Fundamental Units and Concentration

Question

I am working with the Cahn-Hilliard equation which is typically given in terms of a dimensionless concentration, $u$. I generally find myself being confused when dealing with dimensional equations that also have non-dimensional variables. Hence, I wanted to start by reformulating the equation into a totally dimensional form before performing non-dimensionalization to my specific interest. My question is: which units would I use and/or does it even matter?

I come from a mathematics background so I'm not too familiar with concentrations; however, what I gather molarity seems to be the most common unit for concentration. I am wondering how this is related to the other fundamental units if at all. The other units that appear within the equation are only length and time. Further, when I turned the equation fully dimensional it doesn't necessarily matter what I choose because it will cancel out with every other term. Still, I think that should be rather important although perhaps I am wrong.

Finally, I am wondering why the equation is generally given this way in the first place. Is there some advantage to considering $u$ to be dimensionless from the beginning? For reference, Physical, Mathematical, and Numerical Derivations of the Cahn-Hilliard Equation by Lee, et. al. appear to derive this equation with the assumption that $u$ (or $c$ in the case of that paper) be dimensionless from the start - which can be seen when they write $c = \frac{N_B}{N_a}$ before mixing (on pg. $2$).

For reference, the Cahn-Hilliard equation is generally given as: $$ u_t + D\nabla^2(u-u^3+\gamma\nabla^2u) = 0$$ where $D$ is the diffusion coefficient with units length squared over time and $\gamma$ is a constant with dimensions of length squared that dictates the width of the interface once phase separation begins.

Answer 1

"I come from a mathematics background"

In that case, you may find it useful to think of units as of unspecified multipliers.

What follows is, I think, not how physicists usually think of units - there's this whole philosophy about what units mean, physically, that I brush aside here - but I don't think what I state here is particularly controversial. You don't need to dispense of that philosophy; think of this as taking a different point of view.

If you have a measuring stick, and you label its length p, and I have a measuring stick, and I label its length q, and you give me an equation in terms of p, and you let me borrow your measuring stick, I can use q to measure that stick and convert your equation into my units. You can then view dimensional analysis as arising from the rules of factoring in algebra - you can't add two different physical quantities because you've attached two different unspecified multipliers to them, and you can't factor that out.

Now, to understand dimensionless units, consider the definition of a radian: an angle subtended from the center which slices off an arc length equal to the radius. It doesn't matter which measuring stick I use, I can always determine such an angle on a circle of any size. That is, I can always reliably reproduce the unit angle that will serve as my angle measuring standard, without needing anything extra. So there's no unspecified multiplier needed - I just need to be mindful of what the terms in the formula mean, and of how the angle was measured (e.g. I don't want to mistakenly use degrees).

So, to summarize (TLDR):

• for units that have dimensions, you need someone to give you a standard you can measure, so that you can interpret the numerical values in that person's formula
• for dimensionless units, you can, in principle, always recreate the standard on your own, and get the same numbers

I'm not too familiar with concentrations; however, what I gather molarity seems to be the most common unit for concentration.

Here's the age-old problem - human language is messy. There's a bunch of different notions that are all termed "concentration". For example, see this list.

The concentration in Cahn-Hilliard equation is, from what I've been able to gather, an abstract notion of "concentration" which can variously mean volume fraction, mass fraction, or mole fraction (number fraction) depending on what's actually being studied, and on how the person chose to model the system.

E.g., it could denote, in a two-component system, how much volume one of the components takes up in some small volume-neighborhood $dv$ of a point, expressed as a fraction of $dv$.
Its unitless because, in principle, it doesn't matter what you use to measure these volumes (as long as you're consistent).

You can see that this would take a value in the range $[0, 1]$; that is, for components $A$ & $B$, you have $u_A \in [0, 1]$ and $u_B = 1 - u_A$. Sometimes, a rescaled version of that is used; e.g., looks like the Wikipedia article on Cahn-Hilliard equation uses $c = u_A - u_B \in [-1, 1]$

Finally, I am wondering why the equation is generally given this way in the first place.

My guess would be that it's just a way to directly track how much of each of the two components is present in the neighborhood of a point - i.e., you don't care about the actual concentrations, you're interested in the relative concentrations of the components with respect to each other.

Answer 2

One really needs only three units here:

• unit of $u$ - $[u]$ - whatever it is
• unit of time
• unit of length/position/coordinate

All the summed terms should be of the same dimensionality, and the coefficients before each terms should be such, as to render these dimensionalities be the same. On the other hand, differentiation by time/coordinate means dividing by the unit of time/length.