1 Introduction

Precision engineering is crucial for the optimal performance of various manufacturing methods and the parts they produce [1, 2]. Additive manufacturing (AM) techniques such as laser powder bed fusion (LPBF) will result in poor as-built surface quality, requiring extensive post-processing, including traditional machining like milling and non-traditional techniques such as electrochemical machining (ECM). Nickel (Ni) superalloys, for instance, are readily fabricated by LPBF. However, to achieve the required surface quality and geometric precision, ECM is employed to enhance their final surface finish. Their high hardness and strength, a result of LPBF-enabled grain refinement and mechanical strengthening, make ECM an ideal finishing process for these materials [3, 4].

Recognizing the importance of ECM in the post-processing of Ni alloys after LPBF, [5,6,7] the understanding of ECM mechanisms is continuously evolving. ECM utilizes rapid, electrically controlled anodic dissolution to remove excess material [8]. This process is not constrained by the material's hardness or strength, allowing ECM to machine any metallic part given sufficient polarization voltage [6].

The electrochemical nature of ECM, complicated by phenomena of localized corrosion, passivation, and anodic dissolution, is fundamental to understand ECM-related behaviors. For instance, the transition to electrolytes like NaNO3, in place of chlorine-based solutions such as NaCl, has become standard for Ni alloy ECM [9, 10]. This shift is due to NaNO3's electrochemical stability and reduced tendency to cause localized passivation breakdown and corrosion susceptibility [9, 11]. Engineering optimizations, including adjustments to flow rate, pressure, voltage, current density, and other machining parameters, have also been explored to refine the ECM process, aiming for a balance between efficiency and workpiece quality [12]. Such optimizations underscore the necessity of understanding the critical phase interactions. For example, LPBF-produced IN718 Ni superalloy exhibits complex phases that complicate ECM design and efficiency analysis, including γ'-Ni3(Ti,Al,Nb), γ''-Ni3(Nb,Ti), δ-Ni3(Nb,Ti) phases, the Laves phase, and carbides of MC-(Nb,Ti)(C,N) and M23C6-(Cr,Fe)23C6, and this complexity affects the post-ECM surface quality [13, 14].

Despite significant advancements, there remains a gap in establishing a scientific, predictable, and quantitative relationship between ECM conditions, material characteristics, and resultant surface quality [15,16,17,18,19]. Anomalies such as the nonlinear dependency of surface roughness on ECM parameters of voltage, flow rate, and electrolyte concentration have been observed [20, 21]. Furthermore, while there is an ongoing effort to enhance ECM surface finish, the potential existence of a definitive surface roughness limit achievable with ECM remains uncertain. In the context of the rapid evolution of AM techniques and the critical need for high-quality Ni alloys, addressing this ambiguity is imperative for the advancement of ECM design.

In response to these challenges, we propose a robust yet streamlined approach to examining ECM in the context of Ni alloys, particularly focusing on characteristic microstructures such as carbides to anticipate potential surface roughness outcomes. This approach, grounded in the fundamental electrochemical principles of ECM anodic dissolution, aims to establish a theoretical framework and compensate the current ECM modeling based on macroscopic mass transport and machining theories. Through this framework, we summarize several critical paradoxes that serve as a reference for directing future ECM research and development and urge a raised attention and focus on the ECM surface quality understanding.

2 Theoretical deduction of surface roughness in ECM

This communication will take an approach from fundamental surface roughness definition to interpret ECM surface roughness evolution. As shown in Fig. 1a, the definition of surface roughness \({R}_{a}\) is:

$${R}_{a}=\frac{1}{l}{\int }_{0}^{l}\left|z\right|dx$$
(1)

where \(l\) is the measured length of interest, and \(\left|z\right|\) is the absolute distance from rough peaks and valleys to the mean surface line in the direction of \(l\). The main focus of this communication is to answer what causes \(\left|z\right|\) and changes the surface roughness in the ECM setup.

