Sietske Tacoma
email: s.g.tacoma@uu.nl

Bastiaan Heeren
email: bastiaan.heeren@ou.nl

Johan Jeuring
email: j.t.jeuring@uu.nl

Paul Drijvers
email: p.drijvers@uu.nl

Automated feedback on the structure of hypothesis tests

Keywords: Intelligent tutoring systems, domain reasoner, hypothesis testing, statistics education

Understanding the structure of the hypothesis testing procedure is challenging for many first-year university students. In this paper we investigate how providing automated feedback in an Intelligent Tutoring System can help students in an introductory university statistics course. Students in an experimental group (N=154) received elaborate feedback on errors in the structure of hypothesis tests in six homework tasks, while students in a control group (N=145) received verification feedback only. Effects of feedback type on student behavior were measured by comparing the number of tasks tried, the number of tasks solved and the number of errors in the structure of hypothesis tests between groups. Results show that the elaborate feedback did stimulate students to solve more tasks and make fewer errors than verification feedback only. This suggests that the elaborate feedback contributes to students' understanding of the structure of the hypothesis testing procedure.

Introduction

Hypothesis testing is widely used in scientific research and is therefore covered in most introductory statistics courses in higher education [START_REF] Carver | Guidelines for Assessment and Instruction in Statistics Education College Report[END_REF]. This topic is challenging for many students, due to the large number of complex concepts involved (Castro Sotos, Vanhoof, [START_REF] Castro Sotos | Students' misconceptions of statistical inference: A review of the empirical evidence from research on statistics education[END_REF]. Students struggle to understand the logic of hypothesis testing and the role and interdependence of the concepts in the testing procedure [START_REF] Vallecillos | Some empirical evidence on learning difficulties about testing hypotheses[END_REF]. Appropriate feedback might support students in comprehending this structure and understanding the line of argumentation in hypothesis tests [START_REF] Garfield | Learning to reason about statistical inference[END_REF]. Since group sizes in introductory statistics courses are often large, automated feedback in an online system seems a promising direction.

Carrying out a hypothesis test requires several steps, which have to be performed in a particular order. To address the structure of hypothesis tests, feedback should address all aspects of a solution: not only the content of individual steps, but also the relations between steps. An Intelligent Tutoring System (ITS) can provide such sophisticated feedback on the level of steps and can provide detailed diagnostics of student errors [START_REF] Nwana | Intelligent tutoring systems: An overview[END_REF]. Such feedback on the step level is generally more effective than feedback on the level of complete solutions [START_REF] Vanlehn | The relative effectiveness of human tutoring, intelligent tutoring systems, and other tutoring systems[END_REF].

Although ITSs vary considerably in design, they generally contain the following four components: an expert knowledge module, a student model module, a tutoring module, and a user interface module [START_REF] Nwana | Intelligent tutoring systems: An overview[END_REF]. Of these four, the expert knowledge module is the most domain-dependent. It contains information about exercises that students can solve in the ITS and about domain knowledge required to solve these exercises [START_REF] Heeren | Feedback services for stepwise exercises[END_REF], and is therefore also referred to as domain reasoner. Two important paradigms for constructing domain reasoners are modeltracing, in which the ITS checks that a student follows the rules of a model solution, and constraintbased modeling, in which the ITS checks whether a student violates constraints. There exist ITSs that support hypothesis testing based on either of these approaches [START_REF] Kodaganallur | A comparison of model-tracing and constraint-based intelligent tutoring paradigms[END_REF]. We combined the two paradigms in a single ITS supporting hypothesis tests, and in this paper we evaluate the feedback by this ITS. The research question guiding this evaluation is: does the feedback provided by the ITS contribute to student proficiency in producing wellstructured hypothesis tests? Before describing the design of the ITS and the study design, we briefly discuss the two paradigms; for a more extensive overview, see the work of Kodaganallur and colleagues (2005).

Model-tracing

The model-tracing paradigm concentrates on the solution process. The domain reasoner contains a set of expert rules: rules that an expert would apply to construct solutions to exercises in the domain. It can also contain buggy rules: incorrect rules that arise from incorrect domain knowledge. Finally, the domain reasoner contains a model tracer that can identify which expert and buggy rules a student has applied to arrive at a (partial) solution. After this identification, several actions can take place. If a buggy rule is recognized, a diagnosis of the student's error can be given. Otherwise, a hint for a next step can be provided, or just feedback that the current solution path is correct.

Constraint-based modeling

Constraint-based modeling concentrates on partial solutions, rather than on the solution process. The underlying idea is that incorrect knowledge emerges as inconsistencies in students' partial solutions [START_REF] Mitrovic | Constraint-based tutors: A success story[END_REF]. Domain knowledge is represented as a set of constraints, consisting of a relevance condition and a satisfaction condition. For instance, in the domain of hypothesis testing a relevance condition might be that the partial solution contains an alternative hypothesis and a rejection region, and the corresponding satisfaction condition that the position of the rejection region (left side, right side or two sides) matches with the sign of the alternative hypothesis. Errors in student solutions emerge as violated constraints, that is, as constraints for which the relevance condition is satisfied, but the satisfaction condition is not. If a student's solution does not violate any constraint, it is diagnosed as a correct (partial) solution.

Methods

Design of the domain reasoner

The technical design of the domain reasoner evaluated in this study is based on the Ideas framework [START_REF] Heeren | Feedback services for stepwise exercises[END_REF], which uses a model-tracing approach to calculate feedback. However, for the domain of hypothesis testing the constraint-based modeling approach was also considered useful, because of its capability to identify inconsistencies in solution structures. Addressing these inconsistencies might contribute to student understanding of the relations between steps, and hence of the structure of hypothesis tests. Therefore, the domain reasoner designed in this study combines elements of the two paradigms. It contains 36 expert rules, 16 buggy rules, and 49 constraints. Every time a student adds a step to a hypothesis testing procedure, for example defines an alternative hypothesis or calculates the value of the test statistic, the domain reasoner checks the student's solution so far. First, all constraints are checked, and if one is violated, the domain reasoner determines whether a buggy rule was applied. If so, a specific feedback message corresponding to this buggy rule is displayed to the student, and otherwise a general message for the violated constraint is reported. For example, a student solution that does not contain an alternative hypothesis, but does contain a rejection region, violates the constraint with relevance condition "the solution contains a rejection region" and satisfaction condition "the solution contains an alternative hypothesis". The corresponding feedback message addresses the role of the hypotheses: "To which hypotheses does this rejection region correspond? First state hypotheses".

If no constraints are violated, the domain reasoner tries to identify the rule the student applied to arrive at the current partial solution. A feedback message corresponding to the identified rule is shown to the student, for example: "Your rejection region is correct". If no rule is identified, the student's partial solution is marked correct ("This is a correct step"), since no constraints are violated. Besides checking partial solutions, the domain reasoner can also provide hints on what next step to take.

The design of expert rules, buggy rules and constraints was informed by discussions with three teachers of introductory statistics courses in various disciplines about the structure of hypothesis tests and common errors by students. Furthermore, textbooks were consulted. Based on this input, we decided to support two methods for structuring hypothesis tests: the conclusion about the hypotheses can be drawn based on comparison of the test statistic with a critical value, or based on comparison of a p-value with a significance level. In each method, a complete solution should contain four essential steps: (1) state hypotheses, (2) calculate a test statistic, (3) either find a critical value or find a p-value, and (4) draw a conclusion about the hypotheses. Although crucial for the logic of hypothesis testing, stating a significance level was not regarded as an essential step, since it was always specified in the task description. Besides these four essential steps, students could include several other steps in their hypothesis tests, such as a summary of sample statistics and an explicit specification of whether the test was left sided, right sided or two sided.

Participants and study design

The domain reasoner was used in a compulsory Methods and Statistics course for first-year psychology students at Utrecht University. In five weeks of this ten-week course, students received online homework sets consisting of 7 to 13 tasks that were selected by the teachers of the course. These homework sets were designed in the Freudenthal Institute's Digital Mathematics Environment (DME; see [START_REF] Drijvers | Digital design: RME principles for designing online tasks[END_REF], which supports various interaction types, such as formula input and multiple choice tasks. To enable intelligent feedback on hypothesis tests, a connection was set up between the DME and the domain reasoner.

The third, fourth and fifth homework set concerned hypothesis testing. Each of these three homework sets contained two tasks that specifically challenged the students' proficiency in carrying out hypothesis tests, by asking the students to select steps from a drop-down menu and to complete these steps. An example is shown in Figure 1: after selecting a step from the drop-down menu called "Action", it appears as next step in the step construction area. Next, the student can complete the step by filling in the answer boxes and use the check button to check the hypothesis test procedure so far. After finishing the construction of the hypothesis test, the student should state the overall conclusion in the final conclusion area, below the drop-down menu with steps.

