Different exponential models are known to fit charging curves [39]. In this work, the relaxation model initially employed was in the form of a stretched exponential. This model was first proposed by Kohlrausch [40] in 1854 to describe the charge relaxation using a glass Leiden jar. Since then, the model has been widely used in dielectrics [41]. Here, Eq. (1) is similar to the charging model of Greason [42]:

\(Q\left(t\right)={Q}_{f}\left(1-\text{e}\text{x}\text{p}(-\alpha t)\right)\) , Eq. 1

where Q is the charge density at time *t*, *Q**f* is the final charge density, *α* is the charging rate, and *t* is the time. The charging rate *α* is equal to *1/τ*, where *τ* is the time constant. The above-mentioned relaxation model was modified since it is hypothesized that the charging rate is not constant but rather increases proportionally to *τ**β*, where *β* is not the discharging rate but the stretch constant. The equation was then rewritten by Trachenko and Zaccone [2] as a compressed exponential model described by Eq. (2) as:

\(Q\left(t\right)={Q}_{f}\left(1-\text{exp}\left[-{\left(\frac{t}{\tau }\right)}^{\beta }\right]\right)\) , Eq. 2

where the compressed exponential *β* > 1. The compressed exponential model indicates that the charge relaxation is faster than that of the simple exponential model (where *β* = 1) [2]. Eq. (2) was used to fit the respective tribocharging data (charge vs time) as shown in Fig. 12. Data fitting yielded averaged calculated *β* values of ~ 1.5. Interestingly, this value was used previously by other researchers to describe systems that exhibit “jammed” dynamics [43]. Examples of these systems are said to be structural glasses, colloidal gels, entangled polymers, and supercooled liquids, which experience jamming when motion of individual particles become restricted, causing their motion to slow down [44]. This behavior is also typical of granular materials [44] as used in this work.

The constants calculated, *Q**f* and *τ*, which correspond to the final charge density and the time constant, respectively, are presented in Tables 7 to 11. The time constants *τ* are in the range 2.9 to 4.7 for CpTi, 1.7 to 3.5 for Ti6Al4V, 3.80 to 6.20 for AlSi10Mg, 1.1 to 1.9 for IN 738, and 2.0 to 4.6 for SS 316L. The time constant *τ*, which indicates the time it takes for the charge to reach a certain equilibrium, overlaps between CpTi, Ti6Al4V, AlSi10Mg and SS 316L, while it is significantly low for IN 738. This may have something to do with the presence of water on its surface, which acts as a sponge for the ion transfer, allowing it to reach equilibrium faster than other metal powders.

For a specific powder, it was also observed that the time constant *τ* increases proportionally with the magnitude of accumulated charge, while it is inversely proportional to the flow rate (i.e., the faster the flow rate, the smaller the negative charges gained). It is reasonable to presume that the slower flow rates lead to higher number of contacts, consequently leading to a higher accumulated charge, since more charges can be exchanged between the surfaces in contact [45].

Table 7

Calculated constants from the data fitting of CpTi powder.

Feeding time (s) | Flow rate (g/s) | *Q**f* (nC/m2) | *τ* (s) | *α* (1/s) | *R**2* |

15 | 24.3 | -4.96 | 3.60 | 0.72 | 0.98 |

20 | 19.2 | -5.98 | 5.50 | 0.18 | 0.99 |

25 | 15.9 | -6.15 | 4.70 | 0.21 | 0.98 |

30 | 13.4 | -7.15 | 5.90 | 0.17 | 0.98 |

35 | 11.9 | -7.94 | 6.50 | 0.15 | 0.98 |

Table 8

Calculated constants from the data fitting of Ti6Al4V powder.