Fig. 1
figure 1

a Visualization of surface roughness. b Ideal case for surface roughness change during ECM, starting from a smooth surface

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In ECM, if we start with a relatively smooth surface, localized electrochemistry will introduce different anodic dissolution current \({i}_{anode}\) (A/m2), as shown in Fig. 1b. At a certain location, the changed height \(\Delta d\) by anodic dissolution during ECM is:

$$\Delta d=\frac{\Delta t\cdot {i}_{anode}}{k}$$
(2)

where \(\Delta t\) (s) is the ECM time, and \(k\) (C/m3) is the location-specific phase-dependent volumetric dissolved electrons. The parameter \(k\) can be further expressed as:

$$k=\frac{n\cdot {N}_{A}\cdot {k}_{e}}{v}$$
(3)

where \({N}_{A}\) (mol−1) is the Avogadro constant, \(v\) (m3/mol) is the phase-dependent mol volume (6.59 cm3/mol for pure Ni), [22] and \({k}_{e}=1.602\times {10}^{-19} C/e\). \(n\) (unitless) is the characteristic lost electron numbers per metallic atom (\(M\)) during the anodic reaction in Eq. 4 below.

$$M-n{e}^{-}\to {M}^{n+}$$
(4)

With this, the resultant mean height is estimated as:

$$\overline{d }=\frac{1}{l}{\int }_{0}^{l}\frac{\Delta t\cdot {i}_{anode}}{k}dx$$
(5)

The expected surface roughness can then be calculated as:

$${R}_{a}=\frac{1}{l}{\int }_{0}^{l}\left|\frac{\Delta t\cdot {i}_{anode}}{k}-(\frac{1}{l}{\int }_{0}^{l}\frac{\Delta t\cdot {i}_{anode}}{k}dx)\right|dx$$
(6)

To make the case simpler for theoretical comparison, now we focus on a representative situation in ECM-processed Ni superalloys such as IN718, IN738, and Hastelloy X, where carbides (e.g., MC, M23C6, and M7C3 type) are frequently observable. The situation can be simplified and modeled as in Fig. 2a.

Fig. 2
figure 2

a A typical TEM image showing the carbides in the Hastelloy X after ageing heat treatment and the ECM simplified case. The TEM image is reproduced with permission from Ref. [3], © Elsevier B.V. 2023. b Undermining carbides causes an ECM-characteristic materials-dependent periodicity

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This model already considers the non-uniform dissolution reasoning with the mass transport consideration of metal ions [12]. Meanwhile, we have:

$${\Delta d}_{1}=\frac{\Delta t\cdot {i}_{a1}}{{k}_{1}}$$
(7)
$${\Delta d}_{2}=\frac{\Delta t\cdot {i}_{a2}}{{k}_{2}}$$
(8)

Subscript 1 indicates “matrix’, while subscript 2 indicates “carbides”. This non-uniform dissolution is also observed in the authors’ recent study [23] and has been reported in many ECM anodic dissolution studies [12, 24]. As shown in Fig. 3, we could expect carbide-remained or matrix-intact condition during ECM process with different voltages and processing time length, both supporting our simplified dissolution model setup in Fig. 2.

Fig. 3
figure 3

a1-a2 A typical surface morphology image showing the preferential grain boundary dissolution in the IN738 superalloy under 1.3 V for 1500 s in the NaNO3 electrochemical environment. b1-b2 A typical surface morphology image showing (c) the remaining secondary phases (here, residual carbides and intermetallic phases) in in the IN738 superalloy under 12 V for 300 s in the NaNO3 electrochemical environment. The plot is reproduced with permission from Ref. [23], © Springer. 2024

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Considering the volume fraction of the carbides in the ECM-processed matrix as \({v}_{f}\) (unitless), we can calculate the instant mean of the surface height by Eq. 5 as:

$$\overline{d }=\frac{1}{2}\cdot \Delta t\cdot [\frac{\left(1-{v}_{f}\right)\cdot {i}_{a1}}{{k}_{1}}+\frac{{v}_{f}\cdot {i}_{a2}}{{k}_{2}}]$$
(9)

Accordingly, expressing Eq. 9 into Eq. 1, we can obtain the surface roughness for this situation as:

$${R}_{a}=\Delta t\cdot \left|\frac{\left(1-{v}_{f}\right)\cdot {i}_{a1}}{{k}_{1}}-\frac{{v}_{f}\cdot {i}_{a2}}{{k}_{2}}\right|$$
(10)

This is true only when the carbides are not undermined (i.e., the carbides are not corroded or dissolved themselves, but their adjacent dissolution releases them into the electrolyte with the flushing flow), because undermining will take the carbide away with the electrolyte flow, [22] as shown in Fig. 2b.