To evaluate the effects of the domain reasoner feedback, students in the course were randomly divided into an experimental and a control group. This allowed for a comparison of student work between both groups, and differences in student behavior and learning effects could be ascribed to the only difference between both groups, the domain reasoner feedback [START_REF] Cohen | Research methods in education[END_REF]. From the 310 students in the experimental group 226 students worked on the hypothesis testing tasks, of which 154 gave consent for the use of their work in this study. From the 309 students in the control group 216 students worked on the tasks, of which 145 gave consent. The participants were on average 19.3 years old (SD = 1.7 years) and 77% were female. To reduce the Hawthorne effect, i.e. the effect that students might behave differently because they were part of an experiment [START_REF] Cohen | Research methods in education[END_REF], students were not told about the different conditions and whether they were in the experimental or in the control condition. The homework sets were equal in both groups, except for the six tasks in which students constructed complete hypothesis tests by selecting and filling in steps. Students in the experimental group received feedback from the domain reasoner, whereas students in the control group only received verification feedback. Hence students in the experimental group received elaborate feedback on errors in the structure of their hypothesis tests, while students in the control group only received feedback on the correctness of their current step, irrespective of previous steps. As a consequence, for solving a task completely, students in the experimental group needed to include all four essential steps, since otherwise one or more constraints would be violated. For students in the control group, tasks were already marked as solved once they contained a correct conclusion about the null hypothesis. A final difference between the two groups was the hint functionality of the domain reasoner, but in this study we did not take the students' use of hints into account.

Data and data analysis

Data for this study consisted of the students' work on the six hypothesis testing tasks in the DME, that is, all their attempts at constructing steps and drawing final conclusions in these tasks. After exporting log files with the students' attempts from the DME, work from students who did not give consent was deleted and all other attempts were anonymized.

Two measures served as indicators for feedback effectiveness: skipping vs. trying behavior and solving vs. failing behavior [START_REF] Narciss | Exploring feedback and student characteristics relevant for personalizing feedback strategies[END_REF]. Skipping vs. trying concerned the number of students who skipped constructing steps and only filled in a final conclusion versus the number of students who tried constructing steps. This can be regarded as a measure of feedback effectiveness, since students who filled in a final answer without constructing steps apparently did not perceive the feedback on the steps as helpful. Solving vs. failing concerned the number of students who produced complete hypothesis tests versus the number of students who failed to produce complete tests. Since feedback was more elaborate in the experimental group than in the control group, we expected the proportion of solving students to be higher in the experimental group.

To find the number of students who tried constructing steps, for each task the number of students who made at least one step using the drop-down menu was counted. The total number of trying students over all tasks was calculated as the sum of these counts. Next, the total number of all trying and skipping students was found by summing the number of students who used the final answer boxes over all tasks. For both groups the proportion of trying students was found by dividing the number of trying students by the number of trying and skipping students. For solving vs. failing, we considered a task solved if a student had correctly included all four essential steps in the solution. The number of solving students was calculated over all tasks, and divided by the total number of trying students, to find the proportion of solving students. Chi-squared tests were used to evaluate whether the two proportions (trying as proportion of all students and solving as proportion of trying students) were significantly larger in the experimental group than in the control group.

The second part of data analysis concerned errors in the structure of hypothesis tests. Structure errors could be missing information, such as stating a rejection region before stating hypotheses, or inconsistent information, such as a rejection region that did not match with the alternative hypothesis. For the first few tasks, we expected an equal number of structure errors in both groups, since students had not received feedback on their errors yet. In later tasks we expected fewer structure errors in the experimental group than in the control group, and we especially expected that fewer students would make the same structure error in multiple tasks in the experimental group. For each task, the proportion of students who made at least one structure error was calculated and the proportions were compared using Chi-squared tests. Next, for each structure error that the domain reasoner could diagnose, the number of students who made it in only one task and the number of students who made it in multiple tasks were counted and these counts were summed. Again, a Chisquared test was used to compare the proportions between groups.

Results

Table 1 contains the proportion of students who tried using the stepwise structure in the experimental and the control group, and the proportion of students who completely solved the tasks, as proportion of all students who tried the task. As can be seen in the table, students in both groups tried to construct steps almost 80% of the times, and there is no significant difference between these proportions in both groups:  2 (1, N = 1558) = 0.09, p = .759. This implies that the domain reasoner feedback did not impact the students' decision to try or skip using the stepwise structure to solve tasks. The proportion of students who included all essential steps in their solutions does differ significantly between the groups: in the experimental group, this proportion was 0.52 and in the control group it was 0.40 ( 2 (1, N = 1208) = 19.18, p < .001). The effect is regarded small (φ = 0.13). This result suggests that the feedback from the domain reasoner did support students in producing more complete hypothesis tests. Although positive, this result is not very surprising, since the domain reasoner feedback explicitly notified students of missing steps. To find out how much the domain reasoner really contributed to the students' proficiency in producing wellstructured hypothesis tests, we also considered the errors students made in the structure of their hypothesis tests. Figure 2 shows the proportion of students with errors in their solution structure for both groups, compared to the number of students who attempted the tasks. The total number of students who attempted each task is given in Table 2. The graph in Figure 2 shows that for the first two tasks, 3.4 and 3.6, the proportion of students who made structure errors were quite similar in both groups, and above 0.70. This similarity between groups was to be expected, since students just started working with the feedback. For the latter tasks, however, the proportion of students who made structure errors decreased strongly in the experimental group, and remained higher in the control group.

Conclusion and discussion

In this paper we have evaluated the influence of feedback from an Intelligent Tutoring System that supports hypothesis testing on student behavior in six hypothesis testing tasks. Our aim was to evaluate whether this feedback contributes to student proficiency in producing well-structured hypothesis tests.

Students who received domain reasoner feedback did not use the stepwise structure for producing hypothesis tests in more tasks than students who received verification feedback alone, but they did produce more solutions that included all four essential steps in hypothesis testing. Hence, as a consequence of the domain reasoner feedback, students in the experimental group had more opportunities for valuable practice on these essential steps [START_REF] Narciss | Exploring feedback and student characteristics relevant for personalizing feedback strategies[END_REF]. Students in the control group tended to use the steps more as "isolated checkers", rather than as building blocks for producing complete hypothesis tests. By only selecting the steps they wanted to check and omitting the other steps from their solutions, these students missed out on opportunities to practice with the line of argumentation that is important for understanding hypothesis testing [START_REF] Garfield | Learning to reason about statistical inference[END_REF].

Not only did the domain reasoner feedback encourage students to produce more complete hypothesis tests, it also supported students in doing this more independently. In the first two tasks the number of errors in solution structure was similar between groups, and hence students in both groups relied equally on the feedback to help them correct their errors. In the final four tasks, however, students who received domain reasoner feedback made significantly fewer structure errors than students in the control group. This indicates that domain reasoner feedback did more effectively support students in resolving their misunderstandings than verification feedback alone, which is in line with earlier findings that elaborate feedback is more effective than verification feedback (van der [START_REF] Van Der Kleij | Effects of feedback in a computer-based learning environment on students' learning outcomes. A meta-analysis[END_REF].

In this paper, we just examined direct effects of the domain reasoner feedback on student behavior; we have only considered tasks in which domain reasoner feedback was provided and no subsequent work. Future work will focus on longer term learning effects of the domain reasoner feedback; do we see the same effects for other hypothesis testing tasks in which students do not receive feedback on their solution structure? Results from the study discussed in this paper are promising; students demonstrate proficiency in structuring their hypothesis tests and hence seem to have gained understanding of the roles of the different steps in this procedure.

Figure 1 :

 1 Figure 1: Digital Mathematics Environment displaying one of the hypothesis testing tasks (translated)

Figure 2 :

 2 Figure 2: Proportion of students with errors in solution structure in both groups

Table 1 : Proportions of students who tried vs. skipped, and who solved vs. failed, in experimental and control group, over all six tasks

 1

	Experimental	Control	 2	df p-value Effect
	group	group		size

Table 2 : Errors in solution structure in both groups for each of the six tasks

 2

	Task N	 2	df p-value	φ
	3.4	297 0.345	1 .557	-
	3.6	216 0.501	1 .479	-
	4.7	207 5.086	1 .024	0.16
	4.8	194 6.665	1 .010	0.19
	5.3	158 4.936	1 .026	0.18
	5.6	136 7.663	1 .006	0.24

  Table2displays the results of a series of Chi-squared tests which confirmed this finding: in the first two tasks, the number of students who made structure errors did not differ significantly between groups, whereas in the latter four tasks, in the experimental group significantly fewer students made errors in solution structure than in the control group. This suggests that the feedback contributed to the students' proficiency in producing hypothesis tests without making structure errors. Considering the numbers of students who made the various structure errors just once and the number of students who repeated the same structure error in multiple tasks corroborates this finding: a Chi-squared test on these numbers yielded  2 (1, N = 1053) = 38.29, p < .001, φ = 0.19, indicating that students in the experimental group repeated the same structure error in multiple tasks significantly less often than students in the control group. This confirms that the domain reasoner feedback was effective in reducing students' mistakes with respect to the structure of hypothesis tests.

Acknowledgments

We thank teachers Jeltje Wassenberg-Severijnen and Corine Geurts and software developers Noeri Huisman, Martijn Fleuren, Peter Boon and Wim van Velthoven for the valuable collaboration.


