Monday, January 25, 2010

Liquid Crystals and the Buckingham Pi Method





Physicists have at their disposal many mathematical tools to help them solve problems. However, some of the most powerful are dimensional analysis methods.

Dimensional analysis not only helps you check if a formula you derived (or copied down to your test sheet if you are a student) is correct at least dimensionally, but it can help you estimate the answer to many physical systems.

For example, say you are a physicist who is trying to understand how their watch liquid crystal display works.
They have done some reading in the popular press and have found that a watch LCD is probably made of a twisted nematic liquid crystal cell, which can be switched with a magnetic field. (Or electric field).
The physicist reads up on Wikipedia about liquid crystals, how energy efficient they can be, how they are switched on and off, how polarizes are used and that nematic is derived from the Greek nema, which means thread. After some more reading our physicist starts wondering how strong of a magnetic field is needed to switch the LC cell.



Of course one can simply look up the answer online, but our intrepid physicist decides to use dimensional analysis, and more specifically the Buckingham Pi method to determine the general form of the strength of the magnetic field.

Lets start with an aside on the Buckingham Pi method.

Suppose you are attempting to describing a physical system with a physically meaningful equation of n measurable variables.

f\left( {{x_1},{x_2}, \ldots ,{x_n}} \right) = 0

where each {x_n} is expressed in terms of k independent physical units.

The above equation can be restated in terms of p = k - n dimensionless parameters constructed from {x_n} of the form

\pi  = x_1^{{m_1}}x_2^{{m_2}} \ldots x_n^{{m_n}}

where each {m_n} are rational numbers.

From this we can construct the new equation:

F\left( {{\pi _1},{\pi _2}, \ldots ,{\pi _p}} \right) = 0

Note that this solution is not unique, and is only one of a set of dimensionless parameters you can generate with your measurable variables in terms of physical constants. This method does not choose the best form, or even the most physically meaningful form. But it is very useful none the less.
Note that independent physical units are those such as length, time, mass etc. The fundamental SI units (of which there are seven) represent such a set of independent units.

New, lets see how we can use this method to determine a physically meaningful form for an equation to determine the magnetic field strength needed to switch the LCD cell in question.

Consider that we have a cell full of a nematic liquid crystal. The liquid has an elastic constant K, which is measured in Newtons. The LC is also diamagnetic in that it will anti-align with the magnetic field. How strongly the little "threads" or "nemas" in the LC will interact with an external magnetic field has to do with the diamagnetic susceptibility of the LC, {\Delta \mu } .
The "threads" in the LC are twisted, which means that the "threads" in each layer of the LC is rotated, or "twisted" a bit from the direction the previous layer is facing. Picture a DNA molecule, or a helix spiral staircase. This means that the thickness, l, of the cell that is holding the LC is an important variable, as each layer will require a stronger or weaker magnetic field to anti-align the "threads".

In general terms, the little threads like to align along the direction (actually opposite) with the external magnetic field. But the restoring force of the liquid surrounding the threads want to keep this from happening. Sort of like trying to rotate raisins in a bowl of jello. Unless you hold the raisin in place, they spring back to their old alignment due to the elastic constant of the jello.

These little threads act as a polarizer, and you can rotate this polarizer using an applied magnetic field. Crossing the LC polarizer with another fixed polarizer stops light from being transmitted through the LC cell. This is switching the LCD ON.




So, using dimensional analysis, the general form of a dimensionful equation is:

H \propto {l^\alpha }{K^\beta }\Delta {\mu ^\gamma }


We then construct a set of dimensionless quantities from this equation, and equate the exponents to determine the final form.
Lets examine each variable, and determine its dimensional quantities.

H -> magnetic field -> Amperes/meter -> [A][L^-1]

K -> elastic constant -> Newtons -> [M][L][T^-2]

delta_mu -> diamagnetic susceptibility -> Henry/meter -> [M][L^2][T^-2][A^-2][L^-1]

So, equating the fundamental SI units we get:




Equating exponents we get a set of 3 equations and 3 unknowns.




Solving this set of equations we get:




Therefore the dimensionful equation is:





Now, lets do a sanity check; Does this equation make sense?

First, the units check out. So the equation is dimensionally correct. Check.

As the elastic constant K increases ("jello" force increases), it makes sense that you would need a stronger magnetic field to rotate the "threads" in the LC. Check.

If a larger {\Delta \mu } means a greater interaction with the magnetic field. This means that a weaker magnetic field can couple to the "threads" and overcome the "rubberyness" of the LC, and therefore the magnetic field does not have to be as strong to rotate the "threads". Check.

As the thickness of the cell increases, there are more "threads" that are already nearly aligned to the ON position. The more threads that are aligned, the less light makes it through the cross polarizers, and therefore a weaker field is needed to switch ON the LCD. Check.

So, the equation makes physical sense, and is dimensionally correct.

Our physicist uses their google-fu and determines that indeed this is the correct form, up to a multiplicative constant.

Of course, there may be a reliance on other variables, such as temperature, on how the LC behaves. More than likely these are subsumed into K, but can be put into our model later.

This is a very simplistic treatment of the Buckingham Pi theorem, as well as of twisted nematic liquid crystals. However, I hope the power of dimensional analysis is evident in this example.

2 comments:

  1. Very interesting! However, are you sure you did not know a-priory what variables to use? ;)

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  2. What would have happened if there were more variables than equations? I am confused how you would select your dimensionless equations then.

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