physics, forces scale differently with object size, and therefore surprising new phenomena might arise on the microscale compared to what is commonly understood on the macroscale from the sheer fact that different forces can dominate a physical process. A human has no chance of levitating on the surface of water – gravitational forces overcome surface tension – however, water striders have a much smaller size and so seem to break the laws of common physics. In this blog post let’s review scaling laws in a more rigorous way and apply them to some everyday problems.

### Intrinsic (intensive) material properties

The example in the introduction shows well that, with size, properties and forces change magnitude, so it would be nice to be able to describe these somehow using size-invariant quantities. There is a set of properties that are independent of surroundings or external effects, and these are called intrinsic (also called intensive or bulk) properties. To give an example: density of a material is an intrinsic property. If you have a block of material, regardless of its size, its density is always the same. (Here we disregard that density might change with pressure and temperature – this won’t change the concept of size invariance.)

On the other hand, weight is clearly an extrinsic (non-intrinsic) property: based on the object size, it changes magnitude.

Here are some examples of intrinsic properties (focusing on the physics-related ones rather than chemistry): density, permeability, permittivity, electrical conductivity, thermal conductivity, specific heat, temperature, viscosity, surface tension coefficient.

### The pizza crust effect

We’ll start with a simple example to illustrate the importance of scaling laws. Let’s assume you have a pizza with radius *R* and crust width of *c*. We can obtain the areas of the “regular” part, covered with tomato sauce and toppings, as so: *A _{toppings }= (R – c)²Π*, For the crust we use:

*A*, where the approximation is valid as usually

_{crust}= R²Π – A_{toppings}= 2RcΠ – c²Π ≈ 2RcΠ)*2R >> c*. (We can arrive at this even more simply: the perimeter is

*2RΠ*and there is crust all around with

*c*width, so the area is roughly

*2RΠ·c*. .) Now, let’s say that the crust of the average pizza is roughly 3 cm wide, regardless of the size of the pizza. Using the above equations, we can plot the two areas – the one occupied by the toppings and the one for the crust – as a function of the radius:

What does this plot tell us? The crust area scales linearly with the radius, while the topping area scales quadratically. At smaller radii, the linear function is dominant; at larger radii, the quadratic function is dominant. If you love crust, get a smaller pizza; if you are more for the tomato sauce and the real deal, buy a larger one.

This train of thought can be directly extended to 3D: for bread or bagel, the crust volume scales with the surface area (quadratic), while the internal soft part scales with the volume (cubic), so again, you can pick based on your preference.

### Apply the pizza crust effect to a real problem

Let’s get back to the water strider and human comparison. First, assume that a living creature has approximately the same density regardless of size. Its volume depends on the cube of the linear dimension (*L*) – if we double the height/width/depth, the volume change is 8-fold. In other words, the volume scales as *L*^{3}. But the density is intrinsic (fixed, *L*^{0}), so the mass is also going to scale cubically: mass = density X volumes → *L*^{0} *L*^{3 }= *L*^{3}. Moreover, the standard gravity (*g*) at a given geographical elevation is also *L*^{0}, so the gravitational force (F _{g}) scales as *L*^{3} again.

Surface tension is the effect of the cohesive forces within a liquid that try to minimize surface area. When a fluid surface is curved with radius *r*, the surface tension produces pressure as p = y/r , where y is the surface tension constant, in N/m, an intrinsic property. To obtain the force from the phenomenon, pressure must be multiplied by the curved surface. Let’s look at surface areas next. For a sphere of radius, *r*, the total surface area is 4*r*^{2}π. If we double the radius, the surface area changes 4-fold to 16*r*^{2}π. It has a quadratic dependence, or scales as *L*^{2}. A similar argument holds for other shapes. Now, we have everything for surface tension: F = pA = y/rA → *L*^{0} *L ^{-1 }*

*L*

^{2 }=

*L*, so the force from surface tension scales linearly with the object size.

^{1}This is an even more significant difference than the one we saw for the pizza crust! On a microscale the linear force dominates, which is the surface tension, allowing the water strider to roam around happily on the surface of a lake. However, gravity dominates on the macroscale – making us humans sink. (Making a few assumptions, the transition size could be obtained. If you wanted to do this, use a sphere since it has a simple and clear geometry.)

### What are the scaling laws exactly?

The process of finding the scaling laws in general is to write up each force equation as a combination of intrinsic variables and functions of length. Let’s check this out with a more complex example!

In all cases we have to be careful with the first assumptions. These sometimes can lead to seemingly contradictory or ambiguous results. The magnetic force for an electromagnet scales as *L*^{4}, assuming scale-independent maximum current density; however, the magnetic force of a permanent magnet scales as *L*^{3}, assuming scale-independent magnetic strength (see Ref 1). Especially in these situations, be sure to clearly state the assumed size-invariant quantities.

Heating of objects of various sizes can be investigated using scaling laws. The rate of heat flow through a surface is given by Q= hAΔT, where *h* is the heat transfer coefficient, *A* is the surface and ΔT the temperature difference. Moreover, the internal energy change is ΔQ =cmΔT, with *c* the specific heat capacity, *m* the mass and ΔT the temperature difference. Assuming for simplicity that Q = ΔQ / Δ*t*, rearranging from the time difference results in *Δt = cm/ht*. The specific heat capacity and heat transfer coefficient are *L*^{0} quantities (scale invariant), from the mass *L*^{3} scaling that was previously shown, as well as *L*^{2} scaling for the area. In summary Δ*t = cm/hA → L ^{0} L^{3 }L^{0 }L^{-2} = L^{1}*, . The time scale for heat transfer processes scales linearly with size – if boiling a pot of water (in a ~20 cm pot) takes 10 minutes, boiling it in a 20-micron cavity takes 10,000 times less: only 60 ms. This allows for the working principle of inkjet printers. In a fraction of a second, the ink is vaporized to expand and form a droplet to eject from the holder through an opening. Rapid thermal processes are attractive on the microscale for various applications: the polymerase chain reaction (PCR) for DNA sequencing can be carried out at an accelerated rate (Ref 2). Real-life examples include cooking, where you cut the meat and vegetables to cook them more quickly and uniformly, and coffee grounds, where smaller grain size and relatively larger total surface area allows for more flavor to be released.

### Summary

In nature most forces scale with either the volume or the area (and so they’re called the volumetric and surface forces respectively). However, on the microscale surface forces dominate. In fluid mechanics viscosity effects take over inertial effects, leading to low Reynolds number and laminar flow, allowing for the development of microfluidic devices. Electrostatic forces dominate inertia again, making electrostatic actuation feasible. More and more of these new physical phenomena due to scaling laws are discovered and utilized in cutting-edge technologies – one cannot even imagine what the future holds.

References

- S.M. Spearing, in Encyclopedia of Materials: Science and Technology, 2001
- C.D. Ahrberg et al., “Polymerase chain reaction in microfluidic devices”, Lab on a Chip, 2016, 16, 3866–3884