Marianne Van Dijke-Droogers
email: 1m.j.s.vandijke-droogers@uu.nl

Paul Drijvers
email: 2p.drijvers@uu.nl

Arthur Bakker
email: a.bakker4@uu.nl

Repeated sampling in a digital environment: A remix of data and chance

Keywords: TinkerPlots, repeated sampling, (informal) statistical inference, modeling, statistics education

Drawing statistical inferences (SI) is essential in a society where data are of increasing importance. Understanding the relation between data and chance, necessary to understand statistical inference, is however challenging for students. Technological innovationssuch as the Sampler in TinkerPlots (TP)enable students to investigate this relationship by modeling a population and simulating repeated samples. Along this line, the research reported here presents the results of a pilot with fourteen 9th-grade students, inexperienced with sampling, in a Learning Lab. The pilot focuses on how students use TP as a digital environment for exploring data and chancei.e. what strengths and constraints they encounterand how they subsequently use this information for SI. The results suggest that the participating students encountered difficulties in modeling the population, however, they were able to simulate, explore and reason inferentially through repeated sampling in TP.

Drawing statistical inferences (SI) is essential in a society where data play an increasingly important role. However, the handling of variation and uncertainty involved in drawing inferences based on sample data, is challenging for students [START_REF] Castro Sotos | Students' misconceptions of statistical inference: A review of the empirical evidence from research on statistics education[END_REF][START_REF] Konold | Data analysis as the search for signals in noisy processes[END_REF]. New technological innovationssuch as the Sampler-option in TinkerPlotsenable students to model and investigate the interaction between data and chance in order to gain insight into variation and uncertainty [START_REF] Biehler | Editorial: Reasoning about models and modeling in the context of informal statistical inference[END_REF][START_REF] Pfannkuch | Innovations in statistical modeling to connect data, chance and context[END_REF]. Following this approach, Van Dijke-Droogers, Drijvers, and Bakker (2018) suggested an approach of repeated sampling with a black box filled with colored balls to introduce students, inexperienced with sampling, to the key statistical concepts of sample, frequency distribution from repeated samples and the simulated sampling distribution. This approach with a black box seemed promising to invite students to investigate the intertwined relationship between data and chance. As a follow-up to this study, the research reported here investigates how activities in the digital environment of TinkerPlots can strengthen the statistical insights of students, i.e. the interrelation between data and chance.

Theoretical background

Statistical inference

Statistical inference includes making statements about an unknown population based on observed sample results. In contrast to descriptive statistics, which concerns describing the data under investigation, inferential reasoning includes the handling of sampling variation and interpreting the role of chance.

Digital environments for statistical modeling

Recent digital environments that use dynamic visualizations and offer opportunities to create statistical models, enable students to investigate the intertwined relationship between data and chance. Tools such as TinkerPlots that have both data analysis and sampling capabilities, support young students to think about the modeling process. Creating statistical models to simulate data from repeated sampling can be helpful to develop key statistical ideas of distribution and sampling variation (Konold, Harradine, & Kazak, 2007). In an instant of time, a model of the population can be built in a digital environment and then be used to generate a large number of (repeated) sample results. In this way, student can explore sampling variation due to chance, especially when they visualize the results of repeated sampling in a sampling distribution. In this way, students engage in modeling, dealing with variation and thinking about the context [START_REF] Pfannkuch | Innovations in statistical modeling to connect data, chance and context[END_REF]. The work by Garfield, delMas and Zieffler (2012) shows that students can learn to think and reason statistically -or as the authors call it "really cook"by using statistical modeling in digital environments.

Repeated sampling in the black box activity

This learning-or cooking-effect is reflected in the work by [START_REF] Van Dijke-Droogers | Repeated Sampling as a step towards Informal Statistical Inference[END_REF] on repeated sampling from a black box filled with small colored balls ("balletjes" in Dutch). The results of that study suggest that repeated sampling from a black box is a promising approach to introduce students to the concepts of sample, frequency distribution on repeated sampling (resampling) and the simulated sampling distribution, and, therefore, invites reasoning about variation and uncertainty involved in SI. An important characteristic of this 3-step learning approach seems the strong connection between a physical black box experiment in step 1, the visualization of expected sample results from resampling with a black box in a frequency distribution in step 2, and the simulated sampling distribution on resampling in the digital environment of TinkerPlots in step 3. An overview of the learning steps in the black box activity is displayed in Table 1.

Methods

To investigate how students explore and reason with TinkerPlots, we conducted a pilot study in the Teaching and Learning Lab of Utrecht University.

We have chosen to run the pilot in a lab setting. The advantage of this lab setting, over a classroom environment, was that detailed recordings could be made of the actions and conversations of several teams of students working on the same assignment at the same time. During a five-hour session, seven 9th-grade students worked in teams of 2 or 3, on tasks addressing steps 1 to 4 of the learning trajectory. This pilot was repeated in a similar five-hour session, with seven other students. The teams worked on a laptop, the screen of which was displayed on an interactive whiteboard.

Camera recordings of their actions on the screen were made, and student conversations were recorded. Students noted their findings as a team on a student worksheet. We worked with students from pre-university level, the 15% best performing students in our educational system. Figure 1 shows the setup in the Learning Lab.

During the pilot, students went through learning steps 1-4 of the black box activity. Special focus was on how students applied their digital experiences from step 3 to investigate the role of variability and chance in the new context of step 4. In step 3, students used the sampler option in TinkerPlots to simulate repeated samples from the context of the black box. While doing so, they used an instruction sheet with the necessary technical actions in TinkerPlots. In this step, students investigated possible sample results by modeling a black box filled with 750 yellow and 250 orange balls at different sample sizes and different number of repetitions. During a group session in step 3, students discussed the boundaries of common sample results and decided that the middle 80% of the results can be regarded as "most common sample results". They indicated the 10% highest sample results as "exceptionally high" and the 10% lowest sample results as "exceptionally low". It was expected that the students would apply their experiences from step 3 in the new context of step 4, as detailed in Table 2. This table shows an overview of components A to G with statistical insights, TinkerPlots activities in step 3 and expected students' activities in step 4. To make a plot of the sample data, explore the data and choose a visualization that provides a clear overview.

Students simulate one sample of which the data are automatically listed in a table. They use the Plot-options to explore and clearly visualize their data.

E Repeated sampling in a digital environment can produce a large number of sample results. These results can be listed in a table by choosing one specific characteristic.

Use the history button to have repeated samples memorized. To do this, choose a specific characteristic-for example the number of yellow balls-that you want to record from each sample.

Students use the history button to record the number of "students who had breakfast" from each sample. For Task 1 in step 4, students could use a table on their worksheet, as displayed in Table 3. Students used TinkerPlots to determine most commons sample results. For Tasks 2 to 5, more in-depth questions that mainly focused on the role of sample size were asked within the same context, as shown in Table 4. Students were free to choose their own working method within TinkerPlots.

Results

The analysis of the video-recordings shows that the expected students' activities in components A and C to G did occur. However, the students encountered difficulties with component B. Students started with discussing the breakfast context in component A. Students seemed to be involved in this real-life context as they exchanged and discussed all kinds of possibilities and expectations based on their personal experiences and intuition, for example "Only 70% of these students have breakfast, that's way too low for a healthy school. Here, we have a lot more students who have breakfast." After sharing their thoughts, they started working in TinkerPlots. Here, in component B, making a model of the population in TinkerPlots, the difficulties began as 3 out of 6 teams used the sample information for their model. As such, they modeled the population by entering 21 for breakfast and 9 for no breakfast. When simulating samples from this population they got confused and asked the teacher for help. Referring to the black box activity in step 3, made them aware of the difference between population and sample. Another difficulty with the modeling was that 2 teams used a pie chart with percentages. These students entered 70% for breakfast and 30% for no breakfast. Although they used similar proportion as the given population, working with percentages

Sample size 30 Expected number of students having breakfast daily (interval notation)

Most common sample results

Exceptionally high results

Exceptionally low results

Table 3: Table on student worksheet

Task 2: In the past year a lot of attention has been paid to a 'healthy' breakfast at the school. At which sample result, sample size 30, is it likely that the breakfast habits of students have improved? Support your answer with data.

Task 3: At the end of the school year, a sample of 30 students was not feasible. Therefore, a sample will be taken of only 10 students. With which sample result can you now assume that the breakfast habits of students have probably improved?

Task 4: On closer inspection, the school board decides to postpone the sample to the next school year and to draw a sample of 100 students at that time. For which sample result is it likely that the breakfast habits of students have improved?

Task 5: Compare the sample results and corresponding estimates of the population at different sample sizes. What can you conclude about the population estimates from a larger sample size?

Table 4 Tasks 2-5 on Student Worksheet refers to an endless population and in this case the population size is given with 300. So, a finite population of 300 is a better model. After struggling at component B, the other components were performed as expected, although students occasionally needed guidance from the teacher, which mainly consisted of linking the new context in step 4 to the known context of the black box activity in step 3.

The activities in TinkerPlots (re)strengthened the statistical insights of students. As a first strength, each new component offered opportunities for students to discuss and reason about their actions in the digital environment and the implication of their actions. For example, at component A, a student stated "Well, they should take a larger sample to get a better picture of the school" and "Probably 21 students of the sample will have breakfast, or at least about 21 students. 22 or 23 is possible too. Even 30 is possible, but I don't think that will happen". Statistical concepts such as the effect of sample size and possible sample results that were addressed in steps 1-3 of the black box activity, were again considered and discussed by students in a new context.

As a second strength, students referred to terms from steps 1-3 of the black box activity, as they were using terms like "yellow and orange balls" and "content" of the black box. The lay-out and set-up of the digital environment offered opportunities to easily relate both contexts. The following fragment shows an example of students' reasoning while modeling the population in task 1 of learning step 4, where terms related to repeated sampling with the black box are underlined: I think, we should take the 300, because that are all students, like all the balls in the box and the 30 is just a sample, the visible ones… don't you think?

