*4.2. Change in the Corrosion Rate in Di*ff*erent Corrosion Periods*

The PH, Pi, and Rp values of the cast ZK60 alloy generally showed different change tendencies in different immersion periods (Figures 8, 9b and 10b). In the early corrosion period (0–2 h), the corrosion rates (PH and Pi) increased with time (Figures 8b and 9b), while Rt, Rf, and Rp decreased (Table 2 and Figure 10b). In this period, the filiform corrosion was initiated and developed quickly on the cast ZK60 alloy without visible corrosion pits (Figures 12–14), corresponding to the states in Figure 19a–d. In this case, with the development of the corrosion filaments, the broken oxide film increased steadily to accelerate the anodic and cathodic processes, as shown in Figure 9, and made the corrosion rates increase. The increase in the broken oxide film may be the reason why no Ebreak occurred in the anodic polarization curve when t = 2 h (Figure 9). Furthermore, the increase in the broken oxide film enlarged the reactive areas on the cast ZK60 surface to increase CPEdl and CPEf, meanwhile decreasing RL and L in this period (0–2 h, Table 2).

The second corrosion period was determined to be within 2–24 h, according to the above discussion about the corrosion development on the cast ZK60 alloy. In this period, the corrosion filaments gradually occupied most of the active oxide film, and a small number of corrosion pits occurred and developed quickly (Figures 15 and 16), which corresponds to the state in Figure 19e. Because most of the active oxide film was corroded and covered with a black Mg(OH)2 precipitate, the anodic process, Reaction (9), was inhibited significantly to make Ecorr more positive, and Ebreak occurs again with a more positive value (Figure 9). In this case, the corrosion rate (PH and Pi) decreased significantly (Figures 8b and 9b), while Rt, Rf, and Rp increased (Table 2 and Figure 10b). The difference is that Pi and 1/Rp reached the lowest value at t = 24 h, but PH decreased persistently until t = 48 h, while Pw as shown in Figure 7 displayed the highest value at t = 24 h, which may be ascribed to the differences among these test methods. In this period, the decrease in the broken oxide film may also result in a clear increase in RL and L and decrease in CPEdl (Table 2). Meanwhile, CPEf firstly increased and then decreased, which may have been due to the increase in the area and thickness of the corrosion product layer. Even though very few corrosion pits occurred in this period (Figures 15 and 16), their Pdepth increased significantly and reached a large value of about 24 mm y−1 (Figure 17), which was much larger than the general corrosion rates of Pw, PH, and Pi in this period (Figures 7, 8 and 9b). Therefore, the pitting corrosion became more critical than the filiform corrosion in this period.

Over 24–72 h, most of the surface of the cast ZK60 was gradually covered with Mg(OH)2 layer, but the number and depth of the corrosion pits increased (Figures 15–17), corresponding to the state in Figure 19f. Because of the increase in corrosion pits, the anodic process was accelerated again. In this case, the Ebreak and Ecorr of the cast ZK60 alloy became negative, and Pi increased again (Figure 9), while Rt, Rf, and Rp decreased (Table 2 and Figure 10b). However, Pw and Pdepth decreased with time in this period (Figures 7 and 17), indicating that the increase in the corrosion product layer still impeded the growth rate of the pitting depth and the general weight loss, possibly due to the alkalizing effect of the cathodic reaction and the low solubility of Mg(OH)2. In this period, the increase in CPEf may have been due to the dissolution of the corrosion product layer reducing its thickness; meanwhile, the decrease in RL and L may have been related to the acceleration of the pitting corrosion process, similar to in the processes on passive metals [61]. In general, pitting corrosion became more severe in this period and dominated the corrosion behavior of the cast ZK60 alloy. In this case, Pdepth should be adequate to describe the corrosion rate of the cast ZK60 alloy in this period.


**2. Related work**

Prior to the introduction of quantitative genetics to the EC community, research along similar lines was already being conducted. Most notable among these was the discovery that parent-offspring fitness correlation is a good predictor of an algorithm's ability to converge on highly fit solutions [18].

Mühlenbein and Altenberg began to introduce elements of biology theory to EC at roughly the same time. Mühlenbein's work has focused mainly on adapting the equation for the response to selection (also known as the breeder's equation) for use with evolutionary algorithms [19] Initial work involved the development several improved EAs and reproductive operators [21, 23, 24], and progressed to the development of Estimation of Distribution Algorithms (EDAs) [20, 22].

Altenberg's work used Price's Theorem [27] as a foundation for his EC theory. One of his goals was to measure the ability of certain EA reproductive operators to produce high quality individuals, and identify what qualities were important in achieving this [1]. He referred to this as evolvability, and the equations he developed looked similar in some regards to the response to selection equation. In particular he provided a theoretics foundation for why the relationship between parent and offspring fitness (i.e. heritability of fitness) was important.

Another aspect of Altenberg's work involved going beyond a simple aggregation of the relationships between parent and offspring fitness. He focused on the idea that the upper-tail of the distribution was a key element. After all, creating a few offspring that are more fit than their parents can be much more important than creating all offspring with the same fitness as their parents. This is why his equation really became a measure of variance instead of mean, which is what Price's Theorem typically measures. As an indication that his theories were in some sense fundamental to how EAs work, he was able to use them to re-derive the schema theorem [2].

Langdon [14] developed tools based on quantitative genetics for analyzing EA performance. He used both Price's Theorem and Fisher's Fundamental Theorem [26] to model GP gene frequencies, and how they change in the population over time.

Work by Potter et al. [25] also used Price's Theorem as a basis for EA analysis. They also recognized the importance of variance, and developed some approaches to visualizing the distributions during the evolutionary process [5, 6].

The work of Prügel-Bennett & Shapiro [29] [28] is based on statistical mechanics, but it has some important similarities to the methods used in quantitative genetics. Here, populations are also modeled as probability distributions, but the approach taken is more predictive than diagnostic. This means that detailed information about the fitness landscape and reproductive operators is needed in order to analyze an EA. Still, this approach has some interesting capabilities. For example, up to six higher-order cumulants are used to describe the distributions, allowing it to move beyond assumptions of normality, and thus providing much more accurate descriptions of the actual distributions.

Radcliffe [30] developed a theoretical framework that, while not directly related to quantitative genetics, has certain similarities. His formae theory is a more general extension of the schema theorem, and can be applicable at a phenotypic level.