A Yardstick for Measuring Uniform Coating Thickness
by Helmut Fischer
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| Figure 1. Three mm diameter ball bearings coating with Ag. Curve illustrates plating duration in minutes for mean thickness value vs. standard deviation. |
Technical and decorative coatings typically require a uniform coating thickness. Since the thickness of a functional coating may range from a few nm to several hundred µm, it is prudent to use a parameter that is independent of the coating thickness as a quality measure for the uniformity. It can be shown that such a parameter constitutes far more than a quality characteristic; it is also suitable for establishing realistic specification limits. Specifically, this parameter is a measure of the relative variation, which allows not only the evaluation of the effectiveness of measurements for quality improvement but also the evaluation of the quality level of suppliers and production methods. The latter is important in view of the introduction of environmentally friendly technologies, particularly for functional coatings.
Under what conditions does a relative quantity make sense for the designation of the variation of a method? An easy-to-understand experiment that can be applied to the various coating methods shall serve in answering this question. About 5,000 balls for ball bearings are coated in a drum in a silver electrolyte. At various time intervals, 80 balls each are removed as random samples from the total amount and the thickness of their coating is measured. The standard deviation of each random sample is determined according to the known equation:
where xistands for the measurements of the random sample and -x for the arithmetic mean calculated from the measurements.
Plotting the standard deviations calculated for the various exposure times against the respective mean value results in a semi-linear correlation as is apparent from Figure 1. The increase of the standard deviation with the coating thickness is not at all surprising since the coating thickness fluctuation increases with the increase of the mean coating thickness. The functional correlation between the standard deviation s and the mean coating thickness can be described through a linear equation:
where c stands for a process-specific constant. The correlation:
is known in the statistical field as coefficient of variation V1. In English language literature, V is also referred to as "relative standard deviation." By definition, V has no dimension and is often expressed as a percent value:
In Figure 1, V is identical with the incline of the straight line2, in the present example about 0.07, corresponding to:
For the general evaluation, it is important that the inclination factor as the ratio of standard deviation/mean value is essentially independent of the coating thickness, thus allowing for a comparative evaluation between various electrolyte systems (acidic or alkaline baths) and barrels, or degrees of loading.
Production Parameters and Relative Standard Deviation
Two widely known process parameters—Cp and Cpk—are used to determine the specification limits for monitoring a process. The capability index Cp is based on the practical awareness that under the assumption of normal distribution (ND), essentially all individual events (coating thickness measurements) are in the range:
The parameter s indicates that it is a standard deviation secured by a sufficiently large number of single measurements (e.g., n³ 500). The random sample should come from a production that is already under control or is at least essentially under control. The parameter µ should be in good agreement with the desired nominal value xnominal. As a rule, the nominal value is between an upper (USL) and a lower (LSL) specification limit. The quantities USL and LSL are derived from the drawing tolerances; they are explicitly not statistically justified (deviation) limits.
A process is considered absolutely capable if the tolerance is specified such that the condition 8 * s £ USL – LSL is met.
Based on equation 5, the capability index is internationally defined as follows:
With the usual supposition:
the Cp equation can also be defined with the help of V. It is then:
If one selects (the abbreviation) USL/LSL = Z for the tolerance ratio, then equation (7) can be written as follows:
This equation contains only numeric values that express correlation. Z includes only the tolerance specification, and is therefore a "non-statistic" quantity. V, on the other hand, is a process-typical statistical variation quantity. Taking into account the capability condition (8 s £ T) and equation (5), one arrives at:
as a capability index, which in most cases is considered sufficiently appropriate for production. In the same manner, a characteristic Cpk is required as a parameter for fulfilling the specified nominal value (tolerance center) through the process mean value µ, which is written as:
or as:
If it is required to meet specification limits (USL and LSL) on both sides, then the smaller value of the aforementioned correlation is used to evaluate a deviation of the process position from the nominal value. If the production mean is exactly at the nominal value µ, then both Cpk values are equal. However, very often only a minimum value is required for the coating thickness, that is, only LSL is specified. Under such conditions, equation (9) applies, which takes into account the correlation:
(valid for a sufficiently large random sample) will lead to the following correlation:
Often, a numeric value of 1.33 is required for Cpk as well. For practical applications, the numeric value of 1.33 should not be considered an absolute for Cp or for Cpk. Values between 1.1 and 1.33 are often sufficient. However, numeric values less than 1 will often lead to a significant number of outliers in the production.