Student 2: Okay, so then we must enter 210 with breakfast and 90 no breakfast.

A third strength concerned the easy and rich way in which students could explore the data. They could easily create and compare different graphs, like pie charts, dot plots, etc. On the other hand, these explorations took a lot of time and it seemed like students based their decisions, for example about the format of the graph, on personal ideas and less on clarity. Moreover, choosing the right colors and shapes for the balls, was time consuming.

A fourth strength was that every component required action in TinkerPlots. Students could not thoughtlessly enter their values, but they had to be alert and aware of their actions.

In step 4, students were asked to carry out five investigative tasks and to note their answers on student worksheet. An outline of the results on students' worksheet are displayed in Table 5. When more than 80 students (out of 100) are having a daily breakfast, this is extremely high, and therefore it is assumable that more students are having a daily breakfast.

Correct conclusion

A larger sample size gives a more precise estimate of the population

The larger the sample size, the less variation in the corresponding estimate and the bigger the chance of a good estimate

Conclusion and Discussion

The research reported here focused on how activities in the digital environment of TinkerPlots can strengthen students' statistical insights, i.e. the interrelation between data and chance. The main difficulty of students was creating a correct model of the population and, more particularly, distinguishing sample and population characteristics from a given context. However, entering the characteristics into the tool, TinkerPlots, did not cause any problems. The awareness of the difference between sample and population can be improved by spending more time and instruction to modeling a population in a variety of contexts.

A first strength of the activity in the digital environment of TinkerPlots was that students exchanged and discussed statistical information which may lead to a deeper notion of the concepts addressed.

A second strength was that the lay-out in the digital environment could easily be related to the black box activity (balls, sample size). Students explanations contained black box terms several times. A third strength concerned the convenient and rich way in which students explored the data. However, choosing appropriate representations was also time consuming and mainly based on personal preference over clarity. A fourth strength was that every thinking component required action in TinkerPlots, which increased their awareness of the statistical concepts involved.

Although the results seemed promising, a few critical points must be considered. The fact that the students went through the learning steps quite easily may have been caused by the short time frame in which the activities logically followed each other. Although the students were given quite open tasks in step 4, their approach was strongly influenced by their previous activities in step 1 -3. In addition, the students occasionally needed help from the teacher, which consisted of linking the new context to the black box activity in step 1 -3.

The results show that this activity with repeated sampling in the digital environment of TinkerPlots is a promising way of how the interaction between data and chance can be taught in a brief session.

Researcher who would like to repeat this activity should consider that this research highlights the results of a small-scale pilot in a laboratory setting with high performing students.

1.

  Repeated sampling with the physical black box 2. Visualization of expected sample results from repeated sampling with the black box in a frequency distribution, by manually drawing a sketch 3. Simulation of a large number of samples from repeated sampling with the black box and visualization of results in the sampling distribution, by using the Sampler-option in TinkerPlots 4. Using the digital environment of step 3 in a new context

Figure 1 :

 1 Figure 1: An impression of the setup in the Learning Lab

  Components of statistical insightsTinkerPlots activity in step 3 with the black box Expected students' activity in TinkerPlots at step 4 A Sample results vary due to chance Context: A black box is filled with 750 yellow and 250 orange balls. For each sample, the number of yellow balls is counted. Examine the sample results you can expect. Context: At the beginning of the school year 210 out of the 300 students on a specific primary school were used to having breakfast daily.. Task 1: To investigate whether this is still the situation at the end of the school year, a sample of 30 students will be taken. Examine which sample results you can expect if the breakfast habits of students are unchanged. B A model of the population can be entered and examined in a digital environment Model the population in TinkerPlots by using the Sampler-option. The model can be entered as a pie chart, bar graph, histogram, dot plot or curve. Students enter a model of the population in the same way as described in the cell on the left (TP activity B) C Sample size effects sampling variation due to chance Enter the sample size you want to examine. Students enter the sample size they want to examine. In this example, the sample size is 30. D Sample data can be listed in a table A visualization of sample data in a plot provides a clear overview Simulate one sample, the results are automatically displayed in a table.

F

  A visualization of multiple sample results in a plot can be used to estimate possible sample results Visualize the sample results in a sampling distribution. Use dividers to examine possible sample results, i.e. most common sample results, exceptionally high or low Students vizualise the sampling distribution and use dividers to examine possible sample results G A larger sample size reduces the variation in the corresponding estimates of the population and hence, leads to a better outcome. Estimate most common sample results and extrapolate these to the population. Then, try different sample sizes later for comparison. Draw a conclusion about the effect of larger sample sizes to the estimate of the population Students estimate most common sample results and exceptionally high or low results to draw their inferences corresponding to the task (other sample sizes were addressed in tasks 2-4)

Table 1 : Overview of learning steps in the black box activity Van

 1 [START_REF] Van Dijke-Droogers | Repeated Sampling as a step towards Informal Statistical Inference[END_REF] focused on step 2 and showed that students could, based on their experience in step 1, imagine and sketch a frequency distribution from repeated samples with most sample results close to the population proportion and in which strong deviations hardly occur. As a next step, the aim of the research presented here, is to gain insight into how statistical modeling with the Sampler-option in TinkerPlots can strengthen the statistical insights of students concerning sampling variation due to chance. Therefore, we added a fourth step to the black box activity to investigate how students use TinkerPlots to explore data and chance in new situations and how they use this information for SI.

Table 2 : An overview of statistical insights, TinkerPlots activities in step 3 and expected students' activities in step 4

 2

Table 5 : Overview of results on students' worksheet

 5


C Berne

E Andrieu

J Reby

J-M Sobrino

C Blanc

Dissolution Kinetics of α,β -Brass in Basic NaNO 3 Solutions

Corrosion tests at a constant anodic potential, referred to as dissolution tests, were performed to determine the dissolution kinetics of an α,β -brass CuZn40Pb2 (CW617N) in a basic nitrate solution with and without mechanical loading. The corrosion behavior of the α,β -brass was characterized by a two-step mechanism, with an initiation step for which the simultaneous dissolution of all the alloying elements of the β phase occurred only and a propagation step including at first, the afore-mentioned simultaneous dissolution process before a critical time t c and then, both simultaneous dissolution and dezincification of the β phase after t c . The pH and Cu concentration of the electrolyte near the brass surface were determined to be two of the major factors influencing the occurrence of the dezincification process. The dezincification of the β phase extended in depth by a percolation dissolution mechanism. Mechanical loading during the dissolution tests was observed to largely influence the dezincification process. It was assumed to open or close the pores present in the dezincified β phase, which promoted or slowed down the dezincification mechanism depending on the sign of the stress.

[DOI: 10.1149/2.0071602jes]

In various corrosive environments, α,β -brasses are believed to be susceptible to simultaneous dissolution (Cu, Zn) and/or selective dissolution (Zn, dezincification) processes. These phenomena are expected to play a major role during the first stages (incubation and initiation) of stress corrosion cracking (SCC) damage, 1 leading to rupture, that have been observed on gas network components in service. In this framework, it is of interest to develop accelerated tests to evaluate the susceptibility to corrosion and SCC of α,β -brasses, however the development of these accelerated tests requires a good knowledge of brass dissolution mechanisms. Abundant literature can be found for α-brass, for example in nitrite solutions [2][3][4][5][6][7][8][9][10][11] but not for α,β -brasses. [12][13][14][15] In a previous study, 16 the electrochemical behavior of an α,β -brass CuZn40Pb2 (CW617N) was studied in basic nitrate solutions with various basic pHs and nitrate ion concentrations. This specific alloy, CW617N, used principally for its good malleability and machinability (related to the Pb amount) and resistance to corrosion, is principally studied for the development of water taps. [17][18][19] The NaNO 3 solution was selected on the basis of a literature review on α-brass. 20 In the literature, the solutions that are most often used are ammonia, 15,21 sulfuric acid, 22 and nitrite solutions 10 but, according to Fernandez et al., 20 the nitrate ion is far more stable in low pH solution than the nitrite ion. Because the present study was a first step to develop an accelerated SCC test, the nitrite ion was avoided due to its instability in the acid solution that is expected to exist inside a stress corrosion crack. Ammonia solutions, widely used for testing brass corrosion resistance, were also excluded due to the lack of reproducibility of SCC tests performed in this environment. The results from a previous study 16 showed that, in all the chosen experimental conditions, corrosion at the open circuit potential proceeded by the galvanic coupling of the α and β phases, leading to a surface dezincification of the β phase. The extent of the dezincification was affected by the presence of lead in the alloy but the pH was the major parameter. During polarization tests, a pseudo-passive or a passive stage followed by a pseudo-passivity or a passivity breakdown was observed; the corrosion phenomena mainly involved copper and zinc dissolution from the β phase. At pH 11, a Cu 2 O/PbO layer was efficient in achieving passivity of the brass, whereas for potentials higher than 0.03 V/SCE, local dissolution of the β phase was observed. At pH 12, a Cu(OH) 2 -rich surface layer was formed in the pseudo-passivity range; it was not protective enough, and complete dissolution of the β phase was observed for potentials higher than 0.115 V/SCE leading to the removal of lead particles. The results were consistent with those of Heidersbach and Verink 23 who proposed that preferential dissolution of Zn occurred at low potentials, while for higher potentials, both copper and Zn dissolved with or without copper re-deposition depending on the potential value. They were also in good agreement with other authors 20,[24][25] who showed that the precipitation of lead based compounds on the surface of the brass, for potentials higher than the corrosion potential, decreases the extent of corrosion attack. Results from a previous study 16 brought a new insight into the chemical composition of the surface layer depending on the pH and the applied potential. However, tests performed in this previous study were short and knowledge of the dissolution mechanisms and kinetics of α,β -brasses during long tests is important to understand their SCC behavior. Furthermore, in the framework of the development of an SCC test, it was of interest to determine the influence of a mechanical loading on the dissolution mechanism and kinetics. Among the mechanisms proposed in literature 22,[26][27][28][29] that could be used to explain the dissolution behavior of α,β -brasses for long dissolution tests, a volume diffusion mechanism was discussed by Pickering; [26][27][28] Sieradzki et al. [30][31][32][33] proposed another model called "percolation dissolution" while also discussing the major dealloying mechanisms in a recent paper. 34 In the present study, dissolution tests in a basic NaNO 3 solution were performed for the α,β -brass CuZn40Pb2 to contribute to a better understanding of the dissolution processes and kinetics of α,β -brass under a constant anodic potential and under specific mechanical loading in the framework of the development of accelerated SCC tests. Measurements of both the dissolution depths and the weight loss, chemical analyses of the corroded materials, the corrosion products and the solutions were performed to quantify the dissolution processes. The discussion focuses on the dissolution mechanism with reference to the α-brass dissolution processes proposed in the literature. [1][2][26][27][28][29][30][31][32][33][34][35]

