The idea of scaling is the same, regardless of the shape we attach to the edges of the triangle, so we could use pentagons, as shown in Figure 12: The area of the pentagon has plus the area of the pentagon b is equal to the area of the pentagon c. We can generalize this to any form, as long as the three joined forms are similar to each other. If you want an absolutely rigorous proof of this result, you can express the area as an integral and then modify the variable. The factor resulting from b/a then emerges before the integral, since an integral is a linear operator. This means that you don`t even need to evaluate the integral to get the scaling result. As for the area of the ellipse, we expect it to be scaled relative to the circle of a factor λ using the same scale argument. That is, we expect the area of the ellipse λ to be π a2, which is only π a b. This result can be verified if you remember the formula for the area of an ellipse (π times the semi-major axis multiplied by the semi-minor axis). Equation 32b has the advantage of being a power law per se, as is common for scale laws, while equation 32a has the advantage of being slightly more precise when n is not very large. Truly fundamental laws of scale can be obtained by dimensional analysis alone, while quasi-fundamental assumptions involve something beyond. For example, in scaling arguments, it is generally assumed that the mass scales with volume and therefore length are high to the third power. However, strictly speaking, this is only true as long as density can reasonably be considered a scale-independent property – which is usually a good assumption for solids down to the micron scale.
Perhaps the most important quasi-fundamental scaling law that applies to MEMS is cubic squared scaling, which relates volumetric scale dependencies to quantities that change with the surface. The most important structural example is that the inertial forces depend on acceleration and mass and therefore volume, while the resulting stress is staggered with the cross-section. Therefore, at a given strength of the material, the acceleration at which a structure can withstand increases linearly with a decreasing length scale – which justifies microfabricated accelerometers. Table 1 summarizes the near-fundamental scaling of physical parameters down to the size range of interest to MEMS designers. Part of the ability to come up with good scaling arguments is using good terminology and avoiding bad terminology. The scaling laws for volume, area, and length can be expressed in equations: Therefore, the following laws can be obtained for uneven scaling of 3D geometry: There is no way to predict this general scaling behavior based on dimensional analysis. There are certain areas, like fluid dynamics, where you`ll never go beyond square one if you don`t understand non-dimensional scaling. This document is intended as a tutorial that covers the simplest and most widely used scaling laws. Don`t let this discourage you. As a reminder, this document is intended to be a tutorial that covers the simplest and most widely used scaling laws.
You don`t have to be a Nobel laureate to get much value from scaling laws. Non-trivial examples appear in many places, including computer science (e.g., compiler construction) and cryptography. Perhaps the most well-known scale law concerns the ratio between length and area. In Figure 3, each length of the large square is twice as large as the corresponding length of the small square. As for the area, you can see that the area of the large square is not twice as large, but four times as large as the area of the small square. The law of scaling (Eqn. (31)) predicts a rapid increase in nonlinear cubic optical susceptibility with conjugation length. Therefore, we could theoretically expect huge hyperpolarizability for long polymers. However, as is the case with linear polarizability, the second hyperpolarizability of conjugated polymers saturates after a certain number of monomer units, depending on the polymer itself.
The exact determination of the threshold value of monomer units (see Table 3) at which hyperpolarizability saturates is not easy. In fact, as the length of the polymer increases, so does its conformation. Normally, this leads to a decrease in conjugation length and hyperpolarizability. This may artificially lower the above threshold. Among these limitations, the number of conjugate bonds at which saturation of third-order nonlinear optical susceptibility or molecular hyperpolarizability occurs is given in Table 4. Depending on the polymer, saturation usually starts at six to 10 bonds. For these reasons, it is unnecessary to produce very long polymers to optimize their nonlinear optical response. Many researchers have used scale laws to increase the size of their columns to facilitate in situ measurements.7,28,44,47–53 A disadvantage of using scaled columns is that the change in geometry resulting from active resolution has a much smaller impact on spatial distributions than micrometric columns.
For example, suppose two spaces 1 μm and 100 μm apart are exposed to active corrosion, resulting in 1 μm penetration into each crevasse. The deviation for the 1 μm slit will have increased by 200%, while the larger deviation will only result in a magnification of the 1% difference. Another problem arises in Statement 8. Does the statement mean that each dimension of B (length, width and height) is larger by a factor of 3, or does it simply mean that the volume is larger by a factor of 3? When you scale a property, you must specify exactly the property that you are scaling. In principle, any trigonometry is a collection of scale laws based on the idea of similar triangles. If you know that triangle ABC has a right angle (as shown in Figure 10) and you know one of the other angles (for example, α), you can derive the third angle. This triangle is similar to any other triangle with these three angles. All these triangles are connected by a simple scale factor.