4 Scaling laws
Prof. Subhasis Ghosh
2.1.1 Scaling in Mechanics
2.1.2 Scaling in fluids
2.1.3 Scaling in heat transfer
2.1.4 Scaling in Biology
SIZE MATTERS: SCALING DEPENDENCE OF PROPERTIES
As we go from the macroworld to the nanoworld, one comes across two limits. As discussed in first chapter, when the characteristic dimensions of the systems are reduced from the macroscopic to microscopic domain, the effects of gravitational force become negligible as compared to electromagnetic forces. Adhesion, frictional effects, surface tension dominate over gravity. This implies that our reasoning behind the explanation of the properties at the macroscopic level, is no longer valid. However this argument does not hold in case of all the properties. The boundary between the macroscopic and mircoscopic worlds is not sharp and in most cases this boundary depends on the particular effect or property at macroscopic level under consideration. When the characteristic size decreases further to the nanometer range, another limit comes into play. While the macroscopic properties of matter can be explained at the micro to nano level, but the surface effects become dominant at the nanometre scale. Moreover, in nanometer scale, quantum effects start dominating over classical and semiclassical effects.
The scaling laws can be defined as proportionality relations of any parameter associated with a system with its length scale. Different scaling laws reveal the fact that shrinking of a body does not only lead to size reduction, but also leads to modifications of different physical effects and properties of the body. Scaling laws are generally deduced at the macroscopic level, and in some cases may be used to understand the behavior of the microscopic and nanoscopic level. But in many cases, the scaling laws deduced at macroscopic level have to be used carefully due to the fact that several ‘macroscopic’ ‘experimental conditions are not valid particularly at nanoscopic level. To explain different properties at nanoscopic level, there is, thus, a need to modify our understanding and reasoning. This results in different scaling laws to understand the origin of the differences between the macroworld and the nanoworld. In the following sections, we discuss how the properties and effects change with the dimension of the object and how this is incorporated to estimate the surprising behavior of materials at nanoscale.
2.1 SCALING LAWS
Consider a system with all the three dimensions vary proportional to L. Hence surface S and volume V
of the object vary as L2 and L3, respectively. assume that all other material properties such as, density, modulus, strength are constant.
~ 2 and ~ 3 (2.1)
This implies that mass scales to L3,
~ 3 (2.2)
2.1.1 Scaling in Mechanics
First we start with the scaling laws [Ref.1] pertaining to classical mechanical systems. Among the most common forces is gravity. The force of gravity, , on an object on the surface of the earth is = ∝ , where is the acceleration due to gravity. Thus,
The pressure exerted by a body on the ground will be,
~3(2.3)
At the microscopic level, the adhesion forces dominate over gravitational force. The adhesion force between two surfaces either between solid and solid or solid and liquid arises due to forces between atoms and molecules. The adhesion is mainly due to Van der Waals forces, Fvdw. The attractive force experienced by an infinite flat slab placed at a distance x from another infinite flat slab is given by
= ⁄ ~ 3⁄ ~ (2.4)
where H is Hamaker constant [Ref.2], H= π2 C n1 n2, where n1 and n2 are the number of atoms/molecules per unit volume in two slabs and C is the coefficient in the particle-particle Van der Waals pair potential, φ(r)=-/r6. H is of the order of 10−19 J and of 10−20 J in air and water, respectively. This relation is valid for x between 2 and 10 nm. It is obvious that Fvdw(x) varies with the contact area. This implies
( ) ~ 2(2.5)
( ) ~ −1(2.6)
Hence, the adhesion force dominates the gravitational force at nanoscale. Gravity may be neglected at very small scales i.e. at micro and nanoscales.
The frictional force between two sliding surfaces relative to each other is given by = , where is the coefficient of friction. This implies if is constant, then
~ 3 (2.8)
The frictional force generally does not depend on the contact area. The reasoning behind this law is that two rough bodies at macroscopic level touch each other only at very few points and the actual contact area is much smaller than the geometrical contact area. But this is not the case at the microscopic level due to surface roughness. A scanning probe microscope, like atomic force microscope (AFM) can be used to study friction at nanoscopic level by measuring the forces between the probe and the sample as a function of their mutual separation. The tip in AFM is a well-defined sharp pointed contact which enables to measure the interaction forces with nanometer resolution.
It is known for quite some time that continuum mechanics reasoning breaks down at the nanoscale. At this scale, as it has been emphasized the macroscopic physical laws are no longer valid. For example, the friction force (Ffr) is no longer linearly proportional to the applied load, whereas it does depend on the contact area. An understanding of how friction force depends on applied load and contact area at nanoscale is essential for the design of miniaturized mechanical devices and several other applications. It has been shown using massive
Fig. 2.1 A macroscopic contact that appears smooth and continuous is usually composed of multiple contact points between many microasperities. The friction law for a nanoscopic contact, a single or few asperity contacts, is not known. [Ref.3]
molecular dynamics simulations that in nanoscale contacts, friction force depends linearly on the number of atoms that chemically interact across the contact at the interface. If contact area is proportional to the number of interacting atoms, then certain macroscopic law, i.e. linear relationship between friction force and contact area can be extended to the nanoscale. Further it is shown that as the adhesion between the contacting surfaces is reduced, a transition takes place from nonlinear to linear dependence of friction force on load, as in case of macroscopic level. The breakdown of continuum mechanics pertaining to different classical mechanical systems can be understood as a result of the multi-asperity nature of the contact.