Experimental

Material.-The material used is the α,β -brass CuZn40Pb2 (CW617N) removed from extruded rods, as reported in a previous study. 16 Its chemical composition obtained from inductively coupled plasma (ICP) measurements is presented in Table I. Its microstructure was characterized by using optical microscope (OM) observations after abrading, polishing and etching the brass sample surface using Klemm II solution (24 g Na 2 S 2 O 3 , 5 g K 2 S 2 O 5 , 50 mL H 2 O) for 9 minutes. Dissolution tests.-A dissolution test consisted of the immersion of the sample in a nitrate solution with an anodic potential applied on the sample after 15 minutes of immersion at OCP. The duration of the dissolution tests was between 2 h and 48 h. Based on the results from the previous study, 16 a 0.5 M NaNO 3 solution with a pH adjusted to 11 was used. The reference conditions corresponded to dissolution tests performed in the solution without mechanical loading and under an anodic potential equal to 0.05 V/SCE, i.e., a potential after the passivity breakdown, as identified in the previous study. 16 Experiments were performed with a standard device composed of a potentiostat connected to a reference electrode (Saturated Calomel Electrode), a counter electrode (platinum) and a working electrode composed of an α,β -brass sample. The surface exposed to the electrolyte corresponded to the plane perpendicular to the extrusion direction. For the major parts of the dissolution tests, a surface area of 0.2 cm 2 (surface/volume ratio = S/V = 6.7 × 10 -4 cm 2 /mL) was selected by using a varnish. The sample surface was polished with 4000 grit SiC paper before the experiment. Tests were performed with 300 mL of the solution that was open to the air, moderately stirred with a magnetic stirrer and maintained at 23.5 ± 2 • C using a thermostatically-controlled water-bath. A few experiments were performed with 0.02 cm 2 and 4 cm 2 of the surface exposed to 300 mL of the solution (respectively S/V = 6.7 × 10 -5 cm 2 /mL and 1.3 × 10 -2 cm 2 /mL) to study the influence of the S/V ratio on the dissolution kinetics.

To better understand the effect of mechanical loading on the anodic dissolution of α,β -brass CuZn40Pb2, some dissolution tests were performed while applying a mechanical load using a bending device (Figure 1a), allowing for the specimen to be completely immersed in 500 mL of the electrolyte solution. The geometry of the tested specimen is presented in Figure 1b. The stress was perpendicular to the extrusion direction. These conditions were chosen to correspond to the most rigorous requirements according to preliminary tests, not presented here. The surface, either under tensile or compressive loading (Figure 1a), was exposed to the electrolyte using a protective varnish on the non-exposed surface. In each case, the surface exposed to the electrolyte corresponded to a zone in a rectangular shape (10 × 3.5 mm 2 ; surface/volume ratio = 7. 10 -4 cm 2 /mL, close to previous S/V values of 6.7 × 10 -4 cm 2 /mL) in the center of the specimen between the two interior support rollers. The same standard device used in the dissolution tests, without mechanical loading, was used here (Potentiostat, 3 electrodes, magnetic stirring, and thermostatically controlled water-bath). An initial maximal stress of 140 MPa was applied to achieve 70% of the yield strength at 0.2% of strain at the surface of the specimen, which was previously determined by traction tests at 10 -3 s -1 and found to be 205 MPa. The dissolution test was then performed at constant mechanical loading. Displacements during the tests were followed with a sensor (Keyence-GT2-H12K). It is worth noting that some dissolution tests, at the reference conditions, were performed with samples placed on the bending device without applying a mechanical loading to allow comparison between results with and without a mechanical loading by eliminating the effects of the bending device. Furthermore, a preliminary study showed that the OCP values measured for 15 minutes in a 0.5 M nitrate solution at pH 11 for a surface under compressive or tensile loading were similar to those measured for a non-loaded surface; current-potential curves plotted at a scan rate of 0.070 mV.s -1 were similar for the loaded and non-loaded surfaces.

The experiments performed are described in Table II.

Characterization of the corrosion damage.-After dissolution tests at a constant potential were completed, cross-sections of the corroded material were prepared. They were observed using an optical microscope (NIKON) to measure the dissolution depths, and more detailed observations were performed by a scanning electron microscope (SEM, LEO435VP LEO) coupled with an Energy-dispersive X-ray spectrometer (EDS) for chemical analyses. For the dissolution depths, 5 measurements were performed for one sample all along the center of the exposed zone. For each experimental condition, 3 samples were analyzed for reproducibility. Each point on the dissolution curve corresponds to a mean value obtained from 15 measurements (i.e., 5 measurements for one sample × 3 samples). Error bars on the curves take into account the 15 measurements. For each curve, data obtained for different durations correspond to different samples. Weight loss measurements were also performed using a precision balance: after the dissolution tests at a constant potential, the samples were rinsed using distilled water to remove the major part of the corrosion products, and then air-dried. They were weighed and their weight was compared to the weight measured before the dissolution tests. Corrosive solution analyses were performed using inductively coupled plasma atomic emission spectroscopy (ICP-AES -Ultima 2 Horiba). Analyses were performed after rinsing the working electrode and reference electrode with distilled water, as well as dissolving the corrosion products deposited on the platinum electrode with dilute nitric acid leading to a solution called rinsing solution. The rinsing solution and the electrolyte used for the dissolution tests were mixed and the corrosion products present in the mixture were dissolved with dilute nitric acid to perform ICP analyses. X-ray diffraction (XRD) experiments were also performed to analyze the corrosion products removed from the solution. A 2-θ scan was performed from 20 • to 70 • using Cu K-α radiation and SEIFERT-3000TT equipment.

Results

Microstructure.-The metallurgical state of the brass was described in a previous study. 16 It corresponded to that of the center of a 65 mm diameter extruded rod (Figure 2). The α phase (64 at. % Cu, 36 at. % Zn, Table I) was enveloped in β phase (55 at. % Cu, 44 at. % Zn, Table I). A transverse section of the rod (Figure 2a) was characterized by equiaxed grains (α and β phases) with an average size of 24 μm in diameter for the α phase and 20 μm for the β phase. Observations of the longitudinal sections of the rod (Figure 2b) showed textured β grains (their size varied from 20 μm to 120 μm in the longitudinal direction). Moreover, α grains had preferentially germinated at the β grain boundaries, leading to a specific chaining of β and α phases. Because lead is insoluble in Cu-Zn alloys, lead-rich micro-particles were also observed; they were finely dispersed in the alloy but mainly present at the α,β -interphases and in the β phase. Characterization of the damage after the dissolution tests and determination of dissolution kinetics.-In all the chosen experimental conditions, on the basis of optical observations, corrosion phenomena were observed essentially in the β phase, which was more reactive than the α phase due to the higher amount of Zn, as previously observed. 16 This was in agreement with the results obtained by Assouli et al. 15 who showed that the α phase, which contains more copper than the β phase, was more noble than the β phase. Typical observations of the damage caused by 12 h and 48 h dissolution tests at a constant potential at the reference conditions are presented in Figures 3a and3b, respectively. A schematic of the corroded zone is presented in Figure 3c. For short tests, the corroded zone was composed of only non-corroded α grains suggesting a complete dissolution of the β phase, i.e. both Cu and Zn (and a few alloying elements in the β phase as Sn and Ni) dissolution from the β phase. This mechanism was commonly called simultaneous dissolution. [26][27][28] However, for longer dissolution tests, under the outer-layer composed of only non-corroded α grains, an inner layer characterized by pink-colored β grains surrounded by the non-corroded α phase was observed. Chemical analyses by EDX through the sections of the corroded specimens (Figure 3d) confirmed that the α phase was not corroded and showed that the β discoloration was due to a depletion in Zn leading to the formation of a porous Cu-rich phase. In the absence of discoloration, the chemical composition of the β phase was that of a non-corroded sample. The result highlighted that dezincification of the β phase in such a leaded α,β -brass occurred under anodic polarization. Due to the morphology of the corrosion damage, dissolution kinetics were evaluated by measuring two parameters versus the duration of the dissolution tests: the depth of the simultaneous dissolution front, i.e., the average distance between the initial surface of the material and the first β grain not completely dissolved, and the depth of the dezincification front (i.e., selective dissolution), i.e., the average distance between the initial surface of the material and the first intact β grain. With such a definition, the depth of the dezincification front included that of the simultaneous dissolution front. The corroded zone was characterized by an elliptical shape (Figures 3a and3b) corresponding to the β phase dissolution starting at the edges of the exposed zone that was separated from the protected zone by a varnish (Figure 3c). The depth of the corroded zone was observed to be constant in the center of the exposed zone. Therefore, quantification of the corrosion damage, i.e., measurements of the dissolution depths, by OM observations, were conducted in the center of the exposed zone. The dissolution depth vs. time curves (simultaneous and selective dissolution) for the reference conditions (0.5 M NaNO 3 solution, pH 11, polarization at 0.05 V/SCE without mechanical loading) are presented in Figure 4a. They were characterized by two stages separated by a critical time called t c : the second stage the simultaneous dissolution of the β phase had not stopped but continued to proceed with a time lag when compared to the dezincification. When the duration of the dissolution tests increased, the difference in depth between the simultaneous dissolution and dezincification globally increased suggesting that the dezincification extended in depth faster than the simultaneous dissolution. However, due to the difficulty in measuring the dissolution depths, the result was not unambiguously demonstrated in the figures.