Feeding time (s) | Flow rate (g/s) | *Q**f* (nC/m2) | *τ* (s) | *α* (1/s) | *R**2* |

15 | 29.9 | -2.63 | 3.20 | 0.31 | 0.98 |

20 | 23.1 | -2.84 | 4.10 | 0.24 | 0.99 |

25 | 19.4 | -3.44 | 5.70 | 0.18 | 0.97 |

30 | 15.6 | -3.65 | 6.70 | 0.15 | 0.96 |

35 | 13.8 | -3.65 | 5.80 | 0.17 | 0.97 |

Table 9

Calculated constants from the data fitting of AlSi10Mg powder.

Feeding time (s) | Flow rate (g/s) | *Q**f* (nC/m2) | *τ* (s) | *α* (1/s) | *R**2* |

15 | 16.27 | -2.98 | 3.80 | 0.26 | 0.99 |

20 | 12.02 | -3.38 | 3.50 | 0.29 | 0.97 |

25 | 9.85 | -4.29 | 4.90 | 0.20 | 0.99 |

30 | 8.56 | -4.39 | 4.40 | 0.23 | 0.98 |

35 | 7.44 | -6.20 | 6.20 | 0.16 | 0.99 |

Table 10

Calculated constants from the data fitting of IN 738 powder.

Feeding time (s) | Flow rate (g/s) | *Q**f* (nC/m2) | *τ* (s) | *α* (1/s) | *R**2* |

15 | 40.60 | -0.58 | 0.90 | 1.11 | 0.99 |

20 | 26.50 | -0.58 | 0.90 | 1.11 | 0.99 |

25 | 19.36 | -0.77 | 2.00 | 0.20 | 0.99 |

30 | 16.41 | -0.96 | 3.10 | 0.32 | 0.99 |

35 | 13.28 | -1.15 | 4.00 | 0.25 | 0.99 |

Table 11

Calculated constants from the data fitting of SS 316L powder.

Feeding time (s) | Flow rate (g/s) | *Q**f* (nC/m2) | *τ* (s) | *α* (1/s) | *R**2* |

15 | 63.2 | -2.60 | 3.40 | 0.29 | 0.99 |

20 | 45.3 | -4.09 | 5.99 | 0.17 | 0.98 |

25 | 36.6 | -4.83 | 7.10 | 0.14 | 0.97 |

30 | 26.0 | -5.95 | 8.50 | 0.12 | 0.98 |

35 | 23.3 | -6.32 | 9.30 | 0.11 | 0.98 |

The final step in this methodology links the surface charge accumulation and surface potential decay [46, 47]. Considering that the maximum charge is a function of the charging rate, Eq. 3 can be used to determine the transfer efficiency that has to be a specific constant describing the relationship between powder surface composition, surface area and charging rate. Eq. (3) lists:

\({Q}_{f}=k{\left(\frac{dQ}{dt}\right)}^{n}=k{\alpha }^{n}\) , Eq. 3

Where *Q**f* is the absolute charge value, *k* is a constant, *dQ/dt* is the rate of charging with respect to time (*α*), and *n* is an exponent that varies depending on the specific triboelectric charging condition and material involved. Figure 13 represents this analysis using the constants presented in Tables 7 to 11. The charging mechanism, *n*, is reported along with the powder characteristics in Table 12.

Table 12

Powders surface properties and calculated constant *n*.

*Sample* | *State* | *PSD, D**50* | *Surface oxide* | *n* | *R**2* |

CpAl | As received | 34 | TiO2 | 0.75 | 0.88 |

Ti6Al4V | As received | 36 | TiO2, Al2O3 | 0.49 | 0.95 |

AlSi10Mg | As received | 55 | Al2O3 | 1.18 | 0.92 |

IN 738 | As received | 32 | Ni(OH)2 | 0.42 | 0.98 |

SS 316L | As received | 36 | Cr2O3 | 0.89 | 0.89 |

Table 12 thus is the starting of the reporting of a database linking the surface composition of AM powders with their surface area and the charging mechanism. The database will be augmented with different aging treatment (exposure to humidity, oxidation), to yield a comprehensive method permitting to indirectly determine the composition of powder surface scale. This simple method will be an interesting alternative to XPS or Auger techniques.