The rough undermining/ECM period \({t}_{0}\) can be estimated as:

$${t}_{0}=\frac{{D}_{2}\cdot {k}_{2}}{{i}_{a2}}$$
(11)

where \({D}_{2}\) is the characteristic diameter of the carbides (from ~ 1–100 um) [12]. We should note that this \({t}_{0}\) can be shorter than the calculated value, because once the less-corroded secondary phases protrude, the disturbed electric field distribution will dissolve the geometrically protruded part quickly [8]. Taking the typical size of carbides as 1 um, with the total dissolution current of ~ 10–2 A \(\cdot\) cm−2 (see Fig. 4a) at the carbide-participating corrosion range (> 1 V in Fig. 4’s case), \({t}_{0}\cong 290\) s is calculated, matching the experimentally observed periodicity value of ~ 250 s (see Fig. 4b). This clearly shows our model case is valid.

Fig. 4
figure 4

a Potentiodynamic polarization curve of wrought Inconel 738 superalloy in 10 wt.% NaNO3 solution. b Time-dependent current density evolution of wrought Inconel 738 superalloy at the different biased voltage. Reproduced with permission from Ref. [23], © Elsevier B.V. 2024 Reproduced with permission from Ref. [25], © Elsevier B.V. 2023

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Then, taking Eq. 11 into Eq. 10, it is easy to obtain:

$$\begin{array}{ccc}{R}_{a}(t)=(t-N\cdot {t}_{0})\cdot \left|\frac{\left(1-{v}_{f}\right)\cdot {i}_{a1}}{{k}_{1}}-\frac{{v}_{f}\cdot {i}_{a2}}{{k}_{2}}\right|& \text{with}& N\cdot {t}_{0}<t<(N+1)\cdot {t}_{0}\end{array}$$
(12)

where \(N\) is an integer to describe the anodic dissolution time frame, and \(t\) is the actual ECM time. Since the anodic dissolution current density can be quantified with the charge transfer coefficient \(\alpha\), [22, 24] the final format of the roughness expression for this simple case will be:

$${R}_{a}(t)=(t-N\cdot {t}_{0})\cdot FK\cdot \left|\frac{\left(1-{v}_{f}\right)\cdot {n}_{1}\cdot \text{exp}(\frac{{\alpha }_{1}{n}_{1}FE-{\Delta G}_{1}}{RT})}{{k}_{1}}-\frac{{v}_{f}\cdot {n}_{2}\cdot \text{exp}(\frac{{\alpha }_{2}{n}_{2}FE-{\Delta G}_{2}}{RT})}{{k}_{2}}\right|$$
(13)

where \(E\) is (approximately) the applied electric field (when the ECM is using ultra-high voltage off the electrochemical equilibrium), and \(\Delta G\) is the surface activation energy to bring the metallic atoms to the ion state into the electrolyte. Based on the Ref. [26], the reaction constant \(K={c}_{M}\cdot \frac{{k}_{B}\cdot T}{h}\) is estimated, where \({c}_{M}\) (~ 1017 m−2) is the concentration, \({k}_{B}\) is the Boltzmann constant, and \(h\) is the Planck constant).

We should also notice that the charge transfer coefficient \(\alpha\) is defined in Fig. 5 as [27, 28]:

Fig. 5
figure 5

Definition of charge transfer coefficient \(\alpha\) during polarization measurement

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$$\alpha =-\frac{RT}{nF}\frac{dln{i}_{anode}}{dE}$$
(14)

With Eq. 3, Eq. 13 can be simplified as:

$${R}_{a}(t)=(t-N\cdot {t}_{0})\cdot FK\cdot \frac{1}{{N}_{A}\cdot {k}_{e}}\cdot \left|{v}_{1}\cdot \left(1-{v}_{f}\right)\cdot \text{exp}(\frac{{\alpha }_{1}{n}_{1}FE-{\Delta G}_{1}}{RT})-{v}_{2}\cdot {v}_{f}\cdot \text{exp}(\frac{{\alpha }_{2}{n}_{2}FE-{\Delta G}_{2}}{RT})\right|$$
(15)