Figure 4b shows the current density versus immersion time during the dissolution tests. The curve was characterized, in the first 2 h, by a strong current density peak followed by a quick decrease of the current density. Afterwards, the current density slowly increased or tended to stabilize at approximately 1.7 mA.cm -2 . Taking into account the uncertainty in the t c value, previously evaluated from dissolution curves, it should be assumed that a stable value of the current density was reached for time equal to t c . The results showed that the two stages previously defined were clearly observed by recording the currentdensity versus time. Except for the current peak observed during the first 2 hours, the current density during the second stage was higher than that measured during the first; the stabilization of the current density was probably due to the occurrence, after t c , of a dezincification phenomenon along with a simultaneous dissolution. The current peak in the first two hours showed a high rate of dissolution for a few hours followed by a steep fall in current and a slow increase of the current corresponding to a slow increase in the dissolution rate. The current peak in the first two hours was assumed to be due to initiation of dissolution events with the removal of the air-formed film and the formation of a new film on the sample surface. It should be noted also that the air-formed film probably dissolved during the 15 minutes at OCP, before the dissolution tests. Therefore, the results showed that the corrosion damage corresponded to a two-step mechanism: the first step, called as initiation step, was characterized by high current densities and was attributed to β simultaneous dissolution on the extreme surface. The second step, where the dissolution subsequently went through the material, was called as propagation step. For the initiation step, only simultaneous dissolution of the β phase (Figure 3a) had to be considered: the dezincification process that probably occurred during the first 15 minutes at the OCP, before the dissolution test, was neglected on the OM observations due to the duration of the shortest tests (2 h). For the propagation step, only simultaneous dissolution of the β phase was observed before t c , while after t c , both simultaneous dissolution and dezincification of the β phase occurred.

Chemical composition of the electrolyte versus time.-During dissolution tests at a constant potential, a blue coloring of the solution due to the corrosion products was observed after a delay depending on the stirring of the electrolyte. This could correspond to a change in the aggressiveness of the environment related to the critical time, t c , corresponding to the occurrence of a dezincification process. This finding means that the critical time should be correlated with a critical chemical composition of the electrolyte. Therefore, corrosion products in the solution were removed, dried and analyzed by XRD (Figure 5a). These corrosion products were mainly identified as Cu 2 (NO 3 )(OH) 3 , known as rouaite, but there was also evidence of Cu(NO 3 ) 2 . The corrosion products had accumulated on the surface that had been exposed to the electrolyte. In Figure 5b, pH vs. time data points showed a strong decrease of the pH value over time, which could be related to the formation of a very large amount of corrosion products. The cathodic reactions, i.e. oxygen and maybe nitrate reduction, should increase the pH. However, results suggested that this effect was largely compensated by the formation of corrosion products; furthermore, an analysis of the solution showed that nitrate reduction did not occur. Finally, CO 2 should cause the reduction of the pH because the solution was open to air but experiments performed in deaerated conditions showed that the presence of CO 2 was not the major parameter to explain the variation of the pH even though this effect could not be excluded.

To understand further the change of the chemical composition of the electrolyte during the dissolution tests, dissolution kinetics were determined for two additional values of the ratio of the surface exposed to the electrolyte to the volume of the solution (S/V), respectively S/V = 6.7 × 10 -5 cm 2 /mL and 1.3 × 10 -2 cm 2 /mL compared to the ratio used for previous experiments i.e., 6.7 × 10 -4 cm 2 /mL (Figure 6). For all the chosen S/V values, two stages were observed on the dissolution depths vs. time curves as described previously (Figure 6a). For S/V = 6.7 × 10 -5 cm 2 /mL and 1.3 × 10 -2 cm 2 /mL, a limited set of experiments was performed. However, for S/V = 6.7 × 10 -5 cm 2 /mL, no dezincified layer was observed even after a 24 h dissolution test while dezincification was observed after a 48 h test showing that the minimum value of t c would have to lie between 24 h and 48 h. For S/V = 1.3 × 10 -2 cm 2 /mL, a dezincified layer was observed after a 12 h dissolution test showing that the maximum value for t c was 12 h. Therefore, the results clearly show that an increase in the S/V value shifted the critical time, t c , to a lower value, i.e., promoted the occurrence of the dezincification process (Figure 6b). However, mainly for long tests (>40 h), the dezincification depths were smaller for the large S/V value while the simultaneous dissolution depths were quite similar regardless of the ratio (Figure 6a).

Weight loss measurements on the working electrode (α,β -brass CuZn40Pb2 sample) and ICP analyses of the solutions were performed after the dissolution tests for the two higher S/V values, i.e., ratios equal to 6.7 × 10 -4 cm 2 /mL and 1.3 × 10 -2 cm 2 /mL (Figure 7 and Figure 8).

Independent of the S/V value, weight loss measurements (Figures 7a and8a) were coherent with the dissolution depths measured by the OM observations (Figure 6a) with two stage kinetics The area of the surface exposed was equal to 0.2 cm 2 for a solution volume of 300 mL (S/V = 6.7 × 10 -4 cm 2 /mL). The alloying element concentration in solution was expressed in mg.cm -2 to take into account the size of the surface exposed to the electrolyte. characterized by a weight loss that rapidly increased in the first stage followed by, after a critical time (quite similar to t c measured in Figure 6b), a slowing down of the variation in the weight loss versus time curve in the second stage. Again, during the first stage, simultaneous dissolution of the β phase occurred, leading to a rapid increase in the weight loss; after the critical time, t c , the dezincification occurred and was faster than the simultaneous dissolution, leading to a slowing down of the variation in weight loss. Furthermore, the shift of the critical time t c toward lower values when the S/V ratio increased was observed by comparing the weight loss curves plotted for the two S/V values (Figures 7a and8a), which is in agreement with Figure 6b. Figures 7b and8b show the Cu and Zn content in the electrolyte versus time. Regardless of the duration time of the dissolution tests or the S/V ratio, the amount of zinc inside the solution was higher than that of copper, but the differences in the amounts of the two elements were significantly increased during the second stage. This was consistent with the dissolution mechanisms. For short times, before t c , a simultaneous dissolution of the β phase occurred. Due to the chemical composition of the β phase (55 at.% Cu and 44 at.% Zn), the Cu and Zn amounts in solution should be similar. If the errors on the measurements were neglected, the differences could be explained by the precipitation of Cu-rich species that had not been dissolved and were lost during the preparation of the solution for the ICP analyses (see experimental part). For longer times, after t c , the strong increase of the amount of Zn compared to Cu corroborated the occurrence of a dezincification process that was faster than the simultaneous dissolution. Furthermore, a close-up on the Cu versus time curves showed that the rate of leaching of Cu decreased near the critical time, while the amount of Zn continued to increase for longer times during the dissolution tests. This suggested that the transition from the first stage to the second stage is related to the amount of Cu in solution that can be correlated to the accumulation of Cu-rich products on the brass. The weight loss and the amount of the major elements dissolved (by surface unit) for the different S/V experiments did not reflect the difference observed in the kinetics with the different dissolution depths. However, as suggested before, a certain amount of the corrosion products probably remained on the brass sample and had not been taken into account during the ICP analyses.

Results concerning the impurity elements (Figures 7c and8c), mainly Fe that had segregated in the α phase and potentially at the interphases, suggested that, even if the corrosion damage was localized on the β phase, the α phase surface might be slightly dissolved. However, the Fe amount was very low and, as explained previously, corrosion phenomena on α phase was considered as negligible. Finally, Figures 7b and8b showed that Pb ions were also leached into the solution. The influence of the S/V ratio on the extent of leaching of Pb could not be evaluated but, for the two S/V values, the amount of Pb leached into the solution strongly increased after the critical time t c .