2.1.2 Scaling in Fluids
Viscosity is a property of a fluid that offers resistance to flow and arises due to collisions between neighboring particles in a fluid that are moving at different velocities. When a fluid is flowing through a tube, the particles which compose the fluid move more faster along the e tube’s axis, compared to those particles move near the wall of the tube. This results some stress which is the pressure difference between the two ends of the tube) is needed to overcome the friction between particle layers to keep the fluid moving. According to continuum mechanics, the stress required for liquid flow is proportional to the fluid’s viscosity. A fluid that has no resistance to shear stress is known as an ideal fluid. Zero viscosity is observed only at very low temperatures in superfluids. Otherwise, all fluids have positive viscosity, and are technically said to be viscous. Generally, a liquid is said to be viscous if its viscosity is substantially greater than that of water. A fluid with a relatively high viscosity, for example, pitch or glass , may appear to be a solid. As the friction and its effects are quite different at nanoscale, the continuum mechanics based Navier Stokes’ equation that is commonly used for analysis of liquid flow at the macroscopic level can thus not be used as we go to the nanoscale.
The most notable difference between the liquid flow at macro and nanoscale dimensions is the high surface area to volume ratio of the fluid channels. Relatively high surface area of liquid leads to effects such as adsorption of molecules onto the nanochannel surface and changes in effective viscosity. These and other interesting nanoscale effects lead to unique phenomena in this regime. The kinematic viscosity ν is the ratio of the dynamic viscosity η (which defines the resistance to shearing flows) to the density of the fluid ρ.
= –(2.9)
This is a convenient approach in analyzing the Reynolds number, which expresses the ratio of the inertial forces to the viscous forces:
= – =(2.10)
where L is a typical length scale in the system and u is the maximum velocity of the object relative to the fluid. The Reynold’s number encountered by a macroscopic object (say a human) will be much larger as compared to a microscopic body (a bacterium) flowing through the same fluid. This implies that there is a larger degree of viscosity encountered by the bacterium as opposed to its inertial forces. The significance of this is that the bacterium cannot use the same technique for swimming through the fluid as the human. The way a bacterium moves is by “churning” their way through liquid with the corkscrew-type motion of flagella. Re is proportional to volumetric flow, Qvol in a tube. The , through the tube can be given by
Qvol = πr4∆P/8 (2.31)
where r is the radius of the tube, l is the tube length and ∆ is the pressure drop over the length of the tube. As it is already discussed the difference in velocities of the particles of fluid flowing through a tube along the axis of the tube and near tube surface results a pressure difference between the two ends of the tube and needed to overcome the friction between particle layers to keep the fluid moving. In case of shearing flows, adjacent layers move parallel to each other with different velocities. In nanoscale where laminar flow of liquid is important. In case laminar flow of liquid, known as Couette flow, a layer of fluid is trapped between two horizontal plates, one fixed and one moving horizontally at constant speed. In this case, liquid flow is driven by viscous drag and the pressure gradient parallel to the plates. The liquid which shows Couette flow has to be homogeneous in the layer and at different shear stresses. In case of Couette low of liquid , the changes in tube radius, not the length, affect the volumetric flow significantly more than changes to any other variables. So,
Qvol ∝ L4
Fig. 2.2 Fluid flowing through a small circular pipe. Changes in pipe radius affect the volumetric flow significantly more than changes in any of other variables.
That is volumetric flow scales as L4. The volumetric flow can also be expressed as
Qvol = L2(2.13)
Here, u is the average velocity of the fluid and L2 is the cross sectional area of the pipe, L being the radius of the tube. Now the pressure gradient (pressure drop per unit length) can be given by
Therefore scaling law for pressure gradient scales as L−2. Thus, the pressure drop by unit length increases with the decrease in the tube radius, leading to a much larger surface to volume ratio of the flow compared to macroscopic systems. Different techniques of pumping, such as electrohydrodynamic, electroosmotic and electrophoretic techniques, which depend on the surface of the fluid, have been devised for this scale.
Let us now consider the motion of objects falling through a fluid. When any object rises or fall through a fluid it will experience a viscous drag which is essentially the force needed to move a sphere of radius R at a velocity v through a viscous medium and given by Stoke’s law
F = 6πηRv (2.15)
When viscous drag balances the downward force of gravity, the sphere o eventually fall with a constant velocity, known as the terminal velocity. If the flow past the sphere is laminar (for moderate value of Re ), then the terminal velocity is
Here ρsphere is the density of the sphere, ρfluid is the density of the fluid, η is the fluid’s viscosity, g is
gravity and R is the radius of the sphere. So v scales as L2 and viscous force F scales as L3. The time, , that the sphere takes to reach this velocity is equal to / , which means that scales as L2. Thus, a larger object will reach terminal velocity before a smaller one and will fall faster than it.