Now, we can expand this simple situation to consider if when numerous carbides (e.g., they can usually have a 1 vol.% density in the Ni superalloy system) are dispersed on the surface, undermining and anodic dissolution should seamlessly happen all over the surface. Therefore, the average roughness over the whole periodic time on the whole surface should be the achievable surface roughness:

$${R}_{a}=\frac{1}{{t}_{0}}{\int }_{N\cdot {t}_{0}}^{(N+1)\cdot {t}_{0}}{R}_{a}(t)dt=\frac{FK{t}_{0}}{2{N}_{A}{k}_{e}}\cdot \left|{v}_{1}\cdot \left(1-{v}_{f}\right)\cdot \text{exp}(\frac{{\alpha }_{1}{n}_{1}FE-{\Delta G}_{1}}{RT})-{v}_{2}\cdot {v}_{f}\cdot \text{exp}(\frac{{\alpha }_{2}{n}_{2}FE-{\Delta G}_{2}}{RT})\right|$$
(16)

Clearly, Eq. 16 does not show a real time dependence (as expected) and demonstrates the ideal surface roughness value that can be achieved. By substituting Eq. 3 and Eq. 11 into this equation, we can get the final equation format to involve the characteristic physical properties:

$${R}_{a}=\frac{{D}_{2}}{2{v}_{2}}\cdot \text{exp}(\frac{-{\alpha }_{2}{n}_{2}FE+{\Delta G}_{2}}{RT})\cdot \left|{v}_{1}\cdot \left(1-{v}_{f}\right)\cdot \text{exp}(\frac{{\alpha }_{1}{n}_{1}FE-{\Delta G}_{1}}{RT})-{{v}_{2}\cdot v}_{f}\cdot \text{exp}(\frac{{\alpha }_{2}{n}_{2}FE-{\Delta G}_{2}}{RT})\right|$$
(17)

In the real case, \({v}_{f}\ll 1\) and \({v}_{2}={v}_{1}\) are assumed, so we could get a good surface roughness approximation as follows:

$${R}_{a}=\frac{{D}_{2}}{2}\cdot \text{exp}[\frac{{(\alpha }_{1}{n}_{1}-{\alpha }_{2}{n}_{2})\cdot FE-({\Delta G}_{1}-{\Delta G}_{2})}{RT}]$$
(18)

3 Results and case analysis

3.1 Simplified non-disturbed case

Stop** at Eq. 18, we obtain a quantitative surface roughness relationship to the driving force, i.e., applied ECM voltage bias. Assume \({D}_{2}\sim 1 um\), Ni has a molar volume of 6.6 cm3/mol, \({n}_{1}\cong {n}_{2}\cong 2\) (as it is mainly Ni matrix and compounds), We also assume that though the applied voltage in ECM is high enough in the corrosion transpassive range, the charge transfer coefficient has the relationship of \({\alpha }_{1}\cong {\alpha }_{2}\) (can be correctly estimated as ½ [22, 29]) because the (non-equilibrium) surface energy has no clear difference with voltage-induced activation energy. Therefore Eq. 18 can be expressed as:

$${R}_{a}=\frac{{D}_{2}}{2}\cdot \text{exp}(\frac{{\Delta G}_{2}-{\Delta G}_{1}}{RT})$$
(19)

The applied ECM voltage \(E\) is higher to overcome their surface dissolution activation energy \({\Delta G}_{1}\) and \({\Delta G}_{2}\), but clearly, our results indicate that the theoretical surface roughness value does not depend on the external electrical field, etc.

Based on the typical electrochemical impedance spectroscopy (EIS) measurement, \({\Delta G}_{2}-{\Delta G}_{1}\) should have a value of ~ 0.1-1 eV, as after their distinctive passivation and breakdown region, their anodic dissolution happens (see Fig. 6). So, the non-disturbed surface roughness value is \({R}_{a}=\)~20 um.

Fig. 6
figure 6

EIS measurements showing the different phases’ breakdown potential for LPBF Ni superalloy under various (post-)processing conditions.