Influence of a mechanical loading on the dissolution kinetics.-Figure 9 shows the effects of mechanical loading on the dissolution processes. Depths of both dissolution processes vs. time curves without mechanical loading were reported for comparison. It is worth noting that new reference curves were plotted with samples placed on the bending device without applying a mechanical loading to eliminate the effects of the bending device, because, due to the geometry of the bending device, local electrochemical conditions could be different between experiments performed with and without the bending device even though the global surface/volume ratio was the same for experiments using the bending device and others without this device. The bending device corresponded to a cell open to the bulk but, due to the limited volume between the cell and the sample, an evolution of the solution inside the cell could not be excluded. Furthermore, stirring conditions inside the cell could be different than outside. Finally, the local conditions could vary between the tensile and the compressive surfaces. This explained the differences in the reference curves in comparison to those plotted in Figure 4a. In particular, comparison of the curves plotted without the bending device (Figure 4a) and that with the bending device for the tensile surface without applying a mechanical loading (Figure 9a), showed that the use of the bending device induced a delay in the emergence of the dezincification front. On the other side corresponding to the compressive surface, comparison of Figures 4a and9b showed that the use of the bending device did not change the critical time for the reference conditions, i.e. without mechanical loading. However, because the dissolution depths versus time curves were plotted for the reference conditions without a mechanical load though using the same device, it was possible to draw conclusions about the effects of the mechanical loading on the dissolution kinetics. The results clearly showed that all curves exhibited two stages, as previously explained, with a first stage where the dezincification front merged with the simultaneous dissolution front and, after t c , a second stage where both simultaneous dissolution and dezincification of the β phase occurred, dezincification extending faster in depth than the simultaneous dissolution. Due to the previous comments concerning the bending device geometry, there was no comparison of the critical times measured for the tensile and compressive surfaces. Major results were that (i) there was no effect of a tensile mechanical loading on the critical time compared to samples without a mechanical loading; the simultaneous dissolution front was not significantly affected during the entire dissolution test by the loading but, under tensile mechanical loading, during the second stage, i.e. after t c , the dezincification front progress was faster by comparison to the reference conditions without mechanical loading (Figure 9a) (ii) under compressive mechanical loading, the critical time was significantly increased compared to the results without mechanical loading (Figure 9b). However, the kinetics of both dissolution processes were in the range of those measured without mechanical loading.

Discussion

Dissolution mechanisms of the α,β -brass CuZn40Pb2 in nitrate solution.-The results (Figure 4) showed that the corrosion behavior of the α,β -brass CuZn40Pb2 in nitrate solution, under anodic polarization, for long immersion times was described by a two-step mechanism including an initiation step followed by a propagation step.

Concerning the initiation step, the high rate of dissolution in the early stages (current density peak) of the dissolution tests was related to initiation of dissolution events as previously stated, and in agreement with literature data. 31 During the first 15 minutes at OCP before the potential was applied for the dissolution tests, dezincification should occur on the sample surface, due to the high Zn amount in the brass and the galvanic coupling between the α and β phases, as shown in previous work. 16 However, due to the short time at OCP, this phenomenon only concerned the extreme surface layer and only simultaneous dissolution was observed using optical observations during the initiation step of the dissolution tests. Indeed, during immersion at OCP before the dissolution tests, dissolution processes led to the formation of a passive film mainly composed of Cu 2 O with a very low amount of ZnO. 16 When the anodic potential was applied for the dissolution tests, evolution of the oxide layer should occur but copper oxides remained present. The influence of lead was also suspected to be strong with the incorporation of lead in to the passive layer, due to the dissolution of Pb particles, which improved the corrosion resistance. 16,20,24,25 Kumar et al. showed that a high amount of lead in brass could provide an improvement in corrosion resistance under conditions where compounds of lead were precipitated on the surface. 24 The equilibrium of the electrolyte with this Cu 2 O and lead oxidesrich film made it difficult to initiate dezincification as suggested by different authors. [26][27][28][29][30][31][32][33][34] This was in agreement with low dissolution depths for the two first hours and the decay of the current density observed on the current density versus time (Figure 4b) after the initial peak.

After 2 h, the dissolution extended in depth which initiated the propagation step of the dissolution tests, but before t c , only simultaneous dissolution of the β phase occurred. This was in agreement with the corrosion behavior described by Pickering and Kim 27 concerning an alloy susceptible to dealloying. They showed that, initially, selective dissolution could occur and one of three phenomena then happened. The dissolution rate could decrease due to the protective nature of the modified surface layer. The planar surface could become unstable and local penetration could occur, or the dealloying could stop and give way to a non-selective reaction of the alloy. 27 The last phenomenon could happen when the alloy was exposed to conditions which did not produce a protective surface layer. In the present case, it was assumed that when a sufficient drop in the pH value was achieved at the surface (Figure 5b), a part of the hydroxide (rouaite) that was formed, dissolved. In more general terms, the modification of the surface layer should correspond to the formation of non-protective reaction-product layers which would lead to an increase in the simultaneous dissolution kinetics after 2 h. The question that remains is why, after t c , dezincification, i.e., selective dissolution of the β phase, occurred. Sieradzki et al. proposed a percolation mechanism of dealloying based on the concept that above a critical concentration there exist continuous atomic scale conduits of the more reactive component allowing selective dissolution to happen. [31][32][33][34] It could be assumed here that, due to the simultaneous dissolution processes that had occurred for β phase before t c , the percolation threshold had been reached. Furthermore, Sieradzki et al. showed that the transition from simultaneous dissolution to selective dissolution corresponded to the growth of a macroscopic three-dimensional dealloyed structure which defined a roughening transition with a competition between dissolution of the more active component leading to surface roughening and the curvature-driven surface diffusion of the more noble component. [31][32][33][34] It is clear that the dissolution of an atom depends not only on the number of nearest neighbors in the metal but also on their identity, and therefore to the alloy composition. Again, the simultaneous dissolution of β phase before t c should generate a local atomic environment conducive to a stronger reactivity for the more reactive component, i.e. Zn. Furthermore, the propensity to dissolution for an element depends also on the electrolyte composition. Therefore, the transition from simultaneous dissolution to dezincification could also be correlated to further modification of the chemical composition of the electrolyte (Figure 5b). The local dissolution phenomena observed before t c , with simultaneous dissolution of the β phase while the α phase remained uncorroded, should generate an increase in the surface roughness, at a macroscopic scale, with the formation of cavities between the uncorroded α grains. The duration of the simultaneous dissolution process should correspond to the time needed for the electrolyte trapped inside the cavities to reach a critical composition corresponding to the destabilization of β surface. This was consistent with the extent of leaching of Pb before t c when the duration of the tests increased. After t c , the share of Pb in the oxidized metal leached from or accumulated on the surface, was more evenly disturbed as evidenced by the strong increase in the amount of Pb leached in after t c (Figures 7b and8b), because of the removing of Pb particles due to increased β phase dissolution.

Therefore, the results suggested that once the process of dissolution was initiated, any change in the chemical composition of the electrolyte was one of the most obvious parameters that could explain the variations observed in the general rate of dissolution, as well as the occurrence of the dezincification process. In particular, previous studies in the literature demonstrated the major role of the copper concentration in solution on the transition between the simultaneous dissolution and dezincification of the Cu/Zn elements. 35 Sieradzki et al. showed that severe dezincification occurred in concentrated chloride solutions where cuprous ions were highly stabilized by complexing, when compared with cupric ions. 31 Therefore, the Cu concentration in solution, i.e., a solubility limit achievement for Cu, remained a relevant parameter that should explain the transition at t c between the two stages. The transition at t c from the first stage, with the simultaneous dissolution of only the β phase, to the second stage, with Zn dezincification of the β phase faster than the simultaneous dissolution, was identified for each of the experiments performed using a different S/V ratio which corresponded with different Cu and Zn concentrations in solution. Results showed that the dezincification front was observed for: r [Cu] = 7 mg / 300 mL = 5.9 mg.cm -2 .L -1 [Zn] = 30 mg / 300 mL = 25 mg.cm -2 .L -1 for a 4 cm 2 exposed surface r [Cu] = 0.5 mg / 300 mL = 8.3 mg.cm -2 .L -1 [Zn] = 5 mg / 300 mL = 83.3 mg.cm -2 .L -1 for a 0.2 cm 2 exposed surface Taking into account the inherent errors of the chemical analysis and the lack of accuracy for the t c values determined from the dissolution depths vs. time curves, the results showed that a critical concentration of Cu in solution per area of exposed material for a particular potential could be identified as a way to explain the transition between the two stages with the separation of both of the dissolution fronts: it was approximately 6-8 mg.cm -2 .L -1 for the reference conditions. It might have a different value for other experimental conditions (another pH, another nitrate concentration or another applied potential) because these parameters also influenced the stability of the oxide layers and corrosion products and the dissolution kinetics of the more active component, i.e. Zn.