2.1.3Scaling in Heat Transfer
The mechanism of heat transfer is significantly different in small systems. The energy Eth needed to heat a system of characteristic dimension, L, to a given temperature is directly proportional to the system’s mass and therefore its volume
Eth ∝ m ∝ V(2.17)
This means Eth scales as L3. This relationship holds true for the heat capacity, Qcap of the system – also proportional to mass (i.e. volume)
Qcap ∝ ∝(2.18)
That is Qcap scales as L3.
Heat can be transferred as conduction, convection, and radiation, but conduction and convection are the more common modes of heat transmission in smaller systems. The scaling laws are often valid in two distinct regimes: (1) one is the submicrometer scale, which includes systems with L ≤ 1μm, and (2) the other being systems larger than that, with L > 1 . The rate of heat conduction, Qconduct, in a solid is given by
Q conduct = − ∆ /∆ (2.19)
In this equation, heat flows along x direction, while is the thermal conductivity of the solid, A is the cross-sectional area of the heat flow, and ∆ is the temperature difference across the solid. For systems larger than micrometre dimensions, we can assume that is constant throughout the solid and therefore relationship between the heat conduction and the characteristic dimension can be given by
This shows that heat conduction for L > 1 scales asL1.
In submicrometeric systems, however, thermal conductivity varies as the characteristic dimension, ∝ L. This is due to molecular heat transfer mechanisms that varies with the specific heat, molecular velocity and average mean free path. Therefore, for submicrometer systems
That is heat conduction for L ≤ 1μm scales as L2.
To determine the time, ℎ, required to achieve thermal equilibrium in a system, the dimensionless parameter called the Fourier number, 0 = t/L2, can be defined where is thermal diffusivity , which is defined as the thermal conductivity divided by density and specific heat and t is the time it takes heat to flow across the system’s characteristic length, L. If t = ℎ, the characteristic time for thermal equilibrium is proportional to the square of the characteristic dimension:
Heat transferred by the motion of fluid is convection, and is governed by Newton’s cooling law which is essentially the rate of heat convection and given by
Qconvect = ℎ ∆ (2.23)
Here, ℎ , is a heat transfer coefficient, A is the cross sectional area of the heat flow, and ∆ is the difference in temperature between the two point in the fluid. The heat transfer coefficient, ℎ , is a function of flow the fluid moves, which does not significantly impact the scaling of the heat flow. Therefore Qconvect will scale as L2. Since the continuum theory does not hold at submicrometre dimensions, the above scaling law cannot be used for such systems.
2.1.4 Scaling in Biology
Recently, it has been shown that the scaling laws can be applied to living things. In general, smaller animals (with smaller mass) have quick pulse rates and short lives, while larger animals (with higher mass) have slow pulse rates and long lives. And it is also known that the number of heartbeats during an animal’s lifetime is approximately same about one billion beats. Pulse rate and life span are just two of the biological characteristics that vary with body size. The scaling laws governing these
phenomena in biology is represented as quarter-power scaling, and it plays a role in life span, heartbeat, and metabolic rate. The life span (LS), of an animal varies with its mass which is proportional to L3, hence LS scales with mass and length as
∝ 1/4 ∝ L3/4 (2.24)
Also, the heart rate (HR) scales with mass and length as
∝ −1/4 ∝ L−3/4(2.25)
The metabolic rate of animals also scales with its mass. The basal metabolic rate of an animal has to do with its surface-to-volume ratio (= L−1). As size of the living object increases, the less surface area it has to dissipate the heat generated by its mass. One of the many complex factors that control the rate of metabolism is that a larger animal’s metabolism must be slower than a smaller animal’s metabolism to prevent overheating. The metabolic rate, MR, scales with body mass as
∝ 3/4 ∝ L9/4(2.26)
If we consider the mechanical abilities of animals of different sizes, a number of scaling laws emerge. For example, the maximum height to which an animal can jump can be estimated using scaling laws. The energy for an animal to jump high is provided by its muscles. The muscular strength can be assumed to be proportional to its cross-sectional area, and hence
ℎ ∝ 2(2.27)
The mass of the muscles,∝ 3, and thus the ratio of strength to mass,
h=2/3=1
The ratio is a simplified indicator of how high mass living object can jump. For example, an elephant does not have sufficient strength to overcome its weight and cannot jump at all, whereas a grasshopper can jump hundreds of times the length of its body. This is a simplified explanation that does not take into account many factors such as, shape, orientation, and structure of these animals’ muscles and bodies, but does explain quite a bit.
An animal’s ability to fly is curiously follows scaling laws. For an animal to fly, its weight must be balanced by the aerodynamic lift, α. That is,
= α(2.29)
And the lift is proportional to the surface area of the wings, S and the speed of flying, v.
∝ 2(2.30)
Now ∝3 and ∝ 2. Thus, ∝ 1/3 and hence ∝ 2/3. This gives, M ∝ M2/3v2 or
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