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This range is reasonable for post-EIS surfaces, as it is similar to the features’ dimensions by pure corrosion (e.g., pitting with micro-galvanic nature between the matrix and the carbides under \(E\cong 0\), etc.). If we compare this theoretical value with some corrosion study results for Ni superalloys, we can see the predicted surface roughness range and magnitude match, as shown in Fig. 7.

Fig. 7
figure 7

Surface roughness of LPBF (a) AB and (b) HT samples on the Ni superalloy surface (IN718 XOZ plane) after corrosion. Reproduced with permission from Ref. [30], © MDPI under the Creative Commons Attribution (CC BY) license. 2020

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3.2 General ECM case

A more general case is that \({\alpha }_{1}{n}_{1}\) and \({\alpha }_{2}{n}_{2}\) cannot match well, and the influence of ECM potential comes in. With this general potential consideration for ECM, Eq. 18 can be written as:

$${R}_{a}=\frac{{D}_{2}}{2}\cdot \text{exp}[\frac{{\alpha }_{ECM}\cdot FE+({\Delta G}_{2}-{\Delta G}_{1})}{RT}]$$
(20)

Here, with Eq. 14, we have \({\alpha }_{ECM}={\alpha }_{1}{n}_{1}-{\alpha }_{2}{n}_{2}=-\frac{RT}{F}(\frac{dln{i}_{a1}}{dE}-\frac{dln{i}_{a2}}{dE})\).

If we take \(\frac{dln{i}_{a1}}{dE}\) as defined in Figs. 5 and 6 for matrix (\({i}_{a1}\)) and \(\frac{dln{i}_{a2}}{dE}\) as the \(\frac{dlni}{dE}\)-slope difference between the transpassive region and the open current potential region in Fig. 6 for carbides (\({i}_{a2}\)), \({\alpha }_{ECM}<0\) can be expected when the \(\frac{dln{i}_{a}}{dE}\) is taken at the ECM voltage (which is way higher, as the final total \(\frac{dln{i}_{a}}{dE}\to 0\)). According to the Tafel plot study, assume \(\frac{dln{i}_{a1}}{dE}\) dominates and is usually ~ 1/8 V−1 for Ni alloys (note this is the OCP value by Ref [8].) [31] and \(\frac{dln{i}_{a2}}{dE}\) (at ECM voltage) can be neglected, so we could get \({R}_{a}=\)~5 um, clearly setting the ideal possible ECM surface roughness.

From a materials point of view, \({\alpha }_{ECM}<0\) is also reasonable, because \({n}_{2}>{n}_{1}\) (since Nb, Ti, Fe, and Cr in carbides all have elemental covalence > 2 [22], but they are very close, as carbides are MC, M23C7, and M7C3 type to dilute M-ratio inside) and \({\alpha }_{1}\cong {\alpha }_{2}\) may still hold. Given a magnitude consideration, \(E\) is usually chosen at ~ 10 V, and assume \({n}_{2}=2.1\). With this condition, we find \({R}_{a}=\)~400 nm.

With the two case studies here, we can see our theoretical model fits the general quantification purpose of post-ECM surface roughness (see the reported real ECM surface roughness in Fig. 8) by building a clear relationship between the applied ECM voltage bias (as the machining and dissolution driving force) and the final surface roughness.

Fig. 8
figure 8

The reported surface roughness after ECM and unexpected surface roughness dependence on machining time during ECM processing (all the data points are extracted from continuous ECM and pulse-ECM with clearly defined voltage exertion period for Ni based alloys) [16, 17, 19, 23, 32,33,34]. All the roughness is obtained at the ECM voltage above the rapid anodic dissolution range

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4 Electrochemistry-dependent surface roughness paradoxes

While the theoretical model fits the observations and yields a reasonable value for the final expected surface roughness, some paradoxes have been embedded against the solid mathematical and physical deduction.

4.1 Paradox 1: Surface roughness range

Figure 1 gives the solid surface roughness model to lead to the deduction above. However, there is a physically valid yet hidden assumption that this model limits the final surface roughness maximum as ~\({D}_{2}\) (1 um).