Influence of the mechanical loading on the dissolution kinetics.-

The objective is to understand the influence of the mechanical loading on the progress of the dezincification throughout the material. According to Sieradzki, [30][31][32][33][34] dezincification proceeds with a percolation dissolution mechanism which is first characterized by the formation of connected clusters of the more reactive components, i.e. Zn, in the bulk and these percolating clusters of atoms provide a continuous active path for the dissolution process. Further, they provide pathways for the electrolyte to enter the bulk solid by dissolving Zn. Regarding the high Zn proportion in the β phase, the density of these paths inside the β phase after dezincification should be considered strong. According to the literature, it was demonstrated that dezincification induced a tensile stress in the dezincification layer of brass. 21,[36][37] For the βphase in α-β brass, the stress level could be high enough to influence strongly the morphology of the previous paths. They could be very large, and after dissolution they could lead to the formation of pores with a high degree of percolation, i.e. features like tunnels in contact with the electrolyte trapped between the non-corroded α grains. Such a morphology of the porous structure is in good agreement with Sieradzki's observations. 34 The lead particles in the dezincified layer could contribute to this high degree of percolation, in being dissolved or in decreasing the mechanical properties of the dezincified layer while dezincification induced stress. A schematic of the proposed mechanism is shown in Figure 10. Such a mechanism should be in good agreement with the results obtained concerning the influence of the mechanical loading (Figure 9). At a macroscopic scale, the mechanical loading, by inducing a strain in the alloy, should influence the shape of the "tunnels". A compressive mechanical loading should promote the closing of the "tunnels", so that they did not contain enough free volume for the electrolyte to penetrate the evolving porous structure. Therefore, a compressive loading was expected to delay the dezincification of the β phase which was in agreement with Figure 9b; t c was shifted toward longer times showing that a compressive mechanical loading promoted the simultaneous dissolution to the detriment of dezincification. On the contrary, a mechanical tensile loading should contribute to opening the "tunnels". Therefore, this loading was expected to accelerate the dezincification kinetics as shown in Figure 9a. Furthermore, the mechanical loading should influence the dissolution behavior of the α-β brass at an atomic scale. Indeed, as previously stated, the dissolution potential of an atom depends not only on its chemical environment (number and identity of its neighbors) but also on the local stress state. Therefore, mechanical loading should also influence the roughening transition, thereby explaining the transition from simultaneous dissolution to dezincification.

Conclusions

Results for the α,β -brass CuZn40Pb2 in NaNO 3 solutions are summarized in the following points.

r Dissolution tests, in a 0.5 M NaNO 3 solution, at pH 11, at a constant anodic potential, led to the dissolution of the β phase while the α phase remained uncorroded.

r The corrosion behavior of the α,β -brass was characterized by a two-step mechanism: (i) an initiation step for which simultaneous dissolution of only the β phase occurred; (ii) a propagation step, including simultaneous dissolution, before a critical time t c , and both, simultaneous dissolution and dezincification of the β phase, after t c . r The results were in good agreement with the percolation disso- lution mechanism proposed by Sieradzki et al. [30][31][32][33][34] A critical composition of the electrolyte, i.e., a critical concentration of copper per area of exposed material, correlated with a pH value, at the bottom of the corrosion damage contributed to explain the transition from simultaneous dissolution of the β phase to dezincification. The dezinfication led to the development of a porous structure, having a high degree of percolation. The electrolyte could penetrate inside these "tunnels" allowing the corrosion process to continue.

r Depending on its sign (tensile or compressive stress), a mechan- ical loading could close or open the "tunnels" thereby significantly affecting the kinetics of the β phase dezincification.

Figure 1 .

 1 Figure 1. (a) Schematic of the bending device used to determine the influence of mechanical loading on the dissolution kinetics. (b) Geometry of the specimen.

Figure 2 .

 2 Figure 2. Optical microscope observations of the α,β -brass CuZn40Pb2 (CW617N). The sample was observed after polishing followed by Klemm II etching. (a) transverse section (b) longitudinal section.

r

  the first stage (t < t c ) occurred where the dezincification front merged with the simultaneous dissolution front showing that the corrosion phenomena for the β phase corresponded to a simultaneous dissolution process. No dezincification of the β phase was detected under the previously described outer layer;

r

  the second stage (t > t c ) occurred where the dezincification front preceded the dissolution front. At this stage, both the outer layer corresponding to the simultaneous dissolution of the β phase and the inner layer composed of non-corroded α grains and the dezincified β phase were observed. These observations clearly show that during

Figure 3 .

 3 Figure 3. Analysis by optical microscope of a section of the α,β -brass CuZn40Pb2 sample after (a) a 12 h and (b) a 48 h dissolution test in a 0.5 M NaNO 3 solution at pH 11 under anodic polarization at 0.05 V/SCE. (c) Schematic of the corrosion damage and (d) EDX analysis.

Figure 4 .

 4 Figure 4. (a) dissolution depths (b) current density versus time curve during the dissolution tests of the α,β brass CuZn40Pb2 for the reference conditions (0.5 M NaNO 3 solution, with a pH adjusted to 11, and an applied potential equal to 0.05 V/SCE; without mechanical loading).

Figure 5 .

 5 Figure 5. (a) XRD analysis of the blue corrosion products (b) pH vs. time during a dissolution test at the reference conditions, i.e., at 0.05 V/SCE in a 0.5 M NaNO 3 solution at pH 11, without mechanical loading, performed on α,β brass CuZn40Pb2.

Figure 6 .

 6 Figure 6. (a) Dissolution depths of the α,β brass CuZn40Pb2 versus time during the dissolution tests in a 0.5 M NaNO 3 solution, pH 11, with a potential applied of 0.05 V/SCE, without mechanical loading. All of the measurements were performed for three values of the surface/volume ratio. The ratio 6.7 × 10 -4 cm 2 /mL corresponded to a surface exposed to 0.2 cm 2 , i.e., that of the previous experiments; values from Figure 4a were therefore reported for comparison with values obtained for the two other ratios. (b) Critical time t c versus the surface/volume ratio.

Figure 7 .

 7 Figure 7. (a) Weight loss of the α,β -brass CuZn40Pb2 sample (b) Cu, Zn and Pb content of the electrolyte (c) concentrations of the impurity elements (Fe, Ni, Sn) of the electrolyte versus time. Results correspond to the reference conditions (0.5 M NaNO 3 solution, pH 11, potential applied of 0.05 V/SCE, without mechanical loading).The area of the surface exposed was equal to 0.2 cm 2 for a solution volume of 300 mL (S/V = 6.7 × 10 -4 cm 2 /mL). The alloying element concentration in solution was expressed in mg.cm -2 to take into account the size of the surface exposed to the electrolyte.

Figure 8 .

 8 Figure 8. (a) Weight loss of the α,β -brass CuZn40Pb2 sample (b) Cu, Zn and Pb concentrations of the electrolyte (c) concentrations of the impurity elements (Fe, Ni, Sn) of the electrolyte versus time. Results correspond to the reference conditions (0.5 M NaNO 3 solution, pH 11, potential applied of 0.05 V/SCE, without mechanical loading). The area of the surface exposed was equal to 4 cm 2 for a solution volume of 300 mL (S/V = 1.3 × 10 -2 cm 2 /mL).The alloying element concentration in solution was expressed in mg.cm -2 to consider the size of the surface exposed to the electrolyte.

Figure 9 .

 9 Figure 9. Influence of (a) tensile and (b) compressive mechanical loading on the depth of the dissolution of the α,β brass CuZn40Pb2. In each graph, the effect of the mechanical loading is studied in comparison with the results obtained for the reference conditions (0.5 M NaNO 3 , pH 11, applied potential equal to 0.05 V/SCE without mechanical loading).

Figure 10 .

 10 Figure 10. Schematic of the proposed dezincification mechanism (a) before t c (b) after t c .

Table I . Chemical composition of the α,β -brass CuZn40Pb2 (CW617N) and chemical composition of the α and β phases.

 I

		Inductively coupled plasma atomic emission
		spectroscopy			Electron microprobe
		Extruded rod		α Phase		β Phase
	Elements	wt %	at %	wt %	at %	wt %	at %
	Zn	38.35	38.20	36.40	35.80	44.10	43.80
	Al	0.008	0.019	0.008	0.02	0.01	0.03
	Ni	0.051	0.057	0.12	0.13		0.10
	Fe	0.205	0.239	0.26	0.30	0.08	0.09
	Mn	0.004	0.005	-	-		-
	Sn	0.150	0.082	0.08	0.05	0.48	0.26
	Pb	1.875	0.589	0	0	0	0
	Si	0.001	0.002	-	-	-	-
	Cu	59.35	60.81	63.10	63.70	54.10	55.30

Table II . Description of the experimental conditions for the dissolution tests.

 II

		Potential applied		Nitrate ion		Mechanical loading
		(V/SCE)	pH	concentration (M)	S/V cm 2 /mL	(MPa)
	Reference conditions	0.05	11	0.5	6.7 × 10 -4	0
	S/V effect	0.05	11	0.5	6.7 × 10 -5	0
		0.05	11	0.5	1.3 × 10 -2	0
	Mechanical loading effect	0.05	11	0.5	7 × 10 -4	140 (at the surface)
		0.05	11	0.5	7 × 10 -4	0

Acknowledgments

The authors express their sincere thanks to J-C. Salabura for the design of the bending device utilized in the study and R. Mainguy for providing the technical support to establish the experimental device.

This work was performed in the framework of the CETIMAT. The CIRIMAT and the CETIM collaborate for certain aspects of their research activities; this collaboration is performed in a joint laboratory, called CETIMAT.