Influence of erroneous data on the results of calculations from acid-base surface free energy theories. II. Why are negative values of square roots obtained?

The occurrence of negative square roots of the Lifshitz - van der Waals ( LW) component and acid and base ( A and B) parameters calculated from the van Oss - Chaudhury - Good theory was checked for a small artificial set of the work of adhesion input data calculated for 9 solids and 7 liquids. Taking from the literature the data for the LW component and A and B parameters for 7 liquids and the values of such component and parameters for 9 solids ( close to those in the literature), the work of adhesion was calculated and its value was assumed to be error-free ( un-biased). Next, new values of the work of adhesion were obtained by adding a random error having normal distribution belonging to 8 distributions of a mean value equal to the error-free work of adhesion value and standard deviations of 1, 5, 7, 10, 20, 25, 30 and 40% of the mean value. The LW components and A and B parameters for the nine solids were back-calculated for each solid and the error ( bias) level by solving the overdetermined system of equations ( of matrix 7 x 3) for 7 liquids. These 9 solids were grouped in 3 sets of 3 solids in each. It was found that an experimental error caused the work of adhesion data for real systems to be biased. This bias caused the solution of the equation system also to be biased and both biases were linearly dependent. This paper confirms that the appearance of negative roots of A and B parameters is caused by a specific bias in the components of the work of adhesion matrix. If the work of adhesion matrix is negatively biased there is a greater possibility of obtaining a negative value of the square root of gamma(+), and the smaller the value of this parameter the greater is the possibility of obtaining a negative square root for it. Both the negative and positive biases in the work of adhesion matrix almost equally influence the bias in gamma(-). The smaller this parameter the greater is its bias and greater the possibility of obtaining its negative square root.