Therefore, even though our case study gives a good estimation for possible surface roughness in ECM, the matched values by Eq. 19 and Eq. 20 seem to contradict our model’s starting point. The question remains if our description can bind mathematical and physical meanings in a more integrated way.

4.2 Paradox 2: Charge transfer coefficient range

If we look at the “ECM general case” section, the contribution of \({\alpha }_{ECM}\) seems to be highly dependent on the charge transfer coefficient between the matrix and carbides. If we take NbC-type phase as the possible carbide phase, [13] we will have \(\frac{dln{i}_{a2}}{dE}=\sim 10\) V−1 [35]. This will make \({\alpha }_{ECM}>0\). Then, the applied ECM voltage is increasing the final surface roughness (and, of course, this clearly contradicts our common sense). Yet, to solve this paradox and hypothesize a convincing answer, a more systematic individual phase electrochemistry measurement is required, and more dynamic dissolution sequence (e.g., whether matrix-preferential or matrix-maintained is favored) should be considered, as shown in Fig. 3.

4.3 Paradox 3: Material removal efficiency and surface quality

Paradox 2 is important, because it underlines and poses a contradictory insight between ECM material removal efficiency and surface roughness. Assuming we do not consider the singular undermining moment, where a huge chunk of carbide phases is removed away (because \({v}_{f}\ll 1\)), the material removal rate (\(MRR\)), based on our simplified model of Fig. 2, will give us [24]: 

$$MRR=\frac{\left[\left(1-{v}_{f}\right)\cdot {\Delta d}_{1}+{v}_{f}\cdot {\Delta d}_{2}\right]}{\Delta t} = \frac{FK}{{N}_{A}{k}_{e}}\cdot [\left(1-{v}_{f}\right)\text{exp}\left(\frac{{\alpha }_{1}{n}_{1}FE-{\Delta G}_{1}}{RT}\right)+{v}_{f}\text{exp}\left(\frac{{\alpha }_{2}{n}_{2}FE-{\Delta G}_{2}}{RT}\right)]\propto \text{exp}(E)$$
(21)

Obviously, the higher the ECM voltage, the quicker is the material removal to promote the ECM efficiency (see the experimental validation in Fig. 9) [36]. Equation 21 is also reasonable, because the previous study has confirmed a theoretical relationship between current efficiency \(\eta\) (reflecting MRR in the linear electrical response range) and current density or voltage as \(MRR\propto J\cdot \eta \propto J\cdot \text{tanh}(J)\propto E\cdot \text{tanh}(E)\) (see the boxed region in Fig. 9) [37] (we should note that \(\underset{E\to 0}{\text{lim}}E\cdot \text{tanh}\left(E\right)-\tau \cdot \text{exp}(E)=0\) is feasible, if we make the proportional relationship in Eq. 21 better fitted with a correction factor \(\tau\) for \({\alpha }_{1}\) and \({\alpha }_{2}\) difference).

Fig. 9
figure 9

ECM current efficiency for concentration of 250 g/L NaNO3. [37]

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However, if Sect. 3 and Paradox 2 holds, it means we can hardly balance the processing efficiency and final surface quality, and the typical off-equilibrium ECM operation condition seems to play some role (Fig. 9). This is also the ultimate reason we must revisit the electrochemical models for ECM and monitor the real surface evolution (e.g., surface roughness) during ECM.

4.4 Paradox 4: ECM time dependence for surface quality

Many previous studies have been investigating various time duration effects on ECM surface quality [17]. Based on our systematic theoretical analysis, the ECM surface roughness should have a time-independent value, if the pre-ECM materials and ECM conditions are the same. This should also be expected, because machining like traditional milling will only change the machined removal volume but not MRR and surface roughness (Here, we should distinguish that even though some papers may discuss that milling time has an effect on surface roughness, they actually mean that the characteristic time determined by the feed rate and spindle speed instead of the true machining time duration [38, 39]). ECM should have carried the same time-independent surface characteristics of other machining techniques, but this analogy will be contrary to the reported experimental observations (i.e., longer ECM usually gives better surface quality even with the same ECM setup), as summarized in Fig. 8. Then, if both sides are right, what drives this time dependence of surface roughness evolution? How does it make ECM different from traditional machining methods to lose time-independent surface quality characteristics? These important questions remain open to design an efficient and high-quality ECM process, but they are sadly often neglected.

