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Interaction Squares

When we need to show how a quantity f responds to a change in one other quantity x, we usually make a two-dimensional plot of f vs x. But what can we do when f depends on two other quantities, x and y? We might consider a three-dimensional plot, but these are generally difficult for the writer to construct and difficult for the reader to interpret. An alternative is to construct a two-dimensional interaction square. An interaction square is particularly informative when the quantities x and y compete in their contributions to f, and we need to know the conditions under which either x or y dominates the competition. In this document we first describe the standard interaction square, then we show two common variations.

The standard interaction square is obtained when f is linear in the two independent variables x and y,

f = x + y

(1)

We build the square by plotting x vs y, and use the lines x = 0 and y = 0 to divide the plane into four quadrants, as in Figure 1. Next we plot the locus of points that force f = 0. This locus can be called a stationary line. For the standard square it obeys

y = –x

(2)

and appears as a diagonal through the origin, also shown in Figure 1.

Our objective is to identify the regions of the square in which x and y reinforce one another in their contributions to f and to identify those regions in which x and y compete. First note that when x and y have the same signs, then they reinforce one another; this occurs in quadrants 1 and 3. So the more interesting behavior occurs in quadrants 2 and 4, in which the signs of x and y differ. The diagonal (representing f = 0) separates quadrants 2 and 4 into two regions each.

- In quadrant 2, above the diagonal, x dominates and f > 0.
- In quadrant 2, below the diagonal, y dominates and f < 0.
- In quadrant 4, above the diagonal, y dominates and f > 0.
- In quadrant 4, below the diagonal, x dominates and f < 0.

A common application of an interaction square, like that in Figure 1, occurs when x and y represent operating variables (such as temperature and pressure) for some process, and f represents some behavior or property of the process that we wish to control. The interaction square then shows us ranges of x and y values over which we obtain desired values for f.

The linear stationary line (f = 0) in Figure 1 is a consequence of the linearities in the underlying equation (1). Now let us consider situations in which f is a sum of two contributions but one or both independent variables occur in nonlinear forms. In such cases we can still construct an interaction square, but the stationary line will no longer be straight. For example, consider

f = x^{2} + y

(3)

Now the stationary line (locus having f = 0) is a parabola, as shown in Figure 2. Outside the parabola, f > 0, while under it f < 0. When y > 0 (quadrants 1 and 4), x and y reinforce one another. The interesting behavior occurs when y < 0, for then f might be positive or negative. In quadrants 2 and 3, outside the parabola, x dominates and f > 0. But under the parabola, y dominates and f < 0.

Now, what about situations in which f is not a simple sum of two contributions; e.g., what if f is a product of the two? In these cases, we take logarithms to transform the product into a sum, then we construct the interaction square using the logs of the independent variables. For example, consider the ideal gas law:

v = RT/P

(4)

Here the molar volume of the gas depends on absolute temperature and absolute pressure (R is a positive constant). To obtain a sum of temperature and pressure effects we will take the logarithm of this equation, but before doing so, we convert it to a dimensionless form. We divide (4) by the critical volume v_{c}

v_{c} = RT_{c}/P_{c}

(5)

where T_{c} is the critical temperature and P_{c} is the critical pressure. Then (4) becomes

v* = T*/P*

(6)

where v* = v/v_{c}, T* = T/T_{c}, and P* = P/P_{c}. Taking the log of (6) yields

ln v* = ln T* – ln P*

(7)

At the critical point, v* = 1, T* = 1, and P* = 1, so the lines T* = 1 and P* = 1 divide the (logarithmic) interaction square into four quadrants, as in Figure 3.

The stationary line is again a diagonal straight line, but because of the minus sign in (7), this diagonal passes through quadrants 1 and 3. Also because of that sign, when T* and P* have opposite signs, they reinforce one another; this occurs in quadrants 2 and 4. When they have the same sign, they compete. The diagonal divides quadrants 1 and 3 into two regions each.

- In quadrant 1, below the diagonal, T* dominates and v* > 1.
- In quadrant 1, above the diagonal, P* dominates and v* < 1.
- In quadrant 3, below the diagonal, P* dominates and v* > 1.
- In quadrant 3, above the diagonal, T* dominates and v* < 1.

Note that the interaction square in Figure 3 pertains to the ideal-gas law, which is a model of gas behavior. However, near and below the critical point, fluids do not behave like ideal gases, so in those regions Figure 3 does not apply to real gases.