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Figure 2: Criteria for the capability index Cp dependent on the relation of the production variation/% and the ratio of upper/lower specification limit USL/LSL. |
Nomogram 1 (Figure 2) displays the ratio Z = USL/LSL that is the basis for the respective Cp values as well as the ratio mean value µ/LSL to determine the production mean value for a specified lower specification limit. The simple equations that are correlated to the relative unit V are the basis for the nomograms 1 and 2 (Figure 3) and allow the evaluation of the production process with regard to the specification limit requirements or capability indices.
Furthermore, this model should be taken into account in contract negotiations, especially when a production process is to meet certain requirements. With the knowledge that a production process meets given requirements, very high consequential instrumental costs can be avoided. The empirical knowledge has existed in the field of electroplating for a long time, however, only through a quantitative implementation does it become descriptive and an objective basis for the customer and the supplier.
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| Figure 3: Criteria for the capablity index Cpk dependent on the ration of the production mean value/lower specification limit LSL. |
Two examples to explain the use of nomograms 1 and 2:
Example 1: A certain mass product shall be equipped with a Cr coating on a specified functional surface. On the drawing, the specification limits are specified as LSL = 4.5 µm and USL = 10 µm. From pretests, it is known that V = 12%. In this case, the ratio USL/LSL = 2.22. The requirement of Cp = 1.33 cannot be met when V = 12%. The perpendicular above 12% of the V-axis of nomogram 1 intersects the perpendicular of the USL/LSL axis above 2.22 between Cp = 1.1 and Cp = 1.0. The process is not fully optimal in the sense of the typical requirement of Cp = 1.33, however, it is still capable from a practical point of view, as long as it is possible to keep the respective Cpk value above 1.
In the electroplating industry, this is a difficult undertaking. If this goal cannot be met, significant percentages of specification limit violations must be expected. However, exact advance information is not possible because the assumption of normal distribution often does not apply, especially when evaluating large random samples.
Although the error rate can be reduced significantly with a 100% test, a zero error value cannot be achieved. The possibility of outliers must also be considered in this context. A “zero error” requirement may serve as a goal to be accomplished but should never be accepted as a definitive agreement, because it simply cannot be fulfilled in the long run.
Example 2: On small parts, a gold coating with a minimum thickness of 1 µm (= LSL) shall be ensured. A pre-run with 500 parts resulted in a standard deviation of 0.13 µm with an average coating thickness of 0.82 µm. Thus, V = 15.9%. A Cp value of 1.33 is required. When placing a perpendicular in nomogram 2 at 15.9%, then the plot with the parameter value of 1.33 is intersected at µ/ LSL = 2.75.
The mean coating thickness µ that would have to be used for production would be at:
If it were possible to reduce V to 10% by improving the process to achieve a Cp value of at least 1.2, then in similar fashion to above, a production mean value of 1.56 µm would have to be obtained. Thus, a significant savings potential can be realized, especially with expensive coating materials.
Conclusion
A constant, process-typical relative standard deviation can be assumed for coating processes. Meeting specification limits can be verified easily by using two simple nomograms as explained in this paper.
The expenditures for reducing the process variations are often underestimated. As a rule, however, this expenditure will be worthwhile because as experience shows, the often extremely high consequential costs of the process variations are reduced.
Before signing contracts, suppliers and customers should be clear about what is technically possible and should also be clear that obtaining objective production results through measurement is not only possible but also necessary.
Note: The statistical methods for the applied measurement method and related process-typical parameters are described in the FischerscopeMMS Operators Manual.
Footnotes
- "^" indicates the random sample characteristic. For very large random samples V = s/µ.
- When viewed carefully, it is apparent that this straight line does not go exactly through zero. This shall not be further discussed here, although the significant remaining axis section in the fine structure allows for interesting conclusions.
For more information, contact Paul Lomax at (e-mail) plomax@fischer-technology.com.