Besides, from a more fundamental point to correlate the models and the ECM practices, the periodicity consideration from Fig. 2 and Fig. 4 can be useful to explain the benefits from a pulsed ECM (p-ECM) setup [36, 40], and the time match between the multiple phase anodic dissolution behavior and the pulse design of p-ECM might provide answers. Unfortunately, without a systematical mathematical and physical correlation, only processing parameter-related parameters are used for modeling, [36] and reasons from this aligned timeframe angle (with microstructure inputs) are not considered yet [41].

4.5 Paradox 5: Microstructural coupling effects

While most of the studies admit the importance of different phases in ECM and their distinctive dissolution and flattening mechanisms, [12, 22, 24] the coupled microstructural contribution is often neglected when analyzing the ECM outcomes.

For example, one biggest consequence from the microstructural coupling is the (micro-)galvanic corrosion, which accelerates the anode-like part’s corrosion [11, 42]. In the Ni alloy cases, carbide phases will serve as a micro-cathode (this is also expected theoretically, because \(\frac{dln{i}_{a2}}{dE}\) in Sect. 3.2 requires the carbide phases to be stabler), and another corrosion current \({i}_{a12}\) will be coupled to Eq. 12 (see Fig. 10). With this localized dissolution current coupling, Eq. 22 can be expected:

$${R}_{a}(t)=(t-N\cdot {t}_{0})\cdot \left|\frac{\left(1-{v}_{f}\right)\cdot ({i}_{a1}+{i}_{a12})}{{k}_{1}}-\frac{{v}_{f}\cdot ({i}_{a2}-\frac{1-{v}_{f}}{{v}_{f}}{i}_{a12})}{{k}_{2}}\right|$$
(22)

where \({i}_{a12}\) only depends on the micro-galvanic process and carries the form of:

$${i}_{a12}={n}_{1}FK\cdot \text{exp}(\frac{{\alpha }_{12}{n}_{1}FE-{\Delta G}_{12}}{RT})$$
(23)

where \({\alpha }_{12}\) and \({\Delta G}_{12}\) are the micro-galvanic charge transfer coefficient and surface activation energy. We still use \({n}_{1}\), because the matrix as a micro-anode is being corroded (see Fig. 10).

Fig. 10
figure 10

Microstructural coupling during ECM anodic dissolution process

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This will give us the expected surface roughness as:

$${R}_{a}=\frac{{D}_{2}}{2}\cdot \text{exp}[\frac{{(\alpha }_{1}{n}_{1}+{\alpha }_{12}{n}_{1}-{\alpha }_{2}{n}_{2})\cdot FE-({\Delta G}_{1}+{\Delta G}_{12}-{\Delta G}_{2})}{RT}]$$
(24)

In this coupling setup, Paradox 1 regarding surface roughness range and Paradox 2 considering the coupled charge transfer coefficient further stand out. Unfortunately, the high voltage during ECM will blur the clear picture of how the coupling of the corrosion and dissolution process happens and contributes to the final surface roughness, and the explanations for this coupling effect are not well established. A way to distinguish any coupling corrosion process (if any) must be developed to make the ECM surface quality more predictable and understandable.

5 Conclusion

This communication highlights key contradictions in the analysis of surface quality in ECM. Our theoretical models suggest a strong correlation between ECM voltage and surface roughness, challenging prior assumptions, particularly regarding the multiple phases in Ni alloys [22]. After cross-checking the model-predicted surface roughness range and the real ECM cases, the proposed relationship between the ECM voltage and intrinsic electrochemical characteristics (e.g., surface activation energy) to the surface quality is validated. This finding brings to light several paradoxes: the interpretation and variability of surface roughness in models and reality, the uncertainties in surface roughness due to vague charge transfer coefficient quantification, the balance between material removal rates and surface quality, and the unexpected time-dependent nature of ECM surface quality as previously reported. These paradoxes underscore the urgent need for further experimental and theoretical investigation. It is our aim that this communication will foster deeper, more insightful debates on the mechanisms of ECM and bring greater focus to the issues surrounding surface roughness.